The Standard Model from stable intersecting brane …cnedres.org/literature/Nucl.Phys.B/Nucl.Phys.B...

Nuclear Physics B 616 (2001) 3–33www.elsevier.com/locate/npe

The Standard Model from stable intersectingbrane world orbifolds

Ralph Blumenhagen, Boris Körs, Dieter Lüst, Tassilo OttHumboldt-Universität zu Berlin, Institut für Physik, Invalidenstrasse 110, 10115 Berlin, Germany

Received 23 July 2001; accepted 22 August 2001

Abstract

We analyze the perturbative stability of non-supersymmetric intersecting brane world models ontori. Besides the dilaton tadpole, a dynamical instability in the complex structure moduli space occursat string disc level, which drives the background geometry to a degenerate limit. We show that incertain orbifold models this latter instability is absent as the relevant moduli are frozen. We constructexplicit examples of such orbifold intersecting brane world models and discuss the phenomenologicalimplications of a three generation Standard Model which descends naturally from anSU(5) GUTtheory. It turns out that various phenomenological issues require the string scale to be at least of theorder of the GUT scale. As a major difference compared to the Standard Model, some of the Yukawacouplings are excluded so that the standard electroweak Higgs mechanism with a fundamental Higgsscalar is not realized in this set-up. 2001 Elsevier Science B.V. All rights reserved.

PACS: 11.25.-w; 11.10.Kk; 12.10.-g

1. Introduction

During the last years string theory has provided a lot of new insights into fundamen-tal issues of theoretical physics, such as the relation between gauge theories and gravityor geometry, the quantum nature of black holes and the appearance of non-commutativespace–time structures. Also concerning more phenomenological questions strings provedthemselves to be rather fruitful, perhaps most notably in the context of string compactifica-tions with large extra dimensions [1,2] or localized gravity on a four-dimensional domainwall [3,4]. D-branes and non-perturbative duality symmetries always played a key role inall these developments. Nevertheless, still it is a great challenge to derive the observedphysics of the Standard Model of particle physics directly from strings.

E-mail addresses: [email protected] (R. Blumenhagen), [email protected] (B. Körs),[email protected] (D. Lüst), [email protected] (T. Ott).

0550-3213/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved.PII: S0550-3213(01)00423-0

http://www.elsevier.com/locate/npe

4 R. Blumenhagen et al. / Nuclear Physics B 616 (2001) 3–33

Recently a class of string compactifications was investigated which comes relativelyclose to the goal of obtaining just the Standard Model from strings. These models aregiven by type I string compactifications on a six-dimensional torusT 6 with D9-braneswhere internal background gauge fluxes on the branes are turned on [5–12].1 Thus, at treelevel supersymmetry is only broken on the D-branes with the bulk still preserving somesupersymmetry [20–23]. Turning on magnetic flux has the effect that the coordinates ofthe internal torus become non-commutative. In a T-dual picture one is dealing with D6-branes which wrap 3-cycles of the dual torus and intersect each other at certain angles,determined by the original gauge fluxes. In this way it was possible to construct stringmodels with three generations of quarks and leptons and Standard Model gauge groupSU(3) × SU(2)L × U(1)Y , where supersymmetry is broken on the branes by the gaugefluxes or, in the dual picture, by the different intersection angles.

Let us recall in slightly more detail the main features of these type I string models.Following the old ideas of [24] and [25] it was first described in [5] in a pure stringylanguage how type I compactifications with background fluxes or intersecting branes leadto a reduction of the gauge group, to chiral fermions and to broken supersymmetry on thebranes. A nice geometrical feature of such models is that the number of chiral fermionswhich are localized at the intersection points of the D6-branes is simply determined bythe corresponding topological intersection number of the branes. In this way a model withfour generations of quarks and leptons and a Standard Model gauge group was obtained in[5]. Later it was shown in [11] how odd numbers of generations arise, in particular three,if one adds to the gauge fluxes also a quantized background NSNSB-field [8,26–28]. Inthe T-dual picture the torus is then no longer rectangular but tilted by a discrete angle. In[9,10] additional type II models with backgrounds of the formT 2d × (T 6−2d/ZN) wereconsidered where the D(3+ d)-branes wrap onlyd-cycles of the first torus and are point-like on the orbifold. In this way it is possible that the orbifold space, which is transversalto the branes, becomes large, whereas the large extra dimension scenario is in conflict withchirality for the case ofT 6 compactifications [5]. Finally, in [12] a systematic analysiswas provided how to obtainT 6 models with precisely three generations of quarks andleptons and just the Standard Model gauge group without any extension. In this context themass generation forU(1) gauge bosons due to flux-induced Green–Schwarz terms and therelated issue of chiralU(1) anomalies is very important. Furthermore the question how toavoid open string tachyons in a certain range of the toroidal background parameters andother phenomenological issues were also addressed.

For type I compactifications the main consistency restrictions considered so far comefrom the requirement of the absence of massless tadpoles in the Ramond–Ramond(RR) sector of the theory. Having no RR-tadpoles ensures the anomaly freedom in theeffective field theory of the massless modes. As mentioned already, the absence of openstring tachyons is another important constraint for model building, where however some

1 For alternative compactifications with D-branes in type I or type II string theory see [13–17]; a recentdiscussion of heterotic string compactifications with background gauge fluxes and their relation to typeII compactifications with internal H-fluxes can be found in [18]. Non-supersymmetric string models withbackground RR-fluxes were discussed in [19].

R. Blumenhagen et al. / Nuclear Physics B 616 (2001) 3–33 5

‘tachyons’ might be even welcome from the phenomenological point of view, namely thosewhich contribute to the required spontaneous gauge symmetry breaking, in particular ofSU(2)L × U(1)Y → U(1)em. Hence, one may want to identify the standard Higgs fieldwith a tachyon, and also those of other spontaneously broken local gauge groups, such asfor instanceU(1)B–L.

In this paper we like to emphasize that all models considered so far are genericallyunstable due to the existence of NSNS closed string tadpoles. Specifically we will see that,already at the topology of the world sheet disc amplitude, the NSNS-tadpoles related tothe closed string moduliUI , the complex structure deformations ofT 6, and those relatedto the closed string dilatonφ are non-vanishing. This means that in the induced effectivepotential these scalar fields do not acquire a stable minimum but show the typical runawaybehaviour. The explicit form of the potential implies that the internal geometry is drivento a degenerate singular limit, where all D-branes finally lie on top of each other. As aresult, space–time supersymmetry is reenforced. In addition, the dilaton tadpole drivesthe theory to weak coupling. By T-duality this also disproves the existence of partialsupersymmetry breaking in type I vacua after introducing magnetic fluxes into the toroidalN = 4 compactification. Similar partial breaking fromN = 2 toN = 1 has been shown tobe possible in heterotic and type II theories, albeit under very special circumstances only[29–32].

There are essentially two string theoretic methods to cure the problem of the NSNS-tadpoles at least at the next to leading order. First one can employ the Fischler–Susskindmechanism [33,34], by which the back-reaction of the massless fields on the NSNS-tadpoles is taken into account iteratively. As demonstrated in [35,36] solving the stringequations of motions including the one-loop dilaton tadpole in general leads to warpedgeometries and non-trivial profiles of the dilaton and other scalar fields. Moreover, inthe non-supersymmetric type I string theory discussed in [35,36] the phenomenon ofspontaneous compactification occurred due to the NSNS-tadpoles. Of course, at thisstate one is stuck again, as technically the non-linear sigma model in this highly curvedbackgrounds cannot be solved exactly. If it could be solved, one would certainly detect anon-vanishing tadpole at the next order in the string coupling constant. Thus, one mighthope that the non-supersymmetric string theory self-adjust its background order by orderin string perturbation theory until eventually the true quantum vacuum with vanishingtadpoles to all orders is reached [37].

A second less ambitious approach to handle at least some of the tadpoles is simply byfreezing the dangerous closed string scalar fields to fixed values. This can be achieved byperforming appropriate projections in an orbifold theory. In the following we will focuson this second approach. In particular we will construct non-supersymmetric orbifoldintersecting brane models, where the complex structure moduli of the torus are fixed,and hence there are no associated NSNS-tadpoles. However, the dilaton tadpole willstill survive, and it cannot be excluded that new tadpoles will be induced at higherorders in string perturbation theory. As noted in [38], in the M-theory context one caneven contemplate on orbifold actions which freeze the size of the eleventh direction andtherefore of the dilaton in the dual string theory. We will further outline the strategy how


to obtain orbifold models with three generations of quarks and leptons and with StandardModel gauge groupSU(3)× SU(2)L ×U(1)Y in this particular kind of background.

Our work will be organized as follows. In the next section we will review the mainingredients of the toroidal intersecting brane worlds. Next, in Section 3, we will extract forthe toroidal case all NSNS-tadpoles from the infrared divergences in the tree channel Klein-bottle, annulus and Möbius-strip amplitudes, and we will compute the corresponding scalarpotential. In Section 4 we will constructZ3 orbifold intersecting brane models which arefree of geometric NSNS-tadpoles, especially addressing the form of the massless spectrumand the question of anomaly cancellation. These results will be analyzed in Section 5 tofind models which come as close as possible to the Standard Model with three generations.

Unlike the previous toroidal constructions, where the Standard Model fermions originatefrom bifundamental open strings states, we will now be forced to realize the right-handed(u, c, t)-quarks in the antisymmetric representation ofU(3). With this assignment it isindeed possible to get models with three Standard Model generations. We will also discussthe open string tachyons of the theory to see whether the Standard Model Higgs andanother Higgs breakingU(1)B–L can be realized as tachyons. It turns out that all modelswith an appropriate Higgs scalar descend from a GUT theory whereSU(3)× SU(2)L ×U(1)Y ×U(1)B–L can be unified intoSU(5)×U(1) by a deformation which is marginalat tree level. The additional global symmetries prohibit the usual Yukawa couplings ofthe (u, c, t)-quarks and the Standard Model Higgs doublets, so that the standard massgeneration mechanism with fundamental scalar Higgs fields does not work. Furthermore,we analyze the unification behaviour of gauge couplings and the possibility of proton decayin the context of theSU(5)×U(1)GUT model. In an appendix we also include the resultsfor similar six-dimensional models providing an extra consistency check for our formalismvia the six-dimensional anomaly cancellation conditions.

2. Intersecting brane worlds

In this section we review the construction of generically non-supersymmetric open stringvacua with D-branes intersecting at angles. Our starting point is an ordinary type I model,where for simplicity we consider only toroidal orbifold models, which we can write as

(2.1)Type IIB onT 2d

G+ΩG ,

whereG is a finite group acting on the 2d-dimensional torusT 2d . In the following werestrict ourselves to the case that the closed string sector of the orientifold model (2.1)preserves some supersymmetry. Usually, tadpole cancellation requires the introductionof D-branes, which can be chosen to be BPS so that the open string sector preservesthe same supersymmetry as the closed string sector. However, RR-tadpole cancellationalone does not require the open string sector to be supersymmetric. As shown in [5,7]for the toroidal case (G = 1), there exists the possibility of turning on various constantmagneticU(1) fluxes on the D-branes without giving up RR-tadpole cancellation. This


breaks supersymmetry, and one faces the usual problems with non-supersymmetric stringtheories like tachyons, dilaton tadpoles, moduli stabilization and the cosmological constantproblem.

Technically, it turned out to be more appropriate to describe such models in a T-dual language, where the new degrees of freedom are described in a purely geometricmanner. Let us assume that the 2d-dimensional torus can be written as a product ofd

two-dimensional tori

(2.2)T 2d =d⊗I=1

T 2I ,

where on eachT 2I we introduce a complex coordinateZI =XI + iYI . Applying T-duality

TY to the d YI -directions of thed two-dimensional tori, the orientifold model (2.1) ismapped to [39]

(2.3)Type II onT 2d

G+ΩRG ,

whereR is the reflection of theYI andG is the image ofG under T-dualityG= TYGT −1Y .

For the case thatG= ZN the symmetry group acts on each torus by rotations

(2.4)ZLI → e2πivI /NZLI , ZRI → e2πivI /NZRI .

Supersymmetric models have been classified in terms ofvI in [40,41]. The T-dual actionis then given by

(2.5)ZLI → e2πivI /NZLI , ZRI → e−2πivI /NZRI .

Thus, T-duality exchanges left–right symmetric actions with left–right asymmetric actions.Moreover, under T-duality D9a-branes with constant magnetic fluxesFIa are mapped toD(9− d)-branes intersecting at relative angles [42]

(2.6)ϕIab = arctan(FIa)− arctan

(FIb).

They are wrapped around one-dimensional cycles on eachT 2I , so that each branea is

specified by two coprime wrapping numbers(nIa,mIa) for each torus. Moreover, theΩR

symmetry allows two inequivalent choices of the complex structure

(2.7)UI =UI1 − iUI2 = e1

e2=UI1 − i R

I1

RI2

,

of eachT 2, UI1 = 0 or 1/2. The tori are depicted in Fig. 1.In the T-dual picture with magnetic fluxes the tilt of the torus corresponds to turning on

a discrete NSNS two-formB-field. One also has to take into account that for each braneDa there must exist the mirror brane Da′ , which is its image underΩR. For the purelytoroidal case, often called type I′, the RR-tadpole cancellation conditions were derived in[5]. If one introducesK stacks of D6a-branes counted together with theirΩR mirrors,


Fig. 1.

then the four-dimensional RR-tadpole cancellation conditions read

K∑a=1

Na

3∏I=1

nIa = 16,

K∑a=1

Nan1a

∏I=2,3

(mIa +UI1nIa

)= 0,

K∑a=1

Nan2a

∏I=1,3

(mIa +UI1nIa

)= 0,

(2.8)K∑a=1

Nan3a

∏I=1,2

(mIa +UI1nIa

)= 0.

This can be compactly written as

(2.9)K∑a=1

NaΠa =ΠO6,

whereΠa denotes the homological cycle of the wrapped D6a-branes andΠO6 the cycleof the orientifold planes along theXI axes of all threeT 2s. In terms of the T-dual type Itheory (2.8) refers to the cancellation of the D9-brane and O9-plane charges, respectively,the vanishing of the three possible types of D5-brane charges. The tree-level masslessspectrum consists ofN = 4 vectormultiplets in the gauge group

(2.10)G=U(N1)×U(N2)× · · · ×U(NK),equipped with non-supersymmetric chiral matter in bifundamental, symmetric and anti-symmetric representations of the gauge group.2 This chiral matter is localized at the inter-sections of two D-branes and therefore each state appears with a multiplicity given by theintersection number of the two D-branes. Taking these multiplicities into account, the RR-tadpole cancellation conditions guarantee the absence of gauge anomalies in the effectivefour-dimensional low energy theories.

2 More detailed information can be found in the papers [5,7,9,12].


The quite general form of the consistency conditions in terms of the wrapping numbersof the D-branes allows for a bottom-up approach to search systematically for features ofthe Standard Model in the class of these intersecting brane models. In particular, in [11] aleft–right symmetric model with three generations and gauge groupSU(3) × SU(2)L ×SU(2)R × U(1)B−L was constructed. Moreover, in [12] the matter content of a threegenerationSU(3)× SU(2)L × U(1)Y Standard Model was found, containing also right-handed neutrinos and a slightly enlarged Higgs sector. In the first place the model hadgauge groupU(3)× U(2)× U(1)× U(1), but after analyzing mixed anomalies and theappropriate Green–Schwarz mechanism only the Standard Model gauge fields remainedmassless. The broken gauge symmetries including lepton and baryon number survived asglobal symmetries, thus guaranteeing the stability of the proton. Moreover, it was arguedthat at string tree level the radii of the three toriT 2 can be tuned in such a way that openstring tachyons are absent. However, some of the tachyons are welcome, as they can serveas Higgs bosons for breaking the electroweak symmetry.

Even though from a phenomenological point of view the models look quite interesting,we will show in the next section that the string theory is highly unstable.

3. NSNS-tadpoles

So far, for toroidal intersecting brane worlds only the RR-tadpole cancellation conditionswere analyzed in detail. In this section we will compute the NSNS-tadpoles and derive theeffective scalar potential for the closed string moduli at open string tree levele−φ , which isnext to leading order in string perturbation theory. For the purpose of a phenomenologicalapplication we perform the computation for four-dimensional models.

The massless fields in the NSNS-sector are the four-dimensional dilaton and the 21ΩRinvariant components of the internal metric and the internal NS–NS two form flux. In ourfactorized ansatz (2.2) only 9 moduli are evident, which are the six radionsRI1 andRI2related to the size of the internal dimensions and the two-form fluxbI12 on eachT 2. Weextract the NSNS-tadpoles from the infrared divergences in the tree channel Klein-bottle,annulus and Möbius-strip amplitudes, the open string one-loop diagrams. Adding up thelatter three contributions leads to a sum of perfect squares, from which we can read off thedisc tadpoles. The computation is straightforward and the relevant formulas can be foundin [5,11]. By adding up all three contributions we get for the dilaton tadpole

(3.1)〈φ〉D = 1√Vol(T 6)

(K∑a=1

Na Vol(D6a)− 16 Vol(O6)

),

with

(3.2)Vol(D6a)=3∏I=1

LI (D6a)=3∏I=1

√(nIaR

I1

)2 + ((mIa +UI1nIa

)RI2

)2,

and


(3.3)Vol(O6)=3∏I=1

LI (O6)=3∏I=1

RI1.

The result is simply the overall volume of the D6-branes and orientifold planes, the latterones entering with a negative sign. It is just the effective four-dimensional tension inappropriate units. Intriguingly, the dilaton tadpole can be expressed entirely in terms ofthe complex structure moduliUI2 of the threeT 2

I ,

(3.4)〈φ〉D =(

K∑a=1

Na

3∏I=1

√(nIaU

I2

)2 +((mIa +UI1nIa

) 1

UI2

)2

− 163∏I=1

UI2

).

One way to understand this is to realize that the boundary and cross-cap states only coupleto the left–right symmetric states of the closed string Hilbert space. The complex structuremoduli are indeed left–right symmetric, whereas the Kähler moduli appear in the left–rightasymmetric sector, i.e., D-branes and orientifold O6-planes only couple to the complexstructure moduli. This is reversed in the T-dual type I picture, where the tadpole onlydepends on the Kähler moduli.

Besides the dilaton tadpole we also have three tadpoles for the imaginary parts of thecomplex structures, given by

(3.5)⟨UI2⟩D

= 1√Vol(T 6)

(K∑a=1

NaΓI (D6a)LJ (D6a)LK(D6a)− 16 Vol(O6)

),

with I = J =K = I and

(3.6)Γ I (D6a)= (nIaRI1)

2 − ((mIa +UI1nIa)RI2)2LI (D6a)

.

Analogous to (3.4) these tadpoles can also be expressed entirely in terms of the complexstructure moduliUI2 . Concerning type II models which have also been considered in similarconstructions [9] one needs to regard extra tadpoles for the real partsUI1 , which cancel intype I. All NSNS-tadpoles arise from the following scalar potential in string frame

(3.7)

V(φ,UI2

)= e−φ(

K∑a=1

Na

3∏I=1

√(nIaU

I2

)2 +((mIa +UI1nIa

) 1

UI2

)2

− 163∏I=1

UI2

),

with

(3.8)〈φ〉D ∼ ∂V

∂φ,

⟨UI2⟩D

∼ ∂V

∂UI2

.

The type II potential would only change in erasing the term arising from the orientifoldplanes, and a third tadpole would appear due to〈UI1 〉D ∼ ∂V/∂UI1 . Note that this potentialis leading order in string perturbation theory but already contains all higher powers inthe complex structure moduli, though we have only computed their one-point functionexplicitly. One needs to be careful in interpreting it.


In field theory, the presence of a non-vanishing tadpole indicates that the tree-level valuewas not chosen at a minimum of the potential. Even if one can compute higher loopcorrections formally, their meaning is very questionable, as we expect fluctuations to belarge no matter how small the coupling constant may be. The theory is driven away tosome distant minimum anyway, and perturbation theory around the unstable vacuum isimpossible. As a second problem, the open string tachyons which are very often present innon-supersymmetric string vacua would start to propagate at the open string loop level.3

Thus it is mandatory to first shift to a minimum with vanishing tadpoles and withouttachyons before taking perturbations into account.

In string theory the situation is even worse, as higher loop corrections cannot even becomputed because of infinities. If there appears a massless tadpole at genusg in the stringloop expansion, one encounters a divergence at genus 2g from the region in moduli spacewhere a massless mode propagates along a long tube connecting two genusg surfaces. Soeither one finds a new vacuum by regarding the back-reaction of the massless fields alongthe lines of [33–36], or one uses a modification of the model where the tadpoles are absent.We shall pursue the latter strategy in the following section.

Actually, one could have anticipated the result (3.7) immediately, as the source for thedilaton is just the tension of the branes, to first order given by their volumes. The aboveexpression is easily seen to arise from the Dirac–Born–Infeld action for a D9a-brane withconstantU(1) and two-form flux

(3.9)SDBI = −Tp∫

D9a

d10x e−φ√

det(G+ (Fa +B)),

including the Dp-brane tension

(3.10)Tp =√π

16κ0

(4π2α′)(11−p)/2

.

One can take all background fields to be block-diagonal in terms of the two-dimensionaltori. There they take the constant values [39]

Gij = δij , (FIa)ij = mIa

nIaRI1R

I2

εij ,

(3.11)(BI)ij = bI

RI1RI2

εij , with bI = 0 or 1/2.

Integrating out the internal six dimensions, regarding that the brane wraps each torusnIatimes, one only needs to apply the T-duality to arrive at (3.7) except for the negativecontribution of the orientifold tension.

Due to the RR-tadpole cancellation condition and the triangle inequality, the only pointwhere all four tadpoles vanish is atUI2 = ∞. Interestingly, this proves the impossibility of

3 For a discussion of the stability regions of intersecting D-branes with respect to the appearance of tachyonssee [43].


a partial breaking of supersymmetry inN = 4 vacua by relative angles between D6-branes,respectively, magnetic fluxes on D9-branes.4

The potential displays the usual runaway behaviour one often encounters in non-supersymmetric string models. The complex structure is dynamically pushed to thedegenerate limit, where all branes lie along theXI axes and theYI directions shrink,keeping the volume fixed. Put differently, the positive tension of the branes pulls thetori towards theXI -axes. The typical runaway slope being set by the tension (3.10)proportional to the string scale, a ‘slow rolling’ does not appear to be feasible either.Apparently, this has dramatic consequences for all toroidal intersecting brane worldmodels. They usually require a tuning of parameters at tree-level and assume the globalstability of the background geometry as given by the closed string moduli. If at closedstring tree-level one has arranged the radii of the torus such, that open strings stretchedbetween D-branes at angles are free of tachyons, dynamically the system flows towardslarger complex structure and will eventually reach a point where certain scalar fieldsbecome tachyonic and indicate a decay of the brane configuration.

Via T-duality the instability translates back into a dynamical decompactification towardsthe ten-dimensional supersymmetric vacuum. Thus, even if from a heuristic point of viewtoroidal intersecting brane world models look quite promising, the non-supersymmetricstring theory is highly unstable.

4. Orbifold intersecting brane models

One way to avoid this runaway behaviour of the complex structure moduli is to freezethem from the very beginning. This can be achieved by dividing the toroidal model byan appropriate discrete symmetry. For instance, for the left–right symmetric orbifoldZ3

acting as

(4.1)Θ :ZI → e2πi/3ZI ,

on all three complex coordinates, the complex structure on all threeT 2’s is fixed to beeither5

(4.2)UIA = 1

2+ i

√3

2,

or

(4.3)UIB = 1

2+ i 1

2√

3.

In [45,46], where this type of orbifold was considered for the first time, the torus (4.2) withKähler modulus

(4.4)T IA = i√

3

2R2,

4 This possibility has been established inN = 2 type II and heterotic vacua under certain rather specialconditions [29–32].

5 See [44] for a discussion of discrete parameters in type I vacua.


Fig. 2.

was called theA-torus and the torus (4.3) with

(4.5)T IB = i 1

2√

3R2,

the B-torus. Note, that under T-duality these models are mapped to asymmetric typeI orbifolds where the Kähler moduli are frozen. Vice versa, left–right symmetric typeI orbifolds are mapped to asymmetricΩR orientifolds. Therefore, only for left–rightsymmetricΩR orientifolds with intersecting branes the disc scalar potential does notdepend on theUI , preventing the torus from shrinking to degenerate limits.

Thus, we are naturally led to consider the orientifold

(4.6)Type IIA onT 6

Z3 +ΩRZ3 .

This is precisely one example of the supersymmetric orientifolds with D6-branes at anglesintroduced in [45–49].6 In the closed string sector this Z-orbifold has Hodge numbers(n21, n11)= (0,36), where 9 Kähler deformations come from the untwisted sector and theremaining 27 are the blown up modes of the fixed points. As noted before, this manifoldhas frozen complex structure. Due to theZ3 symmetry we have three kinds of O6-planeslocated as indicated in Fig. 2, being identified under the orbifold action.

One can cancel the Klein-bottle tadpoles locally by introducing four D6-branes on top ofthe O6-planes leading to a supersymmetric model with gauge group of rank-2= 16× 2−3,in accord with the rank reduction normally encountered in type I vacua with NSNSB-field of rank-6. In the following we discuss the more general case, where one introducesD6-branes intersecting at angles into such a background and in particular determine thetadpole cancellation conditions and the chiral massless spectrum. In particular, we findthat the phenomenological obstruction of the small rank can be lifted when putting branesat arbitrary angles on the orbifold space. In the dual flux picture this means that addingmagnetic flux to the supersymmetric theory allows to have a larger gauge symmetry. This

6 In [50] extensions toZN × ZM orbifold groups have been considered.


is very surprising in the first place, one would have naively expected the opposite to happen.But, effectively, a D9-brane with additional flux can carry less RR-charge than without.

Note the difference compared to [51] where six-dimensional supersymmetric orien-tifolds of type I′ have been combined with generic brane configurations on an extraT 2.This latter compactification indeed suffers from the very small rank of the gauge group.

An individual D6a-brane is again determined by three pairs of wrapping numbers(nI ,mI ) along the fundamental cycles

(4.7)eA1 = eB

1 =R, eA2 = R

2+ i

√3R

2, eB

2 = R

2+ i R

2√

3,

of eachT 2. Under theZ3 andΩR symmetry in general the branes are organized in orbitsof length six. Such an orbit constitutes an equivalence class[a] of D6a-branes denoted by[(nIa,mIa)]. For theA-torus the six branes contained in the equivalence class[(nI ,mI )] aregiven by

(4.8)

(nI

mI

)Z3⇒

(−nI −mInI

)Z3⇒

(mI

−nI −mI)

ΩR ⇓ ⇓ ⇓(nI +mI

−mI)

Z3⇐(−mI

−nI)

Z3⇐( −nInI +mI

)and for theB-torus by

(4.9)

(nI

mI

)Z3⇒

(−2nI −mI3nI +mI

)Z3⇒

(nI +mI

−3nI − 2mI

)ΩR ⇓ ⇓ ⇓(

nI +mI−mI

)Z3⇐

(nI

−3nI −mI)

Z3⇐(−2nI −mI

3nI + 2mI

)As an example for an orbit on a singleA-torus the equivalence class[(2,1)] is shown inFig. 3. The solid lines represent the images underZ3 and the dashed lines theΩR mirrorbranes. Due to the relation

(4.10)Θ(ΩR)= (ΩR)Θ−1,

only untwisted sector fields couple to the orientifold planes. This is also clear from the factthat the orientifold planes are of codimension one on eachT 2 and therefore can avoid ablown-upP1 from an orbifold fixed points. Similarly, the D6-branes cannot wrap aroundthe blown-up cycles to become fractional branes and thus are not charged under the twistedsector RR-fields. Thus, there are only untwisted tadpoles.

4.1. Tadpoles

Combining the results from [5,46] the computation of the Klein-bottle, annulus andMöbius-strip amplitudes is a straightforward exercise and we will only present the salientfeatures and results of this rather tedious computation. Since the complex structure isfixed we only get one RR- and one NSNS-tadpole cancellation condition. In the annulusamplitude all open string sectors contribute including those from open strings stretched


Fig. 3.

between two branes belonging to the same equivalence class. It turns out to be convenientto define the following two quantities for any equivalence class[(nIa,mIa)] of D6a-branes

Z[a] = 2

3

∑(nIb,m

Ib)∈[a]

3∏I=1

(nIb + 1

2mIb

),

(4.11)Y[a] = −1

2

∑(nIb,m

Ib)∈[a]

(−1)M3∏I=1

mIb,

whereM is defined to be odd for a mirror brane and otherwise even. The sums aretaken over all the individual D6b-branes that are elements of the orbit[a]. The explicitexpressions forZ[a] andY[a] for the four possible tori,AAA, AAB, ABB, BBB, can befound in Appendix A. If we introduceK stacks of equivalences classes[a] of branes, thenthe RR-tadpole cancellation condition reads

(4.12)K∑a=1

NaZ[a] = 2.

Note, that the sum is over equivalence classes of D6-branes. In fact,Z[a] is the projectionof the entire orbit of D6a-branes onto theXI axes, i.e., the sum of their RR-chargeswith respect to the dual D9-brane charge. Therefore the appearance ofZ[a] in the tadpolecancellation condition is very natural, simply meaning that the RR-charges of all D6-braneshave to cancel the RR-charges of the orientifold O6-planes. If theZ[a] are all positive, aswas the case in the supersymmetric solutions of [45,46], then (4.12) implies a very smallrank of the gauge group. However, in the general case theZ[a] may also be negative so thatalso gauge groups of higher rank can be realized.

In the closed string NSNS-sector all scalars related to the complex structure moduliare projected out underZ3, so that only the dilaton itself can have a disc tadpole. This isindeed what we find from the tree-channel one-loop amplitudes, as the only divergences


there comes from the dilaton. The scalar potential for our model is

(4.13)V (φ)= e−φ(∑

a

Na

3∏I=1

LI[a] − 2

),

with the lengths given by

(4.14)LI[a] =√(nIa)

2 + (mIa)2 + nIamIa, for theA-torus,√(nIa)

2 + 13(m

Ia)

2 + nIamIa, for theB-torus.

Thus, similar to the toroidal case discussed in Section 2, whenever the D-branes do notlie on top of the orientifold planes the dilaton tadpole does not vanish. In this way thelocal cancellation of the RR-charge is in one to one correspondence with supersymmetricvacua and the cancellation of NSNS-tadpoles. The only exception to this rule appears tobe a parallel displacement of orientifold planes and D-branes, i.e., a Higgs mechanismbreakingSO(2Na) to U(Na).

4.2. Massless spectrum

Having found the one-loop consistency condition the next step is to determine themassless spectrum and to see whether one can find phenomenologically interesting models.In the closed string sector at string tree levelN = 1 supersymmetry is preserved and weget the same massless spectrum of vector and chiral multiplets as in [46]. However, inthe open string sector we break supersymmetry and get more interesting spectra. For thesupersymmetric brane configurations the massless spectra for these kinds of orientifoldswith D-branes at angles were always non-chiral, which is no longer true in the non-supersymmetric case.

In the following we discuss the most generic situation where all equivalence classescontain six different D6-branes, i.e., there are no dual D9-branes without any magneticflux on their world-volume. A string with both ends on the same individual brane insome equivalences class[a] gives rise to anN = 4 vectormultiplet in the gauge groupU(Na). Open strings stretched between branes belonging to two different classes can breaksupersymmetry and give rise to chiral fermions in the bifundamental representations of thegauge groups. There are 36= 6 × 6 different open string sectors of this kind. Due to theZ3 andΩR symmetry only 6 of them are independent. Thus, we can pick one brane,D6a , from the first stack and determine the massless spectrum with all 6 branes D6bi , i ∈1, . . . ,6, from the second stack. Open strings between D6a andZ3 images of D6b, i.e.,i ∈ 1,2,3, yield chiral fermions in the(Na,Nb) representation and open strings betweenD6a and mirror images, i.e.,i ∈ 4,5,6, in the second stack give rise to chiral fermions inthe(Na,Nb) representation. The multiplicity of these massless states is determined by thetopological intersection number between the branes in question, where intersections withformally negative intersection number have flipped orientation leading to the conjugaterepresentations.

In the end, only the net number of such fermion generations is relevant. For instance,let D6b1 and D6b2 be two different branes in the orbit[b] and D6a another one in the


Fig. 4.

orbit [a]. Assume, that in the D6a–D6b1 sector we have a chiral fermionψa,b1 in the(Na,Nb) representation and in the D6a–D6b2 sector a fermionψa,b2 in the conjugate(Na, Nb) representation. In principle, these can pair up to yield a Dirac mass term withmass of the order of the string scale. Indeed, since in the D6b1–D6b2 sector we get amassless scalarHb1,b2 in the adjoint representation ofU(Nb), the three-point coupling onthe disc diagram as shown in Fig. 4 exists.

Giving a vacuum expectation value to theSU(Nb) singlet in the adjoint ofU(Nb) leavesthe gauge symmetry unbroken and gives a mass to the fermions. From the string point ofview, this deformation is exactly the one studied in [52], which deforms the two intersectingbranes of the orbit[b] into a single brane wrapping a supersymmetric cycle.

For the relevant net number of chiral left-handed bifundamentals one obtains thefollowing simple expressions(Na,Nb)L: Z[a]Y[b] − Y[a]Z[b],

(4.15)(Na,Nb)L : Z[a]Y[b] + Y[a]Z[b].Thus, the combinationsZ[a] andY[a] can be interpreted as effective wrapping numbers.

Finally, we have the open strings stretched between different branes of the sameequivalence class. Open string between the brane D6a and the three images underΩRΘkgive rise to chiral fields in the symmetric and antisymmetric representation. The netnumbers of these massless fields are given by

(Aa)L: Y[a],

(4.16)(Aa + Sa)L: Y[a](Z[a] − 1

2

).

Finally, open strings between the brane D6a and its twoZ3 images yield massless fermionsin the adjoint representation

(4.17)(Adj)L: 3nB

3∏I=1

(LI[a]

)2,

wherenB counts the number ofB-tori in T 6. This latter sector isN = 1 supersymmetric,as theZ3 rotation alone preserves supersymmetry. Before going into the phenomenologicaldetails we first discuss the issue of gauge anomalies in the next section.


4.3. Anomaly cancellation

Since we have chiral fermions, there are potential gauge anomalies, which howevershould be absent due to the string theoretic one-loop consistency of our models. For thespectrum shown in Section 4.2 we obtain that the non-Abelian gauge anomaly of theSU(Na) gauge factor is proportional to

(4.18)∑b =a

2NbZ[b]Y[a] + (Na − 4)Y[a] + 2NaY[a](Z[a] − 1

2

),

which vanishes when we use the RR-tadpole cancellation condition (4.12) in the first termin (4.18). As usual, the Abelian gauge anomalies do not cancel right away. Indeed theU(1)a − g2

µν anomalies are proportional to

(4.19)3NaY[a],

and the mixedU(1)a −U(1)2b anomalies are

(4.20)2NaNbY[a]Z[b].

In order to cancel these anomalies one has to invoke a generalized Green–Schwarzmechanism. It was pointed out in [9] that the relevant axions are among the untwistedsector RR-fields. Using the same notation, we are discussing the couplings in the T-dualtype I language where angles are translated into fluxes. In ten space–time dimensions wehave the RR-fieldsC2 andC6, dC6 = ∗dC2, with world-volume couplings

(4.21)∫

D9a

C6 ∧ Fa ∧ Fa,∫

D9a

C2 ∧ Fa ∧ Fa ∧ Fa ∧ Fa.

Upon dimensional reduction to four dimensions we only get one two-form,B02 = C2. Note,

that the other three type I two-forms

(4.22)BI2 =∫

T 2J ×T 2

K

C6,

are projected out by theZ3 symmetry. The dual four-dimensional axionC00 is given by

(4.23)C00 =

∫T 2

1 ×T 22 ×T 2

3

C6.

From the ten-dimensional couplings (4.21) summed over an entire orbit of the symmetrygroup one obtains the four-dimensional couplings of the RR-forms to the gauge fields

(4.24)NaY[a]∫M4

B02 ∧ Fa, NbZ[b]

∫M4

C00Fb ∧ Fb.

Apparently, these couplings have precisely the right form to cancel the Abelian gaugeanomalies (4.19) and (4.20) via a generalized Green–Schwarz mechanism. In fact there is


only one anomalousU(1)

(4.25)Fmass=∑a

(NaY[a])Fa,

which becomes massive due to the first coupling in (4.24). It can be checked that the mixedU(1)−G2 anomalies are also canceled by this generalized Green–Schwarz mechanism.

4.4. Stability

As the tadpole calculation (4.13) shows, after projecting out the complex structuremoduli, the potential is flat for the remaining moduli at the leading order in the stringperturbation theory. Only the dilaton still runs away to zero coupling. For the backgroundgeometry, we certainly expect to find corrections in higher loop diagrams, when untwistedand twisted Kähler moduli, decoupled only at leading order, can run around the loop.

Definite statements can hardly be made about the higher loop potentials, but at leastqualitatively we can say something about the one-loop potential for the untwisted KählermoduliKIU =RI1RI2. Since the closed string sector is supersymmetric the only dependenceof the one-loop potential onKIU can arise from the annulus and the Möbius-stripamplitudes. Since we assumed that no D6-branes lie along theXI axes, the massless stringsin these sectors are localized at the intersection points of the respective D6-branes and O6-planes. They do not ‘see’ the global geometry of the torus and, hence, there is no explicitdependence of the Möbius-strip amplitude on the radii.

Therefore, the only contribution can come from non-supersymmetric sectors in theannulus amplitude, which depend on the radii. These arise from open strings stretchedbetween two intersecting D6-branes, which are parallel on one or two tori. In the directionsof the latter tori there are both Kaluza–Klein and winding modes for the open strings. Sincethe KK-modes scale like 1/(RI )2 ∼ 1/KIU and the winding modes like(RI )2 ∼KIU thereis a good chance that the one-loop scalar potential stabilizes the untwisted Kähler moduli.Of course, in this argument we have assumed that the Fischler–Susskind mechanismdoes not qualitatively change the picture we derived from the flat tree-level background.Alternatively one can also proceed in a way analogous to the tadpoles associated to thecomplex structuresUI , namely considering orbifold groups where all radii are frozen, liketheZ3 × Z3 orientifold of [39,53].

Another issue concerns the existence of open string tachyons, which also may spoilstability at the open string loop-level. In general, the bosons of lowest energy in a non-supersymmetric open string sector can have negative mass squared. Here one has todistinguish two different cases. Either the two D-branes in question intersect under a non-trivial angle on all three two-dimensional tori or the D-branes are parallel on at least one ofthe toriT 2

I . In the latter case one can get rid of the tachyons at least classically by makingthe distance between the two D-branes on the torusT 2

I large enough. In the former case, itdepends on the three angles,ϕIab, between the branes D6a and D6b whether there appeartachyons or not.

DefiningεIab = ϕIab/π and letPab be the number ofεIab satisfyingεIab > 1/2, to computethe ground state energy in this twisted open string sector one has to distinguish the


following three cases

(4.26)E0ab =

12

∑I

∣∣εIab∣∣− max∣∣εIab∣∣, for Pab = 0,1,

1+ 12

(∣∣εIab∣∣− ∣∣εJab∣∣− ∣∣εKab∣∣)− max

∣∣εIab∣∣,1− ∣∣εJab∣∣,1− ∣∣εKab∣∣, for Pab = 2 and∣∣εIab∣∣ 1

2,

1− 12

∑I

∣∣εIab∣∣, for Pab = 3.

In order for a brane model to be free of tachyons, for all open string sectorsE0ab 0 has to

be satisfied. Since in the orbifold model each brane comes with a whole equivalence classof branes, and the angles between two branes do not depend on any moduli (like in thetoroidal case), freedom of tachyons is quite a strong condition. We will discuss this pointfurther for the concrete three generation models in Section 5. We shall actually find that,even though tree-level stability is a strong condition, it can be satisfied in particular cases.

Even if classically we can avoid tachyons by moving parallel branes far apart, at quantumlevel effective potentials for these open string moduli are generated which might spoilthe stability of the configuration by pulling the branes together until a tachyon reappears.Without knowing the precise scalar potential definite statements cannot be made.

5. Three generation models

In this section we will try to solve the consistency equations for theZ3 orbifold to searchfor models which come as close as possible to the Standard Model, respectively a moderateextension. Amazingly, even in this fairly constrained orbifold set-up it is not too difficult toget three generation models withSU(3)× SU(2)L ×U(1)Y ×U(1)B–L or SU(5)×U(1)gauge group and Standard Model matter fields enhanced by a right-handed neutrino. Onlywhen it comes to the Higgs sector and the Yukawa couplings we encounter some deviationsfrom our Standard Model expectations.

In [9,11,12] three generation intersecting brane worlds were always realized on fourstacks of D6-branes with gauge groupU(3)×U(2)×U(1)×U(1) and chiral matter onlyin the bifundamental representations of the gauge factors. It turns out that such a scenariois not possible in our orbifold case for the following reason. Requiring that there does notexist any matter in the antisymmetric representation ofU(3) forcesY1 = 0. Due to (4.15),this in turn implies that we get the same number of chiral fermions in the(3,2) and inthe(3,2) representation ofU(3)×U(2) leading to an even number of left-handed quarks.Thus, employing only bifundamental fields is not sufficient.

5.1. Extended Standard Model

We are forced to realize the right-handed(u, c, t)-quarks in the antisymmetric represen-tation ofU(3), which, accidentally, is the same as the antifundamental representation3.Moreover, requiring that there does not appear any chiral matter in the symmetric repre-sentation ofU(3) andU(2) forces us to haveZ3 = Z2 = 1/2. After some inspection onerealizes that the best way to approach the Standard Model is to start with only three stacks


of D6-branes with gauge groupU(3)×U(2)×U(1) and

(5.1)(Y1,Z1)=(

3,1

2

), (Y2,Z2)=

(3,

1

2

), (Y3,Z3)=

(3,−1

2

).

Note, that this choice indeed satisfies the RR-tadpole cancellation condition (4.12). Thethree generation chiral massless spectrum is shown in Table 1.

Of course, we now assume that the non-chiral fermions have paired up and decoupledas was described earlier. Note, that the right-handed leptons are realized as open strings inthe antisymmetric representation ofU(2) and the right-handed neutrinos as open stringsin the symmetric representationS of theU(1) living one the third stack of D6-branes. Asexpected from the general analyses of theU(1) anomalies there is one anomalousU(1)gauge symmetry

(5.2)U(1)mass= 3U(1)1 + 2U(1)2 +U(1)3,and two anomaly free ones which can be chosen to beU(1)Y andU(1)B–L

U(1)Y = −2

3U(1)1 +U(1)2,

(5.3)U(1)B–L = −1

6

(U(1)1 − 3U(1)2 + 3U(1)3

).

Analogous to [9,11,12], since the one-loop consistency of the string model requires theformal cancellation of theU(2) andU(1) (non-Abelian) gauge anomalies, the possiblemodels are fairly constrained and require the introduction of right-handed neutrinos.Because the lepton number is not a global symmetry of the model, there exists thepossibility to obtain Majorana mass terms and invoke the see-saw mechanism for theneutrinos mass hierarchy. Since, after introducing the right-handed neutrino into theStandard Model, theU(1)B–L symmetry becomes anomaly-free, it is not too surprisingthat in the string theory this symmetry is gauged. After the Green–Schwarz mechanismthe anomalousU(1)massdecouples, but survives as a global symmetry. Thus, the possibleYukawa couplings are more constrained than in the Standard Model.

It is straightforward but extremely tedious to find realizations of the(Ya,Za) given abovein terms of actual winding numbers[(nIa,mIa)]. We have performed a systematic computersearch and identified 36 solutions for each stack[a], the number being independent ofwhich of the four possible types of the torus had been chosen. Surprisingly, all windingnumbers range between−3 and 3, and only for theBBB-torus from −5 and 5. Theactual number of inequivalent string models with the above mentioned Standard Modellike features then is 4× 363.

5.2. Stability and Higgs scalars

As mentioned already, the primary motivation to study the present class ofΩR orientifoldsof type IIA is their stability. While the closed string sector does not suffer from anymassless disc tadpole apart from the dilaton, and thus all moduli sit at extrema of theirpotential, the open string sector contains tachyonic scalars which indicate an instability.


Table 1Left-handed fermions for the 3 generation model

Matter SU(3)× SU(2)×U(1)3 U(1)Y U(1)B–L

(QL)i (3,2)(1,1,0) 1/3 1/3(ucL)i (3,1)(2,0,0) −4/3 −1/3

(dcL)i (3,1)(−1,0,1) 2/3 −1/3

(lL)i (1,2)(0,−1,1) −1 −1(e+L )i (1,1)(0,2,0) 2 1(νcL)i (1,1)(0,0,−2) 0 1

Assuming the closed string moduli to be qualitatively unaffected by the condensation, theendpoint of this will presumably be a new vacuum, where the isolated cycles themselvesare general supersymmetric ones but still intersects each other in a non-supersymmetricway. Still everything will be stable with respect to tachyons. There are two differentpatterns of gauge symmetry breaking which arise when any two branes condense via thismechanism. When the two branes are of different class[a] and[b], the tachyonic Higgsfield is in the bifundamental representation of theU(Na)× U(Nb) gauge group and thecondensation resembles the Higgs mechanism of electroweak symmetry breaking. On thecontrary, when the two branes are elements of the same orbit[a], the Higgs field will be inthe antisymmetric, symmetric or adjoint representation of theU(Na) and thus affect onlythis factor.

The version of the Standard Model extended by a gaugedB–L symmetry togetherwith right-handed neutrinos requires a two step gauge symmetry breaking. (For theprobably first appearance of such models see [54].) In order to avoid conflicts with variousexperimental facts a hierarchy of Higgs vacuum expectation values is required. First theU(1)B–L has to be broken at a scale at least some 104–6 above the electroweak scale. Thisrequires a Higgs field charged under this group but a singlet otherwise, which can be metwith a tachyon from a sector of strings stretching between two branes in the orbit thatsupports theU(1)3. The second step is the familiar electroweak symmetry breaking whichneeds a bifundamental Higgs doublet.

We have therefore performed a study among all the 4× 363 models looking for sucha suitable tachyon spectrum. In any sector of open strings stretching between two D6-branesa and b the lightest physical state has a mass given by (4.26). By expressingthe angle variables in terms of winding numbers, one can set up a computer programto do the search for models with a Higgs scalar in the(2,1) and/or another one in the‘symmetric’ representation ofU(1)B–L. All other open string sectors need to be free oftachyons. The results are the following: for theAAA and theBBB type tori one can get D6-brane configurations that display only tachyons charged underU(1)B–L, but none of thesemodels does have a suitable Higgs in the(2,1). Vice versa, theAAB andABB models


do have Higgs fields in the(2,1) but no singlets charged underU(1)B–L. 7 Actually,we find a couple of hundred models having either a Higgs in(2,1) or a Higgs in the‘symmetric’ representation ofU(1)B–L. But no model contains both Higgs fields. Thislooks discouraging at first sight. Regarding the necessity to have a hierarchy of a highscale breaking ofU(1)B–L and low scale electroweak Higgs mechanism, we are forced tochoose a model with a singlet Higgs condensing at the string scale but without Higgs fieldin the(2,1) and favour an alternative mechanism for electroweak symmetry breaking. Anexplicit realization is for example given by[(

nI1,mI1

)]= [(−3,2), (0,1), (0,−1)

],[(

nI2,mI2

)]= [(−3,2), (0,1), (0,−1)

],

(5.4)[(nI3,m

I3

)]= [(−3,2), (1,−1), (−1,0)

].

This model has precisely 3 Higgs singlets

(5.5)hi : (1,1)(0,0,−2),

which carry onlyB–L but no hypercharge. They are former ‘superpartners’ of the right-handed neutrinos. Interestingly, it turns out that all solutions to the tadpole conditionswhich display the Higgs singlet charged underU(1)B–L and no tachyons otherwise resultfrom a model with gauge groupSU(5)× U(1) deformed by giving a vacuum expectationvalue to a scalar in the adjoint ofSU(5). Geometrically this is evident in the fact thatthe stacks of branes that support theSU(3) and SU(2)L are always parallel, thus theirdisplacement is a marginal deformation at tree-level. Of course, we have to expect thatquantum corrections will generate a potential for the respective adjoint scalar.

5.3. An SU(5)×U(1) GUT model

In this section we reinterpret the above direct realization of the extended Standard Modelas a GUT scenario. The unified model basically consists in moving the two stacks for theU(2) andU(3) sector on top of each other, thus tuning the adjoint Higgs24 to a vanishingvacuum expectation value. The common GUT gauge groupSU(5) is extended by a singlegaugedU(1) symmetry. On two stacks of branes withN5 = 5 andN1 = 1 the model isrealized by picking again

(5.6)(Y5,Z5)=(

3,1

2

), (Y1,Z1)=

(3,−1

2

).

The task of expressing these effective winding numbers in terms of[(nIa,mIa)] quantumnumbers is identical to that for the previously discussed extended Standard Model. Thenumber of solutions is again 36 per stack, i.e., the total set consists of 4× 362 inequivalentmodels. The resulting spectrum of net chiral fermions is featured in Table 2.

7 For all tori except theAAA type, one can even set up D6-brane configurations without tachyons at all.


Table 2Left-handed fermions for the 3 generationSU(5)×U(1) model

Number SU(5)×U(1)2 U(1)free

3 (5,1)(−1,1) −6/53 (10,1)(2,0) 2/53 (1,1)(0,−2) −2

The anomalousU(1) is given by

(5.7)U(1)mass= 5U(1)5 +U(1)1,in accord with (4.25), and the anomaly-free one is

(5.8)U(1)free= 1

5U(1)5 −U(1)1.

This is the desired field content of a grand unified Standard Model with extra right-handedneutrinos, which then also fits intoSO(10) representations. The usual minimal Higgs sectorconsists of the adjoint24 to breakSU(5) to SU(3)× SU(2)L ×U(1)Y and a(5,1) whichproduces the electroweak breaking. In addition we now also need to have a singlet to breakthe extraU(1)free gauge factor. The adjoint scalar is present as part of the vectormultipletof the formerlyN = 4 supersymmetric sector of strings starting and ending on identicalbranes within the stack [5]. Turning on vacuum expectation values in the supersymmetrictheory means moving on the Coulomb branch of the moduli space, which geometricallytranslates to separating the 5 D65-branes into parallel stacks of 2 plus 3. Actually, the formof the potential generated for this modulus after supersymmetry breaking is not known,and the existence of a negative mass term as required for the spontaneous condensationremains speculative.

Having identified theSU(5) GUT as a Standard Model where two stacks of branesare pushed upon each other, we can refer to the former analysis of the scalar spectrumfor the other two Higgs fields needed. The results of our search for Higgs singlets andbifundamentals done for theSU(3)× SU(2)L ×U(1)Y ×U(1)B–L model in the previoussection apply without modification as the two stacks forSU(3)× SU(2)L are parallel in allcases.

5.4. Yukawa couplings

There is another deviation from the Standard Model which also supports a replacingof the fundamental Higgs scalar by an alternative composite operator. Namely, due tothe additional global symmetries, an appropriate Yukawa coupling giving a mass to the(u, c, t)-quarks is absent. This can be seen from the quantum numbers of the Higgs fieldH resulting from of the relevant Yukawa coupling

(5.9)H QLuR.


Thus,H is forced to have the quantum numbers

(5.10)H : (1,2)(3,1,0).

Apparently, no microscopic open string state can transform in the singlet representation ofU(3) and nevertheless havingU(1) chargeq = 3. Note, that for the(d, s, b)-quarks andthe leptons the relevant Yukawa coupling is

(5.11)H QLdR, H lLeR, H ∗lLνR,

leading to the quantum numbers(1,2)(0,1,1) for the Higgs fieldsH , which at least isnot in contradiction to the open string origin of the model. We conclude, that in openstring models where the(u, c, t)-quarks arise from open strings in the antisymmetricrepresentation ofU(3), there appears a problem with the usual Higgs mechanism.

In a very similar fashion, in theSU(5)×U(1) GUT model theU(1)massdoes not allowYukawa couplings of the type10 · 10 · 5 so that the standard mass generation mechanismdoes not work. Again we are drawn towards a more exotic version of gauge symmetrybreaking and mass generation. The only resolution to this obstacle is to propose that theHiggs fields are not fundamental fields but composite objects with the quantum numbersgiven in (5.10). This possibility is further supported by the analysis in Sections 5.5 and 5.6.

Finally, let us discuss the generation of neutrino masses. A scalarh in the ‘symmetric’representation of theU(1)3 can break theU(1)B–L symmetry via the Higgs mechanism,but does not directly lead to a Yukawa coupling of Majorana type for the right movingneutrinos. However, the dimension five coupling

(5.12)1

Ms

(h∗)2(νc)

LνR,

is invariant under all global symmetries and leads to a Majorana mass for the rightmoving neutrinos. Together with the above mentioned (to be found) composite Higgsmechanism for the standard Higgs field this in principle allows the realization of the see-saw mechanism to generate small neutrino masses. This is in contrast to the neutrino sectoras found in the toroidal models [12] where only neutrino masses of Dirac type could begenerated due to the conservation of the lepton number.

5.5. Gauge couplings

We now comment on the patterns of gauge coupling unification. By dimensionalreduction theU(Na) gauge couplings are given by

(5.13)4π2

g2a

= Ms

gs

3∏I=1

LIa,

wheregs is the string coupling. Using for the Abelian subgroupsU(1)a ⊂U(Na) the usualnormalization tr(Q2

a)= 1/2, the gauge coupling for the hypercharge

(5.14)QY =∑a

caQa,


is given by

(5.15)1

g2Y

=∑a

1

4

ca

g2a

.

Thus, in our case we get with

(5.16)QY = −2

3U(1)1 +U(1)2 = −2

3

√6U(1)1 + 2U(1),

for the Weinberg angle

(5.17)sin2ϑW = 3

6+ 2g2g1

.

The tildedU(1)s in (5.16) denote the correctly normalized ones. Since in all interestingcases theU(3) branes have the same internal volumes than theU(2) branes, (5.17) reducesto the prediction sin2ϑW = 3/8, which is precisely theSU(5) GUT result. Note, thatin contrast to the toroidal intersecting brane world scenario, here the Weinberg angle iscompletely fixed by the wrapping numbers of the D6-branes. Thus, these models are morepredictive and, of course, easier to falsify.

As usual, in order for the gauge couplings and the string coupling to be of order one atthe string scale, the sizes of the tori are forced to be of order the string scale. In principle,by blowing up the 27 orbifold fixed points we can realize a large extra dimension scenariowith arbitrary string scale. The gauge couplings remain of order one as the D6-branes canavoid the blown upP1s, whereas the Planck scale gets large due to the large overall volumeof the compactification manifold. However, the above result for the Weinberg angle rathersuggests that the string scale is close to the GUT scale.

In order to compare the gauge couplings to their experimental values at the weak scale,one has to include in the beta-function all states with masses between the weak and thestring scale. For a detailed analysis we would need the precise masses of all fields. Someof these masses are due to loop correction as, for instance, for the superpartners in theN = 4 vectormultiplet, other masses are already there at tree level like for the lowestenergy scalars in the non-supersymmetric open string NS-sectors.

Thus, the Standard Model gauge couplings run up to the string scale in the sameway as in the non-supersymmetricSU(5) GUT model. However, around 1 or 2 ordersof magnitude below the string scale a lot of new states will begin to contribute to thebeta-function and change the running considerably. Since all the states from theN = 4vectormultiplets might contribute we expect the one-loop beta-function even to changesign. Thus, at least in principle it is not excluded that the non-supersymmetricSU(5)modelwill feature gauge coupling unification.

5.6. Proton decay

In the three generation models in [12] the decay of the proton was prohibited, as thebaryon and lepton numbers survived the Green–Schwarz mechanism as separate global


Fig. 5.

symmetries. In our orbifold models only the combinationB–L appears as a symmetry, sothat there are potential problems with the stability of the proton.

In [10] it was argued that perturbatively the proton is stable in intersecting brane models,as effective couplings with three quarks are forbidden as long as the quark fields appear inbifundamental representations of the stringy gauge group. Apparently, also this argumentdoes not directly apply to our case. Indeed the disc diagram in Fig. 5 generates a dimensionsix coupling

(5.18)L ∼ 1

M2s

(ucLuL

)(e+LdL

),

which preservesB–L but violates baryon and lepton numbers separately. The numbers atthe boundary indicate the D6a-brane to which the boundary of the disc is attached. Thus weconclude, that, as long as we want to work in the large extra dimension scenario withMs 1016 GeV, these models do have serious problems with proton decay. Said differently, alsothe issue of proton decay leads one to chose the string scale rather at the GUT scale thanin the TeV region.

6. Conclusions

In this paper we have discussed the issue of stability of toroidal intersecting braneworlds to the next to leading order in string perturbation theory. The arising instabilityfor the geometric parameters of the internal torus was cured in the case of specific orbifoldmodels where the complex structure is completely frozen. We have studied such a partlystabilizedZ3 orbifold model in great detail and focused on the derivation of the StandardModel. It was possible in this, compared to the toroidal case, fairly constrained set-upto construct a three generation Standard Model extended by right-handed neutrinos anda gaugedU(1)B–L symmetry. A detailed study of the tachyon spectra of these modelsrevealed that it was impossible to realize the entire minimal set of Higgs scalars as openstring tachyons. Thus we had to propose an alternative composite operator instead of thefundamental Higgs field to achieve the electroweak symmetry breaking. Furthermore, itturned out that all the models with the required Higgs to break theU(1)B–L at the string


scale are obtained from anSU(5)× U(1) GUT model by the condensation of an adjointHiggs which is massless at tree level. Thus, the present class of models appears to benaturally unified in terms of the commonSU(5) scenario.

The analysis of more detailed phenomenological issues uncovered some importantdeviations from the ordinary Standard Model due to extra global symmetries, remnantsof a larger gauge symmetry after a Green–Schwarz anomaly cancellation mechanism.The most serious one is surely the absence of appropriate Yukawa couplings for the(u, c, t)-quarks to generate masses via a fundamental Higgs condensation. It appearsto be a generic feature that whenever the Standard Model or GUT matter fields donot exist in the string spectrum as bifundamental fields exclusively, global symmetriesforbid some of the Yukawa couplings required in the standard mass generation process.Therefore, the only resolution seems to consist of a model with composite Higgs whichpresumably circumvents the gauge hierarchy problem simultaneously. Unfortunately, thereis no direct way to prove the existence of such a composite operator with a condensate atthe electroweak scale.

Moreover, without the string scale being of the same order as the GUT scale there wouldappear problems with proton decay and gauge coupling unification. Thus, it appears thatthe natural scale for this intersecting brane model on an orbifold is not the TeV scale butthe GUT scale.

Acknowledgements

We would like to thank Lars Görlich, Axel Krause and Angel Uranga for helpfuldiscussions and correspondence. The group is supported in part by the EEC contractERBFMRXCT96-0045. B.K. and T.O. also like to thank the GraduiertenkollegTheStandard Model of Particle Physics — structure, precision tests and extensions, maintainedby the DFG. In addition this research was supported in part by the National ScienceFoundation under Grant No. PHY99-07949 through the Institute for Theoretical Physics inSanta Barbara. R.B. and D.L. thank the ITP for the hospitality during this work.

Appendix A. Effective wrapping numbers

In this appendix we present the precise form of the effective wrapping numbers,Y[a] andZ[a], in terms of the fundamental wrapping numbers,(nIa,m

Ia).

AAA-torus

Z[a] = n1an

2an

3a + 1

2m1an

2an

3a + 1

2n1am

2an

3a + 1

2n1an

2am

3a − 1

2m1am

2an

3a

− 1

2m1an

2am

3a − 1

2n1am

2am

3a −m1

am2am

3a,

(A.1)Y[a] = n1am

2am

3a +m1

an2am

3a +m1

am2an

3a + n1

an2am

3a + n1

am2an

3a +m1

an2an

3a.


AAB-torus

Z[a] = n1an

2an

3a + 1

2m1an

2an

3a + 1

2n1am

2an

3a + 1

2n1an

2am

3a − 1

2m1am

2an

3a −m1

am2am

3a,

Y[a] =m1am

2am

3a + 2n1

am2am

3a + 2m1

an2am

3a + n1

an2am

3a + 3m1

am2an

3a

(A.2)+ 3n1am

2an

3a + 3m1

an2an

3a.

ABB-torus

Z[a] = n1an

2an

3a + 1

2n1an

2am

3a + 1

2n1am

2an

3a + 1

2m1an

2an

3a + 1

6n1am

2am

3a − 1

6m1am

2am

3a,

(A.3)

Y[a] = 3(m1am

2am

3a + n1

am2am

3a + 2m1

an2am

3a + 2m1

am2an

3a + n1

an2am

3a

+ n1am

2an

3a + 3m1

an2an

3a

).

BBB-torus

Z[a] = n1an

2an

3a + 1

2n1an

2am

3a + 1

2n1am

2an

3a + 1

2m1an

2an

3a + 1

6n1am

2am

3a

+ 1

6m1an

2am

3a + 1

6m1am

2an

3a,

(A.4)

Y[a] = 3(2m1

am2am

3a + 3n1

am2am

3a + 3m1

an2am

3a + 3m1

am2an

3a + 3n1

an2am

3a

+ 3n1am

2an

3a + 3m1

an2an

3a

).

Appendix B. Intersecting branes on the 6D Z3 orbifold

In this appendix we summarize the results for the tadpole cancellation conditions and themassless spectra for the six-dimensionalZ3 orbifolds. The orbifold action on two complexcoordinates is

(B.1)Z1 → e2πi/3Z1, Z2 → e−2πi/3Z2.

As in the four-dimensional case we can distinguish between the two differently orientedtori, A and B, so that in this case we get the three different models,AA, BB and AB.For the six-dimensional closed string spectrum withN = (0,1) supersymmetry one getsbesides the supergravity multiplet

AA: 8× tensors+ 13× hypers,

AB: 6× tensors+ 15× hypers,

(B.2)BB: 21× hypers.

Similar to the four-dimensional case we can define the following quantity

(B.3)Z[a] = 1

3

∑(nIb,m

Ib)∈[a]

2∏I=1

(nIb + 1

2mIb

),


Table B1Chiral fermions for theAA-torus

Representation Number

(Na, Nb)+ c.c. 2Z[a]Z[b] + 32Y[a]Y[b]

(Na,Nb)+ c.c. 2Z[a]Z[b] − 32Y[a]Y[b]

Aa + c.c. 2ZaAa + Sa + c.c. 2Z2[a] −Z[a] −∏

I (LIa)

2

Adja + c.c.∏I (L

Ia)

2

Table B2Chiral fermions for theAB-torus

Representation Number

(Na, Nb)+ c.c. 6Z[a]Z[b] + 2Y[a]Y[b](Na,Nb)+ c.c. 6Z[a]Z[b] − 2Y[a]Y[b]Aa + c.c. 6Z[a]Aa + Sa + c.c. 6Z2[a] − 3Z[a] − 3

∏I (L

Ia)

2

Adja + c.c. 3∏I (L

Ia)

2

which for the three different tori read

AA: Z[a] = n1an

2a + 1

2n1am

2a + 1

2m1an

2a − 1

2m1am

2a,

AB: Z[a] = n1an

2a + 1

2n1am

2a + 1

2m1an

2a + 1

6m1am

2a,

(B.4)BB: Z[a] = n1an

2a + 1

2n1am

2a + 1

2m1an

2a.

Then, the RR-tadpole cancellation condition can be expressed as

(B.5)∑a

NaZ[a] = 4.

Moreover, we define

AA,BB: Y[a] = n1am

2a +m1

an2a +m1

am2a,

(B.6)AB: Y[a] = 1

2n1am

2a + 3

2m1an

2a +m1

am2a,

and theLIa as in Eq. (4.14). Then the chiral massless spectra in the(1,2) representation ofthe little groupSO(4)= SU(2)× SU(2) for the three different four-dimensional tori (seeTables B1–B3) read:

Let us check explicitly the cancellation of theF 4 and R4 anomaly to provide anadditional check of the consistency of the construction. For theF 4 anomaly of theU(Na)


Table B3Chiral fermions for theBB-torus

Rep. Number

(Na, Nb)+ c.c. 18Z[a]Z[b] + 32Y[a]Y[b]

(Na,Nb)+ c.c. 18Z[a]Z[b] − 32Y[a]Y[b]

Aa + c.c. 18Z[a]Aa + Sa + c.c. 18Z2[a] − 9Z[a] − 9

∏I (L

Ia)

2

Adja + c.c. 9∏I (L

Ia)

2

gauge group we get

3nb

(∑b =a

4NbZ[a]Z[b] + 2(Na − 8)Z[a] + 2Na(2Z2[a] −Z[a]

))(B.7)= 3nb

(4Z[a](4−NaZ[a])− 16Z[a] + 4NaZ2[a]

)= 0.

TheR4 anomaly reads

3nb

(1

2

∑b =a

4NaNbZ[a]Z[b] +∑a

Na(Na − 1)

22Z[a] +

∑a

N2a

(2Z2[a] −Z[a]

))

(B.8)

= 3nb

(1

2

∑a

4NaZ[a](4−NaZ[a])−∑a

NaZ[a] +∑a

2N2aZ

2[a]

)= 28· 3nb ,

which is precisely what one needs to cancel theR4 anomaly resulting from the closedstring spectrum in (B.2).

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More exact predictions of SUSYM for string theoryGordon W. Semenoffa, K. Zaremboa,b,c

a Department of Physics and Astronomy, University of British Columbia, Vancouver, BC, V6T 1Z1 Canadab Pacific Institute for the Mathematical Sciences, University of British Columbia,

Vancouver, BC, V6T 1Z1 Canadac Institute of Theoretical and Experimental Physics, B. Cheremushkinskaya 25, 117259 Moscow, Russia

Received 12 June 2001; accepted 13 September 2001

Abstract

We compute the coefficients of an infinite family of chiral primary operators in the local operatorexpansion of a circular Wilson loop inN = 4 supersymmetric Yang–Mills theory. The computationsums all planar rainbow Feynman graphs. We argue that radiative corrections from planar graphswith internal vertices cancel in leading orders and we conjecture that they cancel to all ordersin perturbation theory. The coefficients are nontrivial functions of the ’t Hooft coupling and theirstrong coupling limits are in exact agreement with those previously computed using the AdS/CFTcorrespondence. They predict the subleading orders in strong coupling and could in principle becompared with string theory calculations. 2001 Published by Elsevier Science B.V.

PACS: 12.38.Cy; 12.60.Jv

1. Introduction and summary of main results

The idea that a quantized gauge theory could have a dual description as a string theoryhas a long history. Recently one concrete realization of such a duality has emerged. Ithas been conjectured [1] that there is an exact mapping betweenN = 4 supersymmetricYang–Mills theory (SYM) with gauge groupSU(N) on four-dimensional spacetime andIIB superstring theory on backgroundAdS5 × S5 with N units of RR flux.

This mapping is most useful in the low energy, weakly coupled limit of the string theory.This coincides with the largeN ’t Hooft limit [2] of SYM theory, whereN is takento infinity holding the combination of Yang–Mills coupling constant andN , defined byλ = g2N , fixed and then taking the largeλ limit. This projects onto the strong couplinglimit of the sum of planar Feynman diagrams. In string theory, this coincides with theclassical low energy limit where the string theory is accurately described by type IIB

E-mail address: [email protected] (K. Zarembo).

0550-3213/01/$ – see front matter 2001 Published by Elsevier Science B.V.PII: S0550-3213(01)00455-2


G.W. Semenoff, K. Zarembo / Nuclear Physics B 616 (2001) 34–46 35

supergravity on the background spaceAdS5 ×S5. Some explicit computations can be donethere. The results can then be interpreted in terms of the gauge theory using a well-definedprescription [3,4].

Though it has been used for many computations of the strong coupling limits of gaugetheory quantities (see Refs. [5–9] for reviews), it is difficult to obtain a direct check ofthe Maldacena conjecture. The reason for this is the fact that the correspondence withsupergravity computes gauge theory in the largeλ limit, with corrections from tree levelstring effects being suppressed by powers of 1/

√λ and sometimes computable to the next

order. On the other hand, the only other analytical tool which can be used systematicallyin the gauge theory is perturbation theory which is an asymptotic expansion in smallλ.Generally, the only quantities for which these expansions have an overlapping range ofvalidity is for quantities which are so protected by supersymmetry that they do not dependon the coupling constant.

There is, however, one known example of a quantity which is a nontrivial function ofthe coupling constant and whose largeN limit is computable and is thought to be knownto all orders in perturbation theory in planar diagrams. That quantity is the circular Wilsonloop. Its expectation value was computed in Ref. [10]. The contribution of a subset of allFeynman graphs, the planar rainbow diagrams, were found at each order inλ and the sumof all orders was taken to obtain the result

(1.1)⟨W [circle]⟩ = 2√

λI1

(√λ

) ≈√

2

π

e√λ

λ3/4 asλ → ∞,

whereI1(x) is a modified Bessel function. It was also shown explicitly that the leadingcorrections to the sum of rainbow diagrams cancels identically. It was conjectured that thiscancellation would also occur at higher orders and the result (1.1) was thus the exact sumof all planar diagrams. Some support for this conjecture was developed in Ref. [11]. Theyalso observed that the sum over Feynman diagrams could be obtained for all orders in the1/N expansion and had a beautiful argument that, in the largeλ limit, these higher ordersproduced the expected higher genus string corrections.

The largeλ limit in (1.1) agrees with the expectation value of the circular Wilson loopwhich was computed using the AdS/CFT correspondence in Refs. [12,13]. If (1.1) isindeed an exact result, this provides a nontrivial check on the validity of the AdS/CFTcorrespondence. It gives the further interesting possibility of comparing corrections atsubleading orders in 1/

√λ with string theory computations. Investigations of the relevant

string theory technique appears in Refs. [14–19] but explicit calculations of the 1/√λ

corrections have not yet been done.In this paper we shall report the computation of a series of expansion coefficients which

are related to the circular Wilson loop and chiral primary operators inN = 4 SYM theory.The problem that we pose is the following. When probed from a distance much larger thanthe size of the loop, the Wilson loop operator can be expanded in a series of local operatorswith some coefficients [12,20]:

(1.2)W [C] = ⟨W [C]⟩∑

∆

CAR∆AOA(0),

36 G.W. Semenoff, K. Zarembo / Nuclear Physics B 616 (2001) 34–46

whereOA(0) is a local operator evaluated at the center of the loop,∆A is the conformaldimension ofOA(x) and R is the radius of the loop. The problem is to compute thecoefficientsCA in this operator product expansion (OPE).

We shall concentrate on computing the coefficients for a particular class of chiralprimary operators (CPO). We will be able to compute the contribution of the sum of allplanar rainbow graphs to the coefficientsCA in that case. We are also able to show that theleading order corrections to this sum, which come from diagrams with internal vertices,cancels identically. This leads us to conjecture that the radiative corrections cancel to allorders and the sum of planar rainbow graphs gives the exact result.

We find that the coefficients that we compute are nontrivial functions of the couplingconstant. In the limit of largeλ they coincide with results of the AdS/CFT correspon-dence [12]. This gives a large array of nontrivial functions of the coupling constant whichcould be compared with string computations of the strong coupling limit. There are var-ious reasons why these computations could be simpler than the 1/

√λ corrections to the

expectation value of the Wilson loop itself.The Wilson loop operator inN = 4 SYM theory that is readily computed using

the AdS/CFT correspondence and which has the right transformation properties undersupersymmetry [13,21] contains the scalar fields inside the path-ordered exponential:

(1.3)W [C] = 1

NtrP exp

[∮C

dτ(iAµ(x)xµ +Φi(x)θi |x|)

],

wherexµ(τ) parameterize the contourC andθi are Cartesian coordinates of a point onS5: θ2 = 1. If the size of the contourC is small, the Wilson loop can be expanded in localoperators as in (1.2).

For primary operators, one can choose a basis where

(1.4)⟨OA(x)OB(y)

⟩ = δAB

|x − y|∆A+∆B.

Then their OPE coefficients can be extracted from the large distance behavior of connectedtwo-point correlation functions,

(1.5)〈W(C)OA(L)〉c

〈W(C)〉 = CAR∆A

L2∆A+ · · · ,

whereL R and the omitted terms are of higher order inR2/L2.The coefficient corresponding to the CPO of lowest conformal dimension, which in this

case is∆ = 2, is important as it determines the correlator of two Wilson loops with largeseparation

(1.6)〈W [C1]W [C2]〉c〈W [C1]〉〈W [C2]〉 = C2

2

(R

L

)4

+ · · · .

The coefficients of various CPOs in the expansion of the circular Wilson loop werecalculated in Ref. [12] both perturbatively atλ ∼ 0 and at strong coupling,λ ∼ ∞, using


the AdS/CFT correspondence. Evaluation of the correlators that define coefficients in thestrong-coupling regime involves a hybrid of the supergravity and the string calculations.

In AdS/CFT, the Wilson loop operator (1.3) naturally couples to stringy degrees offreedom. It creates a classical string world-sheet which is embedded inAdS5 × S5 andwhose boundary is the contour of the Wilson loop [21,22].

On the other hand, a local operatorOA(x) emits one of the supergravity fields at pointx.When it contributes to a correlator ofOA(x) with the Wilson loop, this supergravity modepropagates on the backgroundAdS5 × S5 and is then absorbed by a vertex operator whichmust be integrated over the string world-sheet.

1.1. Dimension two operators

Let us begin by considering the CPO with smallest conformal dimension,∆ = 2. It isthe symmetric traceless part of a gauge invariant product of scalar fields,

(1.7)Oij = 8π√2λ

tr

(ΦiΦj − 1

6δijΦ2

).

This operator is the lowest weight component of a short multiplet ofN = 4 super-conformal algebra. Such chiral primary operators have very special properties. The super-conformal algebra guarantees that their conformal dimensions do not receive radiativecorrections, so in this case the conformal dimension is exactly two. Furthermore, it isknown that their two and three-point correlation functions are given by the free field values,that is, that they are independent of the coupling constant,g. It is known that their four-point functions are nontrivial, so they are not free fields in disguise [23–26].

In (1.7), the overall coefficient is chosen to give a canonical normalization of the two-point function:

(1.8)⟨Oij (x)Okl(y)

⟩ = 1

2

(δikδjl + δilδjk − 1

3δij δkl

)1

|x − y|4 .

The small coupling limit of the correlator ofOij with the Wilson loop is straightforwardto obtain. To leading order in perturbation theory:

(1.9)〈W(C)Oij 〉

〈W(C)〉 = 1

N

1

2√

2λ

(θ iθj − 1

6δij

)R2

L4(λ → 0).

The linear dependence onλ is an obvious consequence of the fact that the correlatorcontains two propagators and one power ofλ is cancelled by the normalization.

The AdS dual of the dimension two operator is the negative mass scalar which is alinear combination of the trace of the metric and the Ramond–Ramond four-form field. Itscontribution to the OPE of the circular Wilson loop was calculated in [12]:

(1.10)〈W(C)Oij 〉

〈W(C)〉 = 1

N

√2λ

(θ iθj − 1

6δij

)R2

L4 (λ → ∞).

Comparing OPE coefficients at strong and at weak coupling, we see that the scaling withλ is different. The OPE coefficients are clearly renormalized by radiative corrections. We


shall conjecture that, in the largeN limit, this renormalization is entirely due to planarrainbow diagrams. We shall also obtain the sum of planar rainbows as

(1.11)〈W(C)Oij 〉

〈W(C)〉 = 1

N

√2λ

I2(√λ )

I1(√λ )

(θ iθj − 1

6δij

)R2

L4 ,

whereI2 andI1 are modified Bessel functions. By construction, this expression reduces to(1.9) at smallλ. Since

(1.12)limλ→∞

Ik(√λ)

I1(√λ )

= 1

for anyk, the AdS/CFT prediction (1.10) is also exactly reproduced at largeλ. The sum ofrainbow diagrams thus interpolates between perturbative and strong coupling limits of theOPE coefficient.

1.2. Chiral primary operators

TheOij is the first in an infinite sequence of CPOs. The operator of dimensionk in thissequence is a symmetrized trace ofk scalar fields:

(1.13)OIk = (8π2)k/2

√k λk/2

CIi1···ik trΦi1 · · ·Φik ,

whereCIi1···ik are totally symmetric traceless tensors normalized as

(1.14)CIi1···ikC

Ji1···ik = δIJ .

The AdS duals of CPOs are Kaluza–Klein modes of theAdS5 tachyonic scalar onS5

and each CPO is associated with a spherical harmonic:

(1.15)Y I (θ) = CIi1···ik θ

i1 · · ·θ ik .Here, we are following the notation of Refs. [12,23].

The OPE coefficients depend on how the operators are normalized. When comparingperturbative calculations with the AdS/CFT predictions, we need to use the samenormalization. For operators in short multiplets ofN = 4 supersymmetry, this is easy toachieve, since the two point correlation functions of such operators do not receive radiativecorrections and can be used to fix normalization. The coefficient in (1.13) is chosen to unitnormalize the two point function:

(1.16)⟨OI

k (x)OJk (y)

⟩ = δIJ

|x − y|2k .

The same conventions were used in the supergravity calculations of Ref. [12].At weak coupling, the OPE coefficient of the circular Wilson loop is proportional toλk/2,

wherek is the dimension of the CPO:

(1.17)〈W(C)OI

k 〉〈W(C)〉 = 1

N2−k/2

√k

k! λk/2 Rk

L2k YI (θ) (λ → 0).


It turns out that AdS/CFT correspondence predicts a universal scaling of the OPEcoefficients withλ at strong coupling: all of them are proportional to

√λ independently

of k. This can be easily understood by considering a pair correlator of the Wilsonloops, which is quadratic in OPE coefficients. The Wilson loop correlator is describedby an annulus string amplitude and, therefore, is proportional to the string couplinggs .According to the AdS/CFT dictionary,

gs = g2

4π= λ

4πN.

Hence, OPE coefficients must scale as√λ. An explicit calculation gives [12]:

(1.18)〈W(C)OI

k 〉〈W(C)〉 = 1

N2k/2−1

√kλ

Rk

L2k YI (θ) (λ → ∞).

Our main result is an expression for correlators of the circular Wilson loop with CPOs:

(1.19)〈W(C)OI

k 〉〈W(C)〉 = 1

N2k/2−1

√kλ

Ik(√λ )

I1(√λ )

Rk

L2k YI (θ)

which we expect is exact in the largeN limit. Its expansion inλ reproduces (1.17). Thestrong-coupling limit exactly coincides with the AdS/CFT prediction (using Eq. (1.12)).

2. Resummation of rainbow diagrams

The Euclidean action ofN = 4 supersymmetric Yang–Mills theory is

S =∫

d4xN

λTr

1

2F 2µν + (

DµΦi)2 −

∑i<j

[Φi,Φj

]2

(2.1)+ψT Γ µDµψ − iψT Γ i[Φi,ψ

],

where(Γ µ,Γ i) are ten-dimensional Dirac matrices in the Majorana–Weyl representation.We will work in the Feynman gauge where the gauge field propagator has the form

(2.2)⟨Aab

µ (x)Acdν (y)

⟩0 = λ

δµν

8π2(x − y)2

δadδbc

N.

Our calculation of the OPE coefficients begins with summing all planar rainbowdiagrams of the kind shown in Fig. 1. They containk scalar propagators connecting thepointL to the Wilson loop and propagators of scalars and gauge fields connecting pointsin segments of the loop.

If L lies on the axis of symmetry of the circle, the scalar propagators are constantsequal to

1

8π2(L2 +R2).


Fig. 1. A typical diagram that contributes to the correlator of the circular Wilson loop with thedimensionk CPO.

If the origin is displaced from the axis of symmetry, the propagators will depend onpositions of their endpoints on the circle. In any case, we will be interested in the large-distance asymptotics, and the propagators can be set to 1/8π2L2, up to corrections ofhigher order in 1/L.

The problem thus reduces to resummation of rainbow diagrams for each of the segmentsof the circle. This problem was solved in Ref. [27]. In the following we will review thesalient points involved in finding the solution. We start with the dimension two operator(1.7), when there are only two segments.

2.1. Dimension two operators

In this case we have:

(2.3)

⟨W(C)Oij

⟩ = 1

N

8π√2λ

λ2

(8π2L2)2

(θ iθj − 1

6δij

)R22π

2π∫0

dϕW(ϕ)W(2π − ϕ),

whereW(ϕ′ − ϕ) denotes the sum of rainbow graphs for a segment of the circle betweenpolar anglesϕ andϕ′. To compute this sum, we notice that the sum of the scalar and thegluon propagators between any two points on a circle does not depend on the positions ofthese points:⟨(

iAµ(x)xµ +Φi(x)θ i |x|)ab

(iAµ(y)yµ +Φi(y)θ i |y|)

cd

⟩0

(2.4)= λ

Nδadδbc

|x||y| − x · y8π2|x − y|2 = λ

16π2Nδadδbc.

This observation allows us to replace the field-theory Wick contraction by the matrix-model average defined by the partition function

(2.5)Z =∫

dM exp

(−8π2

λN trM2

).

Upon the replacement ofiAµ(x)xµ + Φi(x)θ i |x| by M, the sum of rainbow diagrams inthe segment of the lengthϕ reduces to the matrix-model counterpart of the Wilson loop:

(2.6)W(ϕ) =⟨

1

Ntr eϕM

⟩.


The matrix model can be viewed as a combinatorial tool which simply counts the numberof planar graphs [10].

The Wilson loop in the matrix model satisfies Schwinger–Dyson identity in the large-N

limit (the loop equation [28]):

(2.7)W ′(ϑ) = λ

16π2

ϑ∫0

dϕW(ϕ)W(ϑ − ϕ).

The solution [10,27,29] of the loop equation is

(2.8)W(ϕ) = 4π√λϕ

I1

(√λϕ

2π

).

The integral we need to compute in order to calculate the correlation function (2.3) isthe right-hand side of the loop equation atθ = 2π . Using properties of the modified Besselfunctions,

I ′k(z) = 1

2

(Ik−1(z)+ Ik+1(z)

),

(2.9)kIk(z) = z

2

(Ik−1(z)− Ik+1(z)

),

we get:

(2.10)

ϑ∫0

dϕW(ϕ)W(ϑ − ϕ) = 32π2

λϑI2

(√λϑ

2π

).

Settingθ = 2π and substituting into (2.3), we obtain:

(2.11)⟨W(C)Oij

⟩ = 1

N2√

2I2(√

λ)(

θ iθj − 1

6δij

)R2

L4.

Dividing by the vacuum expectation value of the Wilson loop,

(2.12)⟨W(C)

⟩ = W(2π) = 2√λI1

(√λ

),

we arrive at the result (1.11) which we quoted earlier.

2.2. Chiral primary operators

The correlator of the Wilson loop with the CPO of dimensionk contains an integral overk − 1 endpoints of the scalar propagators (one integration yields an overall factor of 2π ):⟨

W(C)OIk

⟩= 1

N

(8π2)k/2

√k λk/2

λk

(8π2L2)kCIi1···ik θ

i1 · · ·θ ikRk

(2.13)× 2π

2π∫0

dϕ1 · · ·ϕk−2∫0

dϕk−1W(ϕk−1)W(ϕk−2 − ϕk−1) · · ·W(2π − ϕ1).


It is useful to introduce

(2.14)Fk(ϕ) =ϕ∫

0

dϕ1 · · ·ϕk−2∫0

dϕk−1W(ϕk−1)W(ϕk−2 − ϕk−1) · · ·W(ϕ − ϕ1).

The correlator is expressed in terms ofFk(2π) as

(2.15)⟨W(C)OI

k

⟩ = 1

N

2π√k (8π2)k/2

λk/2Fk(2π)Rk

L2k YI (θ).

To find the functionsFk(ϕ), we again use the loop equation. DifferentiatingFk(ϕ) andusing (2.7), we get the recurrence relations:

(2.16)F ′k(ϕ) = Fk−1(ϕ)+ λ

16π2Fk+1(ϕ),

which are supplemented by initial conditions

(2.17)F1(ϕ) = W(ϕ), F0(ϕ) = 0.

These unambiguously determine allFk . A systematic way to solve these recurrencerelations is to introduce a generating function and then use a Laplace transform to convertdifferential equations into algebraic equations. The result is

(2.18)Fk(ϕ) = k

ϕ

(4π√λ

)k

Ik

(√λϕ

2π

).

It is straightforward to check that this expression solves the recurrence relations with thehelp of (2.9).

Substituting (2.18) into (2.15), we obtain

(2.19)⟨W(C)OI

k

⟩ = 1

N2k/2

√k Ik

(√λ

) Rk

L2kY I (θ).

Normalizing by the Wilson loop expectation value (2.12), we get (1.19).

3. Radiative corrections

The leading radiative corrections come form the Feynman diagrams which are shown inFig. 2.1

Each of these diagrams is separately divergent and regularization is required to definethem properly. We use a regularization by dimensional reduction which was previouslyused in Ref. [10]. The essential observation is thatN = 4 SYM is obtained by dimensionalreduction of ten-dimensional SYM. This dimensional reduction retains sixteen supersym-metries in any dimension. Thus, a supersymmetric-dimensional regularization ofN = 4SYM theory is obtained by dimensionally reducing ten dimensions to 4− ε dimensions.

In this dimensional regularization, the diagram in Fig. 2(a) is of higher order than therelevant leading power,R2/L4, and therefore does not contribute toC2.

1 The diagram similar toe, but with scalar lines replaced by gluon propagators does not contribute because ofR-charge conservation.


Fig. 2. Leading radiative corrections to the correlator of the Wilson loop operator with thek = 2CPO. These are not taken into account by the sum over planar rainbow graphs which were computedin Section 3.

Fig. 3. The contribution to the leading term inR2/L4 of the diagrams in Fig. 2(b)–(e) aregiven by these 2-loop diagrams. This combination of diagrams is known to vanish due to thenonrenormalization theorem of the 2-point function of the CPO.

In the limit L R, using dimensional regularization, the leading,R2/L4, contributionsof the remaining diagrams in Fig. 2 can be seen to be identical to results of computing thediagrams which are displayed in Fig. 3. This sum of diagrams is known to vanish whenthe dimension is exactly four, due to the nonrenormalization theorem for the two-pointfunction of the CPO. This nonrenormalization results from super-conformal invariance.

Similar arguments apply to the higher CPO’s for which similar nonrenormalizationtheorems can be applied [30].

This is by no means a proof that all radiative corrections vanish. But the excellentagreement with strong coupling AdS/CFT results gives optimism that it is indeed the case.

4. Remarks

Our main results (1.11) and (1.19) are valid when the distance from operator insertionto the loop is much larger than the loop’s radius. However, it is not hard to restore the


dependence on the radius and on the orientation of the circle. Consider first the case whenthe operator is inserted at the symmetry axis of the circle. As follows from the discussionin the beginning of Section 2, we can find the correlators of the Wilson loop with CPOsat anyR andL, not only at large distances, simply by replacingL2 with L2 + R2. Asexpected, this is perfectly consistent with the supergravity prediction, which actually isknown in complete generality. If we denote the displacement of the operator insertion fromthe axis of symmetry byr and, as before,L is the distance from the plane of the circle, thecorrelation function is obtained from the large-distance asymptotics by replacing

(4.1)Rk

L2k→ Rk[

(L2 + r2 −R2)2 + 4L2R2]k/2

.

At r = 0, the denominator on the right-hand side indeed coincides with(L2 +R2)k .We were not able to compute the sum of rainbow diagrams forr = 0, but the validity

of the prescription (4.1) can be proved by an indirect argument. The point is that, once weknow the correlators atr = 0, the dependence onr is unambiguously fixed by conformalinvariance, becauser can be always set to zero by a special conformal transformation,which maps a circle onto a circle. This transformation is not anomalous and, therefore,does not affect the dependence of the correlator onλ.

Another remark concerns the dependence of the OPE coefficients onk. It was arguedon general grounds that coupling of a Wilson loop to states of very large spin should befactorially suppressed [31]. In a theory with an exponential density of states, unsuppressedcoupling to such states would lead to a catastrophe in the pair correlator of Wilson loopssimilar to the Hagedorn transition. The CPOs we consider in this paper carry the spin ofSO(6) R-symmetry which forOI

k is equal tok. At weak coupling, the OPE coefficientsare indeed suppressed at largek, as follows from (1.17), but the supergravity result (1.18)seems to suggest that this suppression disappears at strong coupling. Careful inspection ofthe exact OPE coefficients (1.19) shows that limitsλ → ∞ andk → ∞ do not commuteand coupling of the Wilson loop to operators of very high spin is always suppressed, butthe suppression begins with operators of parametrically large spink ∼ λ. At k λ:

(4.2)Ck ∝ 2−k/2−1

√k

k!λ(k+1)/2

I1(√λ )

.

5. Discussion

We have calculated the OPE coefficients of the circular Wilson loops by resummation ofthe planar Feynman graphs without internal vertices. It is likely that other diagrams cancelto all orders of perturbation theory. We have successfully checked this conjecture up totwo loops. Complete agreement of an infinite set of OPE coefficients with the supergravitypredictions at strong coupling strongly suggests that our results are indeed exact in the’t Hooft limit. It would be interesting to see if the arguments of Ref. [11] based on diagram-by-diagram conformal transformations can be invoked to prove that only rainbow diagramscontribute to the OPE coefficients of the circular Wilson loop with CPOs.


Our result for the OPE coefficients, along with the known exact expression for thevacuum expectation value of the circular loop, can be regarded as a prediction for thestring theory inAdS5 × S5. The usualα′ expansion of the world-sheet sigma-model thencoincides with the expansion in 1/

√λ:

〈W(C)OIk 〉

〈W(C)〉(5.1)= 1

N2k/2−1

√kλ

(1− k2 − 1

2√λ

+ k4 − 4k2 + 3

8λ+ O

(1

λ3/2

))Rk

L2kY I (θ).

The calculation of the stringy correction to the expectation of the Wilson loop is a hardproblem analogous to instanton calculations in field theory (see [19], for details). As usual,such problems require delicate treatment of various normalization factors associated withzero modes and with regularization of fluctuation determinants. However, in the ratio of theexpectation values (5.1), these normalization factors cancel. For this reason, a calculationof stringy corrections to the OPE coefficients seems less complicated and perhaps can beaccomplished without tremendous effort.

It would also be interesting to consider similar correlators of Wilson loops with differentcontours [32] or with other operators, where some preliminary results are contained inRefs. [33,34].

Acknowledgements

We are grateful to Jan Plefka, Matthias Staudacher and Arkady Tseytlin for usefulcomments. This work was supported by NSERC of Canada and NATO CollaborativeLinkage Grant SA(PST.CLG.977361)5941.The work of K.Z. was also supported in part bythe Pacific Institute for the Mathematical Sciences and in part by RFBR grant 01-01-00549and RFBR grant 00-15-96557 for the promotion of scientific schools.

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Simple SUSY breaking mechanismby coexisting walls

Nobuhito Marua, Norisuke Sakaib,∗, Yutaka Sakamurab, Ryo Sugisakaba Department of Physics, University of Tokyo, Tokyo 113-0033, Japan

b Department of Physics, Tokyo Institute of Technology, Tokyo 152-8551, Japan


Abstract

A SUSY breaking mechanism with no messenger fields is proposed. We assume that our world ison a domain wall and SUSY is broken only by the coexistence of another wall with some distancefrom our wall. We find anN = 1 model in four dimensions which admits an exact solution of astable non-BPS configuration of two walls and studied its properties explicitly. We work out howvarious soft SUSY breaking terms can arise in our framework. Phenomenological implications arebriefly discussed. We also find that effective SUSY breaking scale becomes exponentially small asthe distance between two walls grows. 2001 Elsevier Science B.V. All rights reserved.

PACS:11.27.+d; 11.30.Pb; 12.60.Jv

1. Introduction

Supersymmetry (SUSY) is one of the most promising ideas to solve the hierarchyproblem in unified theories [1]. It has been noted for some years that one of the mostimportant issues for SUSY unified theories is to understand the SUSY breaking in ourobservable world. Many models of SUSY breaking uses some kind of mediation of theSUSY breaking from the hidden sector to our observable sector. Supergravity providesa tree level SUSY breaking effects in our observable sector suppressed by the PlanckmassMPl [2]. Gauge mediation models uses messenger fields to communicate the SUSYbreaking at the loop level in our observable sector [3].

Recently there has been an active interest in the “Brane World” scenario where our four-dimensional spacetime is realized on the wall in higher-dimensional spacetime [4,5]. In

* Corresponding author.E-mail addresses:[email protected] (N. Maru), [email protected] (N. Sakai),

[email protected] (Y. Sakamura), [email protected] (R. Sugisaka).



48 N. Maru et al. / Nuclear Physics B 616 (2001) 47–84

order to discuss the stability of such a wall, it is often useful to consider SUSY theoriesas the fundamental theory. Moreover, SUSY theories in higher dimensions are a naturalpossibility in string theories. These SUSY theories in higher dimensions have 8 or moresupercharges, which should be broken partially if we want to have a phenomenologicallyviable SUSY unified model in four dimensions. Such a partial breaking of SUSY isnicely obtained by the topological defects [6]. Domain walls or other topological defectspreserving part of the original SUSY in the fundamental theory are called the BPS statesin SUSY theories. Walls have co-dimension one and typically preserve half of the originalSUSY, which are called 1/2 BPS states [7–9]. Junctions of walls have co-dimension twoand typically preserve a quarter of the original SUSY [10,11].

Because of the new possibility offered by the brane world scenario, there has been arenewed interest in studies of SUSY breaking. It has been pointed out that the non-BPStopological defects can be a source of SUSY breaking [8] and an explicit realizationwas considered in the context of families localized in different BPS walls [12]. Modelshave also been proposed with bulk fields mediating the SUSY breaking from the hiddenwall to our wall on which standard model fields are localized [13–16]. The localizationof the various matter wave functions in the extra dimensions was proposed to offera natural realization of the gaugino-mediation of the SUSY breaking [17]. Recentlywe have proposed a simple mechanism of SUSY breaking due to the coexistence ofdifferent kinds of BPS domain walls and proposed an efficient method to evaluate theSUSY breaking parameters such as the boson–fermion mass-splitting by means of overlapof wave functions involving the Nambu–Goldstone (NG) fermion [18], thanks to thelow-energy theorem [19,20]. We have exemplified these points by taking a toy modelin four dimensions, which allows an exact solution of coexisting walls with a three-dimensional effective theory. Although the model is only meta-stable, we were able to showapproximate evaluation of the overlap allows us to determine the mass-splitting reliably.

The purpose of this paper is to illustrate our idea of SUSY breaking due to the coexis-tence of BPS walls by taking a simple soluble model with a stable non-BPS configurationof two walls and to extend our analysis to more realistic case of four-dimensional effectivetheories. We also examine the consequences of our mechanism in detail.

We propose a SUSY breaking mechanism which requires no messenger fields, norcomplicated SUSY breaking sector on any of the walls. We assume that our world is ona wall and SUSY is broken only by the coexistence of another wall with some distancefrom our wall. We find anN = 1 supersymmetric model in four dimensions which admitsan exact solution of a stable non-BPS configuration of two walls and study its propertiesexplicitly. We work out how various soft SUSY breaking terms can arise in our framework.Phenomenological implications are briefly discussed. We also find that effective SUSYbreaking scale observed on our wall becomes exponentially small as the distance betweentwo walls grows. The NG fermion is localized on the distant wall and its overlap with thewave functions of physical fields on our wall gives the boson–fermion mass-splitting ofphysical fields on our wall thanks to a low-energy theorem. We propose that this overlapprovides a practical method to evaluate the mass-splitting in models with SUSY breakingdue to the coexisting walls.

N. Maru et al. / Nuclear Physics B 616 (2001) 47–84 49

In Section 2, a model is introduced that allows a stable non-BPS two-wall configurationas a classical solution. We have also worked out mode expansion on the two-wallbackground, three-dimensional effective Lagrangian, and the single-wall approximationfor the overlap of mode functions to obtain the mass-splitting. Matter fields are alsointroduced. Section 3 is devoted to study how various soft breaking terms arise in thethree-dimensional effective theory. Soft breaking terms in four-dimensional effectivetheory are worked out in Section 4. Phenomenological implications are discussed inSection 5. Additional discussion is given in Section 6. Appendix A is devoted to discussingthe low-energy theorem in three dimensions and the mixing matrix relating the masseigenstates and superpartner states. Low-energy theorems in four dimensions are derivedin Appendix B. In Appendix C, we derive a relation among the order parameters of theSUSY breaking, the energy density of the configuration and the central charge of the SUSYalgebra.

2. SUSY breaking by the coexistence of walls

2.1. Stable non-BPS configuration of two walls

We will describe a simple soluble model for a stable non-BPS configuration thatrepresents two-domain-wall system, in order to illustrate our basic ideas. Here we considerdomain walls in four-dimensional spacetime to avoid inessential complications. Weintroduce a simple four-dimensional Wess–Zumino model as follows:1

(2.1)L = ΦΦ∣∣θ2θ2 +W(Φ)

∣∣θ2 + h.c., W(Φ)= Λ3

g2 sin

(g

ΛΦ

),

whereΦ is a chiral superfieldΦ(Zµ, θ) = A(Zµ) + √2θΨ (Zµ) + θ2F(Zµ), Zµ ≡

Xµ + iθσµθ . A scale parameterΛ has a mass-dimension one and a coupling constantg is dimensionless, and both of them are real positive. In the following, we choosey =X2

as the extra dimension and compactify it onS1 of radiusR. Other coordinates are denotedasxm (m= 0,1,3), i.e.,Xµ = (xm, y). The bosonic part of the model is

(2.2)Lbosonic= −∂µA∗∂µA− Λ4

g2

∣∣∣∣cos

(g

ΛA

)∣∣∣∣2.The target space of the scalar fieldA has a topology of a cylinder as shown in Fig. 1. Thismodel has two vacua atA= ±πΛ/(2g), both lie on the real axis.

Let us first consider the case of the limitR → ∞. In this case, there are two kinds of BPSdomain walls in this model. One of them is

(2.3)A(1)cl (y)= Λ

g

2 tan−1 eΛ(y−y1) − π

2

,

1 We follow the conventions in Ref. [21].


Fig. 1. The target space of the scalar fieldA. The line at ReA= πΛ/g and the line at ReA= −πΛ/gare identified each other.

which interpolates the vacuum atA = −πΛ/(2g) to that atA= πΛ/(2g) asy increasesfrom y = −∞ to y = ∞. The other wall is

(2.4)A(2)cl (y)= Λ

g

−2 tan−1 e−Λ(y−y2) + 3π

2

,

which interpolates the vacuum atA = πΛ/(2g) to that atA = 3πΛ/(2g) = −πΛ/(2g).Herey1 andy2 are integration constants and represent the location of the walls along theextra dimension. The four-dimensional superchargeQα can be decomposed into two two-component Majorana superchargesQ

(1)α andQ(2)

α which can be regarded as superchargesin three dimensions

(2.5)Qα = 1√2

(Q(1)α + iQ(2)

α

).

Each wall breaks a half of the bulk supersymmetry:Q(1)α is broken byA(2)

cl (y), andQ(2)α

byA(1)cl (y). Thus all of the bulk supersymmetry will be broken if these walls coexist.

We will consider such a two-wall system to study the SUSY breaking effects in the low-energy three-dimensional theory on the background. The field configuration of the twowalls will wrap around the cylinder in the target space ofA asy increases from 0 to 2πR.Such a configuration should be a solution of the equation of motion,

(2.6)∂µ∂µA+ Λ3

gsin

(g

ΛA∗)

cos

(g

ΛA

)= 0.

We can easily show that the minimum energy static configuration with unit windingnumber should be real. We find that a general real static solution of Eq. (2.6) that dependsonly ony is

(2.7)Acl(y)= Λ

gam

(Λ

k(y − y0), k

),

where k and y0 are real parameters and the function am(u, k) denotes the amplitudefunction, which is defined as an inverse function of


Fig. 2. The profile of the classical solutionAcl(y). The dotted linesA = −πΛ/(2g) andA= 3πΛ/(2g) are identified.

(2.8)u(ϕ)=ϕ∫

0

dθ√1− k2 sin2 θ

.

If k > 1, it becomes a periodic function with the period 4K(1/k)/Λ, where the functionK(k) is the complete elliptic integral of the first kind. Ifk < 1, the solutionAcl(y) is amonotonically increasing function with

(2.9)Acl

(y + 4kK(k)

Λ

)=Acl(y)+ 2π

Λ

g.

This is the solution that we want. Since the fieldA is an angular variableA = A +2πΛ/g, we can choose the compactified radius 2πR = 4kK(k)/Λ so that the classicalfield configurationAcl(y) contains two walls and becomes periodic modulo 2πΛ/g. Weshall takey0 = 0 to locate one of the walls aty = 0. Then we find that the other wallis located at the anti-podal pointy = πR of the compactified circle. We have computedthe energy of a superposition of the first wallA

(1)cl (y) located aty = y1 in Eq. (2.3) and

the second wallA(2)cl (y) located aty = y2 in Eq. (2.4). This energy can be regarded as a

potential between two walls in the adiabatic approximation and has a peak at|y1 − y2| = 0implying that two walls experience a repulsion. This is in contrast to a BPS configuration oftwo walls which should exert no force between them. Thus we can explain that the secondwall is settled at the anti-podal pointy = πR in our stable non-BPS configuration becauseof the repulsive force between two walls.

In the limit ofR → ∞, i.e.,k → 1,Acl(y) approaches to the BPS configurationA(1)cl (y)

with y1 = 0 neary = 0, which preservesQ(1), and toA(2)cl (y) with y2 = πR neary = πR,

which preservesQ(2). The profile of the classical solutionAcl(y) is shown in Fig. 2. Wewill refer to the wall aty = 0 as “our wall” and the wall aty = πR as “the other wall”.


2.2. The fluctuation mode expansion

Let us consider the fluctuation fields around the backgroundAcl(y),

A(X)=Acl(y)+ 1√2

(AR(X)+ iAI(X)

),

(2.10)Ψα(X)= 1√2

(Ψ (1)α (X)+ iΨ (2)

α (X)).

To expand them in modes, we define the mode functions as solutions of equations:−∂2

y −Λ2 cos

(2g

ΛAcl(y)

)bR,p(y)=m2

R,pbR,p(y),

(2.11)−∂2

y +Λ2bI,p(y)=m2I,p(y)bI,p(y),

−∂y −Λsin

(g

ΛAcl(y)

)f (1)p (y)=mpf

(2)p (y),

(2.12)

∂y −Λsin

(g

ΛAcl(y)

)f (2)p (y)=mpf

(1)p (y).

The four-dimensional fluctuation fields can be expanded as

(2.13)AR(X)=∑p

bR,p(y)aR,p(x), AI(X)=∑p

bI,p(y)aI,p(x),

(2.14)Ψ (1)(X)=∑p

f (1)p (y)ψ(1)

p (x), Ψ (2)(X)=∑p

f (2)p (y)ψ(2)

p (x).

As a consequence of the linearized equation of motion, the coefficientaR,p(x) andaI,p(x)

are scalar fields in three-dimensional effective theory with massesmR,p andmI,p, and

ψ(1)p (x) andψ(2)

p (x) are three-dimensional spinor fields with massesmp , respectively.Exact mode functions and mass-eigenvalues are known for several light modes

of bR,p(y),

bR,0(y)= CR,0 dn

(Λy

k, k

), m2

R,0 = 0,

bR,1(y)= CR,1 cn

(Λy

k, k

), m2

R,1 = 1− k2

k2 Λ2,

(2.15)bR,2(y)= CR,2 sn

(Λy

k, k

), m2

R,2 = Λ2

k2,

where functions dn(u, k), cn(u, k), sn(u, k) are the Jacobi’s elliptic functions andCR,p arenormalization factors. ForbI,p(y), we can find all the eigenmodes

(2.16)bI,p(y)= 1√2πR

eipR y, m2

I,p =Λ2 + p2

R2(p ∈ Z).

The massless fieldaR,0(x) is the Nambu–Goldstone (NG) boson for the breaking of thetranslational invariance in the extra dimension. The first massive fieldaR,1(x) corresponds


Fig. 3. The mode functions for the bosonic modesaR,0 andaR,1. The solid line represents the profileof bR,0(y) and the dashed line is that ofbR,1(y).

Fig. 4. The mode functions for fermionic zero-modesψ(1)0 andψ(2)

0 . The solid line represents the

profile off (1)0 (y) and the dashed line is that off (2)0 (y).

to the oscillation of the background wall around the anti-podal equilibrium point and hencebecomes massless in the limit ofR → ∞. All the other bosonic fields remain massive inthat limit.

For fermions, only zero modes are known explicitly,

f(1)0 (y)= C0

dn

(Λy

k, k

)+ k cn

(Λy

k, k

),

(2.17)f(2)0 (y)= C0

dn

(Λy

k, k

)− k cn

(Λy

k, k

),

whereC0 is a normalization factor. These fermionic zero modes are the NG fermions forthe breaking ofQ(1)-SUSY andQ(2)-SUSY, respectively.

Thus there are four fields which are massless or become massless in the limit ofR → ∞:aR,0(x), aR,1(x), ψ

(1)0 (x) andψ(2)

0 (x). The profiles of their mode functions are shown inFigs. 3 and 4. Other fields are heavier and have masses of the order ofΛ.

In the following discussion, we will concentrate ourselves on the breaking of theQ(1)-SUSY, which is approximately preserved by our wall aty = 0. So we call the fieldψ(2)

0 (x)

the NG fermion in the rest of the paper.


2.3. Three-dimensional effective Lagrangian

We can obtain a three-dimensional effective Lagrangian by substituting the mode-expanded fields Eqs. (2.13) and (2.14) into the Lagrangian (2.1), and carrying out anintegration overy

L(3) = −V0 − 1

2∂maR,0∂maR,0 − 1

2∂maR,1∂maR,1 − i

2ψ(1)0 /∂ψ

(1)0 − i

2ψ(2)0 /∂ψ

(2)0

(2.18)− 1

2m2

R,1a2R,1 + geff aR,1ψ

(1)0 ψ

(2)0 + · · · ,

where /∂ ≡ γm(3)∂m and an abbreviation denotes terms involving heavier fields and

higher-dimensional terms. Hereγ -matrices in three dimensions are defined by(γ m(3)) ≡(−σ 2, iσ 3,−iσ 1). The vacuum energyV0 is given by the energy density of the backgroundand thus

V0 ≡πR∫

−πRdy

(∂yAcl)+ Λ4

g2 cos2(g

ΛAcl

)

(2.19)= Λ3

g2k

2K(k)∫−2K(k)

du(

1+ k2)− 2k2 sn2(u, k),

and the effective Yukawa couplinggeff is

(2.20)

geff ≡ g√2

πR∫−πR

dy cos

(g

ΛAcl(y)

)bR,1(y)f

(1)0 (y)f

(2)0 (y)= g√

2

C20

CR,1

(1− k2).

In the limit of R → ∞, the parametersmR,1 andgeff vanish and thus we can redefinethe bosonic massless fields as

(2.21)

(a(1)0

a(2)0

)= 1√

2

(1 1

−1 1

)(aR,0

aR,1

).

In this case, the fieldsa(1)0 (x) andψ(1)0 (x) form a supermultiplet forQ(1)-SUSY and their

mode functions are both localized on our wall. The fieldsa(2)0 (x) andψ(2)

0 (x) are singletsfor Q(1)-SUSY and are localized on the other wall.2

When the distance between the wallsπR is finite,Q(1)-SUSY is broken and the mass-splittings between bosonic and fermionic modes are induced. The mass squaredm2

R,1 in

Eq. (2.18) corresponds to the difference of the mass squared,m2 betweena(1)0 (x) and

ψ(1)0 (x) since the fermionic modeψ(1)

0 (x) is massless. Besides the mass terms, we canread off the SUSY breaking effects from the Yukawa couplings likegeff .

We have noticed in Ref. [18] that these two SUSY breaking parameters,mR,1 andgeff ,are related by the low-energy theorem associated with the spontaneous breaking of SUSY.

2 The modesa(2)0 (x) andψ(2)0 (x) form a supermultiplet forQ(2)-SUSY.


Fig. 5. The logarithm of the mass-splitting as a function of the distance between the wall. Thehorizontal axis is the wall distance normalized by 1/Λ.

In our case, the low-energy theorem becomes

(2.22)geff

m2R,1

= 1

2f,

wheref is an order parameter of the SUSY breaking, and it is given by the square rootof the vacuum (classical background) energy densityV0 in Eq. (2.19). The low-energytheorem in three dimensions is briefly explained in Appendix A.1. Since the superpartnerof the fermionic fieldψ(1)

0 (x) is a mixture of mass-eigenstates, we had to take into accountthe mixing Eq. (2.21). The mixing in general situation is discussed and is applied to thepresent case in Appendices A.2 and A.3.

Fig. 5 shows the mass-splitting,m2 as a function of the wall distanceπR. As thisfigure shows, the mass-splitting decays exponentially as the wall distance increases. Thisis one of the characteristic features of our SUSY breaking mechanism. This fact can beeasily understood by remembering the profile of each modes. Note that the mass-splitting,m2(= m2

R,1) is proportional to the effective Yukawa coupling constantgeff , which isrepresented by an overlap integral of the mode functions. Here the mode functions ofthe fermionic fieldψ(1)

0 (x) and its superpartner are both localized on our wall, and that

of the NG fermionψ(2)0 (x) is localized on the other wall. Therefore, the mass-splitting

becomes exponentially small when the distance between the walls increases, because ofexponentially dumping tails of the mode functions.

2.4. Single-wall approximation

Next we will propose a practical method of estimation for the mass-splittings. We oftenencounter the case where single-BPS-domain-wall solutions are known but exact two-wallconfigurations are not. This is because the latter are solutions of a second order differentialequation, namely, the equation of motion, while the former are solutions of first orderdifferential equations, namely, BPS equations. We can estimate the mass-splitting by usingonly informations on the single-wall background, even if two-wall configurations are notknown. As mentioned in the previous subsection, the mass-splitting,m2 is related to the


coupling constantgeff and the order parameterf . So we can estimate,m2 by calculatinggeff andf .

When two walls are far apart, the energy of the backgroundV0 in Eq. (2.19) can bewell-approximated by the sum of those of our wall and of the other wall

(2.23)V0 2Λ3

g2

∞∫−∞

du2− 2 tanh2u

= 8Λ3

g2.

Considering the profiles of background and mode functions, we can see that the maincontributions to the overlap integral ofgeff come from neighborhood of our wall andthe other wall. These two regions give the same numerical contributions to the integral,including their signs. Thus we can obtaingeff by calculating the overlap integral ofapproximate background and mode functions which well approximate their behaviors nearour wall, and multiplying it by two.

In the neighborhood of our wall, the two-wall backgroundAcl(y) can be wellapproximated by the single-wall backgroundA(1)

cl (y) with y1 = 0. So,

(2.24)cos

(g

ΛAcl(y)

) cos

(g

ΛA(1)cl (y)

)= 1

cosh(Λy).

Next, we will proceed to the approximation of mode functions. From the mode equationsin Eq. (2.12), we can express the zero-modesf

(1)0 (y) andf (2)

0 (y) as

(2.25)f(1)0 (y)=C

(1)f,0e

− ∫ y0 dy ′Λsin( gΛAcl(y

′)),(2.26)f

(2)0 (y)=C

(2)f,0e

∫ y0 dy ′Λsin

( gΛAcl(y

′)),whereC(1)

f,0 andC(2)f,0 are normalization factors.

Since the functionf (1)0 (y) has its support mainly on our wall, it is simply approximated

near our wall by

(2.27)f(1)0 (y) C

(1)f,0e

− ∫ y0 dy ′Λsin( gΛA

(1)cl (y

′)) = C(1)f,0

cosh(Λy).

Then we can determineC(1)f,0 = √

Λ/2 by the normalization condition.

Similarly, the modef (2)0 (y) can be approximated near our wall by

(2.28)f(2)0 (y) C

(2)f,0e

∫ y0 dy ′Λsin

( gΛA(1)cl (y

′)) = C(2)f,0 cosh(Λy).

Unlike the case off (1)0 (y), however, we cannot determineC(2)

f,0 by using this approximate

expression because the modef(2)0 (y) is localized mainly on the other wall. Here it should

be noted thatf (2)0 (y)= f

(1)0 (y−πR) from Eq. (2.12) and the property of the background:

Acl(y − πR)=Acl(y)− πΛ/g. Thus,

f(2)0 (y)=C

(1)f,0e

− ∫ y−πR0 dy ′Λsin

( gΛAcl(y

′)) = C(1)f,0e

∫ yπR dy ′Λsin

( gΛAcl(y

′))(2.29)=C

(1)f,0e

− ∫ πR0 dy ′Λsin( gΛAcl(y

′))e∫ y0 dy ′Λsin( gΛAcl(y

′)),


and we can obtain a relation:

(2.30)C(2)f,0 = C

(1)f,0e

− ∫ πR0 dy ′Λsin( gΛAcl(y

′)).In the region ofy ∈ [0,πR], the background is well approximated by

(2.31)Acl(y)A(1)cl (y)

(0 y < πR

2

),

A(2)cl (y)

(πR2 < y πR

)with y1 = 0 andy2 = πR, and thus

sin

(g

ΛAcl(y)

)

sin( gΛA(1)cl (y)

)= tanh(Λy)(0 y < πR

2

),

sin( gΛA(2)cl (y)

)= − tanh(Λ(y − πR)

) (πR2 < y πR

)(2.32) tanh(Λy)− tanh

(Λ(y − πR)

)− 1.

Thus the normalization factor can be estimated as

(2.33)C(2)f,0 = C

(1)f,0

eΛπR

cosh2ΛπR 2

√2Λe−ΛπR.

Here we used the fact thatC(1)f,0 = √

Λ/2 andΛπR 1. As a result, the mode function of

the NG fermionf (2)0 (y) can be approximated near our wall by

(2.34)f(2)0 (y)= 2

√2Λe−ΛπR cosh(Λy).

In the limit of R → ∞, theQ(1)-SUSY is recovered and thus the mode function of thebosonic fielda(1)0 (x) in Eq. (2.21),b(1)0 (y), is identical tof (1)

2 (y). However, when the other

wall exist at finite distance from our wall, this bosonic field is mixed with the fielda(2)0 (x)

localized on the other wall. Because the masses of these two fieldsa(1)0 (x) anda(2)0 (x) are

degenerate (both are massless), the maximal mixing occurs. (See Eq. (2.21).)

(2.35)

(bR,0

bR,1

)= 1√

2

(1 −11 1

)(b(1)0

b(2)0

),

where b(2)0 (y) is the mode function ofa(2)0 (y). Thus the mode function of the mass-eigenmodebR,1(y) is approximated near our wall by

(2.36)bR,1(y) 1√2f(1)0 (y)

√Λ

2

1

cosh(Λy).

Then by using Eqs. (2.24), (2.27), (2.34) and (2.36), we can obtain the effective Yukawacoupling constantgeff ,

(2.37)geff 2g√

2Λe−ΛπR.

As a result, the approximate mass-splitting valuem(ap)2R,1 is estimated as

(2.38)m(ap)2R,1 = 2fgeff = 16Λ2e−ΛπR,

by using Eq. (2.23) and the low-energy theorem Eq. (2.22). From this expression, we canexplicitly see its exponential dependence of the distance between the walls. We call thismethod of estimation the single-wall approximation.


Fig. 6. The ratio of the approximate valuem(ap)2R,1 to the exact onem2

R,1 as a function of the walldistanceπR. The horizontal axis is normalized by 1/Λ.

In our model, we know the exact mass-eigenvaluem2R,1. So we can check the validity

of the above approximation by comparing the approximate valuem(ap)2R,1 and the exact

onem2R,1. Fig. 6 shows the ratio ofm(ap)2

R,1 to m2R,1 as a function of the wall distanceπR.

As this figure shows, we can conclude that the single-wall approximation is very well.

2.5. Matter fields

Let us introduce a matter chiral superfield

(2.39)Φm =Am + √2θΨm + θ2Fm,

interacting withΦ in the original Lagrangian (2.1) through an additional superpotential

(2.40)Wint = −hΛg

sin

(g

ΛΦ

)Φ2

m = −hΦΦ2m + · · · ,

which will be treated as a small perturbation.3

Let us decompose the matter fermionΨm(X) into two real two-component spinorsΨ

(1)mα (X) andΨ (2)

mα (X) asΨmα = (Ψ(1)mα + iΨ

(2)mα )/

√2. Then these fluctuation fields can be

expanded by the mode functions as follows:

(2.41)Ψ (1)m (X)=

∑p

f (1)mp(y)ψ

(1)mp(x), Ψ (2)

m (X)=∑p

f (2)mp(y)ψ

(2)mp(x).

The mode equations are defined as−∂y − 2h

gΛsin

(g

ΛAcl

)f (1)

mp(y)=mmpf(2)mp(y),

(2.42)

∂y − 2h

gΛsin

(g

ΛAcl

)f (2)

mp(y)=mmpf(1)mp(y).

3 We can take the interaction likeWint = −hΦΦ2m as in Ref. [18] in order to localize the mode function of the

light matter fields on our wall. The choice ofWint like Eq. (2.40) is completely a matter of convenience.


Thus zero-modes on the two-wall background (2.7) can be solved exactly

f(1)m0 (y)=Cm0

dn

(Λy

k, k

)+ k cn

(Λy

k, k

)2h/g

,

(2.43)f(2)m0 (y)=Cm0

dn

(Λy

k, k

)− k cn

(Λy

k, k

)2h/g

,

the modef (1)m0 (y) is localized on our wall and the modef (2)

m0 (y) is on the other wall.Besides these zero-modes, there are several light modes ofΦm localized on our wall

when the couplingh is taken to be larger thang. Those non-zero-modes can be obtainedanalytically in the limit ofR → ∞. For example, the low-lying mass-eigenvalues arediscrete atm2

mp = p(−p + 4h/g)Λ2 with p = 0,1,2, . . . < 2h/g, and the corresponding

mode functionsf (1)mp(y) for the fieldsψ(1)

mp(x) are

(2.44)

f (1)mp(y)= Cmp

[cosh(Λy)] 2hg −p F

(−p,1− p+ 4h

g,1− p+ 2h

g; 1− tanh(Λy)

2

),

whereF(α,β, γ ; z) is the hypergeometric function andCmp is normalization factors. The

mode functionsf (2)mp(y) for the fieldsψ(2)

mp(x) have forms similar to those off (1)mp(y).

Although we do not know the exact mass-eigenvalues and mode functions in the casethat the wall distance is finite, we can estimate the boson–fermion mass-splittings,m2

p byusing the single-wall approximation discussed in the previous subsection. For example,let us estimate the mass-splitting betweenψ

(1)mp(x) and its superpartnera(1)mp(x). After

including an interaction like Eq. (2.40), the effective Lagrangian has the following Yukawacoupling terms:

(2.45)L(3)int =

∑p

heffpa(1)mpψ

(1)mpψ

(2)0 + h.c.+ · · · ,

(2.46)heffp = √2h

πR∫−πR

dy cos

(g

ΛAcl(y)

)b(1)mp(y)f

(1)mp(y)f

(2)0 (y).

Just like the case ofa(1)0 (x) and a(2)0 (x), the degenerate statesa(1)mp(x) and a(2)mp(x) aremaximally mixed with each other and their mass eigenvalues split into two differentvaluesmmR2p andmmR(2p+1). By calculating the effective couplingheffp in Eq. (2.46)in the single-wall approximation, we can obtain the following mass-splitting (seeAppendix A.3.4)

(2.47),m2p ≡ m2

mR2p +m2mR(2p+1)

2−m2

mp.

Thanks to the approximate supersymmetry,Q(1)-SUSY, we can use the mode functionin Eq. (2.44) as bothf (1)

mp(y) andb(1)mp(y). Then we obtain the mass-splitting in the single


wall approximation

(2.48),m2p = √

2f heffp = 16h

gΛ2e−ΛπR.

This result is independent of the level numberp. However, it is not a general featureof our SUSY breaking mechanism. It depends on the choice of the interactionWint. If wechooseWint = −hΦΦ2

m as an example, we will obtain a different result that,m2p becomes

larger asp increases, just like the result in Ref. [18].

3. Soft SUSY breaking terms in 3D effective theory

In this section, we discuss how various soft SUSY breaking terms in the three-dimensional effective theory are induced in our framework.

Firstly, we discuss a multi-linear scalar coupling, a generalization of the so-calledA-term. Such a “generalized A-term” is generated from the following superpotential termin the bulk theory

(3.1)SA-term=∫

d4XF(Φ(X, θ)/M)

MN−3 Φi1(X, θ) · · ·ΦiN (X, θ)

∣∣∣∣θ2

+ h.c.

(3.2)⊃∫

d4XF ′(Acl(y)/M)Fcl(y)

MN−2Ai1(X) · · ·AiN (X)+ h.c.

⊃ 2

∫dy

F ′(Acl(y)/M)Fcl(y)

2N/2MN−2bRi1,0(y) · · ·bRiN ,0(y)

(3.3)×

∫d3x aRi1,0(x) · · ·aRiN ,0(x),

whereM is the fundamental mass scale of the four-dimensional bulk theory,F(φ) isa dimensionless holomorphic function ofφ, andΦi (i = 1, . . . ,Nm) are chiral mattersuperfields,

(3.4)Φi =Ai +√

2θΨi + θ2Fi.

The equation of motion forFcl is given by

(3.5)Fcl(y)≡ −∂W∗

∂A∗

∣∣∣∣A=Acl(y)

.

Note that the superpotential term Eq. (3.1) is a generalization of Eq. (2.40). In Eq. (3.3),we used the following Kaluza–Klein (KK) mode expansions,

Ai(X)= 1√2

(ARi (X)+ iAIi(X)

),

(3.6)ARi (X)=∑p

bRi,p(y)aRi,p(x), AIi (X)=∑p

bIi,p(y)aIi,p(x).

When the numberN of the matter fields is three, they-integral in Eq. (3.3) becomesanA-parameter in three-dimensional effective theory. WhenN = 2, SA-term in Eq. (3.3)


becomes a so-called B-term and also includes the following Yukawa interactions

(3.7)SA-term⊃ −∫

d4XF ′(Acl(y)

M

)Ψ (X)

(Ai(X)Ψj (X)+Ψi(X)Aj (X)

)+ h.c.

(3.8)

⊃∫

d3xg(B-term)eff ij ψNG(x)aRi,0(x)ψ

(1)j,0(x)

+ g(B-term)eff ji ψNG(x)aRj,0(x)ψ

(1)i,0 (x)

,

where the effective coupling constantg(B-term)eff ij is defined by

(3.9)g(B-term)eff ij = − 1√

2

∫dyF ′

(Acl(y)

M

)fNG(y)bRi,0(y)f

(1)j,0 (y),

and the Weyl fermionΨi(X) is rewritten by Majorana fermionsΨ (1,2)i (X) and mode-

expanded just likeΨ (X) in Eqs. (2.10) and (2.14)

Ψi(X)= 1√2

(Ψ

(1)i (X)+ iΨ

(2)i (X)

),

(3.10)Ψ(1)i (X)=

∑p

f(1)i,p (y)ψ

(1)i,p (x), Ψ

(2)i (X)=

∑p

f(2)i,p (y)ψ

(2)i,p (x).

We now turn to the squared scalar masses. They are generated from the following Kählerpotential term

(3.11)Sscalar mass=∫

d4X G(Φ(X,θ)

M,

Φ(X, θ)M

)Φi(X, θ)Φj (X, θ)

∣∣∣∣θ2θ2

⊃∫

dyGφφ(Acl(y)/M)F 2

cl(y)

2M2bRi,0(y)bRj,0(y)

(3.12)×

∫d3x aRi,0(x)aRj,0(x),

whereG(φ, φ) is a real function andGφφ(Acl/M) ≡ (∂φ∂φG)(Acl/M,Acl/M). We usedthe mode expansion Eq. (3.6) and the fact thatFcl(y) is real.Sscalar massalso involves the following interactions

(3.13)

Sscalar mass⊃ −∫

d4XGφφ(Acl(y)/M)Fcl(y)

M2Ψ (X)

× (A∗i (X)Ψj (X)+Ψi(X)A

∗j (X)

)+ h.c.

(3.14)

⊃∫

d3xg(scalar)eff ij ψNG(x)aRi,0(x)ψ

(1)j,0(x)

+ g(scalar)eff ji ψNG(x)aRj,0(x)ψ

(1)i,0 (x)

,

where the effective coupling constantg(scalar)eff ij is defined by

(3.15)g(scalar)eff ij = − 1√

2M2

∫dy Gφφ

(Acl(y)

M

)Fcl(y)fNG(y)bRi,0(y)f

(1)j,0 (y).

It should be noted that the squared scalar mass terms and the so-called B-term areindistinguishable in three dimensions, because fields in three dimensions are real. We


emphasize that the low-energy theorem (see Appendix A)

(3.16)geff ij = ,m2ij√

2f

relates the mass-splitting,m2ij and the Yukawa coupling constantgeff ij , which in general

receive contributions from various terms like Eq. (3.3) (N = 2) and Eq. (3.12) for,m2ij ,

and Eqs. (3.8) and (3.14) forgeff ij , respectively.Finally, we consider the gauge supermultiplets. The gaugino mass has a contribution

from the following non-minimal gauge kinetic term in the bulk theory.

(3.17)Sgaugino=∫

d4XH(Φ(X,θ)

M

)Wα(X, θ)Wα(X, θ)

∣∣∣∣θ2

+ h.c.,

(3.18)⊃ −∫

d4XH′(Acl(y)/M)Fcl(y)

M

(λ2(X)+ λ2(X)

),

whereH(φ) is a holomorphic function ofφ, andWα is a field strength superfield and canbe written by component fields as

(3.19)Wα = −iλα +δα

βD − i

2

(σµσ ν

)αβVµν

θβ + θ2σ

µαα∂µλ

α,

in the Wess–Zumino gauge. The spinorλ is a gaugino field andVµν is a field strength ofthe gauge field, andD is an auxiliary field.

4. Soft SUSY breaking terms in 4D effective theory

In this section we discuss the soft SUSY breaking terms in four-dimensional effectivetheory reduced from the five-dimensionalN = 1 theory. We will use the superfield for-malism proposed in Ref. [22] that keeps only the four-dimensionalN = 1 supersymmetrymanifest. The four-dimensional SUSY that we keep manifest is the one preserved by ourwall in the limit ofR → ∞, and we call itQ(1)-SUSY. We do not specify a mechanism toform our wall and the other wall. We assume the existence of a pair of chiral supermulti-pletsΦ = A+ √

2θΨ + θ2F andΦc = Ac + √2θΨ c + θ2Fc, forming a hypermultiplet

of the four-dimensionalN = 2 supersymmetry. TheirF -components have non-trivial clas-sical valuesFcl(y) andFc

cl(y). In the following, the background field configurationAcl(y),Ac

cl(y), andFcl(y), Fccl(y) are assumed to be real for simplicity. In this section,X andx

represent five- and four-dimensional coordinates, respectively, andy denotes the coordi-nate of the extra dimension.

The relevant term to generate the generalized A-term is

(4.1)

SA-term=∫

d5XF(Φ(X, θ)/M3/2,Φc(X, θ)/M3/2)

M(3N−8)/2

×Φ1(X, θ) · · ·ΦN(X,θ)

∣∣∣∣θ2

+ h.c.


(4.2)⊃∫

d5X∂F(y)Fcl(y)+ ∂cF(y)F c

cl(y)

M(3N−5)/2A1(X) · · ·AN(X)+ h.c.

⊃∫

dy∂F(y)Fcl(y)+ ∂cF(y)F c

cl(y)

M(3N−5)/2b1,0(y) · · ·bN,0(y)

(4.3)×

∫d4x a1,0(x) · · ·aN,0(x)+ h.c.,

whereM is the fundamental mass scale of the five-dimensional bulk theory. Note thatthe superfieldsΦ, Φc andΦi in five dimensions have mass-dimension 3/2.F(φ,φc) is aholomorphic function ofφ andφc , and

(4.4)

∂F(y)≡ (∂φF)(Acl(y)

M3/2 ,Ac

cl(y)

M3/2

), ∂cF(y)≡ (∂φcF)

(Acl(y)

M3/2 ,Ac

cl(y)

M3/2

).

In Eq. (4.3), we used the following mode expansion,

(4.5)Ai(X)=∑p

bi,p(y)ai,p(x).

The y-integral in Eq. (4.3) is a generalizedA-parameter in four-dimensional effectivetheory. For example, the usualA- andB-parameters have contributions fromN = 3 andN = 2, respectively

(4.6)Aijk =∫


cl(y)

M2 bi,0(y)bj,0(y)bk,0(y),

(4.7)−Bijµ=∫


cl(y)√M

bi,0(y)bj,0(y),

whereµ is the so-calledµ-parameter.WhenN = 2,SA-term in Eq. (4.1) also includes the following Yukawa interaction

(4.8)

SA-term⊃ −∫

d5X∂F(y)Ψ (X)+ ∂cF(y)Ψ c(X)√

M

× (Ai(X)Ψj (X)+Ψi(X)Aj (X)

)+ h.c.

(4.9)

⊃∫

d4xg(B-term)eff ij ψNG(x)ai,0(x)ψj,0(x)

+ g(B-term)eff ji ψNG(x)aj,0(x)ψi,0(x)

+ h.c.,

(4.10)g(B-term)eff ij = −

∫dy

∂F(y)fNG(y)+ ∂cF(y)f cNG(y)√

Mbi,0(y)fj,0(y).

Here we used the mode expansion ofΨ (X), Ψ c(X) andΨi(X),

(4.11)

(Ψ (X)

Ψ c(X)

)=∑p

(fp(y)

f cp(y)

)ψp(x),

(4.12)Ψi(X)=∑p

fi,p(y)ψi,p(x).


In general, the NG fermionψNG(x) is contained in bothΨ (X) andΨ c(X) with modefunctionsfNG(y) andf c

NG(y), respectively. By definition,fNG(y) andf cNG(y) have their

support mainly on the other wall.Next, we discuss the squared scalar masses. The squared scalar masses get contributions

from the term with a real functionG(φ,φc, φ, φc),

(4.13)

Sscalar mass=∫

d5X G(Φ(X,θ)

M3/2 ,Φc(X, θ)

M3/2 ,Φ(X, θ)M3/2 ,

Φc(X, θ)

M3/2

)× Φi(X, θ)Φj (X, θ)

∣∣∣∣θ2θ2

(4.14)⊃∫

d5XG(y)M3∗

A∗i (X)Aj (X)

(4.15)⊃∫

dyG(y)M3∗

b∗i,0(y)bj,0(y)

∫d4x a∗

i,0(x)aj,0(x),

where functionsG(y), Gφφ(y),Gφφc (y), · · · are defined by

G(y)≡ Gφφ(y)F 2cl(y)+ Gφφc (y)Fcl(y)F

ccl(y)

(4.16)+ Gφcφ(y)F ccl(y)Fcl(y)+ Gφcφc (y)

(Fc

cl(y))2,

(4.17)Gφφ(y)≡ (∂φ∂φG)(Acl(y)

M3/2 ,Ac

cl(y)

M3/2 ,A∗

cl(y)

M3/2 ,Ac∗

cl (y)

M3/2

), · · ·

The following Yukawa interactions are also contained inSscalar massin Eq. (4.13),

(4.18)

Sscalar mass⊃ −∫

d5XGφφ(y)Fcl(y)+ Gφφc (y)F c

cl(y)

M3 Ψ (x)

× (A∗i (X)Ψj (X)+Ψi(X)A

∗j (X)

)+ h.c.

(4.19)

⊃∫

d4xg(scalar)eff ij ψNG(x)a

∗i,0(x)ψj,0(x)

+ g(scalar)eff ji ψNG(x)a

∗j,0(x)ψi,0(x)

+ h.c.,

where the effective Yukawa couplingg(scalar)eff ij is defined by

(4.20)

g(scalar)eff ij ≡ − 1

M3

∫dy(Gφφ(y)Fcl(y)+ Gφφc (y)F c

cl(y))fNG(y)b

∗i,0(y)fj,0(y).

Just like the three-dimensional case, the low-energy theorem

(4.21)geff ij = −,m2ij

f

is valid in four dimensions (see Eq. (B.18) in Appendix B.1), wheref is the orderparameter of the SUSY breaking. Both the mass-splittings,m2

ij and the effectivecouplingsgeff ij are the sum of contributions from various terms. However, the squaredmass terms and the B-term are distinguished by chirality of scalar fields in four dimensions,


unlike the three-dimensional case. Therefore, the low-energy theorem should be validseparately for the squared mass terms and the B-term relating to the effective couplingsof the corresponding chirality.

Finally, we consider the gaugino mass. Note that the gauge supermultiplet in five-dimensionalN = 1 theory contains two gauginos in a four-dimensionalN = 1 sense.However, since we are interested only in a four-dimensionalN = 1 SUSY,Q(1)-SUSY,we will consider onlyλ0(x), which is aQ(1)-superpartner of the gauge fieldvµν,0(x),as the gaugino. The gaugino mass has a contribution from the term with a holomorphicfunctionH(φ,φc) of φ andφc

(4.22)Sgaugino=∫

d5XH(Φ(X,θ)

M3/2 ,Φc(X, θ)

M3/2

)Wα(X, θ)Wα(X, θ)

∣∣∣∣θ2.

Performing the mode expansion of the gauge supermultiplet,

(4.23)Vµν(X)=∑p

bv,p(y)vµν,p(x), λ(X)=∑p

fλ,p(y)λp(x),

we can seeSgauginocontains the following term

(4.24)

Sgaugino⊃∫

dy∂H(y)Fcl(y)+ ∂cH(y)F c

cl(y)

M3/2

(fλ,0(y)

)2∫ d4x(λ0(x)

)2,

where

(4.25)

∂H(y)≡ (∂φH)

(Acl(y)

M3/2 ,Ac

cl(y)

M3/2

), ∂cH(y)≡ (∂φcH)

(Acl(y)

M3/2 ,Ac

cl(y)

M3/2

).

Eq. (4.24) contributes to the mass of the gauginoλ0(x). In order to obtain the gauginomass-eigenvalue itself, we have to take account of the derivative term in the extradimensiony, and define a differential operatorOλ like the left-hand side of Eq. (2.12).However, it is very difficult to find eigenvalues ofOλ generally. Therefore the single-wallapproximation explained in Section 2.4 is quite a powerful method to estimatemλ, thanksto the low-energy theorem.

The term (4.22) also includes the following interaction

(4.26)

Sgaugino⊃∫

d5X1√

2M3/2λ(X)σµσ ν

× ∂H(y)Ψ (X)+ ∂cH(y)Ψ c(X)

Vµν(X)+ h.c.

(4.27)⊃ heff

∫d4x λ0(x)σ

µσ νψNG(x)vµν,0(x)+ h.c.,

where the effective coupling constantheff is defined by

(4.28)heff =∫

dy1√

2M3/2fλ,0(y)

(∂H(y)fNG(y)+ ∂cH(y)f c

NG(y))bv,0(y).

This effective coupling constant is related to the mass-splitting of the gauge supermultiplet,which equals the gaugino massmλ, and the order parameter of the SUSY breakingf by


the low-energy theorem

(4.29)heff = mλ√2f

.

This theorem is derived in Appendix B.2.Using Eq. (4.29), we can estimate the gaugino massmλ by calculating the effective

coupling constantheff in Eq. (4.28). For example, if the gauge supermultiplet lives in thebulk, zero-mode wave functionsfλ,0(y) and bv,0(y) become constant: 1/

√2πR in the

single-wall approximation. Thus the gaugino mass is estimated as

(4.30)mλ = 1

4√

2πM3/2R

∫dy(∂H(y)fNG(y)+ ∂cH(y)f c

NG(y)).

5. Phenomenological implications

Here the qualitative phenomenological features in our framework will briefly bediscussed. It is well known that information of fermion masses and mixings can betranslated into the locations of the wave functions for matter fields in extra dimensions [12,17,23–25]. Yukawa coupling in five dimensions is written as

(5.1)

SYukawa=∫

d5X

(yuij√MQi(X, θ)U

cj (X, θ)H2(X, θ)

+ ydij√MQi(X, θ)D

cj (X, θ)H1(X, θ)

+ ylij√MLi(X, θ)E

cj (X, θ)H1(X, θ)

)∣∣∣∣θ2

+ h.c.,

whereyuij , ydij andylij are dimensionless Yukawa coupling constants for up-type quark,

down-type quark and charged lepton sector of order unity, respectively. The fundamentalmass scale of the five-dimensional bulk theory is denoted byM. Notice that additionalcontributions to Yukawa coupling Eq. (5.1) come from terms like Eq. (4.1). If weconsiderM as the gravitational scaleM∗, these contributions are subleading comparedto Eq. (5.1). On the other hand, ifM happens to be the scale of the wall, such asΛ,these contributions will be comparable to Eq. (5.1). Here we simply write down Eq. (5.1),since an analysis of fermion masses and mixings is not the main point of this paper.Performing the mode expansion for each matter supermultiplet, we obtain, for example,up-type Yukawa coupling from Eq. (5.1),

(5.2)

SYukawa⊃∫

dyyuij√MfQi,0(y)fUc

j ,0(y)bH2,0(y)

∫d4x qi,0(x)u

cj,0(x)h2,0(x),

whereqi,0(x) anducj,0(x) are massless fields of fermionic components ofQi(X, θ) andUcj (X, θ), andh2,0(x) is a massless field of a bosonic component ofH2(X, θ), respectively.

fQi,0(y), fUcj ,0(y) andbH2,0(y) are corresponding mode functions. The effective Yukawa


Fig. 7. Schematic picture of the location of the matter fields. 1, 2 and 3 represents the location of thefirst, second and third generation of the matter fields. The solid line denotes the Higgs wave functionand the dotted line denotes the wave function of NG fermion.

coupling in four dimensions is they-integral part of Eq. (5.2). Fermion masses and mixingsare determined by the overlap integral between Higgs and matter fields. For example, thehierarchy of Yukawa coupling is generated by shifting the locations of the wave functionsslightly generation by generation [17]. These shifts are easily achieved by introducingfive-dimensional mass terms in a generation-dependent way. The fermion masses exhibita hierarchym1 < m2 < m3, wherem1,2,3 denote masses of the first, second and thirdgeneration of matter fermions, respectively. Therefore, we can naively expect that thelocations of the wave functions of matter fieldsyi (i = 1,2,3) becomey1 > y2 > y3(> 0)if the Higgs is localized4 aroundy = 0 as shown in Fig. 7.

In the two-wall background configuration, SUSY is broken and fermion and sfermionmasses split. Even though it is difficult to solve mass-eigenvalues directly, we can calculatethe mass-splitting in each supermultiplet thanks to the low-energy theorem Eqs. (4.21) and(4.29). The overlap integral in Eq. (4.19) among the chiral supermultiplets localized onour wall and the NG fermion localized on the other wall determines the mass-splitting andhence sfermion masses. Thus the mass-eigenvalue of the sfermion becomes larger as thelocation is closer to the other wall.

Before estimating the sfermion mass spectrum, we comment on various scales in ourtheory. There are four typical scales in our theory: the five-dimensional Planck scaleM∗,the compactification scale(2πR)−1, the inverse width of the wallΛ and the inverse widtha of zero-mode wave functions. In order for our setup to make sense, we had better keepthe following relation among these scales

(5.3)M∗ > a >Λ> (2πR)−1 > 5000 TeV.

The inequalitya > Λ comes from the requirement that our wall must have enough widthto trap matter modes. The last constraint is required to suppress flavor changing neutralcurrents mediated by Kaluza–Klein gauge bosons [26]. If we consider the flat background

4 Of course, we can also takey1 < y2 < y3(< 0) to realize the fermion mass hierarchy, but these two cases areequivalent since the extra dimension is compactified.


metric,M∗ andR are related by the relationM2pl = (2πR)M3∗ whereMpl is the four-

dimensional Planck scale. Thus the above constraint gives the lower bound forM∗, thatis,

(5.4)M∗ =(M2

pl

2πR

)1/3

>(M2

pl × 5000 TeV)1/3 8× 1014 GeV.

Now, we would like to make a rough estimation of the gravity at the tree level byapplying the results in Section 4 and considering the scaleM as the five-dimensionalPlanck scaleM∗. Let us start with the sfermion masses. We recall that the interactionEq. (4.13) gives Yukawa coupling Eq. (4.20)

(5.5)geff ij = − 1

M3∗

∫dy Fcl(y)fNG(y)b

∗i,0(y)fj,0(y) (i, j = 1,2,3),

where we assumedG = ΦΦ/M3∗ for simplicity. On the other hand, the low-energy theoremfor the chiral supermultiplet (B.18) is

(5.6)geff = −,m2

f.

Assuming the fermion masses are small, we find that the sfermion masses are given by

(5.7)(m2)

ij= f

M3∗

∫dy Fcl(y)fNG(y)b

∗i,0(y)fj,0(y).

The classical configurationAcl(y) is approximately linear iny in the vicinity of thewall, and constant away from the wall. Correspondingly we can approximateFcl(y) bya Gaussian function and the wave function of NG fermion by an exponential function, ifwe consider a large distance between two walls. We also adopt the Gaussian approximationfor the zero-mode wave functions of the matter fields

(5.8)bi,0(y) fi,0(y)Na exp[−a2(y − yi)

2],whereyi is a location of the matter field anda represents a typical inverse width andNa isa normalization constant of the zero-mode wave function for matter fields. Thus we obtainsfermion masses

(5.9)(m2)

ijN2

a

f

M3∗

∫dy(Λ5/2e−Λ2y2)(√

Λe−Λ(πR−y))e−a2(y−yi)2e−a2(y−yj )2

(5.10) f√2

(Λ

M∗

)3Erf [√2πRa]Erf [πRa] e−ΛπR exp

[−a2

2(yi − yj )

2],

where the error function Erf[x] is defined as Erf[x] ≡ 2√π

∫ x0 dy e−y2

and the normaliza-

tion constantsN2a = a√

π Erf[πRa] are substituted. The approximation 2πR yi is used inthe second line. One can see that the sfermion mass matrix is determined by only the rel-ative difference of the coordinates where the matter fields are localized. The dependenceof the distance between the location of the matter and the other wall is subleading. Us-ing the typical example in Ref. [17] which well reproduces the fermion mass hierarchyand their mixingsy1 ∼ 3.05M−1∗ , y2 ∼ 2.29M−1∗ , y3 ∼ 0.36M−1∗ , and diagonalizing the


sfermion mass matrix, we obtain the following results. If we consider the casea ∼ M∗,the overlap between the wave functions of the different generations is small because thewidth of the wave function is small. Hence the hierarchy of the sfermion masses is at mostone order of magnitude. On the other hand, if we consider the casea ∼ 0.1M∗, the over-lap between the wave functions of the different generations is larger, and all the matrixelements of sfermion mass squared matrix are nearly equal. In this case, the rank of thesfermion mass matrix is reduced, then the sfermion mass becomesO(10 TeV),O(1 TeV)andO(100 GeV). Although this result looks like the decoupling solution [27] for FCNCproblem, it has a mixing among the generations too large to be a viable solution for theFCNC problem. Since this result is an artifact of our rough approximation, we expect thata more realistic sfermion masses can be obtained, if we take account of flexibility of themodel, such as the location and shape of the wave functions.

We now turn to the case of gaugino. Let us first consider the case that the gaugesupermultiplet lives in the bulk. Eqs. (4.20) and (4.28) show that the overlap integralfor the chiral supermultiplet receives an exponential suppression but that for the gaugesupermultiplet does not. The gaugino tends to be heavier than the sfermions in this case.There are three ways to avoid this situation. One of them is to tune the numerical coefficientof the term Eq. (4.22) to be small. The second way is to localize the gauge supermultipleton our wall. The third way is to assume that the function∂H and∂cH in Eq. (4.30) haveprofiles which are localized on our wall. Then, even if the gauge supermultiplet lives in thebulk, the gaugino mass is suppressed because of the suppression of the overlap with theNG fermion localized on the other wall.

Next we consider the case that the gauge supermultiplet is localized on the wall. We alsoassume that the wave function of the zero mode of gauge supermultiplet is Gaussian

(5.11)fλ1,0(y)= bv,0(y)∼ exp(−a2y2).

Since Eq. (4.28) gives the gaugino mass through the low-energy theorem Eq. (4.29), wefind by taking the limitπR Λ/(4a2)

(5.12)mλ = f

M3/2∗

∫dy fλ1,0(y)fNG(y)bv,0(y),

(5.13) f

(Λ

2M3∗

)1/2

exp

(−ΛπR + Λ2

8a2

)Erf [√2πRa]

Erf [πRa] ,

where we assumed that the gauge kinetic function isH = Φ/M3/2. Requiring|mλ| ∼O(100 GeV),m2 ∼O(TeV2) and exp[− a2

2 (yi − yj )2] O(0.1)∼O(1), we obtain

(5.14)Λ∼ 101.6∼2(M∗GeV

)3/5

GeV.

Taking Eq. (5.3) into account, we obtain the bounds forM∗ andΛ

(5.15)8× 1014 GeV<M∗ < 3× 1016 GeV,

(5.16)9× 1010 GeV<Λ< 3× 1011 GeV.


SUSY breaking scale can be obtained from the gaugino mass as

(5.17)√f ∼ 2× 1011 GeV,

where we have usedΛ ∼ 1011 GeV andM∗ ∼ 1016 GeV. SUSY breaking scale iscomparable to that of the gravity mediation

√f ∼ 1010∼11 GeV.

Finally, some comments are in order. The above Eqs. (5.10) and (5.13) include onlyeffects of light modes at tree level of gravitational interaction. We would like to comparethese gravity mediated contributions with those induced by coexisting walls (M =Λ). Thebilinear term of the five-dimensional gravitino has a coefficient of ordercgFcl(y)/M

3/2∗in the case of the gravity mediation, and of ordercwFcl(y)/Λ

3/2 in the case of coexistingwalls, wherecg andcw are numerical constants. As long as we have no information aboutthe fundamental theory, we cannot calculate these constantscg, cw in the effective theory.Taking the ratio of these contributions, we obtain

(5.18)non-gravity

gravity∼ cw

cg

(M∗Λ

)3/2

∼ cw

cg

(1016

1011

)3/2

∼ cw

cg× 107.5.

If cw/cg > 10−7, the gravity mediated contribution is smaller than the non-gravitymediated contribution. Ifcw/cg < 10−8, the gravity mediated contribution is larger thanthe non-gravity mediated contribution.

The second comment is on the proton stability in our framework. In the “fat brane”approach, it is well known that the operators which are relevant to the proton decay areexponentially suppressed by separating the quark wave functions from the lepton wavefunctions [23]. This mechanism also works in our model. Noticing that the fifth dimensionis compactified on a circle, it is sufficient for the wave functions of the quark and the leptonto be localized on the opposite side with respect to the planey = 0 where the Higgs field islocalized. This relative location is required to reproduce the quark and lepton masses. Letus suppose that the distance between the locations of the quark and the lepton isr. Then,the dimension-five operators are suppressed by

e−(ar)2

M∗= 1

MP

MP

M∗e−(ar)2,

whereMP is the Planck scale in four dimensions. To keep the proton stable enough asrequired by experiments,MP

M∗ e−(ar)2 ∼ 10−7 is needed. This constraint is indeed satisfied

if we takeM∗ ∼ 1016 GeV andar ∼ O(5–6), and is consistent with Eq. (5.3). Thus, theproton decay process is easily suppressed in our framework.

6. Discussion

In this paper, we proposed a simple SUSY breaking mechanism in the brane worldscenario. The essence of our mechanism is just the coexistence of two different kindsof BPS domain walls at finite distance. Our mechanism needs no messenger fields norcomplicated SUSY breaking sector on any of the walls. The low-energy theorem provides a


powerful method to estimate the boson–fermion mass-splitting. Namely, the mass-splittingcan be estimated by calculating an overlap integral of the mode functions for matter fieldsand the NG fermion. Matter fields are localized on our wall by definition. On the otherhand, since the supersymmetry approximately preserved on our wall is broken due tothe existence of the other wall, the corresponding NG fermion is localized on the otherwall. Thus the mass-splitting induced in the effective theory is exponentially suppressedcompared to the fundamental scaleΛ. This is the generic feature of our mechanism.

Now let us discuss several further issues.As mentioned below Eq. (2.22), the order parameter of the SUSY breakingf is equal

to the square root of the energy density of the wall√V0. From the three-dimensional point

of view, the fundamental theory is anN = 2 SUSY theory withQ(1)- andQ(2)-SUSYs.In general when a BPS domain wall exist, a half of the bulk SUSY, for example,Q(2)-SUSY, is broken. In such a case, an order parameter of the SUSY breakingf2 is equal tothe square root of the energy density of the domain wall

√V0. However, if there is another

BPS domain wall that breaks the other half of the bulk SUSY,Q(1)-SUSY, there is anotherorder parameter of the SUSY breakingf1 and its square is expected to be equal to theenergy density of the additional wall. In the model discussed in Section 2, these two orderparametersf1 andf2 are equal to each other. This is because the two walls are symmetricin this model. However, in the case when our wall and the other wall are not symmetric,two order parametersf1 andf2 can have different values. In Appendix C, we discuss thepossibility of such an asymmetric wall-configuration and the relation amongf1, f2 andV0

and central charge of the SUSY algebra.If we try to construct a realistic model in our SUSY breaking mechanism, a fundamental

bulk theory, which has a five-dimensionalN = 1 SUSY, must have BPS domain walls.Since such a higher-dimensional SUSY restricts the form of the superpotential severely, itis not easy to construct a BPS domain wall configuration. However, a BPS domain wall hasbeen constructed in a four-dimensionalN = 2 SUSY non-linear sigma model [28]. Sincethe non-linear sigma model can be obtained from theN = 1 five-dimensional theory, thisBPS domain wall can be regarded as a BPS domain wall that we desire. It is more difficultto obtain non-BPS configuration of two walls.

Our mechanism can be extended to higher-dimensional cases straightforwardly. Insuch cases, our four-dimensional world is on various kinds of topological defects,such as vortices or intersections of domain walls in six dimensions, monopoles inseven dimensions, etc. Many higher-dimensional theories have BPS configurations ofthese defects. Thus all we need for our mechanism is a stable non-BPS configurationcorresponding to the coexistence of two or more BPS topological defects that preservedifferent parts of the bulk SUSY. We can always use the low-energy theorem likeEqs. (4.21) and (4.29) irrespective of the dimension of the bulk theory, in order to estimatethe mass-splittings between bosons and fermions.

As a future work, we will investigate our SUSY breaking mechanism in the non-trivial metric like the Randall–Sundrum background [5]. To achieve this goal, we needto overcome the technical complexity of dealing with the five-dimensional supergravity.Besides, when we introduce the gravity, the size of the fifth dimension 2πR becomes a


dynamical variable. In the model discussed in Section 2, for example, the force betweenour wall and the other wall is repulsive. Thus the two-wall configuration Eq. (2.7) becomesunstable (2πR goes to infinity) when the gravity is considered. So we must implementan extra mechanism to stabilize the two-wall configuration not only topologically but alsounder the gravity.

Acknowledgements

One of the authors (N.S.) is indebted to useful discussions with Kiwoon Choi, TakeoInami, Ken-ichi Izawa, Martin Schmaltz, Tsutomu Yanagida, and Masahiro Yamaguchi.One of the authors (N.M.) thanks to a discussion with Hitoshi Murayama. This work issupported in part by Grant-in-Aid for Scientific Research from the Ministry of Education,Culture, Sports, Science and Technology, Japan, priority area (#707) “Supersymmetryand unified theory of elementary particles” and No. 13640269. N.M., Y.S. and R.S. aresupported by the Japan Society for the Promotion of Science for Young Scientists(No. 08557, No. 10113 and No. 6665).

Appendix A. Low-energy theorem in three dimensions

In this appendix we will review the low-energy theorem for the SUSY breaking briefly,and apply it to our mechanism.

A.1. SUSY Goldberger–Treiman relation

In general, when the supersymmetry is spontaneously broken, a massless fermion calledthe Nambu–Goldstone (NG) fermionψNG(x) appears in the theory. It shows up in thesupercurrentJmα (x) as follows [20]

(A.1)Jmα = √2if

(γm(3)ψNG

)α

+ Jmφ,α + · · · ,wheref is the order parameter of the SUSY breaking and the abbreviation denotes higher-order terms forψNG(x). Jmφ,α(x) is the supercurrent for matter fieldsφ = (a,ψ) wherea(x) andψ(x) are a real scalar and a Majorana spinor fields, respectively,

(A.2)Jmφ,α = (γ n(3)γ

m(3)ψ

)α∂na + · · · .

In the low-energy effective Lagrangian, there is a Yukawa coupling as follows:

(A.3)LYukawa= geff aψψNG.

Here the effective coupling constantgeff is related to the mass-splitting between the bosonand the fermion,m2 ≡m2

a −m2ψ and the order parameterf by [20]

(A.4)geff = ,m2√

2f.

This is the supersymmetric analog of the Goldberger–Treiman relation.


A.2. Superpartners and mass-eigenstates

When SUSY is broken, a superpartner of a fermionic mass-eigenstate is not always amass-eigenstate. In such a case, we should extend the formula Eq. (A.4) to more genericform.

Let us denote fermionic mass-eigenstates asψ1,ψ2, . . . ,ψN , and their bosonic su-perpartners asa1, a2, . . . , aN . The bosonic mass-eigenstatesa1, a2, . . . , aN are related toa1, a2, . . . , aN by

(A.5)

a1

a2...

aN

= V

a1

a2...

aN

,

whereV is anN ×N unitary mixing matrix.In this case, Eq. (A.4) is generalized to

(A.6)geff i,j = 1√2f

(,M2)

j,i,

wheregeff i,j are Yukawa coupling constants:

(A.7)LYukawa=∑i,j

geff i,j aiψjψNG,

and,M2 is anN ×N matrix defined by

(A.8),M2 ≡ V

m2a1

. . .

m2aN

−m2

ψ1

. . .

m2ψN

V.

A.3. Application to our model

To apply the above low-energy theorem to our mechanism of the SUSY breaking,we should interpret the four-(five-)dimensional bulk theory as a three-(four-)dimensionaltheory involving infinite Kaluza–Klein modes. To illustrate this, let us discuss the low-energy theorem by using the model Eq. (2.1) in the four-dimensional bulk as an example.

A.3.1. Three-dimensional super-transformationThe superpartner ofψ(1)

p (x) for Q(1)-SUSY can be read off from the four-dimensionalsuper-transformation,

(A.9)δξA(X)= √2ξΨ (X),

where ξ is a Weyl spinor which parametrizes the super-transformation. By expandingthe four-dimensional fieldsA andΨ to infinite Kaluza–Klein modes like Eqs. (2.10),


(2.13), (2.14), multiplyingf (1)p (y) and integrating in terms ofy, we can obtain the three-

dimensional super-transformation

(A.10)δζ∑q

(∫dy f (1)

p (y)bR,q(y)

)aR,q(x)

= ζψ(1)

p (x),

where ζ denotes the parameter ofQ(1)-transformation, which is a three-dimensionalMajorana spinor.

Thus the superpartner ofψ(1)p (x) for Q(1)-SUSY, a(1)p (x), is a linear combination of

infinite mass-eigenmodes

(A.11)a(1)p (x)=∑q

(∫dy f (1)

p (y)bR,q(y)

)aR,q(x).

This is becauseQ(1)-SUSY is broken by the backgroundAcl(y). When the distancebetween the walls is infinite,Q(1)-SUSY is recovered anda(1)p (x) becomes a mass-

eigenmode. In this case,Q(2)-SUSY is also recovered anda(2)p (x), which is a superpartner

of the mass-eigenmodeψ(2)p (x), becomes a mass-eigenmode. Since the fieldsa

(1)p (x) and

a(2)p (x) are degenerate, they maximally mix when the wall distance is finite. For example,

(A.12)

(aR,0

aR,1

) 1√

2

(1 −11 1

)(a(1)0

a(2)0

),

that is,

(A.13)a(1)0 1√

2(aR,0 + aR,1).

This can be directly obtained from Eq. (A.11) by settingp = 0.Strictly speaking,a(1)0 (x) has slight but non-zero components of heavier fields

aR,p(x) (p 2). However these components become negligibly small asp increases. Thusby introducing a cutoffN for the Kaluza–Klein level and setting it large enough, we canapply the formula Eq. (A.6) to our case. The mixing matrixV in Eq. (A.5) can be read offfrom Eq. (A.11) as follows:

(A.14)Vp,q =∫

dy f (1)p (y)bR,q(y).

A.3.2. Derivation of Eq. (2.22)Here we will derive the formula Eq. (2.22), as an example. Since the effective coupling

constantgeff in Eq. (2.18) isgeff 1,0 in the notation here, it is related to the element(,M2)0,1 according to Eq. (A.6)

(A.15)(,M2)

0,1 = V0,1mR,1, V0,1 =∫

dy f (1)0 (y)bR,1(y)= k

C0

CR,1,

where normalization factorsC0 andCR,1 are defined by Eqs. (2.15) and (2.17), and

C0 =(∫

dy

dn

(Λy

k, k

)+ k cn

(Λy

k, k

)2)−1/2


(A.16)

=(∫

dy

(1+ k2)− 2k2 sn2

(Λy

k, k

))−1/2

=(V0

g2k2

Λ4

)−1/2

= Λ2

fgk.

Here we used Eq. (2.19) and the relationV0 = f 2. Then we find the low-energy theoremEq. (A.6) using Eq. (2.20)

(A.17)

1√2f

(,M2)

0,1 = V0,1m2R,1√

2f= 1√

2fkC0

CR,1

1− k2

k2 Λ2 = g√2

C20

CR,1

(1− k2)= geff .

When the distance between the walls is large,V0,1 1/√

2 and we obtain Eq. (2.22).In the above calculation, we assumed that the normalization factorsC0, CR,0 andCR,1

are all positive. In fact, we can calculate the boson–fermion mass-splittings including theirsign, irrespective of the sign conventions of these normalization factors. Next, we will showthis fact.

A.3.3. Unambiguity of the sign of the mass-splittingFirstly, we should note that the sign of the normalization factor of the NG fermionC0 is

determined by the convention of the sign of the order parameterf .The supercurrent in Eq. (A.1) can be obtained from that of the bulk theory,

(A.18)Jµα = √2(σνσµΨ

)α∂νA

∗ − i√

2(σµΨ )

α

∂W∗

∂A∗ .

We define the three-dimensional currentsJ(1)mα (x) andJ (2)mα (x) as follows:

(A.19)∫

dy Jmα (X)= 1√2

(J (1)mα (x)+ iJ (2)mα (x)

),

whereJ (1)mα (x) andJ (2)mα (x) are three-dimensional Majorana currents.By substituting the mode expansion of fields:

A(x,y)=Acl(y)+ 1√2

∑p

bR,p(y)aR,p(x)+ i∑p

bI,p(y)aI,p(x)

,

(A.20)Ψ (x, y)= 1√2

∑p

f (1)p (y)ψ(1)

p (x)+ i∑p

f (2)p (y)ψ(2)

p (x)

,

into J (1)mα (x), we can obtain the three-dimensional supercurrent forQ(1)-SUSY

J (1)m(x)= √2 i

∫dy f (2)

0 (y)

(∂yAcl(y)− Λ2

gcos

(g

ΛAcl(y)

))γm(3)ψ

(2)0 (x)

(A.21)+∑p,q

Vp,qγn(3)γ

m(3)ψ

(1)p (x)∂naR,q(x)+ · · · .

Comparing this to Eq. (A.1), we can see that the order parameter of the SUSY breakingf is expressed by

(A.22)f =∫

dy f (2)0 (y)

(∂yAcl(y)− Λ2

gcos

(g

ΛAcl(y)

))= Λ2

gkC0.


Thus if we take a convention off > 0, the normalization factorC0 is set to be positive.Noticing that (,M2)p,q = Vp,q(m

2R,q − m2

p), we obtain the following formula fromEq. (A.6)

m2R,q −m2

p = √2f

geff q,p

Vp,q

(A.23)=√

2Λ2

gkC0

∫dybR,q(y)f

(1)p (y)f

(2)0 (y)∫

dy f (1)p (y)bR,q(y)

.

Therefore, we can calculate the mass-splittingm2R,q −m2

p including its sign, irrespectiveof the sign conventions of the normalization factors.

A.3.4. Estimation in the single-wall approximationFinally, we comment on the estimation of the mass-splitting in the single-wall

approximation (SWA). When we estimate the boson–fermion mass-splitting in SWA, weoften approximate the bosonic mode function by that of its fermionic superpartner in thecalculation of the overlap integral. This means that we estimate the following effectivecoupling asgeff ij in Eq. (A.6):

(A.24)LYukawa= g(SWA)effp a(1)p ψ(1)

p ψ(2)0 + · · · .

As mentioned above, the superpartnera(1)p (x) of the fermionic mass eigenmodeψ(1)

p (x)

is a linear combination of mainly two bosonic mass-eigenmodes

(A.25)a(1)p (x) 1√2

(aR,2p(x)+ aR,2p+1(x)

).

Thus corresponding mode functionb(1)p (y) is

(A.26)b(1)p (y) 1√2

(bR,2p(y)+ bR,2p+1(y)

).

Then by usingb(1)p (y), which is well-approximated byf (1)p (y), as a bosonic mode

function, the formula Eq. (A.23) becomes

√2fg(SWA)

effp = √2f∫

dy b(1)p (y)f (1)p (y)f

(2)0 (y)

f

∫dy(bR,2p(y)+ bR,2p+1(y)

)f (1)p (y)f

(2)0 (y)

f1√2

(geff 2p,p

V2p,p+ geff 2p+1,p

V2p+1,p

)= 1

2

(m2

R,2p −m2p

)+ (m2

R,2p+1 −m2p

)(A.27)= m2

R,2p +m2R,2p+1

2−m2

p,


where we used the fact thatV2p,p V2p+1,p 1/√

2, and the coupling constantg(SWA)effp is

defined by

(A.28)g(SWA)effp ≡

∫dy b(1)p (y)f (1)

p (y)f(2)0 (y)

∫dy(f (1)p (y)

)2f(2)0 (y).

Therefore, what we can estimate in the single-wall approximation is the differencebetween a fermionic mass and an average of squared masses of its bosonic superpartners.

Appendix B. Low-energy theorem in four dimensions

In this appendix we derive the low-energy theorem for chiral and gauge supermultipletsin four dimensions. We will follow the procedure in Ref. [20].

B.1. Low-energy theorem for chiral supermultiplets

Let us denote one-particle state of a scalar boson with the massmB and the momentumpB as |pB〉, and that of a spin 1/2 fermion with the massmF and the momentumpF

as |pF〉, which form a chiral supermultiplet. We perform the Lorentz decomposition of amatrix element for the supercurrentJ

µα (x) between these states.

〈pB|Jµα (0)|pF〉 = [A1(q2)qµ +A2

(q2)kµ +A3

(q2)σµσ νqν]αβχFβ(pF)

(B.1)

+ [A4(q2)σµ +A5

(q2)qµσνqν +A6

(q2)kµσνqν]αβ χF

β (pF),

whereqµ ≡ pµB − p

µF and kµ ≡ p

µB + p

µF . The spinorsχF(pF) and χF(pF) obey the

following equations

(B.2)σ · pFχF(pF)=mFχF(pF), σ · pFχF(pF)=mFχF(pF).

Conservation of the supercurrent leads to a relation among the form factors as

(B.3)q2[A1(q2)−A3

(q2)]=,m2A2

(q2),

where,m2 ≡m2B −m2

F is a mass-splitting between the boson and the fermion.To discuss S-matrix elements, we define an NG fermion sourcejNG

α (x) by using the NGfermion fieldψNG(x) as

(B.4)jNGα (x)= −iσµαα∂µψα

NG(x).

Its matrix element between the boson and the fermion states is decomposed as

(B.5)〈pB|jNGα (0)|pF〉 = B1

(q2)χFα(pF)+B2

(q2)q · σααχF

α(pF),

and thus

(B.6)〈pB|ψαNG(0)|pF〉 = −B1(q

2)

q2 q · σ ααχFα(pF)+B2(q2)χF

α(pF).

Since the combinationJµα − √2if σµααψ

αNG has vanishing matrix element between

the vacuum and the single NG fermion state, all the form factors of〈pB|Jµα −


√2if σµααψ

αNG|pF〉 are regular asq2 → 0. Then comparing Eqs. (B.1) and (B.6), we can

see that the form factorA3(q2) is singular atq2 = 0 unlessB1(0) is zero

(B.7)limq2→0

q2A3(q2)= −√

2ifB1(0).

Substituting it into Eq. (B.3) with the limitq2 → 0, we obtain

(B.8)√

2if B1(0)=,m2A2(0).

To relate the form factorB1(0) to an effective coupling constant of the NG fermionwith the boson and the fermion forming a chiral supermultiplet, we evaluate a transitionamplitude between the in-state|q;pF〉in and the out-state|pB〉out. This S-matrix elementcan be expressed by using an effective interaction LagrangianLint as

out〈pB|q;pF〉in = I〈pB|ei∫

d4xLint(x)|pF〉I

(B.9) i(2π)4δ4(pB − pF − q) I〈pB|Lint(0)|q;pF〉I ,

where|pB〉I and|q;pF〉I denote states in the interaction picture.On the other hand, using the LSZ reduction formula, it can also be written as

in〈pB|q;pF〉out = −i(2π)4δ4(pB − pF − q)χNG(q)qµσµ

I〈pB|ψNG(0)|pF〉I

(B.10)

− i(2π)4δ4(pB − pF − q)χNG(q)qµσµ

I〈pB|ψNG(0)|pF〉I,

whereχNG(q) andχNG(q) are the NG fermion spinors. Since we do not need to distinguishthe interaction picture and the Heisenberg picture for one-particle states, we drop thesubscript I for one-particle states in the following. We obtain a relation between matrixelements of the interaction Lagrangian and the NG fermion field

〈pB|Lint(0)|q;pF〉I

(B.11)= −χNG(q)qµσµ〈pB|ψNG(0)|pF〉 − χNG(q)qµσ

µ〈pB|ψNG(0)|pF〉.At soft NG fermion limitqµ → 0, the S-matrix element Eq. (B.9) should be expressible

by the following nonderivative interaction terms in the effective Lagrangian [19]

(B.12)Lint = geff a∗ψψNG + h.c.+ · · · ,

wherea is a complex scalar field andψ is a two-component Weyl spinor field, which createor annihilate the states|pB〉 and|pF〉, respectively. So its matrix element is written as

(B.13)〈pB|Lint(0)|q;pF〉I = geffχNG(q)χF(pF)+ geff χNG(q)χF(pF).

Substituting Eq. (B.6) into Eq. (B.11), and comparing it with Eq. (B.13) gives a relationbetweenB1(0) andgeff

(B.14)B1(0)= −geff .

Thus Eq. (B.8) becomes the supersymmetric analog of the Goldberger–Treiman relation

(B.15)√

2geff f = i(m2

B −m2F

)A2(0).


Noting that the supercurrent takes the form

(B.16)Jµα = √2if σµααψ

αNG + √

2(σνσµψ

)α∂νa

∗ + · · · ,and substituting it into the left-hand side of Eq. (B.1) with the limitq2 → 0, we candetermine the value of the form factorA2(0) as

(B.17)A2(0)= √2 i.

Thus we obtain the low-energy theorem for the chiral supermultiplets from Eq. (B.15)

(B.18)geff = −m2B −m2

F

f.

B.2. Low-energy theorem for gauge supermultiplets

Next we derive the low-energy theorem for gauge supermultiplets. As the case of chiralsupermultiplets, we consider the Lorentz decomposition of the matrix element for thesupercurrentJµα (x) between one-particle state of the gauge boson|pB〉 with the massmB and the momentumpB, and that of the gaugino|pF〉 with the massmF and themomentumpF

〈pB|Jµα (0)|pF〉= ε∗

ν (pB)[A1(q

2)qνqµ +A2(q2)qνkµ +A3(q

2)qνσµσ ρqρ

+A4(q2)ηµν +A5(q

2)σ νσµ]αβχFβ(pF)

(B.19)

+ ε∗ν (pB)

[A6(q

2)qνσµ +A7(q2)qνqµσρqρ +A8(q

2)qνkµσρqρ

+A9(q2)ηµνσρqρ +A10(q

2)qµσν +A11(q2)σ νσ ρσµkρ

+A12(q2)σµσ ρσ νqρ

]αβχF

β (pF),

whereqµ = pµB −p

µF , kµ = p

µB +p

µF andε∗

ν (pB) is a polarization vector withpB · ε∗(pB)

= 0.Conservation of the supercurrent leads to a relation among the form factors

(B.20)q2[A10(q2)+A11(q

2)−A12(q2)]= −2,m2A11(q

2),

where,m2 ≡m2B −m2

F is the mass-splitting between the gauge boson and the gaugino.A matrix element of the NG fermion sourcejNG

α (x) between the gauge boson and thegaugino states are decomposed as

〈pB|jNGα (0)|pF〉 = ε∗

ν (pB)[B1(q

2)qν +B2(q2)qρσ

ρσ ν]αβχFβ(pF)

(B.21)+ ε∗ν (pB)

[B3(q

2)qνσρqρ +B4(q2)σ ν

]αβχF

β (pF),

and thus

〈pB|ψαNG(0)|pF〉 = ε∗

ν (pB)

[−B1(q

2)

q2 qνσ ρqρ +B2(q2)σ ν

]αβχFβ(pF)

(B.22)+ ε∗ν (pB)

[B3(q

2)qν − B4(q2)

q2 qρσρσ ν

]αβ χF

β (pF).


The regularity of the form factors of the matrix element forJµα − √

2if σµααψαNG as

q2 → 0 leads to the singularity of the form factorA12(q2) atq2 = 0

(B.23)limq2→0

q2A12(q2)= −√

2ifB4(0).

Substituting it into Eq. (B.20) with the limitq2 → 0, we obtain

(B.24)√

2if B4(0)= −2,m2A11(0).

We can relate the form factorB4(0) to an effective coupling constant of the NG fermionwith the gauge boson and the gaugino forming a gauge supermultiplet. By repeating thesame procedure as that in the previous subsection leading to Eq. (B.11), we obtain

〈pB|Lint(0)|q;pF〉I

(B.25)= −χNG(q)qµσµ〈pB|ψNG(0)|pF〉 − χNG(q)qµσ

µ〈pB|ψNG(0)|pF〉.On the other hand, we expect the following nonderivative interaction terms in the

effective Lagrangian [19]

(B.26)Lint = heffψNGσµνλvµν + h.c.+ · · · ,

whereλ is the gaugino field andvµν is the gauge field strength, respectively. So its matrixelement is written as

〈pB|Lint(0)|q;pF〉I = iheffε∗ν (pB)pBµχNG(q)σ

νσµχF(pF)

(B.27)+ iheffε∗ν (pB)pBµχNG(q)σ

νσµχF(pF).

For the case ofmF = 0, comparison between Eq. (B.25) and Eq. (B.27) after substitutionof Eq. (B.22) into Eq. (B.25) gives

(B.28)B4(0)= −imFheff.

Using Eq. (B.24) we obtain the analog of the Goldberger–Treiman relation for gaugesupermultiplets

(B.29)−√2hefff = 2

(m2

B −m2F

)mF

A11(0).

To determine the form factorA11(0), we substitute the following expression of thesupercurrent into Eq. (B.19) with the limitq2 → 0:

(B.30)Jµα = √2if σµααψ

αNG − ivνρ

(σνρσµ

)ααλα + · · · .

Then we find

(B.31)A11(0)= 1

2.

By substituting it into Eq. (B.29), we obtain the low-energy theorem for the gaugesupermultiplets

(B.32)heff = − 1√2f

(m2

B

mF−mF

).


Appendix C. Relation among central charge and order parameters

In the single-wall case, the order parameterf for the SUSY breaking due to the existenceof a BPS domain wall is given by the square root of the energy density of the wall

√V0.

In the two-wall system, however, two different SUSY breakings occur, whose origins areour wall and the other wall, respectively. Thus there are in general two kinds of orderparametersf1 and f2 for different SUSY breakings. Here we shall clarify the relationamongf1, f2 andV0 and the central charge of the SUSY algebra.

Let us begin with the four-dimensional SUSY algebra of the bulk theory. Since weconsider the case of the SUSY breaking, we describe the SUSY algebra in the local form.The three-dimensional SUSY algebra can be derived from the four-dimensional one withthe central charge:

(C.1)Qα, J

νβ(X)

= 2σµαβTµ

ν(X),

(C.2)Qα,J

νβ (X)

= 4i(σµσ ν)αγ εγβ∂µW

∗(A∗(X)),

whereTµν(X) is the energy–momentum tensor

(C.3)Tµν = ∂νA∗∂µA+ ∂νA∂µA

∗ + i

2Ψ σ ν∂µΨ + i

2Ψσν∂µΨ + δµ

νL.

The term containing the superpotentialW(Φ) represents the density of the central charge.In this section, we calculate the SUSY algebra in the following Wess–Zumino model forsimplicity:

(C.4)L = ΦΦ|θ2θ2 +W(Φ)|θ2 + h.c.,

whereΦ =A+ √2θΨ + θ2F is a chiral superfield.

Eqs. (C.1) and (C.2) can be rewritten in terms of the three-dimensional supercurrentdefined by Eq. (A.19) as

(C.5)Q(1)α , J (1)nβ(x)

= 2(γm(3)

)αβ(Ym

n − 2δmn,W∗),(C.6)

Q(2)α , J (2)nβ(x)

= 2(γm(3)

)αβ(Ym

n + 2δmn,W∗),(C.7)

Q(1)α , J

(2)nβ (x)

= 2εαβY2n,

(C.8)Q(2)α , J

(1)nβ (x)

= −2εαβY2n,

where

(C.9)Yµν(x)≡

∫dy Tµν(X), ,W ≡

∫dy ∂yW

(A(X)

).

Note that,W is constant since it depends only on the boundary condition along the extradimension and becomes non-zero on a non-trivial boundary condition. Here we supposethat the background configurationAcl(y) is real, for simplicity. Thus the central charge,W is treated as a real constant in the following discussion.


Since the backgroundAcl(y) is real, four-dimensional fieldsA(X) andΨ (X) are mode-expanded as follows:

(C.10)A(X)=Acl(y)+ 1√2

( ∞∑p=0

bR,p(y)aR,p(x)+ i

∞∑p=1

bI,p(y)aI,p(x)

),

(C.11)Ψ (X)= 1√2

( ∞∑p=0

f (1)p (y)ψ(1)

p (x)+ i

∞∑p=0

f (2)p (y)ψ(2)

p (x)

).

Note the NG bosonaR,0(x) for the broken translational invariance along the extradimension comes from the real part ofA(x) becauseAcl(y) is real. In the fermionicsector, there are the NG fermionsψ(2)

0 (x) andψ(1)0 (x) for brokenQ(1)- andQ(2)-SUSY,

respectively.Yµ

ν(x) can be rewritten in terms of three-dimensional fields as

Ymn(x)= −δmnV0 +

∞∑p=0

∂naR,p∂maR,p +∞∑p=1

∂naI,p∂maI,p

(C.12)+ i

2

∞∑p=0

ψ(1)p γ n(3)∂mψ

(1)p + i

2

∞∑p=0

ψ(2)p γ n(3)∂mψ

(2)p + δm

nL(3)

(C.13)= −δmnV0 + T(3)mn(x),

(C.14)Y2m(x)= fP∂

maR,0 + · · · ,whereV0 is the energy density of the background

(C.15)V0 ≡∫

dy

(∂yAcl)

2 +∣∣∣∣∂W∂A

∣∣∣∣2A=Acl

,

and T(3)mn(x) is the three-dimensional energy–momentum tensor.fP in Eq. (C.14)

corresponds an order parameter for the breaking of the translational invariance along theextra dimension

(C.16)fP = √2∫

dy bR,0(y)∂yAcl(y).

Then, the three-dimensional SUSY algebra becomes as follows:

(C.17)Q(1)α , J (1)nβ(x)

= 2(γm(3)

)αβ−δmn(V0 + 2,W)+ T(3)m

n,

(C.18)Q(2)α , J (2)nβ(x)

= 2(γm(3)

)αβ−δmn(V0 − 2,W)+ T(3)m

n,

(C.19)Q(1)α , J

(2)nβ (x)

= 2εαβ(fP∂

naR,0 + · · ·),(C.20)

Q(2)α , J

(1)nβ (x)

= 2εβα(fP∂

naR,0 + · · ·).On the other hand, the supercurrents have the following forms

(C.21)J (1)mα = √2if1

(γm(3)ψ

(2)0

)α

+ · · · , J (2)mα = √2if2

(γm(3)ψ

(1)0

)α

+ · · · ,wheref1 and f2 are the order parameters of the breaking forQ(1)- andQ(2)-SUSY,respectively.


Then using the commutation relation of the three-dimensional Majorana spinors

(C.22)ψα(x, t),ψβ(x ′, t)

= −(γ 0(3)σ

2)αβδ2(x − x ′),

Eqs. (C.17) and (C.18) are also written as

(C.23)Q(1)α , J (1)nβ(x)

= −2f 21

(γ n(3)

)αβ + · · · ,

(C.24)Q(2)α , J (2)nβ(x)

= −2f 22

(γ n(3)

)αβ + · · · .

By comparing these commutation relations with Eqs. (C.17) and (C.18), we obtain thefollowing relations

(C.25)V0 = f 21 + f 2

2

2, ,W = f 2

1 − f 22

4.

Thus the average of the squares of two different kinds of order parameters gives theenergy density of the background and their difference gives the central charge. From thesecond relation in Eq. (C.25), we can conclude that if the extra dimension is compactified,the superpotentialW must be a multi-valued function, such as those in Ref. [29], in order torealize a situation where the order parameter for the breaking of theQ(1)-SUSY is differentfrom that of theQ(2)-SUSY.

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Superconformal symmetry in 11D superspaceand the M-theory effective action

S. James Gates Jr.Department of Physics, University of Maryland, College Park, MD 20742-4111, USA


Abstract

We establish a theorem about non-trivial 11D supergravity fluctuations that are conformally relatedto flat superspace geometry. Under the assumption that a theory of conformal 11D supergravityexists, similar in form to that of previously constructed theories in lower dimensions, this theoremdemands the appearance of non-vanishing dimension 1/2 torsion tensors in order to accommodatea non-trivial 11D conformal compensator and thus M-theory corrections that break superconformalsymmetry. At the complete non-linear level, a presentation of a conventional minimal superspacerealization of Weyl symmetry in eleven-dimensional superspace is also described. All of our resultstaken together imply that there exists some realization of conformal symmetry relevant for the M-theory effective action. We thus led to conjecture this is also true for the full and complete M-theory. 2001 Published by Elsevier Science B.V.

PACS:03.70.+k; 11.30.Rd; 04.65.+eKeywords:Gauge theories; Supersymmetry; Supergravity

1. Introduction

In a recent note [1], we have resumed studies of a class of problems, the on-shellsuperspace and perturbative description of higher derivative supergravity, that has beenone of several foci of fascination for us since we inaugurated such investigations [2–6] in the middle eighties. Our proposed method uses on-shell superspace to describe,in a perturbative manner, supergravity actions containing higher-order curvature terms.As first stated in the initial efforts, in order for such constructions to make sense theremust be a dimensional parameter with respect to which such a perturbative descriptionis made. One might think that such a parameter is present inall ordinary supergravity

Supported in part by National Science Foundation Grants PHY-98-02551.E-mail address:[email protected] (S.J. Gates Jr.).



86 S.J. Gates Jr. / Nuclear Physics B 616 (2001) 85–105

systems. After all, the Newton constant is always present in such theories. In fact thisis not sufficient. When supergravity theories are written in superspace formulations, it isalways possible to perform certain implicit “re-scalings” of component fields in such a waythat Newton’s constant is effectively absent. One requires a second dimensional constantand this is supplied in superstring theory by the string tension. Similarly, in discussionsof the low-energy M-theory effective action there is also such a constant that we refer toas11. We can choose this parameter to possess the units of length or that of an inversemass. As proposed [2–6] in our inaugural papers on this topic, the superspace torsion,curvature and field strength can be “deformed” perturbatively as power series expansionsin terms of such a dimensional parameter. Although our arguments were made with regardto superstring/heterotic string theories, the same reasoning applies to superstring/M-theory.

Many years ago, we made a conjecture [7] about some of the structure that is required todescribe an eleven-dimensional supergravity theory whose equations of motion is differentfrom those derived from the standard Cremmer–Julia theory. In fact in 1980, it was pointedout that such modification would in all likelihood require the existence of a dimension1/2 and spin-1/2 multiplet of currents (which we now refer to as theJ -tensor). At thattime an equation was given for how this multiplet of currents would begin to modify thesupergeometry of eleven-dimensional supergravity. In a 1996 work [8] we attempted toextend this investigation. However, due to missing terms1 first noted by Howe [9], thiswas not a carried out convincingly. In fact, Howeevenargued that our proposed dimension1/2 and spin-1/2 multiplet could not play a role in 11D supergravity/M-theory. This wasformalized in a result sometimes called “Howe’s Theorem”. Most recently [1], we haveemphasizedthat Howe’s Theorem cannot be valid unless supergravity/M-theory dynamicsadmits a scale-invariance. Since the proposed lowest order M-theory correction violatesthis condition, it is fully our expectation that the dimension 1/2 and spin-1/2 multiplet ofcurrents must play a critical role.

In the mean time, research has been undertaken based upon “Howe’s Theorem” andinvestigating what modifications are allowed within its context. This has led to theappearance of additional multiplets of currents known asX-tensors to be introduced soas to accommodate the higher derivative M-theory corrections. The appearance of theX-tensors does not contradict the appearance of the spin-1/2 J -tensor. In fact, in both our1980 and 1996 discussions of the modified 11D supergeometry we made explicit referenceto our expectation that the spin-1/2J -tensor would most likely beonlypart of an off-shelltheory, i.e., theJ -tensor must be accompanied by other tensors to provide a superspacedescription of the M-theory corrections.

One of the most powerful argument that reveals why theX-tensors alone areinsufficientto describe an off-shell Poincaré supergravity theory is because they are actuallysuperconformal field strengths,2 similar to but distinct from, the superspace Weyl multipletsuperfield. We should also note that the appearance of the superconformalX-tensorswas also foreshadowed in a much overlooked study of conformal symmetry in 4D,

1 Interestingly, these terms were already present in the 1980 work.2 This fact seems not to have been noticed until the recent work in [1].

S.J. Gates Jr. / Nuclear Physics B 616 (2001) 85–105 87

N > 4 superspace geometry [10]. In this work there appears the statement, “The presentanalysis shows that conformal symmetry may still be relevant forN > 4 supergravity.However, it requires at least asecondconformal field strength. . . ” Thus, by this work, therealization that a superconformal superspace field strength not related to the Weyl tensorhas appeared in the literature for some time. Although our comments were directed to 4D,N > 4 superspace geometry, dimensional oxidation implies that this must be true for 11Dsuperspace geometry also. So the significance of our observation has largely been ignoredand has led to much confusion on this subject.

2. Conformal class of bosonic spaces

A special class of infinitesimal fluctuations are those that are conformally related to theflat metric of eleven-dimensional superspace. We begin our study of eleven-dimensionalsupergeometry with this class of theories. However, before we do so, we wish to reviewthis class of structures within the non-supersymmetric case followed by considerations ina well-known supersymmetric context.

In a purely bosonic space, the class of graviton fluctuations that are conformally relatedto flat space are defined by

(1)∇a = ∂a +ψ∂a + k0(∂bψ)Mab,

where the infinitesimal scalar fieldψ (an arbitrary function of the spacetime coordinates)is known as the “scale factor”, “scale compensator”, or “conformal compensator”. Torsionand curvature tensors can be defined by calculating the commutator algebra of thisderivative

[∇a,∇b] = tabc∇c + 12rabc

dMdc,

tabc = (1+ k0)(∂[aψ)δb]c,

(2)rabde = k0

[(∂d∂[aψ

)δb]e − (

∂e∂[aψ)δb]d

].

A Riemannian geometry has no torsion tensor and this is accomplished by demanding,

(3)tabc = 0 → k0 = −1,

thus determining the value of the otherwise undetermined parameterk0 above. As thevanishing torsion determines the Christoffel connection, we see that fixingk0 = −1, isequivalent to choosing the Christoffel connection in (1). Geometries that are conformallyrelated to flat geometry are also compatible with a Riemannian geometry possessingcompletely vanishing torsion. Stated another way, the vanishing of the torsion places nodynamical restriction on the compensatorψ .

For finite values of the conformal compensator, we can integrate (1) to obtain

(4)∇a =ψ[∂a − (∂b lnψ)Ma

b].


3. Minimal conformal class of 4D, N = 1 superspace

We can continue this line of deliberation by looking at the case of 4D,N = 1 superspace.The analog of (1) takes the form

∇α =Dα + 12ΨDα + 0(DβΨ )Mα

β,

∇a = ∂a + 12

(Ψ + Ψ )

∂a + i1(DαΨ )Dα + i1

(DαΨ)Dα

+ i2([Dγ , Dα

]Ψ

)Mα

γ − i2([Dα, Dγ

]Ψ ) Mαγ

(5)+ 3(∂γ αΨ )Mαγ + 3

(∂αγ Ψ ) Mα

γ .

In this expression, the constants0, 1, 2 and3 are the supersymmetric analogs ofk0

in (1) andΨ is acomplexscalar and an infinitesimal superfield. It is an exercise to computeall the dimension 1/2 torsion tensors associated with (5) and we find

Tαβγ = 1

2(0 + 1)(D(αΨ )δβ)γ ,

Tαβγ = (

1 + 12

)(DαΨ )

δβγ ,

(6)Tαbc = 1

2

[Dα

((1− 0)Ψ + Ψ )]

δβγ + [

Dβ(0Ψ + 1Ψ )]

δαγδβγ .

The constraints of 4D,N = 1 supergravity act much as their analog in (3). The condition

(7)∇a = −i[∇α,∇α,determines the constants1, 2 and3 in terms of0. This is equivalent to providing adefinition of Ea andωαb c. The condition

(8)Tαβγ = 0,

determines the constant0 and this is equivalent to providing a definition ofωαβγ . Theform of the fluctuations we are considering also require thatωαβγ = 0.

Demanding that the first two dimension 1/2 torsions should vanish leads to0 = −1 and1 = −1/2. In principle in order for thelast dimension 1/2 torsion to vanish, we are led totwo distinctconditions

(9)Dα[(1− 0)Ψ + Ψ ] = 0, Dβ

[0Ψ + 1Ψ ] = 0,

except for0 = −1 and1 = −1/2 these are proportional to the self-same equation

(10)Dα(2Ψ + Ψ ) = 0.

A solution of this equation introduces the well-known “chiral compensator” superfield,ϕ,of 4D,N = 1 supergravity [11]

(11)Ψ = (2ϕ− ϕ)

.

We see that 4D,N = 1 supergravity fluctuations, conformally related to flat superspace,do not require non-vanishing dimension 1/2 torsionsif and if those fluctuations aredescribed by a chiral compensator. The vanishing of the dimension 1/2 torsion places nodynamical restriction on the chiral compensator. The results of (5)–(11) for the specialvalues0 = −1 and 1 = −1/2 correspond precisely to the infinitesimal limit of the


minimal 4D,N = 1 prepotential formulation of supergravity with the added restrictionthat we are working in a gauge where the conformal prepotential,Um, has also beenset to zero. The property that the full supergravity solution allows us to consistentlyseparate the fluctuations in∇A involving Um from those involvingϕ may be referred toas “separability”. Separability is equivalent to the statement that a Poincaré supergravitymultiplet can be thought of as the combination of a Weyl supergravity multiplet and acompensator multiplet. The compensator multiplet contains all of the Goldstone fieldsrequired to break the superconformal symmetry group to the super-Poincaré group.

4. The conformal class of flat 11D superspace

We now wish to prove a theorem about the eleven-dimensional supersymmetric case.We call this the “11D Torsion–Conformal Compensator Theorem” (11D T–C2 Theorem).Below we will show that 11D supergravity has a behavior that is drastically different fromeither the purely bosonic theory or 4D,N = 1 supergravity discussed above. The formalstatement of the 11D T–C2 Theorem is given below.

Theorem (11D Torsion–Conformal Compensator Theorem).If 11D supergravity iscompletely separable into a Weyl supermultiplet and a compensator supermultiplet, thenthe 11D Poincaré supergeometry mustpossess non-vanishingdimension1/2 torsionsupertensors.

We now prove this by explicit construction. The condition of complete separability meansthat we can set to zero the superfield that contains the Weyl multiplet without affectingthe dynamics of the conformal compensator. This condition is true inall superfieldsupergravity theoriespresentlyknown.

For the class of eleven-dimensional superspaces, we note that the analog of (1) and (5)takes the form

∇α =Dα + 12ΨDα + 0(DβΨ )

(γ de

)αβMde,

∇a = ∂a +Ψ∂a + i1(γa)αβ(DαΨ )Dβ + 2(∂cΨ )Ma

c

(12)+ i3(γade

)αβ(DαDβΨ )Mde,

whereΨ in (12)3 is a real scalar and an infinitesimal superfield and thei ’s are a set ofconstants (essential like those of the 4D,N = 1 theory). Of course the form of Eα is givenby

(13)Eα =Dα + 12ΨDα,

which is appropriate for a supergeometry that is related by a scale transformation to a “flat”eleven-dimensional supergeometry. The constants2 and3 can be eliminated by imposingthe analog of (7) to the 11D theory. This leaves only0 and1 to be fixed.

3 Although we use the same symbol here, this superfield should not be confused with the one that appears in(5).


Computing the dimension 1/2 torsion tensors associated with (12), we find

Tαbc = (1+ 20)(DαΨ )δb

c − (1 + 20)(γ cγb

)αβ(DβΨ ),

Tαβγ = 1

2(D(αΨ )δβ)γ + 1

20(DδΨ )(γ [2])

(αδ(γ[2])β)γ + 1

(γ c

)αβ(γc)

δγ (DδΨ )

(14)

= 132

[(321 − 1− 700)

(γ c

)αβ(γc)

γ δ + 12(380 + 1)

(γ [2])

αβ(γ[2])γ δ

+ 1120(100 − 1)

(γ [5])

αβ(γ[5])γ δ

](DδΨ ).

In reaching the second form of the last equation, we used two Fierz identities in order tomake clear the full content of the equation. Demanding that the dimension 1/2 torsionsvanish leads tofive independent conditions

0 = −1− 700 + 321, 0= 1+ 380,

(15)0 = 1− 100, 0 = 1+ 20, 0 = 1 + 20,

on twoconstants (0 and1). The resulting system is thus overdetermined and inconsistent.Our argument thus far runs exactly parallel to the case of 4D,N = 1 supergravity.

The only consistent solution for completely vanishing dimension 1/2 torsions in thiscircumstance is to also impose the extra conditionDαΨ = 0. But unlike the 4D,N = 1theory discussed above, this has an additional dire consequence,

DαΨ = 0→DαDβΨ = 0 → [Dα,DβΨ = 0

(16)→ i(γ c

)αβ∂cΨ = 0→ ∂cΨ = 0.

This implies thatΨ must be a constant so that (12) reduces to a trivial constant re-scaling of the superframe fields. Thus eleven-dimensional superspace geometry is quiteunlike its bosonic counterpart. Here we see non-constant supergravity fluctuations that areconformally related to the flat superspacenecessarilyproduce dimension 1/2 and spin-1/2torsion tensors that are non-vanishing. At most onlytwo of the five independent structuresthat occur in the dimension 1/2 torsions can be set to zero, if the theory is to possessa non-trivial conformal compensator. This is also seen to be substantially different fromthe case of 4D,N = 1. There it was the case that the existence of the chiral compensatorstill permitted the existence ofnon-trivial fluctuations that are conformally related flatsuperspace even though all dimension 1/2 torsions vanish. This isnotpossible for the 11Dcase since the notion of a chiral superfield is non-existent for an 11D theory!

Within the class of derivatives defined by (2) we next wish to define one that satisfiesthe conventional constraints

(17)∇a = i 132(γa)

αβ [∇α,∇β, Tα[bc] − 255(γbc)α

βTβdd = 0.

The first of these defines Ea andωabc in terms of Eα andωαbc (cf. Eq. (7)). The second ofthese defines the spin-connectionωαbc in terms of the anholonomy (cf. Eq. (8)). We cansatisfy these condition by choosing the constants as

(18)0 = 110, 1 = 1

4, 2 = 15, 3 = 1

160.


Utilizing the fluctuations described by (12) and (18), we calculate the completecommutator algebra associated with this supergravity covariant derivative to find

Tαβc = i(γ c)

αβ,

Tαβγ = 3

40

(γ de

)αβ(γde)

δγ (DδΨ ),

Rαβde = − 1

10

[(γ de

)αγ (D[βDγ ]Ψ )+

(γ de

)βγ (D[αDγ ]Ψ )

+ 18

(γ b

)αβ

(γbde

)γ δ(DγDδΨ )

],

Tαbc = 3

5

[2(DαΨ )δb

c − 34

(γ cγb

)αγ (Dγ Ψ )

],

Tαbγ = [

i 18(D[αDβ]Ψ )(γb)βγ − i 1

2,880

(γ [3])δε(DδDεΨ )(γ[3]γb)αγ

− i 12,880

(γ [3])δε(DδDεΨ )(γbγ[3])αγ − 1

20

(γbγ

c)αγ (∂cΨ )

+ 740

(γ cγb

)αγ (∂cΨ )

],

Rαbde = [−i 1

480

(γbde

)δε(D[α|D|δ|D|ε]Ψ )− 1

80

(γ cγb

de)αγ (∂cDγΨ )

+ 15

(γ de

)αγ (∂bDγΨ )+ 1

5

(∂ [dDαΨ

)δbe]],

Tabc = 1

5

[6(∂[a|Ψ )δ|b]c + i 1

8

(γab

c)αβ(DαDβΨ )

],

Tabγ = −i 1

4(γ[a|)γ δ(∂|b]DδΨ ),

(19)Rabde = 1

5

[(∂[a|∂dΨ

)δ|b]e − (

∂[a|∂eΨ)δ|b]d + i 1

32

(γ[ade

)αβ(∂b]DαDβΨ )

].

These equations emphasize a remarkable fact which is true in all supergeometries. Themost general supergeometry that is conformally related to the flat superspace may beviewed as a geometrical description of a scalar superfield. We have long been aware ofthis fact and have used it previously [19] to derive the first off-shell description of 2D,N = 4 supergravity.Within 2D theories, this result is even more powerful. It impliesthat the supergravity constraints for 2D theories are in one-to-one correspondence withirreducible scalar superfields and thus the supergravity constraints aretotally determinedby the differential equations that define 2D irreducible scalar multipletsand vice versa.

The covariant derivatives in (12) may be “integrated with respect toΨ ” so that in thecase of a finite conformal compensatorΨ we find

∇α = Ψ 1/2[Dα + 110(Dγ lnΨ )

(γ de

)αγMde

],

(20)

∇a = Ψ [∂a + i 1

4(γa)γ δ(Dγ lnΨ )Dδ + 1

5(∂b lnΨ )Mab

+ i 1160

(γade

)γ δ(DγDδ lnΨ )Mde

+ i 27800

(γade

)γ δ(Dγ lnΨ )(Dδ lnΨ )Mde

].

To summarize the main result in this section, we have shown that under the assumptionthat 11D supergravity is separable into a Weyl multiplet superfield (Hα

m) and a conformalcompensator superfield (Ψ ), Howe’s Theorem in the limit of vanish Weyl multiplet forbidsthe appearance of the conformal compensator superfield. This behavior is not congruentwith that (e.g., the discussion from (5)–(11)) found in all previous known cases of off-shellsuperfield supergravity.


5. Traditional approach to Weyl symmetry in 11D superspace

Although Howe’s 1997 paper [9] has described anew formalism to realize thepresence of Weyl symmetry for 11D superspace, in fact there is a traditional manner foraccomplishing this goal. This traditional approach was initiated by Howe and Tucker [12]and then developed by others [13]. We have even been able to extend this traditionaldescription all the way to 10D,N = 1 superspace [14]. In our recent work [1], we began theprocess of extending this traditional approach to the 11D theory. As was shown in [1], thenewer approach requires that there must exist degrees of freedom in addition to those thatreside in Eα . It is thus anon-minimalrealization of Weyl symmetry in 11D superspace. Thetraditional approach [12], does not suffer from this drawback. So it will be the goal of thischapter to extend completely the traditional formalism for Weyl symmetry in superspace tothe 11D case and thereby establish the realization of Weyl symmetry in a minimal mannerwithin this venue.

In our recent work [1], an analysis of the 11D vielbein degrees of freedom wasperformed. This work implies that a solely conventional set of constraints4 can be chosenas

i(γa)αβTαβ

b = 32δab, (γa)αβTαβ

γ = 0, Tα[de] − 255(γde)α

γ Tγ bb = 0,

(γa)αβRαβ

de = 0, (γab)αβTαβ

b = 0, (γ[ab|)αβTαβ|c] = 0,

(21)(γabcde)αβTαβ

e = 0, 16!ε[5]abcdef (γabcde)αβTαβf = 0.

The last four constraints determine the degrees of freedom in Eαµ that correspond to the

elements of the coset

(22)SL(32,R)

SO(1,10)⊗ SO(1,1).

The constraints imply that all 11D supergravity fields are contained in two semi-prepotentials;Ψ (conformal compensator) andHαm (Weyl supermultiplet) and we willuse these constraints in the following.

The last four constraints of our table may be called “coset conventional constraints5”and in fact their existence is an almost universal feature of superspace supergravity theories.The easiest way to understand this is to consider torus compactification of the 11D result.So, for example, in four-dimensionalN -extended supergravity, these coset constraints takethe form

(23)SL(4N,C)

SO(1,3)⊗ SO(1,1)⊗ U(N),

for N = 4 and

(24)SL(16,C)

SO(1,3)⊗ SO(1,1)⊗ SU(4),

4 To our knowledge, this is the first time this particular set of off-shell constraints for 11D superspace has beensuggested. These are different from our previous works (e.g., [1]) but are convenient for many purposes. Thefactor of 2/55 in particular leads to “nice” normalizations in many, many subsequent calculations.

5 The 11D coset conventional constraints are closely related to the constraints discussed in [15].


for N = 4. It is perhaps also useful to point out that an example of coset conventionalconstraints was discussed [16] when we provided the first explicit solution to constraintsfor 2D, N = 1 supergravity. In fact, 4D,N = 1 supergravity is the exception ratherthan the rule when it comes to coset conventional constraints. While 4D,N = 1superspace supergravity theory does not require such constraints, most generic superspacesupergravity theories do require this type of constraint.

A perennial question we often encounter when discussing superspace systems ofconstraints is caused by the fourth entry in our table. The question is, “shouldn’tωabc

be determined by the condition that appears in (3)?” The answer to this is that onecan certainly replace the first constraint in the second row byTabc = 0. However, thisdetermines the connection in an “un-improved manner”. Certain improvement terms (asseen in the 4D,N = 1 case) will occur with our choice.

Motivated by the discussion in the last two chapters and by our work in [1,14], we candefine aminimalrealization of a scale transformation law for the 11D superspace covariantderivative by

δS∇α = 12L∇α + 1

10(∇γ L)(γ bc

)αγMbc,

δS∇a =L∇a + i 14(γa)

αβ(∇αL)∇β + 15(∇cL)Ma

c

(25)+ i 1160

(γabc

)αβ(∇α∇βL)Mbc.

The importance of these equations is that they permit us to calculate the scale variationsof all superspace torsions and curvatures at the full non-linear level. It should also beclearly understood that the derivatives∇A that appear in (25) arenot restricted to describefluctuations conformally related to flat superspace as in (12).In particular the torsions inthis chapter arenotgiven solely by the terms in(19). We find

δSTαβc = 0,

δSTαβγ = 1

2LTαβγ − i 1

4Tαβc(γc)

δγ (∇δL)+ 12(∇(αL)δβ)γ

+ 120(∇δL)

(γ [2])

(αδ(γ[2])β)γ ,

δSTαbc = 1

2LTαbc − i 1

4(γb)γ δ(∇γ L)Tαδc + (∇αL)δbc + 1

5(∇γ L)(γbc)αγ ,

δSTαbγ = LTαbγ − i 1

4(γb)δε(∇δL)Tαεγ − i 1

4Tαbc(γc)

δγ (∇δL)− 1

2(∇bL)δαγ − 110(∇dL)

(γbd)αγ

+ i 14(γb)

δγ (∇α∇δL)− i 1320

(γbde

)δε(∇δ∇εL)(γde)αγ ,

δSTabc = LTabc + i 1

4(γ[a|)δε(∇δL)Tε|b]c + 65(∇[a|L)δ|b]c + i 1

40

(γab

c)δε(∇δ∇εL),

δSTabγ = 3

2LTabγ − i 1

4Tabc(γc)

δγ (∇δL)+ i 14(γ[a|)αβ(∇αL)Tβ|b]γ

(26)− i 14(γ[a|)δγ (∇|b]∇δL),

for the various torsion tensor components and

δSRαβde = LRαβde + 1

5Tαβ[d(∇e]L) + 1

5Tαβδ(∇γ L)

(γ de

)δγ

− 15(∇(α|∇γ L)

(γ de

)|β)γ + i 1

80Tαβc(γcde

)δε(∇δ∇εL),


δSRαbde = 3

2LRαbde − i 1

4(γb)δε(∇δL)Rαεde + 1

5Tαbγ (∇δL)

(γ de

)γδ

+ 15Tαb

[d(∇e]L) + 15

(∇α∇[dL)δbe] + 1

5(∇b∇γ L)(γ de

)αγ

+ i 180Tαb

c(γcde

)δε(∇δ∇εL)− i 1

80

(γcbe

)δε(∇α∇δ∇εL),

δSRabde = 2LRabde + i 1

4(γ[a|)αβ(∇αL)Rβ|b]de + 15Tab

γ(γ de

)γδ(∇δL)

− 15Tab

[d(∇e]L) + i 180Tab

c(γcde

)αβ(∇α∇βL)

(27)+ 15δ[a|

[d(∇b]∇e]L) + i 180

(γ de[a

)αβ(∇b]∇α∇βL),

for the various curvature tensor components.Although these conformal transformation laws may seem quite complicated when

compared to those of [9], an investigation of previous lower-dimensional off-shellsupergravity theories will reveal many similarities.

An especially important superfield occurs at dimension 1/2. We denote this superfieldby Jα where

(28)Jα ≡ 433Tαb

b → δSJα = 12LJα + (∇αL).

This transformation law implies thatJα is not a superconformal invariant superfield.It has been proposed [15] that a rank six and dimension zero field strength denoted by

X[5]a is critical to obtain a superspace description of the M-theory effective action. In ourprevious work [1], we pointed out that there is also an alternate possibility of allowinganother similar tensor,X[2]a . These both can be found in

(29)Tαβc = i(γ c)

αβ+ 1

2

(γ [2])

αβX[2]c + i 1

120

(γ [5])

αβX[5]c,

or alternately we see

X[ab]k ≡ 132(γab)

αβTαβk,

(30)X[abcde]k ≡ i 132(γabcde)

αβTαβk.

It is interesting to note what result is obtained from the quantity defined byTαbb −

(3/4)Tαββ under the action of the scale transformation law in (25). In fact this object is adimension 1/2 spin-1/2 scale covariant. Starting from the third equation in (26) we see

δSTαββ = 1

2LTαββ − i 1

4Tαβc(γc)

δβ(∇δL)+ 12(∇(αL)δβ)β

+ 120(∇δL)

(γ [2])

(αδ(γ[2])β)β

= 12LTαβ

β − i 14

[i(γ c

)αβ

+ 12

(γ [2])

αβX[2]c

+ i 1120

(γ [5])

αβX[5]c

](γc)

δβ(∇δL)+ 1

2(∇(αL)δβ)β + 120(∇δL)

(γ [2])

(αδ(γ[2])β)β

= 12LTαβ

β − 114 (∇αL)+ 33

2 (∇αL)− 5520(∇αL)

(31)→ δS( 1

11Tαββ) = 1

2L( 1

11Tαββ) + (∇αL).


The reason the terms involving theX-tensors have “disappeared” from the final result canbe seen from the following considerations

12

(γ [2])

αβ(γc)

δβX[2]c = −12(γ[2]γc)αδX[2]c

= −12

[(γ[abc])αδ − 2ηca(γb)α

δ]X[ab]c = 0,

(32)

i 1120

(γ [5])

αβ(γc)

δβX[5]c = −i 1120(γ[5]γc)αδX[5]c

= 1120

[ 1120εabcdef

[5′](γ[5′])αδ

− i15ηca(γbdef )αδ]X[abdef ]c = 0.

The conventional constraints imposed upon theX-tensors eliminate their appearance fromthe final result in (31). So that we see an alternate definition of theJ -tensor is givenby Jα = 1

11Tαββ . The appearance of this non-scalar covariant field Poincaré supergravity

strength should not come as a surprise. At least one non-scale invariant field strengthsuperfield can be verified in every known off-shell superspace formulation that has everbeen given.

The reason that this superfield is important is that it allows the definition of a Poincarésuperspace spinorial differentiation operation that acts consistently to remain solely withinthe space of superconformal tensors. To see this, we note that a superconformal tensordenoted byT (w)a1...ap

b1...bq and of weightw can be defined to transform according to

(33)δST (w)a1...apb1...bq =wLT (w)a1...ap

b1...bq ,

under the action of the superspace scale transformation. From this it follows that we find

δS(∇αT (w)a1...ap

b1...bq)

= (w+ 1

2

)L

(∇αT (w)a1...apb1...bq

) +w(∇αL)T (w)a1...apb1...bq

+ 110(∇γ L)

(γ bc

)αγ(MbcT (w)a1...ap

b1...bq),

δS(JαT (w)a1...ap

b1...bq)

= (w+ 1

2

)L

(JαT (w)a1...ap

b1...bq) + (∇αL)T (w)a1...ap

b1...bq ,

δS(Jγ

(γ bc

)αγMbcT (w)a1...ap

b1...bq)

= (w+ 1

2

)LJγ

(γ bc


b1...bq)

(34)+ (∇γ L)(γ bc


b1...bq).

These three equation taken together inform us that the quantity defined by

(35)(∇αT (w)a1...ap

b1...bq) ≡ [(∇α −wJα − 1

10Jγ(γ bc

)αγMbc

)T (w)a1...ap

b1...bq],

possesses a covariant scale transformation law

(36)δS(∇αT (w)a1...ap

b1...bq) = (

w+ 12

)L

(∇αT (w)a1...apb1...bq

),

with scale weightw+ 12.


Examples of 11D scale-covariant supergravity tensors are provided by the “X-tensors”[15]. 6 According to the first result in (26) bothX-tensors are scale-covariant havingw = 0.Thus, we see

δSX[2]c = 0, δSX[5]c = 0

(37)

→ ∇αX[2]c = (∇α − 110Jγ

(γ bc

)αγMbc

)X[2]c,

∇αX[5]c = (∇α − 110Jγ

(γ bc

)αγMbc

)X[5]c,

and consequently

(38)δS(∇αX[2]c

) = 12L

(∇αX[2]c), δS

(∇αX[5]c) = 1

2L(∇αX[5]c

),

and we find the “hatted derivative” of thew = 0 scale-covariantX-tensors arew = 1/2scale-covariant tensors. All of these results are the expected generalization of the onesfound for the 10D,N = 1 superspace [14].

A second especially important superfield occurs at dimension one and is the on-shell11D supergravity field strength. We denote this superfield byWabcd and note that itcontains the usual supercovariantized Weyl tensor7 at second order in itsθ expansion.Given the superspace scale transformation law of the various torsion components (26), itfollows that a superscale-covariant quantity of dimension one and withw = 1 is given by

Wabcd ≡ 132

[i(γ eγabcd

)γαTαe

γ − 13(γabcd)

αβ(∇αTβcc + 14

1,815TαccTβd

d)]

(39)→ δSWabcd = LWabcd .

This 4-form superfield quite properly may be called the 11D Weyl multiplet superfield.However, as we noted above, the component-level supercovariantized Weyl tensor isnotthe leading component of this superfield. Since this superfield contains the Weyl multiplet,it does not vanish when the 11D supergravity multiplet obeys the equations of motionassociated with 11D super-Poincaré action. It also thus corresponds to the on-shell fieldstrength of 11D supergravity. Our discussion above can be applied to solve a small puzzlein the work of [15]. Namely this formula necessarily defines the on-shell field strength,containing the physical degrees of freedom of the 11D theory, as it picks out a particularlinear combination of the two four-forms described in the work of Ref. [15].8

The equation in (35) has an important component implication. If we denote thelocal supersymmetry variation in the putative 11D superconformal theory byδQ, thecorresponding one in the 11D Poincaré theory byδQ, the local scale variation byδS andthe local Lorentz variation byδLL, then Eq. (35) implies

(40)δQ(εα

) = δQ(εα

) − δS(εαλα

) − 110δLL

(εα

(γ bc

)αβλβ

),

whereεα(x) is the local supersymmetry parameter andλα(x) is theθ → 0 limit of Jα .The main results of this section are summarized in Table 1.

6 In their work, Cederwall, Gran, Nielsen and Nilsson only retained theX[5]a tensor.7 We use the non-calligraphic symbolWabcd for the usual Weyl tensor superfield.8 The definition ofWabcd can change slightly depending on the choice of conventional constraints. Our

definition is the one that follows from choosing the constraints as in (21).


Table 111D supergravity field strength superfields

Superfield Conformal Weight Engineering dim.

Wabcd yes 1 1Jα no 1/2 1/2X[abdef ]c yes 0 0X[ab]c yes 0 0

6. Weyl 11D superspace geometry from super-Poincaré geometry

The spinorial “hatted” supercovariant derivative introduced in the last section allowsthe construction of supergeometrical objects that possess superconformal symmetry. Thespinorial Weyl “hatted” supercovariant derivative given in (35)

(41)∇α ≡ ∇α −wJα − 110Jγ

(γ bc

)αγMbc,

may be used to define a bosonic Weyl “hatted” supercovariant derivative,

∇a ≡ i 132(γa)

αβ[∇α, ∇β

= ∇a − i 14(γa)

αβJα∇β − iw 116(γa)

αβ [∇αJβ ]I(42)− i 1

160(γaγde)αβ

[(∇α − 135 Jα

)Jβ

]Mde.

From the results in (25), (28) and (36) we find the following variations,

δS[∇aT (w)]

= (w+ 1)L∇aT (w) + i 14(γa)

αβ(∇αL)(∇βT (w)) +w(∇aL)T (w)

(43)+ 15(∇dL)

(Ma

dT (w)) + i 1

160

(γade

)αβ(∇α∇βL)

(MdeT (w)

),

δS[i(γa)

αβJα(∇βT (w))]

= i(w+ 1)L(γa)αβJα(∇βT (w)) + i(γa)αβ(∇αL)

(∇βT (w))

(44)+ iw(γa)αβJα(∇βL)T (w) + i 110

(γaγ

de)αβJα(∇βL)(MdeT (w)

),

δS[i(γa)

αβ(∇αJβ)T (w)]

= i(w+ 1)L(γa)αβ(∇αJβ)T (w) + 16(∇aL)T (w)(45)− i4(γa)αβJα(∇βL)T (w),

δS[i(γd)

αβ(∇αJβ)(Ma

dT (w))]

= i(w+ 1)L(γd)αβ(∇αJβ)(Ma

dT (w))

(46)− i4(γa)αβJα(∇βL)(Ma

dT (w)) + 16(∇dL)

(Ma

dT (w)),

δS[i(γade

)αβ(∇αJβ)

(MdeT (w)

)]

= i(w+ 1)L(γ dea

)αβ(∇αJβ)

(MdeT (w)

) + i 65

(γade

)αβJα(∇βL)(MdeT (w))

(47)+ i(γade)αβ(∇α∇βL)(MdeT (w)),


δS[i(γade

)αβ(JαJβ)

(MdeT (w)

)]

= i(w+ 1)L(γade

)αβ(JαJβ)

(MdeT (w)

)

(48)+ i2(γade

)αβJα(∇βL)(MdeT (w))

In writing these we have used the short-hand notationT (w) ≡ T (w)a1...apb1...bq . The

definition of∇a insures that the following relation is satisfied,

δST (w)a1...apb1...bq =wLT (w)a1...ap

b1...bq

(49)→ δS(∇aT (w)a1...ap

b1...bq) = (w+ 1)L∇aT (w)a1...ap

b1...bq .

A calculation of the graded commutator algebra of the “hatted” derivatives yield “hatted”torsion and curvature superfields. As well a dilatation field strength superfieldFAB can bedefined from the graded commutator,

(50)[∇A, ∇B ≡ TABγ ∇γ + TABc∇c + 1

2RABdeMe

d −wFABI.

Given the results of (41) and (42) we also find

(51)[∇A, ∇B = [∇A,∇B +KABC∇C + 1

2KABdeMe

d −wFABI.

The quantitiesKABC ,KABde andFAB are expressed9 in terms of the Poincaré superspacecovariant derivative∇A andJα . The two equations immediately above inform us that thesuper-Weyl covariantsTABC andRABde are related to the corresponding Poincaré objectsTAB

C andRABde via,

(52)TABC = TABC + KABC, RABd

e =RABde + KABde.

7. Scale vs. non-scale invariant equations of motion

The analysis of the conformal properties of the full non-linear theory given in a previoussections once again provides an argument against the result known as “Howe’s Theorem”that asserts that there cannot appear a dimension 1/2 and spin-1/2 auxiliary field strengthsuperfield. Thus Howe’s Theorem is equivalent to the imposition of the condition thatJα = 0 be imposed as a kinematic constraint. Let us now impose this condition in additionto those that appear in (21). The scale transformation properties of the superspace are stilldescribed by (25). In turn this observation together with a consistency condition applied to(28) leads to the condition∇αL= 0. This is the full non-linear extension of (16) and leadsto the exact same problem as we found in our prepotential analysis.

Let us use these insights in a different way. In this very short section, we will simplycompare properties of the non-supersymmetric Poincaré gravity equations of motions withtwo possible supersymmetric extensions.

9 Although the calculation of theK-quantities,K-quantities and as wellFAB is straightforward, we will foregogiving explicit expressions for all of these.


For the ordinary 11Dx-space covariant derivative, a scale transformation law is givenby

(53)δS∇a = ∇a − (∇k)Mak,

where(x) is a local scale parameter. This implies that the 11Dx-space Riemann curvaturetensor transforms as

(54)δSrabde = rabde − (∇[a∇[d

)δb]e].

The constraint is, of course, that the torsion tensor vanishestabc = 0. In the absence ofmatter, the usual Einstein–Hilbert action leads to the expected equation of motion

(55)Eab ≡ racbc − 12ηabrcd

cd = 0,

i.e., the Einstein tensor vanishes. It is a simple matter to show that this equation of motionis not a scale-invariant condition since,

(56)δSEab = 2Eab − 9(∇a∇b)+ 9ηab(∇c∇c).

Now in the scheme that was proposed in the works [1] and [8], the equations of motionsare

(57)Jα = 0, X[ab]c = 0, X[abdef ]c = 0.

Due to the observation in (28), we see that the first of these superfield equations is alsonota superscale-invariant condition.

This is to be contrasted with the schemes ([9] and [15]) where the equations of motionsare

(58)X[ab]c = 0, X[abdef ]c = 0,

but the conditionJα = 0 is considered to be a constraint. Thus, in these schemes theequations of motionare superscale covariant. From this point of view, the role of theJ -tensor is to extend the non-scale covariance of the Poincaré equations of motion inx-spaceinto the non-scale covariance of Poincaré equations of motion in superspace. So to acceptthe validity of Howe’s Theorem10 is equivalent to imposing superscale covariance on theequations of motion.

In M-theory, the low-energy effective action is one whose lowest order terms describe11D Poincaré supergravity. Some of the structure of the next order corrections in anexpansion11 have been discussed in the literature. None to date lead to scale-invariantterms appearing in the equations of motion. Thus, it is our position that this alone precludesany superspace obeying Howe’s Theorem from being able to describe the M-theoryeffective action. The non-scale covariance of the M-theory effective action equations ofmotion demand the appearance of theJ -tensor.

10 The authors of [15] have noted the need to check explicit M-theory currents for consistency with Howe’sTheorem.


8. Summary discussion

Under the impetus of the effective M-theory action’s superspace formulation, problemsin 11D supergravity are now being faced. This is welcomed activity. However, we are stillwithout truly fundamental insight into off-shell 11D,N = 1 supergravity, despite the workunderway. It is useful to use 4D,N = 1 supergravity to make this point most sharply. Wecan do this by giving further consideration to the situation of the scale compensator. Thereis another layer to the structure of the theory. The truly irreducible minimal 4D,N = 1 off-shell supergravity compensator satisfies the equationJα = 0. The solution to thisin fourdimensionsimplies thatΨ is a linear combination of a chiral superfield and its conjugate.It is the analog of this irreducibility condition that we still lack in the 11D theory becausechiral superfields cannot exist in 11D superspace.

Since chirality does not exist in 11D, there are no non-trivial conformal fluctuations ofthe flat metric to the equationJα = 0.

The supergravity vielbein in (12) with its 232 d.f. (d.f. ≡ degrees of freedom) isan off-shell but highly reducible supergravity representation. It requires the impositionof differential equations upon it in order to reach aminimal irreducible off-shellrepresentation. Based on some structures observed in an unusual class of algebras [17],we conjecture that such irreducible representation exist. For example, we have foundsome evidence that there may well exist a 32 768 bosonic d.f. and 32 768 fermionic d.f.irreducible representation ofD = 11 supersymmetry in superspace. However, confirmationof this remains for the future. The problem of classifying 11D superspace irreduciblerepresentations remains an important unsolved puzzle.

We again point out for skeptics that non-minimal 4D,N = 1 superspace geometry isthe avatar to set the pattern to be followed by 11D supergravity/M-theory superspace. Itis a demonstrable fact that imposing thesolelyconventional constraints leads to four fieldstrength superfields;Wabcd , Jα , X[ab]c andX[abdef ]c, one on-shell and three off-shellfield strengths. The first of these contains all the usual conformal degrees of freedom,the second allows for the traditional realization of the Weyl symmetry within a Poincarésuperspace and the latter two superfield are roughly speaking the analogs ofGa in thenon-minimal 4D,N = 1 superspace supergravity theory. However,X[ab]c andX[abdef ]cdiffer from Ga in the important respect that they are also conformally covariant likeWabcd . The superfieldWabcd has conformal weight one whileX[ab]c andX[abdef ]c haveconformal weight zero. In terms of semi-prepotentials, all the dynamics are contained intwo superfieldsHαc andΨ . The first of these is the gauge field for the Weyl degrees offreedom and the second is the Goldstone superfield for breaking superconformal symmetryto super-Poincaré symmetry.

Some years ago, we also conjectured [20] that the ultimate formulation of covariantstring field theory must be one in which the fundamental string field functional shouldappear as aprepotentialin an as-yet undiscovered geometrical formulation. In the mostcomplete formulation of open string field theory to date [21], Berkovits has presented aformulation that is hauntingly reminiscent of the 4D,N = 1 prepotential formulation ofYang–Mills theory. This is precisely in accord with our conjecture in [20] and encourages


our belief in a type of universality of prepotential formulations in all theories that containsupergravity.

It has been the suggestion of Cederwall et al. [15] that the quantitiesX[ab]c andX[abdef ]care both equations of motion. In the work of [1] we have pointed out that if one can be setequal to zero as a constraint that the choiceX[abdef ]c = 0 leads to a smaller supergravitymultiplet (if this is a viable option). It also the case that both of theseX-field strengthsuperfieldscannotsimultaneously be set to zero as constraints. To do so would force theconformal semi-prepotentialHαc to be zero up to a pure gauge transformation.

If 11D supergravity is like (i.e., separable) other prepotential formulation of supergravitytheories that possess both a conformal multiplet and a conformal compensator then thelatter will necessarily demand non-vanishing torsion at dimension 1/2. The 11D T–C2 Theorem is the formal statement of the most fundamental reason why it has beenour position that Howe’s Theorem is specious. Stated another way, if Howe’s Theoremwere true and theX-tensors were the only ones required to completely describe thedynamics of the M-theory 11D effective action, then this effective action would possess asuperconformal symmetry. Not possessing a clear understanding of this has led to a numberof confused efforts in the research literature by different research groups working on both11D and 10D superspace problems [22].

In this present work we have found evidence that in 11D supergravity, the canonicalsplit of the fundamental degrees of freedom into a superconformal prepotentialHα

c anda conformal compensatorΨ is valid and that it is possible to conventionally realize asuperconformal symmetry. As 11D supergravity has been proposed as a limit of M-theory,our findings naturally suggest that this canonical split likely carries over to the completeM-theory itself! That is, it appear likely to us that M-theory has a similar split and thatit is possible to realize a superconformal symmetry on whatever may play the role ofthe fundamental degrees of freedom that describe M-theory. Thus, the concept of theprepotential will likely play an important role in the final formulation of M-theory. We thusconjecture that there exist some formulation of M-theory that possesses all the canonicalstructures of superspace supergravity; a conformal prepotential, a conformal compensatingprepotential and a realization of superconformal symmetry.

“Understanding and wisdom only comes to us when our respective muses deign toreceive an audience. Knowing this is the beginning of both.”

– Anonymous

Added note in proof

After the completion of this work, we received communications from B. Nilsson statingthat the issue of superconformal symmetry leads one to “. . . reconsider Weyl superspace assoon as one puts in explicit expressions for theX-tensors.” As well their work requires acomputation of the Weyl-connection to verify the validity of “Weyl superspace”.


A component level discussion of 11D superconformal symmetry may be found in ourfinal reference.

Acknowledgements

We wish to acknowledge discussions with M. Cederwall and P. Howe, B. Nilsson andH. Nishino.

Appendix A. Conventions and notation

The conventions that we use for 11D superspace have been stated in some detail in ourprevious work [8]. In particular, we use real (i.e., Majorana) 32-component spinors for theGrassmann coordinates of superspace. Ourγ -matrices are defined by

(A.1)γ a, γ b

= 2ηabI,

where the signature of the metric is the “mostly minus one”, i.e., diag(+,−, . . . ,−). Thisimplies that our gamma matrices satisfy the complex conjugation conditions

[(γ a

)αβ]∗ = −(

γ a)αβ,

[(γ [2])

αβ]∗ = (

γ [2])αβ,[(

γ [3])αβ]∗ = −(

γ [3])αβ,

[(γ [4])

αβ]∗ = (

γ [4])αβ,

(A.2)[(γ [5])

αβ]∗ = −(

γ [5])αβ .

Our “spinor metric”, with which we raise and lower spinor indices, is denoted byCαβ andsatisfies,

(A.3)Cαβ = −Cβα, [Cαβ ]∗ = −Cαβ.The inverse spinor metricCαβ is defined to satisfy

(A.4)CαβCγβ = δαγ .

We also use superspace conjugation which permits the appearance of appropriate factors ofi even within a theory of involving solely real spinors. A complete discussion of superspaceconjugation can be found in a recent pedagogical presentation [18, p. 13].

We define our gamma matrices with multiple numbers of vector indices through theequations

γaγb = γab + ηab, γbγa = −γab + ηab,γaγbc = γabc + ηa[bγc], γbcγa = γabc − ηa[bγc],γaγbcd = γabcd + 1

2ηa[b|γ|cd], γbcdγa = −γabcd + 12ηa[b|γ|cd],

γaγbcde = γabcde + 16ηa[b|γ|cde], γbcdeγa = γabcde − 1

6ηa[b|γ|cde],γaγbcdef = i 1

120εabcdef[5]γ[5] + 1

24ηa[b|γ|cdef ],(A.5)γbcdef γa = −i 1

120εabcdef[5]γ[5] + 1

24ηa[b|γ|cdef ].


Table 2

[P ] = [1] [P ] = [2] [P ] = [3] [P ] = [4] [P ] = [5]

[Q] = [1] −9 −70 450 2.160 −5.040[Q] = [2] 7 −38 −126 −144 −5.040[Q] = [3] −5 −14 −30 −528 1.680[Q] = [4] 3 2 66 −144 1.680[Q] = [5] −1 10 −30 240 −1.200

The basic identities for non-vanishing traces over the gamma matrices are

132 Tr

(γaγ

b) = δab,

132 Tr

(γa1a2γ

b1b2) = −δ[a1|b1δ|a2]b2,

132 Tr

(γa1a2a3γ

b1b2b3) = −δ[a1|b1δ|a2|b2δ|a3]b3,

132 Tr

(γa1a2a3a4γ

b1b2b3b4) = δ[a1|b1δ|a2|b2δ|a3|b3δ|a4]b4,

132 Tr

(γa1a2a3a4a5γ

b1b2b3b4b5) = δ[a1|b1δ|a2|b2δ|a3|b3δ|a4|b4δ|a5]b5,

(A.6)132 Tr

(γa1 · · ·γa11

) = iεa1...a11.

Using the spinor metric to lower one spinor index of the quantities in (A.2) we find[(γ a

)αβ

]∗ = (γ a

)αβ,

[(γ [2])

αβ

]∗ = −(γ [2])

αβ,

[(γ [3])

αβ

]∗ = (γ [3])

αβ,

[(γ [4])

αβ

]∗ = −(γ [4])

αβ,

(A.7)[(γ [5])

αβ

]∗ = (γ [5])

βα,

and as well the same equations apply to the matrices with two raised spinor indices. Inaddition these satisfy the symmetry relations

(A.8)

(γ a

)αβ

= (γ a

)βα,

(γ [2])

αβ= (γ [2])

βα,

(γ [5])

αβ= (γ [5])

βα,

(γ [3])

αβ= −(

γ [3])βα,

(γ [4])

αβ= −(

γ [4])βα,

where the same equations apply to the matrices with two raised spinor indices.Other identities on the 11D gamma matrices include

(A.9)γ [P ]γ[Q]γ[P ] = c[Q][P ]γ[Q],

where the coefficientsc[Q][P ] are given in Table 2.For example, Table 2 implies

(A.10)γ [3]γ[2]γ[3] = −126γ[2].

Some useful Fierz-type identities include the following

0 = (γ a

)(αβ|(γab)|γ δ),

0 = 5(γ a

)(αβ|(γa)|γ δ) + 1

2

(γ [2])

(αβ|(γ[2])|γ δ),

0 = 6(γ a

)(αβ|(γa)|γ δ) + 1

5!(γ [5])

(αβ|(γ[5])|γ δ),


0 = (γ e

)(αβ|(γabcde)|γ δ) − 1

8(γ[ab|)(αβ|(γ|cd])|γ δ),

0 = 12

(γ ab

)(αβ|(γabcde)|γ δ) + 2(γ[c|)(αβ|(γ|de])|γ δ),

(A.11)0 = (γ a

)(αβ|(γa)|γ )

δ − 12

(γ [2])

(αβ|(γ[2])|γ )δ + 15!

(γ [5])

(αβ|(γ[5])|γ )δ.

Our Lorentz generator is defined to realize[Mab, (γa)αβ

= 0,

[Mab,∇α = 12(γab)α

β∇β,[Mab,∇c = ηca∇b − ηcb∇a,

(A.12)[Mab,Mcd = ηcaMbd − ηcbMad − ηdaMbc + ηdbMac.

One notational device not discussed in our previous paper is the definition of the 11Dsuper-epsilon tensor, which we denote byεA1...A4

B1...B7, and define by

εa1...a4b1...b7 ≡ εa1...a4

b1...b7,

εα1a2...a4β1b2...b7 ≡ (

γ a1b1

)α1

β1εa1...a4b1...b7,

εα1α2a3a4β1β2b3...b7 ≡ (

γ a1b1

)(α1|

β1(γ a2

b2

)|α2)

β2εa1...a4b1...b7,

εα1α2α3a4β1β2β3b4b5b6b7 ≡ (

γ a1b1

)(α1|

β1(γ a2

b2

)|α2|

β2(γ a3

b3

)|α3)

β3εa1...a4b1...b7,

εα1α2α3α4β1β2β3β4b5b6b7

(A.13)≡ (γ a1

b1

)(α1|

β1(γ a2

b2

)|α2|

β2(γ a3

b3

)|α3|

β3(γ a4

b4

)|α4)

β4εa1...a4b1...b7.

The role of this object is that it allows us to convert the components of a super 7-formXA1...A7 into the dual components of a super 4-formXA1...A4 via the definitions

Xa1...a4 ≡ 17! εa1...a4

b1...b7Xb1...b7,

Xα1a2a3a4 ≡ 16! εα1a2a3a4

β1b2...b7Xβ1b2...b7,

Xα1α2a3a4 ≡ 12·5! εα1α2a3a4

β1β2b3...b7Xβ1β2b3...b7,

Xα1α2α3a4 ≡ 13!·4! εα1α2α3a4

β1β2β3b4...b7Xβ1β2β3b4...b7,

(A.14)Xα1α2α3α4 ≡ 14!·3! εα1α2α3α4

β1β2β3β4b5b7b7Xβ1β2β3β4b5b6b7.

As well, it can be used for the reverse purpose of converting the components of a super4-form to those of a super 7-form. The super-epsilon tensor concept has proven very usefulfor 10D theories as we suspect will also be the case for 11D theories.

References

[1] S.J. Gates Jr., H. Nishino, Deliberations on 11D superspace for the M-theory effective action,Univ. of MD preprint UMDEPP 00-032, hep-th/0101037, revised.

[2] S.J. Gates Jr., H. Nishino, Phys. Lett. 173B (1986) 52.[3] S.J. Gates Jr., S.I. Vashakidze, Nucl. Phys. B 291 (1987) 173;

S.J. Gates Jr., H. Nishino, Nucl. Phys. B 291 (1987) 205.[4] S. Bellucci, S.J. Gates Jr., Phys. Lett. 208B (1988) 456.[5] S. Bellucci, S.J. Gates Jr., D. Depireux, Phys. Lett. B 238 (1990) 315.


[6] S. Bellucci, S.J. Gates Jr., B. Radak, P. Majumdar, S. Vashakidze, Mod. Phys. Lett. A 21 (1989)1985.

[7] S.J. Gates Jr., Phys. Lett. 96B (1980) 305.[8] H. Nishino, S.J. Gates Jr., Phys. Lett. B 388 (1996) 504.[9] P. Howe, Phys. Lett. B 415 (1997) 149, hep-th/9707184.

[10] S.J. Gates Jr., R. Grimm, Phys. Lett. 133B (1983) 192.[11] W. Siegel, A derivation of the supercurrent superfield, HUTP-77/A089, December, 1977;

W. Siegel, The superfield supergravity action, HUTP-77/A080, December, 1977;W. Siegel, Nucl. Phys. B 142 (1978) 301.

[12] P. Howe, R. Tucker, Phys. Lett. 80B (1978) 138.[13] W. Siegel, Phys. Lett. 80B (1979) 224;

S.J. Gates Jr., in: P. van Nieuwenhuizen, D.Z. Freedman (Eds.), Supergravity, North-Holland,Amsterdam, 1979, pp. 215–219;S.J. Gates Jr., Nucl. Phys. B 162 (1980) 79;S.J. Gates Jr., Nucl. Phys. B 176 (1980) 397;P. Howe, Phys. Lett. 100B (1980) 389;P. Howe, Nucl. Phys. 199 (1982) 309.

[14] S.J. Gates Jr., H. Nishino, Phys. Lett. B 266 (1991) 14.[15] M. Cederwall, U. Gran, M. Nielsen, B. Nilsson, Manifestly supersymmetric M-theory,

Göteborg ITP preprint, hep-th/007035.[16] S.J. Gates Jr., H. Nishino, Class. Quant. Grav. 3 (1986) 391.[17] S.J. Gates Jr., L. Rana, A primer on supersymmetric quantum mechanics (I), preprint UMDEPP

96-38, in preparation.[18] S.J. Gates Jr., Basic canon inD = 4,N = 1 superfield theory: five primer lectures, in: J. Bagger

(Ed.), Supersymmetry, Supergravity and Supercolliders, TASI 97, World Scientific, Singapore,1999, hep-th/9809064.

[19] S.J. Gates Jr., L. Lu, R. Oerter, Phys. Lett. 218B (1989) 33.[20] S.J. Gates Jr., Strings, superstrings and two-dimensional Lagrangian field theory, in: Z. Haba,

J. Sobczyk (Eds.), Functional Integration, Geometry and Strings, Proceedings of the XXVWinter School of Theoretical Physics in Karpacz, Poland, February, 1989, Birkhäuser, 1989,pp. 140–184.

[21] N. Berkovits, Review of open superstring field theory, hep-th/0105230.[22] L. Bonora, P. Pasti, M. Tonin, Phys. Lett. 188B (1987) 335;

S. Ferrara, P. Fré, M. Porrati, Ann. Phys. 175 (1987) 112.


Cubic interactions of bosonic higher spin gaugefields inAdS5

M.A. VasilievI.E. Tamm Department of Theoretical Physics, Lebedev Physical Institute, Leninsky prospect 53, 119991,

Moscow, Russia


Abstract

The dynamics of totally symmetric free massless higher spin fields inAdSd is reformulated interms of the compensator formalism for AdS gravity. TheAdS5 higher spin algebra is identifiedwith the star product algebra with thesu(2,2) vector (i.e.,o(4,2) spinor) generating elements.Cubic interactions of the totally symmetric bosonic higher spin gauge fields inAdS5, including theinteraction with gravity, are formulated at the action level. 2001 Published by Elsevier ScienceB.V.

PACS:11.10.Kk; 11.15.-q; 11.25.-w; 11.30.Ly

1. Introduction

Irreducible relativistic fields in the flatd-dimensional space–time classify accordingto the finite-dimensional representations of the Wigner little algebral. It is well-knownthat l = o(d − 2) for the massless casem = 0 and l = o(d − 1) for m = 0. From thefield-theoretical viewpoint the difference between the massless and massive cases is that,except for the scalar and spinor fields, all massless fields possess specific gauge symmetriesreducing a number of independent degrees of freedom.

Since the totally antisymmetric symbolεa1...an (a = 1, . . . , n) is o(n) invariant it isenough to consider the representations ofo(n) associated with the Young diagrams havingat most[1

2n] rows. For lower dimensions liked = 4 andd = 5 only the totally symmetricmassless higher spin representations of the little algebra appear, characterized by a singlenumbers. An integer spins massless field is described by a totally symmetric tensorϕn1...ns

subject to the double tracelessness condition [1]ϕrrssn5...ns = 0 which is non-trivial for

s 4. A quadratic action [1] for a free spins field ϕn1...ns is fixed unambiguously [2] up

E-mail address:[email protected] (M.A. Vasiliev).



M.A. Vasiliev / Nuclear Physics B 616 (2001) 106–162 107

to an overall factor in the form

Ss = 1

2(−1)s

∫d4x

∂nϕm1...ms ∂

nϕm1...ms

− 1

2s(s − 1)∂nϕr rm1...ms−2∂

nϕkkm1...ms−2

+ s(s − 1)∂nϕrrm1...ms−2∂kϕnkm1...ms−2

− s∂nϕnm1...ms−1∂rϕrm1...ms−1

(1.1)− 1

4s(s − 1)(s − 2)∂nϕ

rrnm1...ms−3∂kϕ

ttkm1...ms−3

by the requirement of gauge invariance under the Abelian gauge transformations

(1.2)δϕn1...ns = ∂n1εn2...ns

with the parametersεn1...ns−1 being rank(s − 1) totally symmetric traceless tensors,εr rn3...ns−1 = 0. This formulation is parallel [3] to the metric formulation of gravityand is called formalism of symmetric tensors. Fermionic higher spin gauge fields aredescribed analogously [4] in terms of rank(s − 1/2) totally symmetric spinor-tensorsψn1...ns−1/2α subject to theγ -tracelessness conditionγ sαβψr rsn4...ns−1/2β = 0. A progresson the covariant description of generic (i.e., mixed symmetry) massless fields in anydimension was achieved in [5,6].

Higher spin gauge symmetry principle is the fundamental concept of the theory of higherspin massless fields. By construction, the class of higher spin gauge theories consists ofmost symmetric theories having as many as possible symmetries unbroken.1 Any moresymmetric theory will have more lower and/or higher spin symmetries and therefore willbelong to the class of higher spin theories. As such, higher spin gauge theory is of particularimportance for the search of a fundamental symmetric phase of the superstring theory. Thisis most obvious in the context of the so-called Stueckelberg symmetries in the string fieldtheory which have a form of some spontaneously broken higher spin gauge symmetries.Whatever a symmetric phase of the superstring theory is, Stueckelberg symmetries areexpected to become unbroken higher spin symmetries in such a phase and, therefore, thesuperstring field theory has to identify with one or another version of the higher spin gaugetheory.

The problem is to introduce interactions of higher spin fields with some other fields ina way compatible with the higher spin gauge symmetries. Positive results in this directionwere first obtained for interactions of higher spin gauge fields in the flat space with thematter fields and with themselves but not with gravity [8]. In the framework of gravity, thenon-trivial higher spin gauge theories were so far elaborated [9–11] (see also [12,13] for

1 We only consider the case of relativistic fields that upon quantization are described by lowest weight unitaryrepresentations (lowest weight implies in particular, that the energy operator is bounded from below). Beyondthis class some other “partially massless” higher spin gauge fields can be introduced [7] which are either non-unitary or live in the de Sitter space (recall that de Sitter groupSO(d,1) does not allow lowest weight unitaryrepresentations).

108 M.A. Vasiliev / Nuclear Physics B 616 (2001) 106–162

reviews) ford = 4 which is the simplest non-trivial case since higher spin gauge fields donot propagate ifd < 4. As a result, it was found out that

(i) In the framework of gravity, unbroken higher spin gauge symmetries require a non-zero cosmological constant;

(ii) Consistent higher spin theories contain infinite sets of infinitely increasing spins;(iii) Consistent higher spin gauge interactions contain higher derivatives: the higher spin

is the more derivatives appear;(iv) The higher spin symmetry algebras [14] identify with certain star product algebras

with spinor generating elements [15].

Some of these properties, like the relevance of the AdS background and star productalgebras, discovered in the eighties were rather unusual at that time but got their analoguesin the latest superstring developments in the context of AdS/CFT correspondence [16] andthe non-commutative Yang–Mills limit [17]. We believe that this convergency can unlikelybe occasional. Let us note that recently an attempt to incorporate the dynamics of higherspin massless into the two-time version of the non-commutative phase space approach wasundertaken in [18].

The feature that unbroken higher spin gauge symmetries require a non-zero cosmologi-cal constant is of crucial importance in several respects. It explained why negative conclu-sions on the existence of the consistent higher-spin-gravitational interactions were obtainedin [19] where the problem was analyzed within an expansion near the flat background. Alsoit explains why the higher spin gauge theory phase is not directly seen in the M theory (orsuperstring theory) framework prior its full formulation in the AdS background is achieved.The same property makes theS-matrix Coleman–Mandula-type no-go arguments [20] ir-relevant because there is noS-matrix in the AdS space.

A challenging problem of the higher spin gauge theory is to extend the 4d resultson the higher-spin-gravitational interactions to higher dimensions. This is of particularimportance in the context of the possible applications of the higher spin gauge theory tothe superstring theory(d = 10) and M theory(d = 11). A conjecture on the structure ofthe higher spin symmetry algebras in any dimension was made in [21] where the idea wasput forward that analogously to what was proved to be true ind = 4 [15] andd < 4 (see[13] for references) higher spin algebras in any dimension are certain star product algebraswith spinor generating elements.

As a first step towards higher dimensions it is illuminating to analyze the next tod =4 non-trivial case, which isd = 5. This is the primary goal of this paper. The case ofAdS5/CFT4 higher spin duality is particularly interesting in the context of duality of thetype IIB superstring theory onAdS5 × S5 with a constant Ramond–Ramond field strengthto theN = 4 supersymmetric Yang–Mills theory [16]. It was conjectured recently in [22,23] that the boundary theories dual to theAdSd+1 higher spin gauge theories are freeconformal theories ind dimensions. This conjecture is in agreement with the results of[24] where the conserved conformal higher spin currents bilinear in thed-dimensionalfree massless scalar field theory were shown to be in the one-to-one correspondencewith the set of the(d + 1)-dimensional bulk higher spin gauge fields associated with the


totally symmetric representations of the little algebra. In contrast to the regimeg2n→∞underlying the standard AdS/CFT correspondence [16], as conjectured in [22,23], theAdS/CFT regime associated with the higher spin gauge theories corresponds to the limitg2n→ 0. The realization of the higher spin conformal symmetries in the 4d free boundaryconformal theories was considered recently in [25]. It was shown that they indeed possessthe global higher spin conformal symmetries proposed long ago by Fradkin and Linetsky[26] in the context of 4d conformal higher spin gauge theories [27], and some their furtherextensions.2 It was conjectured in [25] that the 4d conformal higher spin gauge symmetriescan be realized as higher spin gauge symmetries ofAdS5 bulk unitary higher spin gaugetheories. Analogous conjecture was made in [28] with respect to the minimal infinite-dimensional reduction of the 4d conformal higher spin algebra.

The AdS5 case is more complicated compared toAdS4. Naively, one might think thatonly one-row massless higher spin representations of the little group characterized by asingle numbers appear. However, there is a catch due to the fact that the classification ofmassless fields inAdSd is different [29] from that of the flat space. As a result, more typesof massless fields appear inAdS5 which all reduce in the flat limit to the some combinationsof the symmetric fields. In theAdS5 case however, they are expected to be described bythe dynamical fields having the symmetry properties of the two row Young diagrams.Unfortunately, so far the covariant formalism for the description of such fields in the AdSspace, that would extend that developed in [30,31] for the totally symmetric higher spinfields, was not worked out. This complicates a formulation of theAdS5 higher spin gaugetheories for the general case. In particular, based on the results presented in the Section 4 ofthis paper, it was argued in [25] that such mixed symmetry higher gauge spin fields have toappear in the 5d higher spin algebras withN 2 supersymmetries. For this reason, in thispaper we confine ourselves to the simplest purely bosonicN = 0 case of theAdS5 higherspin gauge theory. TheN = 1 case will be considered in the forthcoming paper [32]. Toproceed beyondN = 1 one has first of all to develop the appropriate formulation of theAdS5 massless higher spin fields that have the symmetry properties of the two-row Youngdiagrams.

The organization of the paper is as follows. To make it as much selfcontained as possiblewe start in Section 2 with a summary of the general features of the approach developedin [9,30,33] relevant to the analysis of the 5d higher spin gauge theory in this paper. Inparticular, the main idea of the higher spin extension at the algebraic and Lagrangian levelis discussed in Section 2.2. In Section 2.3 we summarize the main results of [30] on theformulation of the totally symmetric bosonic massless fields in any dimension, introducingcovariant notation based on the compensator approach toAdSd gravity explained inSection 2.1. In Section 3 we reformulateAdS5 gravity in thesu(2,2) spinor notations. Thecorrespondence between finite-dimensional representations ofsu(2,2) ando(4,2) relevantto theAdS5 higher spin problem is presented in Section 4.AdS5 higher spin gauge algebrasare defined in Section 5.su(2,2) systematics of the 5d higher spin massless is given in

2 The conformal higher spin gauge theories of [27] generalizeC2 Weyl gravity and are non-unitary because ofthe higher derivatives in the kinetic terms that give rise to ghosts.


Section 6. The unfolded form of the free equations of motion for all massless totallysymmetric tensor fields inAdS5 called Central On-Mass-Shell Theorem is presented inSection 7. The analysis of theAdS5 higher spin action is the content of Section 8 wherewe, first, discuss some general properties of the higher spin action, and then derive thequadratic (Section 8.1) and cubic (Section 8.2) higher spin actions possessing necessaryhigher spin symmetries. The reductions to the higher spin gauge theories that describefinite collections of massless fields of any given spin are defined in Section 9. Conclusionsand some open problems are discussed in Section 10. Appendix A contains a detailedderivation of the the 5d free higher spin equations of motion.

2. Generalities

2.1. AdSd gravity with compensator

It is well-known that gravity admits a formulation in terms of the gauge fields associatedwith one or another space–time symmetry algebra [34–36]. Gravity with the cosmologicalterm in any space–time dimension can be described in terms of the gauge fieldswAB =−wBA = dxnwABn associated with theAdSd algebrah = o(d − 1,2) with the basiselementstAB . Here the underlined indicesm,n, . . .= 0, . . . , d − 1 are (co)tangent for thespace–time base manifold whileA,B = 0, . . . , d are (fiber) vector indices of the gaugealgebrah= o(d − 1,2). Let rAB be the Yang–Millso(d − 1,2) field strength

(2.1)rAB = dwAB +wAC ∧wCB,whered = dxn ∂

∂xnis the exterior differential. One can use the decomposition

(2.2)w =wABtAB = ωLabLab + λeaPa(a, b = 0, . . . , d − 1). HereωLab is the Lorentz connection associated with the Lorentzsubalgebrao(d − 1,1). The frame fieldea is associated with theAdSd translationsPaparametrizingo(d − 1,2)/o(d − 1,1). Provided thatea is non-degenerate, the zero-curvature condition

(2.3)rAB(w)= 0

implies thatωLab andea identify with the gravitational fields ofAdSd . λ−1 is the radiusof the AdSd space–time. (Note thatλ has to be introduced to make the frame fieldea

dimensionless.)One can make these definitions covariant with the help of the compensator field [36]

V A(x) being a time-likeo(d − 1,2) vectorVA normalized to

(2.4)V AVA = 1

(within the mostly minus signature). The Lorentz algebra then identifies with the stabilitysubalgebra ofV A. This allows for the covariant definition of the frame field and Lorentz


connection [36,37]

(2.5)λEA =D(V A)≡ dV A +wABVB,(2.6)ωLAB =wAB − λ(EAV B −EBV A).

According to these definitions

(2.7)EAVA = 0,

(2.8)DLVA = dV A +ωLABVB ≡ 0.

VA is the null vector ofEA = dxnEAn . When the matrixEAn has the maximal rankd itcan be identified with the frame field giving rise to the non-degenerate space–time metrictensor

(2.9)gnm =EAn EBmηAB.The torsion 2-form is

(2.10)tA ≡DEA ≡ λ−1rABVB.

(Note that due to (2.7)DEA =DLEA.) The zero-torsion condition

(2.11)tA = 0

expresses the Lorentz connection via (derivatives of) the frame field in a usual manner.With the help ofVA it is straightforward to build ad-dimensional generalization of the

4d MacDowell–Mansouri–Stelle–West pure gravity action [35,36]

(2.12)S =− 1

4λ2κd−2

∫Md

εA1...Ad+1rA1A2 ∧ rA3A4 ∧EA5 ∧ · · · ∧EAdV Ad+1.

Taking into account that

(2.13)δrAB =DδwAB, δEA = λ−1(δwABVB +DδV A), VAδVA = 0,

along with the identity

(2.14)εA1...Ad+1 = V A1VBεBA2...Ad+1 + · · · + V Ad+1VBε

A1...AdB

one finds

δS =− 1

4λ2κd−2

∫Md

εA1...Ad+1rA1A2 ∧

(2(−1)dλδwA3A4 ∧EA5 ∧ · · · ∧EAd+1

(2.15)+ (d − 4)λ−1rA3A4 ∧ δwA5BVB ∧EA6 ∧ · · · ∧EAdV Ad+1)+ δ1S,

where

δ1S =− 1

4λ2κd−2

∫Md

εA1...Ad+1rA1A2 ∧ tA3 ∧ ((d − 4)

(2δwA4A5 ∧EA6

+ (d − 5)λ−1rA4A5 ∧ δV A6)∧EA7 ∧ · · · ∧EAdV Ad+1

(2.16)+ 4(d − 3)λV A4 ∧EA5 ∧ · · · ∧EAd δV Ad+1)

is the part of the variation that contains torsion.


We shall treat the actionS perturbatively withrAB being small. According to (2.3)this implies a perturbation expansion around theAdSd background. In this framework,the second term in (2.15) and the first term in (2.16) only contribute to the non-linearcorrections of the field equations for the gravitational fieldswAB .

For the part ofδwAB orthogonal toV C

(2.17)δwAB = δξAB, δξABVB = 0

we obtain

δS

δξA1B2= κ2−dεA1...Ad+1t

A3 ∧(EA4 ∧EA5 − d − 4

2λ2 rA4A5

)(2.18)∧EA6 ∧ · · · ∧EAdV Ad+1.

Perturbatively (i.e., forrAB small), (2.18) is equivalent to the zero-torsion condition (2.11).In what follows we will use the so-called 1.5-order formalism. Namely, we will assume thatthe zero-torsion constraint is imposed to express the Lorentz connection via derivatives ofthe frame field. The same time we will use an opportunity to choose any convenient formfor the variation of the Lorentz connection because any term containing this variation iszero by (2.18) and (2.11).

The generalized Einstein equations originating from the variation

(2.19)δωAB = λ(δξAV B − δξBV A), δξAVA = 0

are

κ2−dεA1...Ad+1rA2A3 ∧

(EA4 ∧EA5 − d − 4

4λ2 rA4A5

)(2.20)∧EA6 ∧ · · · ∧EAdV Ad+1 = 0.

The first term is nothing but the left-hand side of the Einstein equations with thecosmological term. The second term describes some additional interaction terms bilinearin the curvaturerAB . These terms do not contribute to the linearized field equations. Inthe 4d case the additional terms are absent because the corresponding part of the actionis topological having the Gauss–Bonnet form. Note that the additional interaction termscontain higher derivatives together with the factor ofλ−2 that diverges in the flat limitλ→ 0. Terms of this type play an important role in the higher spin theories to guaranteethe higher spin gauge symmetries. Let us note that the form of Eq. (2.20) indicates thatbeyondd = 4 the action (2.12) may have other symmetric vacua3 (e.g., with rAB =4λ2

d−4EA ∧ EB ). We shall not discuss this point here in more detail because its analysis

requires the non-perturbative knowledge of the higher spin theory which is still lucking ford > 4.

From (2.16) it follows that the variation ofS with respect to the compensatorVA isproportional to the torsion 2-formtA. This means that, at least perturbatively, there exists

3 I am grateful to K. Alkalaev for the useful discussion of this point.


such a variation of the fields

(2.21)δV A = εA(x), δωAB = ηAB(r, ε)with εAVA = 0 and someηAB(r, ε) bilinear in rAB andεA that S remains invariant. Asa result, there is an additional gauge symmetry that allows to gauge fixVA to any valuesatisfying (2.4). It is therefore shown thatVA does not carry extra degrees of freedom.

The compensator fieldV A makes theo(d − 1,2) gauge symmetry manifest

(2.22)δwAB =DεAB, δV A = εABVB.Fixing a particular value ofVA one is left with the mixture of the gauge transformationsthat leaveV A invariant, i.e., with the parameters satisfying

(2.23)0= δV A = εA(x)+ εABVB.Since the additional transformation (2.21) contains dependence on the curvaturerAB , thisproperty is inherited by the leftover symmetry with the parameters satisfying (2.23).

The fact that there is an additional symmetry (2.21) is not a big surprise in the frameworkof the theory of gravity formulated in terms of differential forms, having explicit invarianceunder diffeomorphisms. That this should happen is most clear from the observation that theinfinitesimal space–time diffeomorphisms induced by an arbitrary vector fieldεn(x) admita representation

(2.24)δwABm = εn∂nwABm − ∂m(εn)wABn = εnrnmAB +DmεAB,

(2.25)δV A = εn∂nV A = εnEAn + εABVB ,where

(2.26)εAB =−εnwnAB.The additional gauge transformation (2.21) withεA = εnEAn can therefore be understoodas a mixture of the diffeomorphisms ando(d − 1,2) gauge transformations.

Another useful interpretation of the formula (2.24) is that, for the vacuum solution sat-isfying (2.3), diffeomorphisms coincide with some gauge transformations. This observa-tion explains why the space–time symmetry algebras associated with the motions of themost symmetric vacuum spaces reappear as gauge symmetry algebras in the “geometricapproach” to gravity and its extensions.

2.2. General idea of the higher spin extension

The approach to the theory of interacting higher spin gauge fields developed originallyin [9,33] for thed = 4 case is a generalization of the “geometric” approach to gravitysketched in Section 2.1. The idea is to describe the higher spin gauge fields in terms ofthe Yang–Mills gauge fields and field strengths associated with an appropriate higher spinsymmetry algebrag being some infinite-dimensional extension of the finite-dimensionalAdSd space–time symmetry algebrah= o(d − 1,2).


Let the 1-formω(x) = dxn ωn(x) be the gauge field ofg with the field strength(curvature 2-form)

(2.27)R = dω+ω ∧ ∗ω,where∗ is some associative product law leading to the realization ofg via commutators.(This is analogous to the matrix realization of the classical Lie algebras with the starproduct instead of the matrix multiplication. A particular realization of the star productrelevant to the 5d higher spin dynamics is given in Section 3.) An infinitesimal higher spingauge transformation is

(2.28)δgω=Dε,whereε(x) is an arbitrary infinitesimal symmetry parameter taking values ing,

(2.29)Df = df + [ω,f ]∗and

(2.30)[a , b]∗ = a ∗ b− b ∗ a.The higher spin curvature has the standard homogeneous transformation law

(2.31)δgR = [R ,ε]∗.The higher spin equations of motion will be formulated in terms of the higher spin

curvatures and therefore admit a zero-curvature vacuum solution withR = 0. Since thespace–time symmetry algebrah is assumed to belong tog, a possible ansatz is with allvacuum gauge fields vanishing except forω0 taking values inh

(2.32)ω0 =wAB0 tAB = ωLab0 Lab + λhaP a.Provided thatha is non-degenerate, the zero-curvature condition

(2.33)R(ω0)=(dωAB0 +ωA0 C ∧ωCB0

)tAB = 0

implies thatωLab0 and ha identify with the gravitational fields ofAdSd . Let us notethat throughout this paper we use notationωLab0 andha for the background AdS fieldssatisfying (2.32) butωLab andea for the dynamical gravitational fields.

Suppose there is a theory invariant under the gauge transformations (2.28). Globalsymmetry is the part of the gauge transformations that leaves invariant the vacuum solutionω0. The global symmetry parameters therefore satisfy

(2.34)0=D0εgl,

where

(2.35)D0f = df + [ω0, f ]∗.The vacuum zero-curvature equation (2.33) guarantees that (2.34) is formally consistent.Fixing a value ofεgl(x0) at some pointx0, (2.34) allows one to reconstructεgl(x) in someneighbourhood ofx0. SinceD0 is a derivation, the star commutator of any two solutions of


(2.34) gives again some its solution. The global symmetry algebra therefore coincides withthe algebra of star commutators at any fixed space–time pointx0, which isg. An importantcomment is that this conclusion remains true also in case the theory is invariant under adeformed gauge transformation of the form

(2.36)δω =Dε +∆(R, ε),where∆(R, ε) denotes someR-dependent terms, i.e.,∆(0, ε)= 0. Indeed, all additionalterms do not contribute to the invariance condition (2.34) once the vacuum solution satisfies(2.33). In fact, as is clear from the discussion in Section 2.1, the deformation of the gaugetransformations (2.36) takes place in all theories containing gravity and, in particular, inthe higher spin gauge theories.

Let us use the perturbation expansion with

(2.37)ω= ω0 +ω1,

whereω1 is the dynamical (fluctuational) part of the gauge fields of the higher spinalgebrag. SinceR(ω0)= 0, we have

(2.38)R =R1 +R2,

where

(2.39)R1 = dω1 +ω0 ∗ ∧ω1 +ω1 ∗ ∧ω0, R2 = ω1 ∗ ∧ω1.

The Abelian lowest order part of the transformation (2.28) (equivalently, (2.36)) has theform

(2.40)δ0ω1 =D0ε.

From (2.31) and (2.33) it follows that

(2.41)δ0R1 = 0.

The idea is to construct the higher spin action from the higher spin curvaturesR in theform analogous to the gravity action (2.12)

(2.42)S =∫UΩΛ ∧RΩ ∧RΛ

with some(d − 4)-form coefficientsUAB built from the frame field and compensator.(Ω,Λ label the adjoint representation ofg.) To clarify whether this is possible or not, onehas to check first of all if it is true for the free field action, i.e., whether some action of theform

(2.43)Ss2 =∫Us

0ΩΛ ∧RΩ1 ∧RΛ1describes the free field dynamics of a field of a given spins. As long asg is not known,a form ofR1 has itself to be fixed from this requirement. In fact, the form ofR1 providesan important information on the structure ofg fixing a pattern of the decomposition ofgunder the adjoint action ofh ⊂ g (up to a multiplicity of the representations associated


with a given spins: it is not a priori known how many fields of a given spin are present ina full higher spin multiplet). For the totally symmetric higher spin gauge fields describedby the action (1.1) this problem was solved first for cased = 4 [33] and then for anydimension both for bosons [30] and for fermions [21]. The results of [30] are summarizedin Section 2.3.

As a result of (2.41) any action of the form (2.43) is invariant under the Abelianfree field higher spin gauge transformations (2.40). However, for generic coefficients, itnot necessarily describes a consistent higher spin dynamics. As this point is of the keyimportance for the analysis of the higher spin dynamics let us explain it in somewhat moredetail. The set of 1-forms contained inω decomposes into subsetsωsn associated with agiven spins. The labeln enumerates different subsets associated with the same spin. (Forthe cased = 4 s is indeed a single number while for generic fields in higher dimensionss becomes a vector associated with the appropriate weight vector of theAdSd algebrao(d − 1,2).) Any subsetωsn forms a representation of the space–time subalgebrah ⊂ g.It further decomposes into representations of the Lorentz subalgebra ofh, denotedωs,tn .For the case of totally symmetric representations discussed in the Introduction, there is asingle integer parametert = 0,1, . . . , s − 1 that distinguishes between different Lorentzcomponents (for definiteness we focus here on the bosonic case of integer spins studiedin this paper). True higher spin field identifies withωs,0n . It is called dynamical higherspin field. The rest of the fieldsωs,tn with t > 0 express in terms of (derivatives of) thedynamical ones by virtue of certain constraints. At the linearized level, the gauge invariantconstraints can be chosen in the form of some linear combinations of the linearized higherspin curvatures

(2.44)Φl(R1)= 0

with the coefficients built from the background frame field. By virtue of these constraintsall fieldsωs,tn turn out to be expressed viat th space–time derivatives of the dynamical field

(2.45)ωs,tn ∼(∂

λ∂x

)t (ωs,0n

)+ pure gauge terms (2.40).

These expressions contain explicitly the dependence on theAdSd radiusλ−1 as a result ofthe definition of the frame field (2.5).

A particular example is provided with the spin-2. Hereω2,0 identifies with the framefield while ω2,1 is the Lorentz connection. (We skip the labeln focusing on a particularspin-2 field.) The constraint (2.44) is the linearized zero-torsion condition.

For s > 2 the fieldsωs,tn with t 2 appear, containing second and higher derivatives ofthe dynamical field. These are called extra fields. From this perspective, the requirementthat the free action contains at most two space–time derivatives of the dynamical field isequivalent to the condition that the variation of the free action with respect to all extrafields is identically zero

(2.46)δSs2

δωs,tn

≡ 0, t 2.


It turns out that thisextra field decoupling conditionfixes a form of the free action (i.e., ofUs

0ΩΛ) uniquely modulo total derivatives and an overall (s- andn-dependent) factor. The

Lorentz-type fieldsωs,1n are auxiliary, i.e., they do contribute into the free action but expressvia the derivatives of the dynamical field by virtue of their field equations equivalent tosome of the constraints (2.44).

Once the extra fields are expressed in terms of the derivatives of the dynamical fields, thehigher spin transformation law (2.28) (and its possible deformation (2.36)) describes thetransformations of the dynamical fields via their higher derivatives. Sincet ranges from 0to s− 1 one finds that the higher spin is the higher derivatives appear in the transformationlaw. Note that this conclusion is in agreement with the general analysis of the structureof the higher spin interactions [8] and conserved higher spin currents [13,38] containinghigher space–time derivatives.

As a first step towards the non-linear higher spin dynamics one can try the action (2.42)with UΩΛ proportional to the coefficientsUs

0ΩΛ in the subsector of each field of spins.This action is not invariant under the original higher spin gauge transformations (2.28)sinceUΩΛ cannot be an invariant tensor ofg. Indeed, since the action is built in terms ofdifferential forms without Hodge star operation, its generic variation is

(2.47)S =−2∫D(UΩΛ)∧ δωΩ ∧RΛ.

If UΩΛ would be ag-invariant tensor,S would be a topological invariant. This cannotbe true since the linearized action (2.43) is supposed to give rise to non-trivial equationsof motion. Therefore,D(UΩΛ) = 0. The trick is that for some particular choice ofUΩΛthere exists such a deformation of the gauge transformations (2.36) that the action remainsinvariant at least in the lowest non-trivial order, i.e., theω2

1ε type terms can be proved tovanish in the variation. (Note that this is just the order at which the difficulties with thehigher-spin-gravitational interactions were originally found [19].) In particular, we showin Section 8 that this is true for theN = 0 5d higher spin theory. This deformation of thegauge transformations is analogous to that resulting via (2.23) from the particular gaugefixing of the compensator fieldV A in the case of gravity which, in turn, is described bythe spin-2 part of the action (2.42) equivalent to (2.12). A complication of the Lagrangianformulation of the higher spin dynamics is that no full-scale extension of the compensatorV A to some representation ofg is yet known. The clarification of this issue is one of thekey problems on the way towards the full Lagrangian formulation of the higher spin theory.Note, that the full formulation of the on-mass-shell 4d higher spin dynamics [11,39] wasachieved by virtue of introducing additional compensator-type pure gauge variables [11,13].

Since the extra fields do contribute into the non-linear action it is necessary to expressthem in terms of the dynamical higher spin fields to make the non-linear action (2.42)meaningful. The expressions (2.45) that follow from the constraints for extra fieldseffectively induce higher derivatives into the higher spin interactions. The same mechanisminduces the negative powers ofλ, the square root of the cosmological constant, into the


higher spin interactions with higher derivatives.4 A specific form of the constraints (2.44)plays a crucial role in the proof of the invariance of the action.

The program sketched in this section was accomplished for the 4d case. The free higherspin actions of the form (2.43) were built in [33]. The 4d higher spin algebrag was thenfound in [14]. In [9] the action (2.42) was found that described properly some cubichigher spin interactions including the gravitational interaction. In this paper we extendthese results to the bosonicN = 0 5d higher spin gauge theory.

2.3. Symmetric bosonic massless fields in AdSd

In this section the results of [30] are reformulated in terms of the compensatorformalism. According to [30], a totally symmetric massless field of spins is describedby a collection of 1-formsdxnωna1...as−1,b1...bt which are symmetric in the Lorentz vectorindicesai andbj separately (a, b= 0, . . . , d − 1), satisfy the antisymmetry relation

(2.48)ωna1...as−1,asb2...bt = 0,

implying that symmetrization over anys fiber indices gives zero, and are traceless withrespect to the fiber indices

(2.49)ωna1...as−3c

c,b1...bt = 0.

(From this condition it follows by virtue of (2.48) that all other traces of the fiber indicesare also zero.)

The higher spin gauge fields associated with the spins massless field therefore takevalues in the direct sum of all irreducible representations of thed-dimensional masslessLorentz groupo(d − 1,1) described by the Young diagrams with at most two rows suchthat the longest row has lengths − 1

(2.50)

ωna1...as−1 is treated as the dynamical spins field analogous to the gravitational frame

(spin-2). The fields corresponding to the representations with non-zero second row (t > 0)are auxiliary (t = 1) or “extra” (t > 1), i.e., express via derivatives of the dynamical fieldby virtue of certain constraints analogously to the Lorentz connection in the spin-2 case.Analogously to the relationship between metric and frame formulations of the linearizedgravity, the totally symmetric double traceless higher spin fields used to describe the higherspin dynamics in the metric-type formalism [1,3] identify with the symmetrized part of the

4 Note that one can rescale the fields in such a way that the corresponding expression (2.45) will not containnegative powers ofλ explicitly. However, as a result of such a rescaling,λ will appear both in positive and innegative powers in the structure coefficients of the algebrag and, therefore, in the non-linear action. From thisperspective, the appearance of the extra fields for higher spins makes difference compared to the case of puregravity that allows the Inönu–Wigner flat contraction.


field ωna1...as−1

(2.51)ϕa1...as = ωa1...as.

The antisymmetric part inωna1...as−1 can be gauge fixed to zero with the aid of thegeneralized higher spin Lorentz symmetries with the parameterεa1...as−1,b. Thatϕa1...as

is double traceless is a trivial consequence of (2.49).The collection of the higher spin 1-formsdxnωna1...as−1,b1...bt with all 0 t

s − 1 can be interpreted as a result of the “dimensional reduction” of a 1-formdxnωn

A1...As−1,B1...Bs−1 carrying the irreducible representation of theAdSd algebrao(d − 1,2) described by the traceless two-row rectangular Young diagram of lengths − 1

(2.52)ωA1...As−1,As B2...Bs−1 = 0, ωA1...As−3CC,B1...Bs−1 = 0.

The linearized higher spin curvatureR1 has the following simple form

RA1...As−1,B1...Bs−11

(2.53)

=D0(ωA1...As−1,B1...Bs−1

)= dωA1...As−1,B1...Bs−11

+ (s − 1)(ωA10 C ∧ωCA2...As−1,B1...Bs−1

1

+ωB10 C ∧ωA1...As−1,CB2...Bs−1

1

),

whereωAB0 is the backgroundAdSd gauge field satisfying the zero-curvature condition(2.33).

In these terms, the Lorentz covariant irreducible fieldsdxnωna1...as−1,b1...bt identify with

those components ofdxnωnA1...As−1,B1...Bs−1 that are parallel toV A in s− t−1 indices andtransversal in the rest ones. The expressions for the Lorentz components of the linearizedcurvatures have the following structure

Ra1...as−1,b1...bt1 =DLω

a1...as−1,b1...bt1 + τ−(ω)a1...as−1,b1...bt

(2.54)+ τ+(ω)a1...as−1,b1...bt

with

(2.55)τ−(ω)a1...as−1,b1...bt = αhc ∧ωa1...as−1,b1...bt c

1 ,

(2.56)τ+(ω)a1...as−1,b1...bt = βΠ(hb1 ∧ωa1...as−1,b2...bt1

),

whereDL is the background Lorentz covariant differential,Π is the projection operatorto the irreducible representation described by the traceless Young diagram of the Lorentzalgebrao(d − 1,1) with s − 1 andt cells in the first and second rows, respectively, andα

andβ are some coefficients depending ons, t andd and fixed in such a way that

(2.57)(τ−)2 = 0, (τ+)2 = 0,(DL

)2 + τ−, τ+ = 0.

For the explicit expressions ofα, β andΠ we refer the reader to the original paper [30].The explicit spinor version of the formula (2.54) ford = 5 will be given in Section 6.


The quadratic action functional for the massless spins field equivalent to theAdSddeformation of the action (1.1) has the following simple form [30]5

Ss2 =1

2χ(s)

∫Md

s−2∑p=0

(p+ 1)(p+ 1)!(d + p− 3)! εc1...cd h

c5 ∧ · · · ∧ hcd

(2.58)∧Rc1a1...as−2,c3b1...bp1 ∧Rc21 a1...as−2,

c4b1...bp .

It is fixed up to an overall normalization factorχ(s) by the conditions that it isP -even andits variation with respect to the “extra fields” is identically zero,

(2.59)δSs2

δωna1...as−1,b1...bt≡ 0 for t 2.

Let us explain how one can derive ao(d − 1,2) covariant form of the same action withthe aid of the compensatorV A. Taking into account the irreducibility properties (2.52) onefinds that the general form of theP -even action written in terms of differential forms is

Ss2 =1

2

∫Md

s−2∑p=0

a(s,p)εA1...Ad+1hA5 ∧ · · · ∧ hAdV Ad+1VC1 · · ·VC2(s−2−p)

(2.60)

×RA1B1...Bs−21 ,

A2C1...Cs−2−pD1...Dp ∧RA31 B1...Bs−2,

A4Cs−1−p...C2(s−2−p)D1...Dp .

Consider a general variation ofSs2 with respect toωA1...As−1,B1...Bs−1. Using that

(2.61)δRA1...As−11 ,

B1...Bs−1 =D0δωA1...As−11 ,

B1...Bs−1,

whereD0 is the background derivative, one integrates by parts taking into account thatD0(V

A) = hA, D0(hA) = 0. With the help of the irreducibility conditions (2.52), the

identity (2.14) and the identity

εA1...A5B6...Bd+1hC ∧ hB6 ∧ · · · ∧ hBd+1

(2.62)

= (d − 3)−1(εA1A2A3A4B5...Bd+1δCA5

− εA1A2A3A5B5...Bd+1δCA4

+ εA1A2A4A5B5...Bd+1δCA3

− εA1A3A4A5B5...Bd+1δCA2

+ εA2A3A4A5B5...Bd+1δCA1

)hB5 ∧ · · · ∧ hBd+1,

which expresses the simple fact that the total antisymmetrization of any set ofd +2 vectorindicesAi is zero, one finds

5 In this paper we use the normalization of fields in terms of the AdS parameterλ different from that of [30].Namely, we assume thatλ enters only via the definition of the frame fieldEA while in [30] the fields werenormalized in such a way that their expressions in terms of the derivatives of the dynamical higher spin field werefree from negative powers ofλ. This difference results in the different form of the dependence of the higher spinaction onλ.


δSs2 =− λ

d − 3

∫Md

s−2∑p=0

((s − p)(d − 7+ 2(s − p))

s −p− 1a(s,p)

− (s −p− 1)a(s,p− 1)

)× VC1 · · ·VC2(s−p)−3εA1...Ad+1V

A4 ∧ hA5 ∧ · · · ∧ hAd+1

∧ (δωA1B1...Bs−21 ,

A2C1...Cs−2−pD1...Dp ∧RA31 B1...Bs−2,

Cs−1−p...C2(s−p)−3D1...Dp

(2.63)

+RA1B1...Bs−21 ,

A2C1...Cs−2−pD1...Dp ∧ δωA31 B1...Bs−2,

Cs−1−p...C2(s−p−3)D1...Dp

).

The idea is to require all the terms in (2.63) to vanish except for the term atp = 0. Thiscondition fixes the coefficientsa(s,p) up to a normalization factora(s) in the form

(2.64)a(s,p)=−a(s)λ−1(d − 3)(d − 5+ 2(s − p− 2))!! (s − p− 1)(s − 2)!

s(d − 3+ 2(s − 2))!! (s − p− 2)! .

As a result the variation (2.63) acquires the form

δSs2 = a(s)∫Md

VC1 · · ·VC2s−3εA1...Ad+1VA4hA5 ∧ · · · ∧ hAd+1

∧ (δωA1B1...Bs−21 ,

A2C1...Cs−2 ∧RA31 B1...Bs−2,

Cs−1...C2s−3

(2.65)+RA1B1...Bs−21 ,

A2C1...Cs−2 ∧ δωA31 B1...Bs−2,

Cs−1...C2s−3).

This formula implies that the free action (2.60), (2.64) essentially depends only on theV A-transversal parts of

(2.66)ωnA1...As−1 = ωnA1...As−1 ,B1...Bs−1VB1 · · ·V Bs−1

and

(2.67)ωnA1...As−1 ,B1 = ωnA1...As−1 ,B1...Bs−1VB2 · · ·V Bs−1.

These fields identify respectively with the frame-like dynamical higher spin fieldωna1...as−1

and the Lorentz connection-like auxiliary fieldωna1...as−1,b expressed in terms of the firstderivatives of the frame-like field by virtue of its equation of motion equivalent to the“zero-torsion condition”

(2.68)0= T1A1...As−1 ≡R1A1...As−1 ,B1...Bs−1VB1 · · ·V Bs−1.

Insertion of the expression forωna1...as−1,b into (2.60) gives rise to the higher spin actionexpressed entirely (modulo total derivatives) in terms ofωn

a1...as−1 and its first derivatives.Since the linearized curvatures (2.54) are by construction invariant under the Abelianhigher spin gauge transformations

(2.69)δω1A1...As−1,B1...Bs−1 =D0ε

A1...As−1,B1...Bs−1

with the higher spin gauge parametersεA1...As−1,B1...Bs−1, the resulting action possessesrequired higher spin gauge symmetries and therefore describes correctly the free field


higher spin dynamics inAdSd . In particular, the generalized Lorentz-like transformationswith the gauge parameter

(2.70)εA1...As−1,B1(x)

guarantee that only the totally symmetric part of the gauge field (2.66) equivalent toϕm1...ms contributes to the action. Analogously, the auxiliary Lorentz-type higher spin fieldhas pure gauge components associated with the generalized Lorentz-type transformationsparameter described by the two-row Young diagram with two cells in the second row.These components do not express in terms of the dynamical higher spin field. However,the invariance with respect to the gauge transformations (2.69) guarantees that these puregauge components do not contribute into the action.

Although the extra fieldsωna1...as−1,b1...bt with t 2 do not contribute to the free action,as we have learned from the four-dimensional case [9] they do contribute at the interactionlevel. To make such interactions meaningful, one has to express the extra fields in terms ofthe dynamical ones modulo pure gauge degrees of freedom. This is achieved by imposingconstraints [30]

(2.71)εa1b1e1...ed−4cf h

e1 ∧ · · · ∧ hed−4 ∧ τ+(R1)ca2...as−1,f b2...bt = 0

(total symmetrizations within the groups of indicesai andbj is assumed). The covariantversion of these constraints is

(2.72)εA1B1E1...Ed−4CFGV

GhE1 ∧ · · · ∧ hEd−4 ∧ τ+(R1)CA2...As−1,FB2...Bs−1 = 0.

The covariant expressions for the operatorsτ± are complicated and will not be given herefor generald . Ford = 5 they are given in Section 6 in the spinor formalism.

An important fact is [30] that, by virtue of these constraints, most of the higher spinfield strengths vanish on-mass-shell according to the following relationship referred to asthe First On-Mass-Shell Theorem

Ra1...as−1,b1...bt1 =Xa1...as−1,b1...bt

(δS2

δωdyn

)for t < s − 1,

(2.73)Ra1...as−1,b1...bs−11 = has ∧ hbsCa1...as ,b1...bs +Xa1...as−1,b1...bs−1

(δS2

δωdyn

).

HereXa1...as−1,b1...bt (δSs2/δωdyn) are some linear functionals of the left-hand sides of thefree field equationsδSs2/δωdyn = 0 for the spins dynamical one-formsωa1...as−1

dyn . The

0-formsCa1...as ,b1...bs are described by the traceless two-row rectangular Young diagramsof length s and parametrize those components of the higher spin field strengths thatcan remain non-vanishing when the field equations and constraints are satisfied. Thesegeneralize the Weyl tensor in gravity (s = 2) that parametrizes the components of theRiemann tensor allowed to be non-vanishing when the zero-torsion constraint and Einsteinequations (requiring the Ricci tensor to vanish) are imposed. The covariant version of(2.73) is

(2.74)RA1...As−1,B1...Bs−11 = hAs ∧ hBsCA1...As ,B1...Bs +XA1...As−1,B1...Bs−1

(δSs2

δωdyn

)


with CA1...As,B1...Bs described by the tracelessV A-transversal two-row rectangular Youngdiagram of lengths, i.e.,

(2.75)CA1...As ,As+1B2...Bs = 0,

(2.76)CA1...As−2CD,B1...Bs ηCD = 0, CA1...As−1C,B1...BsVC = 0.

For completeness, let us present the unfolded equations of motion for all free integer spinmassless higher spin fields inAdSd corresponding to the totally symmetric representationsof the Wigner little group (more precisely, totally symmetric lowest weight vacua of theirreducible representations of theAdSd algebrao(d − 1,2)). The content of the CentralOn-Mas-Shell Theorem is that the equations of motion for massless free fields of all spinscan be written in the form

(2.77)RA1...As−1,B1...Bs−11 = hAs ∧ hBsCA1...As ,B1...Bs ,

(2.78)D0CA1...Au,B1...Bs = 0, u s,

where

(2.79)D0 =DL0 + σ− + σ+.

DL0 is the vacuum Lorentz covariant derivation and the operatorsσ± have the form

σ−(C)A1...Au,B1...Bs = (u− s + 2)ECCA1...AuC,B1...Bs

(2.80)+ sECCA1...AuBs ,B1...Bs−1C,

σ+(C)A1...Au,B1...Bs = uλ2(d + u+ s − 4

d + 2u− 2EA1CA2...Au,B1...Bs

− s

d + 2u− 2ηA1B1ECC

A2...Au,CB2...Bs

− (u− 1)(d + u+ s − 4)

(d + 2u− 2)(d + 2u− 4)ηA1A2ECC

A3...AuC,B1...Bs

(2.81)

+ s(u− 1)

(d + 2u− 2)(d + 2u− 4)ηA1A2ECC

A3...AuB1,CB2...Bs

)

(total symmetrization within the groups of indicesAi andBj is assumed). The set of0-formsCA1...Au,B1...Bs consists of all two-row tracelessVA-transversal Young diagramswith the second row of lengths, i.e.,

(2.82)CA1...Au,Au+1B2...Bs = 0,

(2.83)CA1...Au−2CD,B1...Bs ηCD = 0, CA1...Au−1C,B1...Bs VC = 0.

Eqs. (2.77) (being a consequence of the First On-Mass-Shell Theorem) and (2.78) areequivalent to the free equations of motion of (totally symmetric) massless fields of allspins inAdSd along with some constraints that express an infinite set of auxiliary variablesvia higher derivatives of the dynamical fields of all spins. The proof of the Central On-Mass-Shell Theorem is analogous to that given in thesu(2,2) notation in Section 7 for the5d case. The Central On-Mass-Shell Theorem plays the key role in many respects and, in


particular, for the analysis of interactions as was originally demonstrated in [39] where itwas proved for the 4d case.

Note that, as shown in [40], the equations of motion of massless scalar coincide with thesector of Eqs. (2.78) withs = 0. Analogously, Eqs. (2.78) withs = 1 impose the Maxwellequations on the spin-1 potential (1-form)ω.

3. Compensator formalism insu(2,2) notation

It is well-known that theAdS5 (equivalently, 4d-conformal) algebrao(4,2) is isomorphicto su(2,2) and, as such, admits realization in terms of oscillators [41]

(3.1)[aα, b

β]∗ = δαβ, [aα, aβ ]∗ = 0,

[bα, bβ

]∗ = 0,

α,β = 1, . . . ,4. Here we use the star product realization of the algebra of oscillators thatdescribes the totally symmetric (i.e., Weyl) ordering

(f ∗ g)(a, b)= 1

(π)8

∫d4ud4v d4s d4t f (a + u,b+ t)

× g(a + s, b+ v)exp[2(sαt

α − uαvα)]

(3.2)

= exp

[1

2

(∂2

∂sα∂tα− ∂2

∂uα∂vα

)]f (a + s, b+ u)g(a + v, b+ t)

∣∣∣∣s=u=t=v=0

.

It is straightforward to see that this star product is associative and gives rise to thecommutation relations (3.1) via (2.30). The associative star product algebra with eightgenerating elementsaα andbβ is called Weyl algebraA4. Let us note that the star productalgebras relevant to the higher spin gauge theory (in, particular, the one used throughoutthis paper) are treated as the algebras of polynomials or formal power series thus beingdifferent from the star product algebras of functions regular at infinity that are relevantto the non-commutative Yang–Mills theory [17]. One important difference concerns thedefinitions of the invariant trace operations because, as shown in [15], the star productalgebras of formal power series possess a uniquely defined supertrace operation but admitsno usual trace at all (like the one used in the non-commutative Yang–Mills theory). Itis worth to mention that the superstructure underlying the supertrace of the polynomialstar product algebras is just appropriate in the context of the spinor interpretation of thegenerating elements likeaα andbβ in the 5d higher spin theory studied in this paper.

The Lie algebragl4 is spanned by the bilinears

(3.3)Tαβ = aαbβ ≡ 1

2

(aα ∗ bβ + bβ ∗ aα

).

The central component is associated with the generator

(3.4)N = aαbα ≡ 1

2

(aα ∗ bα + bα ∗ aα

)


while the traceless part

(3.5)tαβ = aαbβ − 1

4δαβN

spanssl4. Thesu(2,2) real form ofsl4(C) results from the reality conditions

(3.6)aα = bβCβα, bα = Cαβaβ,where bar denotes the complex conjugation whileCαβ = −Cβα andCαβ = −Cβα aresome real antisymmetric matrices satisfying

(3.7)CαγCβγ = δβα .

The oscillatorsbα and aα are in the fundamental and the conjugated fundamentalrepresentations ofsu(2,2) equivalent to the two spinor representations ofo(4,2). A o(6)complex vectorVA (A = 0, . . . ,5) is equivalent to the antisymmetric bispinorV αβ =−V βα having six independent components (equivalently, one can useVαβ = 1

2εαβγ δVγ δ

whereεαβγ δ is thesl4 invariant totally antisymmetric tensor (ε1234= 1)). A o(4,2) realvectorVA is described by the antisymmetric bispinorV αβ satisfying the reality condition

(3.8)V γ δCγαCδβ = 1

2εαβγ δV

γ δ.

One can see that the invariant norm of the vector

(3.9)V 2 = VαβV αβhas the signature(++−−−−). The vectors withV 2 > 0 are time-like while those withV 2 < 0 are space-like. To perform a reduction of the representations of theAdS5 algebrasu(2,2) ∼ o(4,2) into representations of its Lorentz subalgebrao(4,1) we introduce asu(2,2) antisymmetric compensatorV αβ with positive square (3.9), which is the spinoranalog of the compensatorV A of Section 2. The Lorentz algebra is identified with itsstability subalgebra. (Let us note thatV αβ must be different from the formCαβ used inthe definition of the reality conditions (3.6) since the latter is space-like and therefore hassp(4;R)∼ o(3,2) as its stability algebra.)

We shall treatV αβ as a symplectic form that allows one to raise and lower spinor indicesin the Lorentz covariant way

(3.10)Aα = V αβAβ, Aα =AβVβα,Using that the total antisymmetrization over any four indices is proportional to theε

symbol, we normalizeV αβ so that

(3.11)VαβVαγ = δαγ , Vαβ = 1

2εαβγ δV

γ δ,

(3.12)εαβγ δ = VαβVγ δ + Vβγ Vαδ + VγαVβδ,(3.13)εαβγ δ = V αβV γ δ + V βγ V αδ + V γαV βδ.

In these terms the Lorentz subalgebra is spanned by the generators symmetric in thespinor indices

(3.14)Lαβ = 1

2(tαβ + tβα)


while theAdS5 translations are associated with the antisymmetric traceless generators

(3.15)Pαβ = 1

2(tαβ − tβα).

The gravitational fields are identified with the gauge fields taking values in theAdS5

algebrasu(2,2)

(3.16)w =wαβaαbβ.The invariant definitions of the frame field and Lorentz connection for ax-dependentcompensatorV αβ(x) are

(3.17)Eαβ =DV αβ ≡ dV αβ +wαγ V γβ +wβγV αγ ,(3.18)ωLαβ =wαβ + 1

2EαγVγβ.

The normalization condition (3.11) implies

(3.19)Eαβ =−DVαβ, Eαα = 0.

The non-degeneracy condition implies thatEαβ spans a basis of the 5d 1-forms. Thebasisp-formsEp can be realized as

(3.20)Eαβ

2 =Eβα2 =Eαγ ∧Eβγ ,(3.21)E

αβ

3 =Eβα3 =Eα2 γ ∧Eβγ ,(3.22)E

αβ4 =−Eβα4 =Eα3 γ ∧Eβγ ,

(3.23)E5 =Eα4 γ ∧Eαγ .The following useful relationships hold as a consequence of the facts that 5d spinors havefour components and the frame field is traceless (3.19)

(3.24)Eαβ ∧Eγδ = 1

2

(V αγE

βδ2 − V βγEαδ2 − V αδEβγ2 + V βδEαγ2

),

(3.25)Eαβ2 ∧Eγδ =−1

3

(V αγE

βδ3 + V βγEαδ3 − V βδEαγ3 − V αδEβγ3 + V γ δEαβ3

),

(3.26)E4αα = 0,

(3.27)Eαβ ∧Eγδ3 =−1

4

(V αγE

βδ4 − V βγEαδ4 + V αδEβγ4 − V βδEαγ4

),

(3.28)Eαβ4 ∧Eγδ =− 1

20

(2V αγ V βδ − 2V αδV βγ − V αβV γ δ)E5.

The gravitational fieldw describes theAdS5 geometry provided thatw = ω0 satisfiesthe zero-curvature equation

(3.29)dω0 +ω0 ∧ ∗ω0 = 0,

and the frame 1-form is non-degenerate. The background frame field and Lorentzconnection will be denotedh= hαβaαbβ andωL0 = ωLα0 βaαb

β , respectively. The vacuum

values of thep-formsEαβp are denotedHαβp .


4. su(2,2)–o(4,2) dictionary

To make contact between the tensor and spinor forms of the higher spin dynamics onehas to identify in terms ofo(4,2) the irreducible finite-dimensional representations ofsu(2,2) described by a pair of mutually conjugated tracelesssu(2,2) multispinors

(4.1)Xα1...αnβ1...βm ⊕Xβ1...βm

α1...αn, Xα1...αn−1γβ1...βm−1γ = 0.

The result is that for evenn+m the representation (4.1) is equivalent to the representationof o(4,2) described by the traceless tree-row Young diagram having two rows of equallengths1

2|n+m| and the third one of length12|n−m|. In other words, theo(4,2) form ofthe representation (4.1) is described by the tensorXA1...Ap,B1...Bp,C1...Cq with p = 1

2|n+m|,q = 1

2|n−m|, which is separately symmetric with respect to the indicesAi ,Bi andCi andsatisfies the conditions

(4.2)XA1...Ap,Ap+1B2...Bp,C1...Cq = 0, XA1...Ap,B1...Bp,Bp+1C2...Cq = 0

and

(4.3)ηD1D2XD1D2A3...Ap,B1...Bp,C1...Cq = 0.

(From these conditions it follows that all other traces vanish as well.) One example of thisidentification is provided by the isomorphism betweenXαβ (with its conjugateXαβ ) andthe 3-form representation ofo(4,2) XA,B,C being totally antisymmetric in its indices.

For the case of half-integer spins with oddn+m, the identification is analogous with thetensor-spinorXA1...Ap,B1...Bp,C1...Cq ;α carrying theo(4,2) spinor indexα, 2p = n+m−1,2q = |n−m| − 1 and theγ -transversality condition with respect to all indices in additionto the tracelessness condition (4.3).

A particular case of a self-conjugated traceless multispinor

(4.4)Xα1...αnβ1...βn, Xα1...αn−1γ

β1...βn−1γ = 0

is most important for this paper. Such a tensor is equivalent to the representation ofo(4,2)described by a length-n rectangular traceless two-row Young diagram, i.e., to

(4.5)XA1...An,B1...Bn,

which is separately symmetric in the indicesAk andBk , has all traces zero and is subjectto the condition that symmetrization of anyn+ 1 indices gives zero. One way to see thisisomorphism is to compare the dimensions of the representations to make sure that they areboth equal to(2n+ 3)(n+ 1)2(n+ 2)2/12. It is easy to see that this formula is true fromthesl4 side. The computation in terms ofo(4,2) is more complicated. The dimensionalityof the representation of the orthogonal algebrao(d) described by the two-row tracelessrectangular diagram of lengths is

(4.6)N (s, d)= (2s + d − 2)! (s + d − 4)! (s + d − 5)!(d − 2)! (d − 3)! s! (s + 1)! (2s + d − 5)! .

For n = s and d = 6 one finds the desired result. Forn = 1 the isomorphism betweenthe adjoint representations ofsu(2,2) ando(4,2) is recovered. Note that the analogous


analysis of the representation (4.4) ofsu(2,2) was done in [28] in terms of representationsof the 5d Lorentz algebrao(4,1)⊂ o(4,2).

In accordance with the analysis of Section 2.3 (and of [28]) we conclude that 5d spins bosonic gauge fields can be described by 1-formsωα1...αs−1

β1...βs−1 which are tracelessmultispinors symmetric in the upper and lower indices. Totally symmetric spins fermionictensor–spinor representations are described by the gauge fieldsωα1...αs−1/2

β1...βs−3/2 andtheir conjugates.

All other representations in the set (4.1) do not correspond to the sets of gauge fieldsassociated with the totally symmetric tensor(–spinor) fields. These are expected to underlythe description of the mixed symmetryAdS5 massless fields to be developed. Accordingto [29], such fields are inequivalent to the totally symmetric higher spin fields in the AdSregime, although reduce in the flat limit to some combinations of the higher spin fieldsassociated with the totally symmetric representations of the flat Wigner little algebra. Asargued in [25], the fieldsωα1...αp

β1...βq with |p − q| 2 necessarily appear in the 5dhigher spin gauge theories withN 2 extended supersymmetry. This raises the importantproblem of the development of the formulation of the corresponding massless fields inAdSd for d > 4. This problem is now under investigation. Prior it is solved, we can onlystudy the purely bosonic theory with totally symmetric higher spin fields, which is thesubject of this paper, and itsN = 1 supersymmetric version, which is the subject of theforthcoming paper [32].

5. 5d higher spin algebra

The AdS5 higher spin algebras are expected to identify with 4d conformal higher spinalgebras studied by Fradkin and Linetsky [26], and their further extensions [25] andreductions [25,28]. One starts with the Lie superalgebra constructed via supercommutatorsof the star product algebra (3.2). In [25] it was argued that this algebra as a whole, calledhu(1,1|8) [42], may play a key role in aAdS5 higher spin gauge theory. The set of thegauge fields corresponding to the algebrahu(1,1|8) is

(5.1)ω(a, b|x)=∞∑

m,n=0

1

m!n!ωα1...αm

β1...βn(x)aα1 · · ·aαmbβ1 · · ·bβn.

The 5d higher spin field strength has the form

(5.2)R(a, b|x)= dω(a, b|x)+ω(a, b|x) ∗ ∧ω(a, b|x).The higher spin gauge fields in (5.1) contain 1-forms in all representations (4.1). Accordingto the analysis of [28] and Section 4 of this paper, only the fields withn=m correspondto usual (i.e., totally symmetric) higher spin fields. Before the free theory of the mixedsymmetryAdS5 higher spin gauge fields is elaborated, we confine ourselves to the higherspin algebra associated with the simplest case of the purely bosonic theory of totallysymmetric higher spin fields.


We therefore want to have only the gauge fields carrying equal numbers of the upper andlowersu(2,2) indices. As a result, the elements of the higher spin algebra should satisfy

(5.3)Naf =Nbf,where

(5.4)Na = aγ ∂

∂aγ, Nb = bγ ∂

∂bγ.

This is equivalent to the condition [26]

(5.5)N ∗ f = f ∗N.Thus, the bosonic 5d higher spin algebra identifies with the Lie algebra built from thestar-commutators of the elements of the centralizer ofN in the star product algebra (3.2).The same algebra (although rewritten in the 4d covariant notations) was interpreted in [26]as the 4d conformal higher spin algebra calledhsc∞(4) and was proved to give rise tothe gauge invariant cubic interactions of the 4d conformal higher spin theory in [27]. Wechange the names of some of the higher spin superalgebras in accordance with the notationof [25,42] to include in our systematics the two-parametric series of matrix extensions ofthe higher spin superalgebras. In particular we will use the namecu(1,0|8)for the algebrahsc∞(4) of [26].

The set of the gauge field corresponding to the algebracu(1,0|8) is

(5.6)ω(a, b|x)=∞∑n=0

1

(n!)2ωα1...αn

β1...βn(x)aα1 · · ·aαnbβ1 · · ·bβn.

Thecu(1,0|8) field strength has the form (5.2) and admits analogous expansion

(5.7)R(a, b|x)=∞∑n=0

1

(n!)2Rα1...αn

β1...βn(x)aα1 · · ·aαnbβ1 · · ·bβn.

So far we considered complex fields. To impose the reality conditions let us define theinvolution † by the relations

(5.8)(aα)† = ibβCβα,

(bα)† = iCαβaβ.

Since an involution is required to reverse an order of product factors

(5.9)(f ∗ g)† = g† ∗ f †

and to conjugate complex numbers

(5.10)(µf )† = µf †, µ ∈ C,

the definition (5.8) contains an additional factor ofi compared to the complex conjugation(3.6). The involution † leaves invariant the defining relations (3.1) of the star productalgebra and has the involutive property(†)2 = Id. By (5.9) the action of † extends to anarbitrary elementf of the star product algebra. Since the star product we use corresponds


to the totally symmetric (i.e., Weyl) ordering of the product factors, the result is simply

(5.11)(f(aα, b

β))† = f (ibγCγα, iCβγ aγ ).

It is elementary to check directly with (3.2) that (5.11) defines an involution of the starproduct algebra.

The reality conditions on the elements of the higher spin algebra have to be imposed ina way consistent with the form of the higher spin curvature. This is equivalent to singlingout a real form of the higher spin Lie algebra. With the help of any involution † this isachieved by imposing the reality conditions

(5.12)f † =−f.This condition defines the real higher spin algebrahu(1,0|8) for four pairs of oscillatorsandcu(1,0|8) as its subalgebra being the centralizer ofN . Note that the operatorN isself-conjugated

(5.13)N† =N.Let us stress that the condition (5.12) extracts a real form of the Lie superalgebra built

from the star product algebra but not of the associative star product algebra itself. Thesituation is very much the same as for the Lie algebrau(n) singled out from the complexLie algebra ofn× n matrices by the condition (5.12) with † identified with the hermitianconjugation. Antihermitian matrices form the Lie algebra but not associative algebra. Infact, the relevance of the reality conditions of the form (5.12) is closely related with thismatrix example because it demonstrates that the spin-1 (i.e., purely Yang–Mills) part ofthe matrix extensions of the higher spin algebras is compact. More generally, these realityconditions guarantee that the higher spin symmetry admits appropriate unitary highestweight representations. Note that in the sector of theAdS5 algebrasu(2,2) the realitycondition (5.12) is equivalent to (3.6).

The higher spin gauge fieldsω(a, b|x) are required to satisfy the condition analogous to(5.12)

(5.14)ω† =−ω,that gives rise to the component form of the reality condition by virtue of (5.11).

For any fixedn the connectionωα1...αnβ1...βn(x) is reducible because it is not traceless.

It decomposes into the set ofn+ 1 irreducible componentsω′α1...αkβ1...αk with all k in the

intervaln k 0 ( ω′α1...αk−1γβ1...βk−1γ = 0). As a result, fields of every spin appear in

infinitely many copies in the expansion (5.6). The origin of this infinite degeneracy can betraced back to the fact that the algebraA0

4 has infinitely many idealsIP (N) associated withvarious central elementsP(N) being star-polynomials ofN , x ∈ IP (N) : x = P(N) ∗ y,y ∗ N = N ∗ y [26]. On the one hand this infinite degeneracy makes 5d higher spingauge theories reminiscent of the superstring theory that contains infinitely many (massive)modes of any given symmetry type. On the other hand a question arises whether it ispossible to consider consistent higher spin models with reduced spectra of spins associatedwith the quotient higher spin algebras. The most interesting reductions are provided with


the algebrahu0(1,0|8) = hu(1,0|8)/IN called hsc0(4) in [26] and its further reductionho0(1,0|8) [25] called hs(2,2) in [28]. (IN is the ideal spanned by the elements of theform g =N ∗f = f ∗N .) The gauge fields of the algebrahu0(1,0|8) correspond to the setof all integer spinss 1 (every spin appears once) whileho0(1,0|8) describes its reductionto the subalgebra associated with even spins. As we show both options are allowed in theframework of the cubic analysis of this paper. We start in Section 6 with the analysis of theunreduced case ofcu(1,0|8) considering the reduced cases afterwards in Section 9. Notethat from this perspective our conclusions are somewhat different from those of [27] whereit was argued that only the unreduced algebracu(1,0|8) admits consistent dynamics inthe framework of the 4d conformal higher spin theory. From the perspective of AdS/CFTcorrespondence the most interesting cases are associated either with the maximally reducedmodels [25,28] and their supersymmetric extensions or with the unreduced models basedon the algebrashu(m,n|8) [25] which, presumably, give rise to all types ofAdS5 masslessfields.

6. 5d higher spin gauge fields

Thecu(1,0|8) linearized higher spin curvature

R1(a, b|x)= dω1(a, b|x)+ω0(a, b|x)∗∧ω1(a, b|x)(6.1)+ω1(a, b|x)∗∧ω0(a, b|x),

with

(6.2)ω0(a, b|x)= ω0αβaαb

β, ω0αα = 0,

satisfying the zero-curvature condition (3.29), provides the 5d spinor version of the formula(2.53). Equivalently,

(6.3)R1(a, b|x)= dω1(a, b|x)+ω0αβ(x)

(∂

∂bαbβ − aα ∂

∂aβ

)∧ ω1(a, b|x).

The component formula reads

R1α1...αs−1

β1...βs−1 = dωα1...αs−1β1...βs−1 − (s − 1)

(ω0

α1γ ∧ ωγα2...αs−1

β1...βs−1

(6.4)−ω0γ β1 ∧ωα1...αs−1

γβ2...βs−1).

The linearized (Abelian) higher spin gauge transformations are

(6.5)δ0ω(a, b|x)=D0ε(a, b|x),where

(6.6)D0 = d +ω0αβ

(∂

∂bαbβ − aα ∂

∂aβ

)is the background covariant derivative. The fact thatω0 satisfies the zero-curvaturecondition implies

(6.7)δ0R1 = 0.


To decompose the representations of theAdS5 algebrasu(2,2) ∼ o(4,2) into rep-resentations of its Lorentz subalgebrao(4,1) we use the antisymmetric compensatorV αβ . A su(2,2) counterpart of the reduction of the tensor higher spin gauge fielddxnωn

A1...As−1,B1...Bs−1 carrying the irreducible representation of theAdSd algebrao(d − 1,2) described by the traceless two-row Young diagram of lengths−1 into a collec-tion of the Lorentz covariant higher spin-1-formsdxnωna1...as−1,b1...bt with all 0 t s−1goes as follows. The fieldV αβ is used to raise and lower spinor indices. Then, the Lorentzalgebra irreducible components correspond to various types of symmetrization betweenthe two types of indices, i.e., again to all two-row traceless Young diagrams but now in thespinor indices,ω′α1...αs−1+q ,β1...βs−1−q with all 0 q s − 1 (all traces withV αβ are zeroand symmetrization with respect to anys + q indices gives zero). The identification withtheo(4,1) tensor notation is

(6.8)ω′α1...αs−1+t ,β1...βs−1−t ∼ ωa1...as−1,b1...bt .

For example, the two-row rectangular diagram of lengths − 1 in tensor notation isdescribed by the one-row diagram of length 2(s − 1) in the spinor notation, whilethe two-row rectangular diagram of lengths − 1 in spinor notation corresponds to theone-row diagram of lengths − 1 in the tensor notation. (Particular manifestations ofthis relationship are those between the vector and traceless antisymmetric second-rankspinor or antisymmetric tensor and symmetric second rank spinor, both underlying theisomorphism between the spinor and vector realization of the 5d space–time symmetryalgebras.) Note that the analogous identification of the representations was discussed inthe recent paper [28], where the spinor version of the linearized higher spin curvatures hasbeen presented. The difference is that in this paper we use the manifestlysu(2,2) covariantcompensator formalism that simplifies greatly the analysis of the interactions.

In what follows we shall use the two sets of the differential operators in the spinorvariables

(6.9)S− = V αβaα ∂

∂bβ, S+ = Vαβbα ∂

∂aβ, S0 =Nb −Na,

and

(6.10)T + = aαbα, T − = 1

4

∂2

∂aα∂bα, T 0 = 1

4(Na +Nb + 4).

They form two mutually commutingsl2 algebras

(6.11)[S0, S±

]=±2S±,[S−, S+

]= S0,

(6.12)[T 0, T ±]=±1

2T ±,

[T −, T +]= T 0,

(6.13)[T i, Sj

]= 0.

(Unusual normalization of the generatorsT i in (6.10), and (6.12) is chosen for the futureconvenience.)


The operatorsT i andS0 are independent of the compensatorV αβ and, therefore, aresu(2,2)-invariant. As a result,

(6.14)D0(T i)= 0, D0

(S0)= 0.

(These relations have to be understood in the sense thatD0(X(f )) = X(D0(f )), whereX is one of the operatorsT i andS0, while f is an arbitrary element of the star productalgebra.) A useful consequence of this fact is

(6.15)R1(T j (ω)

)= T j (R1(ω)).

From (6.11) it also follows

(6.16)[S+,D0(S

−)]+ [D0(S

+), S−]= 0.

According to (5.3) the elements of the higher spin algebracu(1,0|8) satisfyS0(f ) = 0,i.e.,

(6.17)S0ω(a, b|x)= 0, S0R(a, b|x)= 0.

As a result, the operatorsS+ andS− commute to each other on the higher spin gauge fieldsand field strengths ofcu(1,0|8).

TheV αβ -dependent operatorsS± are only Lorentz-invariant. In accordance with (3.17)and (3.19)

(6.18)D0S− = hαβaα ∂

∂bβ, D0S

+ =−hαβbα ∂

∂aβ.

Let us note that the background covariant derivativeD0 (6.6) admits the representation

(6.19)D0 =DL0 + 1

2

[S−,D0S

+]=DL0 − 1

2hαβ

(aα

∂

∂aβ− bβ ∂

∂bα

),

where the background Lorentz derivativeDL0 commutes with all operatorsT i andSi .

From the star product (3.2) it follows that

(6.20)N ∗ f =(T + − T − + 1

2(Nb −Na)

)f.

According to (5.3), forf ∈ cu(1,0|8) this simplifies to

(6.21)N ∗ f = f ∗N = (T + − T −)f.The decomposition intosu(2,2) irreducible fields is

(6.22)ω(a, b)=∞∑

s,n=0

(T +)nvn(T 0)ωsn(a, b),

with

(6.23)T 0ωsn =1

2(s + 1)ωsn,

(6.24)T −ωsn(a, b)= 0.


For the future convenience, we fix the normalization coefficientsvn(T0) in (6.22) in the

form

(6.25)vn

(1

2(s + 1)

)= v

(1

2(s + 1)

)(2i)n

√(2s + 1)!

n! (n+ 2s + 1)! .

Note that the factor ofin in (6.25) implies that the different copies of the fieldsωsn withthe same spin contained in polynomials of degree 4p and 4p+ 2 contribute with oppositesigns. This is appropriate because the coefficients in front of the corresponding parts of theinvariant action will be shown to have opposite signs as well.

Due to (6.15), the linearized curvatures admit the expansion analogous to (6.22)

(6.26)R1(a, b)=∑n

(T +)nvn(T 0)R1,n(a, b)

with

(6.27)T −R1,n(a, b)= 0.

Let us now explain how the invariant version of the Lorentz covariant decompositionused in [28,30] can be defined. Lorentz multispinors associated with the two-row Youngdiagrams havingn1 andn2 cells in the upper and lower rows, respectively, (n1 n2) canbe described as the polynomialsη(a, b) of the spinor variablesaα andbβ subject to theconditions

(6.28)Naη(a, b)= n1η(a, b), Nbη(a, b)= n2η(a, b),

(6.29)S−η(a, b)= 0,

where the latter condition implies that the symmetrization over anyn1 + 1 indices giveszero. The tracelessness condition reads in these terms

(6.30)T −η(a, b)= 0.

Thesu(2,2) irreducible higher spin gauge fieldω admits the following representation interms of the Lorentz-irreducible higher spin fields

(6.31)ω(a, b)=∑t=0

(S+)tηt (a, b),

(Note that the asymmetric form of this formula with respect toaα andbβ is a result of aparticular basis choice.) Sinceω(a, b) has equal numbers ofa andb, we set 2t = n1− n2.For the spins we have 2(s−1)= n1+n2 (cf. (6.23)). Fors fixed,t ranges from 0 tos−1.

One can treat the Lorentz-irreducible 1-formsηt (a, b) as an alternative basis of thehigher spin gauge fields. The linearized higher spin curvature 2-forms (6.1) admit theanalogous expansion

(6.32)R1(a, b)=∑t=0

(S+)tr t1(a, b),

with the Lorentz-irreducible component curvaturesrt1(a, b) satisfying the Young property

(6.33)S−rt1(a, b)= 0


and the tracelessness condition

(6.34)T −rt1(a, b)= 0.

From the definition ofrt1(a, b) it follows that

(6.35)rt1(a, b)=DL0 η

t (a, b)+ τ−(ηt+1(a, b)

)+ τ+(ηt−1(a, b)),

whereDL0 is the Lorentz covariant derivative and the 5d spinor realization of the operators

(2.55) and (2.56) is

(6.36)τ+ = 1

S0 + 1D0(S−),

(6.37)τ− = 1

2

(S+[S−,D0

(S+)]−D0

(S+)S0 − (S+)2 1

S0 + 1D0(S−)).

One can see that the properties (2.57) are satisfied on the space of functionsη satisfying(6.29).

Analogous decomposition

(6.38)D0 =DL0 + τ+ + τ−

exists in the original basis of fieldsω satisfying the condition (6.17). The explicit form ofτ± in this basis is

(6.39)τ± = 1

4

([S−,D0S

+]± 1√1− 4S+S−

([S−,D0S

+]+ 2D0(S+S−

))).

Derivation of this formula is more complicated. It is based on the fact that the operatorS−S+ = S+S− diagonalizes on the vectors with differentt in (6.31) with the eigenvalues−t (t + 1) so that the operatort

(6.40)t = 1

2

(√1− 4S+S− − 1

)has eigenvaluest . The property that

(6.41)[t , τ±] =±τ±turns out to be equivalent to

(6.42)[S+S−, τ±

]= (1∓√1− 4S+S−)τ±.

Taking into account the fact that the decomposition ofD0 into eigenspacesDL0 , τ+ and

τ− of S+S− is unique, the problem is to find such operatorsτ± on the space of functionssatisfying (6.17) that the formulas (6.42) and (6.38) are true. Formula (6.39) solves thisproblem. For (6.38) this is obvious. The verification of the formula (6.42) is also elementarywith the help of identities valid on the subspace of null-vectors ofS0

(6.43)[S+S−,

[S−,D0S

+]]=−2D0(S+S−

),

(6.44)[S+S−,D0

(S+S−

)]= 2(D0(S+S−

)+ S+S−[S−,D0S+]).


Another useful fact is that the operator

(6.45)τ0 = S−D0(S+)− S+D0

(S−)

does not affect the gradationt , i.e.,

(6.46)[S+S−, τ0

]= 0.

It is less trivial to check that(τ±)2 = 0. The simplest way is to use the basis of the fieldsηt , i.e., the operatorsτ± in the form (6.36) and (6.37).

Let us note that the variablesω andη can be interpreted as different representatives of thesame representation of thesl2 algebra spanned by the operatorsSj . Namely, the variablesω are associated by (6.23) with the elements having zero eigenvalue of the Cartan element,while the variablesη are associated by (6.29) with the lowest weight vectors. This suggeststhe idea that there should be some formulation operating in terms of the representations ofthis sl2 algebra as a whole.

Equipped with the operatorsτ± andτ0, one can write the spinor form of the constraints(2.71) either as

(6.47)τ0 ∧ τ+R1 = 0

in the basisω or as

(6.48)τ0 ∧ τ+r1 = 0

in the basisη. To obtain the spinor form of the First On-Mass-Shell Theorem one takesinto account that, as shown in the beginning of this section (see also [28]), the Weyltensor, described in terms of tensors by the lengths two-row traceless Young diagramCa1...as

,b1...bs , is described in terms of spinors by a rank 2s totally symmetric multispinor

Cα1...α2s . Since the First On-Mass-Shell Theorem (2.73) is true for any irreducible higherspin field in the expansion (6.26), it acquires the form

(6.49)R1,n(a, b)|m.s. =H2αβ ∂2

∂aα∂bβResµ

(Cn(µa +µ−1b

)),

where the label|m.s. implies the on-mass-shell consideration modulo terms proportional tothe left-hand sides of the free field equations and constraints (6.47) (equivalently, (6.48)).Resµ singles out theµ-independent part of a Laurent series inµ, i.e.,

(6.50)Resµ

( ∞∑n=−∞

αnµn

)= 1

2πi

∮d logµ

( ∞∑n=−∞

αnµn

)= α0.

Note that a function of one spinor variable

(6.51)Cn(µa +µ−1b

)=∑k,l

µk−l

k! l! Cα1···αkβ1...αln aα1 · · ·aαkbβ1 . . . bβl

has totally symmetric coefficientsCα1...αkβ1...βl while Resµ singles out its part containingequal numbers of the oscillatorsa andb that belongs tocu(1,0|8).


7. Central On-Mass-Shell Theorem

The matter fields and higher spin Weyl tensor can be interpreted as representatives oftheσ− cohomology group associated with the so-called twisted adjoint representation ofthe higher spin algebra. Given automorphismτ of the higher spin algebra (in fact anyassociative algebra used to build a Lie superalgebra via supercommutators), one definesthe covariant derivativeD of a fieldC taking values in the twisted adjoint representation

(7.1)DC = dC +ω ∗C −C ∗ τ (ω).The property thatτ is an automorphism guarantees that this definition is consistent withthe Bianchi identities. (See [13,39] for particular examples and references.) To have aformulation in terms of Lorentz covariant fields (i.e., finite-dimensional representations ofthe Lorentz algebra),τ is required to leave invariant the Lorentz subalgebra of the full AdSalgebra. In terms of the compensator formalism this is automatically achieved by using thecompensator field for definition ofτ . For the problem under consideration, the appropriatedefinition is

(7.2)τ (aα)= bβVβα, τ (bα)= V αβaβimplying

(7.3)τ(f (a, b|x))= f

(τ (a), τ (b)

∣∣x).Let us note that in this section we require the compensatorVαβ to be a constant so that

τ commutes with the exterior differentiald .The linearized covariant derivative (6.19) in the adjoint representation can be written as

(7.4)D0 =DL0 + 1

2

[hαβaαb

β, ·]∗.Analogously to the 4d case [13,39], the twisted linearized covariant derivative results fromthe replacement of the star commutator to star anticommutator in the part of the covariantderivative associated with the frame 1-form

(7.5)D0(C)=DL0 (C)+

1

2

hαβaαb

β,C∗.

In fact, this is not surprising because the only non-trivial Lorentz covariant definition ofthe restriction ofτ to theAdSd algebra in any dimension is to change a sign of the AdStranslations. From the perspective of the higher spin symmetry the problem therefore is tofind an appropriate extension of this automorphism of the AdS algebra to the full higherspin algebra. This is achieved by the definition (7.2) for the case ofAdS5. For some specificchoice of the compensator, this definition reproduces the twisted adjoint representationused in [28].

The twisted covariant derivative (7.5) has the form

(7.6)D0(C)=DL0 (C)+ σ− + σ+,


where

(7.7)σ− =−1

4hαβ

∂2

∂aβ∂bα, σ+ = hαβaαbβ.

The operatorsDL0 andσ± have the properties

(7.8)(σ±)2 = 0,(DL

0

)2 + σ−, σ+ = 0,DL

0 , σ±= 0.

Only the operatorDL0 acts non-trivially (differentiates) on the space–time coordinates

while σ± act in the fiber linear spaceV isomorphic as a linear space to the twistedadjoint representation of the higher spin algebra. Also there is the gradation operatorG=12(Na +Nb) such that

(7.9)[G,DL

0

]= 0, [G,σ±] =±σ±.SinceV is spanned by polynomials in the spinor variablesaα andbβ , the spectrum ofGin V is bounded from below.

The important observation is (see, e.g., [40]) that the non-trivial dynamical equationshidden in

(7.10)D0(C)= 0

are in the one-to-one correspondence with the non-trivial cohomology classes ofσ−. Forthe case withC being a 0-form, the relevant cohomology group isH 1(σ−). For the moregeneral situation withC being ap-form, the relevant cohomology group isHp+1(σ−).From this perspective, the operatorτ− identifies withσ− in the sector of the higher spingauge 1-forms.

Indeed, consider the decomposition of the space of fieldsC into the direct sum ofeigenspaces ofG. Let a field having a definite eigenvaluek of G be denotedCk , k =0,1,2, . . . , . Suppose that the dynamical content of Eqs. (7.10) with the eigenvaluesk kq is found. Applying the operatorDL

0 + σ+ to the left-hand side of Eqs. (7.10) atk kq

we obtain taking into account (7.8) that

(7.11)σ−(DL

0 + σ− + σ+)(Ckq+1)= 0.

Therefore(DL0 +σ−+σ+)(Ckq+1) is σ− closed. If the groupH 1(σ−) is trivial in the grade

kq + 1 sector, any solution of (7.11) can be written in the form(DL0 + σ− + σ+)(Ckq+1)=

σ−Ckq+2 for some fieldCkq+2. This, in turn, is equivalent to the statement that one canadjustCkq+2 in such a way thatCkq+2 = 0 or, equivalently, that the part of Eqs. (7.10) of thegradekq+1 is some constraint that expressesCkq+2 in terms of the derivatives ofCkq+1 (tosay that this is a constraint we have used the assumption that the operatorσ− is algebraicin the space–time sense, i.e., it does not contain space–time derivatives.) IfH 1(σ−) isnon-trivial, this means that Eq. (7.10) sends the corresponding cohomology class to zeroand, therefore, not only expresses the fieldCkq+2 in terms of derivatives ofCkq+1 but alsoimposes some additional differential conditions onCkq+1. Thus, the non-trivial space–timedifferential equations described by (7.10) are classified by the cohomology groupH 1(σ−).


The non-trivial dynamical fields are associated withH 0(σ−) which is always non-zerobecause it at least contains a non-trivial subspace ofV of minimal grade. As follows fromtheH 1(σ−) analysis of the dynamical equations, all fields inV/H 0(σ−) are auxiliary, i.e.,express via the space–time derivatives of the dynamical fields by virtue of Eqs. (7.10).

In the problem under consideration we are interested in the sector of fieldsC(a, b|x)that commute toN (i.e.,Na(C) = Nb(C)). In this sector the representatives ofH 0(σ−)(i.e., fieldsC satisfyingσ−(C)= 0) are described by the fields of the form

(7.12)C0(a, b|x)=Resµ C0(µa +µ−1b, aαb

α∣∣x).

We see that these are just the fields that appeared in the First On-Mass-Shell Theorem(6.49). The additional dependence onaαbα matches the degeneracy of the higher spinfields ofcu(1,0|8) due to traces (i.e., ideals generated byN ).

Application of the same analysis to the higher spin gauge 1-forms with the operatorτ−instead ofσ− leads to the following interpretation of the results of Section 6. The dynamicalfields with spinss 1 belong to the cohomology groupH 1(τ−). τ− exact 1-formsω(a, b|x)= τ−(ξ) describe pure gauge degrees of freedom inω(a, b|x) analogous to theantisymmetric part of the frame field associated with the local Lorentz transformations ingravity. The cohomology groupH 2(τ−) responsible for non-trivial differential conditionson the higher spin gauge fields is a direct sum of two linear spaces

(7.13)H 2(τ−)= VE2 (τ−)⊕ VW2 (τ−).

The spaceVW2 (τ−) called Weyl cohomology is spanned by the 2-forms of the form of theright-hand side of Eq. (6.49), i.e., a generic element ofVW2 (τ−) has the form (to simplifyformulae, in the rest of this section we confine ourselves to the case of irreducible fields ofdifferent spins satisfyingT −ω = 0)

(7.14)H2αβ ∂2

∂aα∂bβResµ

(C(µa +µ−1b

)).

The spaceVE2 (τ−) called Einstein cohomology is spanned by the 2-forms of the form

(7.15)H2αβ

(∂2

∂aα∂bβR(a, b)+ bαaβr(a, b)

),

where the 0-formsR(a, b) and r(a, b) have themselves the properties of the dynamicalfields, i.e.,

(7.16)S±R = 0, T −R = 0, S±r = 0, T −r = 0.

The 0-formsR and r parametrize the right-hand sides of the spins 2 equations ofmotion. They generalize the traceless part of the Ricci tensor and the scalar curvature,respectively. In other words, they correspond to the dynamical equations of motionassociated with the irreducible traceless parts of the double traceless Fronsdal fieldsϕa1...an

in the action (1.1). Eq. (6.49) sends the right-hand sides of the dynamical equationsassociated with theR andr to zero imposing no other conditions on the dynamical fieldsbecause the Weyl cohomology remains arbitrary. This is the content of the First Of-Mass-Shell Theorem that states that (6.49) is equivalent to the free equations of motion for all


spinss 2. Note that the spin-1 Maxwell equations are not contained in Eq. (6.49) whichmerely defines the associated spin-1 field strength as the degree two part of the WeylcohomologyC(a, b). The degree zero partC(0,0) associated with spin-0 field does notshow up in the Weyl cohomology because the scalar fieldC(0,0) is not associated with thegauge fields.

The fact that Eq. (6.49) sew the 0-formsC(a, b|x) to the higher spin curvatures hastwo effects. First, what looked like an independent dynamical spins 1 field in themoduleC(a, b|x) becomes an auxiliary field expressed by (6.49) in terms of the dynamicalfields described by the 1-form gauge fields. Second, the fields on the right-hand sideof (6.49) have to satisfy some differential restrictions as a consequence of the Bianchiidentities. For all spinss 2 these differential restrictions are equivalent to what lookedlike independent equations in the condition that the sectionC(a, b|x) is flat. In other words,the Bianchi identities send to zero the part of the cohomology groupH 1(σ−) associatedwith all spinss 2 (s 3/2 when fermions are included [32]). For spin-1 only a halfof the corresponding part ofH 1(σ−) is sent to zero by the Bianchi conditions. This isassociated with the Maxwell equation that encodes the Bianchi identities for the fieldstrength expressed in terms of the 1-form potential. The dynamical part of the Maxwellequations is imposed by the covariant constancy condition for the spin-1 part ofC(a, b|x),i.e., by setting to zero the rest of the restriction ofH 1(σ−) to the spin-1 sector. The equationfor spin-0 is the condition thatH 1(σ−) = 0 in the spin-0 sector [40] (the situation withspin-1/2 is analogous [32]).

As a result, we arrive at the Central On-Mass-Shell Theorem that states that Eqs. (6.49),(7.10) describe the equations of motion for free massless fields of all spins along withan infinite set of constraints that express some auxiliary fields via higher derivatives of thedynamical fields associated with the cohomology groupH 1(τ−) and the scalar fieldc(x)=C(a, b|x). Let us note that, by construction, the set of fieldsω(a, b|x) andC(a, b|x)provide the complete basis for all combinations of derivatives of massless fields of all spinsthat are allowed to be non-zero by field equations (equivalently, to take arbitrary values atany fixed pointx0 of space–time). The Central On-Mas-Shell Theorem is the starting pointfor the description of the non-linear higher spin dynamics in the unfolded form. Eq. (6.49)also plays the key role in the analysis of cubic higher spin interactions at the action level.

The proof and the meaning of the tensor form of the Central On-Mass-Shell Theorem(2.77) and (2.78) in any dimension is analogous.

8. 5d higher spin action

The aim of this section is to formulate the action for the totally symmetric gaugemassless boson fields inAdS5 that solves the problem of higher-spin-gravitationalinteractions in the first non-trivial order. The results reported here extend the 4d resultsof [9,33] tod = 5.

We shall look for the action of the form

(8.1)S = S2 + S3 + · · ·


within the perturbation expansion (2.37) with the background gravitational field beingof the zero order and the higher spin fields of the first order.S2 is the quadratic actionthat describes properly the free higher spin dynamics.S3 is the cubic part. Higher-ordercorrections do not contribute to the order under investigation. The gauge transformationsare supposed to be of the form (2.36). Equivalently one can expand

(8.2)δω = δ0ω+ δ1ω+ · · · ,whereδ0ω is the linearized Abelian transformation (2.40) whileδ1ω contains terms linearin the dynamical fieldsω1. Recall that the background fieldω0 is chosen in such a waythatR0 = 0 (thus implyingAdS5 background) so thatR starts with the first order part. Asa result, the deformation terms∆(R,ε) in (2.36) contribute toδ1ω.

The free higher spin actionS2 is required to be invariant under the linearized higher spingauge transformations

(8.3)δ0S2 = 0.

This means that the part of the variation of the action, which is linear in the dynamicalfields, is zero. The first non-trivial part is therefore bilinear in the dynamical fields

(8.4)δ1S = δ0S3 + δ1S2 ∼ ω21ε.

Our aim is to find an actionS3 that admits a non-trivial deformation of the gaugetransformation guaranteeing that the gauge variation (8.4) is zero.

Using the decomposition (2.36) for the gauge variation one can rewrite the conditionδ1S = 0 in the equivalent form

(8.5)0= δgS +MS2 +O(ω3

1ε),

whereδg is the original higher spin gauge transformation (2.28) that contains the zero-order part of the variation along with some part of the first-order terms. Other possiblelinear terms in the variation are contained inMω1. Since

(8.6)MS2 = δS2

δωdynMωdyn

a (local) deformationMωdyn fulfilling the invariance condition (8.5) exists iff

(8.7)δgS =−Y(ω1,

δS2

δωdyn, ε

)+O(ω3

1ε),

whereY (ω1, δS2/δωdyn, ε) is some trilinear local functional, i.e., iff the original gaugevariation ofS2 + S3 vanishes on-mass-shellδS2/δωdyn= 0.

Note that a deformation of the gauge variation of the extra and auxiliary fields does notcontribute into the variation to the order under consideration because the variation ofS2

with respect to these fields is either identically zero by theextra field decoupling condition(2.46) for extra fields or zero by virtue of constraints (i.e., by the 1.5-order formalism) forthe Lorentz-type auxiliary fields. This is important because the constraints for the extra andauxiliary fields are not invariant under the original higher spin gauge transformationsδgω.As a result, the higher spin gauge transformation for the extra and auxiliary fields should


necessarily be deformed to be compatible with the constraints. This phenomenon does nothowever affect our consideration because the constraints are formulated in terms of thehigher spin curvatures and therefore are invariant under the linearized higher spin gaugetransformations in the lowest order. As a result, the deformation of the transformation lowfor the extra and auxiliary fields due to the constraints is at least of orderω1ε which wasargued to be irrelevant in the approximation under consideration.

Our analysis of the gauge invariance will be based heavily on the First On-Mass-ShellTheorem (2.73) in its spinor form (6.49). Namely, the variationδgS is some bilinearfunctional of the higher spin curvaturesR which can be replaced by the linearizedcurvaturesR1 at the order of interest. Assuming that the constraints for auxiliary and extrafields are satisfied we can use the representation (2.73) for the linearized curvatures. Allterms contained inX are proportional to the left-hand sides of the free field equationsand, therefore, give rise to some variation of the form (8.7) that can be compensated byan appropriate deformationMω1 (that itself is at least linear in the higher spin curvatures).The terms that cannot be compensated this way are those bilinear in the higher spin WeyltensorsCA1...As−1,B1...Bs−1. Therefore, the condition that the higher spin action is invariantunder some deformation of the higher spin gauge transformations is equivalent to thecondition that the original (i.e., undeformed) higher spin gauge variation of the actionis zero once the linearized higher spin curvaturesR1 are replaced by the Weyl tensorsCaccording to (2.73) i.e., schematically,

(8.8)δgS∣∣R=h∧hC = 0.

Being rather non-trivial, this condition will be shown to admit a solution linking thenormalization coefficients in front of the free higher spin action functionals for differentfields.

Let us now sketch the general procedure for the search of theAdS5 higher spin action.In accordance with (2.42) we shall look for a Lagrangian 5-form bilinear in the higherspin curvatures with some 1-formUΩΛ built from the higher spin gauge 1-forms. As nouseful extension of the compensator formalism to the full higher spin algebra is knownso far, we use a mixed approach with the frame fieldEαβ built from the compensatorV αβ and the gravitational fields associated with theAdS5 subalgebrasu(2,2)⊂ cu(1,0|8).In addition, some explicit dependence on the higher spin gauge fields taking values incu(1,0|8)/su(2,2) will be allowed. Presumably, such an approach is a result of a partialgauge fixing in a full compensator formalism in theAdS5 higher spin theory to bedeveloped. Note that, perturbatively,Eαβ contains the background gravitational field and,therefore, is of the zero order, while the higher spin fields are of the first order.

In our analysis the higher spin gauge fields will be allowed to take values in someassociative (e.g., matrix) algebraω→ ωI

J . The resulting ambiguity is equivalent to theambiguity of a particular choice of the Yang–Mills gauge algebra in the spin-1 sector. Thehigher spin action will be formulated in terms of the trace tr in this matrix algebra (to be notconfused with the trace in the star product algebra). As a result, only cyclic permutationsof the matrix factors will be allowed under the trace operation. Note that the gravitationalfield is required to take values in the center of the matrix algebra, being proportional to the


unit matrix. For this reason, the factors associated with the gravitational field are usuallywritten outside the trace.

Let us consider an action of the form

(8.9)S = SE + Sω,where

(8.10)

SE = 1

2

∫M5

(αEαβ

∂2

∂a1α∂a2β+ βEαβ ∂2

∂bα1∂bβ2

+2∑

ij=1

γ ijEαβ ∂2

∂aiα∂bβj

)

∧ tr(R(a1, b1|x)∧R(a2, b2|x)

)∣∣∣∣a1=b1=a2=b2=0

and

(8.11)Sω = 1

2

∫M5

τ tr(R(a1, b1|x)∧R(a2, b2|x)∧ω(a3, b3|x)

)∣∣∣∣ai=bj=0

.

Here the coefficientsα,β, γ ij and τ are some functions of the Lorentz invariantcombinations of derivatives with respect to the spinor variablesaiα andbαj ,

(8.12)aij = Vαβ ∂2

∂aiα∂ajβ, bij = V αβ ∂2

∂bαi ∂bβj

, cij = ∂2

∂aiα∂bαj

(i, j = 1,2 for (8.10) and 1,2,3 for (8.11)). Functionsα,β, γ ij and τ parametrize theambiguity in all possible contractions of indices of the component higher spin fields andcurvatures. Note that the gravitational field is not allowed to appear among the componentsof the connectionω that enters explicitly the action (8.11). Instead, all terms with thegravitational field in front of the curvature terms are collected in the actionSE (8.10).With this convention,SE contributes both to the quadratic and to the cubic parts of theaction whileSω only contributes to the interaction part of the action.

Below we show that there exists a consistent cubic higher-spin-gravitational interactionfor Sω = 0. Since the aim of this paper is to show that at least some consistent higher-spin-gravitational interaction exists inAdS5, we shall mostly focus on this particular case.Note that it is anyway hard to judge on a full structure of the theory from the perspectiveof the cubic interactions. Indeed, at the cubic level one can switch out interactions amongany three elementary (i.e., irreducible at the free field level) fields without spoiling theconsistency at this order. This is most obvious from the Noether coupling interpretationof the cubic interactions: setting to zero some of the fields is always consistent with theconservation of currents. It is plausible to speculate that the actionSE accounts the termsrelevant to the higher-spin-gravitational interaction but may miss some other higher spininteractions described by the actionSω. Indeed, as a by product of the consideration belowwe shall give an example of a consistent higher spin interactionsSω . Note that even writingdown all terms of the form (8.9) there is little chance to have a fully consistent theorybeyond the cubic order without introducing more dynamical fields because, as we knowfrom the 4d example [39] (see also [12,13]), some lower spin fields (e.g., spin-1 and spin-0)


have to be added. Note that the actions for spin-1 and spin-0 massless fields do not admita formulation in the form (2.42). To simplify the presentation we will assume in this paperthat these fields are set to zero, that is a consistent procedure at the cubic order. By analogywith the 4d case [10] we expect that an extension of the results of this paper to the fullsystem with the lower spin fields will cause no problem. Let us note that an appropriatereformulation of the Lagrangian spin-0 free field dynamics was developed in [40].

Let us now focus on the structure of the actionSE . The ambiguity in the coefficientfunctionsα,β, γij can be restricted by not allowing a contraction of the both of indicesof Eαβ with the same curvature. Another restriction we impose is that a total numberof derivatives ina1 and b1 is equal to the number of derivatives ina2 and b2, i.e., theterms resulting from the products of the polynomials of different powers inR(a1, b1) andR(a2, b2) are required to vanish. (The most important argument for this ansatz is, of course,that it will be proved to work.) We therefore consider the action of the form (8.10) withthe coefficientsγ 11 = γ 22 = 0. Taking into account that the higher spin gauge fields andcurvatures carry equal numbers of lower and upper indices, i.e.,R(µa,µ−1b)= R(a, b),the appropriate ansatz is

(8.13)SE = 1

2AEα,β,γ (R,R),

where the symmetric bilinearAEα,β,γ (f, g) = AEα,β,γ (g, f ) is defined for any 2-formsfandg as

(8.14)

AEα,β,γ (f, g)

∫M5

(α(p,q, t)Eαβ

∂2

∂a1α∂a2βb12+ β(p,q, t)Eαβ ∂2

∂bα1∂bβ

2

a12

+ γ (p,q, t)(Eα

β ∂2

∂a1α∂bβ2

c21−Eαβ ∂2

∂bα1∂a2βc12

))∧ tr

(f (a1, b1|x)∧ g(a2, b2|x)

)∣∣∣∣ai=bj=0

,

where we use notations

(8.15)p = a12b12 q = c12c21, t = c11c22.

The labelsα,β , γ andE in AEα,β,γ (f, g) refer to the functionsα(p,q, t), β(p, q, t),

γ (p,q, t) and the frame fieldEαβ that fix a particular form of the bilinear form. Sometimeswe will write A(f,g) instead ofAEα,β,γ (f, g).

As explained in Section 2.2, non-linear actions of this form cannot have the invarianttrace property, i.e.,A(a ∗ f,g) = A(f,g ∗ a) for generica,f, g ∈ cu(1,0|8). One canrequire however a weaker condition

(8.16)A(N ∗ f,g)=A(f,g ∗N),wheref andg are any elements satisfyingf ∗N =N ∗ f , g ∗N =N ∗ g. From (8.16) itfollows that

(8.17)A(φ(N) ∗ f,g)=A(f,g ∗ φ(N)).


We will refer to the property (8.16) as theC-invariance condition. It will play the key rolein the analysis of the invariance of the cubic action in Section 8.2. The explicit form of therestrictions on the coefficientsα,β, γ due to (8.16) is given in Section 8.1.

The main steps of the rest of the analysis are as follows. First we analyze the quadraticpart of the action choosing the functionsα,β andγ to guarantee that the free actionS2

describes a sum of compatible with unitarity free field actions for the set of the higherspin fields associated with the higher spin algebracu(1,0|8). This is equivalent to thetwo conditions. First, theextra field decoupling conditionrequires the variation of thequadratic action with respect to the extra fields to vanish. Second, the quadratic actionshould decompose into infinite sum of free actions for the different copies of fields of thesame spin associated with the spinor traces as discussed below (6.31). This is referred toas thefactorization condition.

Note that at the free field level there is an ambiguity in the coefficientsα(p,q, t) andβ(p,q, t) due to the freedom in adding the total derivative terms

δS2 = 1

2

∫M5

d(Φ(p,q, t) tr

(R1(a1, b1|x)∧R1(a2, b2|x)

)∣∣a1=b1=a2=b2=0

)

(8.18)

= 1

2

∫M5

∂Φ(p,q, t)

∂p

(hαβ

∂2

∂bα1∂bβ

2

a12− hαβ ∂2

∂a1α∂a2βb12

)∧ tr

(R1(a1, b1|x)∧R1(a2, b2|x)

)∣∣∣∣ai=bj=0

,

where, using the manifestsu(2,2) covariance of our formalism, the differentiald inthe first line is replaced by the backgroundsu(2,2) derivative, and the definition of thebackground frame field (3.17) has been taken into account along with the Bianchi identitiesD0(R1)= 0. As a result, the variation of the coefficients

(8.19)δα(p,q, t)= ε(p, q, t), δβ(p,q, t)=−ε(p, q, t)does not affect the physical content of the quadratic action, i.e., only the combinationα(p,q, t)+ β(p,q, t) has invariant meaning at the free field level. Modulo the ambiguity(8.19) the factorization conditionalong with theextra field decoupling conditionfixthe functionsα,β, γ up to an arbitrary function parametrizing the ambiguity in thenormalization coefficients in front of the individual free actions. The proof of this factis the content of Section 8.1.

In the analysis of the cubic interactions, there are two types of terms to be taken intoaccount. Terms of the first type result from the gauge transformations of the gravitationalfields and the compensatorV αβ that contribute into the factors in front of the higher spincurvatures in the action (8.10). The proof of the respective invariances goes the same wayas in the example of gravity considered in the Section 2.1 as it is based entirely on theexplicit su(2,2) covariance and invariance of the whole setting under diffeomorphisms(recall that the additional invariance (2.21) was identified in Section 2.1 with a mixtureof the diffeomorphisms andsu(2,2) gauge transformations). Also, one has to take intoaccount that the higher spin gauge transformation of the gravitational fields is at least linear


in the dynamical fields and therefore has to be discarded in the analysis ofω2ε type termsunder consideration.

The non-trivial terms of the second type originate from the variation (2.31) of the higherspin curvatures. According to (8.8) the problem is to find such functionsα,β andγ that

(8.20)δgSE(R,R)∣∣E=h,R=h∧hC ≡Ahα,β,γ

(R, [R,ε]∗

)∣∣R=h∧hC = 0

for an arbitrary gauge parameterε(a, b|x). As shown in Section 8.2 this condition fixes thecoefficients in the form

(8.21)

α(p,q, t)+ β(p,q, t)= ϕ0

∞∑m,n=0

(−1)m+n m+ 1

22(m+n+1)(m+ n+ 2)!m!(n+ 1)!pnqm,

(8.22)γ (p,q, t)= γ (p+ q), γ (p)= ϕ0

∞∑m=0

(−1)m+1 1

22m+3(m+ 2)!m!pm,

whereϕ0 is an arbitrary normalization factor to be identified with the (appropriatelynormalized in terms of the cosmological constant) gravitational coupling constant. Letus note that the sign factors in the coefficients (8.21) and (8.22) distinguish between thepolynomials of the oscillatorsaα andbβ of degree 4p and 4p+ 2. Together with the signsdue to the factors ofi in the normalization coefficients (6.22) this implies that fields ofequal spins contribute to the quadratic action with the same sign. The fields of even andodd spins contribute with opposite signs.

As a result, the condition that cubicAdS5 higher spin action possesses higher spin gaugesymmetries fixes uniquely the relative coefficients in front of the free actions of fields of allspins in a way compatible with unitarity. Note that the analysis of the interactions does notfix the ambiguity (8.19). Taking into account (8.18) along with the full Bianchi identities,one observes that the ambiguity inα − β is equivalent to the ambiguity in the interactiontermsSω of the form

(8.23)

Sω = 1

2

∫M5

Φ(p,q, t) tr([ω1,R]∗(a1, b1|x)∧R(a2, b2|x)

+R(a1, b1|x)∧ [ω1,R]∗(a2, b2|x))∣∣ai=bj=0

parametrized by an arbitrary functionΦ(p,q, t).

8.1. Quadratic action

The free field partS2 of the actionS is obtained from (8.13) by the substitution of the lin-earized curvatures (6.1) instead ofR andhαβ instead ofEαβ . The resulting action is mani-festly invariant under the linearized transformations (6.5) because the linearized curvaturesare invariant. We want the free action to be a sum of actions for the irreducible higher spinfields we are working with. This requirement is not completely trivial because of the infi-nite degeneracy of the algebra due to the traces. Thefactorization conditionrequires


(8.24)S2 =∞∑

s,n=0

Ss,n2

(ωsn),

i.e., the terms containing products of the fieldsωsn andωsm should all vanish forn =m. Asfollows from (6.22), (6.23) along with (6.12) this is true if

(8.25)AEα,β,γ

(f,(T +)kg)=AEαk,βk,γk((T −)kf, g), ∀k,

for someαk,βk andγk.An elementary computation shows that

AEα,β,γ(f,T +g

)=AEα1,β1,γ1

(T −f,g

)+∫M5

Q(p,q, t)Eαβ ∂2

∂a1α∂bβ

1

(8.26)∧ tr(f (a1, b1|x)∧ g(a2, b2|x)

)∣∣ai=bj=0,

where

(8.27)Q=(

1+p ∂

∂p

)(α(p,q, t)+ β(p,q, t))+ 2

(1+ q ∂

∂q

)γ (p,q, t)

and

(8.28)

α1(p, q, t)= 4

((2+ p ∂

∂p

)∂

∂p+(

1+ q ∂∂q

)∂

∂q

+(

2p∂

∂p+ 2q

∂

∂q+ t ∂

∂t+ 6

)∂

∂t

)α(p,q, t),

(8.29)

β1(p, q, t)= 4

((2+ p ∂

∂p

)∂

∂p+(

1+ q ∂∂q

)∂

∂q

+(

2p∂

∂p+ 2q

∂

∂q+ t ∂

∂t+ 6

)∂

∂t

)β(p,q, t),

(8.30)

γ1(p, q, t)= 4

((1+ p ∂

∂p

)∂

∂p+(

2+ q ∂∂q

)∂

∂q

+(

2p∂

∂p+ 2q

∂

∂q+ t ∂

∂t+ 6

)∂

∂t

)γ (p,q, t).

Thefactorization conditiontherefore requires

(8.31)Q=(

1+p ∂

∂p

)(α(p,q, t)+ β(p,q, t))+ 2

(1+ q ∂

∂q

)γ (p,q, t)= 0.

Then one observes that from (8.31) it follows that the same is true for the coefficientsα1,β1 andγ1 (8.28)–(8.30), and, therefore, (8.31) guarantees (8.25) for allk. In the sequel,the factorization condition(8.31) is required to be true. Since the operator(1+ q ∂

∂q) is

invertible, it allows to expressγ in terms ofα andβ .


Let us now analyze theC-invariance condition(8.16). Taking into account (6.21) alongwith the factorization condition (8.25), it amounts to

(8.32)AEα,β,γ (T

−f,g)+AEα1,β1,γ1(T −f,g)=AEα,β,γ (f,T −g)+AEα1,β1,γ1

(f,T −g).

Obviously, this is true iff

(8.33)AEα,β,γ (f, g)=−AEα1,β1,γ1(f, g),

i.e.,

α(p,q, t)=−α1(p, q, t), β(p, q, t)=−β1(p, q, t),

(8.34)γ (p,q, t)=−γ1(p, q, t).

This is equivalent to the requirement that the operatorsT + and−T − are conjugated withrespect to the bilinear formAEα,β,γ (f, g)

(8.35)AEα,β,γ(T ±f,g

)=−AEα,β,γ(f,T ∓g

).

Let us note that the original ansatz for the bilinear form (8.14) satisfies

(8.36)AEα,β,γ(T 0f,g

)=AEα,β,γ (f,T 0g).

From (8.28)–(8.30) it is clear that (8.34) reconstructs the dependence ofα(p,q, t),β(p,q, t) and γ (p,q, t) on t in terms of the “initial data”α(p,q,0), β(p,q,0) andγ (p,q,0).

With the help of (8.35) along with (6.12) it is elementary to compute the relativecoefficients of the actions for the different copies of fields in the decomposition (6.22),(6.24). The coefficients (6.25) are fixed so that the linearized actions have the samenormalization for different copies of the higher spin fields parametrized by the labeln

(8.37)S2 =∞∑

s,n=0

Ss2(ωsn).

In the linearized approximation it is therefore enough to analyze the situation for anyfixed n. We confine ourselves to the case ofω = ω0 assuming in the rest of this sectionthat

(8.38)T −ω= 0.

Let us now consider theextra field decoupling condition. Since the generic variationof the linearized higher spin curvature isδR1 =D0δω, whereD0 is theAdS5 backgroundcovariant derivative and because the action is formulated in theAdS5 covariant way with theaid of the compensator fieldV αβ , integrating by parts one obtains for the generic variationof S2

δS2 =∫M5

D0

(α(p,q,0)hαβ

∂2

∂a1α∂a2βb12+ β(p,q,0)hαβ ∂2

∂bα1∂bβ

2

a12

+ γ (p,q,0)(hα

β ∂2

∂a1α∂bβ

2

c21− hαβ ∂2

∂bα1∂a2βc12

))


(8.39)∧ tr(δω(a1, b1|x)∧R1(a2, b2|x)

)∣∣∣∣ai=bj=0

,

where it is taken into account that thet-dependent terms trivialize as a consequence of(8.38). The derivativeD0 produces the frame field every time it meets the compensator.(Recall thatD0(h

αβ) = 0 becauseD20 = R0 = 0.) Taking into account (3.24) and (8.38),

one finds

δS2 = 1

2

∫M5

ρ(p,q)

(∂2

∂a1α∂bβ

2

c21+ ∂2

∂a2α∂bβ

1

c12

)

(8.40)×H2αβ ∧ tr

(δω(a1, b1|x)∧R1(a2, b2|x)

)∣∣∣∣ai=bj=0

,

where

(8.41)ρ(p,q)=(

1+ p ∂

∂p

)(α(p,q,0)+ β(p,q,0)− 2γ (p,q,0)

).

According to (6.8) the extra fields are associated with the multispinors described by thetwo-row Young diagrams of the Lorentz algebra having at least four more cells in the upperrow than in the lower one. As follows from (6.31), generic variation sharing this propertyhas the form

(8.42)δωex(a, b)= (S+)2ξ(a, b).To guarantee(Na − Nb)δω

ex(a, b) = 0, the infinitesimalξ(a, b) is required to satisfy(Na −Nb − 4)ξ(a, b)= 0.

To derive the restriction on the coefficients imposed by the requirement that the extrafields do not contribute to the variation one observes, first, that

(8.43)[S+1 , q

]=−[S+1 ,p]= u, [S+1 , u

]= 0,

where

(8.44)S+1 = b1β∂

∂a1β, u= a12c12

and, second, that the double commutator ofS+1 to the differential operator next toρ(p,q)in (8.40) is zero. As a result, substituting (8.42) into (8.40) one finds that the correspondingvariation of the action vanishes provided that

(8.45)

(∂

∂p− ∂

∂q

)ρ(p,q)= 0.

Therefore, theextra field decoupling conditionrequires

(8.46)ρ(p,q)= ρ(p+ q).From thefactorization condition(8.31) and (8.41), (8.46) it follows that

(8.47)γ (p,q,0)= γ (p+ q,0), ρ(r)=−2

(r∂

∂r+ 2

)γ (r,0).


The functionα(p,q,0)+ β(p,q,0) is fixed in terms ofγ (p + q,0) by thefactorizationcondition(8.31)

(8.48)α(p,q,0)+ β(p,q,0)=−2

1∫0

du

(1+ q ∂

∂q

)γ (up+ q).

The function of one variableγ (p + q,0) parametrizes the leftover ambiguity in thecoefficients (discarding the trivial ambiguity (8.19)) associated with the ambiguity in thecoefficients in front of the free actions of fields with different spins. Indeed, the totalhomogeneity degree in the variablesp andq , telling us how many pairs of indices arecontracted, equals tos− 1. Clearly, this ambiguity cannot be fixed from the analysis of thefree action.

8.2. Cubic interactions

Let us now analyze theon-mass-shell invariance condition(8.8) to prove the existenceof a non-linear deformation of the higher spin gauge transformations that leaves the cubicpart of the actionS = SE invariant to the orderω2ε. As explained in the beginning of thissection this condition amounts to (8.20). Taking into account (6.26), our aim is to provethat there exist such coefficient functionsα,β andγ satisfying theC-invariance condition,factorization conditionandextra field decoupling conditionthat

(8.49)

∑mn

Ahα,β,γ

((T +)mvm(T 0)R1,m(a, b)

∣∣m.s.,

[ε,(T +)nvn(T 0)R1,n(a, b)

∣∣m.s.

]∗)= 0

for any gauge parameterε ∈ cu(1,0|8) and arbitrary Weyl tensorsCn(a) in the spinor form(6.49) of the First On-Mass-Shell Theorem.

To this end, one first of all observes that the dependence ofvn(T0) onT 0 can be absorbed

into (spin-dependent) rescalings of the Weyl tensorsCn(a) which are treated as arbitraryfield variables in this consideration. As a result, it is enough to prove (8.49) for arbitraryconstant coefficientsvn.

Now let us show that, once (8.49) is valid form= n= 0, it is automatically true for allother values ofm andn as a consequence of theC-invariance condition. Indeed, supposethat (8.49) is true form0 m 0, n0 n 0. Consider the term withm=m0 + 1. Then,from (6.21) it follows

(8.50)

(T +)m0+1

R1,m0+1(a, b)=N ∗((T +)m0R1,m0+1(a, b)

)+ T −(T +)m0R1,m0+1(a, b).

The term containingT − gives zero contribution by the induction assumption since, takinginto account (6.27),T − decreases a number ofT +. By virtue of theC-invariance condition(8.16) along with the fact thatN belongs to the center ofcu(1,0|8) so that

(8.51)(N ∗ f ) ∗ g = f ∗ (N ∗ g),


the term containing the star product withN equals to

(8.52)

Ahα,β,γ

((T +)m0vm0

(T 0)R1,m0(a, b)

∣∣m.s.,

[N ∗ ε, (T +)n0vn0

(T 0)R1,n0(a, b)

∣∣m.s.

]∗)

which is zero by the induction assumption valid for anyε. Analogously, one performsinductionn0 → n0 + 1 with the aid of (8.51).

Thus, it suffices to find the coefficients satisfying theC-invariance conditionfor R =R≡R1,0. In other words one has to prove that

(8.53)Ahα,β,γ(R, [ε,R]∗

)= 0

for

(8.54)R(a, b)=H2αβ ∂2

∂aα∂bβResµ

(C(µa +µ−1b

)).

Note that becauseT −(R) = 0 the terms containingc11 (8.12) and, therefore,t (8.15)does not contribute into the condition (8.53).

Using the differential (Moyal) form of the star product (3.2) one finds

Ahα,β,γ (f, η ∗ ε)=∫M5

e12 (c23−c32)

(b12+ b13

)hαβ

(∂2

∂a1α∂a2β+ ∂2

∂a1α∂a3β

)× α((a12+ a13)

(b12+ b13

), (c12+ c13)(c21+ c31),0

)+ (a12+ a13)h

αβ

(∂2

∂bα1∂bβ2

+ ∂2

∂bα1∂bβ3

)× β((a12+ a13)(b12+ b13), (c12+ c13)(c21+ c31),0

)+((c21+ c31)hα

β

(∂2

∂a1α∂bβ2

+ ∂2

∂a1α∂bβ3

)

− (c12+ c13)hαβ

(∂2

∂bα1∂a2β+ ∂2

∂bα1∂a3β

))× γ ((a12+ a13)(b12+ b13), (c12+ c13)(c21+ c31),0

)(8.55)× tr

(f (a1, b1)η(a2, b2)ε(a3, b3)

)∣∣ai=bj=0

provided thatT −f = 0. Let us considerAhα,β,γ (R,R ∗ ε). Rewriting (8.54) as

(8.56)R(ai, bi)=Resµi exp

[µiaiα

∂

∂ciα+µ−1

i bαi∂

∂cαi

]Hαβ ∂2

∂cαi ∂cβi

C(ci)

∣∣∣∣ci=0

,

and using notation

(8.57)k = ∂2

∂c1α∂cα2, ui = ∂2

∂cαi ∂a3α, vi = ∂2

∂ciα∂bα3


along with the identities (3.23) and (3.28) applied to the background fields, one finds

Ahα,β,γ (R,R ∗ ε)=B∫M5

k2H5 Resµ(exp

[12

(µv2 −µ−1u2

)]ϕ(Z)

)(8.58)× tr

(C(c1)C(c2)ε(a3, b3)

)∣∣c1=c2=a3=b3=0,

whereB = 0 is some numerical factor,

(8.59)ϕ(Z)=Z(2γ (Z,−Z)− (α(Z,−Z)+ β(Z,−Z)))and

(8.60)Z = (µk − u1)(µ−1k + v1

).

(Note that the dependence onµ1 in the representation (8.56) for the first factor ofR cancelsout whileµ= µ2 for the analogous representation in the factor ofR in R ∗ ε.)

Analogously, after recycling the product factors under the matrix trace tr and renumer-ating the spinor variables one obtains

Ahα,β,γ (R, ε ∗R)=B∫M5

k2H5 Resµ(

exp[−1

2

(µv1 −µ−1u1

)]ϕ(Y )

)(8.61)× tr

(C(c1)C(c2)ε(a3, b3)

)∣∣c1=c2=a3=b3=0,

where

(8.62)Y = (µk + u2)(µ−1k − v2

).

The problem therefore is to find such a functionϕ(Y ) that

k2 Resµ(

exp[1

2

(µv2 −µ−1u2

)]ϕ(Z)− exp

[−12

(µv1 −µ−1u1

)]ϕ(Y )

)(8.63)× tr

(C(c1)C(c2)ε(a3, b3)

)∣∣c1=c2=a3=b3=0 = 0.

As a first guess let us tryϕ(AB) = Resν(exp[12(ν

−1A+ νB)]). Then the two terms inbrackets in (8.63) amount to

Resµ,ν(

exp[1

2

(µv2 −µ−1u2 + ν−1µk − ν−1u1+ νµ−1k + νv1

)](8.64)− exp

[12

(µ−1u1 −µv1 + ν−1µk + ν−1u2 + νµ−1k − νv2

)]).

These cancel out upon substitutionν ↔ −µ. However, this solution is not completelysatisfactory because the formula (8.59) requiresϕ(Z) to vanish atZ = 0 to have analyticfunctionsα, β , γ .

The following comment is now in order. As discussed in the beginning of this section,throughout this paper we only consider interactions of the higher spin fields with spinss 2. From the perspective of the First On-Mass-Shell Theorem in the form (8.56) thisimplies thatC(c) starts from the fourth-order polynomials in the spinor variablescα , i.e.,

(8.65)C(0)= 0,∂2

∂cα∂cβC(c)

∣∣∣∣cα=0

= 0.


Since the factork2 in (8.63) contains two differentiations both inc1 and inc2, (8.65) meansthat adding a constant toϕ does not affect (8.63). This allows one to cancel out a constantterm inϕ by setting

(8.66)ϕ(A)= ϕ0 Resν(

exp[1

2(ν−1 + νA)]− 1

).

As a result, theon-mass-shell invariance conditionsolves by

2γ (A,−A)− α(A,−A)− β(A,−A)= ϕ0A

−1(

Resν(

exp[1

2

(ν−1 + νA)])− 1

)(8.67)= 1

2ϕ0

1∫0

du(

Resν(ν exp

[12

(ν−1 + νuA)])).

Taking into account (8.47) and (8.48) this is solved by

(8.68)γ (p)= 1

4ϕ0

1∫0

dv vResν(ν exp

[12

(−ν−1 + νvp)])and

α(p,q,0)+ β(p,q,0)

(8.69)= 2γ (p+ q)− 1

2ϕ0

1∫0

duResν(ν exp

[12

(−ν−1 + ν(up+ q))]).Expansion of these expressions forγ (p) andα(p,q,0)+ β(p,q,0) in the power seriesgives (8.21) and (8.22). With aid of these power series expansions one can see that thefollowing identities are true

(8.70)

(p∂2

∂p2 + 3∂

∂p+ 1

4

)γ (p)= 0,

(8.71)

((2+ p ∂

∂p

)∂

∂p+(

1+ q ∂∂q

)∂

∂q+ 1

4

)(α(p,q,0)+ β(p,q,0))= 0.

From (8.28)–(8.30) it follows then that theC-invariance condition(8.34) is satisfied with

α(p,q, t)+ β(p,q, t)= α(p,q,0)+ β(p,q,0),(8.72)γ (p,q, t)= γ (p,q,0).

Thus it is shown that the coefficient functions (8.21) and (8.22) satisfy thefactorizationcondition, C-invariance condition, extra field decoupling conditionand theon-mass-shellinvariance condition. The resulting bilinear form (8.14) defines the action (8.13) thatproperly describes the higher spin dynamics both at the free field level and at the levelof cubic interactions. The leftover ambiguity in the coefficientsα(p,q, t)+ β(p,q, t) andγ (p,q, t) reduces to the overall factorϕ0 that encodes the ambiguity in the gravitationalconstant.


9. Reduced models

So far we discussed the 5d higher spin algebracu(1,0|8) being the centralizer ofN inthe star product algebra. This algebra is not simple as it contains infinitely many idealsIP (N) spanned by the elements of the formP(N) ∗ f for anyf ∈ cu(1,0|8) and any star-polynomialP(N) [26]. Considering the quotient algebrascu(1,0|8)/IP (N) is equivalentto “imposing operator constraints”P(N) = 0. In this section we focus on the algebrahu0(1,0|8) that results fromP(N) =N and its further reductionho0(1,0|8). The algebrahu0(1,0|8) corresponds to the system of higher spin fields of all integer spins with everyspin emerging once.ho0(1,0|8) is its reduction to the system of all even spins. Both ofthese algebras are of interest from the AdS/CFT perspective [25,28].

The explicit construction ofhu0(1,0)= cu(1,0|8)/N via factorization is not particularlyuseful within the star product setup becauseN∗ is the second order differential operator(6.20). A useful approach used in [25] consisted of taking projection by consideringelements of the formf ∗ F wheref was an element ofcu(1,0|8) while F was a certainFock vacuum projector satisfyingN ∗ F = 0. In fact, the left module overcu(1,0|8)generated fromF was shown in [25] to describe 4d conformal fields. In this construction,the factorization ofcu(1,0|8) to cu(1,0|8)/N was automatic. The Fock vacuumF was4d Lorentz invariant and had definite scaling dimension. It is not invariant under theAdS5

Lorentz algebrao(4,1) however, and therefore cannot be used for theAdS5 bulk higherspin gauge theory considered in this paper. On the other hand, from the perspective ofthis paper the Fock module construction is irrelevant. We therefore relax the property thatthe projector is a Fock vacuum for certain oscillators. Instead we shall look for asu(2,2)invariant operatorM satisfying

(9.1)N ∗M =M ∗N = 0,

(9.2)D0(M)= 0.

To satisfy (9.2) we choose a manifestlysu(2,2) covariant ansatzM =M(aαbα). For

any polynomial functionM this would imply that it is a star polynomial ofN . From (9.1)it is clear however thatM cannot be a star product function ofN . Nevertheless, there is aunique (up to a factor) analytic solution forM =M(aαb

α) that solves (9.1). Indeed, from(6.20) it follows that the condition (9.1) has the form

(9.3)−xM(x)+M ′(x)+ 1

4xM ′′(x)= 0, M ′ = ∂M

∂x.

This is solved by

(9.4)M(x)=1∫

−1

dl(1− l2)e2lx,

as one can easily see using(2x − ∂∂l)e2lx = 0 and integrating by parts. Equivalently

(9.5)M(x)=(

1− 1

4

∂2

∂x2

)sh(2x)

x.


Note thatM(x) is even

(9.6)M(−x)=M(x).Having found the operatorM we can write the action for the reduced system associated

with hu0(1,0|8) by replacing the bilinear form in the action with

(9.7)A(f,g)→A0(f, g)=A(f,M ∗ g).Note thatA0(f, g) is well-defined as a functional of polynomial functions (or, formalpower series)f andg for any entire functionM(aαbα) because, for polynomialf andg,only a finite number of terms in the expansion ofM(aαb

α) contributes. The modificationof the bilinear form according to (9.7) with anyM(aαbβ) leads to a new invariant action(8.13). The reason why this ambiguity was not observed in our analysis is that we haveimposed thefactorization conditionin a particular basis of the higher spin gauge fields,thus not allowing the transition to the new bilinear form (9.7).

All other conditions, namely, theC-invariance condition, extra field decouplingconditionand theon-mass-shell invariance conditionremain valid for any entire functionM(aαb

β) inserted into the bilinear form. The factorization condition is relaxed in thissection. Note that theC-invariance conditionguarantees that the bilinear formA0 issymmetric

(9.8)A(f,M ∗ g)=A(f ∗M,g).Inserting a particular functionM(aαbβ) (9.4) we automatically reduce the system to

a smaller subset of fields being linear combinations of the different copies of the fieldsemerged in the originalcu(1,0|8) model. Namely, we can now require all fields in theexpansion (5.6) to be traceless. In other words, the representatives of the quotient algebrahu0(1,0|8) are identified with the elementsg satisfying the traceless condition

(9.9)T −g = 0.

Indeed, by virtue of (6.21) any polynomialg(a, b) ∈ cu(1,0|8) is equivalent to somegsatisfying (9.9) modulo terms containing star products withN which trivialize when actingonM. The star productf ∗ g of any two elementsf andg satisfying the tracelessnesscondition (9.9) does not necessarily satisfy the same condition, i.e.,T −(f ∗ g) = 0(otherwise the elements satisfying (9.9) would form a subalgebra rather than a quotientalgebra). However the difference is irrelevant inside the action built with the help of thebilinear formA0. In particular, the higher spin field strength

(9.10)(dω+ω ∧ ∗ω) ∗Mis equivalent to that of the higher spin algebrahu0(1,0|8).

Thus, the action

(9.11)SEred=1

2AEαβγ (R ∗M,R)

leads to a consistent free field description and cubic interactions for the system of thehigher spin fields associated with the higher spin algebrahu0(1,0|8). The resulting system


describes massless fields of all integer spinss 2, every spin emerges once. The furtherreduction to the subalgebraho0(1,0|8) ⊂ hu0(1,0|8) associated with the subset of evenspins is now trivially obtained by setting to zero all fields of odd integer spins. (For moredetails on the Lie algebraic definition of the corresponding reduction we refer the reader to[25,28].) Note that according to the analysis of Section 8.2 one can consider the dynamicalsystem withn2 fields of each spin, taking values in the matrix algebraMatn. This systemcorresponds to the higher spin algebrahu0(n,0|8). Its reduction toho0(n,0|8) describeshigher spin fields of even spins in the symmetric representation ofo(n) and odd spins inthe adjoint representation ofo(n). (Therefore, no odd spins for the case ofn= 1.)

Note that the conclusions of this section sound somewhat opposite to those of [27] whereit was claimed that the analogous reduction for the 4d conformal higher spin theories isinconsistent.

The following comment is now in order. SinceM is a particular non-polynomial(although entire) function, one has to be careful in treating it as an element of the starproduct algebra which in our setup is regarded either as the algebra of polynomials or offormal power series. SinceM(aαbα) is uniquely defined by the property (9.1) andM ∗Mformally has the same property, one might expect thatM ∗M =mM with some numericalfactor m. Once this would be true, it would be possible to rescaleM to a projectionoperator. However, this is not possible because the parameterm turns out to be infinite.As this issue may be interesting beyond the particular 5d problem studied in this paper,let us consider the general case with the indicesα,β, . . . ranging from 1 to 2n. Eq. (9.3)generalizes to

(9.12)−xM(x)+ 1

2nM ′(x)+ 1

4xM ′′(x)= 0

with the solution

(9.13)M(x)=1∫

−1

dl(1− l2)n−1

e2lx .

An elementary computation then shows that

(9.14)(M ∗M)(x)=1∫

−1

dl

1∫−1

dl′1∫

−1

dk δ

(k − l + l′

1+ ll′)(1− k2)n−1

(1+ ll′)2 e2kx.

From this formula it follows that the expressionM ∗M is ill-defined for anyn because thefactor 1/(1+ ll′)2 gives rise to a divergency at the boundary of the integration region.

Therefore, one cannot treat elements likeM as elements of the star product algebra. Inparticular, this concerns the construction suggested in [28] for the description of the 5dgenerating function for the scalar massless field and higher spin Weyl tensors in terms ofthe fieldsΦ(a,b|x) required to satisfy the conditionN ∗Φ = Φ ∗ N = 0. From what isexplained in this section it is clear that

(9.15)Φ =M ∗ φ = φ ∗M, ∀φ: [φ,N]∗ = 0.


In particularM itself belongs to this class. There is no problem at the linearized level asfar as the variablesΦ are no multiplied, but it is likely to be a problem at the interactionlevel.

Let us stress again that in the construction presented in this section the appearance ofM

in the action functional causes no problem because it is only multiplied with polynomialelements of the higher spin algebra and never with itself.

10. Conclusion

It is shown that 5d higher spin gauge theories admit consistent higher spin interactionsat the action level at least in the cubic order and that, in agreement with the conjectureof [21] and the construction of 4d conformal higher spin algebras of [26], 5d higher spinsymmetry algebra admits a natural realization in terms of certain star product algebras withspinor generating elements. One difference compared to the 4d case is that the 5d higherspin algebracu(1,0|8) contains non-trivial center freely generated by the elementN (3.4).As a result, 5d higher spin algebracu(1,0|8) gives rise to the infinite sets of fields of allspins. That every spin appears in infinite number of copies makes the spectrum of the 5dhigher spin theories reminiscent of the string theory. On the other hand, we have shownthat the factorization of the algebracu(1,0|8) with respect to the maximal ideal generatedbyN , that gives rise to the reduced higher spin algebrahu0(1,0|8) in which every integerspin appears in one copy, admits consistent interactions as well. The same is true for thefurther reduction the algebraho0(1,0|8) discussed in [28], that describes higher spin fieldsof even spins, as well as for the matrix extensionshu0(n,0|8) andho0(n,0|8) that describeeithern2 fields of every integer spin in the case ofhu0(n,0|8) or 1

2n(n+ 1) fields of everyeven spin and12n(n− 1) fields of every odd spin in the case ofho0(n,0|8).

The obtained results are expected to admit a generalization to the supersymmetric case.To this end one extends the set of oscillatorsaα andbβ with the set of Clifford elementsφiandφj (i, j = 1, . . . ,N ) satisfying the commutation relations

(10.1)φi,φj = 0,φi , φj

= 0,φi, φ

j= δij .

The supersymmetric extension of the 5d higher spin algebra is then defined as thecentralizer of

(10.2)NN = aαbα − φiφi .TheN = 1 supersymmetric 5d higher spin theories will be analyzed in [32]. An extensionto N 2 is more complicated because the condition[NN , f ]∗ = 0 allowsf (a, b,φ, φ|x)with |Na −Nb|> 1 that, according to the analysis of Section 4, corresponds to the higherspin potentials with the symmetry properties of theo(4,2) Young diagrams having treerows. Such fields are not related to the totally symmetric tensor and tensor–spinor fieldsdescribed in [21,30] and are expected to correspond to the mixed symmetry freeAdS5 fieldswhich, as shown in [29], are not equivalent to the symmetric fields in theAdS5 backgroundalthough becoming equivalent to some their combinations in the flat limit. Therefore, to


proceed towardsAdS5 supersymmetric higher spin gauge theories it is first of all necessaryto develop an appropriate free field formulation of the mixed symmetry fields in the AdSbackground. This problem is now under study.

Once the formulation of the mixed symmetry fields inAdS5 is developed, it will allowone to consider higher spin theories with allN . These theories are expected to be dual tothe 4d free supersymmetric conformal higher spin theories analyzed in [25]. In [25] it wassuggested that a class of largerCFT4 andAdS5 consistent higher spin theories should existexhibiting manifestsp(8) symmetry. Such theories result from relaxing the condition thatthe AdS5 higher spin algebra is spanned by the elements that commute toNN . Beinganalogous to the 4d higher spin gauge theories based on the algebrashu(n,m|4), thegeneralized higher spin gauge theories based on the algebrashu(n,m|8) are expected tobe dual to thehu(n,m|8) invariant 4d conformal higher spin gauge theories [25]. The5d sp(8) invariant higher spin gauge theories are likely to be generating theories for thereduced models based on the centralizers ofNN in hu(n,m|8) as the ones discussed in thispaper. It is tempting to speculate that the reduction of the higher spin algebrashu(n,m|8)is a result of a certain spontaneous symmetry breaking mechanism with some dynamicalfield ϕ in the adjoint representation of the higher spin algebra that develops a vacuumexpectation valueϕ =NN + · · · .

Acknowledgements

I am grateful to Kostya Alkalaev for careful reading the manuscript and useful commentsand to Lars Brink for the hospitality at the Institute for Theoretical Physics, ChalmersUniversity, where some essential part of this work was done. This research was supportedin part by INTAS, Grant No. 99-1-590, the RFBR Grant No. 99-02-16207 and the RFBRGrant No. 01-02-30024.

Appendix A. Free field equations

In this appendix we give some details on the derivation of the equations of motionthat follow from the quadratic part of the higher spin action. The variation (8.40) can beequivalently rewritten as

δS2 =−1

2

∫M5

ρ(p+ q)([aγ

1∂

∂bγ

1

,∂2

∂a1α∂a2βc12

]+[aγ

2∂

∂bγ

2

,∂2

∂a1α∂a2βc21

])(A.1)×H2αβ ∧ tr

(δω(a1, b1|x)∧R1(a2, b2|x)

)∣∣ai=bj=0.

Taking into account that the Young symmetrizersaγ1 ∂/∂bγ

1 and aγ2 ∂/∂bγ

2 commute top+ q and using the definition of the component fields (6.29), (6.31) we find

δS2 =−∫M5

ρ(p+ q) tr

(∂2

∂a1α∂a2βc12δη

1(a1, b1|x)∧R1(a2, b2|x)


(A.2)+ ∂2

∂a1α∂a2βc21δω(a1, b1|x)∧ r1

1(a2, b2|x))∧H2αβ

∣∣∣∣ai=bj=0

.

Now one observes that the terms containing simultaneouslyη1 andr11 cancel out because

H2αβ is symmetric in the spinor indices. Therefore

δS2 =−∫M5

ρ(p+ q)c12∂2

∂a1α∂a2βH2αβ

(A.3)∧ tr(δη1(a1, b1|x)∧ r0

1(a2, b2|x)+ r11(a1, b1|x)∧ δη0(a2, b2|x)

)∣∣ai=bj=0.

Inserting here the component expansions

(A.4)ηi(a, b|x)=∞∑

u,v=0

1

u!v!δ(2i − u+ v)ηiα1...αu

β1...βv (x)aα1 · · ·aαubβ1 · · ·bβv ,

(A.5)ri1(a, b|x)=∞∑

u,v=0

1

u!v! δ(2i − u+ v)riα1...αu1 β1...βv (x)aα1 · · ·aαubβ1 · · ·bβv ,

(A.6)ρ(p)=∞∑n=0

ρn

n! pn,

where, taking into account (8.22) and (8.47),

(A.7)ρn = (−1)n1

22n+2(n+ 1)! ,and completing the differentiations one gets

δS2 =−∫M5

∞∑u,v=0

ρu+vu!v! (−1)uH2αβ

∧ tr(δη1αγ1...γu+1κ1...κv

,σ1...σuρ1...ρv ∧ r0

1βσ1...σuκ1...κv,γ1...γu+1ρ1...ρv

(A.8)

+ r1αγ1...γu+1κ1...κv1 ,

σ1...σuρ1...ρv ∧ δη0βσ1...σuκ1...κv,γ1...γu+1ρ1...ρv

)∣∣ai=bj=0.

Using (A.7) and the Young properties of the component fields and curvatures one obtains

δS2 =−1

2φ0

∫M5

∞∑n=0

(−1)n2−2n 1

(n− 1)!n!H2αβ

∧ tr(δη1αγ1...γn

,σ1...σn−1 ∧ r0

1γ1...γn,βσ1...σn−1

(A.9)+ r11αγ1...γn

,σ1...σn−1 ∧ δη0

γ1...γn,βσ1...σn−1

)∣∣ai=bj=0.

The free equations of motion corresponding to the variation with respect to Lorentz-typefieldsη1 and the frame-type fieldη0 have the following component form, respectively,

(A.10)0=Hα1σ r01α2...αm+1,

σβ1...βm−1 −

m− 1

2(m+ 1)Vα1β1Hγσ r

01α2...αm+1,β2...βm−1

γ σ


and

0=Hα1γ

(r11β1...βmγ,α2...αm +

m− 1

3r11β1...βmα2,α3...αmγ

)−mHγ

β1

(r11β2...βmα1γ,α2...αm +

m− 1

3r11β2...βmα1α2,α3...αmγ

)+ m

m+ 1Vα1β2H

γσ

(r11β2...βmγ σ,α2...αm + (m− 1)r1

1β2...βmγ α2,σα3...αm

(A.11)+ 1

6(m− 1)(m− 2)r1

1β2...βmα2α3,σγ α4...αm

).

(As usual, the symmetrization of the indices denoted by the same Greek letters is assumed.)

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Classically perfect gauge actions onanisotropic lattices

Philipp Rüfenachta, Urs Wengerb,a Institute for Theoretical Physics, University of Bern, Sidlerstrasse 5, CH-3012 Bern, Switzerland

b Theoretical Physics, Oxford University, 1 Keble Road, Oxford OX1 3NP, United Kingdom

Received 9 August 2001; accepted 13 September 2001

Abstract

We present a method for constructing classically perfect anisotropic actions for SU(3) gaugetheory based on an isotropic Fixed Point Action. The action is parametrised using smeared (“fat”)links. The construction is done explicitly for anisotropyξ = as/at = 2 and 4. The correspondingrenormalised anisotropies are determined using the torelon dispersion relation. The renormalisationof the anisotropy is small and the parametrisation describes the true action well. Quantities such as thestatic quark–antiquark potential, the critical temperature of the deconfining phase transition and thelow-lying glueball spectrum are measured on lattices with anisotropyξ = 2. The mass of the scalar0++ glueball is determined to be 1580(60) MeV, while the tensor 2++ glueball is at 2430(60) MeV. 2001 Published by Elsevier Science B.V.

PACS:11.15.Ha; 12.38.Gc; 12.39.MkKeywords:Lattice gauge theory; Perfect action; Anisotropy; Fat links; Torelons; Static potential; Glueballs

1. Introduction

In lattice QCD the energy of a physical state is measured by studying the decay ofcorrelators of operators which have a non-zero transition matrix element between thevacuum and the state under consideration. If the state is heavy, these correlators decayvery fast in Euclidean time and one has to make sure that the temporal lattice spacing issmall enough such that the signal of the correlator can be accurately traced over a few timeslices before it disappears in the statistical noise. At the same time, one has to pay attentionto choose the physical volume of the lattice large enough so that there are no significantfinite-size effects. Both these requirements together lead to lattices with a large number

Work supported in part by Schweizerischer Nationalfonds.E-mail addresses:[email protected] (P. Rüfenacht), [email protected] (U. Wenger).



164 P. Rüfenacht, U. Wenger / Nuclear Physics B 616 (2001) 163–214

of lattice sites and hence to computationally very expensive simulations. The obvious wayout of this dilemma is to use a smaller lattice spacing in temporal direction compared tothe spatial directions, i.e., the use of anisotropic lattices.

Anisotropic lattices have been widely used over the last few years. Studies comprisingexcited states of nucleons [1], heavy-quark bound states [2–4], heavy–light mesons [5,6],heavy meson semi-leptonic decays [7], long range properties and excited states of thequark–antiquark potential [8–11] as well as states composed purely of gluons (glueballs)[12–15] or gluons and quarks (hybrids) [16–21], have been performed, mainly usingthe standard anisotropic Wilson discretisation or a mean-link and Symanzik improvedanisotropic action [22]. In fact, improved actions turned out to be crucial in anisotropiclattice calculations due to the coarse spatial lattice spacings employed. The discretisationof the continuum action, however, can be done in many different ways leading to differentimprovement schemes. A radical approach of improving lattice actions, based on Wilson’sRenormalisation Group ideas, has been suggested by Hasenfratz and Niedermayer [23],namely the use of actions that are classically perfect, i.e., there are no lattice artifacts on thesolutions of the lattice equations of motion. For SU(3) gauge theory the classically perfectFP action has been constructed and tested in [24–28] and the ansatz has been extended toinclude FP actions for fermions as well [29–33]. In the case of SU(2) gauge theory theFP action has been constructed in [34–36] and its classical properties have been tested onclassical instanton solutions, both in SU(2) and SU(3) [37]. A classically perfect gaugeaction on anisotropic lattices, however, has so far been absent.

It is thus the goal of this work to fill the gap and to present the construction of classicallyperfect gauge actions on anisotropic lattices. The method has been introduced in [38] andit is presented in more detail in [39]. It relies upon a recent parametrisation of the isotropicclassically perfect Fixed Point (FP) action [28,40,41] that includes a rich structure ofoperators as it is based on plaquettes built from simple gauge links as well as from smeared(“fat”) links. We explicitly construct theξ = 2 and 4 actions on coarse configurationstypically occurring in Monte Carlo (MC) simulations. The properties of theξ = 2 actionare studied by performing measurements of the torelon dispersion relation (which servesfor determining the renormalisation of the bare (input) anisotropy), of the static quark–antiquark potential, of the deconfining phase transition and finally of the spectrum of low-lying glueballs in pure gauge theory.

It turns out that the construction of anisotropic classically perfect gauge actions isfeasible. Measuring the renormalised anisotropy using the torelon dispersion relation turnsout to be stable and unambiguous and shows that the renormalisation of the anisotropy issmall and under good control. The measurements of the static quark–antiquark potentialindicate that the violations of rotational symmetry are small if the (spatial) lattice is notexceptionally coarse. This shows that the parametrisation describes accurately the trueclassically perfect action, which is known to have good properties concerning rotationalsymmetry [27]. The study of the glueball spectrum is facilitated a lot due to the anisotropicnature of the action, even for (rather small) anisotropyξ = 2. Results, including continuumextrapolations, are obtained for glueball states having much larger mass than the highest-lying states that could have been resolved with the same amount of computational work

P. Rüfenacht, U. Wenger / Nuclear Physics B 616 (2001) 163–214 165

using the isotropic FP action. However, with the statistics reached so far, it is not yetpossible to conclude whether the scaling properties of the glueball states, i.e., the behaviourof the measured energies as the lattice spacing is changed, are definitely better for theclassically perfect action as compared to the mean-link and Symanzik improved anisotropicaction [12,13]. Especially, this is the case for the lowest-lying scalar glueball where thepresence of a critical end point of a line of phase transitions in the fundamental–adjointcoupling plane, near the fundamental axis, causes large distortions of the scaling behaviour(sometimes called the “scalar dip”). Our action includes in its rich structure operatorstransforming according to the adjoint representation. If their coupling (which we do notcontrol specifically during the construction and parametrisation) lies in a certain region,the effect of the critical end-point on scalar quantities at certain lattice spacings may evenbe enhanced compared to other (more standard) discretisations with purely fundamentaloperators. Concerning the lattice artifacts observed in the glueball simulations, we have toadd that other effects, e.g., due to the finite size of the lattices, may also be present andthat this issue requires further study. Being the first application of the FP action techniqueto anisotropic lattices, however, it goes beyond the scope of this work to systematicallycheck all the possible sources of errors in the glueball mass determinations. Furthermore,we note that the computational overhead of the parametrised anisotropic classically perfectaction compared to the standard Wilson action as well as to the mean-link and Symanzikimproved action is considerable.

The paper is organised as follows. In Section 2 we describe our method of generatinganisotropic perfect gauge actions without having to perform a certain number ofrenormalisation group transformations (RGT) leading from very fine to coarse latticestypical for MC simulations. Instead, we make use of the parametrisation of the isotropicFP action presented in [28,40,41]. We start with theisotropic parametrised FP action onrather large fluctuations and perform a small number of purelyspatialblocking steps. If ascale 2 block transformation is used, the resulting actions have anisotropiesξ = 2,4,8, . . . .The parametrisation ansatz used for these actions is a generalisation of the isotropicparametrisation and is described as well.

Section 3 contains the main body of the present work, that is, the explicit constructionand the application of a classically perfect gauge action onξ = 2 anisotropic lattices. Theresults from measurements of physical quantities are presented there: the anisotropy of theaction is measured using the torelon dispersion relation, the (spatial) scale is determinedby the static quark–antiquark potential, where for some values of the couplingβ , off-axisseparations of the quarks are considered as well, in order to estimate violations of rotationalsymmetry by the parametrised action. Another way of determining the (temporal) scale isthe study of the deconfining phase transition, measuring the critical couplingsβcrit fordifferent temporal extensionsNt of the lattice. Finally, the glueball spectrum is determinedby performing simulations at three different lattice spacings using theξ = 2 action andtaking the continuum limits whenever it is possible.

The feasibility of iterating the procedure, i.e., adding one more spatial blocking stepwhich leads to a classically perfect action with anisotropyξ = 4, is briefly checkedin Section 4. It contains the explicit construction and parametrisation of theξ = 4


action and the determination of the renormalised anisotropy via the torelon dispersionrelation.

Section 5 finally contains a summary of the main results, some conclusions and anoutlook to future work.

The parameters of the classically perfect actions used throughout this work are collectedin Appendix A, while Appendix B contains the detailed results from our simulations.

2. The construction of perfect anisotropic actions

2.1. Introduction

The main idea for the construction of (effective) actions on anisotropic lattices is tostart from an isotropic lattice and to perform one or more purely spatial renormalisationgroup transformations (RGT). Each scale two RGT step doubles the spatial lattice spacingbut preserves the spacing in temporal direction. We are therefore naturally lead from anaction on an isotropic lattice to (effective) actions on anisotropic lattices with anisotropiesξ = 2,4,8, . . . . Of course, choosing the scales of the RGT different from two, we can inprinciple reach any desired anisotropyξ = s, s2, s3, . . . .

A qualitative picture of the construction of classically perfect actions on anisotropiclattices is the following. We start from an isotropic FP action defined as a point in theinfinite-dimensional space of all couplings on the critical surface where the correlationlength is infinite (β → ∞). By performing one (or more) spatial RGT we move awayfrom that point always staying on the critical surface. Since the spatial blocking kernelis zero on all classical configurations, the classical properties of the action are preservedexactly, i.e., it has no lattice artifacts on the solutions of the (anisotropic) lattice equationsof motion. (One can say that the action is an on-shell tree-level Symanzik improved actionto all orders ina.) It is, however, important to note that the resulting anisotropic classicallyperfect (AICP) action is, generally speaking, no longer a FP action, i.e., the fixed point ofa renormalization group transformation. However, it still possesses all the good propertiesof a classically perfect action, in particular it preserves scale invariant solutions of theclassical equations of motion, i.e., classical instanton solutions. Moreover, since the spatialRGT steps are performed on the critical surface atβ → ∞, the transformation reducesto a saddle point problem representing an implicit equation for the anisotropic classicallyperfect action similar to the FP equation in the isotropic case.

In order to check whether this qualitative picture is correct and that the constructionworks in practice, we have analytically calculated the AICP actions ford = 2 scalarfield theory (for earlier studies on this subject, see [42]) as well as for the quadraticapproximation tod = 4 gauge theory for generic anisotropiesξ . In these two cases it turnsout that starting from an anisotropic action for very smooth configurations and performingisotropic RGTs or starting from an isotropic FP action and performing an anisotropic RGTat the end is equivalent and leads to the same classically perfect anisotropic action. It alsoturns out that the quality of the results are as good as what has been obtained with the


isotropic FP action [24,27]. In particular we do not see any deviation from the classicalcontinuum dispersion relation and almost no violation of the rotational symmetry in theperturbative potential forr/a > 1. For further details we refer to [39].

In practice the main advantage of applying one (or more) anisotropic spatial RGT isthat we can directly start from a recent parametrisation of the isotropic FP gauge action[28,40,41] which is valid on rather coarse lattices. We can therefore avoid the cumbersomecascade procedure leading from very fine to coarser and coarser lattices.

In the following we first describe the anisotropic spatial blocking in Section 2.2 and thenpresent an extension of the isotropic parametrisation to include anisotropy in Section 2.3.

2.2. The spatial blocking

The isotropic FP action is defined by the saddle point equation (atβ = ∞):

(1)AFP(V )= minU AFP(U)+ T (U,V )

,

whereT (U,V ) is the blocking kernel connecting the fine gauge configurationsU to thecoarse configurationsV :

(2)T (U,V )=∑nB,µ

(N∞µ (nB)− κ

NRe Tr

[Vµ(nB)Q

†µ(nB ;U)]

),

where the normalisation termN∞µ (nB) is given by

(3)N∞µ (nB)= max

W∈SU(N)

κ

NRe Tr

[WQ†

µ(nB ;U)],

whileQµ(nB;U) is the blocked link,

(4)Qµ(nB ;U)=Wµ(2nB;U)Wµ(2nB + µ;U).HereWµ(n;U) denotes a smeared (fuzzy) fine link, which is constructed as a sum ofsimple staples as well as of diagonal staples along the planar and spatial diagonal directionsorthogonal to the link (RGT 3 transformation) [27], respectively.

To perform the blocking only spatially, doubling the lattice spacing in spatial directionwhile leaving the temporal lattice spacing unchanged, we use a purely spatial blockingkernelTsp(U,V ) obtained by setting in the isotropic definition

(5)Q4(nB ;U)=W4(2nB;U)thus doing only a smearing and no blocking in temporal direction. (In the case of spatialblocking the expression “2nB” stands for(2n1

B,2n2B,2n

3B,n

4B).) In addition, we have the

freedom to choose different values ofκ for the spatial (κs ) and temporal (κt ) blocking.

2.3. The parametrisation

The classically perfect actions, described in principle by an infinite number of couplings,have to be parametrised in order to be suitable for numerical simulations. We have shown


previously [28,40,41] that a parametrisation of the isotropic FP gauge action includingAPE-like smearing behaves much better and is much more flexible compared to commonparametrisations using traces of closed loops with comparable computational cost. We thusdecide to use an extension of this parametrisation to describe the anisotropic classicallyperfect gauge action.

For convenience, let us quickly review the parametrisation of the isotropic action.The parametrisation uses mixed polynomials of traces of simple loops (plaquettes) builtfrom single gauge linksUµ(n) as well as from asymmetrically (APE-like) smeared links

W(ν)µ (n). We introduce the notationS(ν)µ (n) for the sum of two staples of gauge links

connecting two lattice sites in directionµ lying in theµν-plane:

(6)S(ν)µ (n)=Uν(n)Uµ(n+ ν)U†ν (n+ µ)+U†

ν (n− ν)Uµ(n− ν)Uν(n− ν + µ).

To build a plaquette in a planeµν from smeared links it is convenient to introduceasymmetricallysmeared links. First define1

(7)Q(ν)µ = 1

4

( ∑λ =µ,ν

S(λ)µ + ηS(ν)µ

)−

(1+ 1

2η

)Uµ.

Out of these sums of matrices connecting two neighboring pointsn, n + µ, we build theasymmetrically smeared links

(8)W(ν)µ =Uµ +

∑i=1

ciQ(ν)µ

(U†µQ

(ν)µ

)i−1.

The parametersη, ci used for the smearing may depend on local fluctuations measuredby xµ(n) defined as

(9)xµ(n)= Re Tr(Qs

µ(n)U†µ(n)

),

with the symmetrically smeared link

(10)Qsµ(n)= 1

6

∑λ =µ

S(λ)µ (n)−Uµ(n).

This parameter is negative,−4.5 xµ(n) 0, and it vanishes for trivial gaugeconfigurations.

The smearing parameters are chosen to be polynomials ofxµ with free coefficients(determined later by a fit to the FP action):

(11)η= η(0) + η(1)x + η(2)x2 + · · · ,(12)ci = c

(0)i + c

(1)i x + c

(2)i x2 + · · · .

Of course, the asymmetrically smeared linksW(ν)µ built out of a large number of paths

connecting the neighbouring lattice sites are no longer elements of the SU(3) gauge group.They might be projected back to SU(3), however this task increases the computational costin actual numerical simulations. Moreover, our studies have shown that projection reduces

1 The argumentn is suppressed in the following.


the degrees of freedom in defining the action and we are thus using the smeared linksW(ν)µ

as they are.We now build a smeared plaquette variable,

(13)wµν = Re Tr(1−W

plµν

),

as well as the ordinary one,

(14)uµν = Re Tr(1−U

plµν

),

where

(15)Wplµν(n)=W(ν)

µ (n)W(µ)ν (n+ µ)W(ν)†

µ (n+ ν)W(µ)†ν (n)

and

(16)Uplµν(n)=Uµ(n)Uν(n+ µ)U†

µ(n+ ν)U†ν (n),

respectively. Finally, the action is built from these plaquette variables using a mixedpolynomial ansatz of the form

(17)A[U ] = 1

Nc

∑n

∑µ<ν

∑k,l

pkluµν(n)kwµν(n)

l,

where the coefficientspkl are again free parameters defined by a fit to the FP action.In order to adapt this parametrisation of the isotropic FP action to anisotropic actions,

we use three (rather straightforward) extensions. Firstly, the coefficientspkl in Eq. (17)are chosen differently depending on the orientation of the plaquetteµν, i.e., psp

kl forµν ∈ 12,13,23 (spatial plaquettes) andptm

kl for µν ∈ 14,24,34 (temporal plaquettes).Secondly, the parameterη entering in Eq. (7) describing the asymmetry between differentlyoriented staples contributing to a smeared linkW

(ν)µ is generalised to distinguish between

smeared spatial and temporal links:

(18)Q(j)

i = 1

4

( ∑k =i,j

S(k)i + η1S

(j)

i + η3S(4)i

)− 1

2(1+ η1 + η3)Ui,

(19)Q(4)i = 1

4

( ∑λ =i,4

S(λ)i + η4S

(4)i

)−

(1+ 1

2η4

)Ui,

(20)Q(j)4 = 1

4

( ∑λ =4,j

S(λ)4 + η2S

(j)4

)−

(1+ 1

2η2

)U4,

wherei, j, k = 1,2,3 andµ,ν,λ = 1, . . . ,4. The anisotropic parametersη1, . . . , η4 maybe again polynomials in the local fluctuation parameterxµ. These situations are depictedin Fig. 1.

Finally, the construction of the smeared linksW(ν)µ from the matricesQ(ν)

µ , described bythe parametersci in Eq. (8), is generalised such that these parameters are chosen differentlyfor the construction of temporal links (always contributing to temporal plaquettes), spatiallinks contributing to spatial plaquettes and spatial links contributing to temporal plaquettes,ci1, ci2 andci3, respectively, see Fig. 2.


Fig. 1. Asymmetry in the construction of the smeared link matricesQ(ν)µ . From left to right: spatial

staple contributing to a smeared spatial link in a spatial plaquette (η1), temporal staple contributingto a smeared temporal link (η2), temporal staple contributing to a smeared spatial link in a spatialplaquette (η3), temporal staple contributing to a smeared spatial link (η4).

Fig. 2. Asymmetry in the construction of the smeared linksW(ν)µ from the matricesQ(ν)

µ . From left

to right: spatialQ(j)i

contributing to a smeared spatial plaquette (ci1), spatialQ(4)i

contributing to a

smeared temporal plaquette (ci2), temporalQ(j)4 (ci3).

Table 1Comparison of the accuracyχ2

d of the fit on derivatives on 20 configurations atβ = 3.5 and themeasured renormalised anisotropyξR (using the torelon dispersion relation) atβ = 3.3 for differentchoices of the set of non-linear parameters

η ci n max(k + l)sp max(k + l)tm χ2d ξR

4 1 4 4 4 0.0250 1.63(2)2 3 3 4 4 0.02384 3 3 4 4 0.0144 1.912(9)

It is not a priori clear whether all the three extensions are necessary at the same time.Results from fits to different extended parametrisations (Table 1), however, show that it isindispensable to have the full parameter set. The values ofχ2 of the fit to the true classicallyperfect actions as well as the measured renormalised anisotropies of the parametrisedaction (see Section 3.2) indicate that the parametrisation is not flexible enough neither forη3 = η4 ≡ 0 (leading to a positive definite transfer matrix connecting only neighbouringtime slices) nor if the parameterscij , j = 1,2,3, are set to be equal.

Our anisotropic action shall have the correct normalisation and anisotropy in thecontinuum limit. This can be accomplished by demanding the following two normalisationconditions to be exactly fulfilled [39]:


(21)pst01 + pst

10 + 2pst01c13 + pst

01c13η2 + 2pss01c11η3 +pst

01c12η4 = ξ,

(22)pss01 + pss

10 + 2pss01c11 + 2pst

01c12 + 2pss01c11η1 = 1

ξ,

where all parametersη, c denote the constant (0th order) terms in the polynomials inxµ.Let us note that the parametrisation presented here leads to an overhead in the updates

in MC simulations of 66 compared to the Wilson action and of 12 compared to the tadpoleand tree-level improved action.

3. The ξ = 2 perfect action

3.1. Construction

We construct theξ = 2 classically perfect anisotropic action starting from theparametrised isotropic action [28] performing one spatial blocking step as described inSection 2. Studies of the quadratic approximation suggestκt = ξ2κs = 4κs in the RGtransformation for keeping the block transformation as close to the isotropic case aspossible. Furthermore, spatial locality should stay at its optimum for the “isotropic” valueof κs = 8.8 [27], thus we chooseκt = 35.2. Indeed, it turns out that varyingκt awayfrom κt = 35.2, keepingκs = 8.8 fixed, makes it much more difficult to parametrise theresulting perfect action such that the renormalised anisotropyξR stays close to the inputvalueξ = 2. Coarse configurationsV are generated using the ad-hoc anisotropic action(see Appendix A.2), the FP Eq. (1) including the purely spatial blocking kernelTsp(U,V )

is used as a recursion relation with the intermediate isotropic action (see Appendix A.1)on the r.h.s. yielding the classically perfectξ = 2 action on the l.h.s. Examining theblocking, we observe that coarse configurations obtained in a MC run with the ad-hocanisotropic action are mapped to fine configurations that are close to isotropic (concerningspatial and temporal plaquette values and expectation values of the Landau gauge-fixedlink variables) for input anisotropyξad-hoc ≈ 3.2. We thus generate 20 configurationseach atβad-hoc= 2.5,3.0,3.5,4.0 using the ad-hoc anisotropic action2 with ξad-hoc= 3.2.These configurations are spatially blocked using the intermediate isotropic action (seeAppendix A.1) which describes the minimised isotropic configurations on the r.h.s. ofEq. (1) reasonably well.

To construct the action, we perform several non-linear fits for the derivatives of 20configurations atβad-hoc= 3.5, using different sets of parameters. The parametersci arechosen to be non-zero fori = 1,2,3 as adding additional freec4 parameters does notimproveχ2 significantly.

2 Note, that the values ofβad-hoc, used in this section, may not be directly compared to the values ofβperfcorresponding to the parametrised classically perfect action, used in the following sections; but rather thefluctuations (plaquette values, expectation values of Landau gauge-fixed links etc.) of the gauge configurationsproduced in MC runs using the perfect action should lie approximately in the same range as the fluctuations ofthe initial coarse configurations produced with the ad-hoc action.


Having fixed the non-linear parametersη and ci , we include the derivatives of 20configurations atβad-hoc= 4.0 and the action values of all the configurations atβad-hoc=3.5,4.0 in the linear fit of the parameterspkl , keeping the non-linear parameters fixed.The resulting values ofχ2 and the linear behaviour of the actions suggest a linear set withmax(k + l)sp = 4, max(k + l)tm = 3 where the action values are included with weightswact = 0.018 relative to the derivatives. Using this value, the good parametrisation of thederivatives obtained in the full nonlinear fit is preserved, at the same time the mean errorof the action value due to the parametrisation is as small as 0.35%. It is checked that thenumber of data points in the fit is large enough, making sure that the values ofχ2 as well asthe errors of the action values do not increase significantly on configurations independent ofthose used in the fit. Furthermore, these quantities are determined on configurations downto βad-hoc= 3.0 and it turns out that the action works down to this value ofβad-hoc. Finally,it is checked that the linear parameterspkl do not lead to artificial gauge configurations inMC simulations caused by (local) negative contributions to the total action. The parametersof the resulting action are listed in Appendix A.3.

3.2. The renormalised anisotropy

The renormalised anisotropyξR of the action presented above is measured using thetorelon dispersion relation as described in [43,44]. The torelon is a closed gluon flux tubeencircling the periodic spatial boundary of sizeL of a torus. It is created and annihilatedby closed gauge operators winding around this boundary. String models which are a goodapproximation for long flux tubes predict its energy to behave as

(23)E(L)= σL+ π

3L,

whereσ is the string tension and the second term describes the string fluctuation for abosonic string [45–48]. The string models assume no restriction on the transverse modes ofthe string, thus they apply to volumes with large transverse extensionS. Due to this reasonand because we prefer not to have too large transverse momenta, we work onS2 ×L× T

lattices, whereS andT are large, whereas the length of thez-directionL is chosen to berather short such that the torelon energy (see Eq. (23)) is not too heavy. Since the colourflux tube has finite width in physical units, it is useful to employ (iteratively) APE smearedlinks [49] to improve the overlap of the operators with the ground state. One step of APEsmearing acts on a spatial gauge linkUj(x) as

S1Uj(x)≡PSU(3)

Uj(x)+ λs

∑k =j

(Uk(x)Uj (x + k)U

†k (x + j )

(24)+U†k (x − k)Uj (x − k)Uk(x − k + j )

),

the original link variable is replaced by itself plus a sum of the four neighbouring spatialstaples and then projected back to SU(3). The smeared and projected linksS1Uj(x)

have the same symmetry properties under gauge transformations, charge conjugationand reflections and permutations of the coordinate axes as the original gauge links. The


smearing is used iteratively allowing to measure operators on different smearing levels,SnU .

We measure the operators

(25)Tn(x, y, t)= TrL−1∏z=0

SnUz(x, y, z, t),

for all x, y, t , whereSnUz is the n times iteratively smeared link in the longitudinalz direction. By discrete Fourier transform, we project out the state with momentump = (px,py)= (nx, ny)(2π/Sas):

(26)Tn(p, t)=∑x,y

Tn(x, y, t)ei(pxx+pyy)

and build the correlators

(27)Cnn′ (p, t)= 1

Nt

∑τ

⟨Tn(p, τ )Tn′(−p, τ + t)

⟩.

From these correlators of operators measured on the different smearing levels we obtain theenergy valuesatE(p) using variational methods (see Appendix B of [28]). The continuumdispersion relation

(28)E2 = p2 +m2

on the lattice becomes (in temporal units)

(29)(atE)2 = a2

t

(p2 +m2) = (asp)2

ξ2R

+ (mat )2 = 1

ξ2R

(n2x + n2

y

)(2π

S

)2

+ (mat )2,

wherenx , ny are the components of the (transversal) lattice momentum. On an anisotropiclattice, this equation allows for the extraction of the renormalised anisotropyξR = as/at

(measuring the “renormalisation of the speed of light”) as well as the torelon massmat ,which in turn may be used to get an estimate of the scale using Eq. (23) and known valuesof the string tensionσ . The main advantage of this approach is obtaining the renormalisedanisotropyξR as well as an estimate for the scalesas andat by performing computationallyrather inexpensive measurements.

The parameters of the simulations carried out are collected in Table 2.The Polyakov line around the short spatial direction is measured using operators

at smearing levelsSnU with n = 3,6,9,12,15 for β 3.3 andn = 2,4,6,8,10 forβ 3.15, respectively, and smearing parameterλs = 0.1. The measured energiesE(p2)

are given, together with the number of operators used in the variational method, the fitranges and the values ofχ2 per degree of freedom,χ2/NDF, in Table 24 in Appendix B.

Fig. 3 displays an example of a dispersion relation atβ = 3.3 including all values ofp atwhich the energies could be determined on the given lattice. Note, that the energies of thep = 0 torelons (the torelon masses) may be hard to determine because the effective massesdo not reach a plateau within the temporal extent of the lattice, probably due to the small


Table 2Run parameters for the torelon measurements using theξ = 2 perfect action. The lattice extensionsin torelon directionL, the extension in the two transversal spatial directionsS as well as the temporalextensionL are given

β S2 ×L× T Sweeps/Measurements

3.00 82 × 4× 20 72000/144003.15a 82 × 5× 20 54000/108003.15b 52 × 8× 20 47600/95203.30 122 × 6× 30 35600/71203.50a 122 × 8× 24 39400/78803.50b 142 × 6× 30 36800/7360

Fig. 3. Torelon dispersion relation forβ = 3.3. The straight line is the correlated fit toE2(p)=m2

T+p2 in the rangep2 = 1–9.

overlap of the operators used with these states; however, using the non-zero momentumenergies, the masses may still be accurately determined.

To determine the renormalised anisotropyξR as well as the torelon mass in units ofthe temporal lattice spacingmat , we perform fits to the dispersion relation, taking intoaccount the correlations between different operators. The range ofp2 considered is chosendepending onχ2/NDF of the fit and the precision of the dispersion relation data in therespective range. The results are given in Table 3.

Using the finite size relation for the torelon mass corresponding to Eq. (23),

(30)MT (Las)=(σ + D

(Las)2

)Las,

we calculate the string tensionσ and thus the spatial scaleas . The string model predictsD = −π/3 for long strings thence we use this string picture value and stay aware that


Table 3Results of the torelon simulations using theξ = 2 perfect action. The fit range inp2 is given in unitsof (2π/S)2

β Fit range ξR mTat χ2/NDF

3.00 0–5 1.903(81) 1.324(98) 0.393.15a 0–4 1.966(39) 0.700(35) 0.833.15b 0–5 2.022(104) 1.262(84) 0.623.30 1–9 1.912(9) 0.311(5) 0.943.50a 1–10 1.836(9) 0.149(10) 0.943.50b 1–8 1.826(16) 0.208(16) 1.61

Table 4Estimates of the scale determined from torelon results (see Table 3 and text)

β√σ as r0/as as (fm) at (fm)

3.00 0.834(20) 1.43(5) 0.350(12) 0.184(14)3.15a 0.563(8) 2.12(5) 0.236(6) 0.120(5)3.15b 0.579(18) 2.06(8) 0.243(9) 0.120(11)3.30 0.3578(21) 3.33(5) 0.150(2) 0.079(1)3.50a 0.225(5) 5.31(16) 0.094(3) 0.051(2)3.50b 0.304(7) 3.92(12) 0.128(4) 0.070(3)

the estimate gets worse for short strings. To obtain the hadronic scaler0 and the latticespacings we employr0

√σ = 1.193(10) from [28] and use the definitionr0 = 0.50 fm. The

results are collected in Table 4.The two simulations denoted by 3.15b and 3.50b are used to check the stability of the

method. The first one is carried out on a lattice with small transversal spatial size (thetorelon string measured is even longer), the latter is measuring the energies of a rathershort string. The scale is measured accurately using the static quark–antiquark potential(see Section 3.3.1) so that we can compare our torelon estimates to the much more reliablevalues in Table 5. It turns out that the deviation of the torelon estimates for the scaler0/as

from the potential values does not exceed 6% if the length of the string is not too short (sayLas 1 fm). For short strings, the string picture does not apply and we may therefore notexpect good results in this case. But even for very long strings, it is not possible to obtainvery accurate information about the scale as long strings are very heavy and thus difficultto measure (comparable to the long range region of the static quark–antiquark potential),especially when the anisotropyξ is not very large.

There seems to be no problem if the transverse volume is rather small as in theβ = 3.15b

simulation, except of course the large momenta occurring which make the determinationof the energies more difficult. As well, the estimates of the renormalised anisotropyξR do not show significant deviations neither for small transverse volume nor for shortstrings.


Table 5The estimates of the scale obtained from torelon measurements (see Table 4) compared to the scalemeasured using the static quark–antiquark potential (see Table 8). The length of the torelon stringLas in physical units is given as well

β Las (fm) r0/as (torelon) r0/as (potential) Rel. error

3.00 1.48 1.43(5) 1.353(16) +5.7%3.15a 1.23 2.12(5) 2.038(2) +4.0%3.15b 1.96 2.06(8) 2.038(2) +1.1%3.30 0.95 3.33(5) 3.154(8) +5.6%3.50a 0.82 5.31(16) 4.906(21) +8.2%3.50b 0.61 3.92(12) 4.906(21) −20.0%

Finally, we may conclude that determining the renormalised anisotropy using the torelondispersion relation is a stable and apparently sensible procedure. There are no manifestproblems and it works on fine lattices as well as on coarse ones. Checks employing differentlattice extensions indicate that the determination of the renormalised anisotropy is quitestable within a large range of different choices. The estimate of the lattice scale, however,is not very accurate, mainly due to the use of the string picture relation, Eq. (30); the(systematic) error of the torelon scale is about 5% for reasonable torelon lengthsL 1 fmwith a tendency of underestimating the lattice spacingas .

3.3. The static quark–antiquark potential

3.3.1. The scaleTo set the scale of the action at a givenβ-value we measure the static quark–antiquark

potential, which is reasonably well described by the phenomenological ansatz [50],

(31)V (R)= V0 + α

R+ σR.

We determine the lattice spacing using the following definition ofrc [51]:

(32)r2V ′(r)∣∣r=rc

= c,

where the well-known Sommer scaler0 ≈ 0.50 fm corresponds toc = 1.65. For theanisotropic action, we employ the conventional definition ofr0, however,V (r) is nowmeasured in units of the temporal lattice spacing and thus we have

(33)−α + ξ σ

(rc

as

)2

= c,

where σ denotes the fitted dimensionless valueσ = σasat of the string tension. Therenormalised anisotropyξ thus has to be known before the spatial scaler0 may bedetermined. For very fine or coarse lattices it may be advantageous to go to a slightlydifferent separation corresponding to a different value of the constantc on the r.h.s. Forthis purpose, we have collected values forc andrc from high-precision measurements ofthe static potential performed with the Wilson action [52–54] listed in Table 6.


Table 6Parameter values for the determination of the hadronicscalerc throughr2V ′(r)|r=rc = c

rc/r0 c

0.662(1) 0.891.00 1.651.65(1) 4.002.04(2) 6.00

The static quark–antiquark potential is measured on lattices of different scalesas =0.10–0.37 fm, employing Wilson loops built out of APE smeared (see Section 3.2) spatiallinks and simple temporal gauge links. The measurements atβ = 3.00,3.30 include thedetermination of the potential between quarks that are off-axis separated along the latticevectors (1,0,0), (1,1,0), (1,1,1), (2,1,0), (2,1,1), (2,2,1) (and lattice rotations thereof)in order to estimate violations of rotational symmetry. The rest of the measurementsinclude only on-axis separations due to the large computational cost (concerning speed andmemory) of the off-axis measurement. The smearing levels used are as follows (λf = 0.1on all the lattices): on the finest lattice atβ = 3.50, we useSnU , n = 3,6,9,12,15;at β = 3.15 we useSnU , n = 2,4,6,8,10; for the measurements including off-axisseparations, we employ only three different smearing levels each, in order to save memoryand time:SnU , n = 5,10,15 atβ = 3.30 andSnU , n = 2,4,6 on the coarsest lattice atβ = 3.00. The parameters of the simulations are given in Table 7.

The values of the off-axis potentialatV (r) at β = 3.30 are collected in Table 25 inAppendix B and displayed in Fig. 4, the values of the off-axis potential on the coarselattice atβ = 3.00 are collected in Table 26. The potential valuesatV (r) of the on-axissimulations are collected in Table 27.

The parametersα andσ (the string tension) in the phenomenological potential ansatz,Eq. (31), are determined using global fits, the hadronic scaler0 is determined performinglocal fits. The fit ranges and results are given in Table 8. In the global fit to the off-axispotential atβ = 3.30 we exclude the badly measured separations 2(2,1,0) and 2(2,1,1).

Table 7Run parameters for the measurements of the static quark–antiquark potential using theξ = 2 perfectaction

β Off-axis sep. S3 ×T Sweeps/Measurements

3.00 yes 83 ×16 42000/28003.15 no 103 ×20 39800/39803.30 yes 103 ×20 27000/18003.50 no 123 ×24 36400/3640


Fig. 4. Ground state of the static quark–antiquark potential atβ = 3.3. Including a fit to thephenomenological potential ansatz in the ranger = √

2–5.

Table 8Results of the measurements of the static quark–antiquark potential using theξ = 2 perfect action.The parametersα andσ of the potential ansatz Eq. (31) are given together with the global fit rangechosen to determine them. The hadronic scaler0 is given in spatial lattice units together with thelocal fit range used to determine that quantity. Note, that we have chosenc = 6.00 andc = 0.89 forβ = 3.0 andβ = 3.5, respectively (see Section 3.3.1). Additionally, the dimensionless quantityr0

√σ

is given

β Gl. fit α σasat Loc. fit r0/as r0√σ

3.00 1–2√

6 −0.201(18) 0.3595(87)√

6–3 1.353(16) 1.119(51)3.15 1–5 −0.1503(12) 0.1654(6) 1–3 2.038(2) 1.162(15)3.30

√2–5 −0.1539(7) 0.0683(3) 2

√2–2

√3 3.154(9) 1.140(8)

3.50 1–6 −0.1478(4) 0.0320(3) 2–4 4.906(21) 1.189(14)

In principle, it is possible to determine the renormalised anisotropy using the first excitedstate of the potential together with the ground state. The ground state values are again fittedto the ansatzV (r) = V0 + α/r + σr, whereas the values of the excited state are fitted tothe (ad-hoc) ansatzV ∗(r)=A/r +B + Cr +Dr2. As can be seen from Fig. 5, however,for the off-axis potential atβ = 3.3, the energy values of the first excited state have largeerrors. The result for the separationat (V ∗(r0) − V (r0)) = 0.555(47) thus shows a largeerror. Comparing this value tor0(V ∗(r0)− V (r0)) ≈ 3.25(5) of [55] one obtainsr0/at =5.86(59) and thusξR = as/at = 1.86(19) for β = 3.3 which is in agreement with the valuedetermined using the torelon dispersion relation,ξR = 1.912(9). A similar determinationfor the on-axis potential atβ = 3.50 is even more difficult due to the smaller number ofseparations measured. The result isξR = 1.66(32)which does again agree with the torelonresult, ξR = 1.836(9). Due to the large errors (possibly caused by the operators used,


Fig. 5. Static quark–antiquark potential atβ = 3.3 including the first excited stateV ∗(r) atβ = 3.3.The location ofr0 where the determination of the anisotropy is done, using the gap, is marked by thedotted line.

optimised for the ground state) this method of determining the renormalised anisotropyξR is not suitable in the context of this work. However, for the determination of the scaleas well as of the anisotropy on lattices with finer temporal lattice spacing (above all withhigher anisotropies) this way might be feasible at least if off-axis separations are includedin the measurement.

3.3.2. Rotational invariance and scalingThe measurements atβ = 3.0,3.3 including a large number of off-axis separations

provide information about the deviations from rotational invariance. This issue has beenaddressed in [27] for the RGT used in this work and it turned out that violations ofrotational symmetry caused by the blocking are small. That this is still true for theparametrisation employed here is shown by the off-axis potential atβ = 3.30, see Fig. 4.The deviations on the coarse lattice atβ = 3.00 are significantly larger (see Fig. 6) whichis no surprise as this coupling corresponds to a spatial lattice spacingas ≈ 0.37 fm.We present some dimensionless ratios containing information about the deviation fromrotational invariance in Table 9; due to the resulting small values the errors are rather large.One notices that the parametrised perfect action does a good job, forβ = 3.30 the deviationbetween the exactly degenerate separations (2,2,1) and (3,0,0) is zero (within statisticalerrors).

The staticqq-potential is also an effective test of scaling. Expressing the potentialmeasurements performed at different couplingsβ in the RG invariant, dimensionlessratios r/r0 and r0V and subtracting the unphysical constantr0V (r0) should lead topotentials lying exactly on top of each other. Deviations indicate either scaling violationsor ambiguities in the determination ofr0. Fig. 6 includes all potential measurementsand shows that the different curves can hardly be distinguished from each other, exceptsome energies determined on the coarsest lattice atβ = 3.00. They deviate notably from


Fig. 6. The static quark–antiquark potential measurements expressed in RG invariant, dimensionlessunitsr/r0, r0V . The unphysical constantr0V (r0) has been subtracted such that all the curves exactlycoincide atr = r0. The dashed line is a global fit to the phenomenological potential ansatz, Eq. (31).

Table 9The normalised deviations from rotational invariance for the AICP action. The ratios displayed arezero in the case of no rotational symmetry violation

βV (1,1,1)−Vfit(

√3)

V (2,0,0)−V (1,0,0)V (2,2,1)−V (3,0,0)V (3,0,0)−V (1,0,0)

3.00 0.086(4) 0.057(6)3.30 0.029(7) 0.005(6)

the curve which is a fit to the phenomenological potential ansatz, Eq. (31), includingall the measurements that have been included into the global fits of the singleβ

values, see Table 8. The results for the two physical parameters from this global fit areα = −0.27799(5) andσr2

0 = 1.3690(3).

3.4. The critical temperature

The critical temperature of the deconfining phase transition contains information aboutthe temporal scale of the lattice at a given couplingβ as Tc = 1/(Ntat). Comparingthis information to quantities obtained from the measurements of torelons or the staticquark–antiquark potential offers many interesting scaling (and other) tests. However, sincemeasurements of such quantities at the determined values ofβcrit are yet absent and becauseinterpolations inβ are very difficult due to the renormalisation of the anisotropyξ , thesetests are a future project.

The critical temperatureTc may be defined as the location of the peak in thesusceptibility of the Polyakov loopP (Wilson line in temporal direction),


(34)χ ≡N3s

(⟨|P |2⟩ − ⟨|P |⟩2).We determine the values ofβc(Nt ) on rather coarse lattices with temporal extensions

Nt = 3, . . . ,7. Simulations at different values ofβ near the estimated critical couplingsβc are performed and the peaks are determined by employing the Ferrenberg–Swendsenmulti-histogram reweighting [56,57] (see also [28,39,40] for additional details). The runparameters of the simulations are collected in Table 10.

In this first study, we decide not to examine the finite-size scaling ofβcrit but to chooseLs/Lt = ξ · Ns/Nt ≈ 3.5 ∼ 4. Obviously, the anisotropic nature of the action makesthe computational effort of the simulations for exploring the deconfining phase transitionsmaller. The effects of the finite volume are (rather conservatively) estimated by using the

Table 10Run-time parameters ofξ = 2 Tc-simulations. The number of sweeps, the persistence timeτp (ifapplicable), the integrated autocorrelation timeτint and the Polyakov loop susceptibilityχL are givenfor all values ofβ used in the final reweighting procedure, as well as the resultingβc . The systematicerror in the second brackets is a (rather conservative) estimate of the finite volume effects and has tobe considered if the given value should be an estimate of the infinite-volume critical coupling

Lattice size β Sweeps τp τint χL βc

3× 63 2.80 45000 70 1.45(5) 2.863(1)(5)2.85 45000 410 259 11.3(20)2.87 45000 312 21.6(28)2.90 50000 100 2.34(10)

4× 83 3.00 45000 81 1.63(5) 3.032(1)(5)3.02 45000 870 102 3.97(26)3.03 45000 830 372 43.2(22)3.04 45000 188 10.9(18)3.05 55600 144 3.94(21)

5× 93 3.07 44000 117 0.84(4) 3.118(1)(6)3.10 45000 560 154 4.38(63)3.11 20500 640 198 15.0(27)3.12 44500 3700 278 33.2(19)3.15 20000 110 3.26(18)

6× 113 3.17 45000 1440 121 7.4(11) 3.181(1)(6)3.18 45000 1750 201 30.0(15)3.185 45000 232 23.2(24)3.19 45000 123 10.4(14)3.20 45000 100 5.62(36)

7× 133 3.22 32400 700 87 6.7(5) 3.236(1)(6)3.23 44500 750 129 18.4(11)3.24 29000 149 20.7(16)3.25 22400 83 9.2(10)


finite-size scaling relation for a first order phase transition,

(35)βc(Nt ,Ns)= βc(Nt ,∞)− h

(Nt

Ns

)3

.

The parameter|h| is set to the largest value|h| = 0.25 that has been observed with theisotropic action, and the value ofξR used to estimate the spatial lattice volume is set to thelowest value appearing in the whole range ofβ values considered. The extrapolated criticalcouplingsβc are given together with the statistical error and the estimated finite-volumeerror in the last column of Table 10.

3.5. Glueballs

3.5.1. IntroductionThe particles mediating the strong interaction of QCD, the gluons, carry colour charge

and thus interact with each other, unlike, e.g., their counterpart in electromagnetism, thephotons, which have zero electric charge. The spectrum of QCD may thus contain boundstates of (mainly) gluons, called glueballs. These states are described by the quantumnumbersJ denoting the (integer) spin,P denoting the eigenvalue±1 of the state underparity andC denoting the eigenvalue±1 under charge conjugation. Thus the eigenstatesof the Hamiltonian corresponding to glueball states are labeled|JPC〉.

Currently, there is an ongoing debate whether light glueballs (above all the scalar 0++which is the lightest state in pure lattice gauge theory with a mass of about 1.6 GeV) havebeen observed experimentally at about the mass that is predicted by quenched simulationson the lattice, whether the lightest glueball is much lighter (below 1 GeV) and verybroad [58], or whether glueballs have not been observed at all in experiments [59]. Thereare mainly two reasons for this uncertainty. On one hand, the experimental data seem notyet to be accurate and complete enough, despite large efforts in the last years, driven bythe lattice results; on the other hand, lattice simulations with high statistics, measuringglueball states, have been performed only in the quenched approximation, where the quarksare infinitely heavy and thus static. Decreasing the sea (dynamical) quark mass (finallydown to the physical value) will allow to track the glueball states as sea quark effects areincreased. It may turn out, that indeed the glueball mass is lighter than the one measuredin pure gauge theory (for partially quenched results possibly indicating this see [60–62]).It may even happen that by “switching on” the sea quarks the scalar glueball starts todecay (almost) instantaneously toqq states, i.e., it ceases to exist physically. However, the(partially) unquenched results are rather indecisive yet.

3.5.2. Glueballs on the latticeGlueballs in the continuum are rotationally invariant and have a certain (integer) spinJ .

On the lattice, the rotationalO(3) symmetry is broken, only its discrete cubic subgroupOh

survives the discretisation. Therefore, the eigenstates of the transfer matrix are classifiedaccording to the five irreducible representations ofOh: A1, A2, E, T1, T2 with dimensions1, 1, 2, 3, 3, respectively. Their transformation properties may be described by polynomials


Table 11The composition of the subduced representationDJ ↓Oh in terms of the irreducible representationsof the cubic groupOh

Γ p D0 D1 D2 D3 D4 D5 D6

A1 1 0 0 0 1 0 1A2 0 0 0 1 0 0 1E 0 0 1 0 1 1 1T1 0 1 0 1 1 2 1T2 0 0 1 1 1 1 2

in the componentsx, y, z of an O(3) vector as follows:A1 ∼ 1, A2 ∼ xyz, E ∼x2 − z2, y2 − z2, T1 ∼ x, y, z, T2 ∼ xy, xz, yz. Generally, anO(3) representationwith spinJ splits into several representations of the cubic group. SinceOh is a subgroupof O(3), any representationDJ with spinJ in the continuum induces a so-called subducedrepresentationDJ ↓ Oh on the lattice. This subduced representation no longer has to beirreducible but is a direct sum of irreducible representationsΓ p of Oh:

(36)DJ ↓Oh = Γ 1 ⊕ Γ 2 ⊕ · · · .

Table 11 lists the subduced representations ofDJ for J = 1, . . . ,6. The spinJ = 2state for example splits up into the 2-dimensional representationE and the 3-dimensionalrepresentationT2. Approaching the continuum, rotational symmetry is expected to berestored and thus, as a consequence the mass splitting of these two states will disappearand the two representations form together the 5 states of a spinJ = 2 object.

Pure glue physical states on the lattice are created and annihilated by applying gaugeinvariant operators to the pure gauge vacuum. In our simulations, we use space-like Wilsonloops in the fundamental representation of SU(3). Since we do not aim at measuringnon-zero momentum glueballs we consider only translationally invariant operators, i.e.,operators averaged in space.

It is computationally feasible to measure Wilson loop operators up to length 8. Thecomposition of the irreducible representationsΓ PC of the cubic group in terms of these22 loop shapes has been done already in Ref. [63]. Operators contribute to the two- andthree-dimensional representations with two and three different polarisations, respectively,in analogy to different magnetic quantum numbersm for a given angular momentuml intheO(3) group. Measuring all these polarisations may suppress statistical noise more thanjust increasing statistics since the different polarisations of a loop shape are expected to beanti-correlated.

Due to mixing with other states present in pure gauge theory on a periodic lattice, suchas torelons or glueball pairs, and due to the breaking of the continuum rotational symmetryby the lattice, the identification and the continuum spin assignment of single glueball statesare additional vital questions which will be addressed in Sections 3.5.5, 3.5.7.


Table 12Run parameters for the glueball simulations using theξ = 2 perfect action

β (as/r0)2 S3 ×T V Sweeps/Measurements

3.15 0.2408(5) 83 ×16 (1.96 fm)3 241000/241003.30 0.1005(6) 103 ×20 (1.59 fm)3 99000/99003.50 0.0415(4) 123 ×24 (1.22 fm)3 115000/11500

3.5.3. Simulation parametersTo allow for a continuum extrapolation, at least for the lighter states, we decide to

perform simulations at three different lattice spacings in the range 0.10 fm as 0.25 fmin volumes between (1.2 fm)3 and (2.0 fm)3. The simulation parameters are given inTable 12. The gauge fields are updated by performing compound sweeps consisting of4 pseudo over-relaxation and 1 Metropolis sweep, and after every 2 compound sweeps wemeasure all 22 loop shapes up to length 8. The measurement is performed on APE-smearedconfigurations (see Section 3.2) in order to spatially enlarge the operators, thus improvingthe overlap with the glueball states and reducing unphysical high-momentum fluctuations.On the coarse lattice atβ = 3.15 we use smearing levelsSn, n= 2,4,6,8,10, on the otherlattices we use levelsSn, n= 3,6,9,12,15, always with smearing parameterλs = 0.1.

3.5.4. Determination of the energiesThe masses of the lowest states and the first excited states (if possible) in all the

representations are determined using the variational techniques described in Appendix Bof [28]. The results in units of the temporal lattice spacingat are listed in Tables 28–30 inAppendix B together with the numberN of operators entering into the fitting procedure,the time slices on which the initial generalised eigenvalue problem is solved (usually 1/2),the number of operatorsM in the final fit and the corresponding value ofχ2/NDF. Theseresults are then multiplied byξR · r0/as from Tables 3 and 8 to obtain the glueball massesin units of the hadronic scaler0, listed in Table 13.

In order to obtain reliable estimates of the glueball masses from the variational method,one has to pay attention that among the large number of operators (up to 145 from fivesmearing levels) there are no correlators entering the process that are measured excep-tionally bad, i.e., with large errors even at small time separation, as this may destabilisethe determination. To filter out such data, we look at the relative errors of the singleoperators depending on the temporal separation of the creation and annihilation opera-tors and drop all correlators whose signal falls below a certain threshold value. Usually,this is repeated with different threshold values, yielding different sized sets of operators(2–4 sets per representation). We performO(100) mass determinations on each operatorset, varying the analysis parameters such asM (see Appendix B of [28]) or the fittingranget = tmin · · · tmax. It turns out that generally the variational method used is stable,provided that the badly measured operators are absent from the beginning and providedthat no important operators are missing. Patently, to find such a window of the number


Table 13Final glueball mass estimates in terms of the hadronic scaler0, mGr0 from the measurements usingthe perfectξ = 2 action. The continuum spin assignmentJ is given as well

Channel J β = 3.15 β = 3.30 β = 3.50

A++1 0 2.58(9) 3.55(13) 3.65(15)

A++1

∗0 5.47(53) 6.83(37) 6.49(148)

E++ 2 5.63(23) 5.93(15) 6.08(28)

E++∗ 2 7.95(73) 8.76(48) 10.66(59)

T++2 2 5.67(25) 5.82(13) 6.13(16)

T++2

∗2 7.97(69) 10.16(30)

A++2 3 6.71(73) 9.11(64) 10.92(48)

A++2

∗14.12(130)

T++1 3 7.54(68) 9.00(44) 11.21(33)

T++1

∗8.69(91)

A−+1 0 6.36(59) 6.13(28) 6.79(27)

A−+1

∗10.27(110)

E−+ 2 7.41(48) 8.23(28) 8.48(29)

E−+∗ 8.35(125) 12.66(84)

T−+2 2 7.41(49) 8.18(21) 8.57(42)

T−+2

∗15.08(67)

T+−1 1 7.20(39) 7.69(26) 9.12(24)

A+−2 3 11.02(68)

A+−2

∗15.55(147)

T+−2 7.75(87) 10.29(118)

T+−2

∗14.98(139)

E+− 12.42(42)

T−−1 10.81(83)

T−−2 10.85(83)

A−−2 12.18(45)

A−−1 15.34(103)


of initial operators,N , may be difficult if there is a rather small total number of opera-tors.

For some of the heavy states, such as thePC = −− glueballs, whose masses can onlybe determined from theβ = 3.50 measurements, we have to resort to using the correlatorsat t0 = 0 andt1 = 1 for the solution of the first generalised eigenvalue problem. This hasthe advantage that the correlators from separationt0 = 0 are positive definite by definition(which is not true fort0 1), thus rendering the second generalised eigenvalue problemwell defined even if all the initial operators are kept,M = N . However, the correlatorsC(t = 0) are under suspicion of containing rather little physical information about thecorrelation lengths (which correspond to thedecay of the correlators), actually theyalmost only provide information about the relative normalisation of operators. Obviously,operators with a large signal att = 0 might be preferable because their signal/noise ratiois probably better on average, however in general they do not correspond to the operatorshaving the largest overlap with the lowest-lying states. It turns out that one has to pay muchmore attention in choosing the fit ranget = tmin · · · tmax when usingt0 = 0, t1 = 1 becausein this case at smallt the contamination of the lowest masses due to higher states is muchmore significant.

During the analysis, it turns out that the mass determinations for the scalar representationA++

1 are rather challenging. The ground state receives a larger relative error thancomparable states where the operators are measured equally well; for the first excited stateit is very hard to obtain a stable determination, and the error of its mass may turn out to behuge (e.g., atβ = 3.50). Probably, it is the underlying vacuum, having the same quantumnumbers as the glueball state, that is responsible for these troubles. We treat the vacuumjust like another state in this representation, so that the glueball ground state is effectivelya first excited state and the glueball first excited state is effectively a second excited stateof the representation. Attempts of using other ways of getting rid of the vacuum such asthe usual v.e.v. subtraction or the subtraction of larget correlators (that are assumed tocontain solely noise, however correlated to the noise at lowert) or even more sophisticatedmethods (like solving a generalised eigenvalue problem using larget correlators to dig outthe vacuum state) do not at all succeed in improving the situation.

3.5.5. Torelons and multi-glueball statesBesides single glueballs, the spectrum of pure gauge theory on a lattice with periodic

boundary conditions also contains states consisting of several glueballs, torelons or mixedstates of glueballs and torelons. Although we expect that the operators used to measureglueball energies couple most strongly to the single glueball states, other states with similarmasses and compatible quantum numbers might mix with them and distort the result, i.e.,the energy determined by the analysis of the correlation matrix may be dominated bya multi-glueball or torelon state with smaller energy than the single-glueball state to bemeasured.

In principle, there are several means of determining the nature of a state that has beenmeasured. Firstly, the simulation may be repeated on lattices of different physical size,keeping the lattice spacing fixed. As multi-glueball and torelon states show a finite-volume


scaling behaviour very different from single glueballs, such states stand out and may bedropped from further analysis. Secondly, one may measure additional operators that couplestrongly to torelon or multi-glueball states. Including or excluding these operators in thevariational method and studying the coefficients obtained from the variational method, themixing strength is determined and states which do not mix considerably with any of theadditional operators may be safely considered to be single-glueball states.

Finally, using the mass estimates for the low-lying glueballs, one may determine theapproximate locations of the lowest-lying multi-glueball states. Also, the minimal energyof mixing torelons may be estimated using the string formula, Eq. (23).

The first two of the methods mentioned above require additional work and computertime and go beyond the scope of this work. The last method, however, can be done rathereasily. Let us first calculate the energy of the lowest-lying torelon states that may interferewith our measurements. Single torelons (see Section 3.2) transform non-trivially underZ3 symmetry operations; our operators, closed Wilson loops, however, are invariant underthese transformations. This means that they cannot create single torelon states, howeverthe creation of two torelons of opposite center charge is possible. If we assume that theydo not interact considerably and if our lattice extension is rather large then we can use thesimple formulaE2T ≈ 2σL to estimate the energy of a torelon pair with momentum zero,whereσ is the string tension (measured by the staticqq potential, see Section 3.3.1) andL is the spatial extent of the lattice. Table 14 lists the minimum energies of torelon pairs tobe expected on our lattices.

Note, that a state composed of two opposite center charge torelons and with totalzero momentum is symmetric under charge conjugation. The operators forC = − statestherefore do not create such torelon pairs.

To estimate the lowest energies of multi-glueball states present on our lattices, we followthe method used by Morningstar and Peardon [13], described in great detail in Appendix Fof [39]. We assume that the energy of multi-glueball states is approximately given by thesum of the energies of the individual glueballs, i.e., that there is no substantial interaction.It is therefore clear that the lowest-lying multi-glueball states with zero total momentumare the two-glueball states, with energy

(37)E2G ≈√

p2 +m21 +

√p2 +m2

2,

Table 14The minimum energies of momentum zero torelon pairson the lattices used in the glueball simulations, given inunits ofr−1

0

β r0E2T

3.15 10.603.30 8.243.50 6.92


wherem1 denotes the rest mass of the first glueball with momentump andm2 denotesthe rest mass of the second glueball with momentum− p. The masses are taken from ourdeterminations at the respective values ofβ and the (lattice) momenta are chosen such thatthe energyE2G of the glueball pair contributing to some representationΓ PC is minimised.Note, that glueball pairs contribute to different representations of the cubic group, not onlydepending on the representations according to which the single glueballs transform, butalso on the single glueball momentump.

The lower bounds of the multi-glueball energy region as well as the lower bound fortorelon pairs are indicated in Figs. 7–9, together with the determined energies of all thestates in the respective representations. It turns out that all of the excited states measuredand even some of the lowest states in a given representation could be affected by torelonpairs or multi-glueball states. Of course, it is not at all ruled out that these states are indeedsingle glueballs, however one has to be very careful with the interpretation and has to keepin mind that these issues require further study.

3.5.6. The continuum limitIn order to remove discretisation errors from glueball massesr0mG obtained at finite

lattice spacing, we have to perform the continuum limit, i.e., to extrapolate the resultsto as = 0. Having obtained results at three values of the couplingβ , one of themcorresponding to a rather coarse lattice spacing, and having no accurate information (e.g.,from perturbation theory) about the behaviour of the energies depending on the cut-off

Fig. 7. The results of the glueball measurements atβ = 3.15 converted into units ofr0. The lowerbounds of the energies of multi-glueball states (solid lines) or torelon pairs (dotted line) are given aswell.


(about the form of the curve used in the extrapolation) this is not an easy task and ratherambiguous.

We perform two different kinds of continuum limits. Firstly, we extrapolate the massesin terms of the hadronic scaler0, r0mG =mGat · ξ · r0/as , secondly, we extrapolate massratiosmG1/mG2 of different glueball representations. Generally, the procedure we resortto is the following: we include all the three energies or mass ratios, obtained atβ =3.15,3.30,3.50, into the fit and use the form

(38)r0mG|as = r0mG|as=0 + c2

(as

r0

)2

,

thus including the correction of the smallest power of the (spatial) lattice spacingas to beexpected to occur. Additionally, we perform fits including only the results from the twofiner lattices atβ = 3.30,3.50 using a constant (if this fits the results reasonably well) orthe form stated above. This value of the continuum mass (or mass ratio) is then comparedto the result of the fit of all the three data points, which is only accepted if the two resultscoincide within their errors.

Masses in terms ofr0For the extrapolation of masses in terms of the hadronic scaler0, this procedure works

very well for the representationsA++2 , E−+, E++, E++∗, T −+

2 andT ++2 where we may

always fit the results from the two finer lattices to a constant which agrees with the resultof the extrapolation using Eq. (38) on all three data points. For theA−+

1 representation, weobtain the best results from fits with a constant to two as well as to three data points.

Much more difficult is the extrapolation for the scalar glueball and its excited state,A++

1 andA++1

∗. It turns out that the fits have bad values ofχ2 and for the ground state

determinations the result obtained performing a fit to a constant mass using the results fromthe two finer lattices does not correspond to the fit including the third, coarser lattice and aquadratic term inas . We decide to keep the (usual) result, including the quadratic term andkeep in mind that there may be present some ambiguities which will be discussed later.

The results of the continuum extrapolations for various glueball states are collected inTable 31. In Figs. 10 and 11 the fitting curves are shown together with the measured valuesfor thePC = ++ and thePC = −+ sectors where extrapolations are possible with ourdata.

Other representations where energies can be determined at several values ofβ , such asT +−

1 , T ++1 or T ++

2∗, do not yield consistent results using the procedure presented above

and we decide to abstain from performing continuum extrapolations in terms ofr0mG forthese states.

Mass ratiosExtrapolating glueball mass ratiosmG1/mG2 to the continuum has the advantage that

uncertainties present in the determination ofr0/as and the renormalised anisotropyξR

cancel out. Furthermore, we may assume that finite size effects are similar for glueballstates measured on the same lattice as opposed to completely different quantities used,e.g., to obtainr0/as andξR.


Fig. 10. Mass estimates of thePC = ++ glueballs in terms of the hadronic scaler0 against(as/r0)2.

The curves are the continuum limit extrapolations of the forms as indicated in the text. Circles:A++

1 , boxes:A++2 , diamonds:E++, upward triangles:T++

1 , downward triangles:T++2 ; solid

symbols: ground states, open symbols: first excited states. Note that the continuum results of therepresentationsT++

1 andT++2

∗have been obtained using a fit of glueball mass ratios.

Fig. 11. Mass estimates of thePC = −+ glueballs in terms of the hadronic scaler0 against(as/r0)

2. The curves are the continuum limit extrapolations of the formr0mG+c2(as/r0)2. Circles:

A−+1 , diamonds:E−+, downward triangles:T−+

2 ; solid symbols: ground states, open symbols: firstexcited states.


The common way of extrapolating glueball mass ratios to the continuum is using ratiosmG/m0++ , i.e., ratios to the scalar glueball mass, which is the lightest mass present. Doingthis, we observe that the extrapolation is difficult, the value ofχ2 of the fit is rather badand the errors are large. Furthermore, the resulting continuum mass ratios are not in goodagreement to ratios of the continuum masses extrapolated in terms ofr0. This is not asurprise, because the behaviour of the scalar glueball as a function of the lattice spacing israther complicated, i.e., there are large cut-off effects that cannot be described too easily.Additionally, the errors of the mass estimates of the scalar glueball are quite large. Due tothese reasons, we decide to use the well measured mass of the tensorT ++

2 representation,which show small errors and seem to scale rather well.

The procedure of the extrapolation is the same as in the case ofr0mG described in theprevious section; the unambiguous results are listed in Table 32. Again, for theA−+

1 states,as well as for theE++, the fit to a constant works out best. Note, that in this way we canobtain reliable continuum results for the representationsT +−

1 , T ++1 andT ++

2∗

which isnot possible performing the extrapolation using masses in terms ofr0. Furthermore, all theother results listed are in agreement to the ratios of the continuum massesr0mG within onestandard deviation.

In the following, we will use the finite lattice spacing results from the simulation onthe finest lattice atβ = 3.50 for the energies of states that cannot be extrapolated tothe continuum in one of the two ways, e.g., to study degeneracies or to draw a (ratherqualitative) picture of the low-lying glueball spectrum. In tables listing continuum results,the values not extrapolated toas = 0 will be stated in square brackets.

3.5.7. Continuum spin identificationOnce the extrapolation of the measured glueball energies to the continuum,as = 0, has

been performed, what remains is the assignment of continuum spinJ to the different repre-sentations of the cubic groupΓ . From Table 11 we know from which lattice representationsΓ PC a continuum glueball with quantum numbersJPC may obtain contributions, wherethe assignment of parityP and charge conjugationC is simply one to one. Degeneracies(in the continuum limit) between several representationsΓ PC contributing to the samecontinuum stateJPC are a strong indication for the correctness of the assignment of allthe lattice states involved to the same continuumJPC glueball. Furthermore, we make theassumption that the mass of the glueballs increases with the spin getting larger.

In thePC = ++ sector, we observe a single low-lying state,A++1 , which is thus as-

signedJ = 0. TheE++ andT ++2 states are degenerate to a very high precision as it should

happen if these representations correspond to the five polarisations of aJ = 2 glueball. Theexcited stateA++

1∗

again has no degenerate partner and is assigned to an excited state of thecontinuumJ = 0 glueball. The excited statesE++∗ andT ++

2∗

again turn out to be degen-erate and so they are assigned to an excitation of theJ = 2 glueball. The same situation ismet with the representationsA++

2 andT ++1 which are thus assignedJ = 3. Since the first

excitationT ++2

∗is part of theJ = 2 state, the missing 3 polarisations should come from

the second excitation ofT ++2 which, however, cannot be measured by our simulations.


In thePC = −+ sector, the situation of theA−+1 and theE−+ andT −+

2 is very similarto the one of their partners in thePC = ++ sector; these states are thus assignedJ = 0andJ = 2, respectively. The remaining three, excited states have no apparent degeneracies,there is thus no safe assignment of continuum spin to these states. The facts that the excitedstateA−+

1∗

is rather light and comes with no degenerate partner, raises the presumption thatit is the excitedJ = 0−+ glueball.

In the other two sectors, continuum extrapolations of our measured energies are notpossible, except for the representationT +−

1 using mass ratios. However, looking atthe degenerate representations in thePC = ++,−+ sectors, it is noticeable that thedegeneracies are apparent even for finite lattice spacing atβ = 3.50,3.30. Assuming thatthis behaviour persists forPC = +−,−−, at least for energies not too high, we may assigncontinuum spin even to some of the remaining states.

In the PC = +− sector, we notice the lowest-lyingT +−1 state having no degenerate

partner thence suggesting aJ = 1 interpretation. Next, there is theA+−2 which is

likely to correspond toJ = 3, which is the smallest possible spin corresponding to thisrepresentation.

Finally, in thePC = −− sector, there are different possible scenarios. The almost exactdegeneracy ofT −−

1 andT −−2 suggests them contributing to theJ = 3 continuum state. The

rather heavier stateA−−2 could still be degenerate with the two latter states (note the high

mass and the finite lattice spacing!) and carry the remaining polarisation. Admittedly, itcould as well correspond to an excitedJ = 3 state or even toJ = 6. The finalA−−

1 state isvery heavy and indicates that glueballs with even spin andPC = −− have large energiesas the representationA1 contributes solely to even numbered spin states up toJ = 8.

Let us remark that a new method of improving the overlap of operators to glueballs ofa given continuum spinJ thus rectifying the continuum spin identification is presented in[64,65].

The well supported continuum spin assignments presented above are given together withthe glueball masses in Tables 13, 31, 32, while the final results on the masses of these statesare collected, together with the masses in MeV, in Table 15.

3.5.8. DiscussionIn order to discuss the results obtained from the glueball measurements let us first list

the sources of problems that may affect our results. Firstly, there are finite size effects;our finest lattice (β = 3.50) has got a spatial volume of (1.22 fm)3 which is rather smallcompared to other volumes used in glueball measurements. The magnitude of finite-sizeeffects depends largely on the quantity studied, more precisely on the form of the glueballwave functions (above all their extension). Due to the three volumes employed having suchdifferent size, finite-size effects may not only systematically shift the lattice results but theycan also make reliable continuum extrapolations much more difficult or even impossible.

Secondly, the mass of the scalar glueballA++1 (above all the ground state) shows very

large cut-off effects. These could be due to the presence of a critical end point of aline of phase transitions in the fundamental–adjoint coupling plane. Our parametrisationincludes in its rich structure operators transforming in the adjoint representation. If the


Table 15Final results for the masses of continuum glueballs with quantum numbersJPC , obtained fromthe lattice representationsΓ PC . For the conversion to MeV,r−1

0 ≈ 0.5 fm ≈ 395 MeV has beenused. Quantities in brackets have not been extrapolated to the continuum but denote results from thesimulation on the finest lattice atβ = 3.50 (corresponding toas = 0.102 fm)

JPC Γ PC mGr0 mG (MeV)

0++ A++1 4.01(15) 1580(60)

0++∗A++

1∗

7.66(71) 3030(280)

2++ E++, T++2 6.15(15) 2430(60)

2++∗E++∗, T++

2∗

10.52(51) 4160(200)

3++ A++2 , T++

1 11.65(49) 4600(190)

0−+ A−+1 6.46(18) 2550(70)

2−+ E−+, T−+2 8.75(26) 3460(110)

0−+∗A−+

1∗ [10.27(110)] [4060(430)]

1+− T+−1 9.45(71) 3730(280)

3+− A+−2 [11.02(68)] [4350(270)]

net coupling of the parametrised action (which we do not control during the constructionand the parametrisation) lies in a certain region, the effect of the critical end-point onscalar quantities at certain lattice spacings (sometimes called the “scalar dip”) may evenbe enhanced compared to other (more standard) discretisations with purely fundamentaloperators. It is important to note that the true classically perfect action is not expected tobe sensitive to this critical point, but it is rather the parametrisation which may incidentallycause the sensitivity close to it.

Furthermore, our analysis shows that states consisting of several glueballs or torelonpairs could mix with most of the higher-lying states occurring in our measurements. Thismakes the interpretation of the measured states more difficult, additionally, systematiceffects (e.g., different mass shifts at different values ofβ because of the different latticesizes or due to the presence of operators having larger overlap with unwanted states) mayagain complicate the continuum extrapolation.

There are several possibilities of improving the situation. Firstly, and most simple,improving the statistics may help. This can be seen, e.g., by looking at the results of thesimulation on the coarsest lattice atβ = 3.15. Although the lattice is coarser, more energiescan be determined than on the finerβ = 3.3-lattice, mainly due to the larger statistics.Additionally, a lot of problems stated previously could be kept under control if for eachvalue ofβ , we determine the glueball energies on several lattices of different physical size.A way of improving the measurements is systematically studying the operators employedto create and annihilate single glueball states in order to increase their overlap with thesingle glueballs to be measured and to decrease their overlap with all the other states.This includes the study of other smearing techniques, such as, e.g., Teper fuzzing [66].


Conversely, the introduction of operators coupling most strongly to the unwanted multi-glueball or torelon-pair states allows for the calculation of mixing strengths in order toexclude all the unwanted states from further analysis. Another technically rather easybut computationally expensive way of improving the results is performing simulationson additional values ofβ . This allows for a more detailed study of the continuum limit,especially in the case of non-perturbatively improved actions, where there is no clearcutinformation about the way how the continuum is approached. Naturally, one has to takecare not to go too deep into the strong coupling region as there is not much informationabout the continuum.

There is one error which is not cured by the measures recommended in the previousparagraph, namely the effects on the scalar states, coming from the “scalar dip”.Morningstar and Peardon improve the situation by adding a two-plaquette adjoint term witha negative coefficient which results in an approach to the continuum on a trajectory alwaysfar away from the dangerous “dip” region [14]. In principle, the classically perfect actioncould be treated the same way: extract all the operators with adjoint contributions presentin the parametrisation, determine their sign and add another (non-linear) constraint to thefit, namely that the action of all these operators together corresponds to an adjoint operatorwith a certain (negative) coupling. Because of the large freedom in the fit, the inclusion ofthis single criterion should not impair the quality of the parametrisation considerably.

Tables 16 and 17 compare our continuum glueball masses obtained with the anisotropicclassically perfect action (AICP) to the results of the isotropic FP action as well as to resultsobtained by other collaborations. There is reasonable agreement between the differentdeterminations of the 0++, 2++, 0++∗, 0−+, 2−+ glueball masses. The situation for the1+− state is different, however. This is the heaviest state that occurs in our analysis andcorrespondingly difficult to determine on the lattices used, having a rather small anisotropyξ = 2. Thinking of all the possible sources of errors stated above, which are (partly) alsopresent in the analyses of the other groups, we tend to explain this discrepancy withunderestimated (or disregarded) systematic errors as discussed previously. In particular,by looking at the finite lattice spacing results forT +−

1 in Table 13, we observe that themasses determined on the coarser (and larger) lattices atβ = 3.15,3.30 coincide withthe (continuum) results obtained by other groups and do not even show a significantdiscrepancy, despite the considerable difference between the lattice spacings. The massdetermined on the fine (and small) lattice atβ = 3.50 however, is much higher. Thistendency is even amplified by the continuum extrapolation. We suspect that this stateexhibits strong finite-size effects pushing up its mass in small volumes.

Figs. 12 and 13 compare our measurements of theA++1 , E++ and T ++

2 states tomeasurements obtained by other groups, using different actions, as well as to the isotropicFP action results. Concerning the scalar glueball, the artifacts of the isotropic FP actionat moderate lattice spacings corresponding to(as/r0)

2 0.15 (as 0.19 fm) seem tobe rather smaller than the ones of the tadpole and tree-level improved action (M&P)and certainly much smaller than the artifacts of the Wilson action. For the AICP action,however, the situation is less clear. The mass obtained on the coarsest lattice atβ = 3.15(corresponding toas ≈ 0.24 fm) exhibits large cut-off effects of about 35% (compared to


Table 16Comparison of the lowest-lying glueball masses in units ofr0. Values in brackets denote massesobtained at a lattice spacinga = 0.10 fm and are not extrapolated to the continuum

Collab. r0m0++ r0m2++

UKQCD [67] 4.05(16) 5.84(18)Teper [68] 4.35(11) 6.18(21)GF11 [69] 4.33(10) 6.04(18)M&P [13] 4.21(15) 5.85(8)Liu [15] 4.23(22) 5.85(23)

FP action 4.12(21) [5.96(24)]AICP 4.01(15) 6.15(15)

Table 17Comparison of glueball masses in units ofr0. Values in brackets denote masses obtained at a latticespacinga = 0.10 fm and are not extrapolated to the continuum

Collab. r0m0−+ r0m0++∗ r0m2−+ r0m1+−

Teper [68] 5.94(68) 7.86(50) 8.42(78) 7.84(62)M&P [13] 6.33(13) 6.50(51) 7.55(11) 7.18(11)

FP action [6.74(42)] [8.00(35)] [7.93(78)]AICP 6.46(18) 7.66(71) 8.75(26) 9.45(71)

Fig. 12. Lattice results and continuum extrapolation of the scalar (0++) glueball mass for theanisotropic classically perfect action, together with results obtained with different other actions. Thesolid and dotted lines are the continuum extrapolations of the AICP and M&P data, respectively. Thedifferent continuum values for the Wilson action stem from different groups.


Fig. 13. Lattice results and continuum extrapolation of the tensor (2++) glueball mass for theanisotropic classically perfect action, together with results obtained from simulations employingdifferent other actions. The continuum extrapolations of theE++ and T++

2 representations areaveraged in order to get the mass of the single continuum glueball. The solid and dotted lines arethe continuum extrapolations of the AICP and M&P data, respectively. The different continuumvalues for the Wilson action stem from different groups.

about 20% for the M&P action), probably due to the scalar dip. Concerning the tensorglueball, the picture is not clear, mainly due to the considerable statistical errors of themeasurements, except for the ones obtained using the tadpole and tree-level improvedanisotropic action.

4. Repeating the spatial blocking step

As noted in Section 2, the spatial blocking step used to obtain a classically perfectξ = 2 action may be repeated straightforwardly in order to generate actions with higheranisotropies. Using another spatial scale 2 blocking step we create a classically perfectξ = 4 gauge action and parametrise it in the same way as theξ = 2 action. Furthermore,we measure the renormalised anisotropy and show that it is indeed feasible to constructactions suitable for MC simulations of heavy states such as glueballs. The classicallyperfectξ = 4 action is not yet examined thoroughly, however the parametrisation is ready(see Appendix A.5) and the analyses of theξ = 2 action (see Section 3) may be repeatedfor ξ = 4.


4.1. Construction

Each application of the spatial blocking step (slightly) renormalises the anisotropy ofthe action; the renormalised anisotropies of the final actions thus have to be measured inthe end. However, this is necessary anyhow if one is interested in comparing most results(other than, e.g., mass ratios, where the renormalisation of the anisotropy cancels out) toother collaborations or to the experiment.

In order to be able to repeat the spatial blocking step getting fromξ = 2 to ξ = 4we construct aξ = 2 action which is valid on minimisedξ = 4 configurations includinginto the fit configurations at different values ofβ over a large range. Theξ = 2 actionpresented in Section 3 is not suitable for this task as it is dedicated to be used solely onlargely fluctuating configurations aroundβ = 3.0. The construction and the parameters ofa suitable intermediateξ = 2 action are described in Appendix A.4.

Once the fine action to be used on the r.h.s. of the renormalisation group equation,Eq. (1), is ready, we may proceed along the same lines as for theξ = 2 action. We performthe non-linear fit on 2ξad-hoc= 6 configurations each atβad-hoc= 4.0,3.5,3.0,2.5,2.0. Itturns out that in order to obtain parametrisations that are free of dangerous “traps” in theu–w plane (see Section 3.1) we have to include the conditionA(u,w) > 0 at ten points(u,w) into the non-linear fit. After studying the values ofχ2 as well as the linear behaviourof the parametrisations we decide to use a set with max(k+ l)sp= 3 and max(k+ l)tm = 2.The parameters are given in Appendix A.5.

4.2. The renormalised anisotropy

To check whether the construction of the classically perfectξ = 4 action works well andwhether the parametrisation reproduces the input anisotropy, we measure the renormalisedanisotropy at one value ofβ = 3.0, using the torelon dispersion relation, following themethod described in Section 3.2. The simulation parameters are collected in Table 18.

Due to the (expected) coarser spatial lattice spacing we do not need as many smearingsteps as for theξ = 2 action and thus perform our measurements on smearing levelsSn,n= 2,4,6,8,10 keepingλs = 0.1 fixed.

Fig. 14 displays the dispersion relation so obtained, while Table 33 collects the measuredenergiesE(p2) determined using variational methods (see Appendix B of [28]), togetherwith the number of operators used in the variational method, the fit ranges and the valuesof χ2 per degree of freedom,χ2/NDF. The renormalised anisotropy as well as the torelon

Table 18Run parameters for the torelon measurements using theξ = 4 perfect action. The lattice extension intorelon directionL, the extension in the two transversal spatial directionsS as well as the temporalextensionL are given

β S2 ×L× T Sweeps/Measurements

3.0 82 × 4× 32 54000/10800


Fig. 14. Torelon dispersion relation forβ = 3.0. The straight line is the correlated fit toE2(p)=m2

T+p2 in the rangep2 = 1–5.

Table 19Results of the torelon simulations using theξ = 4 perfect action. The fit range inp2 is given in unitsof (2π/S)2

β Fit range ξR mTat χ2/NDF

3.0 1–5 3.71(8) 0.235(13) 0.82

mass are determined in exactly the same way as for theξ = 2 action. The results are listedin Table 19. Furthermore, we may again evaluate an estimate of the lattice scale using thestring picture relation, Eq. (30), yielding

√σ as = 0.532(16), and thusr0/as = 2.241(86)

which corresponds toas = 0.22 fm, at = 0.060 fm. Having a torelon of lengthLas ≈0.89 fm the error of the scale estimate is expected to be about 5–10% (see Table 5).

Concerning the repeated application of the spatial blocking method presented in thiswork to yield classically perfect gauge actions for higher anisotropies, we may concludethat the construction including the blocking as well as the parametrisation works in exactlythe same way as the construction of theξ = 2 action. The resulting action shows arenormalisation of the anisotropy of about 7% at the value ofβ = 3.00 (correspondingroughly toas 0.2 fm). Calculations of physical observables such as the static quark–antiquark potential or glueball masses using theξ = 4 classically perfect action arepresently absent.


5. Conclusions

In this work, we have presented the construction and parametrisation of a classi-cally perfect anisotropic SU(3) gauge action based on the Fixed Point Action tech-nique. The recently parametrised isotropic FP action and its parametrisation ansatz us-ing mixed polynomials of plaquettes built from single gauge links as well as fromsmeared links have been the starting point. Performing one and two purely spatial block-ing (RG) steps, respectively, starting from the isotropic action, and adapting the para-metrisation appropriately, we have obtained parametrised classically perfect actions withanisotropiesξ = 2 and 4. Theξ = 2 action has been tested extensively in measure-ments of the torelon dispersion relation, of the static quark–antiquark potential, of thedeconfining phase transition as well as of the low-lying glueball spectrum of pure gaugetheory.

The results of the torelon measurements show that the renormalisation of the anisotropy(due to quantum corrections and parametrisation artifacts) is small (below 10% forξ = 2and 4). The rotational invariance of the action has been examined by measuring the staticquark–antiquark potential, including separations corresponding to a large set of latticevectors, again with reassuring results. The glueball measurements, including some ratherheavy states, confirm that the use of anisotropic lattices facilitates spectroscopy whenheavy states are present. Compared to the isotropic simulations, with the same amountof computational work it is possible to resolve states with considerably larger energy,allowing continuum extrapolations from larger ranges of the lattice spacing. The massof the scalar glueball with quantum numbersJPC = 0++ is measured to be comparable(1580(60) MeV) to masses obtained by other groups (around 1670 MeV); however itshows large cut-off effects when measured on a lattice with spatial lattice spacingas ≈0.25 fm. This could be caused by a large sensitivity of the action to a critical point in thefundamental–adjoint coupling plane, due to adjoint terms in the action. Additionally, onehas to be aware of the manifold sources of possible errors in measurements of the glueballspectrum. Having at hand the statistics reached so far, no ultimate statement about thegoodness of the parametrised classically perfect actions can be made yet.

During our studies of both isotropic and anisotropic classically perfect actions, we havenoticed that the examination of scaling properties of lattice gauge actions is a very delicateproblem: quantities which can be reliably measured and which are not very sensitive tosystematic factors (like the volume, other states mixing with the observed ones or themethod used for extracting the mass etc.), e.g., the critical temperature or the hadronicscaler0 (for moderate lattice spacings) generally exhibit rather small cut-off effects andthus demand very large statistics, if the differences between the actions should be explicitlyidentified. On the other hand, some glueball states show large cut-off effects which makesthem interesting objects to study scaling violations. As pointed out in this work, however,there are a lot of systematic factors making the extraction of lattice artifacts difficult. Thetask of comparing different actions is even more difficult if the measurements and analysesare performed independently by different groups as this introduces additional systematicdiscrepancies and ambiguities.


Let us therefore conclude that despite the rich parametrisation which allows for all thebeautiful properties of classically perfect actions like scale invariance of instanton solutionsor rotational invariance, there is no conclusive evidence of the parametrised classicallyperfect actions (isotropic as well as anisotropic) behaving significantly better than otherimproved actions (such as the tadpole and tree-level improved anisotropic gauge action),however, there is also no evidence for the converse. Certainly, one has to consider the largeoverhead of the parametrisation as compared to other actions if one is about to plan puregauge simulations.

For the future, very accurate scaling tests comparing the classically perfect actions toother improved gauge actions are desirable. On one hand, these can include large statisticsmeasurements of the critical temperature and the hadronic scale, on the other hand, onemight perform extensive simulations of the glueball spectrum, including all the limits to betaken and systematically excluding all known sources of errors.

With respect to the glueball measurements another promising plan would be to gaincontrol about the adjoint operators present in the parametrisation of the classically perfectactions in order to circumvent the critical point in the adjoint coupling plane such that theinfluence of the scalar dip is minimised, similarly to what has been done for the tree-leveland tadpole improved actions by Morningstar and Peardon [14].

Acknowledgements

We would like to thank Ferenc Niedermayer and Peter Hasenfratz for valuablediscussions. Further thanks go to Simon Hauswirth, Thomas Jörg and Michael Marti forcomputer support. U.W. acknowledges support by a PPARC special grant.

Appendix A. Action parameters

In this appendix we collect the parameters of the FP and the classically perfect actionsthat have been used or constructed throughout this work. The isotropic FP action for coarseconfigurations has been presented already in [28]. The isotropic action in Appendix A.1is an intermediate parametrisation valid on configurations typical for MC simulations thatare minimised once, parametrised during the cascade process for the isotropic action. It isused in the spatial blocking (see Section 3.1) to minimise the coarseξ = 2 configurations,constructed using the ad-hoc anisotropic action presented in Appendix A.2. The resultingξ = 2 perfect action for coarse configurations is presented in Appendix A.3.

To repeat the blocking step, we need aξ = 2 action which is valid forξ = 4configurations minimised once in a purely spatial blocking step. This action is describedin Appendix A.4. The resultingξ = 4 action, finally, is presented in Appendix A.5.


Table 20The linear parameterspkl of the parametrised intermediate isotropic FP action

pkl l = 0 l = 1 l = 2 l = 3 l = 4

k = 0 0.629227 −0.556304 0.186662 −0.010110k = 1 −0.368095 0.852428 −0.199034 0.031614k = 2 0.389292 −0.207378 −0.010898k = 3 −0.054912 0.039059k = 4 −0.000424

A.1. The intermediate isotropic action

The intermediate isotropic action has been parametrised during the construction of theisotropic FP action. It is supposed to be valid on configurations that are obtained byminimising configurations typical for MC simulations. It fulfills theO(a2) Symanzikconditions (see [28,40]) and uses polynomials in the fluctuation parameterxµ(n), thence itsmoothly approaches the continuum limit and is expected to interpolate between the rathercoarse configurations mentioned above and the smooth limit. It is not intended to be usedin MC simulations, since its linear behaviour (see Section 3.1) is not checked.

This action has been used for the (spatial) minimisation of coarseξ = 2 configurations,i.e., to describe the isotropic configurations on the r.h.s. of Eq. (1).

The non-linear parameters describing polynomials of order 3 are (cf. formula (8), (7)and (11), (12)):

η(0) = 0.082, η(1) = 0.292353,

η(2) = 0.115237, η(3) = 0.011456,

c(0)1 = 0.282, c

(1)1 = −0.302295,

c(2)1 = −0.302079, c

(3)1 = −0.052309,

c(0)2 = 0.054, c

(1)2 = 0.298882,

c(2)2 = −0.081365, c

(3)2 = −0.023762,

c(0)3 = −0.201671, c

(1)3 = 0.022406,

c(2)3 = 0.004090, c

(3)3 = 0.014886,

c(0)4 = −0.008977, c

(1)4 = 0.245363,

c(2)4 = 0.140016, c

(3)4 = 0.028783.

The linear parameters are collected in Table 20.

A.2. “Ad-hoc” anisotropic actions

In the spatial blocking procedure described in Section 2 one needs coarse anisotropic(ξ = 2,4,6, . . .) gauge configurations which are spatially minimised leading toξ ′ = ξ/2


configurations. As the goal of this step is to obtain a perfect anisotropic action withanisotropyξ these coarse anisotropic configurations have to be produced using some otheraction that is already present. This requirement might seem to endanger the whole ansatz,however it is not crucial how the coarse configurations exactly look like as the perfectnessof the resulting coarse action comes from the perfect action on the fine configuration as wellas from the exactness of the RG transformation. Still, we try to create coarse configurationsthat might look similar to future ensembles produced using the perfect anisotropic actionon the coarse level and whose minimised configurations appear to have an anisotropyapproximatelyξ ′ = ξ/2.

In order to achieve this, we modify the isotropic FP action by adding a term(ξ2 − 1)pst10

(wherepst10 denotes the simple temporal plaquette). This modification turns the isotropic

Wilson action into the Wilson action with bare anisotropyξ and is expected to workapproximately also for our FP action. The main argument of using this coarse action andnot, e.g., the anisotropic Wilson action is that due to the spatial lattice spacingas beinglarger than the temporal spacingat , theO(a2

s ) artifacts are also expected to be larger thantheO(a2

t ) effects. The modification described above should (approximately) preserve theFP properties corresponding toas and is thus expected to be considerably better than thenaive anisotropic Wilson action.

Using this ad-hoc modification of the isotropic FP action (“ad-hoc” action) in MCsimulations shows (as expected) that (at least for small anisotropiesξ 5) its properties onthe glueball spectrum are not comparable with the ones of the isotropic FP action, howeverthe generated ensembles resemble to the ones generated with true perfect anisotropicactions much more than ensembles generated with the Wilson action.

The anisotropiesξad-hoc that have to be used to generate coarse anisotropic configura-tions turning into minimised configurations with anisotropyξ ′ areξad-hoc≈ 3.2 for ξ ′ = 1andξad-hoc≈ 6 for ξ ′ = 2. However, this value ofξad-hoc varies considerably withβ—butas stated at the beginning of this section, the exact form of the coarse configurations is notessential.

A.3. Theξ = 2 perfect action

Theξ = 2 perfect action uses the parametrisation described in Section 2.3. The numberof non-zero asymmetry valuesη(0)i is 4, the parametersc(0)i (i = 1, . . . ,3) are splitted into 3parameters depending on the contribution to the smeared plaquette. The linear parameterspkl are non-zero for 0< k + l 4 for spatial plaquettes and 0< k + l 3 for temporalplaquettes.

The non-linear parameters (constants inxµ) have the values

η1 = −0.866007, η2 = −0.884110, η3 = 2.212499, η4 = 1.141177,

c11 = 0.399669, c12 = 0.519037, c13 = −0.071334,

c21 = −0.076357, c22 = −0.031051, c23 = −0.282800,

c31 = 0.032396, c32 = −0.015844, c33 = −0.046302.



Table 21The linear parameters of theξ = 2 parametrised classically perfect action

psskl

l = 0 l = 1 l = 2 l = 3 l = 4

k = 0 0.433417 0.098921 −0.116251 0.023295k = 1 0.217599 −0.272668 0.248188 −0.045278k = 2 0.316145 −0.180982 0.028817k = 3 −0.039521 0.003858k = 4 0.005443

pstkl l = 0 l = 1 l = 2 l = 3

k = 0 −0.190195 0.554426 −0.121766k = 1 1.521212 −0.328305 0.086655k = 2 0.011178 0.020932k = 3 0.022856

A.4. Theξ = 2 intermediate action

In order to be able to repeat the spatial blocking step constructing aξ ≈ 4 action basedon theξ = 2 perfect action we need a parametrisation of theξ = 2 action which is validon (ξ ≈ 2) configurations that are obtained by spatially minimising coarseξ = 4 config-urations once. To construct such an action, we perform a non-linear fit to the derivativesof 5 sets of two configurations each atβad-hoc= 6,10,20,50,100. The non-linear para-meters are chosen to be linear in the fluctuation parameterxµ(n). Having four differentparametersη and splitting upci into three parameters (as it is done for all anisotropic para-metrisations), this makes 20 non-linear parameters to be fitted which is quite at the edge ofwhat is still possible on our computers and that is why we restrict the total number of con-figurations to 10. Rough checks performed on a larger number of configurations, with aneven larger number of parameters show however, that the resulting non-linear parametersare stable and describe the data accurately.

The derivatives and action values of 5 sets of 10 configurations each are included in thelinear fit (where the relative weight of the action values is chosen to be 1.9× 10−2 for theconfigurations atβad-hoc= 50,100 and 7.6 × 10−4 at β = 6,10,20). A linear set wherethe parameterspkl are non-zero for 0< k + l 3 for spatial plaquettes and 0< k + l 4for temporal plaquettes describes the full action very well concerning this data. Again, thisparametrisation is not intended to be used in MC simulations, thus the linear behaviour ofthe action is not checked.

The non-linear parameters (up to first order inxµ) have the values

η(0)1 = −1.861267, η

(1)1 = −0.327466,

η(0)2 = −1.075610, η

(1)2 = −0.550398,

η(0)3 = 2.750293, η

(1)3 = 0.089874,


Table 22The linear parameters of the intermediateξ = 2 parametrised classically perfect action

psskl

l = 0 l = 1 l = 2 l = 3

k = 0 0.088016 0.002225 −0.000285k = 1 0.341850 −0.015888 −0.004087k = 2 −0.053007 0.010121k = 3 0.010500

pstkl l = 0 l = 1 l = 2 l = 3 l = 4

k = 0 0.280043 5.077727 −13.714872 12.739964k = 1 1.343946 −6.934825 27.673937 −32.288928k = 2 2.069084 −17.392027 28.248910k = 3 3.691733 −9.584760k = 4 0.712244

η(0)4 = 1.107017, η

(1)4 = 0.265817,

c(0)11 = 0.520960, c

(1)11 = 0.006339,

c(0)21 = −0.075219, c

(1)21 = 0.059506,

c(0)12 = 0.266240, c

(1)12 = 0.121035,

c(0)22 = −0.080771, c

(1)22 = −0.021515,

c(0)13 = 0.159372, c

(1)13 = 0.039564,

c(0)23 = −0.043901, c

(1)23 = 0.009672.


A.5. Theξ = 4 perfect action

Theξ = 4 perfect action uses the parametrisation described in Section 2.3. The numberof non-zero asymmetry valuesη(0)i is 4, the parametersc(0)i (i = 1, . . . ,3) are splitted into 3parameters depending on the contribution to the smeared plaquette. The linear parameterspkl are non-zero for 0< k + l 3 for spatial plaquettes and 0< k + l 2 for temporalplaquettes.

The non-linear parameters (constants inxµ) have the values

η1 = −1.491457, η2 = −1.115141,

η3 = 1.510985, η4 = 7.721347,

c11 = 2.014408, c12 = 0.128768, c13 = 0.162296,

c21 = −0.915620, c22 = 0.134445, c23 = −0.013383,

c31 = 1.166289, c32 = 0.061278, c33 = 0.000759,



Table 23The linear parameters of theξ = 4 parametrised classically perfect action

psskl

l = 0 l = 1 l = 2 l = 3

k = 0 0.027625 0.000052 0.000000k = 1 0.072131 −0.016852 −0.000054k = 2 0.036818 0.003558k = 3 −0.007413

pstkl

l = 0 l = 1 l = 2

k = 0 0.795779 0.621286k = 1 2.130563 −0.286602k = 2 0.076086

Appendix B. Collection of results

Table 24Collection of results of the torelon measurements using theξ = 2 perfect action. For eachβ-valueand momentump2 = p2

x + p2y we list the number of operatorsM kept after the first truncation

in the variational method, the plateau region on which the fit of the correlators to the formZ(p2)exp(−tE(p2)) is performed (fit range), as well as the extracted energy together with theχ2

per degree of freedom (χ2/NDF)

β p2 M Fit range atE(p2) χ2/NDF

3.0 0 2 1–5 1.368(30) 0.19

1 3 1–6 1.372(17) 0.49

2 3 1–6 1.451(17) 0.28

4 3 1–8 1.570(21) 0.46

5 3 1–6 1.610(19) 0.34

8 2 1–3 1.707(29) 0.05

3.15a 0 3 2–5 0.688(16) 0.11

1 4 1–5 0.806(5) 0.06

2 4 1–5 0.902(6) 0.59

4 3 1–5 1.059(8) 2.35

5 4 1–5 1.151(7) 0.16

8 4 1–5 1.310(15) 0.71

9 4 1–5 1.356(23) 1.58

10 4 1–3 1.470(17) 0.42

13 3 1–3 1.621(23) 0.01

18 3 1–3 1.762(47) 0.56

(continued on next page)


Table 24 — continued


3.15b 1 3 1–5 1.418(23) 0.05

2 3 1–5 1.541(37) 0.53

4 3 1–4 1.751(67) 1.20

5 3 1–5 1.922(104) 1.08

3.3 0 3 1–10 0.318(6) 1.39

1 3 1–10 0.417(3) 1.20

2 3 1–5 0.493(3) 0.13

4 4 1–4 0.633(4) 1.26

5 5 1–10 0.688(3) 0.84

8 5 1–8 0.830(5) 1.21

9 5 1–6 0.877(6) 0.42

10 5 1–10 0.921(4) 0.77

13 5 1–10 1.036(6) 0.30

18 4 1–6 1.205(12) 0.15

3.5a 1 5 3–8 0.305(6) 0.10

2 5 1–10 0.407(4) 1.65

4 4 1–10 0.538(4) 0.96

5 5 1–10 0.586(3) 1.16

8 5 1–7 0.721(4) 0.09

9 5 2–6 0.733(10) 0.11

10 4 1–8 0.795(4) 0.96

13 5 1–6 0.901(4) 1.26

18 5 1–6 1.058(9) 1.45

3.5b 1 4 3–7 0.321(5) 1.00

2 5 1–8 0.432(3) 0.81

4 5 1–8 0.586(4) 0.29

5 5 1–7 0.656(3) 1.24

8 5 2–7 0.806(14) 0.36

9 5 1–7 0.859(7) 0.74

10 5 1–8 0.913(4) 0.68

13 5 2–5 1.010(16) 0.32

18 3 1–5 1.215(12) 0.17


Table 25Collection of results of the off-axisqq measurements using theξ = 2 perfect action atβ = 3.30. Foreach separation vector we list the length of the vector in spatial lattice units, the number of operatorsM kept after the first truncation in the variational method, the plateau region on which the fit of thecorrelators to the formZ(r)exp(−tE(r)) is performed (fit range), as well as the extracted energytogether with theχ2 per degree of freedom (χ2/NDF)

r |r| M Fit range atV (r) χ2/NDF

(1,0,0) 1 3 3–10 0.3006(1) 0.20

(1,1,0) 1.414 3 3–10 0.3814(2) 0.21

(1,1,1) 1.732 3 3–10 0.4250(3) 0.65

(2,0,0) 2 3 6–10 0.4469(6) 0.22

(2,1,0) 2.236 3 4–10 0.4752(4) 0.41

(2,1,1) 2.449 3 4–10 0.4971(5) 0.83

(2,2,0) 2.828 3 4–10 0.5309(7) 0.59

(3,0,0) 3 3 4–10 0.5445(7) 1.71

(2,2,1) 3 3 5–10 0.5456(8) 0.62

(2,2,2) 3.464 3 4–10 0.5852(9) 1.53

(4,0,0) 4 3 4–10 0.6269(1) 0.51

(3,3,0) 4.243 3 5–10 0.6455(16) 0.94

(4,2,0) 4.472 3 5–10 0.6630(14) 0.86

(4,2,2) 4.899 3 5–10 0.6964(18) 0.34

(5,0,0) 5 2 5–10 0.7044(22) 1.24

(3,3,3) 5.196 3 4–10 0.7215(17) 1.31

(4,4,0) 5.657 2 6–10 0.7509(41) 1.23

(4,4,2) 6 3 5–10 0.7783(25) 0.77


Table 26Collection of results of the off-axisqq measurements using theξ = 2 perfect action atβ = 3.00. Foreach separation vector we list the length of the vector in spatial lattice units, the number of operatorsM kept after the first truncation in the variational method, the plateau region on which the fit of thecorrelators to the formZ(r)exp(−tE(r)) is performed (fit range), as well as the extracted energytogether with theχ2 per degree of freedom (χ2/NDF)

r |r| M Fit range atV (r) χ2/NDF

(1,0,0) 1 3 2–7 0.5053(3) 0.70

(1,1,0) 1.414 3 3–5 0.7322(7) 0.01

(1,1,1) 1.732 3 3–8 0.8902(14) 0.60

(2,0,0) 2 3 2–8 0.9407(10) 0.89

(2,1,0) 2.236 3 2–8 1.0642(11) 0.36

(2,1,1) 2.449 2 2–5 1.1676(13) 0.27

(2,2,0) 2.828 3 2–6 1.3082(18) 0.17

(3,0,0) 3 3 2–6 1.3392(25) 0.16

(2,2,1) 3 3 2–5 1.3867(20) 0.26

(2,2,2) 3.464 3 2–6 1.5675(55) 0.84

(4,0,0) 4 2 1–5 1.7429(24) 0.14

(4,2,0) 4.472 3 2–6 1.9328(93) 1.47

(4,2,2) 4.899 3 1–4 2.1197(34) 0.15

Table 27Collection of results of the on-axisqq measurements using theξ = 2 perfect action atβ = 3.15,3.50. For each separation along the axes we list the number of operatorsM kept after the firsttruncation in the variational method, the plateau region on which the fit of the correlators to theformZ(r)exp(−tE(r)) is performed (fit range), as well as the extracted energy together with theχ2


β r M Fit range atV (r) χ2/NDF

3.15 1 5 3–10 0.3703(2) 0.456

2 5 3–7 0.6110(4) 0.735

3 5 3–10 0.8019(8) 1.209

4 5 3–7 0.9775(16) 1.238

5 5 4–9 1.129(8) 0.519

3.50 1 5 3–12 0.25910(5) 1.163

2 5 3–10 0.36490(12) 0.769

3 5 7–12 0.42168(38) 0.664

4 5 5–11 0.46587(51) 0.205

5 5 5–12 0.50388(76) 1.139

6 5 5–11 0.54051(96) 0.853


Table 28Results from fits to theξ = 2, β = 3.15 glueball correlators on the 83 × 16 lattice in units of thetemporal lattice spacingat : t0/t1 are used in the generalised eigenvalue problem,N is the number ofinitial operators measured andM denotes the number of operators kept after the truncation inC(t0)

Channel N t0/t1 M Fit range χ2/NDF Energies

A++1 87 1/2 12 1–4 0.69 0.645(10)

A++1

∗59 1/2 10 1–3 1.23 1.365(104)

E++ 47 1/2 7 1–3 0.10 1.405(29)

E++∗ 47 1/2 5 1–3 0.17 1.985(140)

T++2 47 1/2 10 1–4 0.10 1.416(32)

T++2

∗22 1/2 6 1–3 1.20 1.990(131)

A++2 21 1/2 2 1–3 0.03 1.675(147)

T++1 36 1/2 4 1–3 0.34 1.881(131)

T++1

∗36 1/2 6 1–3 0.31 2.170(182)

A−+1 25 1/2 3 1–3 0.001 1.588(114)

E−+ 25 1/2 2 1–3 0.01 1.849(81)

E−+∗ 25 1/2 2 1–3 2.19 2.084(268)

T−+2 64 1/2 4 1–3 0.02 1.850(83)

T+−1 61 1/2 4 1–3 0.24 1.798(59)

T+−2 18 1/2 4 1–3 0.41 1.935(177)

Table 29Results from fits to theξ = 2, β = 3.3 glueball correlators on the 103 × 20 lattice in units of thetemporal lattice spacingat : t0/t1 are used in the generalised eigenvalue problem,M denotes thenumber of operators kept after the truncation inC(t0)

Channel t0/t1 M Fit range χ2/NDF Energies

A++1 1/2 9 2–7 0.90 0.590(17)

A++1

∗1/2 11 1–3 1.57 1.133(53)

E++ 1/2 11 1–5 0.97 0.983(17)

E++∗ 1/2 8 1–3 0.90 1.453(68)

T++2 1/2 12 1–4 1.09 0.965(15)

T++2

∗1/2 10 1–4 1.68 1.386(51)

A++2 1/2 4 1–3 0.01 1.511(94)

T++1 1/2 8 1–5 0.84 1.492(62)

A−+1 1/2 6 1–4 4.38 1.017(39)

E−+ 1/2 3 1–4 0.74 1.365(36)

T−+2 1/2 4 1–5 1.16 1.357(25)

T+−1 1/2 8 1–4 0.95 1.276(33)


Table 30Results from fits to theξ = 2, β = 3.5 glueball correlators on the 123 × 24 lattice in units of thetemporal lattice spacingat : t0/t1 are used in the generalised eigenvalue problem,N is the number ofinitial operators measured andM denotes the number of operators kept after the truncation inC(t0)

Channel N t0/t1 M Fit range χ2/NDF Energies

A++1 91 1/2 24 2–7 0.31 0.405(13)

A++1

∗110 0/1 110 4–6 0.30 0.720(158)

E++ 104 1/2 23 2–5 0.16 0.675(25)

E++∗ 104 1/2 14 1–3 0.70 1.183(55)

T++2 53 1/2 19 2–5 0.31 0.681(12)

T++2

∗53 1/2 22 1–3 0.69 1.128(23)

A++2 15 1/2 6 1–3 0.05 1.212(42)

A++2

∗15 1/2 6 1–3 1.12 1.568(130)

T++1 23 1/2 11 1–4 0.05 1.245(25)

A−+1 15 1/2 9 1–3 0.17 0.754(23)

A−+1

∗15 1/2 9 1–3 0.61 1.140(112)

E−+ 21 1/2 14 1–3 0.27 0.942(24)

E−+∗ 21 1/2 11 1–3 1.20 1.406(80)

T−+2 75 1/2 9 2–5 0.32 0.952(38)

T−+2

∗75 1/2 6 1–3 1.70 1.675(59)

T+−1 53 1/2 14 1–3 0.69 1.013(17)

A+−2 11 1/2 5 1–3 0.07 1.224(64)

A+−2

∗11 1/2 5 1–3 0.79 1.726(147)

T+−2 22 1/2 9 2–4 0.33 1.142(121)

T+−2

∗12 1/2 9 1–3 1.78 1.663(139)

E+− 50 0/1 50 1–3 1.01 1.379(34)

T –1 80 0/1 80 2–4 0.57 1.200(67)

T –2 80 0/1 80 2–5 0.20 1.205(81)

A–2 35 0/1 35 1–3 0.16 1.352(37)

A–1 30 0/1 30 1–4 0.14 1.703(99)


Table 31Results of the continuum extrapolations of selected glueball representations in terms of the hadronicscaler0. The continuum spin assignmentJ , the terms in the fit, constant (c) or (as/r0)

2 (c2), andthe goodness of the fit,χ2/NDF, are also given

Channel J Terms in the fit χ2/NDF r0mG

A++1 0 c, c2 2.72 4.01(15)

A++1

∗0 c, c2 1.43 7.66(71)

E++ 2 c, c2 1.06 6.16(24)

E++∗ 2 c, c2 3.85 10.59(63)

T++2 2 c, c2 1.98 6.14(18)

A++2 3 c, c2 1.65 11.65(57)

A−+1 0 c 2.50 6.46(18)

E−+ 2 c, c2 1.06 8.73(34)

T−+2 2 c, c2 1.07 8.77(39)

Table 32Results of the continuum extrapolations of selected glueball ratios,mG/mT ++

2. The continuum spin

assignmentJ , the terms in the fit, constant (c) or (as/r0)2 (c2), and the goodness of the fit,χ2/NDF,

are also given. The last column lists the masses converted to units ofr−10 using the continuum result

for the mass of the tensor glueballT++2 , values in bold face will be used further on

Γ PC J Terms in the fit χ2/NDF mΓ PC /mT ++2

r0mG

E++ 2 c 1.20 1.01(2) 6.20(30)

E++∗ 2 c, c2 2.20 1.74(12) 10.68(105)

T++2

∗2 c, c2 3.01 1.70(8) 10.44(80)

A++2 3 c, c2 1.11 1.90(11) 11.66(102)

T++1 3 c, c2 2.60 1.90(8) 11.66(83)

A−+1 0 c 1.39 1.09(4) 6.69(44)

T−+2 2 c, c2 1.23 1.45(8) 8.90(75)

T+−1 1 c, c2 2.29 1.54(7) 9.45(71)


Table 33Collection of results of the torelon measurement atβ = 3.0, using theξ = 4 perfect action. Foreach momentump2 = p2

x + p2y we list the number of operatorsM kept after the first truncation

in the variational method, the plateau region on which the fit of the correlators to the formZ(p2)exp(−tE(p2)) is performed (fit range), as well as the extracted energy together with theχ2



3.0 1 5 3–10 0.317(5) 1.34

2 3 4–10 0.375(8) 0.31

4 3 3–11 0.487(7) 0.65

5 5 4–11 0.520(11) 0.35

8 5 3–7 0.663(17) 0.09

9 3 4–11 0.754(37) 0.87

10 5 4–11 0.678(32) 1.56

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Torsion, topology and CPT anomaly intwo-dimensional chiralU(1) gauge theory

F.R. Klinkhamer, C. MayerInstitut für Theoretische Physik, Universität Karlsruhe, D-76128 Karlsruhe, Germany

Received 1 June 2001; accepted 17 September 2001

Abstract

We consider the CPT anomaly of two-dimensional chiralU(1) gauge theory on a torus withtopologically nontrivial zweibeins corresponding to the presence of spacetime torsion. The resultingchiral determinant can be expressed in terms of the standard chiral determinant without torsion, butwith modified spinor boundary conditions. This implies that the two-dimensional CPT anomaly canbe moved from one spin structure to another by choosing appropriate zweibeins. Similar results applyto higher-dimensional chiral gauge theories. 2001 Elsevier Science B.V. All rights reserved.

PACS:04.20.Gz; 02.40.Pc; 11.15.-q; 11.30.ErKeywords:Torsion; Topology; Chiral gauge theory; CPT violation

1. Introduction

Recently, it has been shown that there is a violation of CPT invariance in certain(non-)Abelian chiral gauge theories defined on nonsimply connected spacetime manifolds[1]. The well-known CPT theorem [2,3] is evaded by the breakdown of Lorentz invariancedue to the quantum effects of the chiral fermions [1,4].

The Abelian CPT anomaly is particularly obvious on the two-dimensional torus [5]where the chiral determinant can be computed exactly (see Ref. [6] and references therein).For this reason, we will consider in this paper primarily two-dimensional chiralU(1) gaugetheory defined over the torus. More precisely, we study the CPT anomaly on the torus inthe presence of spacetime torsion. That is, we consider the effects of a nontrivial configu-ration of zweibeins, which gives rise to a nonvanishing torsion tensor [7,8]. (Zweibeins arethe two-dimensional analogs of vierbeins or tetrads in four dimensions.)

The main goal of the present paper is then to understand better the role of topologyand spin structure for the two-dimensional CPT anomaly, by studying the response of the

E-mail addresses:[email protected] (F.R. Klinkhamer),[email protected] (C. Mayer).



216 F.R. Klinkhamer, C. Mayer / Nuclear Physics B 616 (2001) 215–232

chiral determinant to the introduction of topologically nontrivial zweibeins on the torus.For a general discussion of spinors over nonsimply connected spacetime manifolds, werefer the reader, in particular, to Refs. [9,10].

The paper is organized as follows. In Section 2, we discuss some aspects of the geometryof the two-dimensional torus with torsion and establish our notation. Specifically, wemention two consequences of torsion at the level of the spacetime structure. Namely,parallelograms need not close and extremal and autoparallel curves need not coincide.We also comment on some interesting properties of topologically nontrivial zwei- (orvier-)beins on the torus and their possible origin.

In Section 3, we show that one can relate the fermionic Lagrangian with nontrivialzweibeins to the Lagrangian with trivial zweibeins by a simple spinor redefinition. Thisfield redefinition can, however, change the spinor boundary conditions.

In Section 4, we use this property of the fermionic Lagrangian to express the chiraldeterminant for topologically nontrivial zweibeins in terms of the chiral determinantfor trivial zweibeins but modified spinor boundary conditions. We also give a heuristicargument for the result. A similar calculation is done in Appendix A for the Diracdeterminant of a vector-likeU(1) gauge theory.

In Section 5, we discuss the role of torsion for the two-dimensional chiral CPT anomalyand find that only the topologically nontrivial part of the zweibeins affects the anomaly.

In Section 6, we summarize our results and briefly comment on the four-dimensionalcase.

2. Topology, geometry and torsion

2.1. Zweibeins on the torus

The Cartesian coordinatesxµ ∈ [0,L], µ= 1,2, are taken to parameterize a particulartwo-dimensional torusT 2[i], with modulus (Teichmüller parameter)τ = i. This toruscan be thought of as a square with flat Euclidean metricgµν(x) = δµν ≡ diag(1,1) andopposite sides identified; see Fig. 1.

Zweibeins locally define an orthonormal basis of one-forms

(2.1)ea(x)= eaµ(x)dxµ.

Here, Latin indices (a, b, . . .) refer to the local frame and Greek indices (µ,ν, . . .) tothe base space. Throughout this paper, summation over equal upper and lower indices isunderstood. The inverse zweibeinseµa (x) are defined by

(2.2)δab = eµa (x)e

bµ(x).

In terms of the zweibeins, the metric can be expressed as follows (see, for example,Refs. [7,8]):

(2.3)gµν(x)= eaµ(x)ebν(x)δab.

F.R. Klinkhamer, C. Mayer / Nuclear Physics B 616 (2001) 215–232 217

Fig. 1. The torusT 2[i] is represented as a square with opposite sides identified. Two distinctnoncontractible curves are labeleda andb.

Inversely, this equation defines the zweibeins, but only up to a space-dependent orthogonaltransformation.

In the following, we consider zweibeinseaµ(x) taking values in a matrix representationof the groupSO(2):

(2.4)

(e1µ(x)

e2µ(x)

)≡ δ1

µ

(cosϕ(x)sinϕ(x)

)+ δ2

µ

(−sinϕ(x)cosϕ(x)

),

parametrized by the real functionϕ(x). The corresponding metric is flat,gµν(x) = δµν .Obviously, the choiceϕ(x)= 0 yields the trivial zweibeinseaµ(x)= δaµ. The extra degreeof freedomϕ(x) in the zweibeins (2.4) can, however, generate torsion, as will be shown inthe next subsection.

The zweibeins (2.4) need to be defined in a consistent way onT 2[i], namely

(2.5)eaµ(0, x2)= eaµ

(L,x2), eaµ

(x1,0

)= eaµ(x1,L

).

This requirement naturally leads to a decomposition of theSO(2) rotation angleϕ(x) intotopologically trivial and nontrivial parts:

(2.6)ϕ(x)≡ ω(x)+ χ(x),

where the real functionω(x) is taken to be strictly periodic inx1 andx2, with periodL.The other real functionχ(x) is associated with the two generating curvesa andb of thehomology groupH1(T 2,Z)= Z ⊕ Z; cf. Ref. [8]. Specifically, the functionχ(x) is givenby

(2.7)χ(x)≡ (2π/L)(mx1 + nx2), m,n ∈ Z,

for the two distinct noncontractible curvesa and b shown in Fig. 1. (The notationχ(x;m,n) would, of course, be more accurate.)

2.2. Connection

The condition for parallel transport of an arbitrary vector fieldCµ(x) along theinfinitesimal path(x, x + δx) is

(2.8)Cλ(x)eaλ(x)=(Cλ(x)+ δCλ(x)

)eaλ(x + δx).


With the bilinearAnsatz[11]

(2.9)δCλ(x)≡−Γ λµν(x)C

µ(x)δxν,

condition (2.8) enables us to express the Riemann–Cartan connectionΓ λµν(x) in terms of

the zweibeins:

(2.10)Γ λµν(x)= eλa(x)e

aµ,ν(x)=

(δλ2δ

1µ − δλ1δ

2µ

)∂νϕ(x),

for the particular zweibeins (2.4) which have vanishing metric Christoffel symbol [8,9]. Asusual, the notationφ,ν stands for∂φ/∂xν . The result (2.10) demonstrates that the freedomin choosing a particular connection onT 2[i] is a consequence of the fact that for a fixedmetricgµν(x)= δµν the zweibeins are defined only up to an orthogonal transformation.

Since the zweibeins considered take values in a two-dimensional matrix representationof the groupSO(2), we can write the transformation of the connection under anSO(2)transformationeµa (x) as

(2.11)Γ ′aµb = eaλΓ

λµν e

νb +

(∂µe

aν

)eνb.

While the connection itself does not transform as a tensor, its antisymmetric component inthe two lower indices does. This object is called the torsion tensor [7,8]

(2.12)T λµν(x)≡ Γ λ

µν(x)− Γ λνµ(x).

For the connection (2.10), we obtain

(2.13)T λ12(x)=−T λ

21(x)= ∂λϕ(x),

where the functionϕ(x) parameterizes the zweibeins (2.4).Parallel transport enables us to give an “operational definition” of torsion (see also

Refs. [7,11]). Consider the parallelogram spanned by the line elements(x,E1 · x) and(x,E2 · x), where the infinitesimal displacements ofx are given by

(E1 · x)µ = xµ + εµ1 (x), ε

µ1 (x)≡ εa1e

µa (x),

(2.14)(E2 · x)µ = xµ + εµ2 (x), ε

µ2 (x)≡ εa2e

µa (x),

with real infinitesimal coefficientsεa1 andεa2 . There is then torsion if the parallelogramdoes not close, i.e.,[E2E1 −E1E2] · x = 0. In fact, a short calculation yields

(2.15)

([E2E1 −E1E2] · x)µ = [

Γ µρσ (x)− Γ µ

σρ(x)]ερ2 (x)ε

σ1 (x)= T µ

ρσ (x)ερ2 (x)ε

σ1 (x),

where Eq. (2.9) has been used for the parallel transport ofεµ1 (x) andεµ2 (x).

2.3. Extremal and autoparallel curves

Apart from the fact that parallelograms do not close in the presence of torsion, thereis a further consequence of torsion at the level of the spacetime structure: extremal andautoparallel curves do not necessarily coincide (see also Ref. [7]).


The equations for an extremal curve (shortest or longest line) on the two-dimensionalflat spacetime manifoldT 2[i] are given by

(2.16)x1(τ )= 0, x2(τ )= 0,

where the dot denotes differentiation with respect to the affine parameterτ . Extremalcurves onT 2[i] may or may not close. An example of an extremal curve that doesnotclose is given byx1 = τ andx2 =√

2τ , since√

2 is an irrational number. Recall that theratio of the periodicities inx1 andx2 is exactly 1 for the particular torusT 2[i] considered;see Fig. 1.

The equations for an autoparallel curve (straightest line) can be deduced by requiringthat autoparallel curves are always tangent to the zweibeins. The substitutionCµ =dxµ(τ)/dτ ≡ xµ(τ ) in Eq. (2.9) gives the following result:

(2.17)xλ + Γ λµν(x)x

µxν = 0.

Note that the Riemann–Cartan connectionΓ λµν in Eq. (2.10) also has a symmetric part in

µν. For the special case ofω(x)= 0 in Eq. (2.6), the equations become

x1 − (2π/L)(mx1 + nx2)x2 = 0,

(2.18)x2 + (2π/L)(mx1 + nx2)x1 = 0.

Since the general solution of these coupled differential equations can be quite involved, weonly discuss the class of solutions satisfying

(2.19)mx1 + nx2 = 0.

In this case, the Eq. (2.18) reduce to Eq. (2.16), but with the additional constraint (2.19)on the constant velocitiesxµ. One easily recognizes that Eq. (2.19) describes autoparallelcurves by noting that solutions to this equation are curves of constantχ ; see Eq. (2.7).

In contrast to extremal curves, which may or may not close on the torusT 2[i], theautoparallel curves given by Eqs. (2.18) and (2.19) always close, since their slopes arerational (x2/x1 = −m/n). More specifically, the extremal curve(x1, x2) = (τ,

√2τ )

mentioned a few lines below Eq. (2.16) doesnot solve Eq. (2.18) and is, therefore, notautoparallel, as long as the torsion parameter function is topologically nontrivial,ϕ(x)=χ(x) = 0. This clearly shows the difference of the two types of curves in the presence oftorsion.

2.4. Topologically nontrivial vielbeins

In this last subsection on geometry, we elaborate on the special nature of topologicallynontrivial zweibeins (2.4)–(2.7) with(m,n) = (0,0) andω(x) = 0. For these zweibeins,namely, the torsion tensor (2.13) would be constant (and nonzero) over the whole spacetimemanifold T 2[i]. This would then correspond to a new local property of spacetime. Onemanifestation would be the nonclosure of parallelograms as discussed in Section 2.2.

For topologically nontrivial zweibeins (2.4)–(2.7) with(m,n)= (1,1) andω(x)= 0, atypical parallelogram obtained by parallel transport, with lengths|εµ1 | = |εµ2 | = ", would


fail to close by a distance (2.15) of order

(2.20)(2π/L)"2 ∼ 10−25 m(1010 lyr/L

)("/m)2,

which would still be 10 orders of magnitude above the Planck lengthlP ≡√hG/c3. As will

be discussed in Section 6, a similar torsion effect may occur in four (or more) dimensions,for appropriate vier- (or viel-)beins. The level of accuracy indicated by Eq. (2.20) might be,in principle, within reach of experiment. (We have in mind a rapidly rotating (∼ 100 Hz)experimental setup in a free-fall environment (e.g., in a drag-free satellite). See, forexample, Eq. (3.35) in Ref. [12] for the optimal sensitivity of a resonant-bar detector forperiodic gravitational waves.)

Throughout this paper, we consider the zwei- or vierbeins as fixed classical backgroundfields. Let us, however, briefly remark on the possible origin of translation-invariant torsionresulting from topologically nontrivial vierbeins (see also Section 6). The crucial pointis that this type of torsion would not have to be generated dynamically by a local spindensity, but could perhaps arise as a kind of boundary condition (most likely, set at thebeginning of our universe).1 Moreover, the spin density can only be expected to give anegligible contribution to the torsion tensor for the present cosmological number densitiesn of protons or electrons. In fact, the order of magnitude to be compared with Eq. (2.20) is

(2.21)(Gc−3)(hn)"2 ∼ n l2P"

2 ∼ 10−70 m(n/m−3)("/m)2.

(See also Section V A 3 of Ref. [7].) Hence, the translation-invariant torsion fromtopologically nontrivial vierbeins may at present be an extremely weak effect, but theeffect is still many orders of magnitude above that expected from the ordinary matter ofthe universe.

3. Fermionic Lagrangian

We use a “chiral” basis for the two-dimensional Dirac matrices

(3.1)

γ 1 ≡(

0 +1+1 0

), γ 2 ≡

(0 +i−i 0

), γS ≡ iγ 1γ 2 =

(+1 00 −1

),

whereγS anticommutes withγ 1 and γ 2. (The suffix S stands for “strong reflection”,originally introduced by Pauli in the proof of the CPT theorem [2,3].)

For trivial zweibeinseaµ(x)= δaµ ≡ 1, the manifestly Hermitian Lagrangian is given by

(3.2)L[Ψ ,Ψ,A,1

]= (i/2)Ψγµ( →∂µ +iAµ

)Ψ − (i/2)Ψ ( ←

∂µ −iAµ

)γ µΨ,

with constant gamma matrices

(3.3)γ µ ≡ δµa γa.

1 Note that the two-dimensional gravitational Einstein–Cartan action [7], based on the Ricci scalar defined interms of the connection (2.10), gives the field equation∂2ϕ = 0, which is trivially solved by the configuration(2.7). The situation in four dimensions is less satisfactory, as will be discussed in Section 6.


This Lagrangian is invariant under globalSO(2) transformations (withγS as definedabove),

(3.4)Ψ (x)→ e−iκγS/2Ψ(x ′), Ψ (x)→ Ψ (x ′)eiκγS/2,

and localU(1) gauge transformations,

(3.5)Ψ (x)→ eiξ(x)Ψ (x), Ψ (x)→ Ψ (x)e−iξ(x),

supplemented by the usual transformations of the gauge fieldAµ(x). For the basis ofgamma matrices (3.1), the two (independent) Dirac spinors can be decomposed into fourone-component Weyl spinors:

(3.6)Ψ (x)≡(ψR(x)

ψL(x)

), Ψ (x)≡ (

ψR(x) ψL(x)),

where(1∓ γS)/2 projects on the left- and right-moving subspaces of solution space.Throughout this paper, we consider only topologically trivial gauge potentialsAµ(x).

We therefore take theU(1) gauge potentialAµ(x) to be periodic inx1 andx2, with periodL. The spinors are allowed to have either periodic or antiperiodic boundary conditions:

Ψ(x1 +L,x2)=−e2πiθ1Ψ

(x1, x2),

(3.7)Ψ(x1, x2 +L

)=−e2πiθ2Ψ(x1, x2).

(The adjoint spinorsΨ (x) obey the same boundary conditions.) The variablesθ1, θ2 ∈0,1/2 then fix the spinor boundary conditions, with(θ1, θ2) = (1/2,1/2) correspondingto doubly-periodic boundary conditions and(θ1, θ2) = (0,0) to doubly-antiperiodicboundary conditions. Mixed spinor boundary conditions correspond to(θ1, θ2) = (1/2,0)or (0,1/2). The four possible combinations of(θ1, θ2) are said to define the four spinstructures over the torus.

For the general zweibeins (2.4), the Lagrangian (3.2) becomes

(3.8)L[Ψ ,Ψ,A, e

]= (i/2)Ψ γ µDµΨ + h.c.,

with space-dependent gamma matrices

(3.9)γ µ(x)≡ eµa (x)γa,

and covariant derivatives

(3.10)DµΨ ≡ (∂µ + iAµ + iΩµ)Ψ.

The Lagrangian (3.8) is invariant under gaugedSO(2) transformations (3.4) due to thepresence of the spin connection [8,9]

Ωµ(x)≡ Γ abµ (x) σab/2= eaν,µe

bνσab/2=−γS∂µϕ(x)/2,

(3.11)σab ≡ i[γa, γb]/4,provided Eq. (2.11) is used for the transformations. Here, the generator ofSO(2)transformations is given byσ12 =−σ21 = γS/2, for the representation of gamma matriceschosen.


It is now possible to rewrite the Lagrangian (3.8) as follows:

(3.12)L[Ψ ,Ψ,A, e

]= L[Ψ eiϕγS/2,e−iϕγS/2Ψ,A,1

],

for zweibeins eaµ(x) and parameter functionϕ(x) given by Eqs. (2.4) and (2.6),respectively. The effects of the nontrivial zweibeins (2.4) can therefore be absorbed bythe simple spinor redefinition

(3.13)Ψ ′(x)≡ e−iϕ(x)γS/2Ψ (x), Ψ ′(x)≡ Ψ (x)eiϕ(x)γS/2.

Obviously, this field redefinition changes the spinor boundary conditions according to thevalues ofm and n in the functionϕ(x); see Eqs. (2.6) and (2.7). Note, however, thatthis field redefinition is not a properSO(2) spinor redefinition onT 2[i], in the sensethat contractible loops of spinor rotations need not correspond to contractible loops ofcoordinate rotations (see Ref. [9] for further details). Physical consequences of changedspinor boundary conditions are, for example, the difference of the vacuum energy density[10] and the occurrence of the CPT anomaly (see Section 5).

4. Chiral determinant with torsion

In this section, we express the two-dimensional chiral determinant with torsion in termsof the standard chiral determinant without torsion. This can be done by use of the identity(3.12) and field redefinition (3.13).

The chiral determinant with torsion is then given by the following path integral:

(4.1)Dθ1,θ2[A,e] =∫ [

DψRDψL

](θ1,θ2)

exp−S[ψR eiϕ/2, eiϕ/2ψL,A

],

in terms of the standard action for the one-component Weyl spinorψL(x) and its conjugateψR(x),

(4.2)S[ψR,ψL,A

]= ∫T 2[i]

d2x ψR(x)σµ(∂µ + iAµ(x)

)ψL(x),

with σµ ≡ (1, i). The parametersθ1 and θ2 in Eq. (4.1) denote the spinor boundaryconditions for the compact dimensions; see Eq. (3.7). Recall also that the zweibeinseaµ(x)

are given by Eqs. (2.4)–(2.7) and the corresponding torsion tensor by Eq. (2.13).The easiest way to calculate the chiral determinant with torsion is to perform the field

redefinition (3.13):

(4.3)

Dθ1,θ2[A,e] =∫ [

D(ψ ′R e−iϕ/2

)D(e−iϕ/2ψ ′

L

)](θ ′1,θ ′2)

exp−S[ψ ′

R,ψ′L,A

],

whereθ ′1 and θ ′2 indicate the boundary conditions of the transformed spinor fields (seebelow). Now, we only need to compute the relevant Jacobians and the next subsectionreviews a convenient method.


4.1. Jacobians for infinitesimal phase transformations

We propose to use Fujikawa’s method [13] to compute the Jacobians of the spinorredefinition (3.13) for the case of an infinitesimal phaseϕ(x) = α(x). Note, however,that our spinor redefinition isnot a chiral transformation, as was the case in Fujikawa’soriginal calculation.

The relevant Hermitian Dirac operator(i/D)= (i/D)† is given by

(4.4)i/D ≡ iγ µ(x)Dµ = ieiϕγS/2γ aδµa (∂µ + iAµ)e−iϕγS/2 ≡(

0 i/dR

i/dL 0

),

with

(4.5)i/dR ≡ (i/dL)† ≡ eiϕ/2 iσµ(∂µ + iAµ)eiϕ/2.

Since we intend to compute the Jacobians for the left- and right-moving chiral fermionsseparately, we can work with the following Hamiltonian:

(4.6)H ≡ (i/D)(i/D)† =(−/dR/dL 0

0 −/dL/dR)≡(H+ 00 H−

),

which has the advantage of being Hermitian for each chirality separately,H± = (H±)†.Explicitly, its components are given by

(4.7)H± =−Dµ±D±µ ∓ F,

with the further definitions

(4.8)D±µ ≡ ∂µ + iAµ ∓ i∂µϕ/2, F ≡ ∂1A2 − ∂2A1.

Following Ref. [13], we introduce normalized eigenfunctions ofH±,

(4.9)H± φ±,k(x)= λ2kφ±,k(x),

∫T 2[i]

d2x φ†±,k(x)φ±,l(x)= δkl,

for k, l ∈ Z. The four independent Weyl spinors are then expanded as follows:

ψR(x)=∑k

akφ+,k(x), ψR(x)=∑k

akφ†+,k,

(4.10)ψL(x)=∑k

bkφ−,k(x), ψL(x)=∑k

bkφ†−,k,

with Grassmann numbersak, ak , bk, bk . Note that the eigenfunctionsφ±,k(x) have beenassigned to the Weyl spinors in order to diagonalizeH :

(4.11)⟨Ψ ∣∣H |Ψ 〉 =

∑k

λ2k

(akak + bkbk

).

For the field redefinition used in Eq. (4.3) and with the definition

(4.12)ψ ′R(x)≡

∑k

a′kφ+,k(x),


the Grassmann variableak changes to

(4.13)a′k =∑l

( ∫T 2[i]

d2x φ†+,k(x)e

−iϕ/2φ+,l(x))al ≡

∑l

V+,kl[−ϕ/2]al.

The changes for the Grassmann variablesak, bk andbk are analogous, but with matricesV T+[ϕ/2], V−[ϕ/2] andV T−[−ϕ/2] replacingV+[−ϕ/2] in Eq. (4.13). The superscript Tindicates the transpose of the matrix.

Formally, the functional measure can be written as a product over the differentials dak

and dbl :

(4.14)DψRDψL ≡∏k∈Z

dak∏l∈Z

dbl.

Under a spinor redefinition, the change of the functional measure is then given by thecorresponding Jacobians,

(4.15)Dψ ′RDψ ′

L = JRJLDψRDψL,

with

(4.16)JR ≡ (detV T+[ϕ/2])−1

, JL ≡ (detV−[ϕ/2]

)−1.

For the moment, the Jacobians in Eq. (4.15) are only considered as formal expressions.The regularized determinant arising from a phase transformation (3.13) with an

infinitesimal parameterϕ(x) = α(x) can be calculated using the plane-wave method ofRef. [13]. The result is

(4.17)detV±[α] = expiA±[α,M],

with

(4.18)A±[α,M] ≡ M2

4π

∫T 2[i]

d2x α(x)e±F(x)/M2,

where the regulator massM is to be taken to infinity at the end of the calculation. Eq. (4.17)will be adopted as the proper definition of the determinant of the matricesV±[α] forinfinitesimalα(x). For later convenience, we establish two further identities:

(4.19)detV±[α]detV±[β] = detV±[α + β],and

(4.20)detV T±[α] = detV ∗±[−α] =(detV±[−α]

)∗ = detV±[α],for infinitesimalα(x) andβ(x).


4.2. Chiral determinant

The method used in the previous subsection holds for infinitesimal phase transforma-tions. Here, we simplydefinethe determinant of a finite phase transformation (3.13) tobe

(4.21)detV±[ϕ/2] ≡ limN→∞

(detV±

[ϕ/(2N)

])N.

For topologically trivial functionsϕ(x) = ω(x) as given in Eq. (2.6), this definition isunproblematic. For topologically nontrivial functionsϕ(x) = χ(x) as given in Eq. (2.7),on the other hand, the result turns out to break translation invariance; cf. Eq. (2.5). Thecorresponding phase factor is, nevertheless, well-behaved for the appropriate limitM →∞, as will become clear shortly.

From Eqs. (4.17) and (4.21), the combined regularized Jacobian (4.15) for a left-movingfermion and its conjugate is found to be given by

(4.22)JRJL = (detV T+[ϕ/2])−1(

detV−[ϕ/2])−1 = exp

−iW [ϕ,F,M],with

(4.23)W [ϕ,F,M] ≡ M2

4π

∫T 2[i]

d2x ϕ(x)cosh(F(x)/M2),

andF as defined in Eq. (4.8). For the topologically nontrivial partχ(x)= (2π/L)(mx1 +nx2) of ϕ(x), the corresponding phase factor (4.22) approaches 1 forM2 = 8πN/L2 withintegerN → ∞. The remaining (translation-invariant) phase factor depends only on thetopologically trivial partω(x) of ϕ(x).

We can now express the chiral determinant (4.1) for nontrivial zweibeins (eaµ = 1) interms of the chiral determinant for trivial zweibeins (eaµ = 1):

(4.24)Dθ1,θ2[A,e] = expiW [ϕ,F,M]Dθ ′1,θ ′2[A,1],

with the definitions

(4.25)2θ ′1 ≡ (2θ1 +m) mod 2, 2θ ′2 ≡ (2θ2 + n) mod 2,

and the understanding thatM has to be taken to infinity in the way discussed in theprevious paragraph. In fact, the regulator dependence drops out in the limitM → ∞ forthe physically relevant ratio of chiral determinants:

(4.26)Dθ1,θ2[A,e]Dθ1,θ2[B,e] =

Dθ ′1,θ ′2[A,1]Dθ ′1,θ ′2[B,1] ,

with modified spinor boundary conditions given by the parametersθ ′1 andθ ′2 of Eq. (4.25).Here, B is considered to be a fixed reference field, for example,B1(x) = B2(x) =(2π/L)/

√2 (this particular choice is motivated by Eq. (4.32) below). Eq. (4.26) is the

main result of this section.


Two remarks on the result (4.26) are in order. The first remark is that, in the end, only thetopologically nontrivial partχ(x) of ϕ(x) contributes to this ratio of chiral determinants,i.e., the dependence on the functionω(x) from Eq. (2.6) drops out. The second remarkis that the torsion does not affect the translation invariance of the normalized Euclideaneffective action (defined as minus the logarithm of the chiral determinant), because theright-hand side of Eq. (4.26) is translation-invariant by construction [6].

4.3. Heuristic argument

We now present a heuristic argument [5] for the change of the chiral determinant due tothe presence of torsion, restricting ourselves to the case of a constant gauge potential

(4.27)Aµ = (2π/L)hµ

and constant torsion tensor (2.13) determined by

(4.28)ϕ(x)= χ(x)= (2π/L)(mx1 + nx2).

The path integral (4.1) to be calculated is then

(4.29)D(θ1,θ2)[hµ,m,n] ≡∫ [

DψRDψL

](θ1,θ2)

exp−S

[ψR,ψL,hµ,m,n

],

with the simplified action

S[ψR,ψL,hµ,m,n

](4.30)=

∫T 2[i]

d2x ψR(x)eiχ(x)/2σµ(∂µ + i2πhµ/L)eiχ(x)/2ψL(x).

The one-component spinorsψL andψR with boundary conditions determined byθ1 andθ2 can be expressed in a Fourier basis as follows:

ψL ≡∑

p1,p2∈Z

bp1p2 exp

+2πi

L

((p1 + 1

2 + θ1)x1 + (p2 + 1

2 + θ2)x2),

(4.31)ψR ≡∑

q1,q2∈Z

aq1q2 exp

−2πi

L

((q1 + 1

2 + θ1)x1 + (q2 + 1

2 + θ2)x2),

with Grassmann variablesbp1p2 andaq1q2. The measure of the path integral (4.29) can bewritten as in Eq. (4.14), but withk andl replaced by pairs of integers(q1, q2) and(p1,p2).

Using this plane wave decomposition, the path integral of Eq. (4.29) is formally givenby the following infinite product:

(4.32)∏

p1,p2∈Z

(p1 + 1/2+ θ1 +m/2+ h1 + i(p2 + 1/2+ θ2 + n/2+ h2)

),

up to a constant overall factor. Although this expression needs regularization, we canalready infer that the introduction of torsion only changes the spinor boundary conditions.Namely,m/2 appears together withθ1 andn/2 with θ2. Hence, the chiral determinant of


aU(1) gauge theory onT 2[i], with torsion determined byϕ(x)= χ(x) and with constantgauge potentials, is proportional to the chiral determinant of the theory without torsion andnew spinor boundary conditionsθ ′1 andθ ′2 given by Eq. (4.25). This explains the resultfound in the previous subsection, at least for the particular gauge potentials (4.27) andtorsion parameter function (4.28).

In Appendix A, we discuss the zeta-function regularization of a product similar to theone of Eq. (4.32), which occurs for the vector-likeU(1) gauge theory. Again, the spinorboundary conditions are found to be changed according to Eq. (4.25).

5. CPT anomaly with torsion

The calculation of Section 4.2 has shown how to relate the chiral determinant withtorsion to the standard chiral determinant without torsion. The result (4.26) demonstratesthat the introduction of the topologically nontrivial zweibeins (2.4)–(2.7) can effectivelychange the spinor boundary conditions according to the constantsm andn appearing in thetorsion tensor (2.13).

It has been shown in Ref. [5] that the CPT anomaly of chiralU(1) gauge theory onthe torus without torsion appears only fordoubly-periodicspinor boundary conditions, atleast for a particular class of regularizations that respect modular invariance. Under a CPTtransformation of the gauge potential,

(5.1)Aµ(x)→ACPTµ (x)≡−Aµ(−x),

the CPT anomaly onT 2[i] manifests itself as a sign change of the chiral determinant,

(5.2)D1/2,1/2[ACPT,1]=−D1/2,1/2[A,1],

for the case of trivial zweibeins (eaµ = δaµ ≡ 1).In this paper, we only consider asinglecharged chiral fermion. The chiralU(1) gauge

anomaly needs, however, to be cancelled between different species of chiral fermions.There is then the CPT anomaly (5.2), as long as thetotal numberNF of charged chiralfermions isodd. Note that even if there is no net CPT anomaly (that is, forNF even), theremay still be Lorentz noninvariance; see Ref. [5] for further details.

A consequence of our result (4.24) is that the CPT anomaly can be moved to differentspinor boundary conditions by choosing appropriate zweibeins. (The additional phasefactor (4.22) is CPT-even.) For example, we can now have the CPT anomaly fordoubly-antiperiodicspinor boundary conditions,

(5.3)D0,0[ACPT, e]=−D0,0[A,e],

provided the zweibeinseaµ(x) have topologically nontrivial torsion determined byoddconstantsm andn in Eq. (2.7).

According to the heuristic argument of Section 4.3, the chiral determinant is formallyproportional to the infinite product (4.32). It is then easy to understand that there is a CPTanomaly if both 2θ1 +m and 2θ2 + n are odd. Start, for example, with purely antiperiodic


spinor boundary conditions(θ1 = θ2 = 0). Now, the introduction of torsion with oddm andoddn formally leads to the infinite product

(5.4)∏

p′1,p

′2∈Z

(p′

1 + h1 + ip′2 + ih2

),

which equals the chiral determinant of a torsionless theory with doubly-periodic spinorboundary conditions. Under a CPT transformation,hµ →−hµ, the single factor withp′

1 =p′

2 = 0 is CPT-odd, whereas the other factors combine into a CPT-even product (whichstill needs to be regularized). Hence, for torsion determined by oddm and oddn, the CPTanomaly has been moved to the doubly-antiperiodic spin structure. Analogous argumentsapply to the other cases.

In short, the CPT anomaly of chiralU(1) gauge theory with an odd number of chargedchiral fermions on the torusT 2[i] occurs only if the following conditions hold:

(5.5)(2θ1 +m)= 1 mod 2, (2θ2 + n)= 1 mod 2,

at least for the regularizations used in Refs. [5,6]. Here,θ1 andθ2 determine the fermionboundary conditions (3.7) andm and n are constants appearing in the topologicallynontrivial zweibeins (2.4)–(2.7).

6. Discussion

For two-dimensional chiralU(1) gauge theory, we have presented in this papera calculation of the chiral determinant on the torusT 2[i] with nontrivial zweibeinscorresponding to the presence of torsion.

In Section 4.2 we have shown how to relate the chiral determinant with torsion to thechiral determinant without torsion by the spinor redefinition (3.13). The Jacobian of thisredefinition turns out to be a gauge-invariant and CPT-even phase factor (4.22), whichcancels in the ratio of the chiral determinants (4.26). The chiral determinant with torsionis then proportional to the chiral determinant without torsion, but with spinor boundaryconditions changed according to Eq. (4.25). This result was confirmed in Section 4.3 bya heuristic argument for a particular choice of gauge potentials and torsion. Hence, theCPT anomaly can effectively be moved from one spin structure to another by choosingtopologically nontrivial zweibeins.

The calculations of the present paper demonstrate that the two-dimensional CPTanomaly is a genuine effect for chiralU(1) gauge theory on the torus. The CPT anomalycan be moved around between the different spin structures by taking appropriate zweibeins;see Eq. (5.5). But the anomaly cannot be removed completely from the general theory,which is a sum over all spin structures [9].

Alternatively, we can fix the spinor boundary conditions (for example, antiperiodicboundary conditions (3.7) withθ1 = θ2 = 0) and consider different classes (m,n ∈ Z) ofzweibeins (2.4)–(2.7), with the corresponding torsion tensor (2.13). It is quite remarkablethat topologically nontrivial spacetime torsion, which is not visible in the metric and


the curvature, can affect the local physics of chiralU(1) gauge theory in the same wayas different spinor boundary conditions would do for the case of trivial zweibeins (i.e.,vanishing torsion).

But, as we have shown in Section 2, there are more consequences of torsion than justmodified spinor boundary conditions. There is, for example, the fact that parallelogramsdo not close and that extremal and autoparallel curves need not coincide if torsion ispresent. Moreover, these local manifestations of torsion can already occur for topologicallytrivial zweibeins withm = n = 0 in Eq. (2.7), whereas the boundary-like effects requiretopologically nontrivial zweibeins (m = 0 or n = 0). Still, topologically nontrivialzweibeins may have a special status, as discussed in Section 2.4.

In this paper, we have focused on two-dimensional chiralU(1) gauge theory, becausethe chiral determinant is known exactly [6]. But our discussion of the effects of torsion canbe readily extended to higher-dimensional orientable manifolds. Consider, for example,the flat spacetime manifoldR2 × T 2, with noncompact coordinatesx0, x3 ∈ R andperiodic coordinatesx1, x2 ∈ [0,L]. The zweibeins (2.4)–(2.7) can then be embedded inthe vierbeinseAM(x) as follows:

(6.1)eAM(x)=eaµ(x), for A= a ∈ 1,2, M = µ ∈ 1,2,δAM, otherwise,

with indicesA andM running over 0, 1, 2, 3. TheSO(2) angleϕ(x) which enters thenontrivial zweibein part of Eq. (6.1) is taken to be purely topological, namelyϕ(x) =χ(x1, x2) with χ as given by Eq. (2.7).

The metric resulting from the vierbeins (6.1) is flat,gMN(x) = δMN . Note, however,that the vierbeins (6.1) withχ = 0 do not solve the vacuum field equations of theEinstein–Cartan theory [7], in contrast with the situation in two dimensions as mentionedin footnote 1. These vierbeins could play the role of prior-geometric fields (that is, non-dynamical fields); see, for example, the discussion in Ref. [14]. For the moment, let us justcontinue with the particular vierbeins (6.1), regardless of their origin.

In order to be specific, we also take the particular chiral gauge theory correspondingto the well-knownSO(10) grand-unified theory with three families of quarks and leptons.The CPT anomaly now gives two Chern–Simons-like terms [1] for the hyperchargeU(1)gauge field in the effective action, again provided condition (5.5) holds.2 These Chern–Simons-like terms affect the local physics, making the propagation of photons birefringent[19,20]. This last phenomenon is all the more remarkable, since at tree level torsion doesnot couple to the photons because of gauge invariance [7].

To summarize, a topological component of a (prior-geometric) torsion field couldmodify the propagation of photons via the CPT anomaly. Inversely, the propagation ofphotons could perhaps inform us about the structure of spacetime.

2 It has been claimed in Ref. [15] that a cosmic torsion fieldSµ(x) ≡ εµνρσ Tνρσ (x) could also generate a

Chern–Simons-like term for the photon field via the quantum effects of Dirac fermions coupled to both photon andtorsion fields. This radiatively induced Chern–Simons-like term must, however, vanish according to an argumentbased on gauge invariance and analyticity [16,17] or, for constantSµ in particular, causality [18]. Note that theCPT anomaly necessarily involves chiral (Weyl) fermions, not Dirac fermions [1,5].


Appendix A. Dirac determinant for two-dimensional U (1) gauge theory with torsion

In this appendix, we evaluate the regularized fermionic determinant of a two-dimension-al U(1) gauge theory with a single Dirac fermion, i.e., the vector-likeU(1) gauge theory.The spacetime manifold considered is the torusT 2[i] shown in Fig. 1. In order to simplifythe calculation, we take, as in Section 4.3, constant gauge potentialsAµ(x)= (2π/L)hµ,with hµ ∈ R, and constant torsion tensor components (2.13) determined byϕ(x)= χ(x)=(2π/L)(mx1 + nx2), with m,n ∈ Z.

For a single Dirac fermion, the fermionic determinant (exponent of minus the Euclideaneffective action) is the infinite product of the following eigenvalues:

(A.1)λp1p2 ≡ (2π/L)2((p1 + a1)

2 + (p2 + a2)2),

with quantum numbersp1,p2 ∈ Z and (noninteger) parameters

(A.2)a1 ≡ 1/2+ θ1 +m/2+ h1, a2 ≡ 1/2+ θ2 + n/2+ h2.

Compare with the product (4.32) for a single chiral fermion.This product of eigenvalues can be regularized using zeta-function techniques (see

Refs. [21,22] and references therein). For a ≡ (a1, a2), we define the regularized Diracdeterminant as follows:

(A.3)Dθ1,θ2Dirac [hµ,m,n] ≡ exp

−ζ ′E

(s, a)∣∣

s=0

,

with the generalized Epstein zeta function

(A.4)ζE(s, a)≡ ∑

p1,p2∈Z

((pi + ai)g

ij (pj + aj ))−s

,

for gij ≡ (2π/L)2 δij and Re(s) > 1. The prime in Eq. (A.3) denotes differentiation withrespect to the variables (which is set to 0 afterwards).

Our evaluation of the sum (A.4) essentially repeats the calculation of Ref. [22], to whichthe reader is referred for further details. In the rest of this appendix,gij will stand for theinverse of the matrixgij and we will setpi ≡ pi .

By writing the generalized Epstein zeta function (A.4) as a Mellin transform,

(A.5)ζE(s, a)= 1

D(s)

∑p1,p2∈Z

∞∫0

dt ts−1 exp−tλp1p2,

we can apply the generalized Poisson resummation formula,∑p1,p2∈Z

exp−π(pi + ai)g

ij (pj + aj )

(A.6)=√

det(gij )∑

p1,p2∈Z

exp−πpigijpj + 2πipjaj

,

to the integrand of Eq. (A.5). The result is given by

(A.7)ζE(s, a)= D(1− s)

D(s)πs−1

√det(gij )

′∑p1,p2∈Z

(pigijp

j)s−1 exp

2πipjaj

,


with the prime on the sum indicating that the modespi = 0 are excluded, since they do notcontribute for the region Re(s) > 1 where the original sum (A.4) is convergent. Analyticcontinuation tos = 0 then yields

ζE(0, a)= 0,

(A.8)ζ ′E

(0, a)= π−1

√det(gij )

′∑p1,p2∈Z

(pigij p

j)−1 exp

2πipjaj

.

It is a remarkable fact [22] that one can express thisζ ′E(0, a) in terms of the Riemann

theta function and Dedekind eta function:

(A.9)ζ ′E

(0, a)=− log

∣∣∣∣∣ 1

η(τ)ϑ

[1/2− a1

1/2+ a2

](0, τ )

∣∣∣∣∣2

,

with modulusτ = i for the particular matrixgij ∝ δij of Eq. (A.4). Here, the Riemanntheta function with characteristicsa andb is defined as in Ref. [23],

(A.10)ϑ

[a

b

](z, τ )≡

∑n∈Z

expiπτ(n+ a)2 + 2πi(n+ a)(z+ b)

,

and the Dedekind eta function is given by

(A.11)η(τ)≡ eiπτ/12∞∏m=1

(1− e2πiτm).

Note that the regularization method used has eliminated theL-dependence present inEq. (A.1) and produced the result (A.9) which does not depend onL; cf. Refs. [21,22].

The theta functions (A.10) obey the following identity:

(A.12)ϑ

[a +N

b+M

](z, τ )= e2πiaMϑ

[a

b

](z, τ ),

for arbitrary integersN andM. In addition, there are some further properties for the specialcase ofz= 0 andτ = i, which allow us to write Eqs. (A.3) and (A.9) as

(A.13)logDθ1,θ2Dirac [hµ,m,n] = −ζ ′

E

(0, a)= log

∣∣∣∣∣ 1

η(i)ϑ

[θ ′1 + h1

θ ′2 + h2

](0, i)

∣∣∣∣∣2

,

with

(A.14)2θ ′1 ≡ (2θ1 +m) mod 2, 2θ ′2 ≡ (2θ2 + n) mod 2.

This shows that the effect of torsion (parametersm and n) for the regularized Diracdeterminant can be entirely absorbed by a change of spinor boundary conditions, as givenby Eq. (A.14). Note also that the identity (A.12) implies the gauge invariance of (A.13)underhµ → hµ + nµ, for nµ ∈ Z.


References

[1] F.R. Klinkhamer, Nucl. Phys. B 578 (2000) 277.[2] W. Pauli, in: W. Pauli, L. Rosenfeld, V. Weisskopf (Eds.), Niels Bohr and the Development of

Physics, Pergamon, London, 1955, p. 30.[3] G. Lüders, Ann. Phys. (NY) 2 (1957) 1.[4] F.R. Klinkhamer, Nucl. Phys. B 535 (1998) 233.[5] F.R. Klinkhamer, J. Nishimura, Phys. Rev. D 63 (2001) 097701.[6] T. Izubuchi, J. Nishimura, JHEP 10 (1999) 002.[7] F.W. Hehl, P. von der Heyde, G.D. Kerlick, J.M. Nester, Rev. Mod. Phys. 48 (1976) 393.[8] T. Eguchi, P.B. Gilkey, A.J. Hanson, Phys. Rep. 66 (1980) 213.[9] S.J. Avis, C.J. Isham, Nucl. Phys. B 156 (1979) 441.

[10] R. Banach, J.S. Dowker, J. Phys. A 12 (1979) 2545.[11] A. Einstein, The Meaning of Relativity, 5th edn., Princeton Univ. Press, Princeton, 1955,

pp. 70, 145.[12] D.H. Douglass, V.B. Braginsky, in: S.W. Hawking, W. Israel (Eds.), General Relativity: An

Einstein Centenary Survey, Cambridge Univ. Press, Cambridge, 1979, p. 90.[13] K. Fujikawa, Phys. Rev. D 21 (1980) 2848;

K. Fujikawa, Phys. Rev. D 29 (1984) 285.[14] C. Will, Theory and Experiment in Gravitational Physics, 2nd edn., Cambridge Univ. Press,

Cambridge, 1993, p. 17.[15] A. Dobado, A.L. Maroto, Mod. Phys. Lett. A 12 (1997) 3003.[16] S. Coleman, S.L. Glashow, Phys. Rev. D 59 (1999) 116008.[17] M. Pérez-Victoria, JHEP 04 (2001) 032.[18] C. Adam, F.R. Klinkhamer, Phys. Lett. B 513 (2001) 245.[19] S.M. Carroll, G.B. Field, R. Jackiw, Phys. Rev. D 41 (1990) 1231.[20] C. Adam, F.R. Klinkhamer, Nucl. Phys. B 607 (2001) 247.[21] S.W. Hawking, Commun. Math. Phys. 55 (1977) 133.[22] S.K. Blau, M. Visser, A. Wipf, Int. J. Mod. Phys. A 6 (1991) 5409.[23] D. Mumford, Tata Lectures on Theta, Birkhäuser, Boston, 1983.


The zeros of the QCD partition functionA.D. Jacksona, C.B. Langb, M. Oswalda, K. Splittorff a

a The Niels Bohr Institute, Blegdamsvej 17, DK-2100 Copenhagen Ø, Denmarkb Institut für Theoretische Physik, Universität Graz, A-8010 Graz, Austria


Abstract

We establish a relationship between the zeros of the partition function in the complex massplane and the spectral properties of the Dirac operator in QCD. This relation is derived within thecontext of chiral Random Matrix Theory and applies to QCD when chiral symmetry is spontaneouslybroken. Further, we introduce and examine the concept of normal modes in chiral spectra. Usingthis formalism we study the consequences of a finite Thouless energy for the zeros of the partitionfunction. This leads to the demonstration that certain features of the QCD partition function areuniversal. 2001 Published by Elsevier Science B.V.

PACS: 11.30.Rd; 05.45.-a

1. Introduction

In recent years the spectral correlators of the Dirac operator in QCD have been theobject of intense study using both numerical and analytic means. These correlators containvaluable information regarding both the chiral properties of the QCD vacuum and thetopological structure of the gauge fields. The relation to the chiral properties of the QCDvacuum was established by Banks and Casher [1]: the eigenvalue density of the QCDDirac operator at eigenvalue zero is proportional to the chiral condensate and is, therefore,an appropriate order parameter for chiral symmetry. As a complement to the Banks–Casherrelation, one has the Yang–Lee picture [2] of a phase transition. In the attempt to analyzephase transitions in statistical spin models Lee and Yang [2] introduced the concept ofthe zeros of the finite volume partition function in the thermodynamic limit. The volumedependence of these zeros allows finite size scaling studies and subsequent identificationof universality classes. In the case of chiral symmetry, one focuses on the zeros of thepartition function in the complex quark mass plane. If these zeros pinch the real axis andexhibit a constant density, a discontinuity in the partition function arises at the pinch, and

E-mail address: [email protected] (M. Oswald).

0550-3213/01/$ – see front matter 2001 Published by Elsevier Science B.V.PII: S0550-3213(01)00459-X


234 A.D. Jackson et al. / Nuclear Physics B 616 (2001) 233–246

chiral symmetry is spontaneously broken. The mass zeros of the partition function andthe low lying eigenvalues of the Dirac operator thus contain similar information about thechiral phase transition. The relation between the two is, however, involved. The partitionfunction and its zeros are obtained by averaging over all gauge field configurations. Bycontrast, the eigenvalues of the Dirac operator are given for each gauge field configuration,and only a density of eigenvalues is well-defined after averaging.

The challenge of deriving relations between the zeros of the partition function and theeigenvalues of the Dirac operator was first taken up by Leutwyler and Smilga [3]. Theystudied QCD in a Euclidean 4-dimensional box with side lengthL subject to the constraintthat

(1)1

Λ L 1

mπ

.

Heremπ is the pion mass andΛ is the typical QCD scale. They computed the QCDpartition function for equal quark masses using the effective chiral Lagrangian and foundthat quark masses enter only in the rescaled combinationmL4Σ , whereΣ is the chiralcondensatein the chiral limit. They further observed that the partition function zeros couldbe thought of as average positions of the eigenvalues. While highly suggestive, these resultswere not completely quantitative. The situation has changed dramatically since then. Themain break through came with the introduction [4] of random matrix concepts in QCDwhich permits the study of the correlations of the eigenvalues of matrices drawn on ageneral weight constructed to ensure the chiral structure of each eigenvalue spectrum. Therelation of random matrix theory to QCD in the limit (1) has been established through anumber of universality studies [5], lattice QCD simulations [6,7], and direct calculationsusing the effective chiral Lagrangian [8]. (For a review of random matrix theory in QCDsee [9].) In terms of the spectral correlation functions, the universal limit in which QCDand chiral random matrix theory (χRMT) coincide is the limit

(2)N → ∞, λ→ 0, m→ 0,

in which the microscopic variables

(3)ζ ≡ 2Nλ, µ≡ 2Nm

are kept fixed andN is identified as the dimensionless volume. (Here,λ denotes aneigenvalue of the Dirac operator andm is the dimensionless quarkmass parameter.)The determination of the individual eigenvalue distributions and their most importantcorrelators now permits direct comparison of partition function zeros and eigenvaluepositions. The suggestion of Leutwyler and Smilga is remarkably accurate. The zeros andthe average positions of the eigenvalues are intimately connected.

Here we shall demonstrate that this relationship can be understood as a fundamentalproperty of the chiral ensembles. We show that the zeros are uniquely trapped by themaxima of the joint eigenvalue distribution function. This trapping appears on all scalesand is thus relevant for any finiteN as well as in the largeN limit. To obtain a betterunderstanding of the relation between the maxima of the joint eigenvalue distributionfunction and the zeros of the partition function, we introduce and determine the spectral

A.D. Jackson et al. / Nuclear Physics B 616 (2001) 233–246 235

“normal modes” of the chiral unitary ensemble. This provides us with a simple tool todescribe the fluctuations of the eigenvalues about the maximum of the joint eigenvaluedistribution.

As suggested in (1), chiral random matrix theory is not expected to describe all aspectsof QCD. Only correlations below a certain energy length are expected to be in agreementwith chiral random matrix theory [10]. In solid state physics, this energy is denoted as theThouless energy. Recently [11] it was realized that the effects of a finite Thouless energycan be studied naturally using the language of spectral normal modes [12]. We thus performa normal mode analyses of the chiral ensemble to formulate and establish certain universalfeatures of the partition function zeros. This argument is independent of standard proofs ofuniversality, and its general nature can shed some light on the way universality is realized.

In Section 2 we show that forNf = 1 the zeros of the partition function ofχRMT aretrapped between the maximum positions of the joint distribution function. This result holdsfor all scales. In Section 3 we derive the normal modes of the chiral unitary ensembles andfind that they are Chebyshev polynomials in the largeN limit. We discuss the effects ofthe Thouless energy in Section 4. In Section 5 we make the connection to the familiarmicroscopic spectral density. Our conclusions are contained in Section 6.

2. Zeros of the partition function in χRMT

2.1. Chiral random matrix theory

We start with the partition function of chiral random matrix theory (χRMT) for Nfflavors, which is given by [4,13]

(4)ZNf ,ν

N,β (mf )=∫DW

Nf∏f=1

det(D +mf )exp

[−NβΣ2

2Tr(W†W

)],

where β denotes the Dyson index andDW is the Haar measure over the Gaussiandistributed random matricesW . D is the analogue of the Dirac operator which has thechiral structure

(5)D =(

0 iW

iW† 0

).

HereW is aN ×M matrix with ν = |N −M| playing the rôle of the topological charge.Without loss of generality we assumeν to be positive. The chiral condensate in the chirallimit, Σ , is related to the eigenvalue density ofD, ρ(λ), via the Banks–Casher relation [1]

(6)Σ = limλ→0

limmf→0

limN→∞

πρ(λ)

N.

The partition function is invariant under transformationsW → U†WV , whereU is aN × N matrix andV aM ×M matrix. Following the diagonalizationW = U†ΛV , the


partition function can be expressed in terms of the eigenvalues ofW (with Σ ≡ 1),

ZNf ,ν

N,β (mf )

(7)=( Nf∏f=1

m|ν|f

) +∞∫−∞

· · ·+∞∫

−∞

N∏k=1

[dλk

Nf∏f=1

(λ2k +m2

f

)λβν+β−1k e−

Nβ2 λ2

k

]∆(λ2)β.

The Vandermonde determinant,∆(λ2), which is the nontrivial Jacobian of the transforma-tion from the matrices to the eigenvalues, has the form

(8)∆(λ2)=

∏k<l

(λ2k − λ2

l

).

The partition function (7) can now be written as an integral over the joint probability

densityPNf ,ν

N,β (λ1, . . . , λN ; mf ) as

(9)ZNf ,ν

N,β (mf )=+∞∫

−∞· · ·

+∞∫−∞

N∏k=1

dλk PNf ,ν

N,β

(λ1, . . . , λn; mf )

with

PNf ,ν

N,β

(λ1, . . . , λN ; mf )

(10)=( Nf∏f=1

m|ν|f

)N∏k=1

[ Nf∏f=1

(λ2k +m2

f

)λβν+β−1k e−

Nβ2 λ2

k

]∆(λ2)β.

Unlike real QCD,χRMT has the special feature that the partition function can be expressedin terms of the eigenvalues of the Dirac operator. This enables us to derive a number ofstatements regarding the zeros of the partition function. We now focus on the caseβ = 2—the universality class of QCD with 3 colours and quarks in the fundamental representationof the gauge group. (The choiceβ = 1 corresponds to QCD with two colours in thefundamental representation;β = 4 describes QCD with any number of flavours and quarksin the adjoint representation of the gauge group.)

Eq. (10) expresses an evident duality between flavor and topology: the joint probabilitydensity forNv massless flavours andNf massive flavours depends only onν +Nv . Thisrelation was proven for the QCD partition function independent ofχRMT in [14].

We now wish to determine the maximum of the joint probability distribution. This willallow us to put a tight bound on the zeros of the partition function. The chiral normalmodes, to be discussed in Section 3, describe fluctuations about the maximum of the jointprobability distribution.

2.2. Extremum of the joint probability distribution

In order to determine the maximum of the joint eigenvalue probability distribution,

we consider variations of logPNf ,ν

N with respect to the eigenvalues. (We assume theeigenvalues to be ordered withλi < λi+1.) We introduce the coordinatesyi = λ2

i and


evaluate the equations

(11)∂ logP

Nf ,ν

N

∂yi= 0.

ForNf = 0 and topological sectorν this yields

(12)

(ν + 1

2

)1

Nyi− 1+ 1

N

∑j =i

2

yi − yj= 0.

We now choose to focus on the quenched, i.e.,Nf = 0, joint eigenvalue probability

distribution. The solution to this equation reveals that the maximum of logPNf =0,νN is

obtained for

(13)Lν−1/2N

(Nλ2

i

)= 0,

whereLαN denotes the generalized Laguerre polynomials. This result follows from theobservation that Laguerre’s differential equation,

(14)zLαN(z)′′ + (α + 1− z)LαN(z)

′ +NLαN(z)= 0,

reduces to (12) at the zeros ofLν−1/2N . (The proof follows from considerations similar to

those made in Appendix A.6 in [15].) Forν = 0 we can use the fact that

(15)L−1/2n

(z2)= (−1)n

n!22n H2n(z)

to see that the density of eigenvalues in theN → ∞ limit is precisely that of the usualGaussian ensembles, i.e., a semicircle with support−2 λ +2. The partition function

for β = 2 is the average of a product of fermionic determinants overPNf =0,νN

(16)ZNf ,ν

N (mf )=⟨ Nf∏f=1

m|ν|f

N∏i=1

(λ2i +m2

f

)⟩.

This can be readily evaluated using orthogonal polynomials, and the result forν = 0 agreeswith the one presented in [16]

(17)ZNf ,ν=0N (mf )=

Nf∏f=1

(N + f − 1)!NN+f−1

CNfN (mf )

∆Nf (−m2f ) ,

where

(18)CNfN = det

[L(0)N+f−1

(−Nm2f ′)]f,f ′=1,...,Nf

,

and∆Nf (−m2f ) is the Vandermonde determinant with the negative square of theNf

masses as arguments. For the special case ofNf = 1 this yields the result

(19)ZNf =1,ν=0N (m)= N !

NNL(0)N

(−Nm2),which is now ready for investigation. The expression (19) coincides (up to a constant) withthe series expansion for the partition function derived in [17]. From the expression found


there we find that

(20)ZNf =1,νN (m)∼m|ν|L(ν)N

(−Nm2).2.3. Trapping of the zeros

The closed forms given above allow us obtain information about the zeros of the partitionfrom the spectral correlators. Specifically, we now show that the locations of the partitionfunction zeros inχRMT are trapped by the maxima of the joint distribution function.

The Laguerre polynomialsLαn(z) are polynomials orthogonal on the interval[0,∞] withweight functionw(z) = zα exp(−z). They have three properties which are useful for ourpurpose:

• For orthogonal polynomials in general, the zeros of theN th order polynomial andthe end-points of the weight function defineN + 1 intervals. Exactly one zero of theorthogonal polynomial of order(N + 1) lies in each of these intervals.

• For fixedN , theith zero of the Laguerre polynomial,LaN(x)= 0, is a monotonicallyincreasing function ofa, thus

(21)dxi

da> 0.

• The generalized Laguerre polynomials are related to one another via

(22)La+1N (x)= −dLaN+1(x)

dx.

In the last section we saw that the massless joint distribution function has its maxima whenthe eigenvalues are located at the zeros ofL

−1/2N (Nλ2) and that the partition function for

Nf = 1 andν = 0 is proportional toL0N(−Nm2). We are interested in relating the zeros

of the partition function to the position of the eigenvalues at the maximum of the jointdistribution function. Since the zeros ofL0

N(−Nm2) follow from those ofL0N(Nm

2) by arotation from the real to the imaginary axes inm, we restrict our attention in the followingconsiderations to Laguerre polynomials of positive argument.

It follows from (21) that the zeros ofZNf=1,ν=0N are trapped between the corresponding

zeros ofL−1/2N andL+1/2

N . According to (22), the zeros ofL1/2N lie at the extrema ofL−1/2

N+1 .

The extrema ofL−1/2N+1 are evidently trapped by the zeros ofL−1/2

N (and the end points, ifnecessary). The result is that theith zero ofL0

N is trapped between theith and(i + 1)st

zero ofL−1/2N and, given the nature of the argument, is expected to be closer to the lower

value. In other words, the zeros ofZNf =1,ν=0N (m) are trapped by the most probable values

for the eigenvalues ofPNf =0,ν=0N . This result is an exact property of the chiral unitary

ensembles and is consequently valid on every scale including the microscopic scale.The trapping just derived relates the zeros of theNf = 1 partition function and the

location of the maximum in the joint eigenvalue distribution of the quenched(Nf = 0)ensemble. We can also relate the zeros of theNf = 1 partition function to the maximum

of its own integration kernel,PNf =1,νN (λi,m). Form→ 0 we haveP

Nf =1,νN (λi,m)=


PNf =0,ν+1N (λi) by flavor-topology duality. The other limit,m→ ∞, decouples one flavor

and leavesPNf =0,νN . So, with increasingm, the extrema ofP

Nf =1,νN (m) move smoothly

from those ofPNf =0,ν+1N to those ofP

Nf =0,νN , i.e., from the zeros ofLν−1/2

N to the

zeros ofLν+1/2N . By (21), they must pass the zeros of the partition functionLνN for some

intermediates values of the massm. The relation between the collective maximum of logP

and the average eigenvalue positions will be reconsidered in Section 5.

3. Normal modes in χRMT

We have seen that the maximum of the massless joint distribution function is obtainedwhen the eigenvalues are located at the zeros of the Laguerre polynomials. In order tostudy the properties of fluctuations about this maximum, it is useful to make a Gaussian

approximation toPNf =0,νN which leads to the form

(23)logPNf =0,νN ≈ logP

Nf =0,νN;(0) + 1

2δλiCij δλj ,

whereδλi is the position of theith eigenvalue relative toλi , its value at the collective

maximum of logPNf =0,νN . The matrixC is defined as

(24)Cij = ∂2

∂λi∂λjlogP

Nf =0,νN ,

evaluated at the maximum. Concentrating again on the caseNf = 0, we find that thediagonal elements ofC are

(25)Cii = −2N − 2ν + 1

λ2i

− 4∑j =i

(λ2i + λ2

j )

(λ2i − λ2

j )2,

and that the off-diagonal elements are

(26)Cij = 8λiλj(λ2i − λ2

j )2.

We now consider the eigenvalue equation for the real symmetric matrixC:

(27)N∑i=1

Cij φ(k)j = ωkφ

(k)i .

The eigenvectors,φ(k), are the (normalized) normal modes of theχRMT spectrum. Theydescribe the statistically independent correlated fluctuations of the eigenvalues of therandom matrix about their most probable values. The normal modes provide an alternatedescription of the eigenvalues of any given random matrix since

(28)δλi =N∑k=1

ckφ(k)i with

N∑i=1

φ(k)i φ

(k′)i = δkk′ .


We can locate the eigenvalues by specifying either theδλi or the amplitudesck asconvenient. The eigenvalues,ωk , provide a measure of the magnitude of these fluctuations.

The derivation of these eigenvalues and eigenvectors can be performed as in [12]. Theresulting eigenvalues are

(29)ωk = −4kN.

As in [12], we find a linear dispersion relation valid for allk andN . The linearity of (29) isa reflection of the well-known rigidity of random matrix spectra. Furthermore, this result isindependent ofν. Just as in the Gaussian case, the eigenvectors are found to be Chebyshevpolynomials in the largeN limit (i.e., with corrections of order 1/N ):

(30)φ(k)i =

√2

NU2k−1

(λi

2

).

The normalization of the eigenvectors is

(31)∫dx ρ(x)φ(k)(x)φ(l)(x)= δkl,

whereρ(x) is again the semicircle

(32)ρ(x)= N

π

√4− x2.

Note that only odd normal Chebyshev polynomials appear. This is a consequence of chiralsymmetry, which ensures that all nonzero eigenvalues come in pairs(−λ,λ). Eq. (23) cannow be written as

(33)log

(PNf =0,νN

PNf =0,νN;(0)

)= 1

2

N∑k=1

|ck|2ωk.

Following (28), the coefficientsck are constructed as

(34)ck =N∑i=1

δλiφ(k)i ,

and are statistically independent. It is obvious from (33) that that the mean squareamplitude for thekth normal mode is

(35)⟨c2k

⟩= 1

|ωk| .We can use the normal modes to construct a Gaussian approximation to the partitionfunction as

(36)ZNf =1,νN (m)=

∞∫0

N∏k=1

dck exp

[−1

2

N∑k=1

|ck|2|ωk|](m2 + (λk + δλk)

2),with the λk given by (13) andδλk given by (28). The Gaussian approximation (33) alsopermits a simple approximate calculation of the number variance [12]. This calculation


reveals that the familiar logarithmic behaviour of the number variance (i.e., the “spectralrigidity” of the random matrix ensembles) is a direct consequence of the linearity of thedispersion relation (29) for allk.

We have chosen to consider the normal modes for the caseNf = 0 andν = 0. Thischoice is some what arbitrary; it would be equally sensible to start with theNf = 0 andν = 1 normal modes. We will employ this Gaussian approximation below to consider thesensitivity of the partition function zeros to the effects of a Thouless energy. Since theresulting shifts are small, this arbitrariness will be of no consequence.

4. Effects of a Thouless energy

Normal modes describe the correlated fluctuations of eigenvalues about their mostprobable values. As we have seen (29), the normal modes forχRMT obey a lineardispersion relation. By contrast, uncorrelated eigenvalues obey a quadratic dispersionrelation, and the mean square amplitude of the lowest mode withk = 1 is larger by afactor ofN/k [12]. In QCD, it is expected that spectral correlations in a sufficiently smallenergy domain will follow the predictions ofχRMT. On larger energy scales, spectralcorrelations die out. The characteristic energy which divides these regions is the Thoulessenergy, usually denoted byEc. In 4-dimensional QCD the Thouless energy is estimated tobe [10]

(37)Ec/D ∼ √N,

whereD is the mean level spacing. This behaviour has been verified in lattice studies [18].The connection between the Thouless energy and the normal modes of the eigenvaluespectrum has been investigated in [11] for the case of sparse matrices. There it was foundthat “almost all” normal modes obey the linear dispersion relation discussed above withremarkable accuracy. Reflecting the presence of a Thouless energy, a small fraction (i.e.,approximately 1/

√N ) of long wave length modes are strongly enhanced. The mean square

amplitude of the longest wave length mode withk = 1 approaches the value appropriatefor uncorrelated eigenvalues. Since such enhancement can be readily incorporated in ourGaussian approximation to the partition function (36), normal modes provide us with anatural and convenient tool to study the effects of a Thouless energy on the zeros of thepartition function. So far we have seen that the most probable eigenvalues are located atthe zeros of the Laguerre polynomialL−1/2

N and that the zeros of the partition functionare given by the zeros ofL0

N . The Gaussian approximation (36) allows us to see howthis result is modified by the presence of a Thouless energy. To mimic the effects of theThouless energy, we shall enhance the long wavelength modes in the partition function.

Our aim is to demonstrate that the zeros of the partition function in the microscopicregion remain virtually unaffected even if the enhancement of the soft modes is substantial.Since we are concerned only with the microscopic zeros, every long wavelength modecontributes to a “breathing” of the spectrum. In order to investigate the influence of thesoft modes, it is sufficient to evaluate the strength (i.e., mean square amplitude) of thiseffective breathing.


For concreteness we start with√N longest wavelengths modes with fluctuations as

given by the Gaussian approximation. Letλi denote the values at the maximum of

PNf =0,ν=0N . The fluctuations introduced in the Gaussian approximation by the

√N longest

wavelengths modes are

(38)λi −→ λi

(1+

√N∑

k=1

η(k)ck

)≡ λi s.

From (30) we know that the normal modes are√

2/N U2k−1(λi/2). This gives for thelinear coefficient

(39)η(k) =√

2

Nk(−1)k+1.

Additionally we know from (26) that the normal modes are linearly independent. With thisinformation the variance ofs becomes in leading order of 1/N

(40)⟨s2⟩− 〈s〉2 = ⟨

s2⟩− 1 ∼√N∑

k=1

k2

N

1

kN= 1

N2

√N∑

k=1

k = 1

N2

√N(√N + 1

)∼ 1

N.

Here we used the facts that the linear terms vanish and that〈c2k〉 = 1/|ωk| with (29) for the

quadratic terms. The result is simply aO(1/N) correction.In order to study the effects of a Thouless energy, we now enhance the mean square

amplitudes of these√N soft modes by a factor ofN/k2. This factor provides a smooth

interpolation from the behaviour of uncorrelated soft modes (fork = 1) to that ofχRMT(for k = √

N ). This interpolation is completely consistent with the results of [11]. We nowfind that

(41)⟨s2⟩− 〈s〉2 ∼

√N∑

k=1

k2

N

1

kN

N

k2∼ log(N)

2N.

The result still shows strong suppression inN . The decision to single out√N soft modes

for enhancement is not essential. We could declare any fraction of the long wave lengthmodes which vanishes in theN → ∞ limit as “soft”. A similar interpolation between thelimits of uncorrelated andχRMT modes will always lead to a value of〈s2〉 − 1 whichvanishes asN → ∞. In short, the effects of a Thouless energy are expected to have anegligible effect on〈λ2

i 〉 over the entire microscopic spectrum.The question is now how we can evaluate the effect on the zeros from the enhanced long

wave length modes. To this end we introduce the distribution function of the fluctuations

(42)PN(s)≡ 1

N sγ e− s2

2σ2

and fixγ andσ by the value of〈s2〉 found above.N is the normalization. In the RMT casewe haveγ =N/2−1 andσ 2 = 2(N+1)/N2, while in the case where the long wavelengthmodes are enhanced we haveγ =N/ log(N)− 1 andσ 2 = log(N)(1+ log(N)/(2N))/N .


The effect of the first√N normal modes is different for large and small eigenvalues.

Whereas for the smallest√N eigenvalues it amounts to a breathing it means incoherent

fluctuations for the larger ones. We now evaluate the effect of this breathing on themicroscopic zeros. Recall that the partition function forNf = 1 andν = 0 is the averageof the fermionic determinant with respect to the joint probability distribution, and that∏Ni=1(m

2 − λ2i )= L

−1/2N (Nm2). In the microscopic limit where the quantityµ≡ 2Nm is

fixed we have

(43)L−1/2N

(µ2

4Ns2

)= (−1)N

N !22NH2N

(µ

2√N s

)−→ 1√

Nπcos

(µ

s

).

To investigate the correction to the microscopic partition function zeros under the influenceof the

√N longest wavelength normal modes we thus have to consider the following

integral

(44)Z(µ)∼ 1

NK

∞∫0

ds exp

(γ logs − s2

2σ 2

)cos

(µ

s

)s2

√N,

where the normalization factorK is

(45)K =∞∫

0

ds PN(s)s2√N.

The factors2√N comes from the rescaling in the fermionic determinant. For the evaluation

of (44) we make use of the series expansion of the cosine

(46)cos

(x

s

)= cos(x)+ x sin(x)(s − 1)+O(s − 1)2.

This yields

(47)Z(µ)∼ cos(µ)+µsin(µ)

√2σΓ

((γ + 2

√N + 2)/2

)Γ((γ + 2

√N + 1)/2

) + · · · .

Sinceγ √N , we can use that [19]

(48)Γ (z+ a)

Γ (z+ b)∼ za−b

[1+ (a − b)(a + b− 1)

2z+O

(1

z2

)],

and finally find that in the case where we consider√N RMT long wavelength modes the

correction term to cos(µ) is of order 1/√N , and in the case of

√N enhanced wavelength

modes it is of order logN/√N . In both cases the zeros of the partition function are

unaffected in the microscopic limit.These results suggest a new kind of “universality” of the microscopic partition function.

As a specific example this universality shows that the microscopic zeros are unaffectedeven if the

√N longest wavelength normal modes are enhanced in such a way that they

interpolate between Poissonian and RMT statistics.


5. The microscopic limit

So far we have been discussing the joint distribution function, the zeros of the partitionfunction, and theN positions that specify the collective maximum. Here we link this tothe more familiar microscopic-eigenvalue density. For a finiteN the eigenvalue density isfound from the joint distribution function as

(49)ρNf ,ν

N (λ, mf )=∞∫

−∞· · ·

∞∫−∞

dλ2 · · ·dλN PNf ,ν

N

(λ,λ2, λ3, . . . , λN ; mf ).

The double-microscopic spectral density is then defined as [4,20,21]

(50)ρNf ,νs (ζ ;µ1, . . . ,µNf )≡ lim

N→∞1

NρNf ,ν

N

(ζ

N,µ1

N, . . . ,

µNf

N

),

and similarly for all other spectral correlations. The functional form of the microscopiceigenvalue density has been derived in [21] and forNf = 1 and topological chargeν itreads

ρNf =1,νs (ζ ;µ)= |ζ |

2

(Jν−1(ζ )

2 + Jν−2(ζ )Jν(ζ ))

(51)− |ζ |Jν−2(ζ )[µIν(µ)Jν−2(ζ )+ ζ Iν−2(µ)Jν(ζ )](ζ 2 +µ2)Iν−2(µ)

.

The partition function in the microscopic limit for one flavor is proportional toIν(µ). Theextrema of (51) are given by

(52)∂ρ

(Nf =1,ν)s (ζ ;µ)

∂ζ= 0,

and are obviously functions ofµ. If the derivative in (52) is evaluated atζ = µ then the

solutions of (52) corresponding to the maxima ofρ(Nf =1,ν)s coincide with the zeros of

theNf = 1 microscopic partition function,ZNf =1,ν(m), for an imaginary argument. Toconclude: the peaks of the microscopic eigenvalue density are in one to one correspondencewith the zeros of the partition function and by the trapping proven in Section 2.3 also tothe positions which maximize the joint probability distribution.

6. Conclusions and outlook

In the present paper we have established an intimate relationship between zeros ofthe partition function and the spectral properties of the Dirac operator. The relation isderived within chiral random matrix theory and applies to QCD Dirac spectra and partitionfunction zeros near the origin (in the microscopic regime). Through the introduction ofspectral normal modes we have tested the validity of the relationship when a finite Thoulessenergy is introduced. The observed independence of the microscopic zeros complimentsthe existing universality studies. The present study treats one flavour. For more flavourswith degenerate masses, it is known that the zeros of the microscopic partition function


are not confined to the imaginary axis [22]. While this makes the relation between theeigenvalues and the zeros somewhat less direct, there is no reason to expect that the normalmode analysis should not apply for any number of flavours.

We remark that the normal mode analysis is generically applicable and not a specialfeature of random matrix theory. In particular, the normal mode analysis lends itself to astudy of the spectral properties of the Dirac operator in lattice QCD. Such a study wouldbe truly interesting in that it would shed new light on the role of random matrix likecorrelations in lattice gauge theories. Almost all of the normal modes are expected to begiven by random matrix theory while only the very long wavelength modes are determinedby the detailed dynamics.

In a broader perspective this study may also be seen as the first step towards theestablishment of a one-to-one correspondence between the zeros and the most likelyeigenvalue positions whenever the short range spectral correlations are random matrix like.If such a general relation were established then it could be used to argue that the criticalexponents for the Yang–Lee edge in QCD must coincide with the ones for the gap in thespectral density of the Dirac operator. Such relationship, if true, would allow for substantialsimplifications when trying to determine critical exponents by lattice techniques.

Acknowledgements

The authors wants to thank Poul Henrik Damgaard and Thomas Wilke for usefuldiscussions.

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The structure of large logarithmic corrections atsmall transverse momentum in hadronic collisions

Daniel de Floriana,d,1, Massimiliano Grazzinib,c,d,2a Departamento de Física, FCEYN, Universidad de Buenos Aires, (1428) Pabellón 1, Ciudad Universitaria,

Capital Federal, Argentinab Dipartimento di Fisica, Università di Firenze, I-50019 Sesto Fiorentino, Florence, Italy

c INFN, Sezione di Firenze, I-50019 Sesto Fiorentino, Florence, Italyd Institute for Theoretical Physics, ETH-Hönggerberg, CH 8093 Zürich, Switzerland

Received 4 September 2001; accepted 13 September 2001

Abstract

We consider the region of small transverse momenta in the production of high-mass systems inhadronic collisions. By using the current knowledge on the infrared behaviour of tree-level and one-loop QCD amplitudes atO(α2

S), we analytically compute the general form of the logarithmically-enhanced contributions up to next-to-next-to-leading logarithmic accuracy. By comparing the resultswith qT -resummation formulae we extract the coefficients that control the resummation of thelarge logarithmic contributions for both quark and gluon channels. Our results show that within theconventional resummation formalism the Sudakov form factor is actually process-dependent. 2001Published by Elsevier Science B.V.

PACS:12.38.Bx; 12.38.Cy

1. Introduction

The transverse-momentum distribution of systems with high invariant mass produced inhigh-energy hadron collisions is important for QCD studies and for physics studies beyondthe Standard Model (see, e.g., Refs. [1–4]).

We consider the inclusive hard-scattering process

(1)h1(p1)+ h2(p2) → F(Q2, q2

T ;φ) + X,

This work was supported in part by the EU Fourth Framework Programme “Training and Mobility ofResearchers”, Network “Quantum Chromodynamics and the Deep Structure of Elementary Particles”, contractFMRX-CT98-0194 (DG 12-MIHT).

E-mail address:[email protected] (M. Grazzini).1 Partially supported by Fundación Antorchas and CONICET.2 Partially supported by the Swiss National Foundation.



248 D. de Florian, M. Grazzini / Nuclear Physics B 616 (2001) 247–285

where the final-state systemF is produced by the collision of the two hadronsh1 andh2

with momentap1 andp2, respectively. The final stateF is a generic system of nonstronglyinteracting particles, such asoneor morevector bosons(γ ∗,W,Z, . . .), Higgs particles(H ) and so forth. We denote by

√s the center-of-mass energy of the colliding hadrons

(s = (p1 + p2)2 2p1p2), and byQ2 and q2

T the invariant mass and total transversemomentum of the systemF , respectively. The additional variableφ in (1) denotes thepossible dependence on the kinematics of the final state particles inF (such as rapidities,individual transverse momenta and so forth).

We assume that at the parton level the systemF is produced with vanishingqT (i.e., withno accompanying final-state radiation) in the leading-order (LO) approximation. SinceF

is colourless, the LO partonic subprocess is eitherqf qf ′ annihilation, as in the case ofγ ∗,W andZ production, orgg fusion, as in the case of the production of a Higgs bosonH .

When the transverse momentum of the produced systemq2T is of the order of its

invariant massQ2 the fixed order calculation is reliable.3 In the regionq2T Q2

large logarithmic corrections of the formαnS/q

2T log2n−1Q2/q2

T appear, which spoil theconvergence of fixed-order perturbative calculations. The logarithmically-enhanced termshave to be evaluated at higher perturbative orders, and possibly resummed to all ordersin the QCD coupling constantαS. The all-order resummation formalism was developedin the eighties [5–14]. The structure of the resummed distribution is given in terms of atransverse-momentum form factor and of process-dependent contributions.

The coefficients that control the resummation of the large logarithmic contributions fora given process in (1) can be computed at a given order if an analytic calculation at largeqT at the same order exists. At first order inαS the structure of the large logarithmiccontributions is known to be universal and depends only on the channel in which the systemis produced in the LO approximation. At second relative order inαS, only a few analyticalcalculations are available, like the pioneering one for lepton-pair Drell–Yan production,performed by Ellis, Martinelli and Petronzio in Ref. [15]. Using the results of Ref. [15]Davies and Stirling [14] (see also [16]) were able to obtain the complete structure of theO(α2

S) logarithmic corrections for that process.The analysis performed by Davies and Stirling is by far nontrivial because it requires the

integration of the analyticqT distribution in the smallqT limit. Moreover, the calculationcannot tell anything about the dependence of these coefficients on the particular processin (1) and should in principle be repeated for each process.

In this paper we address this problem with a completely independent and generalmethod. Our basic observation is that the large logarithmic corrections are of infrared (softand collinear) nature, and thus their form can be predicted once and for all in a general(process independent) manner.

The structure of the logarithmically-enhanced contributions atO(αnS) is controlled by

the infrared limit of the relevant QCD amplitudes at the same order. The infrared behaviourof QCD amplitudes atO(αS) is known since long time [17]. Recently, soft and collinearsingularities arising in tree-level [18,19] and one-loop [20–23] QCD amplitudes atO(α2

S)

3 It is assumed that all other dimensionful invariants are of the same orderQ2.

D. de Florian, M. Grazzini / Nuclear Physics B 616 (2001) 247–285 249

have been extensively studied and the corresponding kernels have been computed [18–25].By using this knowledge, and exploiting the relatively simple kinematics of the process (1),we will construct general approximations of the relevant QCD matrix elements that are ableto control all singular regions corresponding toqT → 0 avoiding double counting. By usingthese approximations we will compute the general structure of theO(α2

S)-logarithmically-enhanced contributions both forqq- and forgg-initiated processes.

The results provide an important check of the validity of the resummation formalismand allow to extract the general form of the resummation coefficients. In particular, in thequark channel we can confirm the results of Ref. [14] in the case of Drell–Yan and in thegluon channel we can give the coefficients in the important case of Higgs boson productionthrough gluon–gluon fusion.

The universality of our method relies on the fact that the infrared factorization formulaewe use depend only on the channel (qq or gg) in which the systemF is produced at LOand not on the details ofF .

Our main results were anticipated in a short letter [26]. This paper is organized asfollows. In Section 2 we review the framework of the resummation formalism and presentthe strategy for the calculation. In Section 3 we perform the calculation explicitly for theO(αS) corrections and extract the first order coefficients. Sections 3 and 4 are devoted tothe calculation atO(α2

S) for the quark and the gluon channel and constitute the main partof this work. Finally in Section 6 we present our final results and discussion.

2. Resummation formula

The transverse momentum distribution for the process in Eq. (1) can be written as:

(2)dσF

dQ2dq2T dφ

=[

dσF

dQ2dq2T dφ

]res.

+[

dσF

dQ2 dq2T dφ

]fin.

.

Both terms on the right-hand side are obtained as convolutions of partonic cross sectionsand the parton distributionsfa/h(x,Q2) (a = qf , qf , g is the parton label) of the collidinghadrons.4

The partonic cross section that enters in the resummed part (the first term on the right-hand side) contains all the logarithmically-enhanced contributionsαn

S/q2T logmQ2/q2

T .Thus, this part has to be evaluated by resumming the logarithmic terms to all orders inperturbation theory. On the contrary, the partonic cross section in the second term on theright-hand side is finite (or at least integrable) order-by-order in perturbation theory whenqT → 0. It can thus be computed by truncating the perturbative expansion at a given fixedorder inαS.

4 Throughout the paper we always use parton densities as defined in theMS factorization scheme andαS(q2)

is the QCD running coupling in theMS renormalization scheme.


Since in the following we are interested in the small-qT limit we will be concerned onlywith the first term in Eq. (2). The resummed component is5[

Q2 dσF

dQ2 dq2T dφ

]res.

=∑a,b

1∫0

dx1

1∫0

dx2

∞∫0

dbb

2J0(bqT )fa/h1

(x1, b

20/b

2)fb/h2

(x2, b

20/b

2)(3)× sWF

ab(x1x2s;Q,b,φ).

The Bessel functionJ0(bqT ) and the coefficientb0 = 2e−γE (γE = 0.5772. . . is the Eulernumber) have a kinematical origin. To correctly take into account the kinematics constraintof transverse-momentum conservation, the resummation procedure has to be carried out inthe impact-parameterb-space. The resummed coefficientWF

ab is

WFab(s;Q,b,φ)

=∑c

1∫0

dz1

1∫0

dz2CFca

(αS

(b2

0/b2), z1

)CF

cb

(αS

(b2

0/b2), z2

)δ(Q2 − z1z2s

)

(4)× dσ(LO)Fcc

dφSFc (Q,b),

wheredσ(LO)cc /dφ corresponds to the leading order cross section for the production of

the large invariant mass systemF in the cc channel, withc representing either a quarkq or a gluong. The resummation of the large logarithmic corrections is achieved byexponentiation, that is by showing that the Sudakov form factor can be expressed as

(5)Sc(Q,b) = exp

−

Q2∫b2

0/b2

dq2

q2

[Ac

(αS

(q2)) log

Q2

q2+ Bc

(αS

(q2))]

.

The functionsAc(αS),Bc(αS), as well as the coefficient functionsCab(αS, z) in Eqs. (4),(5) are free of large logarithmic corrections and have perturbative expansions inαS as

(6)Ac(αS) =∞∑n=1

(αS

2π

)n

A(n)c ,

(7)Bc(αS) =∞∑n=1

(αS

2π

)n

B(n)c ,

(8)Cab(αS, z) = δabδ(1− z)+∞∑n=1

(αS

2π

)n

C(n)ab (z).

5 This expression can be generalized to include the dependence on the renormalization and factorization scalesµR andµF , respectively (see, e.g., Ref. [27]).


The coefficients of the perturbative expansionsA(n)c , B(n)

c andC(n)ab (z) are the key of

the resummation procedure since their knowledge allows to perform the resummationto a givenLogarithmic order: A(1) leads to the resummation of leading logarithmic(LL) contributions,A(2),B(1),C(1) give the next-to-leading logarithmic (NLL) terms,A(3),B(2),C(2) give the next-to-next-to-leading logarithmic (NNLL) terms, and soforth.6 The coefficient functionsC(n)

ab (z) depend on the process, as it has been confirmed

by calculations ofC(1)ab (z) for several processes. The Sudakov form factorSc(Q,b) that

enters Eq. (4) is oftensupposedto be universal. However, as we will show, this is not thecase, and anticipating our results we label all process-dependent coefficients by the upperindexF . The coefficientsA(1), B(1), A(2) are universal and are known both for the quark[10] and for the gluon [13] form factors

A(1)q = 2CF , A(1)

g = 2CA,

B(1)q = −3CF , B(1)

g = −2β0,

(9)A(2)q = 2CFK, A(2)

g = 2CAK,

where

(10)β0 = 11

6CA − 2

3nf TR

and

(11)K =(

67

18− π2

6

)CA − 10

9nf TR.

The NNLL coefficientB(2) was computed by Davies and Stirling [14] for the case ofDrell–Yan (DY):

B(2)DYq = C2

F

(π2 − 3

4 − 12ζ(3)) + CFCA

(119 π2 − 193

12 + 6ζ(3))

(12)+ CFnf TR

(173 − 4

9π2),

whereζ(n) is the Riemannζ -function (ζ(3) = 1.202. . .). It is also worth noticing that,even though there is no analytical result available for it, the coefficientA

(3)q,g has been

extracted numerically with a very good precision in Ref. [29].As anticipated in the introduction, a direct way to obtain the coefficients in Eqs. (6), (7)

at a given order involves the computation of the differential cross sectiondσ/dq2T dQ2dφ

at small qT at the same order. A comparison with the power expansion inαS of theresummed result in Eq. (3) allows to extract the coefficients that control the resummationof the large logarithmic terms. However, it has been shown by Davies and Stirling that is itmore convenient to takez = Q2/s moments7 of the differential cross section defining thedimensionless quantity

(13)Σ(N) =1−2qT /Q∫

0

dz zNQ2q2

T

dσ0/dφ

dσ

dq2T dQ2dφ

.

6 In a different classification the coefficientC(1) enters only at NNLL [28].7 Here we follow Ref. [14] in the unconventional definition of the moments:f (N) = ∫ 1

0 dz zNf (z).


Notice that in the definition ofΣ the cross section has been normalized with respectto the lowest order partonic contributiondσ0/dφ and multiplied byq2

T to cancel its1/q2

T singular behaviour in the limitqT → 0. The upper limit of integrationz = 1 −2qT /Q(

√1+ q2

T /Q2 − qT /Q) ∼ 1 − 2qT /Q has been approximated to a first order

expansion inqT /Q and corresponds to the kinematics for the emission of soft particles(i.e, when the center of mass energys is just enough to produce the system withinvariant massQ and transverse momentumqT ). Working with moments allows to avoidcomplicated convolution integrals implicit in (3) and makes possible to factorize the partondensities from the partonic contribution to the cross section. In this way, the correspondingexpression from the resummed formula (3) reads

(14)Σ(N) =∑i,j

fi/h1

(N,µ2

F

)fj/h2

(N,µ2

F

)Σij (N),

where

Σij (N) =∑a,b

∞∫0

b dbq2T

2J0(bqT )C

Fca

(N,αS

(b2

0/b2))CF

cb

(N,αS

(b2

0/b2))

(15)

×exp

−

Q2∫b2

0/b2

dq2

q2

[Ac

(αS

(q2)) log

Q2

q2+ BF

c

(αS

(q2))]

−µ2F∫

b20/b

2

dq2

q2(γai + γbj )

(N,αS

(q2))

and an ordered exponential is understood. Notice that the appearance of an extra terminvolving the anomalous dimensionsγab in the exponential in (15) is due to the evolutionof the parton densities from the original scaleb2

0/b2 in (3) to the arbitrary factorization

scaleµF at which they are now evaluated.In order to extract the resummation coefficients, we can directly study the partonic

contributionΣij . Furthermore, since we want to perform a calculation ofΣij to O(α2S)

and our main interest is the second order coefficientB(2), it is clear that only the diagonalcontribution Σcc can give the desired information. Each possible “flavour changing”contribution in Eq. (15) would add at least one extra power ofαS in the perturbativeexpansion. ‘Nondiagonal’ contributions toΣij , which can be evaluated in a simpler way,might be used to check the structure and consistency of the resummation framework at agiven perturbative order but do not provide any additional information on the coefficients.

In order to have transverse momentumqT = 0 at least one gluon has to be emitted and,therefore, the perturbative expansion ofΣcc begins atO(αS)

(16)Σcc(N) = αS

2πΣ

(1)cc (N) +

(αS

2π

)2

Σ(2)cc (N)+ · · · .


From the expansion of the resummed formula (15) it is possible to obtain the expressionfor the first two coefficients in (16) as8

(17)Σ(1)cc (N) = A(1)

c logQ2

q2T

+ B(1)c + 2γ (1)

cc (N)

and

Σ(2)cc (N) = log3 Q2

q2T

[−1

2

(A(1)

c

)2]+ log2 Q2

q2T

[−3

2

(B(1)c + 2γ (1)

cc (N))A(1)

c + β0A(1)c

]+ log

Q2

q2T

[A(2)

c + β0(B(1)c + 2γ (1)

cc (N)) − (

B(1)c + 2γ (1)

cc (N))2

+ 2A(1)c C(1)F

cc (N) − 2∑j =c

γ(1)cj (N)γ

(1)jc (N)

]+ B(2)F

c + 2γ (2)cc (N) + 2

(B(1)c + 2γ (1)

cc (N))C(1)F

cc (N) + 2ζ(3)(A(1)

c

)2

(18)− 2β0C(1)Fcc (N) + 2

∑j =c

[C

(1)Fcj (N)γ

(1)jc (N)

].

The computation ofΣ(1)cc (N) can provide information on the first order coefficientsA

(1)c

(the logarithmic term in (17)) andB(1)c (the constant term in (17)) as well as on the one-

loop anomalous dimensionsγ (1)cc (N) (theN -dependent term in (17)9). In the same way,

the coefficientsA(2)c andB(2)F

c can be extracted from the second order result (18). At thisorder, also the coefficient functionsC(1)F

ij (N) contribute to the logarithmic and constantterms and therefore should be known in order to be able to proceed with the extractionof A

(2)c andB

(2)Fc . Fortunately, there is another related quantity which allows to obtain

the coefficient functionsC(1)Fij (N) from a first order calculation. This is theqT -integrated

cross section

(19)

p2T∫

0

dq2T

q2T

Σic.

Whenp2T Q2 the perturbative expansion toO(αS) reads (neglecting again terms that

vanish whenpT → 0)

p2T∫

0

dq2T

q2T

Σcc= αS

2π

[−1

2A(1)

c log2 Q2

p2T

− (B(1)c + 2γ (1)

cc (N))log

Q2

p2T

+ 2C(1)Fcc (N)

],

(20)

p2T∫

0

dq2T

q2T

Σic= αS

2π

[−γ

(1)ci (N) log

Q2

p2T

+ C(1)Fci (N)

], i = c.

8 For the sake of simplicity in the presentation, and unless otherwise stated, we fix the factorization andrenormalization scales toµ2

F = µ2R = Q2.

9 Notice that all moments but one can actually be extracted. The remaining one can be obtained by imposingquark number and momentum conservation rules.


The integration overqT adds one power in the logarithm, with the coefficient functionsC

(1)Fij (N) appearing now in the constant term. It is worth noticing that at variance with the

calculation ofΣ the configuration withqT = 0 now contributes to Eq. (19).In the quark channel(c = q), for the sake of simplicity and in order to compare directly

with the calculation performed in [14], we will concentrate on thenonsingletcontributionto the cross section defined by

(21)σNS =∑ff ′

(σqf qf ′ − σqf qf ′ ).

The second order expansion forΣNSqq (N) in terms of the resummation coefficients reads

like the one in Eq. (18) but without the ‘singlet’ contributions involving∑

j =c and withthe corresponding nonsinglet anomalous dimension. In the following the label NS will bealways understood inΣqq .

3. The calculation at O(αS)

The calculation atO(αS) is not difficult and the results are rather well known.Nevertheless, we will give in this section the details on the computation as a way to presentthe main ideas of the method developed to obtain the resummation coefficients.

At this order only one extra gluon of momentumk can be radiated and the kinematicsfor the processcc → g + F is (see Fig. 1)

(22)p1 + p2 → k + q.

We denote the corresponding matrix element byM(0)cc→g F (p1,p2, k,φ) and the usual

invariants are defined as

(23)s = (p1 + p2)2, u = (p2 − k)2, t = (p1 − k)2, z = Q2/s.

The differential cross section can be written as

dσcc→g F

dq2T dQ2dφ

=∫ ∣∣M(0)

cc→gF (p1,p2, k,φ)∣∣2

8s(2π)2

(4π)εq−2εT

Γ (1− ε)

du

u

(24)× δ

(1

u(u− umax)(u− umin)

),

Fig. 1.O(αS) contribution to the process (1).


where the two roots of the equation(p1 +p2 − q)2 = 0 are given by

umin = Q2z − 1−

√(1− z)2 − 4z q2

T /Q2

2z,

(25)umax= Q2z − 1+

√(1− z)2 − 4z q2

T /Q2

2z.

In order to regularize both ultraviolet and infrared divergences we work in the conventionaldimensional regularization scheme (CDR) with 4−2ε space–time dimensions, consideringtwo helicity states for massless quarks and 2− 2ε helicity states for gluons. The lowest-order cross section (atqT = 0) needed to constructΣ in Eq. (13) is given by

(26)dσ0

dφ=

∣∣M(0)cc→F (p1,p2, φ)

∣∣22s

,

in terms of the Born matrix element|M(0)cc→F (p1,p2, φ)|2.

As has been stated, we want to obtainΣ(1)cc by using our knowledge on the behaviour

of QCD matrix elements in the soft and collinear regions atO(αS). The starting point isthe observation that, whenq2

T is small, the additional gluon is constrained to be eithercollinear to one of the incoming partons or soft. Thus there are three singular regions ofM(0)

cc→F (p1,p2, k,φ) in theqT → 0 limit:

• first collinear region:p1k → 0;• second collinear region:p2k → 0;• soft region:k → 0.

It is clear that, sinceq2T is small but does not vanish, these regions do not produce any real

singularity, i.e., poles inε, but are responsible for the appearance of the logarithmically-enhanced contributions. Whenp1k → 0 the matrix element squared factorizes as follows:

(27)∣∣M(0)

cc→gF (p1,p2, k,φ)∣∣2 4παSµ

2ε

z1p1kPcc(z1, ε)

∣∣M(0)cc→F (z1p1,p2, φ)

∣∣2,where

(28)Pqq (z, ε)= CF

[1+ z2

1− z− ε(1− z)

],

(29)Pgg(z, ε) = 2CA

[z

1− z+ 1− z

z+ z(1− z)

]are theε-dependent real Altarelli–Parisi (AP) kernels in the CDR scheme. In the left-handside of Eq. (27) the matrix element squared is obtained replacing the two collinear partonsc andg by a partonc with momentumz1p1.

Notice that in the gluon channel there are additional spin-correlated contributions andEq. (27) is strictly valid only after azimuthal integration. Since here and in the followingwe will always be interested in azimuthal integrated quantities, Eq. (27) can be safely usedalso in the gluon channel.


In the limit p2k → 0 the singular behaviour is instead

(30)∣∣M(0)


2ε

z2p2kPcc(z2, ε)

∣∣M(0)cc→F (p1, z2p2, φ)

∣∣2.Let us now consider the limit in which the gluon becomes soft. As it is well known soft-factorization formulae usually involve colour correlations, that make colour and kinematicsentangled. In general colour correlations relate each pair of hard momentum partons in theBorn matrix element. In this case the hard momentum partons are only two and colourconservation can be exploited to obtain:

(31)∣∣M(0)


2εCc4S12(k)∣∣M(0)

cc→F (p1,p2, φ)∣∣2,

where

(32)S12(k) = p1p2

2p1kp2k

is the usual eikonal factor and we have defined

(33)Cq = CF , Cg = CA.

In Eq. (31) colour correlations are absent and factorization is exact. This feature will persistalso atO(α2

S).In each of the singular regions discussed above, Eqs. (27), (30) and (31) provide an

approximation of the exact matrix element that can be used to compute the cross sectionin the smallqT limit. In principle it might be possible to split the phase space integrationin regions where only soft or collinear configurations can arise, and use in each region thecorresponding approximation. Unfortunately, such method probes to be very difficult to beextended toO(α2

S), where the pattern of singular configurations is much more complicated.Thus our strategy is to unify the factorization formulae in order to obtain an approximationthat it is valid in the full phase space.

As can be easily checked, if we identify the momentum fractionsz1 andz2 with z, thecollinear factorization formulae in Eqs. (27), (30) contain the correct soft limit in Eq. (31).Therefore, the unification of soft and collinear limits is rather simple: the usual collinearfactorization formula already contains both. Strictly speaking, one can use the symmetry inthe initial states in order to perform the integration in Eq. (24) only over half of the phasespace (i.e., by taking for instance onlyu = umax) and multiplying the result by two. In thisway only one possible collinear configuration can occur and Eq. (27) provides the neededapproximation for the matrix element.

At this order it is even possible to write down a general factorization formula for the threeconfigurations that shows explicitly the 1/q2

T singularity of the matrix element squared as

(34)∣∣M(0)

cc→gF (p1,p2, k,φ)∣∣2 → 4πµ2εαS

q2T

2(1− z)

zPcc(z, ε)

∣∣M(0)cc→F (φ)

∣∣2,where we have used Lorentz invariance in order to write|M(0)

cc→F (φ)|2 only as a functionof the final state kinematics. We can now use this formula to compute the smallqT

behaviour ofΣcc(N) in a completely process independent manner. In fact the process


dependence, given by the Born matrix element, is completely factored out and cancelsin Σ . By replacing Eq. (34) in Eq. (24) and using the definition ofΣ we obtain, keepingfor future use itsε dependence:

Σ(1)cc (N, ε) = 1

Γ (1− ε)

(4πµ2

q2T

)ε1−2qT /Q∫

0

dz zN2(1− z)Pcc(z, ε)√(1− z)2 − 4z q2

T /Q2

(35)≡ 1

Γ (1− ε)

(4πµ2

q2T

)ε

CcFcc(N, ε).

Explicitly, settingε to 0, we have

Σ(1)qq (N) = 2CF log

Q2

q2T

− 3CF + 2γ (1)qq (N),

(36)Σ(1)gg (N) = 2CA log

Q2

q2T

− 2β0 + 2γ (1)gg (N),

for the quark and gluon channels. Comparing to Eq. (17) we see thatA(1)c = 2Cc is the

coefficient of the leading 1/(1−z) singularity in the AP splitting functions whereasB(1)c =

−2γ (1)c is given by the coefficient of the delta function in the regularized AP kernels

(37)γ (1)q = 3

2CF , γ (1)

g = β0.

Finally, in order to obtain the coefficientC(1)ab , we have to evaluate the integrals in

Eq. (19) and compare to the results from Eq. (20). As far as the diagonal contribution isconcerned, one has to take into account also the one-loop correction to the lowest ordercross section, a contribution formally proportional toδ(q2

T ). The interference betweenthe one-loop renormalized amplitude with the lowest order one depends of course on theprocess. Nevertheless, its singular structure is universal and allows to write in general [30]

M(0)†cc→FM

(1)cc→F + c.c.

(38)= αS

2π

(4πµ2

Q2

)εΓ (1− ε)

Γ (1− 2ε)

(−2Cc

ε2 − 2γcε

+AFc (φ)

)∣∣M(0)cc→F

∣∣2.The finite partAF

c depends (in general) on the kinematics of the final state noncolouredparticles and on the particular process in the class (1) we want to consider. In the case ofDrell–Yan we have [31]:

(39)ADYq = CF

(−8+ 23π

2),whereas for Higgs production in themtop → ∞ limit the finite contribution is [32]:

(40)AHg = 5CA + 2

3CAπ

2 − 3CF ≡ 11+ 2π2.

The diagonal term in Eq. (19) can be evaluated integrating Eq. (35), from 0 top2T , keeping

into account the contribution in Eq. (38) and subtracting the following factorization


counterterm in theMS scheme:

(41)R(FCT)cc (N) = −2

ε

Γ (1− ε)

Γ (1− 2ε)

(4πµ2

µ2F

)ε

γ (1)cc (N).

As for the nondiagonal contribution, one needsΣic(N), that can be computed, analogouslyto Eq. (35) as

Σic(N) = 1

Γ (1− ε)

(4πµ2

q2T

)ε1−2qT /Q∫

0

dz zN(1− z)Pci (z, ε)√

(1− z)2 − 4z q2T /Q

2

(42)→ 1

Γ (1− ε)

(4πµ2

q2T

)ε1∫

0

dz zN Pci (z, ε),

where the functionsPci(z, ε) are the nondiagonal AP splitting kernels

(43)Pgq(z, ε) = CF

[1+ (1− z)2

z− εz

],

(44)Pqg(z, ε) = TR

[1− 2z(1− z)

1− ε

],

and the absence of singularities asz → 1 has been exploited to setqT → 0 in the integral.The factorization counterterm to be subtracted in this case is

(45)R(FCT)ic (N) = −1

ε

Γ (1− ε)

Γ (1− 2ε)

(4πµ2

µ2F

)ε

γ(1)ci (N).

Comparing the total results to Eq. (20) we obtain forC(1)ab :

(46)C(1)Fab (z) = −P ε

ab(z)+ δabδ(1− z)

(Ca

π2

6+ 1

2AF

a (φ)

),

whereP εab(z) represent theO(ε) term in the APPab(z, ε) splitting kernels in Eqs. (28),

(29), (43), (44) and are given by:

P εqq (z) = −CF (1− z), P ε

gq(z) = −CF z,

(47)P εqg(z) = −2TRz(1− z), P ε

gg(z) = 0.

As can be observed, the coefficient function contains both ahard process dependentcontribution (proportional toAF

a (φ)) originated in the one-loop correction as well as a‘residual’ collinear contribution proportional theε part of the splitting functions whichhas origin in the particularities of theMS scheme (see Eq. (41)), where only theε = 0 (andnot the full) component of the splitting functions is factorized. The general expressionin Eq. (46) reproduces correctly the coefficientC

(1)ab computed for Drell–Yan [14], Higgs

production in themtop → ∞ limit [33,34], γ γ [35] andZZ [36] production.

Summarizing theO(αS) results, the coefficientsA(1)c andB

(1)c are fully determined by

theuniversalproperties of soft and collinear emission. The functionC(1)ab depends instead

on the process through the one-loop corrections to the LO matrix element.


4. The calculation at O(α2S): the quark channel

At O(α2S) Σqq(N) receives two contributions:

• Real emission of two partons recoiling against the final state systemF(Q2, q2T ,φ);

• Virtual corrections to single-gluon emission.

In the following we compute these contributions in turn.

4.1. Real corrections

The computation of the double real corrections toΣqq(N) represents the most involvedpart of the complete calculation. The difficulties arise both from the fact that the additionalparton in the final state implies three more phase space integrals, and from the appearenceof many more singular configurations that contribute to the limitqT → 0.

The kinematics for the double real emission processcc → i + j + F is (see Fig. 2)

(48)p1 + p2 → k1 + k2 + q,

and the corresponding matrix element is denoted byM(0)cc→ij F (p1,p2, k1, k2, φ). The

usual invariants are defined as

(49)s = (p1 + p2)2, t = (p1 − q)2, u = (p2 − q)2, s2 = (k1 + k2)

2,

and fulfill the following relations

(50)s + t + u = Q2 + s2, q2T = ut − s2Q

2

s.

In terms of these invariants, the real contribution to the cross section at fixedq2T is given

by

(51)

dσcc→ij F

dq2T dQ2dφ

=∫ ∣∣M(0)

cc→ij F (p1,p2, k1, k2, φ)∣∣2

2s

(s2q2T )

−ε

(4π)4−2εΓ (1− 2ε)

du

Q2 − u

dΩ

2π,

wheredΩ is

(52)dΩ = sinθ−2ε2 dθ2 sinθ1−2ε

1 dθ1,

Fig. 2.O(α2S) contribution from double real emission.


with the angles defined in the frame where the partons corresponding to momentumk1 andk2 are back-to-back [15].

We see from Eq. (51) that the first step of the calculation involves the integration overthe two angles. The integrals needed here are typical of heavy quark production at NLOand most of them can be found in Refs. [37,38]. The results of the angular integrals containpoles up to 1/ε while terms that develop an extra additional singularity ass2 → 0 have tobe computed up toO(ε).

The second step is the integration overu (or s2). The integration limits are given by thetwo rootsumax andumin in Eq. (25). At this point, it is convenient to define the ‘symmetric’value for whichu = t

(53)u0 = Q2 −√s(q2T + Q2

).

This value ofu corresponds also to the maximum ofs2

(54)smax2 ≡ A = Q2

1+ z − 2√

1+ q2T /Q

2 √z

z

and, in the CM frame ofp1 andp2, to the configuration whereqz = 0. The singularity ins2 is made manifest by use of the identity

(55)

s−1−ε2 = −1

εδ(s2)

(1− ε logA + 1

2ε2 log2A

)+ 1

(s2)A+− ε

(logs2

s2

)A+

+O(ε2),

with the distributions defined as:

(56)

A∫0

ds2f (s2)

(s2)A+=

A∫0

ds2

s2

(f (s2)− f (0)

),

(57)

A∫0

ds2f (s2)

(logs2

s2

)A+

=A∫

0

ds2logs2

s2

(f (s2)− f (0)

).

In order to obtainΣ(N) one finally has to integrate overz, keeping only the termsthat do not vanish in the small-qT limit. At the beginning we consider only theN = 0moment.10 We will later show how to perform the calculation for generalN , once onemoment is known, in a simpler way. Notice that after implementing the regularization ofthe s2 = 0 singularities using Eq. (55), the last two integrals can be performed directlyin four dimensions, since the small transverse momentumqT acts as a regulator of otherpossible singularities.

10 Notice that the calculation of a single moment is enough to obtain the resummation coefficientsA(2)

andB(2).


The double real contributions to (the nonsinglet part of )Σ(2)qq (N) fall into three classes,

according to the possible different final states:

• q + q → q + q + F ;• q + q → q + q + F ;• q + q → g + g + F .

Notice that theq + q → q + q + F is needed to form the nonsinglet combination.As we did atO(αS), to study the smallqT behaviour ofΣ we will rely on the structure

of soft and collinear singularities of the corresponding QCD matrix element. In principlethere are, of course, configurations where the two final state partons are hard and emittedback-to-back with small total transverse momentum. Nevertheless, these configurations donot produce any singularities whenqT → 0 and thus may be neglected. Finally, noticethat we consider onlydoublesingularities, i.e., configurations where two extra partons areeither collinear or soft, without caring aboutsinglesingularities. Configurations with onlyone collinear or soft parton (and the other hard) do not contribute toΣ(2) since the systemF is not emitted with smallqT in such case.

4.1.1. Contribution fromqq andqq emissionFor theq + q → q + q + F contribution we have three singular regions atO(α2

S) [19]:

• first triple-collinear region:k1p1 ∼ k2p1 ∼ k1k2 → 0;• second triple-collinear region:k1p2 ∼ k2p2 ∼ k1k2 → 0;• double-soft region:k1, k2 → 0.

In the first region the singularity is controlled by the following collinear factorizationformula [18,19,24]∣∣M(0)

qq→qqF (p1,p2, k1, k2, φ)∣∣2

(58) (8πµ2εαS)2

u2 Pq→q1q2(q3)

∣∣M(0)qq→F (z3p1,p2, φ)

∣∣2,wherePq→q1q2(q3) is the splitting function that controls the collinear decay of an initialstate quark of momentump1 into a final state quark–antiquark pairq1q2 of momentak1

andk2 and the ‘off-shell’ quarkq3 that participates in the hard cross section. The explicitexpression ofPq→q1q2(q3) is obtained from the one ofPq1q2q3, the splitting function for thedecay of a (‘off-shell’) quark into a final state quark–antiquark pair plus a quark, given inEq. (A.1), with the following definitions

s12 = s2, s13 = −2p1k1, s23 = −2p1k2,

(59)x1 = −z1/z3, x2 = −z2/z3, x3 = 1/z3,

wherez1 andz2 are the momentum fractions ofq1 andq2 (z3 = 1 − z1 − z2). Notice thatEq. (59) corresponds to the following transformation:

(60)r1 → k1, r2 → k2, r3 → −p1,

applied to the expression in Eq. (A.1) to cross the ‘off-shell’ parton to the final state.A formula similar to Eq. (58) can be written in the second collinear region.


In the double-soft region the factorization formula is instead [19]:∣∣M(0)qq→qqF (p1,p2, k1, k2, φ)

∣∣2(61) (

4πµ2εαS)2CFTR(I11 + I22 − 2I12)

∣∣M(0)qq→F (φ)

∣∣2,where

(62)Iij = Iij (k1, k2) = pik1pjk2 + pj k1pik2 − pipjk1k2

(k1k2)2pi(k1 + k2)pj (k1 + k2).

The reader can easily check that by defining the momentum fractions in Eq. (59) as11

(63)z1 = k1p2

p1p2, z2 = k2p2

p1p2.

Eq. (58) correctly keeps into account also the double-soft limit in Eq. (61). Thus, at leastoutside the second collinear region, the factorization formula (58) with the definitions (63)correctly gives the full singular behaviour in this channel.

The strategy to perform the calculation is the following. We use Eq. (58) to approximatethe matrix element in its region of validity and compute its contribution toΣ

(2)Rqq (0) by

integrating only in half of the phase space, that is fromu0 to umax. The remaining region,which is obtained by exchangingu ↔ t , will give, due to the symmetry of the initial state,exactly the same contribution and it is taken into account by multiplying the computedresult by 2. As it happens at leading order, the information on the process, embodied in theBorn matrix element is completely factored out in the calculation and disappears inΣ . Infact the Born matrix element can be fully written in terms of the (fixed) kinematics of thefinal state particles|M(0)

qq→F (z3p1,p2, φ)|2 ≡ |M(0)qq→F (φ)|2.

For theq + q → q + q + F contribution, needed to form the nonsinglet contribution inEq. (21), there are only two singular configurations:

• first triple-collinear region:k1p1 ∼ k2p1 ∼ k1k2 → 0;• second triple-collinear region:k1p2 ∼ k2p2 ∼ k1k2 → 0.

For the first collinear region we can write:∣∣M(0)qq→qqF (p1,p2, k1, k2, φ)

∣∣2(64) (8πµ2εαS)

2

u2 Pq→q1q2(q3)

∣∣M(0)qq→F (p1, z3p2, φ)

∣∣2.Here Pq→q1q2(q3) is now the splitting function which controls the collinear decay of aninitial state quark into a final stateqq pair. The explicit expression forPq→q1q2(q3) can beobtained from the expression ofPq1q2q3 in Eq. (A.1) with the following definitions

s12 = −2p1k2, s13 = −2p1k1, S23 = s2,

(65)x1 = 1/z3, x2 = −z2/z3, x3 = −z1/z3,

11 To parameterize the triple-collinear limit it is necessary to introduce an additional light-cone vectorn. Thisdefinition corresponds to the choicen = p2. Notice that a similar definition can be adopted also atO(αS) toreobtain Eq. (34) in the smallqT limit.


i.e., corresponding to the crossing transformation:

(66)r1 → −p1, r2 → k2, r3 → k1,

and similarly for the second collinear configuration (withp1 ↔ p2). There is a partialcancellation between theCFTR contribution toΣ(2)

qq from Eqs. (58) and (64), due to thenonsinglet combination. Once this cancellation is carried out, the part corresponding tothe production of ‘nonidentical’ partons in theqq channel gives the followingCFTR

contribution toΣ(2)qq :

(67)

Σ(2)Rqq(nid)(0) = CFnf TRK

[−2

3

1

εFqq (0, ε)− 4

3log2 Q2

q2T

− 2

9log

Q2

q2T

+ 1+ 2

9π2

],

where

(68)K = 1

Γ (1− 2ε)

(4πµ2

q2T

)ε(4πµ2

Q2

)ε

,

and the explicit expression of functionFqq (0, ε), defined in Eq. (35) is

(69)Fqq (0, ε) = 2 logQ2

q2T

− 3− ε.

At the beginning of Eq. (67) we have isolated a divergent term which will be cancelled bya similar one appearing in the virtual contribution.

The part corresponding to the production of ‘identical’ partons in theqq channelgives also aCF (CF − CA/2) contribution, which does not contain any logQ2/q2

T term.Therefore, there is a great simplification in the calculation sinceqT can be set to zero justafter performing the angular integrations. We find:

(70)Σ(2)Rqq(id)(0)= CF

(CF − 1

2CA

)(−6+ 2π2 − 16ζ(3)).

The calculation of theqq contribution can be performed with exactly the same strategy asfor theqq channel.12 After theCFTR contribution has been cancelled with a similar onein theqq channel only a contribution proportional toCF (CF − CA/2) remains13

(71)Σ(2)Rqq(qq)(0) = −CF

(CF − 1

2CA

)( 132 − π2 + 4ζ(3)

).

4.1.2. Contribution fromgg emissionThe calculation of the double-gluon emission correction toΣ

(2)R is more difficult,

because it is not possible to keep into account all possible singular configurations by usingonly the triple-collinear splitting functions. We will divide the calculation in two parts,according to the corresponding colour factors. First we will consider the non-Abelian,CFCA term, which turns out to be simpler, and finally the Abelian,C2

F part.

12 A factor 1/2 has been included to account for the two identical particles in the final state.13 The overall minus sign here is due to the fact that this quantity must be subtracted in order to construct the

nonsinglet combination.


CFCA contributionFor this colour structure there are three singular regions to be considered [19]:


We point out that, as discussed in Ref. [19], thanks to the coherence properties of soft-gluon radiation, the soft-collinear region does not give any contribution proportional toCFCA (see later).

In the first collinear region the singularity is controlled by the following factorizationformula:∣∣M(0)

qq→gg F (p1,p2, k1, k2, φ)∣∣2nab

(72) (8πµ2εαS)2

u2 CFCAP(nab)q→g1g2(q3)

∣∣M(0)qq→F (z3p1,p2, φ)

∣∣2,where P

(nab)q→g1g2(q3)

is the non-Abelian part of the splitting function that controls thecollinear decay of an initial state quark into a final state gluon pair. This function canbe obtained from Eq. (A.7) with the replacement in Eq. (59).

A similar formula to Eq. (72) can be written in the second collinear region (byp1 ↔ p2

exchange).In the double-soft region the factorization formula is instead (see Eq. (A.3) of Ref. [19]):∣∣M(0)

qq→ggF (p1,p2, k1, k2, φ)∣∣2nab

(73)

(4πµ2εαS

)2CFCA

(2S12(k1, k2)− S11(k1, k2)− S22(k1, k2)

)∣∣Mqq→F (φ)∣∣2,

where the non-Abelian double-soft function reads

Sij (k1, k2) = (1− ε)

(k1k2)2

pik1pjk2 + pik2pj k1

pi(k1 + k2)pj (k1 + k2)

− (pipj )2

2pik1pj k2pik2pj k1

[2− pik1pjk2 + pik2pj k1

pi(k1 + k2)pj (k1 + k2)

]+ pipj

2k1k2

[2

pik1pjk2+ 2

pj k1pik2

(74)− 1

pi(k1 + k2)pj (k1 + k2)

(4+ (pik1pjk2 +pik2pj k1)

2

pik1pj k2pik2pjk1

)].

As it happens in theqq andqq channels, it turns out that by defining the momentumfractions of the gluons as in Eq. (63), the factorization formula in Eq. (72) correctlyaccounts also for the double-soft configuration. Furthermore, we have verified that Eq. (72)does not introduce any additional spurious singularities in the other infrared configurations.Thus for this colour structure the situation is similar to the one in theqq andqq channelsand we can follow the same strategy. We approximate the non-Abelian part of the matrixelement in the region fromu0 to umax using Eq. (72). We first perform the angular integrals


and then, exploiting thep1 ↔ p2 symmetry, do the remainingu andz integrations onlyover half of the phase space, i.e., withu from u0 to umax.

In order to perform the last two steps, that are considerably more complicated than in thecase ofqq emission, we developed MATHEMATICA [39] programs that are able to handlethe cumbersome intermediate expressions in the smallqT limit.

The result is14

Σ(2)Rqq(ggnab)(0)= CFCAK

[(1

ε2 + 1

ε

(11

6+ log

Q2

q2T

))Fqq (0, ε)+ log3 Q2

q2T

(75)

+ 13

6log2 Q2

q2T

+(

35

18− 2

3π2

)log

Q2

q2T

+ 4ζ(3)− 2+ 7

18π2

],

in agreement with Ref. [40]. The first line of Eq. (75) comes from the singularδ(s2) termsand will be exactly cancelled by a contribution appearing in the virtual correction.

C2F contribution

For this colour structure there are six singular regions (plus the ones generated frompermutations likek1 ↔ k2) to be considered [18,19]:

• first triple-collinear region:k1p1 ∼ k2p1 ∼ k1k2 → 0;• second triple-collinear region:k1p2 ∼ k2p2 ∼ k1k2 → 0;• double-soft region:k1, k2 → 0;• first soft-collinear region:k1 → 0, k2p1 → 0;• second soft-collinear region:k1 → 0, k2p2 → 0;• double-collinear region:k1p1 → 0, k2p2 → 0.

In the first region the singularity is controlled by the collinear factorization formula:∣∣Mqq→gg F (p1,p2, k1, k2, φ)∣∣2ab

(76) (8πµ2εαS)2

u2 C2F P

(ab)q→g1g2(q3)

∣∣M(0)qq→F (z3p1,p2, φ)

∣∣2,whereP (ab)

q→g1g2(q3)is the Abelian part of the splitting function that controls the collinear

decay of an initial state quark into a final state gluon pair. This function can be obtainedfrom Eq. (A.6) with the replacement in Eq. (59).

In the double-soft region the factorization formula is obtained by factorizing the twoeikonal factors for independent gluon emissions (see Eq. (A.3) of Ref. [19]):∣∣M(0)

qq→ggF (p1,p2, k1, k2, φ)∣∣2ab

(77) (4πµ2εαS

)216C2

FS12(k1)S12(k2)∣∣M(0)

qq→F (φ)∣∣2,

with S12(k) defined in Eq. (32).

14 A factor 1/2 has been included to account for the two identical particles in the final state.


In the soft-collinear region, say whenk2 → 0 and k1p1 → 0 we have instead (seeEq. (A.5) of [19]):∣∣M(0)

qq→ggF (p1,p2, k1, k2, φ)∣∣2

(4πµ2εαS

)2 2CF

(1− z1)p1k1

(p1 − k1)p2

(p1 − k1)k2p2k2Pqq (1− z1, ε)

(78)× ∣∣M(0)qq→F

((1− z1)p1,p2, φ

)∣∣2,wherez1 is the momentum fraction of the collinear gluon of momentumk1 and can beidentified with the one parametrizing the triple collinear splitting in Eq. (76). Notice that,since the soft gluon of momentumk2 does not resolve the pair of collinear partons, thereis no non-Abelian contribution in Eq. (78).

In the double-collinear region we have, when, e.g.,k1p1 → 0 andk2p2 → 0:∣∣M(0)qq→ggF (p1,p2, k1, k2, φ)

∣∣2 (4πµ2εαS)

2

(1− z1)p1k1(1− z2)p2k2Pqq(1− z1, ε)Pqq(1− z2, ε)

(79)× ∣∣M(0)qq→F

((1− z1)p1, (1− z2)p2, φ

)∣∣2,wherez1 and z2 here represent the momentum fractions (see below) involved in the twocollinear splittings.

As it happens for theCFCA contribution, Eq. (76) supplemented with the definitions(63) is able to approximate correctly also the double-soft and soft-collinear regions in halfof the phase space. But, at variance with theCFCA case, the same formula cannot describecorrectly the double-collinear region, since that one corresponds to the emission of gluonsfrom different legs, i.e., with a kinematical configuration completely different from thetriple-collinear case. Therefore, the strategy followed for the other colour factors does notwork in this case.

In order to overcome this problem there are in principle two strategies. The first oneis to split the phase space in order to isolate the double-collinear region and perform thecalculation separately for its contribution using the expression in Eq. (79). The second isto modify Eq. (76) in order to enforce the correct singular behaviour in all possible limits.We decided to follow the second strategy and for that we have first studied Eq. (76) withthe definitions (63) and isolated the terms that do, incorrectly, contribute (terms withx2 inthe denominator) when the collinear gluons are emitted from the different legs. In this way,we were able to find a slight modification ofP (ab)

g1g2q3 in Eq. (A.6) that allows to take intoaccount the double-collinear region without spoiling the behaviour in the other regions as

D(ab)g1g2q3

=

s2123

2s13s23x3

[(1+ x2

3

x1x2− ε

x21 + x2

2

x1x2

)fq(z1)fq(z2)− ε(1+ ε)

]+ s123

s13

[(x3(1− x1)+ (1− x2)

3

x1x2− ε

(x2

1 + x1x2 + x22

)1− x2

x1x2

)× fq(z2) + ε2(1+ x3)

]


(80)+ (1− ε)

[ε − (1− ε)

s23

s13

]+ (1 ↔ 2).

With respect to the expression ofP (ab)g1g2q3 of Eq. (A.6), the only difference is due to the

introduction of the extra factorsfq(z). The functionfa(z), anticipating that a similarapproach will be followed in the gluonic channel, is defined by

(81)fa(z) = z

2Ca(1− z)Paa(1− z, ε), a = q,g,

wherePaa are the collinear splitting kernels in Eqs. (28), (29).The functionD(ab)

g1g2q3 depends on the new momentum fractionsz1 andz2 of the gluonswith respect to the incoming antiquark of momentump2. These momentum fractionsshould be the ones relevant for the double-collinear limit. Our improved factorizationformula is, outside the second triple-collinear region given by∣∣M(0)

qq→gg F (p1,p2, k1, k2, φ)∣∣2ab

(82) (8πµ2εαS)2

u2C2

F D(ab)q→g1g2(q3)

∣∣M(0)qq→F (z3p1,p2, φ)

∣∣2,whereD(ab)

q→g1g2(q3)is obtained fromD(ab)

g1g2q3 in Eq. (80) with the definitions in Eqs. (59),(63) and by setting

(83)z1 = p1k1

p1p2, z2 = p1k2

p1p2.

With Eq. (82) we can consistently approximate the relevant matrix element in the regionfrom u0 to umax as we did in the other channels, keeping into account all the singularregions. In fact in the triple-collinear regionz1, z2 → 0 andfq(z1), fq(z2) → 1. Therefore,in this limit Eq. (82) reduces to Eq. (76). The factorsfq(z) become relevant in the double-collinear region, since they ensure that the correct limit is recovered whenp1k1 → 0 andp2k2 → 0 (and the same fork1 ↔ k2).

Notice that the modification of the triple-collinear formula does not spoil the processindependence of our calculation: it just allows to write an ‘improved’ formula that correctlyinterpolates all possible (double-) collinear and soft singularities in the region of phasespace where we have to integrate it. Therefore, with this approach we can avoid to split thephase space in regions where different approximations should be applied.

It is worth noticing that the modification in Eq. (80) makes the calculation more involvedalready at the level of the angular integrals, mostly due to the introduction of the ‘new’momentum fractionsz1 andz2.

For this colour structure we have to subtract the contribution from the factorizationcounterterm, which can be written as

dσFCT

dq2T dQ2dφ

= αS

2π

∫dx

xR

(x,µ2

F

)(dσ(p1,p2, φ, k)

dq2T dQ2 dφ

)p1→xp1

(84)+ αS

2π

∫dx

xR

(x,µ2

F

)(dσ(p1,p2, φ, k)

dq2T dQ2 dφ

)p2→xp2

,


wheredσ(p1,p2, φ, k) corresponds to the cross section for the production ofF and onlyone extra gluon (see Eq. (24)) and

(85)R(x,µ2

F

) = −1

εPAPqq (x)

Γ (1− ε)

Γ (1− 2ε)

(4πµ2

µ2F

)ε

,

with PAPqq (z) = (Pqq(z,0))+ = CF

( 1+z2

1−z

)+ the regularized AP splitting function andµF

the factorization scale. In the limit of smallqT and after taking moments with respect toz,the contribution from the counterterm factorizes as

(86)Σ(2)qq(FCT)(N) = 2CFFqq (N, ε)

[−1

εK

(Q2

µ2F

)ε

γ (1)qq (N)

],

whereFqq (N, ε) and K are defined in Eqs. (35) and (68), respectively. Therefore, inthe qT → 0 limit also the contribution of the factorization counterterm becomes processindependent.

Our final result for theN = 0 moment of the factorized contributionΣ(2)Rqq(ggab)(0) ≡

Σ(2)Rqq(ggab)(0)− Σ

(2)qq(FCT)(0) is:

Σ(2)Rqq(ggab)(0)= CFK

[(2

ε2 + 3

ε

)CFFqq(0, ε)+ 2

ε

1∫0

2Pqq(z, ε) logz

]

(87)

+ C2F

[−2 log3 Q2

q2T

+ 9 log2 Q2

q2T

−(

2+ 2

3π2

)log

Q2

q2T

+ 16ζ(3)− π2 − 43

4

],

in agreement with the result of Ref. [40] for Drell–Yan. Notice that sinceγ(1)qq (0) = 0 there

is no contribution from the factorization counterterm toΣ(2)Rqq(ggab)(0). As we did for the

other colour factors, we have isolated in the first line of Eq. (87) the part that will becancelled by a similar term in the virtual contribution.

A comment to the results obtained so far is in order. The formulae in Eqs. (67), (70),(71), (75), (87) show that the contribution toΣqq(0) from double real emission are actuallyindependent on the specific process in (1). This feature of the double real emission, whichis due to the universality of soft and collinear radiation, will persist also in the gluonchannel. The explicit results obtained so far all agree with the ones obtained for Drell–Yan in Ref. [40].

4.2. Virtual corrections

The second part on the calculation ofΣqq(N) involves the (one-loop) virtual correctionsto single-gluon emission. The corresponding soft and collinear limits have been recentlystudied in Refs. [21–23]. The kinematics is the same as atO(αS) and the singularitiesoriginated by the same configurations discussed above Eq. (27). In the first collinear regionthe interference between the tree-level and one-loop contributions to single gluon emission


behaves as [21,22]

M(0)†qq→g F (p1,p2, k,φ)M(1)u

qq→g F (p1,p2, k,φ)+ c.c.

(88)

4παSµ2ε

z1p1k

[Pqq (z1, ε)

(M(0)†

qq→F (z1p1,p2, φ)M(1)qq→F (z1p1,p2, φ)+ c.c.

)+ 2g2

S

(4πµ2

2p1k

)ε

P(1)q→(q)g(z1, ε)

∣∣M(0)qq→F (z1p1,p2, φ)

∣∣2].In Eq. (88) there are two terms. In the first one the tree-level splitting kernelPqq(z1, ε) is

factorized with respect to the interference of therenormalizedone-loop amplitudeM(1)qq→F

and the tree level oneM(0)qq→F .

The second term contains instead theunrenormalizedone-loop correction to the splittingkernelP (1)

q→(q)g(z1, ε) times the Born matrix element squared. The functionP(1)q→(q)g(z1, ε)

controls the one-loop collinear splitting of an initial state quark into a final state quark withmomentum fractionz1, in the CDR scheme. Its explicit expression can be derived from theresults of Refs. [21,22] and is up toO(ε0):

P(1)q→(q)g(x, ε)

= (1− x)−ε CΓ

(4π)2

(89)

×[CAPqq(x, ε)

(− 1

ε2 + 1

2log2(1− x)+ Li2

1

1− x− Li2(1− x)

)+ CF Pqq(x, ε)

(−2

εlog(x)− 2 log(x) log(1− x)+ 2 Li2(1− x)

)+ CF (CF − CA)x

],

where

(90)CΓ = Γ (1+ ε)Γ 2(1− ε)

Γ (1− 2ε).

A factorization formula similar to Eq. (88) holds when the gluon is radiated by the initialstate antiquark.

Let us now consider the soft region. At one-loop order, for a general amplitude withn

hard partons, soft factorization formulae involve colour correlations between two and threehard momentum partons in the matrix element squared [23]. Nevertheless, in the case ofonly two hard partons the soft singularity is controlled by a simpler factorization formula(see Eq. (57) of Ref. [23])

M(0)†qq→g F (p1,p2, k,φ)M(1)u


16παSµ2εCF

(91)×(S12(k)

(M(0)†

qq→F (φ)M(1)qq→F (φ) + c.c.

) + S(1)12 (k)

∣∣M(0)qq→F (φ)

∣∣2),


where

(92)S(1)12 (k) = − αS

2πCAS12(k)

1

ε2

Γ 4(1− ε)Γ 3(1+ ε)

Γ 2(1− 2ε)Γ (1+ 2ε)

[4πµ2S12(k)

]εis the unrenormalizedone-loop correction to the tree-level eikonal factor. Likewise atO(αS) (see Eq. (31)), in Eq. (91) colour correlations are absent, and the factorizationformula is similar in structure to the collinear one in Eq. (88).

One can verify that, as it happens at leading order, the factorization formula Eq. (88)with z1 = z = Q2/s correctly reproduces also the behaviour in the soft-region, given byEq. (91).

Furthermore, by expressingkp1 in terms ofqT and using Lorentz invariance as we didat leading order, a single factorization formula in the smallqT limit is obtained:

M(0)†qq→g F (p1,p2, k,φ)M(1)u


(93)

4παSµ2ε

q2T

2(1− z)

z

[Pqq (z, ε)

(M(0)†

qq→F (φ)M(1)qq→F (φ)+ c.c.

)+ 2g2

S

(4πµ2

q2T

)ε

(1− z)εP(1)q→(q)g(z, ε)

∣∣M(0)qq→F (φ)

∣∣2],and this formula can be used to approximate the virtual contribution in the full phase space.The same formula can be obtained by defining the collinear momentum fraction in Eq. (88)as:

(94)z1 = 1− kp2

p1p2,

and similarly whenp1 ↔ p2. It is important to point out that, at variance with what happensin the double real emission contribution, here a process-dependent information appears,i.e., the one-loop matrix elementM(1)

qq→F (φ). The most general structure of the product

M(0)†qq→FM

(1)qq→F + c.c. is, according to Eq. (38):

M(0)†qq→FM

(1)qq→F + c.c.

(95)= αS

2π

(4πµ2

Q2

)εΓ (1− ε)

Γ (1− 2ε)

(−2CF

ε2 − 3CF

ε+AF

q (φ)

)∣∣M(0)qq→F

∣∣2.In Eq. (95) the structure of the poles inε is universal and fixed by the flavour of theincoming partons, whereas, as discussed in Section 3 the finite part is parameterized by ascalar functionAF

q (φ) depending on the kinematics of the final state particles.

The contribution from the UV counterterm in theMS scheme is:

(96)Σ(2)UVCT(N) = CFFqq (N, ε)K

(Q2

µ2

)ε(−1

ε

)β0.

By approximating our matrix element with Eq. (93), the calculation can now be performedquite easily as in Eq. (35). Using Eq. (95), and adding the contribution of the UV


counterterm in Eq. (96) we find:

Σ(2)Vqq (0)

(97)

= CFK[

− 1

ε2 (2CF + CA)− 1

ε

(CA log

Q2

q2T

+ 3CF + β0

)]Fqq (0, ε)

− 2

ε

1∫0

2Pqq(z, ε) logz

+ CA

(− log3 Q2

q2T

+ 3

2log2 Q2

q2T

+ π2

3log

Q2

q2T

+ π2

2− 15

2− 8ζ(3)

)+ CF

((−5+ 4

3π2

)log

Q2

q2T

+ 39

2− 2π2

)

+(

2 logQ2

q2T

− 3

)AF

q (φ)+ β0 logQ2

µ2R

(3− 2 log

Q2

q2T

),

whereµ2R is the renormalization scale at whichαS is now evaluated. The terms involving

Fqq (0, ε) andPqq (z, ε) in Eq. (97) are the ones that cancel against the corresponding termsin Eqs. (67), (75) and (87). In the case of Drell–Yan, by using Eq. (39), our result agreeswith the one of Ref. [40].

4.3. Total result for theqq channel

After adding the real and virtual contributions in

Σ(2)qq (0)= Σ

(2)Rqq(nid)(0)+ Σ

(2)Rqq(id)(0)+ Σ

(2)Rqq(qq)(0)+ Σ

(2)Rqq(ggnab)(0)

(98)+ Σ(2)Rqq(ggab)(0)+ Σ

(2)Vqq (0)

all divergent terms inε cancel out and we find:

Σ(2)qq (0)= log3 Q2

q2T

[−2C2F

] + log2 Q2

q2T

[9C2

F + 2CFβ0]

+ logQ2

q2T

[C2

F

(2

3π2 − 7

)+ 2CFAF

q (φ)

+ CFCA

(35

18− π2

3

)+ CFnf TR

(−2

9

)]

(99)

+[C2

F

(−15

4− 4ζ(3)

)− 3CFAF

q (φ)

+ CFCA

(−13

4− 11

18π2 + 6ζ(3)

)+ CFnf TR

(1+ 2

9π2

)],

where we have set againµ2F = µ2

R = Q2. It is worth noticing that the process dependencein Eq. (99) is fully contained in the functionAF

q (φ).


Once one moment (theN = 0 in this case) has been computed, it is quite simple toextend the calculation to a general value ofN by studying the combination [14]

(100)Σ(N) − Σ(0) =∫

dz(zN − 1

) q2T Q

2

dσ0/dφ

dσ

dq2T dQ2dφ

.

Here, the factor(zN − 1) eliminates singularities in the integrand whenz → 1 and allowsto setqT = 0 once the integral over the variableu has been done (in most of the cases it ispossible to setqT = 0 even before integrating overu). In that sense the complexity of thecalculation is considerably reduced and the result can be expressed as

Σ(2)qq (N) = log3 Q2

q2T

[−2C2F

] + log2 Q2

q2T

[9C2

F + 2CFβ0 − 6CFγ(1)qq (N)

]+ log

Q2

q2T

[C2

F

(2

3π2 − 7

)+ CFCA

(35

18− π2

3

)− 2

9CFnf TR + 2CFAF

q (φ)+ (2β0 + 12CF )γ(1)qq (N)

− 4(γ (1)qq (N)

)2 + 4C2F

(1

(N + 1)(N + 2)− 1

2

)]

(101)

+[C2

F

(−15

4− 4ζ(3)

)+ CFCA

(−13

4− 11

18π2 + 6ζ(3)

)− 3CFAF

q (φ)+ CF nf TR

(1+ 2

9π2

)+ 2γ (2)

(−)(N)

+ 2CFγ(1)qq (N)

(π2

3+ 2

1

(N + 1)(N + 2)

)+ 2γ (1)

qq (N)AFq (φ)

− 2CF (β0 + 3CF )

(1

(N + 1)(N + 2)− 1

2

)],

where γ(2)(−)(N) is the nonsinglet space-like two-loop anomalous dimension [41]. The

extraction of the resummation coefficients for theqq channel from Eq. (101) will beperformed, along with the corresponding one for thegg channel, in Section 6.

5. The calculation at O(α2S): the gluon channel

The strategy for the computation of theO(α2S) contributions in the gluon channel is the

same as the one developed for theqq case. In a similar way, we first consider the doublereal emission contribution and then the virtual correction.

Let us first discuss the contribution coming from the factorization counterterm, that willbe subtracted from the real corrections in the next subsection. By following the same stepsthat lead to Eq. (86) we obtain

Σ(2)gg(FCT)(N) = 2CAFgg(N, ε)

[−1

εK

(Q2

µ2F

)ε

γ (1)gg (N)

]


(102)+ 2CFFgq(N, ε)

[−1

εK

(Q2

µ2F

)ε

γ (1)qg (N)

].

Eq. (102) contains two terms. The first one, due to the subtraction of one collinear gluon,is analogous to the one in Eq. (86) and contributes to bothC2

A andCATR colour factors.The second term is due to the subtraction of a quark (antiquark) collinear to the initial stategluons, and contributes to theCFTR part. The functionFgq in Eq. (102) is defined as inEq. (35) by

(103)CFFgq(N, ε) ≡1−2qT /Q∫

0

dz zN2(1− z)Pgq(z, ε)√(1− z)2 − 4z q2

T /Q2.

TheqT → 0 limit can be safely taken andFgq(N, ε) gives

(104)Fgq(N, ε) → 2

1∫0

dz zN((1+ (1− z)2

z− εz

)= 2γ (1)

gq (N) − 2ε1

N + 2.

5.1. Real corrections

The contributions toΣ(2)gg from double real emission fall in two classes:

• g + g → q + q + F ;• g + g → g + g + F .

The kinematics is the same as discussed at the beginning of Section 4.1. As we didfor the quark channel, we will first perform the calculation for a fixed moment and thenextend it for generalN . Since theN = 0 moment is divergent for the gluon channel (see,e.g., Eq. (104)), we start fromN = 1. Furthermore, as it happens at LO, spin-correlationsappear in the collinear decay of a gluon. Nevertheless, since the correlations cancel out afterintegration, we will use in the collinear factorization formulae directly the spin-averagedsplitting functions.

5.1.1. Contribution fromqq emissionFor this contribution the strategy followed for theCFTR andCFCA terms in the quark

channel applies. The singular regions are:


In the first triple-collinear region the factorization formula reads∣∣M(0)gg→qq F (p1,p2, k1, k2, φ)

∣∣2(105) (8πµ2εαS)

2

u2Pg→q1q2(g3)

∣∣M(0)gg→F (z3p1,p2, φ)

∣∣2,wherePg→q1q2(g3) is the splitting function that controls the decay of an initial state gluoninto a final state quark–antiquark pair and a gluon. It can be obtained from the expression


of Pq1q2g3 that describes the decay of an (off shell) gluon into a final state quark–antiquarkpair plus a gluon, given in Eq. (A.8), with the crossing transformation (59). In the double-soft region the factorization formula is the same as in Eq. (61) withCF → CA and, likewisein the quark channel, the soft behaviour is correctly taken into account by Eq. (105) withthe definitions (63). Therefore, we can follow the strategy successfully applied in the quarkchannel to obtain

(106)Σ(2)Rgg(qqab)(1) = CFnf TR K

(− 8

27− 16

9log

Q2

q2T

+ 16

9log

Q2

µ2F

)and

Σ(2)Rgg(qqnab)(1) = CAnf TRK

(−2

εFgg(1, ε)+ 25

27− 2π2

9+ 2

9log

Q2

q2T

− 4

3log2 Q2

q2T

(107)− 4

3log

Q2

µ2F

(−11

3+ 2 log

Q2

q2T

)).

In Eqs. (106), (107) we have already subtracted theCFTR and CATR terms from thefactorization counterterm (withN = 1) in Eq. (102). Furthermore, in Eq. (107) we haveisolated a divergent term that will be cancelled by a similar term in the virtual contribution.The explicit expression of the functionFgg(1, ε), defined in Eq. (35), reads

(108)Fgg(1, ε)= −11

3+ 2 log

Q2

q2T

.

5.1.2. Contribution fromgg emissionThe calculation of thegg C2

A contribution toΣ(2)gg parallels the one for theC2

F partin the quark channel since the singular configurations have the same complicated patternas described above Eq. (76). The triple-collinear region is controlled by the factorizationformula∣∣M(0)

gg→ggF (p1,p2, k1, k2, φ)∣∣2

(109) (8πµ2εαS)2

u2 Pg→g1g2(g3)

∣∣M(0)gg→F (z3p1,p2, φ)

∣∣2,where the functionPg→g1g2(g3) is now obtained by applying the crossing transformation(59) to the splitting functionPg1g2g3 that controls the collinear decay of a gluon into threefinal state gluons, given in Eq. (A.11).

The factorization formulae in the soft-collinear and double-collinear regions areanalogous to Eqs. (78), (79), and can be obtained from them by conveniently changing thecolour factors (CF → CA) and splitting functions (Pqq → Pgg). The factorization formulain the double-soft region receives two contributions analogous to the ones in Eqs. (73), (77).

Likewise in the quark channel, Eq. (109) with the momentum fractions defined as inEq. (63) approximates correctly, again in half of the phase space, all possible infraredconfigurations but the double-collinear one.

In order to proceed further, we use the technique developed for the quark channel. Asbefore, we first study the behaviour of Eq. (109) with the definitions (63) in the double


collinear limit and identify the terms that do (incorrectly) contribute in that limit. Thoseterms have to be modified in order to enforce the correct double-collinear limit withoutaffecting the singular behaviour in the other regions. The modified splitting function weobtain is:

Dg1g2g3 = Pnon-singg1g2g3 +

[P

sing-1g1g2g3 + (

Psing-1g1g3g2 + P

sing-1g3g1g2

)fg(z2)

(110)+ (P

sing-2g1g2g3 + P

sing-2g1g3g2

)fg(z2)+ P

sing-2g3g2g1fg(z1)fg(z2)

] + (1↔ 2).

The first term

Pnon-singg1g2g3 = C2

A

(1− ε)

4s212

t212,3 + 3

4(1− ε)+ s123

s12

[4x1x2 − 1

1− x3+ 3

2+ 5

2x3

]+ s2

123

s12s13

[x2x3 − 2+ x1(1+ 2x1)

2+ 1+ 2x1(1+ x1)

2(1− x2)(1− x3)

](111)+ (5 permutations),

contains the part ofPg1g2g3 in Eq. (A.11) that does not contribute to the double-collinearlimit. Therefore, this part of the splitting function does not need any modifications. Thevariabletij,k is defined in Eq. (A.4).

The second part is instead modified with the introduction of the functionfg(z), defined

in Eq. (81). The functionsP sing-1g1g2g3 andP sing-2

g1g2g3 are

(112)Psing-1g1g2g3 = C2

A

s123

s12

(x1x2 − 2

x3+ (1− x3(1− x3))

2

x3x1(1− x1)

),

(113)Psing-2g1g2g3 = C2

A

s2123

s12s13

(x1x2(1− x2)(1− 2x3)

x3(1− x3)+ 1− 2x1(1− x1)

2x2x3

).

Our improved factorization formula is, therefore,∣∣M(0)gg→ggF (p1,p2, k1, k2, φ)

∣∣2(114) (8πµ2εαS)

2

u2 Dg→g1g2(g3)

∣∣M(0)gg→F (z3p1,p2, φ)

∣∣2.As for the quark channel, the expression ofDg→g1g2(g3) is obtained from the one ofDg1g2g3

in Eq. (110) by using Eqs. (59), (63) and definingz1 andz2 through Eq. (83).In the triple-collinear limitfg(z1), fg(z2) → 1 and the various contributions in Eq. (110)

reconstruct the triple-collinear splitting functionPg1g2g3. The role of the functionsfg isagain to enforce the correct behaviour in the double-collinear region. It is worth stressingthat there are in principle many ways to conveniently modify the splitting function and thatwe have tried to find the simplest one that fulfills all the requirements and can be integratedafterwards.

The functionPg1g2g3 in Eq. (A.11) has by itself the most complicated expression amongthe variousPa1a2a3 because one has to sum over six permutations. Besides that, themodification in (110) makes the angular integration very involved. Since many of theensuing terms have an additional singularity ass2 → 0 some of the angular integrals in


Ref. [37] have to be evaluated one order higher inε. Once the angular integrals havebeen performed, one has to face an additional complication: ‘spurious’ 1/q2

T and 1/q4T

singularities appear in the intermediate steps, which of course cancel in the final result, butcreate additional problems to take theqT → 0 limit. The final (factorized) result is

Σ(2)Rgg(gg)(1)

= CAK[(

3

ε2 + 3

ε

11

6+ 1

εlog

Q2

q2T

)CAFgg(1, ε)+ 2

ε

1∫0

2zPgg(z, ε) logz

]

(115)

+C2A

[−log3 Q2

q2T

+ 77

6log2 Q2

q2T

−(

82

9+ 4π2

3

)log

Q2

q2T

− 533

27+ 11π2

9+ 10ζ(3)

],

where we have isolated in the first line the terms that will be cancelled by analogous virtualcontributions. Notice that, sinceγ (1)

gg (1) = −23nf TR , there is no contribution to Eq. (115)

from the factorization counterterm in Eq. (102).

5.2. Virtual corrections

We finally compute the small-qT behaviour of the virtual contribution toΣ(2)gg . The

calculation parallels the one for the quark in Section 4.2, and the singular configurationsare the same as at leading order. For the collinear limit, say whenkp1 → 0, we can write aformula similar to Eq. (88)

M(0)†gg→gF (p1,p2, k,φ)M(1)u

gg→gF (p1,p2, k,φ)+ c.c.

(116)

4παSµ2ε

z1p1k

[Pgg(z1, ε)

(M(0)†

gg→F (z1p1,p2, φ)M(1)gg→F (z1p1,p2, φ)+ c.c.

)+ 2g2

S

(4πµ2

2p1k

)ε

P(1)g→(g)g(z1, ε)

∣∣M(0)gg→F (z1p1,p2, φ)

∣∣2],where Pgg(z1, ε) is the tree-level splitting kernel in Eq. (29) andP (1)

g→(g)g(z1, ε) is theunrenormalizedone-loop correction to the AP kernel for the collinear splitting of an initialstate gluon into a final state gluon with momentum fractionz1, in the CDR scheme. Itsexplicit expression can be derived from the results of Ref. [21] and is up toO(ε0):

P(1)g→(g)g(x, ε)

(117)

= (1− x)−ε CΓ CA

(4π)2

[Pgg(x, ε)

(− 1

ε2− 2

εlog(x)− 2 log(1− x) log(x)+ π2

3

)− 1

3(CA − 2nf TR)x

].

A similar formula holds when the gluon is radiated by the initial state antiquark. In thesoft region, the factorization formula is the same as in Eq. (91) withCF → CA and one


can verify that, as it happens in the quark channel, Eq. (116) withz1 = z = Q2/s correctlyreproduces also the behaviour in the soft-region [21,23]. In the same way as for the quarkchannel we can write down a single factorization formula in the smallqT limit:

M(0)†gg→gF (p1,p2, k,φ)M(1)u

gg→gF (p1,p2, k,φ)+ c.c.

(118)

4παSµ2ε

q2T

2(1− z)

z

[Pgg(z, ε)

(M(0)†

gg→F (φ)M(1)gg→F (φ)+ c.c.

)+ 2g2

S

(4πµ2

q2T

)ε

(1− z)εP(1)g→(g)g(z, ε)

∣∣M(0)gg→F (φ)

∣∣2].According to Eq. (38) therenormalized amplitudeM(1)

gg→F can be written, up toO(ε0)

as:

M(0)†gg→FM

(1)gg→F + c.c.

(119)= αS

2π

(4πµ2

Q2

)εΓ (1− ε)

Γ (1− 2ε)

(−2CA

ε2 − 2β0

ε+AF

g (φ)

)∣∣M(0)gg→F

∣∣2.In the case of Higgs production, in themH mtop limit and including also the finiterenormalization to the effectiveggH vertex, the functionAH

g (φ) is given in Eq. (40).

The contribution from the UV counterterm (in theMS scheme) needed to renormalizethe splitting kernelP (1)

g→(g)g is: 15

(120)Σ(2)ggUVCT(N) = CAFgg(N, ε)K

(Q2

µ2

)ε(−1

ε

)β0.

By approximating our matrix element with Eq. (118), using Eq. (119), and adding thecontribution from Eq. (120) we find

Σ(2)Vgg (1)= CAK

[− 3

ε2CA − 3

εβ0 − 1

εCA log

Q2

q2T

]Fgg(1, ε)

− 2

ε

1∫0

2zPgg(z, ε) logz

+ CA

(−log3 Q2

q2T

+ 11

6log2 Q2

q2T

+(

−65

18+ 5π2

3

)log

Q2

q2T

− 11π2

6+ 389

27− 8ζ(3)

)+ 4

9nf TR

(121)

+AFg (φ)

(−11

3+ 2 log

Q2

q2T

)− β0 log

Q2

µ2R

(−11

3+ 2 log

Q2

q2T

).

15 The total UV counterterm in the case of Higgs production would be three times this one, but we have includedpart of it in the renormalized amplitude (119).


The terms involvingFgg(1, ε) and Pgg(z, ε) in Eq. (121) cancel the correspondingdivergent contributions in Eqs. (107) and (115).

5.3. Total result for the gluon channel

After adding all the contributions in

(122)Σ(2)gg (1)= Σ

(2)Rgg(qqab)(1)+ Σ

(2)Rgg(qqnab)(1)+ Σ

(2)Rgg(gg)(1)+ Σ(2)V

gg (1),

all divergent terms cancel out and we obtain

Σ(2)gg (1)= log3 Q2

q2T

[−2C2A

] + log2 Q2

q2T

[8CAβ0 + 4CAnf TR]

+ logQ2

q2T

[C2

A

(−229

18+ π2

3

)+ 2

9CAnf TR

− 16

9CFnf TR + 2CAAF

g (φ)

]

(123)

+[C2

A

(2ζ(3)− 16

3− 11

18π2

)− 8

27CFnf TR

+ CAnf TR

(37

27− 2

9π2

)− 11

3CAAF

g (φ)

],

where we have set againµ2F = µ2

R = Q2.The contribution for generalN can be computed as explained in the previous section for

the quark channel. The total result is:

Σ(2)gg (N) = log3 Q2

q2T

[−2C2A

] + log2 Q2

q2T

[8CAβ0 − 6CAγ

(1)gg (N)

]+ log

Q2

q2T

[C2

A

(67

9+ π2

3

)− 20

9CAnf TR + 2CAAF

g (φ)

+ 2β0(γ (1)gg (N) − β0

) − 4(γ (1)gg (N)− β0

)2

− 4nf γ(1)gq (N)γ (1)

qg (N)

]

(124)

+[C2

A

(−16

3+ 2ζ(3)

)+ 2CFnf TR + 8

3CAnf TR

− 2β0

(AF

g (φ)+ CAπ2

6

)+ 2γ (2)

gg (N)

+ 2γ (1)gg (N)

(AF

g (φ)+ CAπ2

3

)+ 4CFnf γ

(1)qg (N)

1

(N + 2)

].

Hereγ (2)gg (N) is the singlet space-like (gluon–gluon) two-loop anomalous dimension [42]

whereas the coefficient 1/(N +2) has origin on theN moments of−P εgq(z), see Eq. (104).


6. Final results and discussion

We can now compare the results obtained in the previous sections with the second orderexpansion of the resummation formula in Eq. (18). As for theN -dependent contributionsin Eqs. (101), (124), they fully agree with the ones in Eq. (18).16 This agreement can beconsidered as a nontrivial check of the validity of the resummation formalism, because theexpressions in Eqs. (101), (124) are completely general and the process dependence is fullyembodied in the functionsAF

c (φ). As an alternative, given the resummation formalism forgranted, the result in Eqs. (101), (124) can be considered as an independent re-evaluationof the two-loop anomalous dimensions.

As far as theN -independent part is concerned, it can be used to fix the coefficientsA(2)

andB(2). By comparing the single-logarithmic contributions in Eqs. (101), (124) with theone in Eq. (18) we obtain for the coefficientA

(2)a :

(125)A(2)a = KA(1)

a , a = q,g,

whereK is given in Eq. (11), thus confirming the results first obtained in Ref. [10,13]. Bycomparing the nonlogarithmic terms we find that the coefficientB(2) can be expressed aswell by a single formula for both channels:

(126)B(2)Fa = −2γ (2)

a + β0(2

3Caπ2 +AF

a (φ)), a = q,g,

whereγ(2)a are the coefficients of theδ(1 − z) term in the two-loop splitting functions

P(2)aa (z) [41,42], given by

γ (2)q = C2

F

(3

8− π2

2+ 6ζ(3)

)+ CFCA

(17

24+ 11π2

18− 3ζ(3)

)− CFnf TR

(1

6+ 2π2

9

),

(127)γ (2)g = C2

A

(8

3+ 3ζ(3)

)− CFnf TR − 4

3CAnf TR.

From Eq. (126) we see thatB(2), besides the−2γ(2)a term which matches the expectation

from theO(αS) result, receives aprocess-dependentcontribution controlled by the one-loop correction to the LO amplitude (see Eq. (38)). Thus, as anticipated at the beginning,although the Sudakov form factor in Eq. (5) is usually considered universal we find that itis actually process-dependent beyond next-to-leading logarithmic accuracy.

However, by using the general expression in Eq. (126) it is possible to obtainB(2) fora given process just by computing the one-loop correction to the LO amplitude for thatprocess. For the Drell–Yan case, by using Eq. (39), our result forΣ

(2)qq (N) agrees with the

one of Ref. [14], confirming the coefficientB(2)DYq in Eq. (12).

16 We have checked that the results in the quarksingletchannel are also in agreement with Eq. (18).


In the interesting case of Higgs production in themtop → ∞ limit, by using Eq. (40) wefind:17

(128)

B(2)Hg = C2

A

(236 + 22

9 π2 − 6ζ(3)) + 4CFnf TR − CAnf TR

( 23 + 8

9π2) − 11

2 CFCA.

In particular, this result allows to improve the present accuracy of the matching betweenresummed predictions [44] and fixed order calculations [45].

Notice that in this case, the coefficientB(2)Hg turns out to be numerically large. Actually,

for nf = 5 we haveB(2)Hg /B

(1)g ≈ −14, whereas for Drell–Yan the same ratio leads to

B(2)DYq /B

(1)q ≈ −1.9, i.e., about 7 times smaller than for Higgs production. Both the

appearance of aC2A term (compared toC2

F in the quark case) and the size of the one-loop corrections to Higgs production are the reasons for the large coefficient. Clearly, theuse ofB(2)H

g in the implementation of the resummation formula will have an important

phenomenological impact [46]. Actually, one can expect that the inclusion ofB(2)Hg , which

will tend to reduce the resummed cross section, will partially compensate the increase inthe normalization produced by the (also) large coefficientC

(1)Hgg [33,34].

The fact that the Sudakov form factor is process-dependent is certainly unpleasant. Usu-ally it is called the quark or gluon form factor, since it should be determined by the universalproperties of soft and collinear emission. With the result in Eq. (126), instead we find, forexample, that the form factor forγ γ production is different from the one for Drell–Yan.Moreover, since the hard functionAF

c depends in general on the details of the kinematicsof F (in case ofγ γ production it would depend, e.g., on the rapidities of the photons), thesame happens to the coefficientB(2) and thus to the Sudakov form factor in Eq. (5).

However, the results in Eqs. (46) and (126) suggest a simple interpretation [27]. We cansee in Eq. (46) that the process-dependent coefficients functionsC

(1)Fab (z) have two contri-

butions. The first has acollinearorigin and is driven by theO(ε) part of thePab(z, ε) ker-nel (see Eq. (47)). The second has instead ahard origin, and contains the finite part of theone-loop correction to the leading order subprocess. As a consequence, the scale at whichαS should be evaluated is different for these two terms. In the collinear contributionαS

should be evaluated at same scale as the parton distributions are, i.e.,b20/b

2. By contrast, thecorrect scale at whichαS should be evaluated in the hard contribution is the hard scaleQ2.

As discussed in Ref. [27] this mismatch, that affects the resummation formula in itsusual form Eq. (4), can be solved by introducing a new process-dependent hard functionHF

c (αS(Q2)). The ensuing resummation formula is [27]

WFab(s;Q,b,φ)

=∑c

1∫0

dz1

1∫0

dz2Cca

(αS

(b2

0/b2), z1

)Ccb

(αS

(b2

0/b2), z2

)δ(Q2 − z1z2s

)

17 Actually, using the results of Ref. [43] for the two-loopgg → H amplitude, one can also obtainB(2)Hg for

arbitrarymtop.


(129)× dσFcc(Q

2, αS(Q2),φ)

dφSc(Q,b),

where

(130)dσF

cc(Q2, αS(Q

2),φ)

dφ= dσ

(LO)Fcc (Q2, φ)

dφHF

c

(αS

(Q2), φ)

.

As discussed in Ref. [27], this modification is sufficient to make the Sudakov form factorSc(Q,b) and the coefficient functionsCab(αS(b

20/b

2), z) process-independent, withCab

andHFc being dependent on the introduced ‘resummation-scheme’. We point out that this

modification is not only a formal improvement, since, once a resummation scheme is fixed,the resummation coefficients in Eq. (129) are now universal and it is enough to computethe functionHF

c at the desired order for the process under consideration.Summarizing, in this paper we have exploited the current knowledge on the in-

frared behaviour of tree-level and one-loop QCD amplitudes atO(α2S) to compute the

logarithmically-enhancedcontributions up to next-to-next-to-leading logarithmic accuracy,in an general way, for both quark and gluon channels. Comparing our results with theqT -resummation formula we have extracted the coefficients that control the resummationof the large logarithmic contributions. We have presented a result that allows to computethe resummation coefficientB(2)F for any process, by simply knowing the one-loop (vir-tual) corrections to the lowest order result. In particular, we have obtained the result for thecase of Higgs production in the largemtop approximation, which turns out to be numeri-cally relevant for phenomenological analyses.

The results of our calculation clearly show that the Sudakov form factor is actuallyprocess dependent within the conventional resummation approach. An improved versionof the resummation formula where this problem is absent has been presented in Ref. [27].

Acknowledgements

We thank Stefano Catani for a fruitful Collaboration and helpful discussions, ZoltanKunszt, James Stirling, Luca Trentadue and Werner Vogelsang for discussions, andChristine Davies for providing us with a copy of her Ph.D. thesis, where the details ofthe calculation for Drell–Yan are shown.

This work has been almost entirely performed at the Institute for Theoretical Physics atthe ETH-Zurich. We thank Zoltan Kunszt and the staff of ETH for the warm hospitalityand for the pleasant time we spent in Zurich.

Appendix A. Triple-collinear splitting functions

In this appendix we collect the various expressions of the triple-collinear splittingfunctions. Denoting byr1, r2 andr3 the momenta of the final state partons that becomecollinear, the triple-collinear splitting functions depend on the invariantssij = (ri + rj )

2,s123 = s12 + s13 + s23 that parameterize how the collinear limit is approached, and on


the momentum fractionsxi (i = 1,2,3) involved in the collinear splitting. The splittingfunction for the collinear decay of a quarkq in qq pair plus a quark is

(A.1)Pq1q2q3 = [Pq ′

1q′2q3

+ (2 ↔ 3)] + P

(id)q1q2q3

,

where

Pq ′1q

′2q3

= 1

2CFTR

s123

s12

(A.2)×[− t212,3

s12s123+ 4x3 + (x1 − x2)

2

x1 + x2+ (1− 2ε)

(x1 + x2 − s12

s123

)],

P(id)q1q2q3

= CF

(CF − 1

2CA

)(1− ε)

(2s23

s12− ε

)+ s123

s12

[1+ x2

1

1− x2− 2x2

1− x3− ε

((1− x3)

2

1− x2+ 1+ x1 − 2x2

1− x3

)− ε2(1− x3)

]

(A.3)

− s2123

s12s13

x1

2

[1+ x2

1

(1− x2)(1− x3)− ε

(1+ 2

1− x2

1− x3

)− ε2

]+ (2 ↔ 3),

and the variabletij,k is defined as

(A.4)tij,k ≡ 2xisik − xj sik

xi + xj+ xi − xj

xi + xjsij .

The splitting function for theq → qgg decay can be decomposed according to the differentcolour coefficients:

(A.5)Pg1g2q3 = C2F P (ab)

g1g2q3+CFCAP

(nab)g1g2q3

,

and the Abelian and non-Abelian contributions are

P (ab)g1g2q3

=

s2123

2s13s23x3

[1+ x2

3

x1x2− ε

x21 + x2

2

x1x2− ε(1+ ε)

]+ s123

s13

[x3(1− x1)+ (1− x2)

3

x1x2+ ε2(1+ x3)

− ε(x2

1 + x1x2 + x22

)1− x2

x1x2

]+ (1− ε)

[ε − (1− ε)

s23

s13

](A.6)+ (1 ↔ 2),

P (nab)g1g2q3

=(1− ε)

(t212,3

4s212

+ 1

4− ε

2

)+ s2

123

2s12s13

[(1− x3)

2(1− ε)+ 2x3

x2+ x2

2(1− ε)+ 2(1− x2)

1− x3

]


− s2123

4s13s23x3

[(1− x3)

2(1− ε)+ 2x3

x1x2+ ε(1− ε)

]+ s123

2s12

[(1− ε)

x1(2− 2x1 + x21)− x2(6− 6x2 + x2

2)

x2(1− x3)

+ 2εx3(x1 − 2x2)− x2

x2(1− x3)

]

(A.7)

+ s123

2s13

[(1− ε)

(1− x2)3 + x2

3 − x2

x2(1− x3)

− ε

(2(1− x2)(x2 − x3)

x2(1− x3)− x1 + x2

)− x3(1− x1)+ (1− x2)

3

x1x2

+ ε(1− x2)

(x2

1 + x22

x1x2− ε

)]+ (1 ↔ 2).

When a gluon decays collinearly, spin-correlations are present. Here we are concerned onlywith spin-averaged splitting functions. When the gluon decays in aqq pair plus a gluonthe splitting function is

(A.8)Pg1q2q3 = CFTRP(ab)g1q2q3

+ CATRP(nab)g1q2q3

,

where

P(ab)g1q2q3

= −2− (1− ε)s23

(1

s12+ 1

s13

)+ 2

s2123

s12s13

(1+ x2

1 − x1 + 2x2x3

1− ε

)

(A.9)

− s123

s12

(1+ 2x1 + ε − 2

x1 + x2

1− ε

)− s123

s13

(1+ 2x1 + ε − 2

x1 + x3

1− ε

),

and

P(nab)g1q2q3

=− t223,1

4s223

+ s2123

2s13s23x3

[(1− x1)

3 − x31

x1(1− x1)− 2x3(1− x3 − 2x1x2)

(1− ε)x1(1− x1)

]+ s123

2s13(1− x2)

[1+ 1

x1(1− x1)− 2x2(1− x2)

(1− ε)x1(1− x1)

]+ s123

2s23

[1+ x3

1

x1(1− x1)+ x1(x3 − x2)

2 − 2x2x3(1+ x1)

(1− ε)x1(1− x1)

](A.10)− 1

4+ ε

2− s2

123

2s12s13

(1+ x2

1 − x1 + 2x2x3

1− ε

)+ (2 ↔ 3).

In the case of a gluon decaying into three collinear gluons we have:

Pg1g2g3 = C2A

(1− ε)

4s212

t212,3 + 3

4(1− ε)

+ s123

s12

[4x1x2 − 1

1− x3+ x1x2 − 2

x3+ 3

2+ 5

2x3 + (1− x3(1− x3))

2

x3x1(1− x1)

]


+ s2123

s12s13

[x1x2(1− x2)(1− 2x3)

x3(1− x3)+ x2x3 − 2+ x1(1+ 2x1)

2

+ 1+ 2x1(1+ x1)

2(1− x2)(1− x3)+ 1− 2x1(1− x1)

2x2x3

](A.11)+ (5 permutations).

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Next-to-next-to-leading logarithms in four-fermionelectroweak processes at high energy

J.H. Kühna, S. Mocha, A.A. Peninb,c, V.A. Smirnovd

a Institut für Theoretische Teilchenphysik, Universität Karlsruhe, 76128 Karlsruhe, Germanyb II. Institut für Theoretische Physik, Universität Hamburg, 22761 Hamburg, Germany

c Institute for Nuclear Research of Russian Academy of Sciences, 117312 Moscow, Russiad Nuclear Physics Institute of Moscow State University, 119899 Moscow, Russia

Received 5 July 2001; accepted 13 September 2001

Abstract

We sum up the next-to-next-to-leading logarithmic virtual electroweak corrections to the highenergy asymptotics of the neutral current four-fermion processes for light fermions to all orders inthe coupling constants using the evolution equation approach. From this all order result we derivefinite order expressions through next-to-next-to leading order for the total cross section and variousasymmetries. We observe an amazing cancellation between the sizable leading, next-to-leading andnext-to-next-to-leading logarithmic contributions at TeV energies. 2001 Elsevier Science B.V. Allrights reserved.

PACS:12.38.Bx; 12.38.Cy; 12.15.Lk

1. Introduction

Experimental and theoretical studies of electroweak interactions have traditionallyexplored the range from very low energies, e.g., through parity violation in atoms, up toenergies comparable to the masses of theW - andZ-bosons, e.g., at LEP or the Tevatron.The advent of multi-TeV-colliders like the LHC or a future linear electron–positron colliderduring the present decade will give access to a completely new energy domain.

Once the characteristic energiess are far larger than the masses of theW - andZ-bosons,MW,Z , multiple soft and collinear gauge boson emission is kinematically possible.Conversely, exclusive reactions like electron–positron (or quark–antiquark) annihilationinto a pair of fermions or gauge bosons will receive large negative corrections from virtualgauge boson emission. These double logarithmic “Sudakov” corrections [1,2] proportional

E-mail address:[email protected] (S. Moch).



J.H. Kühn et al. / Nuclear Physics B 616 (2001) 286–306 287

to powers ofg2 ln2(s/M2W,Z) are dominant at high energies and thus have to be controlled

in higher orders to arrive at reliable predictions.The importance of large logarithmic corrections for electroweak reactions at high

energies in one-loop approximation which may well amount to ten or even twenty percentwas noticed already several years ago [3,4]. The need for a resummation of higher ordersof these double-logarithmic terms in the context of electroweak interactions was firstemphasized in [5] which also contains a first discussion of this resummation. In particular,it was shown that the double logarithms do not depend on the details of the mechanism ofthe gauge boson mass generation. The issue is complicated by the appearance of massive(W,Z) and massless (γ ) gauge bosons in theSUL(2)× U(1) theory, which necessarilyhave to be treated on a different footing. A complete analysis of this problem in the doubleor leading logarithmic (LL) approximation by the systematic separation of soft (ωγ M)and hard (ωγ M) photons was given in [6]. In two loops the results of this approachessentially based on the concept of infrared evolution equations have been confirmed byexplicit calculations in [7].

The large coefficient in front of the single logarithmic term in the one-loop corrections tothe electroweak amplitudes (see, e.g., [8]) suggests that subleading terms play an importantrole also in higher orders, as long as realistic energies of order TeV are under consideration.Motivated by this observation a systematic evaluation of the next-to-leading logarithmic(NLL) terms for the neutral current massless four fermion process was performed inRef. [9]. Indeed one finds sizeable two-loop effects both for the total cross section, forthe left-right and for the forward–backward asymmetry. A subclass of NLL corrections forgeneral electroweak processes was subsequently evaluated in [10] without, however, thevery importantangular dependentcontributions.

Various authors have also extracted the double and single logarithmic corrections fromthe complete one-loop calculations [11,12]. The analysis for the general electroweakprocesses given in [12] is in full agreement with [9], whereas a different prescription toseparate the QED contribution is adopted in [11]. Higher order heavy fermion mass effectson the asymptotic high energy behavior of the electroweak amplitudes were discussedin [13]. The incomplete cancellation of the real and virtual electroweak double logarithmiccorrections in the inclusive cross sections was investigated in [14].

Following the approach of [9] which in turn is based on the investigations in the contextof QCD [15–23], this paper is devoted to the derivation of the next-to-next-to-leadinglogarithmic (NNLL) terms for the massless neutral current four-fermion cross sections.In Section 2 we present as a first step the NNLL form factor which describes the scatteringamplitude in an external Abelian field for theSU(N) gauge theory. The derivation is basedon the evolution equation derived in Refs. [18–20]. In Section 3 we then generalize theresult to the four fermion process in theSU(N) gauge theory. After factoring off thecollinear logarithms we use an evolution equation for the remaining amplitudes whichis governed by an angular dependent soft anomalous dimension matrix [20,21,23].

Finally we apply this result to electroweak processes in Section 4. To identify the pureQED infrared logarithms which are compensated by soft real photon radiation we combinethe hard evolution equation which governs the dependence of the amplitudes ons with the

288 J.H. Kühn et al. / Nuclear Physics B 616 (2001) 286–306

infrared evolution equation [6]. The latter describes the dependence of the amplitude on afictitious photon mass which serves as an infrared regulator and drops out after includingthe effect of the soft photon emission. The hard and infrared evolutions are matched byfixing the initial conditions at the scaleMW,Z . Section 4 also contains a discussion ofthe numerical implications of our result. A brief summary and conclusions are given inSection 5.

2. The Abelian form factor in the Sudakov limit

Let us first consider the vector form factor which determines the fermion scatteringamplitude in an external Abelian field for theSU(N) gauge model. In the Bornapproximation,

(1)FB = ψ(p2)γµψ(p1),

wherep1 denotes the incoming andp2 the outgoing momentum.There are two “standard” regimes of the Sudakov limits = (p1 − p2)

2 → −∞: (i) on-shell massless fermions,p2

1 = p22 = 0 and gauge bosons with a small non-zero massM2

Q2 [2], or (ii) slightly off-shell fermionsp21 = p2

2 = −M2, and massless gauge boson [1].Let us consider the first case and choose, for convenience,p1,2 = (Q/2,0,0,∓Q/2) sothat 2p1p2 =Q2 = −s. The asymptoticQ-dependence of the form factor in this limit isgoverned by the evolution equation [18,19]

(2)∂

∂ lnQ2F =[ Q2∫M2

dx

xγ(α(x)

) + ζ (α(Q2)) + ξ(α(M2)

)]F .

Its solution is

(3)F = F0(α(M2)

)exp

Q2∫M2

dx

x

[ x∫M2

dx ′

x ′ γ(α(x ′)

) + ζ (α(x)) + ξ(α(M2))]FB.

The LL approximation includes all the terms of the formαn log2n(Q2/M2) and isdetermined by the one-loop value ofγ (α). The NLL approximation includes all the termsof the formαn log2n−m(Q2/M2) with m= 0,1. This requires the one-loop values ofγ (α)andζ(α)+ ξ(α) and using the one-loop running ofα in γ (α). The NNLL approximationincludes all the terms of the formαn log2n−m(Q2/M2) with m= 0,1,2. In this caseγ (α)is required up toO(α2), ζ(α), ξ(α) andF0(α) up to O(α) together with the two-looprunning ofα in γ (α) and one-loop running ofα in ζ(α).

The functions entering the evolution equation can be determined by comparing Eq. (3)expanded in the coupling constant to the asymptotic, i.e., leading inM2/Q2, fixed orderresult for the form factor. To compute this fixed order asymptotic result we apply theexpansion by regions approach formulated in [24] and discussed using characteristic two-loop examples in [25]. It consists of the following steps: (i) consider various regions of a


loop momentumk and expand, in every region, the integrand in Taylor series with respectto the parameters that are there considered small, (ii) integrate the expanded integrand overthe whole integration domain of the loop momenta, (iii) put to zero any scaleless integral.In step (ii) dimensional regularization [26] withd = 4− 2ε spacetime dimensions is usedto handle the divergences. The following regions are relevant in the considered version (i)of the Sudakov limit [27]:

hard (h): k ∼Q,1-collinear (1c): k+ ∼Q, k− ∼M2/Q, k ∼M,2-collinear (2c): k− ∼Q, k+ ∼M2/Q, k ∼M,

(4)soft (s): k ∼M.Herek± = k0 ± k3, k = (k1, k2). By k ∼Q, etc. we mean that any component ofkµ is oforderQ. In one loop this leads to the following decomposition [9]

(5)F (1) = (∆h +∆c +∆s)FB,

∆(1)h = CF

(− 2

ε2 + 1

ε

(2 ln

(Q2) − 3

) − ln2(Q2) + 3 ln(Q2) + π2

6− 8

),

∆(1)c = CF(

2

ε2− 1

ε

(2 ln

(Q2) − 4

) + 2 ln(Q2) ln

(M2)

− ln2(M2) − 4 ln(M2) − 5π2

6+ 4

),

(6)∆(1)s = CF(

−1

ε+ ln

(M2) + 1

2

),

where CF = (N2 − 1)/(2N) is the quadratic Casimir operator of the fundamentalrepresentation of theSU(N) group and the subscriptc denotes the contribution of bothcollinear regions. The ’t Hooft scaleµ has been dropped in the argument of the logarithmsas well as the factor(4πe−γEε(µ2))ε per loop. For a perturbative functionf (α) we define

(7)f (α)=∑n

(α

4π

)nf (n).

The contribution of all the regions add up to the well known result

(8)F (1) = −CF(

ln2(Q2

M2

)− 3 ln

(Q2

M2

)+ 7

2+ 2π2

3

)FB.

On the other hand the one-loop form factor can be written as

(9)F (1) =(

1

2γ (1) ln2

(Q2

M2

)+ (ξ(1) + ζ (1)) ln

(Q2

M2

)+ F (1)0

)FB.

The expansion by regions is very efficient for determination of the parameters of theevolution equation. Indeed, it not necessary to compute the complete asymptotic result inorder to obtain the functions parameterizing the logarithmic contributions. In the process of


the scale separation through the expansion by regions the logarithmic contributions showup as the singularities of the contributions from different regions which cancel in the totalresult. Thus one can identify the regions relevant for determining a given parameter of theevolution equation and compute them separately up to the required accuracy. For example,the anomalous dimensionsγ (α) and ζ(α) are known to be independent on the infraredcutoff and are completely determined by the contribution from the hard loop momentum[18,19]. If dimensional regularization is used for the infrared divergences of the hard loopmomentum contribution, as in our approach, the anomalous dimensionsγ (α) and ζ(α)are given by the coefficients of the double and single poles of the hard contribution tothe exponent (3), respectively [19,22]. On the contrary, the functionsξ(α) andF0(α) fixthe initial conditions for the evolution equation. They are not universal and depend on theinfrared sector of the model. Furthermore, the values ofξ(α) andF0(α) depend on thedefinition of the lower integration limits in Eq. (3). To determine the functionξ(α) one hasto know also the singularities of the collinear region contribution whileF0(α) requires thecomplete information on the contributions of all the regions.

From the first line of Eq. (6) we find the one-loop anomalous dimensions

γ (1) = −2CF ,

(10)ζ (1)= 3CF .

With the above values ofγ (1) andζ (1) it is straightforward to obtain the one-loop resultfor the remaining functions

ξ(1) = 0,

(11)F(1)0 = −CF

(7

2+ 2π2

3

)by comparing Eqs. (9) and (8). Note that in the Born approximationF

(0)0 = 1.

A similar decomposition can be performed in two loops

(12)F (2) = (∆hh +∆hc +∆cc + · · ·)FB.Only the hard–hard part is now available [28]. However, this information is sufficient todetermine the two-loop valueγ (2). Beside the running of the coupling constant,γ (2) is theonly two-loop quantity we need for the NNLL approximation. It reads [29]

(13)γ (2) = −2CF

[(67

9− π2

3

)CA − 20

9TFnf

],

for α defined in theMS scheme. HereCA = N is the quadratic Casimir operator of theadjoint representation,TF = 1/2 is the index of the fundamental representation andnf isthe number of light (Dirac) fermions.

Let us consider the two-loop corrections. The LL, NLL and NNLL approximations aregiven, respectively, by

(14)F (2)LL = 1

8

(γ (1)

)2 ln4(Q2

M2

)FB,


(15)F (2)NLL = 1

2

(ζ (1)− 1

3β0

)γ (1) ln3

(Q2

M2

)FB,

(16)F (2)NNLL = 1

2

(γ (2) + (

ζ (1) − β0)ζ (1)+ F (1)0 γ (1)

)ln2

(Q2

M2

)FB,

whereβ0 = 11CA/3 − 4TFnf /3 is the one-loopβ-function provided the normalizationpoint ofα isM. The two-loop running of the coupling constant in the leading orderγ (α)

starts to contribute in the three-loop NNLL approximation.The presence of a scalar particle in the fundamental representation with no Yukawa

coupling to fermions leads only to a modification of theγ - andβ-functions. One scalarboson with the mass much less thanQ gives the additional contribution of 10CFTF /9to γ (2) and the additional contribution of−TF/3 toβ0.

For the standard model inspired case of theSU(2)L andU(1) gauge groups withnf = 6and one charged scalar boson either in the fundamental representation ofSU(2) or of theunitU(1) charge up to NNLL approximation we have

F (1) =[−3

4ln2

(Q2

M2

)+ 9

4ln

(Q2

M2

)−

(21

8+ π2

2

)]FB,

(17)F (2) =[

9

32ln4

(Q2

M2

)− 19

48ln3

(Q2

M2

)−

(469

48− 7π2

8

)ln2

(Q2

M2

)]FB,

and

F (1) =[− ln2

(Q2

M2

)+ 3 ln

(Q2

M2

)−

(7

2+ 2π2

3

)]FB,

(18)F (2) =[

1

2ln4

(Q2

M2

)− 52

9ln3

(Q2

M2

)+

(619

18+ 2π2

3

)ln2

(Q2

M2

)]FB,

respectively. The relatively small coefficient of the LL terms and the large coefficient ofthe NNLL terms in the form factor are clearly indicative of the importance of the NNLLcorrections and, as we will see, reflect the general structure of the logarithmically enhancedelectroweak corrections.

3. The four fermion amplitude

Let us now investigate the four-fermion scattering at fixed angles in the limit when all theinvariant energy and momentum transfers of the process are far larger than the gauge bosonmass,|s| ∼ |t| ∼ |u| M2. The analysis of the four fermion amplitude is complicated bythe extra kinematical variable and the presence of different “color” and Lorentz structures.We adopt the following notation

Aλ = ψ2taγµψ1ψ4t

aγµψ3,

Ad5 = ψ2γµγ5ψ1ψ4γµγ5ψ3,

AλLL = ψ2Ltaγµψ1Lψ4Lt

aγµψ3L,

(19)AdLR = ψ2Lγµψ1Lψ4Rγµψ3R,


etc. Hereta denotes theSU(N) generator,pi the momentum of theith fermion andp1, p3

are incoming, andp2, p4 outgoing momenta, respectively. Hencet = (p1 − p4)2 = −sx−

andu = (p1 + p3)2 = −sx+ wherex± = (1 ± cosθ)/2 andθ is the angle between the

spatial components ofp1 andp4. The complete basis consists of four independent chiralamplitudes, each of them of two possible color structure. For the moment we consider aparity conserving theory, hence only two chiral amplitudes are not degenerate. The Bornamplitude is given by

(20)AB = ig2

sAλ.

The collinear divergences in the hard part of the virtual corrections and the corresponding“collinear” logarithms are known to factorize. They are responsible, in particular, for thedouble logarithmic contribution and depend only on the properties of the external on-shellparticles but not on a specific process [15–20]. This fact is especially clear if a physical(Coulomb or axial) gauge is used for the calculation. In this gauge the collinear divergencesare present only in the self energy insertions to the external particles [16,19,20]. Thus, foreach fermion–antifermion pair of the four-fermion amplitude the collinear logarithms arethe same as for the form factorF discussed in the previous section. Let us denote byA theamplitude with the collinear logarithms factored out. For convenience we separate fromAall the corrections entering Eq. (3) so that

(21)A = ig2

sF2A.

The resulting amplitudeA contains the logarithms of the “soft” nature corresponding to thesoft divergences of the hard region contribution and the renormalization group logarithms.It can be represented as a vector in the color/chiral basis and satisfies the followingevolution equation [20,21,23]:

(22)∂

∂ lnQ2A = χ(

α(Q2))A,

whereχ(α) is the matrix of the soft anomalous dimensions. Note that we do not includeto Eq. (22) the pure renormalization group logarithms which can be absorbed by fixing thenormalization scale ofg in the Born amplitude (20) to beQ. The solution of Eq. (22) reads

(23)A =∑i

A0i(α(M2))exp

[ Q2∫M2

dx

xχi

(α(x)

)],

whereχi(α) are eigenvalues ofχ(α) andA0i (α) areQ-independent eigenvectors ofχ(α)which determine the initial conditions for the evolution equation atQ =M. Similar tothe functionF0(α) they get contributions from all the regions while the matrix of thesoft anomalous dimensions is given by the coefficients of the single pole of the hardregion contribution to the exponent (23) [9,20]. Strictly speaking the matricesχ(α(Q2))

for different values ofQ do not commute and the solution is given by the path-orderedexponent [20]. The NLL approximation is given by the one-loop value ofχ(α) while the


NNLL approximation requiresA0i (α) up toO(α) together with the one-loop running ofαin χ(α).

In one loop the elements of the matrixχ(α) do not depend on chirality and read [9]

χ(1)λλ = −2CA(ln(x+)+ iπ)+ 4

(CF − TF

N

)ln

(x+x−

),

χ(1)λd = 4

CFTF

Nln

(x+x−

),

χ(1)dλ = 4 ln

(x+x−

),

(24)χ(1)dd = 0.

In higher orders the matrixχ(α)may be non-degenerate for the different chiral componentsof the basis. In the Abelian case, there are no different color amplitudes and there is onlyone anomalous dimension

(25)χ(1) = 4 ln

(x+x−

).

In terms of the functions introduced above the one-loop correction reads

A(1) = ig2

s

[(γ (1) ln2

(Q2

M2

)+ (

2ξ(1) + 2ζ (1)+ χ(1)λλ)ln

(Q2

M2

)+ 2F (1)0

)Aλ

(26)+ χ(1)λd ln

(Q2

M2

)Ad + A(1)0

],

whereA(1)0 = ∑i A

(1)0i = A(1)|Q2=M2 has the following decomposition

(27)A(1)0 = A(1)0λLLAλLL + A(1)0

λLRAλLR + · · · .

For the present two-loop analysis of the annihilation cross section only the real part of thecoefficientsA(1)0 is needed,

Re[A(1)0λLL

] =(CF − TF

N

)f (x+, x−)+CA

(85

9+ π2

)− 20

9TFnf ,

Re[A(1)0λLR

] = −(CF − TF

N− CA

2

)f (x−, x+)+CA

(85

9+ π2

)− 20

9TFnf ,

Re[A(1)0dLL

] = CFTF

Nf (x+, x−),

(28)Re[A(1)0dLR

] = −CFTFN

f (x−, x+),

where

(29)f (x+, x−)= 2

x+lnx− + x− − x+

x2+ln2x−.

A scalar particle in the fundamental representation with no Yukawa coupling to fermionsgives the additional contribution of−5TF/9 to the first two lines of Eq. (28).


The two-loop correction is obtained by the direct generalization of the form factoranalysis. The only complication is related to the matrix structure of Eq. (23):

(30)A(2)LL = ig2(Q2)

s

(γ (1))2

2ln4

(Q2

M2

)Aλ,

(31)A(2)NLL = ig2(Q2)

s

[(2ζ (1)+ χ(1)λλ − 1

3β0

)Aλ + χ(1)λd Ad

]γ (1) ln3

(Q2

M2

),

A(2)NNLL = ig2(Q2)

s

[(γ (2)+ (

2ζ (1)− β0)ζ (1)+ 2F (1)0 γ (1)

+ 1

2

((4ζ (1)− β0

)χ(1)λλ + χ(1)λλ

2 + χ(1)dλ χ(1)λd))

Aλ

+ 1

2

((4ζ (1)− β0

)χ(1)λd + χ(1)λd χ(1)λλ + χ(1)λd χ(1)dd

)Ad + γ (1)A(1)0

]

(32)× ln2(Q2

M2

).

The structure of the infrared singularities of the hard part of the two-loop correctionspresented here is in full agreement with the result of [30] which was confirmed by explicitcalculation [31].

To illustrate the significance of the subleading contributions let us again discuss thestandard model inspired example considered in the previous section. Having the result forthe amplitudes it is straightforward to compute the one- and two-loop corrections to thetotal cross section of the four-fermion annihilation process using the standard formulae.For the annihilation process one has to make the analytical continuation of the above resultto the Minkowskian region of negativeQ2 = −s according tos+ i0 prescription. Althoughthe above approximation is formally not valid for small anglesθ <M/

√s we can integrate

the differential cross section over all angles to get a result with the logarithmic accuracy.In this way we obtain for the case ofSU(2)L group

σ (1) =[−3 ln2

(s

M2

)+ 80

3ln

(s

M2

)−

(22

9+ 3π2

)]σB,

(33)σ (2) =[

9

2ln4

(s

M2

)− 449

6ln3

(s

M2

)+

(2414

9+ 37π2

3

)ln2

(s

M2

)]σB,

and

σ (1) =[−3 ln2

(s

M2

)+ 26

3ln

(s

M2

)+

(221

9− 3π2

)]σB,

(34)σ (2) =[

9

2ln4

(s

M2

)− 125

6ln3

(s

M2

)−

(1625

18− 37π2

3

)ln2

(s

M2

)]σB,

for the initial and final state fermions of the same or opposite isospin, respectively. HereσB is the Born cross section with theMS couplings constant normalized at the scale

√s.


For theU(1) group we have

σ (1) =[−4 ln2

(s

M2

)+ 12 ln

(s

M2

)−

(376

9− 4π2

3

)]σB,

(35)σ (2) =[8 ln4

(s

M2

)− 532

9ln3

(s

M2

)+

(1130

3+ 16π2

3

)ln2

(s

M2

)]σB.

Similar to the form factor, we observe a relatively small coefficient of the LL terms and alarge coefficient of the NNLL terms.

4. NNL logarithms in electroweak processes at high energies

We are interested in the processf ′f ′ → f f . In the Born approximation, its amplitudeis of the following form

(36)AB = ig2

s

∑I,J=L,R

(T 3f ′T 3

f + t2WYf ′Yf

4

)Af ′fIJ ,

where

(37)Af ′fIJ = f ′

I γµf′I fJ γµfJ ,

tW = tanθW with θW being the Weinberg angle andTf (Yf ) is the isospin (hypercharge)of the fermion which depends on the fermion chirality.

To analyze the electroweak correction to the above process we use the approximationwith theW - andZ-bosons of the same massM, the Higgs boson of the massMH ∼M and massless quarks and leptons. A fictitious photon massλ has to be introduced toregularize the infrared divergences. We insert the mass into the gauge boson propagators“by hand” to investigate the leading ins−1 behavior of the amplitudes, leaving aside theHiggs mechanism of the gauge boson mass generation. This approach is gauge invariant asfar as power unsuppressed terms are considered. The NNLL approximation is not sensitiveto the fine details of the mass generation because we need only the hard part of thepotentially dangerous self-energy insertion to the gauge boson propagator. Indeed, the onlyeffect of the virtual Higgs boson is the modification of the functionsβ0, γ (2) andA(1)0

λ.The functionγ (2) as well as the running of the coupling constant inγ (1) andχ(1) aredetermined by the singularities of the hard part of the corrections. On the other hand,the vacuum polarization of the off-shell gauge boson in the Born amplitude contributingto A(1)0

λ is infrared safe and can be computed in the massless approximation, i.e., in theleading order ins−1 it receives only the contribution of the hard region. The only effectof the Higgs mechanism is that we have two different massesMZ andMW = cosθWMZ .Since cosθW ∼ 1 we neglect this difference in our calculation. The correction due to theheavy gauge boson mass splitting will be discussed at the end of the section.

Let us first consider the equal mass caseλ =M, where we can work in terms of thefields of unbroken phase. In the massless quark approximation the Higgs boson couplesonly to the gauge field. Therefore the result of Sections 2 and 3 for theSU(2)L gauge


group with the couplingg and theU(1) gauge group with the couplingtWg can be directlyapplied to the electroweak processes. For the standard electroweak model with one chargedHiggs doublet one has to replace in the above expressionsnf → 2Ng + 1/4 for SU(2)Lparameters andTFnf → 5Ng/3+1/8 forU(1) parameters withNg = 3 being the numberof generations. For example we have

γ (2) = 10

3Ng − 263

12+ π2,

(38)β0 = −4

3Ng + 43

6,

for SU(2)L and

γ (2) =(

200

27Ng + 5

9

)t2W

Y 2f

4,

(39)β0 = −20

9Ng − 1

6,

for U(1).The result for the amplitudes is obtained by projecting on a relevant initial/final state with

the proper assignment of isospin/hypercharge. For example, the projection of the basis (19)

on the states corresponding to the neutral current processes readsAλIJ → T 3f T

3f ′Af

′fIJ ,

AdIJ → Af′fIJ . The only complication in combinatorics is related to the fact that now we

are having different gauge groups for the fermions of different chirality. In particular, thedouble logarithmic approximation is given by the exponential factor

(40)exp

[−

(Tf (Tf + 1)+ t2W

Y 2f

4+ (f ↔ f ′)

)L

(Q2)],

where

(41)L(Q2) = g2

16π2 ln2(Q2

M2

),

andTf (Tf + 1)= CF .The photon is, however, massless and the corresponding infrared divergent contributions

should be accompanied by the real soft photon radiation integrated to some resolutionenergyωres to get an infrared safe cross section independent on an auxiliary photonmass. In practice, the resolution energy is much less than theW -(Z-)boson mass and themassive gauge bosons are supposed to be detected as separate particles. To study the virtualcorrections in the limit of the vanishing photon mass we follow a general approach of theinfrared evolution equations developed in [6] (see also references therein). It is convenientto use the auxiliary photon massλ as a variable of the infrared evolution equation below theelectroweak scaleM. The dependence of the virtual corrections onλ in the limit λM

is canceled by the contribution of the real soft photon radiation. Forωres M, the softphoton emission is of the pure QED nature. Therefore, the kernel of the infrared evolutionequation which governs theλ dependence of the virtual corrections to the amplitudes isAbelian. This dependence is given by the QED factorU which can be directly obtained


from the general formulae given above:

U =U0(αe)exp

−αe(λ

2)

4π

[(Q2f +Q2

f ′ −(

76

27

(Q2f +Q2

f ′) + 16

9QfQf ′

)αe

πNg

)× ln2

(Q2

λ2

)−

(3(Q2f +Q2

f ′) + 4QfQf ′ ln

(x+x−

))ln

(Q2

λ2

)(42)+ 8

27

(Q2f +Q2

f ′)αeπNg ln3

(Q2

λ2

)]+O

(α3e

),

whereαe is theMS QED coupling constant and we use the following expressions for theQED functions

ζ (1)e = 3Q2f ,

χ(1)e = 4Qf ′Qf ln

(x+x−

),

βe0 = −32

9Ng,

(43)γ (2)e = 320

27NgQ

2f .

The expressions forβe0 andγ (2)e can be obtained by substitutingTFnf → 8Ng/3 to thegeneral formulae. The coefficientU0(αe) in Eq. (42) is a two-component vector in thechiral basis.

In a full analogy with the renormalization group all the information on the non-Abeliangauge dynamics above the electroweak scale up to power suppressed contributions iscontained in the initial condition for this Abelian infrared evolution equation at the pointλ=M. To fix a relevant initial condition for the evolution inλ below the electroweak scaleone has to subtract the QED virtual correction (42) computed with the photon of the massM from the complete result withλ =M [6]. This leads, in particular, to the modificationof the functionγ (α) so that the double logarithmic exponential factor becomes

(44)exp

[−

(Tf (Tf + 1)+ t2W

Y 2f

4− s2WQ2

f + (f ↔ f ′))L

(Q2)],

wheresW = sinθW . In the NNLL approximation after the subtraction we get

γ (2) = −2

[(−20

9Ng + 263

18− 2π2

3

)Tf (Tf + 1)−

(100

27Ng + 5

18

)t2W

Y 2f

4

(45)+ 160

27Ngs

2WQ

2f

].

A similar subtraction should be done for the parametersζ (1) andχ(1) which take the form[9]

(46)ζ (1) = 3

(Tf (Tf + 1)+ t2W

Y 2f

4− s2WQ2

f

),


and

χ(1)λλ = −4(ln(x+)+ iπ)+

(t2WYf ′Yf − 4s2WQf ′Qf + 2

)ln

(x+x−

),

χ(1)λd = 3

4ln

(x+x−

),

χ(1)dλ = 4 ln

(x+x−

),

(47)χ(1)dd = (

t2WYf ′Yf − 4s2WQf ′Qf)ln

(x+x−

).

For I or J =R the matrixχ(1) is reduced to

(48)χ(1) = (t2WYf ′Yf − 4s2WQf ′Qf

)ln

(x+x−

).

At the same time we have some freedom in the definition of the coefficientsF0(α),A0(α) andU0(αe). If we use the one-loop normalization conditionU |Q2=M2 = 1, then

U(1)0 = 0 and no QED subtraction is necessary forF (1)0 and A(1)0 . In this case the QED

factorU is universal and has no matrix structure. To summarize, we have two evolutionequations and corresponding initial conditions. The coefficientsF0(α) andA0(α) give theinitial condition for the hard evolution of the amplitudes inQ atQ =M while the abovesubtraction of the QED contribution gives the initial condition for the infrared evolution ofthe amplitudes inλ atλ=M.

The result for then-loop correction to the amplitude (36) can be decomposed as

(49)A(n) =A(n)LL +A(n)NLL +A(n)NNLL + · · · .Explicit expressions forA(1)LL andA(1)NLL can be found, for example, in [9]. In the NNLLapproximation one has to take into account also the one-loop constant contributioncorresponding toF (1)0 andA(1)0 terms of Eq. (26). It reads

aA(1)NNLL

= ig2

s

∑I,J=L,R

−

(7

2+ 2π2

3

)[Tf (Tf + 1)+ t2W

Y 2f

4+ (f ↔ f ′)

]

×[T 3f ′T 3

f + t2WYf ′Yf

4

]+

(2T 3f ′T 3

f + t2WYf ′Yf

4

)t2WYf ′Yf

4

[f (x+, x−)(δIRδJR + δILδJL)

− f (x−, x+)(δIRδJL + δILδJR)] −

[(20

9Ng + 5

18

)T 3f ′T 3

f

+(

100

27Ng + 5

18

)t2WYf ′Yf

4

](50)+

[1

2f (x+, x−)+ 170

9+ 2π2

]T 3f ′T 3

f + 3

16f (x+, x−)δILδJL

aA

f ′fIJ ,


wherea = g2/16π2 and we keep only the real part ofA(1)0 .Let us consider the two-loop corrections. The two-loop LL corrections to the chiral

amplitudes were obtained in [6]

a2A(2)LL

= ig2

s

∑I,J=L,R

1

2

(Tf (Tf + 1)+ t2W

Y 2f

4− s2WQ2

f + (f ↔ f ′))2

(51)×[T 3f ′T 3

f + t2WYf ′Yf

4

]L2(Q2)Af ′f

IJ ,

and the two-loop NLL corrections with the exception of the trivial corrections proportionalto β0 can be found in [9]

a2A(2)NLL

= − ig2

s

∑I,J=L,R

[Tf (Tf + 1)+ t2W

Y 2f

4− s2WQ2

f + (f ↔ f ′)]

×

3

[Tf (Tf + 1)+ t2W

Y 2f

4− s2WQ2

f + (f ↔ f ′)][T 3f ′T 3

f + t2WYf ′Yf

4

]+

[−4(ln(x+)+ iπ)+ ln

(x+x−

)(2+ t2WYf ′Yf

)]T 3f ′T 3

f + 3

4ln

(x+x−

)δILδJL

+ ln

(x+x−

)[t2WYf ′Yf − 4s2WQf ′Qf

][T 3f ′T 3

f + t2WYf ′Yf

4

](52)×L(

Q2)l(Q2)Af ′fIJ ,

where

(53)l(Q2) = g2

16π2 ln

(Q2

M2

),

with δIL = 1 for I = L and zero otherwise. The second line of Eq. (52) corresponds to theζ (1) term of Eq. (31) while the third and forth lines correspond to theχ(1) terms in Eq. (31).A part of theβ0 NLL corrections is absorbed by choosing the normalization point of thecoupling constants in Eq. (36) to beQ. The rest is due to the running of the couplingconstant in the double logarithmic integral and corresponds to theβ0 term in Eq. (31). It isof the form

a2A(2)NLL

∣∣β0

= − ig2

s

∑I,J=L,R

1

3

[(4

3Ng − 43

6

)Tf (Tf + 1)+

(20

9Ng + 1

6

)t2W

Y 2f

4

(54)− 32

9Ngs

2WQ

2f + (f ↔ f ′)

][T 3f ′T 3

f + t2WYf ′Yf

4

]l(Q2)L(

Q2)Af ′fIJ ,

provided the normalization point of the coupling constants isM with the exception of thecoupling constants entering the Born amplitude (36) normalized at the scaleQ.


Let us consider the NNLL contribution. For convenience we split it in four parts

(55)A(2)NNLL =∆1A

(2)NNLL +∆2A

(2)NNLL +∆3A

(2)NNLL +∆4A

(2)NNLL ,

where the trivialβ20 renormalization group logarithms which can be absorbed into the

running of the coupling constants in Eq. (36) are not included. The correction∆1A(2)NNLL

corresponding to theγ (2), ζ (1)2, β0ζ

(1) andF (1)0 γ (1) terms of Eq. (32) is

a2∆1A(2)NNLL

= ig2

s

∑I,J=L,R

−

[(−20

9Ng + 263

18− 2π2

3

)Tf (Tf + 1)

−(

100

27Ng + 5

18

)t2W

Y 2f

4+ 160

27Ngs

2WQ

2f + (f ↔ f ′)

]+ 9

2

[Tf (Tf + 1)+ t2W

Y 2f

4− s2WQ2

f + (f ↔ f ′)]2

+ 3

2

[(4

3Ng − 43

6

)Tf (Tf + 1)+

(20

9Ng + 1

6

)t2W

Y 2f

4

− 32

9Ngs

2WQ

2f + (f ↔ f ′)

]+

(7

2+ 2π2

3

)[Tf (Tf + 1)+ t2W

Y 2f

4− s2WQ2

f + (f ↔ f ′)]

×[Tf (Tf + 1)+ t2W

Y 2f

4+ (f ↔ f ′)

]

(56)×[T 3f ′T 3

f + t2WYf ′Yf

4

]l2

(Q2)Af ′f

IJ .

The correction∆2A(2)NNLL corresponding to theζ (1)χ(1) andβ0χ

(1) terms of Eq. (32) reads

a2∆2A(2)NNLL

= ig2

s

∑I,J=L,R

[(−4(ln(x+)+ iπ)+ ln

(x+x−

)(2+ t2WYf ′Yf

))T 3f ′T 3

f

+ 3

4ln

(x+x−

)δILδJL

]×

[(2

3Ng − 43

12

)+ 3

(Tf (Tf + 1)+ t2W

Y 2f

4− s2WQ2

f + (f ↔ f ′))]

+ 4 ln

(x+x−

)[((10

9Ng + 1

12

)t2WYf ′Yf

4− 16

9Ngs

2WQf ′Qf

)+ 3

(t2WYf ′Yf

4− s2WQf ′Qf

)


×(Tf (Tf + 1)+ t2W

Y 2f

4− s2WQ2

f + (f ↔ f ′))]

(57)×[T 3f ′T 3

f + t2WYf ′Yf

4

]l2

(Q2)Af ′f

IJ .

The correction∆3A(2)NNLL corresponding to theχ(1)

2terms of Eq. (32) reads

a2∆3A(2)NNLL

= ig2

s

∑I,J=L,R

1

2

(−4

(ln(x+)+ iπ

)+ ln

(x+x−

)(2+ t2WYf ′Yf − 4s2WQf ′Qf

))2

T 3f ′T 3

f + ln

(x+x−

)×

[−4

(ln(x+)+ iπ

) + 2 ln

(x+x−

)(1+ t2WYf ′Yf − 4s2WQf ′Qf

)]×

[3

4δILδJL + t2WYf ′Yf T

3f ′T 3

f

]+ ln2

(x+x−

)[3

(T 3f ′T 3

f + t2WYf ′Yf

4δILδJL

)(58)+ (

t2WYf ′Yf − 4s2WQf ′Qf)2t2WYf ′Yf

4

]l2

(Q2)Af ′f

IJ .

The correction∆4A(2)NNLL corresponding to the real part of theγ (1)A(1)0 term of Eq. (32)

reads

a2∆4A(2)NNLL

= − ig2

s

∑I,J=L,R

(Tf (Tf + 1)+ t2W

Y 2f

4− s2WQ2

f + (f ↔ f ′))

×(

2T 3f ′T 3

f + t2WYf ′Yf

4

)t2WYf ′Yf

4

× [f (x+, x−)(δIRδJR + δILδJL)− f (x−, x+)(δIRδJL + δILδJR)

]−

[(20

9Ng + 5

18

)T 3f ′T 3

f +(

100

27Ng + 5

18

)t2WYf ′Yf

4

]

(59)

+[

1

2f (x+, x−)+ 170

9+ 2π2

]T 3f ′T 3

f + 3

16f (x+, x−)δILδJL

l2

(Q2)Af ′f

IJ .

With the expression for the chiral amplitudes at hand, we can compute the leading andsubleading logarithmic corrections to the basic observables fore+e− → f f .

In the NNLL approximation one has to take into account also the effect of analyticalcontinuation to the physical positive real value of the invariants. For the annihilationprocesses it is more natural to normalize the QED factor at the Minkowskian points =M2


to U |s=M2 = 1 so that after the expansion inαe it reads

U =

1− αe(λ2)

4π

[(Q2f +Q2

f ′)ln2

(s

λ2

)−

((3+ 2iπ)

(Q2f +Q2

f ′) + 4QfQf ′ ln

(x+x−

))(60)× ln

(s

λ2

)]+O

(α2e

).

Let us consider the total cross sections of the quark–antiquark/µ+µ− production in thee+e− annihilation. The LL, NLL and NNLL corrections to the cross sections to one andtwo loops read

RQQ = 1− 1.66L(s)+ 5.31l(s)− 15.53a+ 1.93L2(s)

− 9.43L(s)l(s)+ 28.79l2(s),

Rqq = 1− 2.18L(s)+ 20.58l(s)− 36.00a+ 2.79L2(s)

− 50.06L(s)l(s)+ 293.95l2(s),

Rµ+µ− = 1− 1.39L(s)+ 10.12l(s)− 30.99a+ 1.42L2(s)

(61)− 18.43L(s)l(s)+ 99.04l2(s),

whereQ= u, c, t , q = d, s, b, RQQ = σ/σB(e+e− →QQ) and so on. TheMS couplingsin the Born cross section are normalized at

√s. Numerically, we haveL(s)= 0.07 (0.11)

and l(s) = 0.014 (0.017) for√s = 1 TeV and 2 TeV, respectively. HereM =MW has

been chosen for the infrared cutoff anda = 2.69×10−3, s2W = 0.231 for theMS couplingsnormalized at the gauge boson mass. The small difference between the two-loop NLLcoefficients in Eq. (61) and the result of [9] is due to theβ0 contribution (54).

To get the infrared safe result for the semi-inclusive cross sections one has to add tothe expressions given above the standard QED corrections due to the soft photon emissionand thepureQED virtual correction which is determined for massless or light fermions ofthe massmf λM by Eqs. (42), (60). To derive the QED factor forλ far less thanthe fermion massλ mf M one has to change the kernel of the infrared evolutionequation and match the new solution to Eq. (42) at the pointλ=mf . The sum of the realand virtual QED corrections depends ons, ωres and on the initial/final fermion massesbut not onMZ,W . Note that our analysis implies the resolution energy for the real photonemission to be smaller than the heavy boson mass. If the resolution energy exceedsMZ,W

the analysis is more complicated due to the fact that the radiation of real photons is not ofPoisson type because of its non-AbelianSU(2)L component [6]. In the case of the quark–antiquark final state the strong interaction also produces the logarithmically growing terms.They can be read off the results of Section 1 for the form factor. For massless quarks thecompleteO(α2

s ) corrections including the bremsstrahlung effects can be found in [28].For completeness we give a numerical estimate of corrections to the cross section

asymmetries. In the case of the forward–backward asymmetryAFB (the difference of thecross section averaged over forward and backward semispheres with respect to the electron


beam direction divided by the total cross section) we get

RFBQQ = 1− 0.09L(s)− 1.23l(s)+ 3.58a+ 0.12L2(s)

+ 0.65L(s)l(s)+ 2.56l2(s),

RFBqq = 1− 0.14L(s)+ 7.15l(s)− 7.93a+ 0.02L2(s)

− 1.27L(s)l(s)− 36.55l2(s),

RFBµ+µ− = 1− 0.04L(s)+ 5.49l(s)− 10.77a+ 0.27L2(s)

(62)− 6.29L(s)l(s)+ 6.64l2(s),

whereRFB = AFB/AFBB . For the left–right asymmetryALR (the difference of the cross

sections of the left and right particles production divided by the total cross section) weobtain in the same notation

RLRQQ = 1− 2.34L(s)+ 8.98l(s)+ 16.96a− 0.46L2(s)

+ 8.37L(s)l(s)− 5.97l2(s),

RLRqq = 1− 1.12L(s)+ 11.86l(s)− 7.28a− 0.81L2(s)

+ 18.05L(s)l(s)− 120.67l2(s),

RLRµ+µ− = 1− 13.24L(s)+ 113.77l(s)− 60.36a− 0.79L2(s)

(63)+ 32.40L(s)l(s)− 293.36l2(s).

Finally, for the left–right asymmetryALR (the difference of the cross sections for the leftand right initial state particles divided by the total cross section) which differs fromALR

for the quark–antiquark final state we have

R LRQQ = 1− 2.75L(s)+ 10.07l(s)+ 8.13a− 0.91L2(s)

+ 12.19L(s)l(s)− 30.86l2(s),

R LRqq = 1− 1.07L(s)+ 11.56l(s)− 7.04a− 0.77L2(s)

(64)+ 17.17L(s)l(s)− 116.97l2(s).

In the 1–2 TeV region the two-loop LL, NLL and NNLL corrections to the cross sectionscan be as large as 1–4%, 5–10%, and 5–9%, respectively. However, we observe asignificantcancellationbetween different terms and the sum of the known two-loop correctionsamounts of approximately 1–2%. The sum of the two-loop correction to the asymmetriesis even smaller and does not exceed 1% level with the exception of theRLR

µ+µ− . For thisquantity the relatively large corrections are the consequence of the numerically small Bornapproximation.

Let us discuss the accuracy of our result. At TeV energies the LL, NLL and NNLLcorrections of Eqs. (61)–(64) provide asymptotic expressions for the cross section and theasymmetries to one and two loops. The complete one-loop corrections are known exactly(see [32] for the most general result) and we have included the dominant one-loop termsin Eqs. (61)–(64) to demonstrate the structure of the expansion rather than for precise


numerical estimates. For physical applications, the mass difference between theW and theZ gauge boson, power suppressed terms and also top quark mass effects can be important.The effect of theW andZ gauge boson mass difference on the coefficients of the NLL andNNLL terms is suppressed as(MZ −MW)/M ∼ 0.1 while the leading power correctionscan be as large asM2/s < 0.01. Thus, except for the production of third generation quarks,the above expressions approximate the exact one-loop result with 1% accuracy in theTeV region. At the same time both effects can be neglected in two-loop approximation.Therefore, the only essential deviation of the complete two-loop NLL and NNLL resultfrom Eqs. (61)–(64) for the production of the third generation quarks is due to the large topquark Yukawa coupling. Numerically, the corresponding corrections can be as importantas the generic non-Yukawa ones.

Finally, let us emphasize that the angular dependent NLL and NNLL terms arequite important for the cross section and dominate in particular the forward–backwardasymmetry.

5. Summary

In the present paper we employed the evolution equation approach to analyze the highenergy asymptotic behavior of the four-fermion amplitudes in the non-Abelian gaugemodels. The results were used to compute the NNLL electroweak corrections to the neutralcurrent four-fermion processes at high energy in the massless quark approximation to allorders in the coupling constants. We have shown the NNLL approximation to be insensitiveto the details of the gauge boson mass generation as well as to the Higgs boson mass andself-coupling.

We have calculated the explicit expressions for the one- and two-loop terms whichsaturate the NNLL corrections to the basic observables in the TeV region. In general,the two-loop NLL and NNLL corrections exceed the LL contribution in the TeV regiondue to the numerically small coefficient in front of the double logarithmic terms. Hencethe truly asymptotic behavior sets in only at an significantly higher energy. At the sametime the two-loop NNLL corrections are numerically of the same magnitude but slightlysmaller than the NLL ones, both being in the range of 1–10%. This could be consideredas a signal of convergence of the logarithmic expansion at TeV energies. Indeed, the two-loop coefficients in front of ln2

(s/M2

)is (a few units)×α2. This is not an unusually large

value in a non-Euclidean regime where the expansion parameter isα, rather thenα/(4π)as can be seen in Eqs. (33)–(35) (see also [22,29,31]). Moreover, we have observed asignificant cancellation between the two-loop LL, NLL and NNLL terms. As a result ofthis cancellation the sum of these two-loop corrections to the cross sections is of order1–2% for all the processes.

Thus, if we assume no further growth of the coefficient for the single logarithmic andnon-logarithmic two-loop terms and the observed pattern of cancellation to hold, we wouldargue that our NNLL result approximates the exact cross sections with 1% accuracy. Theaccuracy is less for the production of third generation quarks where we cannot neglect the


large Yukawa coupling to the Higgs boson that modifies the NLL and NNLL terms in ourformulae.

Acknowledgements

S.M. would like to thank P. Uwer for fruitful discussions. The work of J.H.K. andS.M. was supported by the DFG under contract FOR 264/2-1, and by the BMBF undergrant BMBF-05HT1VKA/3. The work of A.P. was supported by the DFG through grantKN 365/1-1, by the BMFB through grant 05 HT9GUA 3, by the EC through contractERBFMRX-CT98-0194, and by INTAS through grant 00-00313. The work of V.S. wassupported by the Russian Foundation for Basic Research through project 01-02-16171,and by INTAS through grant 00-00313.

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Supernova neutrinos:Earth matter effects and neutrino mass spectrum

C. Lunardinia,b, A.Yu. Smirnovc,d

a SISSA-ISAS, via Beirut 2-4, 34100 Trieste, Italyb INFN, sezione di Trieste, via Valerio 2, 34127 Trieste, Italy

c The Abdus Salam ICTP, Strada Costiera 11, 34100 Trieste, Italyd Institute for Nuclear Research, RAS, Moscow, Russia


Abstract

We perform a detailed study of the Earth matter effects on supernova neutrinos. The dependencesof these effects on the properties of the original neutrino fluxes, on the trajectory of the neutrinosinside the Earth and on the oscillation parameters are described. We show that, for a large fraction(∼ 60%) of the possible arrival times of the signal, the neutrino flux crosses a substantial amountof the matter of the Earth at least for one of the existing detectors. For oscillation parameters fromthe LMA solution of the solar neutrino problem the Earth matter effect consists in an oscillatorymodulation of theνe and/orνe energy spectra. The relative deviation with respect to the undistortedspectra can be as large as 20–30% forE 20 MeV and 70–100% forE 40 MeV. For parametersfrom the SMA and LOW solutions the effect is localized at low energies (E 10 MeV) and is notlarger than∼ 10%. The Earth matter effects can be revealed (i) by the observation of oscillatorydistortions of the energy spectra in a single experiment and (ii) by the comparison between thespectra at different detectors. For a supernova at distanceD = 10 Kpc, comparing the results ofSuperKamiokande (SK), SNO and LVD experiments one can establish the effect at (2–3)σ level,whereas larger statistical significance ((4–5)σ ) is obtained if two experiments of SK-size or larger areavailable. Studies of the Earth matter effect will select or confirm the solution of the solar neutrinoproblem, probe the mixingUe3 and identify the hierarchy of the neutrino mass spectrum. 2001Published by Elsevier Science B.V.

1. Introduction

Neutrinos from gravitational collapses of stars stop to oscillate in vacuum long timebefore they reach the surface of the Earth. For mass squared differences implied by theexperimental results on solar and atmospheric neutrinos, the supernova neutrinos arrive

E-mail address:[email protected] (C. Lunardini).



308 C. Lunardini, A.Yu. Smirnov / Nuclear Physics B 616 (2001) 307–348

at Earth in mass eigenstates. The reason is either the loss of coherence between masseigenstates on the way from the star to the Earth or the adiabatic conversion from flavourto mass eigenstates inside the star [1].

After a travel of thousands of years, the neutrinos can still be forced to oscillate again:it is enough to put on the way of the neutrinos a filter consisting of flavor non-symmetricmatter, like the Moon or the Earth. The neutrinos will oscillate both in the filter and invacuum after it. Indeed, the eigenstates of the Hamiltonian in matter do not coincide withthe mass eigenstatesνi , i = 1,2,3. So the latter turn out to be mixed and therefore willoscillate:νi ↔ νj . As a consequence, oscillations in the flavour content of the flux arerealized.

The possibility of oscillations of supernova neutrinos in the matter of the Earth hasbeen discussed long time ago [2]. It was marked that the effect of oscillations can besignificant for values of parameters:m2 ∼ 10−6–6× 10−5 eV2 and sin2 2θ > 2× 10−2.The effect is different for detectors with different trajectories of the neutrinos inside theEarth, and studying the oscillation effects in these detectors one can restore the directionto the supernova [2].

The detection of the neutrino burst from SN1987A triggered a number of new studiesof oscillations of supernova neutrinos and, among them, of oscillations inside the Earth.It was suggested that the difference in the Kamiokande-2 (K2) and IMB spectra of eventscould be related to oscillations ofνe in the matter of the Earth and to the different positionsof the detectors at the time of arrival of the burst [3]. For this mechanism to work one needsm2 ∼ 10−5 eV2 and large mixing of the electron neutrinos.

The first two K2 events showed some directionality with respect to the supernova.The interpretation of these events as due to the scattering of electron neutrinos from theneutronization peak on electrons put strong bounds on the oscillation parameters, excludinga large part of the region which could be relevant for the solar neutrinos [4]. In this regionone expects the disappearance of theνe flux due to conversionνe → νµ, ντ inside the star. Itwas marked however that oscillations inside the Earth can regenerate theνe flux, so that thebound is absent in the region of the large mixing angle (LMA) solution of the solar neutrinoproblem [5,6] (see also the discussions in [7,8]). In this context, it was shown that theexistence of the Earth matter effect depends on the conversion in the high density resonanceassociated with the largem2 which is responsible for the oscillations of atmosphericneutrinos [9–11]. In particular, the adiabatic conversion in the higher resonance suppressesthe Earth matter effects associated with the lowerm2 which governs solar neutrinos.

At the cooling stage, when the fluxes of all neutrino species are produced, the conversioninside the star leads to partial or complete permutation of theνe andνµ, ντ spectra [12,13].This causes the appearance of an high energy tail in theνe spectrum which contradicts theSN1987A observations [13] and therefore implies the exclusion of some range of the largemixing angles. However in the range ofm2 ∼ 10−5 eV2 the bound is absent due to theEarth matter effect.

Further studies of the Earth matter effect on supernova neutrinos have been performed inRef. [14] in connection with the role the supernova neutrinos can play in the reconstructionof the neutrino mass spectrum. The three neutrino schemes both with normal and inverted

C. Lunardini, A.Yu. Smirnov / Nuclear Physics B 616 (2001) 307–348 309

mass hierarchy which explain the atmospheric and the solar neutrino data were considered.

General formulas have been derived for neutrino and antineutrino fluxes at different

detectors in presence of the Earth matter effect. The main qualitative features of the effect

have been discussed and some examples of modification of the spectrum due to the Earth

matter effect were given for schemes with LMA and SMA solutions of the solar neutrino

problem. The role of the detection of the Earth matter effect in the identification of the

neutrino mass scheme was clarified. In particular, it was shown that the very fact of the

detection of the Earth matter effect in the neutrino and/or antineutrino channels will allow

to establish the type of the mass hierarchy and to restrict the elementUe3 of the mixing

matrix.

In connection with the fact that the LMA gives the best global fit of the solar neutrino

data, the interpretation of the SN1987A data has been revisited [15–18]. The Earth matter

effects on the antineutrino fluxes in the LMA range of the oscillation parameters have been

studied in details (the two layer approximation of the Earth density profile has been used).

The regions of the oscillation parameters have been found [16] in which the Earth matter

effects can explain the difference of the K2 and IMB spectra. Such an interpretation also

favors the normal mass hierarchy case or very small values ofUe3 for the inverted mass

hierarchy [16,17,19].

Recently, the Earth matter effects were considered also in Ref. [20] where the expected

spectra of events at SuperKamiokande (SK) and SNO have been calculated in three

neutrino context with a two-layers approximation for the Earth profile.

In this paper we perform a detailed study of the Earth matter effect on the supernova

neutrino fluxes in the three neutrino schemes which explain the solar and atmospheric

neutrino data. Using the realistic Earth density profile, we study the dependence of the

regeneration effect on the oscillation parameters, on the trajectory of the neutrinos inside

the Earth and on the properties of the original neutrino fluxes. The effects are calculated

for both neutrinos and antineutrinos.

We consider the possible arrival directions of the neutrino burst at existing detectors

and estimate the probability to observe the Earth matter effect at least in two detectors.

The possibility to identify the regeneration effect by existing and future experiments is

discussed.

The paper is organized as follows. In Section 2 we discuss the features of the neutrino

fluxes originally produced in the star and the possible trajectories of the neutrinos inside

the Earth. The general properties of the Earth matter effects are summarized in Section 3; a

more specific discussion for LMA oscillation parameters is given in Section 4. In Section 5

we briefly discuss the regeneration effects for neutrino parameters from the LOW and SMA

solutions of the solar neutrino problem. The possibilities of observation of the Earth matter

effects are studied in Section 6. Discussion and conclusions follow in Sections 7 and 8.


2. From the star to the Earth: trajectories and spectra

2.1. Inside the star

In this section we summarize the features of the neutrino fluxes as they are producedinside the star and the properties of the supernova that will be used in our calculations.

A supernova is source of fluxes of neutrinos and antineutrinos of all the three flavours,e,µ, τ . These fluxes,F 0

α andF 0α (α = e,µ, τ ), are characterized by the hierarchy of their

average energies,

(1)〈Ee〉< 〈Ee〉< 〈Eµ〉,and by the equality of fluxes of the non-electron neutrinos (which will be denoted asνx ):

(2)F 0µ = F 0

µ = F 0τ = F 0

τ ≡ F 0x .

In absence of neutrino mixing,θ = 0, the neutrino flux at Earth is determined by theoriginal flux produced in the star. If the decrease of the average energy and of the neutrinoluminosity with time occurs over time scales larger than the duration of the burst the fluxof the neutrinos of a given flavour,να , can be described by a Fermi–Dirac spectrum as:

(3)F 0α (E,Tα,Lα,D)= Lα

4πD2T 4α F3

E2

eE/Tα + 1,

whereE is the energy of the neutrinos,Lα is the total energy released inνα andTα isthe temperature of theνα gas in the neutrinosphere. HereD represents the distance of thesupernova from the Earth; typicallyD ∼ 10 Kpc for a galactic supernova. The quantityF3 is given byF3 = 7π4/120 5.68. According to the hierarchy (1) the indicative valuesTe = 3.5 MeV,Te = 5 MeV andTx = 8 MeV will be taken as reference in our calculations;results will be presented also for other choices of the temperatures and their dependence onthe specific values ofTe, Te andTx will be studied. We consider equipartition of the energybetween the various flavours, so thatLα EB/6, with EB the binding energy emitted inthe core collapse of the star:EB 3× 1053 ergs.

In presence of neutrino mixing and masses the neutrinos undergo flavour conversionon their way from the production point in the star to the detector at Earth. Matter effectsdominate the conversion inside the star, where a wide range of matter densities is met. Theconversion effects depend on the distribution of matter in the star; the radial profile

(4)ρs(r)= 1013C

(10 Km

r

)3

g cm−3,

with C 1–15 provides a good description of the matter distribution forρs 1 g cm−3

[8,10,21,22]. Forρs 1 g cm−3 the exact shape of the profile depends on the details ofthe composition of the star. For the neutrino parameters we will consider the resonanttransitions in the star occur at densities larger than 1 g cm−3, where the profile (4) applies;the valueC = 4 will be used unless differently stated.


2.2. Crossing the Earth

If the neutrinos cross the Earth before detection, regeneration effects can take place dueto the interaction with the matter of the Earth, analogously to what is expected for solarneutrinos [23].

What is the chance that one, two or even three detectors situated in different places ofthe Earth will detect the Earth matter effect on supernova neutrinos?

Due to the short duration of the burst, and the spherical symmetry of the Earth, for agiven detector the trajectories of neutrinos (and therefore the regeneration effect) can becompletely described by the nadir angleθn of the supernova with respect to the detector:if cosθn > 0 the detector is shielded by the Earth. The angleθn depends (i) on the locationof the supernova in Galaxy, (ii) on the timet of the day at which the burst arrives at Earthand (iii) on the position of the detector itself.

We first consider a supernova located in the galactic center (declination1 δs = −28.9)and three detectors [24]: LVD [25], SNO [26] and SK [27]. The positions of these detectorson the Earth are given in the Appendix A.

The Fig. 1a shows the dependence of cosθn on the timet for the three detectors. We fixedt = 0 to be the time at which the star lies on the Greenwich meridian. The horizontal line atcosθn = 0.83 corresponds to the trajectory tangential to the core of the Earth (θn = 33.2),so that trajectories with cosθn < 0.83 are in the mantle of the Earth. For cosθn > 0.83 thetrajectories cross both the mantle and the core.

From the figures it appears that:

1. For most of the arrival times the supernova is seen with substantially different nadirangles at the different detectors, so that one expects different Earth matter effectsobserved.

2. At any timet the neutrino signal arrives at Earth, at least one detector is shielded bythe Earth (cosθn > 0) and therefore will see the regeneration effect. Earth shieldingis verified even for two detectors simultaneously for a large fraction of the times.

3. At any possible arrival timet one of the detectors is not shielded by the Earth. Sothat, once the direction to the supernova is known, one can identify such a detectorand use its data to reconstruct the neutrino energy spectrum without regenerationeffect.

4. For a substantial fraction of the times for one of the detectors the trajectory crossesthe core of the Earth.

In Fig. 1b,c, we show similar dependences of cosθn on the timet for other locationsof the star in the galactic plane. Notice that in the case (b) two detectors are not shieldedby the Earth for most of the times and no one trajectory crosses the core. In contrast, forthe position (c) all the three detectors are always shielded by the Earth. In general forthe detectors we are considering, which are placed in the northern hemisphere, the Earthshielding is substantial for a supernova located in the southern hemisphere, as it is the casefor stars in the region of the galactic center. If a supernova event occurs in the northern

1 We defineδs as the the angle of the star with respect to the equatorial plane of the Earth.


Fig. 1. The cosines of the nadir anglesθn of SuperKamiokande, SNO and LVD detectors with respectto the supernova as functions of the arrival time of neutrino burst. The three panels refer to threedifferent locations of the star in the galactic plane (given by the declination angleδs). We fixedt = 0as the time at which the star is aligned with the Greenwich meridian.

hemisphere (δs> 0), corresponding to some peripherical regions of the galactic disk, theEarth coverage of northern detectors will be scarce, or even null in the limitδs = 90. Inthis case southern detectors would be more promising for the observation of Earth mattereffects.

Clearly the determination of the position of the supernova is important for predictionsand the experimental identification of the regeneration effect. The localization of the starcan be done either by direct optical observations or by the experimental study of theneutrino scattering on electrons [28], which has substantial directionality. Triangulationtechniques and neutron recoil methods have also been discussed [28,29]. As we havementioned in the introduction, the study of Earth matter effects by high-statisticsexperiments can allow to reconstruct the direction to the star.


3. Neutrino conversion in the star and in the Earth

In this section we summarize the general properties of the Earth matter effect onsupernova neutrinos. We will focus on the 3ν-schemes which explain the atmospheric andthe solar neutrino data.

3.1. Neutrino mass and mixing schemes

We assume that the atmospheric neutrinos have the dominant mode of oscillationsνµ ↔ντ with parameters [30]:

(5)∣∣m2

3 −m22

∣∣ ≡m2atm= (1.5–4)× 10−3 eV2, sin2 2θµτ > 0.88.

The solar neutrino data are explained either by vacuum oscillations (VO solution) or byone of the MSW solutions (LMA, SMA or LOW). The latter are based on the resonantconversion driven by the oscillation parameters

(6)∣∣m2

2 −m21

∣∣ ≡m2, sin2 2θ.

Moreover, we considerm2atmm2.

The electron flavor is distributed in the mass eigenstatesν1 and ν2 with admixturesUe1 ≈ cosθ, Ue2 ≈ sinθ. We will call ν1 andν2 the solar pair of states.

Three features of the neutrino schemes, which are important for the supernova neutrinoconversion, are still unknown:

1. The admixtureUe3 of the νe in the third eigenstate. Only an upper bound on thisparameter is known from the CHOOZ and Palo Verde experiments [31,32]:

(7)|Ue3|2 0.02.

2. The type of mass hierarchy. In the case ofnormal mass hierarchy the solar pair ofstates is lighter thanν3: m3 > m2,m1. In the case of inverted mass hierarchy thestates of the solar pair are heavier thanν3: m3<m2 ≈m1.

3. The values of the solar parametersm2, sin2 2θ. Different solutions correspond tosubstantially different values of the oscillation parameters (see, e.g., [33]).

The masses and mixings determine the pattern of level crossings in the star [14]. Thereare two resonances (level crossings) in the schemes under consideration:

• The high density (H) resonance, determined by the parametersm2atm andUe3. The

conversion in the region of this resonance is described by the Landau–Zener typeprobability,PH, of transition between the matter eigenstatesν2m andν3m.

• The low density (L) resonance with parameters of the solar pair:m2, sin2 2θ. Wedenote asPL the probability ofν2m → ν1m transition associated to this resonance.

Depending on the type of mass hierarchy and on the value ofθ, the resonances appearin different channels. There are four possibilities [14]:

1. Normal mass hierarchy andθ < π/4: both the resonances are in the neutrinochannel.


2. Normal mass hierarchy andθ > π/4: the H resonance is in the neutrino channel,whereas the L resonance is in the antineutrino channel. This possibility is disfavoredby the present data on solar neutrinos.

3. Inverted mass hierarchy andθ < π/4: the H resonance is in the antineutrinochannel, the L resonance is in the neutrino channel.

4. Inverted mass hierarchy andθ > π/4: both the resonances are in the antineutrinochannel.

These different schemes correspond to different conversion effects both inside the starand in the matter of the Earth. As it was shown in [14], the Earth effects in the 3ν contextdepend on (i) the type of mass hierarchy, (ii) the adiabaticity in the high density resonance,which is determined by|Ue3| and by the density profile of the star, (iii) the oscillationparameters in the low resonance which are determined by the solution of the solar neutrinoproblem.

In what follows we will consider the various possibilities in order. We will takeoscillation parameters from one of the regions of the solutions of the solar neutrinoproblem, and also assume that the mixing parameter|Ue3| is small, so that oscillationsinside the Earth are reduced to 2ν problem.

3.2. Antineutrino channels

Let us first consider the scheme with normal mass hierarchy.As discussed in Section 3.1, in this case there is no level crossing in the high resonance

region in the antineutrino channel, so that the antineutrino flux at the detector does notdepend on the jump probabilityPH. We get:

(8)FDe = Fe + (

F 0e − F 0

x

)(1− 2PL

)(P1e − |Ue1|2),

where

(9)Fe ≈ F 0e − (

F 0e − F 0

x

)[(1− PL

) − (1− 2PL

)|Ue1|2],is theνe flux arriving at the surface of the Earth (without Earth matter effect) and the fluxesF 0α are defined in Eq. (3). HereP1e denotes the probability ofν1 → νe conversion inside

the Earth andPL is the jump probability in the L resonance. The forms (8) and (9) are theconsequence of the approximate factorization of the dynamics and reduction of the threeneutrino problem to an effective two neutrino conversion (see [14] for details).

Let us consider the relative Earth effect expressed by the ratio:

(10)R ≡ FDe − Fe

Fe.

From Eqs. (8) and (9) we find

(11)R = r(1− 2PL

)freg,

wherer is the (“reduced”) flux factor:

(12)r ≡ F 0e − F 0

x

F 0e [PL + (1− 2PL)|Ue1|2] + F 0

x [(1− PL)− (1− 2PL)|Ue1|2],


andfreg the regeneration factor:

(13)freg ≡ (P1e − |Ue1|2).

The three factors present in Eq. (11) describe the initial conditions (initial fluxes) anddifferent stages of the evolution of the antineutrino state. Let us consider the properties ofthese factors in order.

1. The L-resonance factor,(1 − 2PL), is close to the adiabatic value 1 (i.e.,PL 0)especially if the resonance is in the neutrino channel [34]. So, in this case from Eqs. (11)and (12) we get the simplified expressions:

(14)R = r freg,

(15)r = F 0e − F 0

x

F 0e |Ue1|2 + F 0

x (1− |Ue1|2).

2. The flux factor,Eq. (15), determines the sign and the size of the effect. Due to thehierarchy of energies, Eq. (1), a critical energyEc exists at whichr = 0. Furthermorewe haver > 0 below the critical energy,E < Ec, and r < 0 for E > Ec. For realistictemperatures of the neutrino fluxes (see Section 2.1) one gets:

(16)Ec = (25–28) MeV.

At E Ec the flux factor (15) is dominated by the harder fluxF 0x , so that one finds the

asymptotic behavior:

(17)r(E Ec

) = − 1

(1− |Ue1|2) − 1

|Ue2|2 .

Similarly, atE Ec the fluxF 0e dominates, giving the limit:

(18)r(E Ec

) = 1

|Ue1|2 .

From Eqs. (17) and (18) it follows that at very high, as well as at very low energies, therelative regeneration effect (14) becomes independent of the original fluxes.

3. The regeneration factor,Eq. (13), describes the propagation effect inside the Earthand is analogous to the regeneration factor which appears for solar neutrinos. Notice thatfreg corresponds to genuine matter effect: it is zero in vacuum.

The dynamics of propagation and properties of the regeneration factor (13) are differentfor oscillation parameters from different solutions of the solar neutrino problem.

In what follows we perform numerical calculations of the Earth regeneration factor usinga realistic density profile of the Earth [35]. We compare these results with results of the twolayers approximation in Appendix B.


3.3. Neutrino channels

If the hierarchy of the neutrino mass spectrum is normal, the H resonance is in theneutrino channel and theνe flux at the detector depends onPH [14]:

(19)FDe Fe + (

F 0e − F 0

x

)PH

(1− 2PL

)(P2e − |Ue2|2

),

where theνe flux arriving at the surface of the Earth equals:

(20)Fe F 0e − (

F 0e − F 0

x

)[1− PHPL − PH(1− 2PL)|Ue2|2

].

HereP2e is the probability of the transitionν2 → νe inside the Earth.From Eqs. (19) and (20) one finds the relative Earth matter effect,R ≡ (FD

e − Fe)/Fe ,and the flux factor,r:

(21)R = rPH(1− 2PL)freg,

(22)r = F 0e − F 0

x

F 0e PH[PL + (1− 2PL)|Ue2|2] + F 0

x [1− PHPL − PH(1− 2PL)|Ue2|2] .

The regeneration factor,freg, is given by:

(23)freg ≡ (Pe2 − |Ue2|2

) = −(P1e − |Ue1|2

).

Let us comment on the features of the ratioR:1. From Eq. (21) it follows that if the adiabaticity in the high density (H) resonance

inside the star is fulfilled,PH → 0, the Earth matter effect disappears. The reason is thatin the adiabatic case the original electron neutrinos convert almost completely intoνµ andντ fluxes in the H resonance. Then the electron neutrinos detected at Earth result fromthe conversion of the originalνµ andντ fluxes. Since these fluxes are equal, Eq. (2), nooscillation effect will be observed due to conversion in the low density resonance.

The Earth matter effect is maximal in the limit of strong violation of the adiabaticity inthe H resonance:PH → 1, when the dynamics is reduced to a two neutrino problem withoscillation parameters of the L resonance.

The jump probabilityPH is determined by the density profile of the star and theoscillation parameters|Ue3|2 ≈ tan2 θ13 andm2

atm. In Fig. 2 we show the lines of equalPH in the (m2

atm–tan2 θ13)-plane, together with the exclusion region from the CHOOZexperiment. We use the density profile (4); the error inPH due to the uncertainty in thedensity profile is estimated to be within a factor of 2 [14]. The Fig. 2 shows that as|Ue3|2 decreases in the range allowed by the bound (7) the transition in the H resonancevaries from perfectly adiabatic (PH 0), for |Ue3|2 5× 10−4, to strongly non-adiabatic(PH 1), for |Ue3|2 10−6. The intervals of adiabaticity and strong adiabaticity violationchange only mildly asm2

atm varies in the presently allowed range. Notice that futureatmospheric neutrino studies and the long base-line experiments will sharpen the allowedregion ofm2

atm.2. The low density resonance factor,(1− 2PL), is zero ifPL = 1/2, which corresponds

to a situation when the neutrino beam arriving at Earth consists in incoherent and equal


Fig. 2. Lines of constant flip probabilityPH in them2atm–tan2 θ31-plane. The solid lines refer to

E = 50 MeV and correspond, from right to left, toPH = 0.05,0.1,0.2,0.4,0.8,0.95. The dashedlines correspond to the same values ofPH with E = 5 MeV. The exclusion region from the CHOOZexperiment is shown.

Fig. 3. Lines of constant jump probability in the low density resonance,PL , in thetan2 θ–m2-plane. The solid lines correspond toPL = 0.05, E = 50 MeV (upper line) andE = 5 MeV (lower line). The dashed lines correspond toPL = 0.5 and, from the upper to the lower,E = 40,20,10,5 MeV. The plot of the lines is restricted to the region of parameters for which the Lresonance inside the star occurs at densities larger than 1 gcm−3 (see Section 2.1 of the text). The al-lowed regions for the SMA, LMA and LOW solutions of the solar neutrino problem are represented.

fluxes ofν1 andν2. In this case the effects ofν1 → νe andν2 → νe oscillations cancel eachother.

In Fig. 3 we show the lines ofPL = 1/2, calculated with the density profile (4) inthe (tan2 θ–m2)-plane for different values of the neutrino energy. The lines cross theallowed region of the SMA solutions for the energy intervalE = 5–15 MeV. In Fig. 3 we


show also the linesPL = 0.05 for two different energies:E = 5 MeV andE = 50 MeV.These lines determine the lower edge of the adiabaticity region in the (tan2 θ–m2)-plane. Notice that the LMA region is in the adiabaticity domain for all the relevant energies,whereas the SMA solution region is in the domain of significant adiabaticity violation.Depending on the details of the density profile of the star the LOW solution lies either inthe adiabaticity region or in the region of partial adiabaticity breaking.

A qualitative treatment does not depend on whether the low density resonance is inthe neutrino or antineutrino channel (dark side of the parameter space). Quantitatively theresults are different.

3. The flux factor,r, Eq. (22), changes sign at lower critical energy with respect to thecase of antineutrinos, since the originalνe spectrum is softer than theνe spectrum. We get:

(24)Ec = (16–24) MeV.

Similarly to what was discussed forνe , the flux factor becomes independent of the originalfluxes in the low and high energy limits:

(25)r(E Ec)= − 1

1− PHPL − PH(1− 2PL)|Ue2|2 ,

(26)r(E Ec)= 1

PH[PL + (1− 2PL)|Ue2|2] .4. The Earth regeneration factor, Eq. (23), depends on the mixing and mass squared

difference of the solar pair and on the nadir angleθn. It will be described in detail in thefollowing sections.

3.4. Schemes with inverted mass hierarchy

If the hierarchy of the mass spectrum is inverted the high density resonance is in theantineutrino channel (see Section 3.1) and the Earth matter effect forνe depends onthe jump probabilityPH . The expressions (8) and (9) for theνe fluxes are immediatelygeneralized to:

(27)FDe = Fe + (

F 0e − F 0

x

)PH

(1− 2PL

)(P1e − |Ue1|2),

(28)Fe ≈ F 0e − (

F 0e − F 0

x

)[1− PH PL − PH

(1− 2PL

)|Ue1|2],in analogy with Eqs. (19) and (20).

Taking PL = 0 (see Section 3.2), we find the relative deviation,R, and the reduced fluxfactor:

(29)R = rPHfreg,

(30)r = F 0e − F 0

x

F 0e PH|Ue1|2 +F 0

x (1− PH|Ue1|2),

with the asymptotic limits:

(31)r(E Ec

) = − 1

1− PH|Ue1|2 ,


(32)r(E Ec

) = 1

PH|Ue1|2 .

Clearly, the conversion ofνe is independent ofPH. The expressions of the neutrinofluxesFD

e andFe can be obtained from Eqs. (19) and (20) by the replacementPH → 1;they become analogous to Eqs. (8) and (9). With the same prescription, from Eqs. (21) and(22) one gets the expressions of the ratiosR andr.

Summarizing the results of Sections 3.2–3.4 we can say that the mass hierarchy and theadiabaticity in the H density resonance (and thusUe3) determine the channel (νe or νe) inwhich the Earth matter effects appear, which is:

• Both theνe and νe channels if the H resonance is strongly non-adiabatic,PH = 1,regardless to the hierarchy.

• The νe channel for adiabatic H resonance,PH = 0, and normal hierarchy.• Theνe channel for adiabatic H resonance,PH = 0, and inverted hierarchy.

The possibility of probingUe3 and the mass hierarchy by the study of Earth effects onsupernova neutrinos will be discussed in Section 7.

4. The Earth matter effects for the LMA parameters

4.1. Antineutrino channels

Let us now study the features of the relative Earth matter effect, Eq. (14), forνe withmixing and mass squared difference in the LMA region.

The dynamics of the conversion inside the Earth is described by the regeneration factorfreg, Eq. (13). For LMA parameters the Earth matter effect consists in an oscillatorymodulation of the neutrino energy spectrum.

The Fig. 4 shows the ratioR as a function of the neutrino energy for various values ofθn. For mantle crossing trajectories,θn > 33.2, the effect is mainly due to the interplayof oscillations and adiabatic evolution. That is, oscillations in medium with slowly varyingdensity. Small density jumps produce only rather weak effects. As a result, the features ofthe regeneration factor,freg, are similar to what is predicted in the case of propagation inmedium with constant density (see Appendix B). The factor is positive in the whole energyspectrum, so that the sign of the matter effect is determined by the flux factor (15): we haveR > 0 forE < Ec andR < 0 forE > Ec.

The energy spectrum shows regular oscillations in energy with period

(33)E ≈ 2π

φ(θn,E)E,

where the oscillation phaseφ is determined by the integral

(34)φ(θn)= 2π

r(θn)∫0

dx

lm(n(x),E),


Fig. 4. The relative Earth matter effect inνe channel,R, as function of the antineutrino energyfor LMA oscillation parameters and various values of the nadir angleθn. We have takenm2 = 5 × 10−5 eV2, sin2 2θ = 0.75; Te = 5 MeV, Tx = 8 MeV. The figure refers to normalmass hierarchy (or inverted hierarchy withPH = 1).

over the neutrino trajectory. Herelm is the (instantaneous) oscillation length in matter, andn(x) is the electron density along the trajectory. As follows from Eqs. (33) and (34) theperiod of oscillations decreases with the nadir angle and increases with the energy. Thedependence on the energy appears inE, (see Eq. (33)) explicitly, and implicitly via theoscillation length.

As a result of adiabatic evolution, the depth of oscillations of the regeneration factor isdetermined by the electron number density at the surface of the Earth (see Appendix B),n0e :

(35)Df ≈ 2√

2GFn0e

E

m2sin2 2θ0

m.

Hereθ0m is the mixing angle of the solar pair in matter at the surface.


Fig. 5. The same as Fig. 4 forθn = 60 and various values ofm2 .

The depthDf has a resonant dependence on the quantityx ≡ 2E|V |/m2, with V

being the matter potential (see the Appendix B for details). BothDf and lm increase asthe system approaches the resonance; correspondingly, the periodE/E increases. Forneutrinos propagating in the mantle andm2 = 5 × 10−5 eV2 (which is used in theFig. 4) the resonance is realized atE = ER 150 MeV. Thus the Earth effect is largerin the highest energy part of the spectrum.

For core crossing trajectories the behavior of the Earth effect becomes irregular due tothe interference of the oscillations in the core and in the mantle. The modulations in theenergy spectra have smaller period both due to presence of large densities and larger lengthof the trajectory. Moreover, now the effect can change the sign both below and above thecritical energy. That is, for some energies the Earth effect is negative at low energies andpositive at high energies.

As m2 decreases, as shown in Fig. 5, the regeneration factor increases, since theresonance of the system is realized at lower energies. In particular, form2 = 2 ×10−5 eV2, the resonance energy equalsER 60 MeV. The period (in the energy scale)of the oscillatory modulation of the spectrum increases with the decrease ofm2.

The change of the mixing parameter|Ue1|2 ≈ cos2 θ (within the LMA region)influences the regeneration factor rather weakly. However variations of|Ue1|2 change thespectrum arriving at the surface of the Earth. According to the Eq. (9), with the increaseof |Ue1|2 (i.e., decrease of the mixingθ), the contribution of the hard component in the


Fig. 6. The same as Fig. 4 forθn = 60 and various values ofTe.

spectrum decreases: the composite spectrum becomes softer.Given the mixingUe1 the factorr , Eq. (15), depends only on the original fluxes,F 0

e andF 0x . The Fig. 6 shows the relative Earth effect,R, as a function of the energy for different

values ofr determined by different temperatures of the originalνe spectrum. As it appearsin the Fig. 6, the critical energyEc decreases withTe, so that the region in which the fluxfactor suppresses the regeneration effect shifts to lower energies. ForE Ec andE Ec

the depth of oscillations depends only very weakly on variations ofTe , according to thelimits (18) and (17), which do not depend on temperatures. The values ofTe andTx onlyaffect the rapidity of the convergence to these limits: the convergence is faster for the largerdifferenceTx−Te. This appears in Fig. 7b, in which the flux factorr is plotted as a functionof the antineutrino energy with the same parameters as in the Fig. 6. For sin2 2θ = 0.75,as used in the Figs. 6 and 7, one findsr(E Ec)= 1.33 andr(E Ec)= −4. Thus theEarth effect has stronger enhancement at high energies.

If the hierarchy is inverted the Earth matter effect onνe is affected by the adiabaticityin the high density resonance. Such dependence is illustrated in Fig. 8, which shows the


Fig. 7. The flux factorsr and r for various values of the temperaturesTe andTe. We have takenTx = 8 MeV,m2 = 5× 10−5 eV2, sin2 2θ = 0.75 andPH = 1.

relative effectR as a function of the energy for various values ofPH. 2 According to Eq.(29) the effect is proportional toPH and is maximal atPH = 1. The dependence ofR onPH is transparent in the limits of high and low energies. Combining Eqs. (29) and (31) wefind that at high energiesR depends onPH as:

(36)R(E Ec

) = − PH

1− PH|Ue1|2 freg,

that is, R ∝ PH for PH 1. The weak dependence ofR on PH in the softer part of thespectrum in Fig. 8 is explained by the fact that the dependence onPH cancels in the lowenergy limit,E Ec, as can be seen from Eqs. (29) and (32).

4.2. Neutrino channels

Let us discuss the properties of the Earth regeneration effect in theνe channel.As it was shown in Fig. 3, for LMA oscillation parameters the adiabaticity in the L

resonance inside the star is satisfied, so thatPL = 0 and Eqs. (21) and (22) reduce to:

(37)R = rPHfreg,

(38)r = F 0e − F 0

x

F 0e PH|Ue2|2 +F 0

x (1− PH|Ue2|2) .

2 Rigorously,PH is energy dependent, however this dependence is weak (see, e.g., [14]). Therefore, forillustrative purpose we consideredPH as a constant.


Fig. 8. The same as Fig. 4 for inverted mass hierarchy and various values ofPH. We have takenθn = 60.

The regeneration factorfreg, Eq. (23), and thereforeR, have similar dependence onθn andm2 as in the case of antineutrinos. These dependences are illustrated in the Figs. 9 and10, wherePH = 1 was taken. The oscillation length and the period of the modulations inthe energy spectrum increase with the increase of the energy and the decrease ofm2(Fig. 10). The depth of the oscillations of the regeneration factorfreg is larger than forantineutrinos since (if the L resonance is in the neutrino channel) matter enhances theνe

mixing and suppresses the mixing ofνe :

(39)sin2 2θm(ν)< sin2 2θ < sin2 2θm(ν).

The depth of oscillations has a resonant character (see Appendix B), increasing as theresonance energy is approached. According to Eq. (35) the depth gets larger for smallerm2 (Fig. 10).

The dependence of the Earth matter effect on the flux factor,r, Eq. (38), is illustratedin Fig. 11, where the ratioR is plotted for different values of the temperatureTe, Tx =8 MeV andPH = 1. According to Sections 3.2 and 3.3 the flux factor suppresses the Earthmatter effect at energies close to the critical energy,Ec, which is slightly lower than forνe: Ec 22 MeV for Te = 3.5 MeV. At high and low energies the asymptotic limits (25)and (26) are realized. This is shown in Fig. 7a, with the same values of the temperaturesas in Fig. 11 andPH = 1. From Eqs. (25) and (26) (withPL = 0) and from the Fig. 7 it


Fig. 9. The relative Earth matter effect inνe channel,R, as function of the neutrino energy for LMAoscillation parameters and various values of the nadir angleθn. We have takenm2 = 5×10−5 eV2,

sin2 2θ = 0.75;Te = 3.5 MeV, Tx = 8 MeV; PH = 1 (or inverted hierarchy).

follows that, in contrast to the case ofνe, the Earth effect has stronger enhancement at lowenergy: for sin2 2θ = 0.75 one getsr(E Ec)= 4 andr(E Ec)= −1.33. Notice thatthe convergence to these limits is faster than forνe due to larger difference of theνe andνx temperatures.

According to Eq. (37), the Earth matter effect is larger for largerPH, i.e., for maximaladiabaticity breaking in the high density resonance inside the star. From Eqs. (37) and (38)the following asymptotic limit follows:

(40)R(E Ec)= − PH

1− PH|Ue2|2freg,

similarly to Eq. (36). The combination of Eqs. (37) and (26) implies thatR becomesindependent ofPH atE Ec. These features of the dependence ofR onPH are shown inFig. 12.


Fig. 10. The same as Fig. 9 forθn = 60 and various values ofm2.

As discussed in Section 3.4, the results for inverted hierarchy of the spectrum areobtained from the description given for normal hierarchy by the replacementPH → 1.Therefore the results shown in the Figs. 9–11, in whichPH = 1 was used, apply to the caseof inverted hierarchy.

5. The case of oscillation parameters in the LOW and SMA regions

5.1. LOW parameters

For oscillation parameters in the region of the LOW solution of the solar neutrinoproblem the mass squared difference is smaller by at least two orders of magnitude withrespect to the LMA solution:m2 ∼ 10−8–10−7 eV2. Therefore the resonance energyin the Earth is very small,ER < 1 MeV, and in the whole energy spectrum of supernovaneutrinos the neutrino system is far above the resonance. Both theνe andνe mixings aresuppressed in the matter of the Earth and the oscillation lengths approach the refractionlength,lm ≈ l0 ∼ 8000 Km. As a consequence, the regeneration inside the Earth has onlyweak dependence on the neutrino energy and no oscillatory distortions appear in the energyspectra at the detectors. In contrast, the regeneration effect depends strongly on the nadirangleθn: for E = 10 MeV the regeneration factor has a maximum,freg 0.04, for θn 25 (see, e.g., [36]).

The Fig. 13 shows the relative Earth effects forνe and νe as functions of the energyfor m2 = 10−7 eV2, sin2 2θ = 0.9, θn = 25 andPH = 1. We have also consideredperfect adiabaticity in the L resonance inside the star,PL = 0. The Fig. 13 shows that


Fig. 11. The same as Fig. 9 forθn = 60 and various values ofTe .

for E > 5 MeV the effect is not larger than 20% forνe and 10% forνe; it decreaseswith the increase of energy with a∼ 1/E behavior. In the neutrino channel the relativedeviationR, Eq. (21), is larger than in the antineutrino channel, especially in the very softpart of the spectrum,E 10 MeV. This is explained (i) by the fact that the neutrino systemapproaches the resonance at low energies, and therefore theνe mixing is enhanced, and (ii)by the larger flux factor in the low energy limit,E Ec (see Fig. 7). We get∼ 50% effectatE = 2 MeV and∼ 100% atE = 1 MeV.

5.2. SMA parameters

For oscillation parameters in the SMA region the fluxes at the detector,FDe andFD

e ,Eqs. (8) and (19), are substantially different with respect to the case of LMA parameters,due to differences in the L resonance factor,(1− 2PL), and in the regeneration factor.

In what follows we describe the Earth effect forνe ; effects on νe conversion areextremely small due to the suppression of the mixing in matter and will not be considered.


Fig. 12. The same as Fig. 9 forθn = 60 and various values ofPH.

Fig. 13. The relative deviationsR (solid line) andR (dashed line) as functions of the neutrino(antineutrino) energy for LOW oscillation parameters. We have takenm2 = 10−7 eV2,

sin2 2θ = 0.9; Te = 3.5 MeV, Te = 5 MeV, Tx = 8 MeV andθn = 25. We have also assumedPH = 1 andPL = 0.

Let us first discuss the factor(1 − 2PL). Form2 and sin2 2θ in the SMA region theconversion in the L resonance inside the star occurs in the adiabaticity breaking regime(Fig. 3). The jump probabilityPL differs from 0 significantly and, in particular, as shownin Fig. 3,PL is close to 1/2, thus suppressing the matter effect (Eq. (21)).

In Fig. 14 we show the factor(1− 2PL) as a function of the neutrino energy form2 =6× 10−6 eV2 and two values of sin2 2θ in the SMA region. We used the profile (4) with


Fig. 14. The quantity 1− 2PL as a function of the neutrino energy form2 = 6 × 10−6 eV2 and

different values of sin2 2θ in the SMA region. We have takenC = 4 in the profile (4).

C = 4. For the largest possible mixing, sin2 2θ ∼ 5 × 10−3, the factor is negative above10 MeV; for the best fit point, sin2 2θ ∼ 2.4× 10−3, it is negative in the whole detectablepart of the spectrum (E > 5 MeV). These results, however, strongly depend on the modelof the star. For massive stars (M > 30M) the density profile may have smaller gradient,so that the adiabaticity breaking is weaker. In this case the energy at whichPL = 1/2 islarger, and in a significant part of the spectrum the low resonance factor can be positive.Notice that in the high energy part of the spectrum the factor can be as large as 0.7–0.8 inabsolute value.

The regeneration factor,freg, Eq. (23), has a peculiar resonant behaviour in dependenceon the energy and on the nadir angleθn. For mantle crossing trajectories,θn > 33.2,the matter effect consists in resonantly enhanced oscillations, with oscillation lengthcomparable or larger than the radius of the Earth. A peak appears in theνe spectrum atE ∼ 15 MeV.

For core crossing trajectories,θn < 33.2, the regeneration factor exhibits a narrowpeak atE 7 MeV due to parametric resonance, with two smaller peaks at higher andlower energy due to MSW resonances in the mantle and in the core, respectively. Thesefeatures are shown in the Fig. 15, which represents the relative deviationR as a functionof the energy forPH = 1, θn = 0 and various values of the factorC in the density profile(4), corresponding to different values ofPL . With increase ofC the jump probabilityPL

decreases, so that the region where(1 − 2PL) > 0 expands to higher energies. With thedecrease ofPL the Earth matter effect changes from negative to positive and the size of theeffect increases; it can be as large as∼ 30%.

As in the case of the LMA solution, the effect decreases withPH and disappears forperfectly adiabatic transition in the H resonance,PH = 0. If the mass hierarchy is invertedthe matter effect exists, in the neutrino channel, independently of the character of the Hresonance.

6. Observation of Earth effects

Let us first summarize the results of the previous sections.


Fig. 15. The relative Earth matter effect inνe channel,R, as function of the neutrino energy forSMA oscillation parameters and various values of the density profile factorC. We have takenm2 = 6 × 10−6 eV2, sin2 2θ = 5 × 10−3; Te = 3.5 MeV, Tx = 8 MeV; PH = 1 (or invertedhierarchy);θn = 0.

The regeneration effects in the matter of the Earth produce a distortion of the neutrinoenergy spectrum. The character of the distortion depends (i) on the properties of the originalneutrino and antineutrino fluxes, (ii) on the nadir angleθn and (iii) on the features of theneutrino masses and mixings, in particular on the solution of the solar neutrino problem.

For oscillation parameters from the LMA solution the distortion of the spectrum hasoscillatory character with larger oscillation depth and period in the high energy part ofthe spectrum. The effect exists in the neutrino or in the antineutrino channel or in bothdepending on the adiabaticity in the H resonance and on the mass hierarchy.

For the LOW solution there is no oscillatory behaviour in the spectra. The effect is verysmall in the whole observable part of the energy spectrum and decreases with the increaseof the energy.

For parameters in the region of the SMA solution the Earth effect exists for neutrinosonly and consists in the appearance of one or three (for core crossing trajectories) resonancepeaks (or dips) in the low energy part of the spectrum.

In this section we discuss the possibility of detecting the Earth matter effects at presentand future detectors.


6.1. Detection of supernova neutrinos. Numbers of events

As follows from the results of the previous sections, the observation of the Earthmatter effect requires: (i) separate detection of neutrinos of different flavours, (ii) separatedetection of neutrinos and antineutrinos, (iii) the reconstruction of the neutrino energyspectrum.

In what follows we concentrate on events from charged current (CC) scattering onnucleons or nuclei at real-time detectors. These events better satisfy the requirements (i)–(iii); in particular CC processes have high sensitivity to the neutrino energy spectrum.

We consider:

1. The detection ofνe at water Cerenkov detectors (SuperKamiokande and the outervolume of SNO) via the reaction:

(41)νe + p→ e+ + n.

Other CC reactions (e.g., the scattering ofνe and νe on oxygen nuclei) havesubstantially smaller cross section so that they contribute to the total number of eventsat few per cent level; they will not be considered further.

2. Heavy water detectors (the inner volume of SNO experiment) with the detectionreactions:

(42)νe + d → e+ p+p,

(43)νe + d → e+ + n+ n,

which represent the dominant channel of CC detection. Events from the process(43) will be distinguished by those from (42) if neutrons are efficiently detected incorrelation with the positron.

3. Liquid scintillator detectors (LVD), which are mostly sensitive toνe via the reaction(41) with only little sensitivity to absorption processes on carbon nuclei.

Besides pure CC processes, the reactions

(44)νi + e→ νi + e, i = e,µ, τ,

(45)νi + e→ νi + e,

and the NC breakup of deuterium:

(46)νi + d → νi + n+ p, i = e,µ, τ,

(47)νi + d → νi + n+ p,

allow to reconstruct the total neutrino flux; moreover, due to its good directionality, thescattering ofνe on electrons is relevant to the location of the supernova.

Radiochemical experiments could provide information on the totalνe flux above acertain threshold.

The number of CC events with lepton having the observed kinetic energyEe is given by

(48)dNα

dEe=NT

+∞∫−∞

dE′e R

(Ee,E

′e

)E(E′e

)∫dEFα(E)

dσ(E′e,E)

dE′e

,


whereE′e is the true energy of the electron (or positron),NT is the number of target particles

in the fiducial volume andE represents the detection efficiency. Heredσ(E′e,E)/dE

′e is

the differential cross section andR(Ee,E′e) is the energy resolution function, which can

be described by a Gaussian form:

(49)R(Ee,E

′e

) = 1

√

2πexp

[− (Ee −E′

e)2

22

].

The energy resolution and the other parameters of the detectors (volume, efficiency, etc.)are summarized in Appendix A.

The energy spectrum (48) of the charged leptons reflects the spectrum of the neutrinos,with the following differences:

• The energy dependence of the cross section,3 σ ∝E2, substantially enhances the highenergy part of the spectrum.

• The integration over the neutrino energy and the convolution with the energyresolution function, Eq. (48), lead to averaging out the fast modulations in low energypart of the spectrum (appearing for LMA oscillation parameters). Conversely, thelarge-period oscillations at high energies will appear in the lepton spectrum (48).

6.2. Identification of the Earth matter effects

The Earth matter effects can be identified:

1. At a single detector, by the observation of deviations of the energy spectrum withrespect to what expected from conversion in the star only.

2. By the comparison of energy spectra from different detectors.

In the Figs. 16–19 we show examples of the spectra expected at SK, SNO and LVDfor oscillation parameters from the LMA solution,PH = 1 and various arrival times of theneutrino burst. We considered a supernova located in the direction of the galactic center(Fig. 1a) at a distanceD = 10 Kpc and releasing a total energyEB = 3 × 1053 ergs. Thehistograms represent the numbers of events from the reaction (41) for SK (panels (a)) andLVD (panels (b)); the panels (c) show the sum of the numbers of events from the reactions(43) and (41) at SNO. In (d) we plot the numbers of events in the inner volume of SNOfrom the scattering (42).

As can be seen in Fig. 1a, fort = 1 hour the neutrinos arriving at SK have corecrossing trajectory (cosθn = 0.93). For SNO the trajectory crosses the mantle only andis rather superficial (cosθn = 0.10); the LVD detector is not shielded by the Earth. Thecorresponding spectra are shown in Fig. 16 form2 = 5× 10−5 eV2 and sin2 2θ = 0.75and the same temperatures as in the Figs. 4 and 9 (see Section 2.1). The spectrum ofSK events exhibits deviations from the undistorted spectrum in some isolated bins, which

3 Theσ ∝ E2 dependence constitutes a good approximation at low energies; in the highest energy part of thesupernova neutrino spectrum deviations due to weak magnetism and recoil effects are relevant, see, e.g., [37–39].In our calculations we used the cross sections in Ref. [39] for the scattering (41) and in Ref. [40] for the reactions(42), (43).


Fig. 16. The energy spectra expected at SK, SNO and LVD with (solid lines) and without (dottedlines) Earth matter effect, for the same parameters as in Figs. 4 and 9 andt = 1 hour of Fig. 1a.A distanceD = 10 Kpc from the supernova and binding energyEB = 3×1053 ergs have been taken.In this specific configuration LVD is not shielded by the Earth, thus observing undistorted spectrum.The histogram (c) refers to the sum of events fromνe + p → e+ + n and νe + d → e+ + n + n

scatterings, while the panel (d) shows the events fromνe + d → e + p + p. In (a) and (b) only theevents fromνe + p→ e+ + n are shown.

correspond to the minima in the antineutrino spectrum. At SNO the Earth effect producesa narrowing of the spectrum. The difference with respect to the SK spectrum is attributedto the smaller oscillation phase (shorter trajectory in the Earth) of the neutrinos arriving atSNO.

The comparison of the results of the three experiments can be performed in variousways. If the direction to the supernova is determined, the LVD spectrum is known to befree from regeneration effects. Therefore it can be used to predict the energy distributionsat SK and SNO without Earth matter effects. Such predictions can be compared to theobservations of SNO and SK. Due to the relatively small statistics of the LVD events,however, the accuracy of the reconstruction will not be high and the deviation from theundistorted spectrum (e.g., in the range 40–65 MeV) will not be larger than 2σ .

Higher statistical significance is obtained if data from a second large volume detectorare available. Kilometer-scale neutrino telescopes, though primarily devoted to the studyof high-energy neutrinos, are expected to be sensitive to supernova neutrinos. In particular,


Fig. 17. The same as Fig. 16 for different values of some parameters:m2 = 3 × 10−5 eV2,

sin2 2θ = 0.9; Te = 3 MeV, Te = 4 MeV.

for a supernova at distanceD = 10 Kpc the ice Cerenkov detector of AMANDA [41]would observe more than 2× 104 events fromνe scattering on protons, Eq. (41) [42,43].Unfortunately, the presence of a relatively large background and the absence of sensitivityto the neutrino energy spectrum [42–44] strongly restrict the potential of the study of thissignal. We mark, however, that substantial upgrades of the experimental apparatus arepossible [44] and the optimization of the detector for supernova neutrino observation wouldbe of great interest. Besides the present neutrino telescopes, the detection of supernovaneutrinos is among the goals of future large volume detectors, like UNO [45] and NUSL[46]. We find that the comparison of the energy spectra observed by SK and by anotherdetector with comparable or larger statistics could establish the Earth matter effects atmore than∼ 5σ level.

Even larger regeneration effect can be realized due to specific features of the originalneutrino spectra and of the solution of the solar neutrino problem. The Fig. 17 shows thesame spectra as Fig. 16 form2 = 3 × 10−5 eV2, sin2 2θ = 0.9, Te = 3 MeV andTe =4 MeV. In this case the regeneration effect is substantially larger (see also Figs. 5 and 6)and can be established by SK–LVD comparison with high ( 3σ ) statistical significance.Notice also that the statistics of each experiment, and therefore the power of the comparisonbetween different detectors, is higher for smaller distanceD to the supernova and/or larger


Fig. 18. The same as Fig. 16 fort = 8 hours of Fig. 1a. For this configuration SNO is unshielded bythe Earth.

binding energyEB. For instance forD = 3 Kpc andEB 4.5 × 1053 ergs the statisticsis ∼ 17 times higher and the differences between the spectra of SK and LVD (unshielded)can be as large as (6–10)σ .

Besides the comparison of the spectra, more specific criteria of identification of the Eartheffect can be elaborated if the location of the supernova and the solar neutrino oscillationsparameters are known. For instance, for LMA parameters and rather superficial trajectoryin the mantle the effect consists in a narrowing of the spectrum (see, e.g., Figs. 16, 17panels (c) and (d)). Thus the comparison of the widths of the spectra at different detectorsmay establish the Earth effect.

As a further illustration, in Figs. 18 and 19 we show the expected spectra for the sameparameters as in Fig. 16 but different arrival times of the signal (see Fig. 1a). Fort = 8hours SNO is unshielded while SK and LVD have deep trajectories in the mantle. Fort = 17 hours SNO has core crossing trajectory (cosθn 1) and LVD observes effects ofregeneration in the mantle only; SK is unshielded.

For neutrino parameters from the SMA solution the Earth effects on the spectra aresmaller than∼ 5–10%. This is explained by the fact that the narrow resonance peaks appearin the low energy part of the spectrum where the smoothing effect of the integrations (48) isstronger, thus suppressing the effect. Moreover, in the low energy region the detection cross


Fig. 19. The same as Fig. 16 fort = 17 hours of Fig. 1a. For this configuration SK is unshielded bythe Earth.

sections and efficiencies are small. Therefore, regeneration effects with SMA parametersappear difficult to be observed. Similar conclusions hold for the case of LOW parameters,for which the effect is small and localized at low energies.

As we have mentioned in Section 3.4, already the very fact of establishing the Earthmatter effect in the neutrino and/or in antineutrino channel will have important implicationsfor the neutrino mass and flavor spectrum. For this it will be enough to study some integraleffect of regeneration. Let us consider the following possibility. As we have discussed inSection 4, for LMA parameters the regeneration effect is negative above the critical energyand moreover the relative size of the effect increases withE. The Earth matter effect atlow energy is small mainly due to the dependence of the regeneration factorfreg ∝ E

(Eq. (35)). Therefore to identify the regeneration effect one can compare the signals fromthe process (41) in the various detectors at low,E <Es, and high,E >Es, energies [47],whereEs is some separation energy. Let us introduce the numbers of events

(50)NL ≡Es∫

Eth

dEedN(νp)

dEe, NH ≡

∞∫Es

dEedN(νp)

dEe,


Fig. 20. The ratio of the numbers of low and high energy events fromνe + p → e+ + n reactionat SK, SNO and LVD, as a function of the separation energyEs. The bars represent 1σ statisticalerrors. The panels (a) and (b) refer to the spectra shown in Figs. 16 and 17, respectively. We havetaken a minimum energyEth = 20 MeV for the calculation of the numbers of low energy events.

and the ratio

(51)R ≡ NL

NH.

In absence of Earth effectsR has the same value for every experiment provided thatthe detection efficiencies are independent of energy atE > Eth, or in the particularcase of equal efficiencies. Therefore, differences in the quantity (51) are entirely dueto regeneration effects: stronger effect corresponds to largerR. In Fig. 20 we show thedependence ofR on the separation energyEs for different detectors; we have takenEth =20 MeV. The panels (a) and (b) of the figures refer to the situations illustrated in Figs. 16and 17, respectively. In agreement with the analyses of the spectra, in the latter case (panel(b)) the effect is stronger: the deviation of the ratioR for SK from the valueR0 in absenceof regeneration, given by LVD, can be as large as (2–3)σ . Larger deviation ((4–5)σ ) isrealized ifR0 is provided by an experiment with volume comparable or larger than the oneof SK.


7. Discussion

Within 2–3 years the solution of the solar neutrino problem can be identified by theresults of the SNO, KamLAND and BOREXINO experiments. In particular, KamLANDwill be able to establish the LMA solution and to measurem2 and sin2 2θ with 10–20% accuracy [48–51]. This will enormously sharpen the predictions for the Earth mattereffects.

The possibility exists thatUe3 will be determined by MINOS [52] provided that itsvalue is not too far from the present upper bound, Eq. (7). In this case,Ue3 is certainly inthe adiabatic range. In a long perspective, a neutrino factory [53] will be able to cover thewhole the range ofUe3 relevant for supernova neutrinos. Thus, eitherUe3 will be measuredor the upper bound onUe3 will be so strong that the transition in the supernova will becertainly non-adiabatic.

Let us consider possible implications of the supernova neutrino results depending on thesolution of the solar neutrino problem.

1. Suppose that the LMA solution will be identified with parameters close to the presentbest fit point (m2 = (3–6)× 10−5 eV2). As we have seen, in this case the regenerationeffect can be observed.

The features of the Earth matter effects depend on the value ofUe3 and on the type ofmass hierarchy. For normal mass hierarchy andUe3 in the adiabatic range (which appearsas the most plausible scenario) we expect regeneration effects in the antineutrino channeland no effect in the neutrino channel (see Section 3.4). In the supernova data furtherconfirmations of such a possibility are (i) the absence of the neutronization peak inνe andappearance of theνµ/ντ neutronization peak, (ii) hard spectrum ofνe during the coolingstage:〈Ee〉> 〈Ee〉.

The relative size of the effect inνe channel,R, is determined by the regeneration factorand the flux factor, according to Eq. (14). At high energies, when the flux factor reachesthe asymptotic value (17), Eq. (14) gives:

(52)R(E Ec

) ∼ − freg

sin2 θ.

That is, the effect is completely predicted in terms of solar oscillation parameters.In the case of inverted mass hierarchy the Earth matter effect should be observed in the

neutrino channel and no effect is expected in the antineutrino channel if|Ue3|2 > 10−5 (seeFig. 2). This possibility will be confirmed by the observation of theνe-neutronization peakand of an hard spectrum of theνe during the cooling stage. The relative size of the Earthmatter effect,R, is given by Eq. (37) (with the replacementPH → 1) with the high energyasymptotic (see Eq. (40))

(53)R(E Ec)∼ − freg

cos2 θ.

In the limit |Ue3|2 10−5 the high density resonance is inoperative, so that the result isinsensitive to the mass hierarchy. Oscillations appear in both the neutrino and antineutrino


channels, and, at high energies, they are determined by the solar oscillation parameters(Eqs. (52) and (53)). The ratio of the relative effects at high energies equals:

(54)R

R tan2 θfreg

freg.

So, possible checks of this equality would be the confirmation of the neutrino scheme withvery smallUe3.

If Ue3 is in the intermediate region:|Ue3|2 ∼ 10−6–10−5 the situation is morecomplicated. One expects to observe oscillations both in the neutrino and antineutrinochannels; the regeneration effect depends on the mass hierarchy and on the specific valueof Ue3. In the case of normal mass hierarchy, the relative effect in the neutrino channel isproportional toPH = PH(Ue3) (Eq. (37)) and at high energies, when the flux factor reachesthe asymptotic value we get, (Eq. (40)):

(55)R(E Ec)∼ − PH

1− PH sin2 θfreg.

Sincefreg is determined by the solar parameters, by measuring the relative deviationR wecan determine the value ofPH via the Eq. (55) and therefore get information aboutUe3.

For antineutrinos with inverted hierarchy we find the high energy asymptotic (see Eq.(36)):

(56)R(E Ec

) ∼ − PH

1− PH cos2 θfreg.

In practice, the observation of the Earth matter effect in theνe channel and absence ofthe effect inνe channel will testify for normal mass hierarchy and|Ue3|2 > 10−5. In theopposite situation, effect in theνe channel and absence of the effect inνe channel, theinverted hierarchy will be identified with|Ue3|2> 10−5. However the present experimentshave lower sensitivity toνe fluxes with respect to the fluxes ofνe, so that it may be difficultto establish “zero” regeneration effect with high enough accuracy.

If the Earth matter effect is observed in both channels, one should compare the sizeof the effect with that predicted in the absence of the high resonance in a given channel.Thus, if the observed signal in the neutrino channel is smaller than what is predicted inthe assumption ofPH = 1, whereas in the antineutrino channel prediction and observationcoincide, we will conclude that the hierarchy is normal and the ratio of the observed topredicted signals in the neutrino channel can give the value ofPH. The opposite case ofcoincidence of the predicted and observed signals in the neutrino channel and suppressedobserved signal in the antineutrino channel will testify for the inverted mass hierarchy.

Besides the probing of the neutrino mass spectrum and mixing, a study of the propertiesof the original neutrino fluxes can be done with Earth matter effects. In principle, a detailedstudy of the observed energy spectra will allow to reconstruct the flux factor as well as todetermine the critical energyEc.

2. Suppose that the future solar neutrino experiments will identify the SMA solution. Inthis case, the Earth matter effect is expected in the neutrino channel only, and only if the


high density resonance is inoperative. This requires the inverted mass hierarchy or verysmallUe3 in the case of normal mass hierarchy. As discussed in Section 5.2, the effect issmall and difficult to be observed even in the most favorable situations.

For the rather plausible case of the normal mass hierarchy and|Ue3|2 > 10−5 no Earthmatter effect should be seen.

The observation of the Earth matter effect in the SMA case will allow to conclude thatthe mass hierarchy is inverted or the hierarchy is normal butUe3 is very small:|Ue3|2 10−5. In principle, the intermediate case|Ue3|2 ∼ 10−5 can be identified if the observedsignal will be smaller that the expected one forPH = 1. Notice that from Eqs. (21) and (22)one gets the low energy asymptotic for the flux factor:

(57)R(E Ec)= PH(1− 2PL)freg1

PHPL= freg(1− 2PL)

1

PL,

which does not depend onPH. So it will be difficult to disentangle the effect ofPH fromuncertainties in the original neutrino fluxes.

The situation can be much more complicated if a sterile neutrino exists. This can beclarified in 2–3 years: the MiniBooNE experiment and further searches for sterile neutrinosin the solar and atmospheric neutrino experiments will allow to establish the existence ofsterile neutrinos.

Negative results of the searches will strongly favor the 3ν schemes discussed in thispaper. Still some uncertainty will remain: sterile neutrinos, unrelated to the LSND result,may exist and weakly mix with active neutrinos. Even a very small mixing (unobservableby other means) of sterile states with masses in the wide range from sub eV up to 10 keVcan strongly modify the properties of the neutrino burst.

8. Conclusions

1. There is a big chance that at least one of the existing detectors (SK, SNO, LVD) willbe shielded by the Earth at the moment of arrival of a supernova neutrino burst, so that theEarth matter effect on the neutrino flux will be observed. For supernova in a region close tothe galactic center the most plausible configuration is that for two detectors the trajectoriesof the neutrino burst cross the Earth, whereas the third detector is unshielded.

We found that the detectors considered can register Earth matter effects (cosθn> 0) fora significant fraction ( 60%) of the possible arrival times of the signal and core effect for∼ 20% of the times. These fractions may be even larger depending on the specific locationof the star in the galaxy.

The comparison of the signals from different detectors allows to identify the Earth mattereffect and to get information on the neutrino mass spectrum, substantially reducing theuncertainties related to the model of the star and to the original neutrino fluxes.

2. We studied the effects of the matter of the Earth on supernova neutrinos in theframework of three flavours with either normal or inverted hierarchy.


3. The strongest regeneration effect is expected for oscillation parameters in the region ofthe LMA solution of the solar neutrino problem, especially for the lowest values ofm2in this region:m2 = (2− 5)× 10−5 eV2.

In the νe channel the effect exists in the scheme with normal mass hierarchy (orderingof the states) or in the scheme with inverted mass hierarchy for|Ue3|2 < 10−5, whenthe conversion in the high density resonance is non-adiabatic. The effect consists in anoscillatory modulation of the energy spectra and is negative (except in small energyintervals for core crossing trajectories) above the critical energyEc ∼ 25 MeV, thussuppressing the signal. The relative size of the effect increases with energy and atE ∼60–70 MeV it can reach 50–70%. The period of modulation increases with energy andaboveE ∼ 40 MeV no averaging occurs in the energy spectrum of events. At low energiesthe effect is small and the modulations are averaged out. Thus, in the LMA case the mostsensitive region to the Earth matter effect is aboveE ∼ 40 MeV.

The oscillatory picture (position of minima and maxima) is very sensitive tom2. Fortrajectories in the mantle only the modulation of the spectrum is rather regular. For thecore-crossing trajectories the structures become narrower and irregular.

The Earth matter effect decreases with the increase ofm2 and form2 > 10−4 eV2

it will be difficult to be observed.

4. For theνe channel substantial Earth matter effect is expected in the case of normalmass hierarchy, provided that|Ue3|2 < 10−5, or in the case of inverted mass hierarchy. Theeffect has oscillatory character, similarly to theνe case, and can be as large as 100% atE > 50–60 MeV.

Thus, as in the case of antineutrinos, one should search for an oscillatory modulation ofthe signal in the energy rangeE > 40 MeV.

The size of the effect depends on the properties of the original neutrino fluxes via theflux factor. This dependence, however, disappears in the high energy limit,E 50 MeV.

5. For oscillation parameters from the SMA solution the effect appears in theνe channelonly, and only if the H resonance is inoperative (very smallUe3 or inverted mass hierarchy).The relative effect can be as large as 30% for core crossing trajectories due to the parametricenhancement of oscillations. The effect is localized in the low energy part of the spectrum:E = 5–10 MeV. It can be further suppressed by the L-resonance factor(1− 2PL) which inturn strongly depends on the density profile of the star in the outer region. The effect willbe strongly smoothed in the spectrum of observed events by integrations over the neutrinoenergy and the true energy of electron. Practically no distortion is seen and the effect isreduced to an increase of the number of events in the region of the peak. Effectively thiswill make the spectrum narrower. One can probably identify this effect by comparing thesignals from two high statistics experiments.

6. In the case of LOW solution the Earth matter effect is significant at very low energieswith a smooth 1/E behaviour. If the H resonance is inoperative, in theνe channel theeffect (excess of flux) can reach 100% atE = 1 MeV and 20% in theνe channel (the


difference is mainly due to the difference in the flux factors). AtE = 5 MeV, the effectin the flux is below (10–20)%. The nadir angle dependence is determined by the lengthof the trajectory: the oscillation length almost equals the refraction length and dependsonly weakly on the neutrino energy. For the core crossing trajectories the effects can beenhanced due to parametric effects.

7. The identification of the Earth matter effect is possible in a single detector (in theLMA case) by the observation of the oscillatory modulation of the energy spectrum inthe high energy part:E > 40 MeV. Another method consists in the comparison of signalsfrom two (or several) different detectors. If one of the detectors is unshielded by the Earthits result can be used to reconstruct the spectrum of the neutrinos arriving at Earth andmake predictions of the signal expected in the other detectors in absence of matter effect.For a supernova at distanceD = 10 Kpc and with energy releaseEB = 3 × 1053 ergs weestimated that the Earth matter effects can be established at (2–3)σ level by comparisonbetween SK, SNO and LVD results, and at (4–5)σ by comparison between the spectra fromtwo large volume detectors (of SK size or larger).

Another method to identify the Earth matter effect is to study the ratio of numbers ofevents in the high and in the low parts of the spectrum in different detectors.

8. Studies of the Earth matter effect will allow to establish or most probably to confirmthe solution of the solar neutrino problem, to get information aboutUe3 and to identify thetype of hierarchy of the neutrino mass spectrum.

Acknowledgements

The authors are grateful to P. Antonioli and W. Fulgione for discussions and forproviding material on the LVD experiment. Thanks also to H. Robertson for givinginformation on the SNO detector and to J. Beacom for helpful comments. C.L. wishes tothank A. Friedland, P. Krastev, C. Burgess, A. Bouchta, C. De Los Heros and R. Hubbardfor useful discussions.

Appendix A. Parameters of detectors

We summarize here the characteristics of the SK, SNO and LVD detectors that havebeen used in the analysis of Section 6.2.

For each experiment we consider:

1. The position of the detector on the Earth, which is relevant for determining thetrajectory of the observed neutrinos inside the Earth (see Section 2.2). The locations ofthe three experiments are given in Table 1 in terms of northern latitude,δ, and easternlongitudeα.


Table 1Summary of the characteristics of the detectors we consider, in particular their locations on the Earth(in terms of northern latitudeδ and eastern longitudeα), the fiducial masses and the coefficientsA

andB which appear in the expression of the energy resolution, Eq. (A.2)

Detector δ α Material Mass (ktons) A B

SK 3621′ 13718′ H2O 32 0 0.5SNO 4630′ −8101′ H2O 1.4 0 0.35

D2O 1.0LVD 4225′ 1331′ (CH2)10 1.0 0.07 0.23

2. The material which the detector is made of and its fiducial mass. These quantities arequoted in Table 1.

3. The detection efficiencyE(E′e) (see Eq. (48)).

The SNO efficiency is high, so that the shape of the energy spectrum and the totalnumber of events are determined by the detection cross section [54]. Therefore, we havetakenE = 1 in Eq. (48).

The efficiency of the LVD detector has been provided by the dedicated collaboration[55]. ForEth 10 MeV, it is given by the gaussian integral function:

(A.1)E(Em,Eth)= 1√2π

x∫−∞

e−y2

2 dy,

wherex ≡ (Em −Eth)/σth andEm is the measured energy of the lepton. The values ofthe parametersEth andσth are:Eth = 4 MeV andσth = 0.74 MeV for the core counters(massM = 0.57 ktons);Eth = 7 MeV andσth = 1.1 MeV for the external counters (massM = 0.43 ktons).

For SK we adopted the efficiency published for the Kamiokande-2 experiment [56].

4. The energy resolution, which appears in the resolution function (49). We followedRef. [57] for SK and SNO experiments, and Refs. [55,58] for LVD. The parameter

depends on energy as follows:

(A.2)

MeV=A

Ee

MeV+B

√Ee

MeV.

The values of the coefficientsA andB are given in Table 1.

Appendix B. The regeneration factor: step-like and realistic Earth profile

The analytical expressions for the regeneration factors (23) and (13) can be obtainedin the two layers approximation of the Earth density profile. In this approximation themantle and the core of the Earth are considered as layers with constant densities. Therefore


the neutrinos experience a constant matter potential along trajectories in the mantle anda step-like profile mantle–core–mantle along core crossing trajectories. In what followswe summarize the analytical results obtained in the two-layers approximation, whichcorrectly describe the general features of the Earth matter effects. The comparison withexact numerical calculations will be given at the end of this appendix.

For νe propagating in the mantle only the regeneration factor,freg, Eq. (23), has theform:

(B.1)freg =D(E,θ, θn, ρm)sin2(πd

lm

),

whered = 2R⊕ cosθn is the length of the trajectory in the Earth,R⊕ 6400 Km is theradius of the Earth andρm the matter density in the mantle. The depthD of oscillationsequals [13]:

(B.2)D = sin 2θm sin(2θm − 2θ),

with θm andlm being the mixing angle and the oscillation length in matter andθ the mixingangle in vacuum.

Since the mixing is enhanced in matter,θm > θ , the depthD is positive as well as thewhole regeneration factor (B.1). From Eq. (B.2) one gets:

(B.3)D = xsin2 2θ

(x − cos2θ)2 + sin2 2θ,

where

x = 2EV

m2,

andV is the matter potential. The expression (B.3) vanishes in the limits of low (x 1)and high (x 1) energies. It reaches the maximum

(B.4)Dmax = cos2 θ,

atx = 1, which corresponds to the energy:

(B.5)ER = m2

2√

2GFne.

HereGF is the Fermi constant andne the electron number density in the medium.Thus, the depth of oscillations,D, has a resonant character withE = ER being the

resonance condition. The width of the resonance is given by the intervalE−–E+ in whichD Dmax/2. One finds:

(B.6)E±ER

= 2− cos2θ ± √(1− cos2θ)(3− cos2θ),

which shows that the peak atE ∼ ER is wide for LMA oscillation parameters and getsnarrower asθ decreases. Notice that in the limitθ → 0 the maximal depth increases,Dmax → 1, but the oscillations disappear due to the decrease of the oscillation phase andthe vanishing of the resonance width (see Eq. (B.6)).


For antineutrinos, and trajectory in the mantle only, the regeneration factorfreg, Eq.(13), has the same form as in Eq. (B.1), with the oscillation depth

(B.7)D = −sin2θm sin(2θm − 2θ)= xsin2 2θ

(x + cos2θ)2 + sin2 2θ,

where we defined

x ≡ 2E|V |m2 .

Similarly to the case ofνe the depthD, and therefore the regeneration factorfreg, ispositive, sinceθm< θ ; moreover it has a similar resonant behaviour with maximum

(B.8)Dmax = sin2 θ,

and the same resonance energy, Eq. (B.5). The bordersE−, E+ of the resonance intervalare given by Eq. (B.6) with the replacement cos 2θ → −cos2θ . In the limit θ → 0 theEarth effect disappears due to the vanishing of the oscillation depth (B.8). For maximalmixing, cos2θ = 0, we get

D = D = x

x2 + 1.

If the neutrino trajectory crosses both the mantle and the core the analytical treatmentof the regeneration factors,freg and freg, is more complicated [59]. The interplay ofoscillations in the mantle and in the core determine irregular oscillations of the factorswith energy. The depth of oscillations is larger in the region of the energy spectrum closeto the resonance energies in the mantle and in the core; for SMA oscillation parametersparametric effects appear (see Fig. 15). In contrast to the propagation in the mantle only(i.e., in uniform medium), the regeneration factors have negative sign in some intervals ofenergy.

Once a realistic density profile of the Earth is considered, the calculation of theregeneration factors requires a numerical treatment. The results are presented in Fig. 21together with the analytical curves obtained with the two-layers approximation. The figureshows the factorsfreg and freg as functions of the neutrino (antineutrino) energy form2 = 5 × 10−5 eV2, sin2 2θ = 0.75 andθn = 0. We used the realistic profile in Ref.[35] and chose a step-like (two-layers) profile with densitiesρm = 4.51 andρc = 11.95,corresponding to the average densities of the profile of Ref. [35] in the mantle and in thecore along the diameter of the Earth.

From the Fig. 21 it follows that the position of the oscillation maxima and minima onthe energy axis are well reproduced by the step profile. This good approximation of theoscillation phase is ascribed to the choice ofρm andρc to be the average densities of thetwo layers along the trajectory of the neutrinos.

In contrast, the depth of oscillations given by the numerical calculation deviatessignificantly, up to∼ 50%, from the result of the analytical (two-layers) approximation.As a general tendency, the depth of oscillations in the realistic density profile appearssmaller with respect to the case of the two-layers profile. If the density jumps along the


Fig. 21. The regeneration factors for neutrinos,freg, and antineutrinos,freg, calculated with step-like(two layers) and realistic profiles. We have takenm2 = 5× 10−5 eV2, sin2 2θ = 0.75,θn = 0 andthe densitiesρm = 4.51 andρc = 11.95 for the two-layers profile.

trajectory are not very large (e.g., for trajectories in the mantle only) this feature can beinterpreted, qualitatively, by considering the density as smoothly varying along the pathof the neutrinos. In this case the neutrino conversion occurs adiabatically and the depthof oscillations is determined by the matter density at the surface of the Earth. Since thesurface density is smaller than the average density along the trajectory and the latter in turnis smaller than the resonance densityρR in the relevant range of energies, a smaller depthof oscillations is expected.

A better approximation of the numerical results can be obtained by using the averagedensity in the determination of the oscillation phase and the surface density in thedetermination of the depth of oscillations.

For SMA parameters the adiabaticity is broken and the two layers model gives a goodapproximation of the numerical results.

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therein.

http://www.sns.ias.edu/~jnb/


Diffusion corrections to the hard pomeron

Marcello Ciafalonia,1, A.H. Muellerb,2, Martina Taiutica Theoretical Physics Division, CERN, CH 1211 Geneva 23, Switzerland

b LPT, Université de Paris-Sud, 91405 Orsay, Francec Dipartimento di Fisica dell’Università, Firenze, Italy


Abstract

The high energy behaviour of two-scale hard processes is investigated in the framework of small-x

models with running coupling, having the Airy diffusion model as prototype. We show that, in someintermediate high energy regime, the perturbative hard Pomeron exponent determines the energydependence, and we prove that diffusion corrections have the form hinted at before in particularcases. We also discuss the breakdown of such regime at very large energies, and the onset of thenon-perturbative Pomeron behaviour. 2001 Published by Elsevier Science B.V.

PACS: 12.38.Cy

1. Introduction and outline

High energy hard scattering has received considerable attention in recent years. Theessential problem is to determine the Green’s functionG(Y ; k, k0) for gluon–gluon forwardscattering, wherek andk0 are the mass scales of the gluons andY = log(s/kk0) is therapidity corresponding to a center-of-mass energy squareds. The classic calculation doneby Balitsky, Fadin, Kuraev and Lipatov (BFKL) [1] several years ago corresponds to anapproximation (the leading series of powers inαY ) where the QCD running couplingαs(t)is treated as a constant,α. In this case the high-energy behaviour ofG is determined bythe rightmost singularity in theω plane, whereω is the variable conjugate toY . Thissingularity atωs = αχm = αNc

πχm is given in terms of the saddle pointχm of the function

χ(γ ) which gives the eigenvalues of the BFKL kernel.

Work supported in part by EU QCDNET Contract FMRX-CT98-0194, by MURST (Italy), and by the USDepartment of Energy.

E-mail address: [email protected] (M. Ciafaloni).1 On sabbatical leave from Dipartimento di Fisica and INFN, Firenze, Italy.2 On sabbatical leave from Columbia University, New York, USA.



350 M. Ciafaloni et al. / Nuclear Physics B 616 (2001) 349–366

When higher-order corrections [2,3] to the BFKL kernel are taken into account, thesituation changes conceptually due to the running of the QCD coupling [4,5]. Runningcoupling effects mean that the saddle point ofχ is now a function of the scalet = log(k2/Λ2), ωs(t) = αs (t)χm. Furthermore,ωs(t) is not a point of singularity of theGreen’s functiongω(t, t0), although the valueωs(t) does control theY -dependence ofG over a limited region of moderately largeY -values [5]. The rightmost singularity ofgω(t, t0) is atω = ωP , is independent oft andt0 and determines the asymptotically largeY -dependence ofG, although fort andt0 sufficiently large this asymptotic behaviour willnot set in untilY is quite large. The singularity atωP is determined by non-perturbativephysics.

The fact that running coupling effects can change the character of theY -dependenceof G is easy to see. In the fixed coupling limit the BFKL kernel gives a contributionproportional toαY each time it acts. In the running coupling case, the contribution isproportional toαs(t ′)Y αs(t)Y +b log(t ′/t)α2

s (t)Y , when expressed in terms of the fixedexternal scalet of the scattering. However, the contribution of a single running couplingterm vanishes, because the average value of log(t ′/t) is zero since the probabilities fort ′ > t and for t ′ < t are equal in fixed coupling BFKL evolution. At the level of tworunning coupling contributions, one gets the average ofα4

s (t)Y2 log2(t/t ′) α5

s Y3 [6–9],

and this contribution exponentiates. This simple perturbative argument is valid so long as

α5s Y

3 1, but is difficult to extend to values ofY > t53 . And it is exactly in the region

t53 < Y < t2 where the most dramatic running coupling effects on BFKL evolution take

place.In the present paper we calculate, starting from some small-x models, the diffusion

and running coupling corrections to the hard-Pomeron exponent. Basically, we prove thevalidity of the leading running coupling corrections hinted at before [6–9], in the full ranget Y t2, and we discuss some features of the regimeY t2.

Because of the conceptual complexity caused by the running of the coupling, it is usefulto have a simple model where the essence of running coupling effects are present and yet arather explicit discussion of theY -dependence, in the various regimes ofY , can be given.Such a model was introduced some time ago [5], which takes into account the running ofthe coupling as well as diffusion in momentum scales in terms of the quadratic dependenceof χ(γ ) about the saddle point atγ = 1

2 (Section 2). Here we study the perturbativebehaviour of this (Airy) model and of a simple generalization whereχ(γ ) is represented asa sum of two poles inγ [9]. These two models give identical results forG in the regiont Y t2, and we believe they represent QCD accurately in this region, as outlined below.

The solution of the Airy model is given in terms of Airy functions for the perturbativepart of the evolution and in terms of a reflection coefficient, determined by the way therunning coupling is regularized in the infrared, for the non-perturbative part (Section 2).The singularity atωP resides in the reflection coefficientS(ω) [5].

When t Y t53 the behaviour of the perturbative part,GP (Y ; t, t0), of G, is

determined by a saddle point of theω-integral atωs(t) givingGP exp(ωs(t)Y ). In this

region ofY running coupling effects play a minor role. WhenY reachest53 the saddle

M. Ciafaloni et al. / Nuclear Physics B 616 (2001) 349–366 351

point atωs(t) no longer gives the dominant contribution toGP . Nevertheless, the largestterm in the exponent governing theY -dependence ofGP is still ωs((t + t0)/2)Y , withthe next largest corrections being given by the “diffusion” term and by theY 3/t5 termdiscussed above (Sections 3 and 4). This result is summarized by Eqs. (21) and (50) ofthe text, which give the dominant behaviour throughout the regiont Y t2. WhenY reachest2 the character of the solution changes drastically. ForY t2 the magnitudeof GP goes as exp(ω

√Y ) whereω is a pure,t-independent, number (Section 5). This

behaviour comes from two complex-valued saddle points havingω ∼ 1/√Y , whose exact

magnitude is model dependent (Section 6). Because the saddle points are complex thereis an oscillating prefactor inGP , which cannot be given a sensible physical interpretation,and calls for non-perturbative contributions to take over.

The physical interpretation of our main results, Eqs. (21) and (50), seems clear. For

t Y t53 , GP behaves—with good approximation—as in the fixed coupling case,

having a magnitude proportional to exp(ωs(t)Y ), with a spread int − t0 = t given by

(t)2 ∼ ωs(t)Y . At Y ∼ t 53 running coupling effects become more important. Fort fixed,

GP (Y ; t, t0) reaches a maximum value proportional to exp(ωs(t)+ η3/3) with η∼ Y/t 53 ,

whilet is well fixed at a value∼ Y 2/t3, with only small fluctuations of size√Y/t from

that value. Thus the values oft are no longer given by pure diffusion, but now (for afixed t) t0 is being pulled in the infrared where the coupling is large. WhenY gets as largeast2 the preferred value oft becomes as large ast and the perturbative part of the modelceases to make physical sense.

Finally, a comment on the accuracy to which we have calculatedGP . It is convenientto write logGP = ωsYf (Y 2/t4,t/t) + small terms. We have calculated terms inf ofsize 1,Y 2/t4, t/t , and (t)2t2/Y 2, as given in Eq. (21). In the dominant region oft ∼ Y 2/t3 all these corrections are of sizeY 2/t4. We certainly expect further correctionsto f of sizeY 4/t8, as discussed in Sections 3 and 4. So long asY t2 we believe wehave the dominant terms in the exponent and so have the essence of the growth ofGP withY and its dependence ont . However, we only have some preliminary ideas on how tocalculate corrections beyond this region (Section 5), and thus we likely do not have all thelarge terms in the exponent ofGP .

2. The gluon density in small-x models

We consider in this paper a particular form of the 2-scale gluon Green’s function whichhas been established for the Airy model [5] and for the truncated BFKL models [9]. Bydefining

(1)G(Y ; k, k0)= 1

kk0

∫dω

2πieωY gω(t, t0),

whereY ≡ log(s/(kk0)) andti ≡ log(k2i /Λ

2), gω takes a factorized form fort > t0, namely

(2)gω(t, t0)=Fω(t)(Fω(t0)+ S(ω)Fω(t0)

)(t > t0),

where the various terms are defined as follows:


– Fω(t) is the “regular” solution of the homogeneous small-x equation being consid-ered, which vanishes exponentially for larget values;

– Fω(t0) is an “irregular” solution, which is instead exponentially increasing witht0;– S(ω) is a “reflection” coefficient, which has been explicitly constructed in some cases

[5,9] and is dependent on the strong coupling region, e.g., on how thet0 = 0 Landaupole is smoothed out or cutoff.

While the explicit form and size of the non-perturbativeS(ω) part is dependent on themodel—in particular on the number of poles taken into account in the effective eigenvaluefunction [9], the perturbative term is unambiguously defined in the large-t region, and issupposed to be calculable in a realistic small-x framework [10]. For this reason, most ofour analysis will concern the perturbative term.

In order to understand better the meaning of Eq. (2), let us derive it explicitly in the Airymodel. The defining equation for the Green’s function is, in operator notation,

(3)(ω− αsK)gω = 1,

whereK is in general an integral kernel, andαs (t) ≡ NCαs(t)/π ≡ 1/bt is the runningcoupling. The Airy model obtains by assuming thatK is scale invariant, and described bya quadratic eigenvalue function

(4)χ(ω) χm + 1

2χ ′′m

(γ − 1

2

)2

(χm,χ′′m > 0),

which is a reasonable approximation around the minimum ofχ , which is taken to be atγ = 1

2. Hereγ is a variable conjugated tot by Fourier transform. Therefore, by usingEq. (4) int-space, Eq. (3) becomes

(5)[ω−ωs(t)

(1+D∂2

t

)]gω(t, t0)= δ(t − t0),

whereωs(t)= χmαs(t) is the hard Pomeron exponent, andD = 12χ

′′m/χm is the diffusion

coefficient.In other words, the Airy Green’s function satisfies a second order differential equation

in the t variable and has, therefore, the well-known form

(6)gω(t, t0)= t0(FRω (t)FLω (t0)Θ(t − t0)+ (t ↔ t0)

),

whereFRω (t)(t0FLω (t0)) is the regular solution of the homogeneous equation in (5) fort → ∞ (of the adjoint homogeneous equation fort0 → −∞). Eq. (6) is the basisfor Eq. (2), but should be better specified by smoothing out or cutting offαs (t) in aregion t t , wheret > 0 defines the boundary of the perturbative regimeαs (t) = 1/bt .Depending on such procedure, the left-regular solution can be evaluated for larget0 valuesin the form

(7)FLω (t0)=F Iω(t0)+ S(ω)FRω (t0),whereF I is irregular fort0 → ∞, andS(ω) is a well-defined reflection coefficient. Wehave thus derived Eq. (2) withF = FR andF = t0F I . A similar derivation holds for thetruncated BFKL models withn poles, and in particular for the 2-pole model (Section 6).


In the Airy model, Eqs. (6) and (7) can be given in explicit form. In the perturbativeregiont > t we can set, by Eq. (5),

(8)Fω(t)=FRω (t)= Ai(ξ)=+i∞∫

−i∞

dν

2πieνξ−

13ν

3 =√ξ

π√

3K 1

3

(2

3ξ

32

),

and furthermore,

(9)Fω(t0)= t0F Iω(t0)=πt0

ω

(2bω

χ ′′m

) 23

Bi(ξ0), S(ω)= −Bi(ξ )

Ai(ξ ),

where Ai(ξ) is the Airy function,

(10)Bi(ξ)= e iπ6 Ai(ξe

2iπ3)+ e −iπ

6 Ai(ξe

−2iπ3)

is the irregular Airy solution,

(11)ξ =(

2bω

χ ′′m

) 13(t − χm

bω

)=D− 1

3 t23

(ωs(t)

ω

) 23(ω

ωs(t)− 1

)is the Airy variable, andξ is its value fort = t , at the boundary of the perturbative region.Note that, given the delta-function source in (5), the regular and irregular solutions inEqs. (8) and (9) must have a well defined Wronskian.

From the explicit expressions (8)–(10) it follows thatFω andFω are analytic functionsof ω, showing an essential singularity atω = 0 only. Instead, the reflection coefficientS(ω)—quoted in Eq. (9) in the caseαs(t) is cutoff at t = t—showsω singularitiesin Reω > 0 at the zeros of the Airy function, meaning that the leading PomeronsingularityωP actually occurs in the non-perturbative term in Eq. (2). However, the latteris suppressed, at larget0, by the ratio Ai(ξ0)/Bi(ξ0), meaning that the perturbative termmay actually be more important at large scales and intermediate energies.

By using the decomposition in Eq. (2) we can rewrite Eq. (1) as a sum of two terms

(12)G(Y ; k, k0)=GP +GR,where

(13)GR(Y ; k, k0)= 1

kk0

∫CR

dω

2πieωY S(ω)Fω(t)Fω(t0)

carries the (power behaved) Regge contributions (Fig. 1), the leading one being the (non-perturbative) Pomeron, while

(14)GP (Y ; k, k0)= 1

kk0

∫Cs

dω

2πieωYFω(t)Fω(t0) 1

kk0exp

(ωs(kk0)Y + · · ·)

corresponds to the “background integral” and is characterized by the two-scale exponentωs that will turn out to be determined byαs(kk0) (Section 3). Our goal in the followingis to analyze in more detail theY -dependence in Eq. (14), by determining the regime inwhich the exponentωs is relevant, and the magnitude and form of diffusion correctionsto it.


3. Diffusion features of the Airy model: a heuristic approach

We have just clarified that the two terms in the decomposition (2) generate theY -dependence inG(Y ; k, k0) in a quite different way. The non-perturbative (Pomeron)term provides just a Regge-pole behaviour∼ exp(ωP Y ), while the perturbative one isanalogous to a “background integral” (Fig. 1) and will generate a non-trivial exponentonly if the small-ω oscillations ofF and F are kept in phase by theω-integration. Bywriting, for the Airy model,

(15)gP (Y ; t, t0)≡ kk0GP =∫dω

2πieωY

πt0

ω

(2bω

χ ′′m

) 23

Ai(ξ)Bi(ξ0),

we expect, fort = t − t0 t that phase relations are kept only ifξ andξ0 are kept finite

for largeY . By the definition (11), this implies thatω− ωs(t) ω− ωs(t0)= 0(t− 53 ) are

small parameters. Furthermore, in this region, by Eq. (11),

(16)ξ D− 13 t

23

(ω

ωs(t)− 1

)is just linear inω − ωs . By replacing the linearized expression (16) into Eq. (15), we canrewrite it in the form

(17)gP (Y ; t, t0) π(Dt)− 13 eωs(t)Y

+i∞∫−i∞

dξ

2πieξηAi(ξ)Bi(ξ − δ),

where we have introduced the parameters

(18)η=D 13Yωs(t)t

− 23 ∼ Y t− 5

3 , δ =t(Dt)− 13 .

The integral in (17) can now be evaluated in a heuristic way by introducing the integralrepresentations (Fig. 2(a))

(19)Ai(ξ)=∫CR

dνA

2πieνAξ−

13ν

3A, Bi(ξ)=

∫C+I +C−

I

dνB

2πeνBξ−

13ν

3B

Fig. 1. Integration contourCR + Cs for the Green’s function decomposition in Eq. (12).CR(Cs)refers to the Pomeron (perturbative) contribution.


Fig. 2. Contours in theν = γ − 12 plane for (CR) the regular and (CI ) the irregular solution in the

case of (a) the Airy model and (b) the two-pole collinear model.

and by performing theξ -integration under the integral, to yield a delta-function. We thusobtain

I (η, δ)≡∫dξ

2πieξηAi(ξ)Bi(ξ − δ)

=+i∞∫

−i∞

dνA

2π2iexp

(ην2A + (

η2 + δ)νA + ηδ+ η3

3

)

(20)= 1

2π√πη

exp

(η3

12+ ηδ

2− δ2

4η

).

By inserting Eqs. (20) and (18) into Eq. (17), we finally get

(21)gP (Y, t, t0)= 1√4πDωs(t)Y

exp

[ωs(t)Y

(1+ t

2t

)− (t)2

4Dωs(t)Y+ η3

12

]which provides the diffusion and running coupling corrections to the hard Pomeronbehaviour.

There are two features worth noting in Eq. (21), in comparison with the customaryexpression with frozen coupling. Firstly, the exponentωs is corrected by a term linearint = t − t0 which provides the symmetrical argumentωs(

t+t02 ) in the running coupling,

as is appropriate for the factorized scale∼ kk0 already introduced in Eq. (12). Furthermore,the exponent carries the diffusion correctionη3 ∼ Y 3/t5, which is of relative orderY 2/t4

compared to the leading termωs(t)Y . This correction was obtained as running couplingeffect in [6–8] and confirmed [9] in the models considered here under the assumption

η 1, orY t53 .

The question then arises of the boundary of validity of the heuristic argument presentedabove. We shall see in Section 4 that the assumptionη 1 can be relaxed, and replaced

by η t13 , or Y t2. ForY t2 on the other hand, the linearization in Eq. (14) breaks

down, and the integral in Eq. (13) enters a new regime in which it first decreases, and thenstarts oscillating (Section 5), so that the phase relations are lost.


A first hint at such behaviour is obtained by replacing the ansatz in Eq. (21) in thediffusion equation

(22)∂

∂YgP (Y, t, t0) ωs(t)

(1+D∂2

t

)gP (Y, t, t0).

In fact, theY -derivative of the exponentE(Y, t,t)= loggP is given by

(23)∂E(Y, t,t)

∂Y= − 1

2Y+ωs(t)

(1+ t

2t

)+ (t)2

4DωsY 2+ ∂

∂Y

(η3

12

)and should match the right-hand side of Eq. (22), which is given by

ωs(t)

[1+D

((∂

∂tE

)2

+ ∂2E

∂t2

)],

(24)∂E

∂t= − 1

2tωs(t)Y

(1+ t

t

)− t

2DωsY

(1+ 2

t

t

)+ ∂

∂t

(η3

12

)+ 1

2t.

If we keep terms of relative ordert/Y ,t/t , Y 2/t4 (such terms are all of the same orderfor fixed values ofη), we find that Eqs. (23) and (24) are indeed consistent, provided

(25)∂

∂Y

η3

12=Dω

3s Y

2

4t2,

thus reproducing the expression (18) forη(Y, t). But, because of (25), we also generatefrom Eq. (22) terms of relative orderY 4/t8 which are subleading only ifY t2, and donot check with Eq. (23). Therefore, the heuristic argument breaks down forY = 0(t2).

4. Detailed analysis of the regime t < Y t2

In this section we give a more complete evaluation of the integral (15) which definesgP

and in doing so we confirm the result (21) and its validity boundary, obtained in a heuristicway in the previous section.

4.1. Choice of integration contour

We begin with theω-integration contour in (13) being parallel to the imaginaryω-axis with Reω = ωs (Fig. 3). In order to effectively separate leading from non-leadingbehaviours in (13) it is convenient to use different forms of the product Ai(ξ)Bi(ξ0) forImω > 0 and for Imω < 0. To that end we write

(26)Ai(ξ)Bi(ξ0)= ∓2i√ξξ0

3π2 K 13

(2

3ξ

32

)[K 1

3

(2

3ξ

32

0 e∓iπ

)− 1

2K 1

3

(2

3ξ

32

0

)],

where the upper (lower) sign will be the form used in (13) when Imω > 0 (< 0). Eq. (26)follows from (8) and (10) along with

(27)Kν(eiπζ

)+Kν(e−iπ ζ

)= 2 cosπνKν(ζ ).


Fig. 3. Integration contours in theξ plane, cut atξc ∼ t 23 . Cs corresponds to Reω= ωs . The dashed

(dotted) contourC1 (C2) corresponds to the deformed contour defined in the text. The correspondingones in Imξ < 0 are not shown.

Thus

(28)gP (Y ; t, t0)= gP (Y ; t, t0)+R(Y ; t, t0)with

gP (Y ; t, t0)=∫

∓ t0dω

3π2ωD

√(ω

ωs(t)− 1

)(ω

ωs(t0)− 1

)(29)× eωYK 1

3

(2

3ξ

32

)K 1

3

(2

3ξ

32

0 e∓iπ

),

and

R(Y ; t, t0)=∫

± t0dω

6π2ωD

√(ω

ωs(t)− 1

)(ω

ωs(t0)− 1

)(30)× eωYK 1

3

(2

3ξ

32

)K 1

3

(2

3ξ

32

0

),

where the upper (lower) sign is to be used when Imω > 0 (< 0).We shall first show thatR can be chosen small compared tog0 by a judicious contour

deformation. Since the integrand of (24) decreases for positive realξ , we are led to deformCs around the realξ axis to reach a pointξ1> 1 to be defined below. In estimating the size


of R we shall takeξ0 ≈ ξ so that the full exponent appearing in (30), in the region where|ξ | 1 where the asymptotic form ofK 1

3can be used, is

(31)E = ωs(t)Y + (ω−ωs(t)

)Y − 4

3ξ

32 .

In the region whereξ t23

(32)E −ωsY ξη− 4

3ξ

32 .

Thus we can make the second term on the right-hand side of (32) dominate the first termat ξ = ξ1 if

(33)ξ1/η2 1.

We note that (33) is possible, keepingξ1 t23 , so long asη t

13 or, equivalently, so

long asY t2. We anticipategP being of sizeeωsY so that if we are able to choose an

integration path in (24) having Re43ξ32 (ω− ωs)Y , thenR can be neglected. A contour

of this kind isC2 in Fig. 3. It is basically a deformation of the contourCs at Reω =ωs in order to haveξ1 as starting point and to depart from it with Arg(ξ − ξ1) < π

3 . Itsbasic property is to lie completely within the regions|Argξ | < π

3 , such that the product

of regular Airy functions is damped (Reξ32 > 0). It is then easy to convince oneself that

Re(43ξ

32 ) Re(ω− ωs)Y on a contour of this kind, so that the contributionR in Eq. (30)

can be neglected compared togP so long asY t2.

4.2. From the ω-integral to the ξ -integral

Our task is now to evaluate (29) and to specify once again a convenient path ofintegration inξ , that will turn out to be different from that chosen before. We are ableto evaluate (29) only in a linear approximation ofξ andω − ωs(t), and we first turn to amore complete justification of this approximation. When|ξ0| and|ξ | are large, the termsappearing in the exponent in (29) are

(34)E = ωY − 2

3ξ

32 + 2

3ξ

32

0 .

This can be written as

(35)E = ωs(t)Y + ξη(ω

ωs

) 23 − 2

3ξ

32 + 2

3

(ξ − δ

(ω

ωs

) 13) 3

2

,

with δ as given in (18). The linear approximation gives

(36)Elin = ωs(t)Y + ξη− 2

3ξ

32 + 2

3(ξ − δ) 3

2 ,

In order to see how closeE andElin are, we deformCs to the contourC1, in this case(Fig. 3). The latter, departing fromξ1 and reaching the origin, is chosen in such a way as to

run over the imaginaryξ axis in the whole linearization region| Imξ |< t 23 , and to approach


Cs later on in the Reω < ωs region. It is straightforward to see that the regions ofC1 onthe real and imaginary axis are the dominant ones in the integral. In fact, from Eq. (36) onesees that Re(Elin − ωsY ) vanishes atξ = 0 and decreases steadily below zero forξ on theimaginary axis. Furthermore, in the asymptotic region whereC1 starts approachingCs theAiry phase is negligible and the integrand in Eq. (29) is damped because of Reω < ωs .

Now we want to replaceE by Elin in the exponent. We can estimate the error in theexponent by comparing (35) and (36) in the region 0 ξ ξ1. The maximum deviationoccurs forξ = ξ1 and the size of the deviation is

(37)E = E − Elin ξ1η[(

ω

ωs

) 23 − 1

]−√

ξ1 δ

[(ω

ωs

) 13 − 1

].

From (11) one sees[(ω/ωs) 13 − 1] and[(ω/ωs) 2

3 − 1] are of sizeξ1t−23 , so thatE has

terms of sizeξ21ηt

− 23 and ξ

32

1 t/t . Taking ξ1 = Nη2 for N large, the smallest we are

allowed to takeξ1, and takingt (Dt) 13η2 as region oft wheregP , as given in (19),

take its maximum value we find

(38)E ≈ η5

t23

, ∼ ωsY Y4

t8.

Thus we can expect our final result and have errors of sizeE in the exponent.In changing the exponentE in (29) toElin we take the same contour as indicated in Fig. 3,

but now the integration is continued along the imaginaryξ -axis up toξ = i∞. Using thelinearized form fordξ also induces an error in the prefactor of the dominant exponential of

size(ω − ωs)/ωs ∝ (η/t 13 )2 ∼ Y 2/t4 compared to 1. Thus, finally, the integration which

we need to evaluate is

(39)gP (Y ; t, t0)= ∓[Dt]− 13 e±i

π3 eωs(t)Y

∫C1

dξ eηξ Ai(ξ)Ai((ξ − δ)e∓2i π3

),

where we have used (8) to express theK 13

functions appearing in (23) by Airy functions

for reasons which will become clear in a moment. The integration is taken with the contourC1 extended to±i∞, with, as usual, the upper (lower) sign referring to Imξ greater (less)than zero.

4.3. Evaluation of the ξ -integrals

In order to evaluate the integral in (39) it is convenient to define

(40)I± =±i∞∫

ξ1±iεdξ eηξP±(ξ), J± =

±i∞∫ξ1±iε

dξ eηξQ±(ξ),

(41)P±(ξ)=U(ξ)V±(ξ − δ), Q±(ξ)= ∂

∂ξU(ξ)V±(ξ − δ),

with

(42)U(ξ)= Ai(ξ), V±(ξ)= ∓e±i π3 Ai((ξ − δ)e∓2i π3

).


Usingd2U(ξ)/dξ2 ≡ U ′′(ξ)= ξU(ξ) andV ′′±(ξ − δ)= (ξ − δ)V±(ξ − δ) one can showthat

(43)P ′′′ = [4ξP − δP ]′ − 2P − 2δQ, Q′′ = 2(ξP )′ − P − δQfrom which, by integration by parts, one finds

2I = −η[4∂

∂η− δ− η2

]I − 2δJ + eηξ1P2(ξ1, η),

(44)J(δ+ η2)+ I = −2η

∂I

∂η+ eηξ1Q2(ξ1η),

where the± indices have been suppressed in (43)–(45). Also

P2 = P ′′(ξ1)− ηP ′(ξ1)+ η2P(ξ1)− (4ξ1 − δ)P (ξ1),(45)Q2 =Q′(ξ1)− 2ξ1P(ξ1)− ηQ(ξ1).

Using Eq. (44), one easily finds

(46)

[η∂

∂η+ 1

2− (δ+ η2)2

4η

]I± = eηξ1

4η2

[(δ+ η2)P2± − 2δQ2±

]= S±.

The solution to (46) is

(47)

I± = 1√η

exp

[η3

3− (δ− η2)2

4η

](C± +

η∫η0

dη′√η′ exp

[−η

′3

3+ (δ− η′2)2

4η′

]S±(η′)

)

with C± a constant to be determined. Referring back to (39) one can write

(48)gP (Y ; t, t0)=(D2t

)− 13 eωs(t)Y (I+ − I−).

Now the combinationS+ − S− occurring in the integrand of the particular solution of(47) which enters in (48) always involves the combination

K 13

(2

3(ξ1 − δ) 3

2 e−iπ)

+K 13

(2

3(ξ1 − δ) 3

2 e+iπ)

=K 13

(2

3(ξ1 − δ) 3

2

)∝ exp

[−2

3(ξ1 − δ) 3

2

]so that the particular solution can be ignored. Although we have focused here onη largethe procedure also should be valid whenη 1. For η 1, the first factor in (47),1/

√ηexp[−δ2/(4η)], gives the usual BFKL diffusion which means thatC+ − C− can

be determined by evaluatingI+ − I− whenδ = 0 and whenη is small. One finds

(49)C+ −C− = 1√4π

giving exactly the result (21)

(50)gP (Y ; t, t0)= 1√4πDωs(t)Y

eωs(t)Y− 1

4η (δ−η2)2+ η3

3 .


In arriving at this result we have chosen to expand the integrand of (29) aboutξ andtand then integrate overξ . Alternatively, if the expansion had been aboutξ0 andt0 with theintegration overξ0 the result

(51)gP (Y ; t, t0)= 1√4πDωs(t0)Y

eωs(t0)Y− 1

4η0(δ0−η2

0)2+ η30

3

would have been obtained withη0 = Yωs(t0)D 13 t

− 23

0 and withδ0 = (t0 − t)(Dt0)− 13 . The

difference of these exponents is

(52)δ0η0 − δη

2− 1

4

(δ2

η− δ2

0

η0

)+ η3 − η3

0

12=.

Whentt

1,

(53) δ(Dt) 13d

dt

(δη

2− δ2

4η+ η3

12

).

Usingδ ∝ η2 as the important values ofδ in (50), one easily finds

(54)∝ η5

t23

∼ ωsY(Y 4

t8

)which is exactly the size of the error estimated in (32). This conclusion is consistent withthe estimate

ωsY= 0(Y 4/t8) obtained from the heuristic argument based on Eq. (22).

5. The very large Y regime

In the previous section we have exploited the decomposition (28) of the Green’sfunction in two terms, one (Eq. (29)) with product of Airy functions out of phase(∼ K(ξ)K(e∓iπ ξ0)), and the other in phase(∼ K(ξ)K(ξ0), Eq. (30)), corresponding tothe exponents

(55)E(ω,Y )= ωY − 3

2ξ

32 ± 3

2ξ

32

0 .

We have shown that, forY t2, the integral is dominated by Reω ωs , the contribution(30) being negligible on a contour with a slight deformation in Reω > ωs . On the otherhand, forY t2, it becomes profitable to distort the contour in Reω < ωs , where the

contribution (30) becomes actuallydominant, because Reξ32 may take negative values.

In fact, for small enoughω in Imω><0, the exponent with (−) sign in Eq. (55) takes theform

(56)E± ωY ± 4i

3

tωs(t)

Dω

with derivatives

(57)dE±dω

= Y ∓ 4i

3

χm

b√Dω2

,d2E±dω2

= ±8i

3

χm

b√Dω2

.


Therefore, there are saddle points of the exponent (56) at

(58)ω± =√

2

3(1± i)

(χm

bY√D

) 12

with

(59)d2E±dω2±

= (1∓ i)(

3b√D

2χm

) 12

Y32 .

Since(d2E±/dω2±)−12 ω± the Gaussian integration, in (24), about the saddle points

should be an accurate evaluation of the integral. One finds

(60)gP (Y ; t, t0)= t0

3√π

(3b

4χmD5/6Y

) 34[cos

(ω

√Y + π

8

)]eω

√Y ,

with ω2 = 8χm3b

√D

.It then appears that the perturbative solution becomes smaller than predicted by the

exponentωs(kk0)Y (because√Y ωsY for Y t2) and furthermore it starts oscillating,

thus loosing positivity. It is no surprise that, at such largeY values, our perturbativecalculation no longer has physical sense. After all, whenY approachest2 from belowit is clear from (21), or (50), thatt approachest at which point one expects the singularpotential evident in (5) to become troublesome. That is, whenY approachest2 the cutoffat t is clearly necessary. Two questions then arise: firstly, whether we can still describe thebehaviour ofgP in the intermediate regionY ∝ t2; and, secondly, at whichY values doesthe non-perturbative Pomeron part really take over.

We are unable to answer either question in detail. However, for the pure Airy modeldescribed by Eq. (22), we can provide a partial resummation of the corrections to theexponent of relative order(Y/t2)2n as follows. Referring to Eq. (24) we first neglect, in thelargeY regime, the term∂2E/∂t2 compared to(∂E/∂t)2; then we consider the exponentE(Y ; t,t) around its maximum int so as to neglect itst derivative, and we take theansatz

(61)E(Y ; t,tmax)= ωs(t)Yf (z), z=DχmY2

t4= η3

ωsY,

which is supposed to describe theY 2/t4 dependence. Finally, by replacing Eq. (61) intoEq. (22), we get the non-linear differential equation

(62)f (z)+ 2zf ′(z)= 1+ z[f (z)+ 4zf ′(z)]2,

which is a sort of Hamilton–Jacobi limit of the diffusion equation (22).For z 1 (Y t2), the iterative solution to Eq. (62) isf (z)= 1 + z/3 + 2z2/3 + · · · ,

which yields theη3/3 term of Eq. (50) and the first non-trivial correction to it. On theother hand, forz 1 (Y t2), Eq. (62) still makes sense, with a solutionf (z) ∼z− 1

4 + O(z− 12 ), in order to compensate the term 1 in the r.h.s. Therefore, sincez∝ Y 2/t4,

the exponent in Eq. (61) turns out to be in agreement with the saddle point estimate in


Eq. (60). Note, however that while the behaviour (50) is supposed to be universal forY t2 (cf. Section 6), the resummation in Eq. (62) and the largeY regime in Eq. (60)are typical of the Airy model, because they involve largeξ properties of Eq. (22).

We wish now to compare the largeY behaviour of the perturbative term in Eqs. (21)and (50)—with its intricate transition to the behaviour (60) just discussed—to the non-perturbative Pomeron term. The latter can only be discussed in a model dependent way. Inthe Airy model defined in Eqs. (8) and (9) it is straightforward to show that

(63)gR(Y ; t, t0) eωP Y πt0ωP

(2bωPχ ′′m

) 23

Ai2(ξP ) exp

[ωP Y − 4

3t

32

(bωP

χmD

) 12],

where ξP ∼ t (bωP /χmD)13 and we have assumedωP ωs(t), for definiteness. It is

clear from Eq. (63) that the Pomeron term is exponentially suppressed with respect tothe perturbative one for larget . However, it takes over at very large energies, such that

(64)Y > Yc 4t32

3ωP

(bωP

χmD

) 12

,

that is, even before the regionY t2 is reached, as already pointed out for collinear modelsin [9].

The above estimate ofYc is model dependent in several ways. In fact the weight of thePomeron term depends on the small-x equation being used (cf. Section 6) and, within thegiven equation, on the way the strong coupling region is smoothed out or cutoff [10].Furthermore, unitarity corrections may affect it, and there is no consensus on how toincorporate them. However, the basic qualitative feature underlying all models is that“tunneling” to the Pomeron behaviour occurs at someYc < t

2, i.e., before the perturbativecalculation looses sense, thus insuring cross-section positivity.

6. Extension to collinear small-x models

The above evaluation of perturbative diffusion properties of the Airy model (Eqs. (21)and (50)) can be generalized to other small-x models, for which the expression (12) of theperturbative Green’s function is sufficiently explicit.

For instance, a simple two-pole collinear model [9] is provided by the effectiveeigenvalue function

(65)χ(γ,ω)= 1

γ+ 1

1+ω− γ ,

where we have introduced theω-shift [12], but no further subleading terms. Thecorresponding Green’s function is defined by [11]

(66)

[ω− χ

(1

2+ ∂t ,ω

)1

bt

]gω(t, t0)= δ(t − t0),


which basically provides a second order differential equation forgω. Therefore, thesolution is of type in Eq. (2), which, fort > t0> t , reads

gω(t, t0)= 1

ωδ(t − t0)+ 1

ω2bt0W 1

bω, 1

2

[(1+ω)t]

(67)× [W− 1

bω ,12

(−(1+ω)t0)+ S(ω)W 1

bω ,12

((1+ω)t0

)].

HereWk,m(z) is the Whittaker function, satisfying the Coulomb-like differential equation

(68)

[− d2

dz2− k

z+ m2 − 1

4

z2+ 1

4

]Wl,m(z)= 0

and its “irregular” counterpart “W−k,m(z)” is more precisely defined by the real analyticcontinuation

(69)W−k(−z)= 1

2

(eiπkW−k

(eiπ z

)+ e−iπkW−k(e−iπ z

)).

Such solutions admit also theγ -representation [10]

(70)W± 1bω,1/2

(±(1+ω)t)=∫

CR,CI

dγ

2πie(γ− 1

2 t− 1bωX(γ,ω)),

whereCR(CI ) is the contour for the regular (irregular) solution shown in Fig. 2(b), andχ(γ,ω)= ∂

∂γX(γ,ω).

The energy dependence of the perturbative functionGP (Y ; t, t0) in Eq. (12) was alreadyinvestigated in Refs. [10,11] and is characterized by various regimes.

(1) In the collinear limitt t0 > 1, andY 1, GP (Y ; t, t0) is dominated by thecustomary anomalous dimension saddle point, which exists in the energy region

(71)

√logt/t0bY

ω > ωs(t0) ωs(t)

(Y ≡ log

s

k2

),

whereωs(t)= 4(1+ωs)bt 4

btis the saddle point exponent.

(2) In the diffusion regiont = t − t0 t , andY t 1, the importantω valuesdrift towardsωs(t) ωs(t0), and the asymptotic behaviour of the Green’s functionmatches a properly chosen Airy model.

In fact, if t and 1ω

are both large, butω − ωs(t) ω − ωs(t0) is small, the phase in

Eq. (70) is finite only if(γ − 12)∼ (bω)

13 is small also. This means that we are probing the

region close to the minimum ofχ(γ ), in such a way that

(72)

(2bω

χ ′′m

) 13(t − χm

bω

)= ξ =

(tωs(t)√Dω

) 23(ω

ωs(t)− 1

)is finite. This is precisely the region where theW ’s become asymptotically Airy functions,with parameters

(73)χm = 4

1+ωs , D = χ ′′m

2χm= 4

(1+ωs)2 , ωs(1+ωs)= 4

bt.


For this reason, the linearized expression (15) holds in the present case also, under thesame conditions.

The above argument shows that, in the regiont Y t2, and (t − t0) t , thediffusion corrections of Eq. (19) are valid for the two-pole model also, and for anytruncated model where theγ -representation (70) holds for both the regular and the irregularsolution.

Of course, forY t2, the importantω values become much smaller thanωs(t), andthe phases in Eq. (70) become stationary atχ(γ ,ω) = 0. For the Airy model this occursat ν = γ − 1

2 ∼ ± i√D

, and provides the asymptotic behaviour used in Eq. (56). For thetwo-pole model, instead,γ drifts to±i∞, and this provides the asymptotic formulas

gω ∼ exp

(±2πi

bω

),

(74)gP (Y ; t, t0)∼ expω√Y cos

(ω

√Y + π

8

),

with ω = 4πb

. Although the precise exponent is different, the qualitative behaviour (74)is the same as for the Airy model. This follows from the analogous structure of theγ -representation (69), with its exp( const

ω) behaviour.

In conclusion, in the ranget Y t2, the perturbative calculation shows a universalbehaviour, described by Eqs. (21) or (50), where the only model dependence lies in theparametersχm andD, describing the hard Pomeron and the diffusion coefficient. ForY t2 on the other hand, the perturbative behaviour is more model dependent and finallystarts oscillating, thus loosing physical sense.

At some intermediate valueY = Yc (t Yc t2), the non-perturbative Pomeron takesover. We do not have a reliable model for such transition. Is it an abrupt “tunneling” effect[9], as in the two-pole model (withωP ωs(t)), or is it instead mediated by a longdiffusion regime, as perhaps expected in unitarized models (ωP 1)? This is an openquestion, which deserves further investigation.

References

[1] L.N. Lipatov, Sov. J. Nucl. Phys. 23 (1976) 338;E.A. Kuraev, L.N. Lipatov, V.S. Fadin, Sov. Phys. JETP 45 (1977) 199;Ya. Balitskii, L.N. Lipatov, Sov. J. Nucl. Phys. 28 (1978) 822.

[2] V.S. Fadin, L.N. Lipatov, Phys. Lett. B 429 (1998) 127, and references therein.[3] G. Camici, M. Ciafaloni, Phys. Lett. B 412 (1997) 396;

G. Camici, M. Ciafaloni, Phys. Lett. B 430 (1998) 349.[4] J.C. Collins, J. Kwiecinski, Nucl. Phys. B 316 (1989) 307.[5] G. Camici, M. Ciafaloni, Phys. Lett. B 395 (1997) 318.[6] Yu.V. Kovchegov, A.H. Mueller, Phys. Lett. B 439 (1998) 428.[7] N. Armesto, J. Bartels, M.A. Braun, Phys. Lett. B 442 (1998) 459.[8] E.M. Levin, Nucl. Phys. B 445 (1999) 481.[9] M. Ciafaloni, D. Colferai, G.P. Salam, JHEP 9910 (1999) 017;

M. Ciafaloni, D. Colferai, G.P. Salam, JHEP 0007 (2000) 054.


[10] M. Ciafaloni, D. Colferai, G.P. Salam, Phys. Rev. D 60 (1999) 114036.[11] M. Taiuti, Thesis, University of Firenze, 2000, unpublished.[12] G.P. Salam, JHEP 9807 (1998) 19;

M. Ciafaloni, D. Colferai, Phys. Lett. B 452 (1999) 372.


Gluon-fusion contributions toH + 2 jet productionV. Del Ducaa, W. Kilgoreb, C. Olearic, C. Schmidtd, D. Zeppenfeldc

a INFN, Sezione di Torino, via P. Giuria, 1, 10125 Torino, Italyb Physics Department, Brookhaven National Laboratory, Upton, NY 11973, USA

c Department of Physics, University of Wisconsin, Madison, WI 53706, USAd Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48824, USA


Abstract

Real emission corrections to Higgs production via gluon-fusion, at orderα4s , lead to a Higgs plus

two-jet final state. We present the calculation of these scattering amplitudes, as induced by top-quarktriangle-, box- and pentagon-loop diagrams. These diagrams are evaluated analytically for arbitrarytop massmt . We study the renormalization and factorization scale-dependence of the resultingH +2jet cross section, and discuss phenomenologically important distributions at the LHC. The gluon-fusion results are compared to expectations for weak-boson fusion cross sections. 2001 ElsevierScience B.V. All rights reserved.

PACS: 14.80.Bn; 13.85.-t; 12.38.Bx; 13.85.Hd

1. Introduction

Gluon fusion and weak-boson fusion are expected to be the most copious sources ofHiggs bosons inpp-collisions at the large hadron collider (LHC) at CERN. Beyondrepresenting the most promising discovery processes [1,2], these two production modesare also expected to provide a wealth of information on Higgs couplings to gauge bosonsand fermions [3]. The extraction of Higgs boson couplings, in particular, requires precisepredictions of production cross sections.

Next-to-leading order (NLO) QCD corrections to the inclusive gluon-fusion crosssection are known to be large, leading to aK-factor close to two [4]. Because thelowest order process is loop induced, a full NNLO calculation would entail a three-loop evaluation, which presently is not feasible. In the intermediate Higgs mass range,which is favored by electroweak precision data [5], the Higgs boson massmH is smallcompared to the top-quark pair threshold and the largemt limit promises to be an adequate

E-mail address: [email protected] (C. Oleari).



368 V. Del Duca et al. / Nuclear Physics B 616 (2001) 367–399

approximation. Consequently, present efforts on a NNLO calculation of the inclusivegluon-fusion cross section concentrate on themt → ∞ limit, in which the task reduces toan effective two-loop calculation [6]. In order to assess the validity of this approximation,gluon-fusion cross-section calculations, which include all finitemt corrections, are needed.Of particular interest are phase space regions where one or several of the kinematicalinvariants are of the order of, or exceed, the top-quark mass, i.e., regions of large Higgsboson or jet transverse momenta, or regions where dijet invariant masses become large. Forlarger Higgs boson masses, top-mass corrections become important and a full calculationof H + 2 jet production is needed.

A key component of the program to measure Higgs boson couplings at the LHC isthe weak-boson fusion (WBF) process,qq → qqH via t-channelW or Z exchange,characterized by two forward quark jets [3]. QCD radiative corrections to WBF areknown to be small [7] and, hence, this process promises small systematic errors.H + 2jet production via gluon-fusion, while part of the inclusive Higgs signal, constitutes abackground when trying to isolate theHWW andHZZ couplings responsible for theWBF process. A precise description of this background is needed in order to separatethe two major sources ofH + 2 jet events: one needs to find characteristic distributionswhich distinguish the weak-boson fusion process from gluon-fusion. One such featureis the typical large invariant mass of the two quark jets in WBF. A priori, this largekinematic invariant,m2

jj 4m2t , invalidates the heavy top approximation and requires

a full evaluation of all top-mass effects. We will find, however, that even in this phase-space region the largemt limit works extremely well, provided that jet transverse momentaremain small compared tomt .

In a previous letter [8] we presented first results of our evaluation of the real-emission corrections to gluon-fusion which lead toH + 2 parton final states, at orderα4

s .The contributing subprocesses include quark–quark scattering which involves top-quarktriangles, quark–gluon scattering processes which are mediated by top-quark triangles andboxes, and gluon scattering which requires pentagon diagrams in addition. The purpose ofthis paper is to provide details of our calculation and to give a more complete discussionof its phenomenological implications. In Section 2, we start with a brief overview ofthe calculation. Full expressions for the quark–quark and the quark–gluon scatteringamplitudes are given in Section 3. Expressions for thegg → ggH amplitudes, whichwere obtained by symbolic manipulation, are too long to be given explicitly. Instead wedescribe the details of the calculational procedure in Section 3.3. The matrix elements forall subprocesses have been checked both analytically and numerically. The most importantof these tests are described in Section 4. We then turn to numerical results, in particularto implications for LHC phenomenology. In Section 5, we first compare overallH + 2jet cross sections from weak-boson fusion and from gluon-fusion and determine thesubprocess decomposition of the latter. QCD uncertainties are assessed via a discussionof the scale dependence (renormalization and factorization) of our results. We investigatevarious distributions, searching for characteristic differences between gluon-fusion andWBF. Our final conclusions are given in Section 6.

V. Del Duca et al. / Nuclear Physics B 616 (2001) 367–399 369

A number of technical details are collected in Appendixes A–E. Scalar integrals, inparticular the evaluation of scalar five-point functions, are discussed in Appendix A.Appendix B gives useful relations among Passarino–VeltmanCij and Dij functions.Finally, in Appendixes C, D and E, we provide expressions for the color decompositionand the tensor integrals encountered in triangle, box and pentagon graphs.

2. Outline of the calculation

The production of a Higgs boson in association with two jets, at orderα4s , can proceed

via the subprocesses

(2.1)qq → qqH, qQ→ qQH, qg → qgH, gg → ggH,

and all crossing-related processes. Here the first two entries denote scattering of identicaland non-identical quark flavors. In Fig. 1 we have collected a few representative Feynmandiagrams which contribute to subprocesses with four, two and zero external quarks. In ourcalculation, the top-quark is treated as massive, but we neglect all other quark masses, sothat the Higgs boson only couples via a top-quark loop. Typically we have aggH couplingthrough a triangle-loop (Fig. 1(a)), agggH coupling mediated by a box-loop (Fig. 1(b))and aggggH coupling which is induced through a pentagon-loop (Fig. 1(c)). The numberand type of Feynman diagrams can be easily built from the simpler dijet QCD productionprocesses at leading order. One needs to insert the Higgs-gluon “vertices” into the tree-leveldiagrams for 2→ 2 QCD parton scattering in all possible ways.

In the following counting, we exploit Furry’s theorem, i.e., we are counting as one thetwo charge-conjugation related diagrams where the loop momentum is running clockwiseand counter-clockwise. This halves the number of diagrams. In addition, the crossedprocesses are not listed as extra diagrams, but are included in the final results. Three distinctclasses of processes need to be considered.

1. qq → qqH and qQ → qQH There are only 2 diagrams obtained from theinsertion of a triangle-loop into the tree-level diagrams forqq → qq . One of them isdepicted in Fig. 1(a), while the other is obtained by interchanging the two identicalfinal quarks. In the case ofqQ→ qQH , whereQ is a different flavor, there is onlyone diagram, i.e., Fig. 1(a).

Fig. 1. Examples of Feynman graphs contributing toH + 2 jet production via gluon-fusion.


2. qg → qgH At tree level, there are 3 diagrams contributing to the processqg → qg:one with a three-gluon vertex and two Compton-like ones. Inserting a triangle-loopinto every gluon line, we have a total of 7 different diagrams. In addition, we caninsert a box-loop into the diagram with the three-gluon vertex, in 3 different ways:the 3! permutations of the 3 gluons are reduced to 3 graphs by using Furry’s theorem.In total we have 10 different diagrams for theqg → qgH scattering amplitude.

3. gg → ggH Four diagrams contribute to the tree-level scattering processgg → gg:a four-gluon vertex diagram and 3 diagrams with two three-gluon vertices each.Inserting a triangle-loop in any of the gluonic legs gives rise to 19 different diagrams.The insertion of the box-loop in the 3 diagrams with three-gluon vertices yieldsanother 18 diagrams. Finally, there are 12 pentagon diagrams (corresponding to 4!permutations of the external gluons, divided by 2, according to Furry’s theorem).

The amplitudes for these processes are ultraviolet and infrared finite inD = 4dimensions. Nevertheless, we keptD arbitrary in several parts of our computationbecause some functions are divergent inε = (4 − D)/2 at intermediate steps. Obviously,these divergences cancel when the intermediate expressions are combined to give finalamplitudes. An example of this behavior is given by Eqs. (C.9) and (C.10), where thedivergent part of theB0 functions cancels among the different contributions to the trianglegraphs.

Given the large number of contributing Feynman graphs, it is most convenient togive analytic results for the scattering amplitudes for fixed polarizations of the externalquarks and gluons. These amplitudes are then evaluated numerically, instead of using tracetechniques to express polarization averaged squares of amplitudes in terms of relativisticinvariants. We proceed to derive explicit expressions for these amplitudes.

3. Notation and matrix elements

Within the SM, the effective interaction of the Higgs boson with gluons is dominatedby top-quark loops because the top Yukawa coupling,ht = mt/v with v = 246.22 GeV,is much larger than theHbb coupling. In the following we only consider top-loopcontributions. All theH + 4 parton amplitudes, at lowest order, are then proportional tohtg

4s , wheregs = √

4παs is theSU(3) coupling strength. It is convenient to absorb thesecoupling constants into an overall factor

(3.1)F = htg4s

16π24mt = 4m2t

vα2s ,

where we have anticipated the loop factor 1/16π2 and the emergence of an explicit factor,4mt , from all top-quark loops, which results from the compensation of the chirality flip,induced by the insertion of a single scalarHtt vertex.


3.1. qQ→ qQH and qq → qqH

The simplest contribution toH + 2 jet production is provided by theqQ → qQH

process depicted in Fig. 1(a). Other four-quark amplitudes are obtained by crossing. Weneglect external fermion masses and use the formalism and the notation of Ref. [9] for thespinor algebra. For a subprocess like

(3.2)q(p1, i1

) + Q(p4, i4

) → q(p2, i2

) + Q(p3, i3

) +H(P),

with

(3.3)p21 = p2

2 = p23 = p2

4 = 0, P 2 =m2H ,

each external (anti-)fermion is described by a two-component Weyl-spinor of chiralityτ = σi = Si σi ,

(3.4)ψ(pi , σi

)τ

= Si

√2 p0

i δσiτ χσi(pi

).

Herepi , σi andii denote the physical momentum, the helicity and the color index of thequark or anti-quark, and the sign factorSi allows for an easy switch between fermions(Si = +1) and anti-fermions (Si = −1). The quark–gluon vertices of Fig. 1(a), includingthe attached gluon propagators, are captured via the effective quark currents

Jµ21 = δσ2σ1χ

†σ2

(p2

)(σµ

)τχσ1

(p1

) 1

(p1 − p2)2

(3.5)= δσ2σ1〈2|(σµ)τ|1〉 1

(p1 − p2)2,

and

(3.6)Jµ43 = δσ4σ3〈4|(σµ

)τ|3〉 1

(p3 − p4)2.

Here we have used helicity conservation via the assignmentsτ = σ1 = σ2 andτ = σ3 = σ4,respectively, and the sign factors for the fermions provide an easy connection between thegluon momentaq1 = p2 −p1 andq2 = p4 −p3 going out of the top-quark triangle and thephysical quark momentapi = Sipi . Finally we have used the shorthand notation

(3.7)|1〉 = χσ1

(p1

), 〈2| = χ†

σ2

(p2

), |3〉 = χσ3

(p3

), 〈4| = χ†

σ4

(p4

),

and(σµ)± = (1,±σ ) is the reduction of Dirac matrixesγ µ into the two-component Weylbasis. Since the quark currentsJµ

12 andJµ43 are conserved, we have

(3.8)Jµ12(p1 − p2)µ = 0, J

µ34(p3 − p4)µ = 0.

The Weyl spinors and the currentsJ21 andJ43 are easily evaluated numerically [9]. Thescattering amplitude for different flavors on the two quark lines is then given by

(3.9)AqQ = FqQJµ21J

ν43Tµν(q1, q2)t

ai2i1

tai4i3 =AqQ

2143tai2i1

tai4i3,

where the overall factor

(3.10)FqQ = S1S2S3S44√p0

1p02p

03p

04F,


includes the normalization factors of external quark spinors. Thetaij = λaij /2 are the colorgenerators in the fundamental representation ofSU(N),N = 3. As detailed in Appendix C,the tensorTµν(p, q) can be written as

T µν(p, q)

= FT(p2, q2, (p + q)2

)(p · qgµν − pνqµ

)(3.11)+ FL

(p2, q2, (p + q)2

)(q2p2gµν − p2qµqν − q2pµpν + p · qpµqν).

Analytic expressions for the scalar form factorsFT and FL are given in Eqs. (C.9)and (C.10).

The scattering amplitude for two identical quarks is obtained from the result above byincluding Pauli interference, which results from interchanging quarks 2 and 4,

(3.12)Aqq =Aqq2143t

ai2i1

tai4i3 −Aqq4123t

ai4i1

tai2i3.

The squared amplitude, summed over initial- and final-particle color, becomes

(3.13)∑col

∣∣Aqq∣∣2 = (|A2143|2 + |A4123|2

)N2 − 1

4+ 2 Re

(A2143A∗

4123

)N2 − 1

4N.

The squared amplitude for theqQ→ qQH process can be read from Eq. (3.13) by puttingA4123= 0.

3.2. qg → qgH

Twenty distinct Feynman graphs contribute to the process

(3.14)g(q1, a1

) + q(p1, i1

) → g(q2, a2

) + q(p2, i2

) +H(P),

and crossing related processes (the color index of the external gluons is indicated withai).However, pairs with opposite directions of the top-quark fermion arrow in the loop arerelated by charge conjugation and will not be counted separately in the following (seeAppendixes C and D).

The resulting ten Feynman graphs are depicted in Fig. 2. Following Ref. [9], all gluonmomenta are treated as outgoing. For the specific process of Eq. (3.14) we then setq1 = −q1 andq2 = +q2, with

(3.15)p21 = p2

2 = q21 = q2

2 = 0, P 2 =m2H .

Feynman graphs with a triangle insertion in an external gluon line, i.e., with one light-like gluon attached to the top-quark triangle, receive contributions from a single formfactor,FT , only (see Eq. (C.12)). These simplifications are best captured by replacing thepolarization vectorsεµi = ε

µi (qi) with the effective polarization vectors

(3.16)eµiH = FT

(0, (P + qi)

2,P 2) 1

(P + qi)2

(qµi εi · P − ε

µi qi · P

),

for gluonsi = 1,2, withFT given in Eq. (C.10).


Fig. 2. Feynman graphs contributing to the processqg → qgH . Graphs (3) and (4) are the same as(1) and (2), but with gluon labels interchanged. No distinction is made between the two orientationsof the fermion arrow on the top-quark loop, because they are related by Furry’s theorem.

External quark lines are handled as for theqQ → qQH amplitudes. Feynman graphs5 through 10 are proportional to the quark currentJ

µ21 defined in Eq. (3.5). Spinor

normalization factors are absorbed into the overall factor

(3.17)Fqg = −S1S22√p0

1p02Fδσ1σ2.

Using the shorthand notation [9]

(3.18)〈2, qi | = χ†σ2

(p2

)(/εi)σ2(/p2 + /qi)−σ2

1

(p2 + qi)2,

(3.19)|qi,1〉 = (/p1 − /qi)−σ1(/εi)σ1χσ1

(p1

) 1

(p1 − qi)2,

for the 2-component Weyl spinors describing emission of a gluon next to an external quark,we arrive at a compact notation for the contributions to theqg → qgH scattering amplitude


Aqg =Aqgµ1µ2

εµ11 ε

µ22

= Fqg

(ta1ta2

)i2i1

[〈2|(/e1H)σ1|q2,1〉 + 〈2, q1|(/e2H)σ1|1〉]+ (

ta2ta1)i2i1

[〈2|(/e2H)σ1|q1,1〉 + 〈2, q2|(/e1H)σ1|1〉]+ [

ta1, ta2]i2i1

[+2

(J21 · q2e1H · ε2 − J21 · e1H(p2 − p1) · ε2

− J21 · ε2q2 · e1H)

− 2(J21 · q1e2H · ε1 − J21 · e2H(p2 − p1) · ε1

− J21 · ε1q1 · e2H)

−[(

(q1 + q2) · (p1 − p2)

(q1 + q2)2FT − (p1 − p2)

2FL

)Jµ21

+ J21 · P(q1 + q2)2

FT Pµ

]× [

ε1 · ε2(q2 − q1)µ − 2q2 · ε1ε2µ + 2q1 · ε2ε1µ]

(3.20)−Bµ1µ2µ3εµ11 ε

µ22 J

µ321

],

whereFT = FT ((q1 + q2)2, (p1 − p2)

2,P 2), and analogously forFL.The contributions of the box diagrams enter in the last line of Eq. (3.20) via the tensor

Bµ1µ2µ3 = Bµ1µ2µ3(q1, q2, q3) (with q3 = p2 −p1). Gauge invariance and Bose symmetryof the gluons limit the relevant structure of these box contributions to just two independentscalar functions, as we will now show.

It is easy to see that theqg → qgH amplitudeAqg is invariant under the replacementsεµ1 → ε

µ1 + κ1q

µ1 andεµ2 → ε

µ2 + κ2q

µ2 , for arbitrary constantsκi . 1 By proper choice of

these constants, the polarization vectors of the two on-shell gluons can be made orthogonalto bothq1 andq2. This can be seen by introducing a convenient basis of Minkowski space,composed of the vectorsqµ1 , qµ2 and

(3.21)xµ = q2 · q3qµ1 + q1 · q3q

µ2 − q1 · q2q

µ3 ,

(3.22)yµ = εµαβρq1αq2βq3ρ.

The vectoryµ is orthogonal to all momenta occurring in the boxes (q1, q2 andq3) whilexµ is orthogonal toqµ1 , qµ2 andyµ. More precisely

(3.23)x · x = detQ3 ≡ q1 · q2[q2

3q1 · q2 − 2q1 · q3q2 · q3],

(3.24)y · y = detQ3,

(3.25)x · q1 = 0, x · q2 = 0, x · q3 = −detQ3

q1 · q2,

(3.26)y · q1 = 0, y · q2 = 0, y · q3 = 0, y · x = 0,

1 These gauge invariance conditions provided a stringent test of our numerical programs.


where detQ3 denotes the Gram determinant of the box, i.e., the determinant of the 3× 3matrix with elements(Q3)ij = −qi · qj . Obviously detQ3 is symmetric under interchangeof the three gluon labels. The non-symmetric form given in Eq. (3.23) pertains to ourparticular situation where onlyq3 may be off-shell (q2

1 = q22 = 0).

In the following we take

(3.27)εµ1 = xµ√−detQ3

,

(3.28)εµ2 = yµ√−detQ3

,

as the two independent polarization vectors of each of the on-shell gluons. With this choice,we eliminate the twoBc contributions in the box tensorBµ1µ2µ3 (see Eq. (D.6)) since theycontain a factorqµ2

1 or qµ12 , which vanishes upon contraction with the polarization vectors.

We can then write the squared element forqg → qgH , summed over the polarizationvectors of the external gluons in the form

(3.29)∑pol

∣∣Aqg∣∣2 =

(1

detQ3

)2[∣∣Aqgxx

∣∣2 + ∣∣Aqgxy

∣∣2 + ∣∣Aqgyx

∣∣2 + ∣∣Aqgyy

∣∣2],where the shorthandAqg

xy = Aqgµ1µ2x

µ1yµ2, etc. has been used. Expressions for thecontracted tensor integrals for the boxes that appear in Eq. (3.20) are given in Eqs. (D.15)–(D.20).

In addition, the color structure of theqg → qgH amplitude is given by (see Eq. (3.20))

(3.30)Aqg = (ta1ta2

)i2i1

Aqg

12 + (ta2ta1

)i2i1

Aqg

21,

so that the resulting color-summed squared amplitude takes the form

(3.31)∑col

∣∣Aqg∣∣2 = (∣∣Aqg

12

∣∣2 + ∣∣Aqg

21

∣∣2) (N2 − 1)2

4N− 2 Re

[Aqg

12

(Aqg

21

)∗]N2 − 1

4N.

3.3. gg → ggH

For the process

(3.32)g(q1, a1

) + g(q2, a2

) → g(q3, a3

) + g(q4, a4

) +H(P),

we introduce the outgoing momentaqi , so thatq1 = −q1, q2 = −q2, q3 = +q3 andq4 = +q4, where

(3.33)q21 = q2

2 = q23 = q2

4 = 0, P 2 =m2H ,

andai are the color indices in the adjoint representation carried by the gluons.Due to the large number of diagrams and the length of the results, we are not going to

write explicitly the expressions for the amplitude, but we describe in detail the procedurewe follow.


We used QGRAF [10] to generate the 49 diagrams for this process. As detailed inSection 2, these diagrams are obtained by the insertion of a triangle-, a box- or a pentagon-loop into the tree-level diagrams forgg → gg scattering, so that we can write the un-contracted amplitude in the “formal” way

(3.34)(Agg

)αβγ δ

≡19∑i=1

cTi Tiαβγ δ +

18∑i=1

cBi Biαβγ δ +

12∑i=1

cPi Piαβγ δ,

where the tensor functionsT i , Bi andP i are Feynman diagrams that contain a triangle,a box and a pentagon fermionic loop, whilecTi , cBi andcPi are the respective color factors.This amplitude will be contracted with the external polarization vectors of the gluons,εi ,to give

(3.35)Agg = εα1 εβ2 ε

γ

3 εδ4

(Agg

)αβγ δ

≡ (Agg

)ε1ε2ε3ε4

.

We used Maple to trace over the Diracγ matrixes and to manipulate the expressions. Sincethe Higgs couples as a scalar to the massive top-quark in the loop, the resulting trace inthe numerator of the genericn-point function has at mostn− 1 loop-momentum factors.Using the tensor-reduction procedure described by Passarino and Veltman [11], we canexpress the triangle and box one-loop tensor integrals in terms of the external momentaq

µi

and in terms of the scalar functionsCij (i = 1, . . . ,2, j = 1, . . . ,4) andDij (i = 1, . . . ,3,j = 1, . . . ,13). For speed reasons, we preferred not to express directly the final amplitudein terms of scalar triangles and boxes (theC0 andD0 functions in Passarino–Veltmannotation), in our Monte Carlo program. Expressions for the amplitude written in terms oftheC0 andD0 integrals are considerably larger than the result where we keep theCij andDij functions.

In dealing with diagrams with a pentagon loop, we worked directly with the dot productsin the numerator. In fact, the generic tensor pentagon appearing ingg → ggH scatteringhas the form

E(p1,p2,p3,p4)α,αβ,αβγ,αβγ δ

(3.36)

=∫

dDk

iπD/2

kα, kαkβ, kαkβkγ , kαkβkγ kδ[k2 −m2

t ][(k + p1)2 −m2t ][(k + p12)2 −m2

t ]× ([

(k + p123)2 −m2

t

][(k + p1234)

2 −m2t

])−1,

wherepij = pi + pj , and similar ones forpijl and pijln, and where the set ofpi ,p1,p2,p3,p4, is one of the 24 permutations of the external gluon momentaq1, q2,

q3, q4. These permutations are reduced to 12, once Furry’s theorem is taken into account.The generic scalar five-point function, that is Eq. (3.36) with a 1 in the numerator, will beindicated withE0(p1,p2,p3,p4).

The tensor indices appearing in the numerator are always contracted with one of thefollowing:

1. The metric tensorgµν .2. One of the external momentaqµi .3. One of the external polarization vectorε

µi .


The procedure we have used in these three cases is the following.

1. Every time there is ak2 = kαkβgαβ product in the numerator, we write it as

(3.37)k2 = [k2 −m2

t

] +m2t ,

and the first term is going to cancel the first propagator, giving rise to a four-pointfunction, while the last one will multiply the rest of the tensor structure in thenumerator, that now has been reduced by two powers of the loop momentumk.

2. We rewrite every scalar product of the type(k · qi) in the numerator as a differenceof two propagators, using the identity

(3.38)k · qi = 1

2

[(k + p + qi)

2 −m2t

] − [(k + p)2 −m2

t

] − 2qi · p,

wherep is an arbitrary momentum. The first two terms in the sum are going to cancelthe two propagators adjacent to the external gluon leg with momentumqi , whilethe last one will contribute to the rest of the tensor structure in the numerator, nowreduced by one power ofk.

3. When the dot product(k · εi) appears in the numerator, we choose the four externalgluon momenta as our basis of Minkowski space. Hence we can expand the externalpolarization vectorεi as

(3.39)εµi =

4∑j=1

εij qµj , i = 1, . . . ,4,

where the coefficientsεij are computed by inverting the system of equations

(3.40)εi · qk = −4∑

j=1

εij (Q4)jk, i, k = 1, . . . ,4,

where the elements of the matrixQ4 are given by

(3.41)(Q4)jk = −qj · qk.Using Eq. (3.39), we can rewrite every scalar product of the form(k · εi) in thenumerator of the Feynman diagrams as a sum over(k · qj ), that we handle in thesame way as shown in Eq. (3.38).The matrixQ4 is singular when the four gluons become planar, i.e., when the fourgluon momenta cease to be linearly independent. This can easily be seen in the center-of-mass frame of partons 1 and 2, where we can write

q1 = E(1,0,0,1),

q2 = E(1,0,0,−1),

q3 = E3(1,sinθ,0,cosθ),

q4 = E4(1,sinθ ′ cosφ,sinθ ′ sinφ,cosθ ′),

and the determinant of the matrixQ4 becomes

(3.42)detQ4 = −4E4E23E

24

(1− cos2 θ ′)(1− cos2 θ

)(1− cos2φ

).


The cuts imposed on the final-state partons (see Eq. (5.1)) avoid the singular regionwhen one of the final-state partons is collinear with the initial-beam direction(singularities inθ andθ ′).The singularity inφ is un-physical, and the final amplitude should be finite nearthe singular pointsφS = 0,π . In our FORTRAN program, when the Monte Carlointegration approaches the singular points inφ, we interpolate the value we needfrom the values of the amplitude in the pointsφ = φS ± 0.01π . We have checkedthat the interpolated amplitude differs from the exact value of the amplitude by lessthan 1%, in the non-singular region.

By iterating this reduction procedure, we can write the contracted amplitude forgg →ggH scattering in terms of

– twelveE0(pa,pb,pc,pd) functions, i.e., the scalar five-point functions computed indifferent kinematics;

– twelve permutations ofDij functions with argumentDij (pa,pb,pc), with a < c, sixDij (pa,pb + pc,pd) with a < d and twelveDij (pa,pb,pc + pd), together with thecorrespondingD0 functions;

– threeCij (pa + pb,pc + pd), and fourCij (pa,pb + pc + pd), together with thecorrespondingC0 functions.

As usual, the setpa,pb,pc,pd is chosen in the group of permutations of the gluonmomentaq1, q2, q3, q4. To reduce the number of theDij and Cij functions to thisindependent set, we made use of some identities between these functions, that we collectin Appendix B.

As stated in Eq. (3.34), we can study the color factors of thegg → ggH process,by dividing the full amplitude into three different classes, according to the number offermionic propagators in the loop. We first discuss the color factors of the diagramscontaining a pentagon-loop.

Diagrams with a pentagon-loopThe contribution from the sum of charge-conjugated pentagon diagrams is proportional

to the sum of two color traces with fourt matrixes (see Eq. (E.3)). From the invarianceproperty of the trace under cyclic permutations, we have only(4 − 1)! = 6 independenttraces, from the permutation of the four gluon indices. These six color traces combinetogether as in Eq. (E.3) to give rise to three independent color structures

c1 = Tr(ta1ta2ta3ta4

) + Tr(ta1ta4ta3ta2

),

c2 = Tr(ta1ta3ta4ta2

) + Tr(ta1ta2ta4ta3

),

(3.43)c3 = Tr(ta1ta4ta2ta3

) + Tr(ta1ta3ta2ta4

).

The ci color coefficients are real. In fact, using the identity (the sum over the repeatedindex is understood)

(3.44)Tr(ta1ta2ta3ta4

) = 1

4Nδa1a2δa3a4 + 1

8

(da1a2l + if a1a2l

)(da3a4l + if a3a4l

),


where f is the (totally antisymmetric)SU(N) structure constant andd is the totallysymmetric symbol, we can write, for example,

(3.45)c1 = 1

4

[2

Nδa1a2δa3a4 + da1a2lda3a4l − f a1a2lf a3a4l

],

and similar ones forc2 andc3. Sincef andd are real constants, theci are real too.A few useful identities can be derived if we take the differences of theci

c1 − c2 = −1

2f a1a2lf a3a4l ⇒ f a1a2lf a3a4l = 2(c2 − c1),

c3 − c1 = −1

2f a1a4lf a2a3l ⇒ f a1a4lf a2a3l = 2(c1 − c3),

(3.46)c2 − c3 = −1

2f a1a3lf a4a2l ⇒ f a1a3lf a4a2l = 2(c3 − c2).

Note that the differences of theci in the system (3.46) automatically embodies the Jacobiidentity: by summing the three expressions, we have

(3.47)f a1a2lf a3a4l + f a1a4lf a2a3l + f a1a3lf a4a2l = 0.

Diagrams with a box-loopThese diagrams all contain a three-gluon vertex together with the quark loop. Since

the sum of the charge-conjugated boxes is proportional to the structure constantf (seeEq. (D.3)), the final color factors accompanying these diagrams are a product of twof ’s,such asf a1a2lf a3a4l .

With the help of Eq. (3.46), we can express these products in terms of differences of theci color factors.

Diagrams with a triangle-loopThe same argument can be used to show that the color structure of all the diagrams with

a three-point function insertion are proportional to the product of two structure constants,that are then converted to differences ofci color factors, using the identities of Eq. (3.46).

Since all the color structures of the diagrams contributing to thegg → ggH process canbe written in terms of theci color structures of Eq. (3.43), we can then decompose the fullamplitudes in the following way

(3.48)Agg =3∑

i=1

ciAggi .

The sum over the external colored gluons of the squared amplitude becomes

(3.49)∑col

∣∣Agg∣∣2 =

3∑i,j=1

Aggi

(Aggj

)∗ ∑col

cicj ,

where we have taken into account the fact that theci are real (see Eq. (3.45)). UsingEq. (3.43), one finds


(3.50)C1 ≡∑col

cici = (N2 − 1)(N4 − 2N2 + 6)

8N2, no summation overi,

(3.51)C2 ≡∑col

cicj = − (N2 − 1)(N2 − 3)

4N2 , i = j,

and we finally get

(3.52)∑col

∣∣Agg∣∣2 = C1

3∑i=1

∣∣Aggi

∣∣2 + C2

3∑i,j=1i =j

Aggi

(Aggj

)∗.

4. Checks

We were able to perform two different kinds of checks on the analytic amplitudes wecomputed: a gauge-invariance and a large-mt limit check.

4.1. Gauge invariance

Gauge invariance demands that the amplitudes should be invariant under the replacementεi → εi + κiqi , for arbitrary values ofκi . This implies that for theqg → qgH process wemust have (see Eq. (3.20))

(4.1)qα1 ε

β

2

(Aqg

)αβ

= 0

εα1qβ2

(Aqg

)αβ

= 0

whenqi · εi = 0, i = 1, . . . ,2,

and forgg → ggH (see Eq. (3.35))

(4.2)

qα1 εβ2 ε

γ

3 εδ4

(Agg

)αβγ δ

= 0

εα1qβ

2 εγ

3 εδ4

(Agg

)αβγ δ

= 0

εα1 εβ

2 qγ

3 εδ4

(Agg

)αβγ δ

= 0

εα1 εβ2 ε

γ

3 qδ4

(Agg

)αβγ δ

= 0

whenqi · εi = 0, i = 1, . . . ,4.

We checked the gauge invariance in Eqs. (4.1) and (4.2) both analytically and numerically(in the final FORTRAN program).

Using Eq. (3.20), it is straightforward to check that the system (4.1) is satisfied. Tocheck gauge invariance for thegg → ggH process, we wrote the contracted amplitudeof Eq. (3.35) in terms of scalar pentagon (E0), box (D0), triangle (C0) and bubble (B0)integrals, keeping the space–time dimensionD arbitrary. This means that we expressed theCij andDij functions in terms ofB0, C0 andD0 “master” integrals. The coefficients ofthese scalar integrals are then functions of scalar products(qi · qj ), (qi · εj ), (εi · εj ) andof theεij coefficients introduced in Eq. (3.39).

Since the two-point functionsB0 are divergent inε = (D − 4)/2, and since the totalamplitude must be finite, these poles must cancel. In fact, the factors multiplying theB0


contributions are proportional to(D−4), so that only the pole coefficient ofB0 contributesto the finite amplitude (see comment after Eq. (C.10)).

To implement the gauge invariance check with respect to the polarization vectorεi , asdescribed in the system (4.2), we make the replacement

(4.3)εi → qi ⇒ εii = 1, εij = 0, j = i,

in Eq. (3.35), and we impose the orthogonality conditionεk · qk = 0, that constrains theεkjcoefficients to satisfy the identity (see Eq. (3.39))

(4.4)εk · qk =4∑

j=1

εkj (qj · qk)= 0, k = i.

We have checked gauge invariance in two different ways.

1. Suppose that instead of considering the QCD processgg → ggH , we considerthe QED analogue,γ γ → γ γH . In this scenario, all the diagrams with a three-or a four-gluon vertex are no longer present: the only surviving diagrams are theones containing a pentagon-loop, with no color structure associated. The amplitude,not contracted with any external photon polarization vectors, is (see Eq. (3.34) forcomparison)

(4.5)(Aγ γ→γ γH

)αβγ δ

≡12∑i=1

P iαβγ δ.

The gauge invariance of this expression allows us to check the correctness of thetensor reduction of the pentagon diagrams only. We have contracted Eq. (4.5) withthe polarization vectors of the photons and we have applied the tensor reductionprocedure previously described. Instead of expressing the results in terms ofCij

andDij functions, we have expressed these coefficient functions in terms ofB0,C0, D0 andE0 scalar integrals, keeping the space–time dimensionD arbitrary. Sincethese scalar integrals form a set of independent functions, we expect the coefficientsof these integrals to be zero, in order to fulfill the gauge-invariance test. Note thatwe have considered the twelve scalar five-point functionsE0 as independent fromthe four-point functionsD0, that is we have not used Eq. (A.1). This is indeed thecase, since, in arbitraryD dimensions, the scalar pentagon cannot be expressed as acombination of scalar boxes only, so that it is really an independent integral.

2. Finally, we have checked that our full QCD amplitude satisfies the four identities inthe system (4.2). Since the amplitude can be split into three different contributionsaccording to the three independent color factorsci (see Eq. (3.48)), this means thatnot only the full amplitude is gauge invariant, but that the three sub-amplitudesAgg

i

are separately gauge invariant, and satisfy a system of equations similar to (4.2).

4.2. Large-mt limit

The amplitudes for Higgs plus two partons agree in the large-mt limit with thecorresponding amplitudes obtained from the heavy-top effective Lagrangian [12]. This


check was done numerically by settingmt = 3 TeV. We found good agreement with themt → ∞ results, within the statistical errors of the Monte Carlo program, for Higgs bosonmasses in the range 100<mH < 700 GeV.

5. Applications to LHC physics

The gluon-fusion processes atO(α4s ), together with weak-boson fusion (qq → qqH

production viat-channel exchange of aW orZ), are expected to be the dominant sourcesof H + 2 jet events at the LHC. The impact of the former on LHC Higgs phenomenologyis determined by the relative size of these two contributions. However, the gluon-fusioncross sections forH + 2 jet events diverges as the final-state partons become collinearwith one another or with the incident beam directions, or as final-state gluons become soft.A minimal set of cuts on the final-state partons, which anticipates LHC detector capabilitiesand jet finding algorithms, is required to define anH + 2 jet cross section. Our minimalset of cuts is

(5.1)pTj > 20 GeV, |ηj |< 5, Rjj > 0.6,

wherepTj is the transverse momentum of a final-state parton andRjj describes theseparation of the two partons in the pseudo-rapidityη versus azimuthal angle plane

(5.2)Rjj =√@η2

jj + φ2jj .

ExpectedH + 2 jet cross sections at the LHC are shown in Fig. 3, as a function ofthe Higgs boson mass,mH . The three curves compare results for the expected SM gluon-fusion cross section atmt = 175 GeV (solid line) with the large-mt limit (dotted line),and with the WBF cross section (dashed line). Error bars indicate the statistical errors of

Fig. 3.H + 2 jet cross sections in pp collisions at√s = 14 TeV as a function of the Higgs boson

mass. Results are shown for gluon-fusion processes induced by a top-quark loop withmt = 175 GeVand in themt → ∞ limit, computed using the heavy-top effective Lagrangian, and for weak-bosonfusion. The two panels correspond to two sets of jet cuts: (a) inclusive selection (see Eq. (5.1)) and(b) WBF selection (Eqs. (5.1) and (5.3)).


the Monte Carlo integration. Cross sections correspond to the sum over all Higgs decaymodes: finite Higgs width effects are included.

In all our simulations, we used the CTEQ4L set for parton-distribution functions [13].Unless specified otherwise, the factorization scale was set toµf = √

pT 1pT 2. Since thiscalculation is a LO one, we employ one-loop running of the strong coupling constant. InFig. 3 we fixαs = αs(MZ)= 0.12.

The left panel in Fig. 3 shows cross sections within the minimal cuts of Eq. (5.1). Thegluon-fusion contribution dominates because the cuts retain events with jets in the centralregion, with relatively small dijet invariant mass. In order to assess background levels forWBF events, it is more appropriate to consider typical tagging jet selections employed forWBF studies [14]. This is done in Fig. 3(b) where, in addition to the cuts of Eq. (5.1), werequire

(5.3)|ηj1 − ηj2|> 4.2, ηj1 · ηj2 < 0, mjj > 600 GeV,

i.e., the two tagging jets must be well separated, they must reside in opposite detectorhemispheres and they must possess a large dijet invariant mass. With these selection cutsthe weak-boson fusion processes dominate over gluon-fusion by about 3/1 for Higgs bosonmasses in the 100 to 200 GeV range. This means that a relatively clean separation of weak-boson fusion and gluon-fusion processes will be possible at the LHC, in particular whenextra central-jet-veto techniques are employed to further suppress semi-soft gluon radiationin QCD backgrounds. We expect that a central-jet veto will further suppress gluon-fusionwith respect to WBF by an additional factor of three [14].

A conspicuous feature of theH + 2 jet gluon-fusion cross sections in Fig. 3 is thethreshold enhancement atmH ≈ 2mt , an effect which is familiar from the inclusive gluon-fusion cross section. Near this “threshold peak” the gluon-fusion cross section rises toequal the WBF cross section, even with the selection cuts of Eq. (5.3). Well below thisregion, the large-mt limit provides an excellent approximation to the totalH + 2 jet ratefrom gluon-fusion, at least when considering the total Higgs production rate only. Neartop-pair threshold the large-mt limit underestimates the rate by about a factor of 2.

A somewhat surprising feature of Fig. 3(b) is the excellent approximation provided bythe large-mt limit at Higgs boson masses below about 200 GeV. Naively one might expectthe large dijet invariant mass,mjj > 600 GeV, and the concomitant large parton center-of-mass energy to spoil themt → ∞ approximation. This is not the case, however. As shownin Ref. [8], the large-mt limit works well in the intermediate Higgs mass range, as long asjet transverse momenta stay small:pTj mt .

In Fig. 4 we have plotted the individual contributions to the gluon-fusion cross sectionwhich are coming from thegg → ggH , qg → qgH and qq → qqH sub-processes,including all crossed processes for each of the three subgroups. Results are shown afterimposing the inclusive cuts of Eq. (5.1) (left panel) and the WBF cuts of Eqs. (5.1) and (5.3)(right panel), withmt = 175 GeV, so that the sum of the three curves in each panel addup to the solid line curve in Fig. 3. External gluons dominate in the inclusive-cut case (leftpanel): final-state gluons tend to be soft and initial gluons preferably lead to soft events dueto the rapid fall-off of the gluon parton distribution function,g(x,µf ), with increasingx.


Fig. 4.H + 2 jet contributions to the cross section in pp collisions at√s = 14 TeV as a function of

the Higgs boson mass. Results are shown for the different contributions to the gluon-fusion process(gg,qg andqq amplitudes) using (a) the inclusive cuts of Eq. (5.1) and (b) the WBF cuts of Eqs. (5.1)and (5.3).

When themjj > 600 GeV constraint of the WBF cuts is imposed, the gluon contributiondies rapidly, as is shown in Fig. 4(b).

The results shown in Fig. 3 raise two questions, which we intend to answer in thefollowing: (i) what are the uncertainties in the prediction of theH + 2 jet cross section,and (ii) which distributions are the most effective in distinguishing gluon-fusion and weak-boson fusion contributions toH + 2 jet events?

In order to asses the sensitivity of the gluon-fusion cross section to higher order QCDcorrections, we have plotted in Fig. 5 the total cross section for several choices of therenormalization scale (the factorization scale has been kept atµf = √

pT 1pT 2). We havefixed Λ5

MS= 254 MeV, so that,αs(MZ) = 0.12, with nf = 5 active flavors. We have

chosen five different scales:µ0 = √pT 1pT 2, MZ , mjj ,

√mHj1mHj2 and

√s, i.e., the

geometric average between the transverse momenta of the two jets, theZ mass, theinvariant mass of the two jets, the geometric mean of the two invariant masses of the Higgsand the jets and the partonic center-of-mass energy. For every event generated by our MonteCarlo, we have computed the running of the coupling constantαs(µr) at the valuesµr =ξµ0, whereξ was allowed to vary from 1/5 to 5. We can see that the renormalization-scaledependence is very strong, mainly due to the fact that this is a leading-order calculation, atorderα4

s .What is the “most natural” scale forαs is an unresolved issue. The good agreement [8]

between the complete result and themt → ∞ one (away from threshold) implies that thecross section is dominated by Feynman diagrams with a gluon exchange in thet channel.These diagrams contain a triangle-loop that couples thet-channel gluon with the Higgs.For this reason, it seems reasonable to make the replacement

(5.4)α4s → αs(pT 1)αs(pT 2)α

2s (mH ).

With this choice for the strong coupling constant we have a total cross section (within thecuts of Eq. (5.1)) of aboutσ = 9.6 pb, which sits in between the two values computed using


Fig. 5. Renormalization-scale dependence of the total cross section forH plus two jet productionwith the inclusive cuts of Eq. (5.1). The renormalization scaleµr = ξµ0 is varied in the range1/5 < ξ < 5. The five curves correspond, from top to bottom, to the following choice ofµ0: thegeometric mean of the transverse momenta of the two jets, theZ mass, the invariant mass of thetwo jets, the geometric mean of the two invariant masses of the Higgs and the jets, and the partoniccenter-of-mass energy.

α4s (MZ) andα4

s (mjj ) (see Fig. 5). Dismissing the extreme choiceµr = √s (which is ill

defined at higher orders), and allowing for the conventional factor of 2 variation ofξ , Fig. 5suggests an uncertainty of the gluon-fusionH + 2 jet cross section of about a factor 2.5 ascompared to the central value of 9.6 pb obtained with the central choice of Eq. (5.4).

Keeping fixed the renormalization scale atMZ , we have collected in Fig. 6 the resultsfor the factorization-scale dependence of the total cross section. The factorization scaleµf = ξµ0 was allowed to vary in the range described by 1/5< ξ < 5, whereµ0 was takenequal to the geometric average of the transverse momenta of the two jets, the invariant massof the two jets and the partonic center-of-mass energy. Compared with the variation withrespect to the renormalization scale, we see that the dependence on the factorization scaleis almost negligible: in Fig. 6 theH + 2 jet cross section varies between 9.2 and 10.2 pb.

In the following we use the renormalization scale choice of Eq. (5.4) and setµf =√pT 1pT 2. We takemH = 120 GeV throughout, as a characteristic Higgs boson mass.Turning now to the issue of differentiating between gluon-fusion and WBF processes,

the prominent characteristics to be considered here are the jet properties. Fig. 7 showsthe dijet-mass distribution for gluon-fusion and WBF processes, using the inclusive andthe WBF cuts. In this last case, we have suppressed the constraintmjj > 600 GeV, inorder to access the region of small dijet invariant mass. In both panels, the high dijet massregion (mjj > 1 TeV) is dominated by WBF. The significantly softer dijet-mass spectrumof the gluon-fusion processes is characteristic of QCD processes, which are dominatedby external gluons, as compared to quarks in WBF processes (see Fig. 4 and comments


Fig. 6. Factorization-scale dependence of the total cross section forH plus two jet production with theinclusive cuts of Eq. (5.1). The factorization scaleµf = ξµ0 is varied in the range 1/5< ξ < 5. Thethree curves correspond to the following choice ofµ0: the geometric average between the transversemomenta of the two jets, the invariant mass of the two jets and the partonic center-of-mass energy.

Fig. 7. Dijet invariant-mass distribution of the two final jets for gluon-fusion (solid) and WBF(dashes) processes. Left panel (a) inclusive cuts of Eq. (5.1); right panel (b) WBF cuts of Eqs. (5.1)and (5.3), where we have suppressed the constraintmjj > 600 GeV.

about it). Fig. 7 also shows themt → ∞ dijet-mass distributions (dotted curves), that arealmost indistinguishable from themt = 175 GeV result: large dijet invariant masses donot invalidate themt → ∞ limit as long as the Higgs boson mass and the jet transversemomenta are small enough, less thanmt in practice.

A characteristic of WBF events is the large rapidity separation of the two tagging jets,a feature which is not shared byH + 2 jet events arising from gluon-fusion. The rapidityseparation of the jets is shown in Fig. 8, for both gluon-fusion (solid) and WBF (dashes)processes. The two panels correspond to the inclusive cuts of Eq. (5.1) and to the stricter


Fig. 8. Rapidity separation of the two final jets for gluon-fusion (solid) and WBF (dashes) processes.Left panel (a) inclusive cuts of Eq. (5.1); right panel (b) WBF cuts of Eqs. (5.1) and (5.3), where wehave suppressed the constraint|ηj1 − ηj2|> 4.2.

Fig. 9. Azimuthal-angle distribution between the two final jets, with the WBF cuts of Eqs. (5.1)and (5.3). Results are shown for gluon-fusion processes induced by a top-quark loop withmt = 175 GeV and in themt → ∞ limit, computed using the heavy-top effective Lagrangian, andfor weak-boson fusion.

WBF cuts of Eq. (5.3), where we have suppressed the constraint|ηj1 −ηj2|> 4.2, in orderto have access to the entire@ηjj range. The jet separation cut,|ηj1 − ηj2| > 4.2, is oneof the most effective means of enhancing WBF processes with respect to gluon-fusion.The small dip in the gluon-fusion distribution at small@ηjj is a consequence of the cutRjj > 0.6.

A second jet-angular correlation, which allows to distinguish gluon-fusion from weak-boson fusion, is the azimuthal angle between the two jets,φjj . The distributions for gluon-fusion and WBF processes are shown in Fig. 9. In the WBF processqQ → qQH , thematrix element squared is proportional to


Fig. 10. Transverse-momentum distribution of the Higgs from gluon-fusion (solid) and from WBF(dashes) processes with the inclusive selection of Eq. (5.1).

(5.5)|AWBF|2 ∝ 1

(2p1 · p2 +M2W)2

1

(2p3 · p4 +M2W)2

sm2jj ,

and is dominated by the contribution in the forward region, where the dot products inthe denominator are small. Since the dependence ofm2

jj on φjj is mild, we have the flatbehavior depicted in Fig. 9. The azimuthal-angle distribution of the gluon-fusion processis instead characteristic of the CP-even operatorHGµνG

µν , whereGµν is the gluon fieldstrength tensor [15]. This effective coupling can be taken as a good approximation for theggH coupling in the high-mt limit. Note that the large-mt limit (dotted line) is almostindistinguishable from themt = 175 GeV result (solid line).

Finally, in Fig. 10, we show the transverse-momentum distribution of the Higgs boson ingluon-fusion (solid lines) and in WBF (dashes lines) processes, with the inclusive selectionof Eq. (5.1). Within these cuts, both differential cross sections peak around a value ofpTH ≈ 50 GeV. Note, however, that, while the peak position of the WBF distribution islargely tied to the mass of the exchanged intermediate weak-bosons, the peaking of thegluon-fusion processes occurs just above 40 GeV, which is a direct consequence of thepTj > 20 GeV cut of Eq. (5.1).

6. Conclusions

In the previous sections, we have provided the results of theO(α4s ) calculation of

H + 2 jet cross section, including the full top-mass dependence. For the quark–quark andquark–gluon scattering amplitudes we have found very compact analytic expressions. Thegg → ggH amplitudes, which include pentagon-loops, are more complex and availableanalytically, as Maple output, and numerically, in the form of a FORTRAN program.


Numerical investigations of the resulting cross sections at the LHC provide manyinteresting insights. With minimal jet-selection cuts (parton separation ofRjj > 0.6 and jettransverse momenta in excess of 20 GeV) the gluon-fusion induced cross section is sizable,of order 10 pb formH = 120 GeV, which corresponds to about 30% of the inclusive Higgsproduction rate. Since our calculation gives theH + 2 jet rate at leading order, it exhibitsthe large renormalization-scale dependence to be expected of an orderα4

s process.As expected, the large-mt limit provides an excellent approximation to the fullmt

dependence when the Higgs mass is small compared to the top-pair threshold. The large-mt

limit is found to break down formH >mt and when jet transverse momenta become large(pTj mt ). However, large dijet invariant masses do not invalidate themt → ∞ limit, aslong as the Higgs boson mass and the jet transverse momenta are small enough, less thanthe top-quark mass in practice. This observation opens the possibility of NLO correctionstoH +2 jet production from gluon-fusion. Performing the calculation in the large-mt limitwould correspond to a 1-loop calculation of a 2→ 3 process. Such a calculation might bedesirable to reduce systematics errors in the extraction ofHZZ andHWW couplings fromthe competing weak-boson fusion processes at the LHC.

Consideration of the gluon-fusionH + 2 jet rate as a background to WBF studiesconstitutes an important application of our calculation. While the overallH + 2 jet rateis dominated by gluon-fusion at the LHC, kinematic properties of the two processes aresufficiently different to allow an efficient separation. Gluon-fusion events tend to be soft,with a relatively small separation of the two jets. In contrast, the two tagging jets of weak-boson fusion events have very large dijet invariant mass, and are far separated in rapidity.Using rapidity and invariant-mass cuts, the gluon-fusion cross section can be suppressedwell below the WBF rate. In addition, the azimuthal angle between the two jets showsa dip at 90 degrees which is characteristic for loop-induced Higgs couplings to gaugebosons [15]. Based on our calculation we conclude that a relatively clean separation ofweak-boson fusion and gluon-fusion Higgs plus two-jet events will be possible at the LHC.

Acknowledgements

We thank E. Richter-Was for insisting on the importance of this calculation at an earlystage. C.S. acknowledges the US National Science Foundation under grant PHY-0070443.W.K. acknowledges the DOE funding under Contract No. DE-AC02-98CH10886. Thisresearch was supported in part by the University of Wisconsin Research Committee withfunds granted by the Wisconsin Alumni Research Foundation and in part by the USDepartment of Energy under Contract No. DE-FG02-95ER40896.

Appendix A. Scalar integrals: C0, D0 and E0 functions

All the scalar integrals needed for the calculation are finite inD = 4 dimensions, dueto the presence of the top-quark mass. No further regulator is required. Scalar triangles(C0) and boxes (D0) have been known for a long time in the literature [16] and efficient


computational procedures are available [17]. Following the procedure outlined in Ref. [18],we can express all scalar pentagons as linear combinations of scalar boxes

(A.1)E0(p1,p2,p3,p4)=5∑

i,j=1

FijDj

0,

where

D10 =D0(p2,p3,p4), D2

0 =D0(p4,p3,p1 + p2),

D30 =D0(p1,p2 + p3,p4), D4

0 =D0(p1,p2,p3 + p4),

(A.2)D50 =D0(p1,p2,p3),

and the matrixF = C−1, with

(A.3)Cij = (ri − rj )2 − 2m2

t ,

and

r1 = 0, r2 = p1, r3 = p12 = p1 + p2,

(A.4)r4 = p123= p1 + p2 + p3, r5 = p1234= p1 + p2 + p3 + p4.

Appendix B. Relations among Cij and Dij functions

In this section, we collect a few identities between theCij andDij functions. Theirdefinition can be found in Ref. [11]. Please note that we use a(+,−,−,−) metrictensor, so that Passarino–Veltman recurrence relations are the same as ours if we makethe substitutionp · q → −p · q andδµν → −gµν in their formulae.

Starting with a three-point vertex function

(B.1)∫

dDk

iπD/2

f (k)

[k2 −m2t ][(k + p)2 −m2

t ][(k + p + q)2 −m2t ],

wheref (k) is an arbitrary function, that, for our purpose, will take the valuesf (k) =1, kα, kαkβ , and imposing the equality between this integral and the same integral wherethe integration variable has been shifted according tok → −k − p− q , we have∫

dDk

iπD/2

f (k)

[k2 −m2t ][(k + p)2 −m2

t ][(k + p + q)2 −m2t ]

(B.2)=∫

dDk

iπD/2

f (−k − p − q)

[(k + p + q)2 −m2t ][(k + q)2 −m2

t ][k2 −m2t ].

If f (k)= 1, this identity gives

(B.3)C0(p, q)= C0(q,p),

while if f (k)= kα , it gives

C11(p, q)= −C12(q,p)−C0(q,p),

(B.4)C12(p, q)= −C11(q,p)−C0(q,p).


In a similar way, iff (k)= kαkβ we get

C21(p, q)= C22(q,p)+ 2C12(q,p)+C0(q,p),

C22(p, q)= C21(q,p)+ 2C11(q,p)+C0(q,p),

C23(p, q)= C23(q,p)+C12(q,p)+C11(q,p)+C0(q,p),

(B.5)C24(p, q)= C24(q,p).

Starting with a four-point function, we derive, in the same fashion,

D0(p, q, l)=D0(l, q,p),

D11(p, q, l)= −D13(l, q,p)−D0(l, q,p),

D12(p, q, l)= −D12(l, q,p)−D0(l, q,p),

D13(p, q, l)= −D11(l, q,p)−D0(l, q,p),

D21(p, q, l)=D23(l, q,p)+ 2D13(l, q,p)+D0(l, q,p),



D24(p, q, l)=D26(l, q,p)+D13(l, q,p)+D12(l, q,p)+D0(l, q,p),



D27(p, q, l)=D27(l, q,p),

D31(p, q, l)= −3D13(l, q,p)− 3D23(l, q,p)−D33(l, q,p)−D0(l, q,p),



D34(p, q, l)= −2D13(l, q,p)−D12(l, q,p)− 2D26(l, q,p)−D39(l, q,p)

−D23(l, q,p)−D0(l, q,p),


−D23(l, q,p)−D0(l, q,p),

D36(p, q, l)= −D13(l, q,p)− 2D12(l, q,p)− 2D26(l, q,p)−D38(l, q,p)

−D22(l, q,p)−D0(l, q,p),


−D21(l, q,p)−D0(l, q,p),


−D22(l, q,p)−D0(l, q,p),


−D21(l, q,p)−D0(l, q,p),


D310(p, q, l)= −D13(l, q,p)−D12(l, q,p)−D11(l, q,p)−D26(l, q,p)

−D310(l, q,p)−D25(l, q,p)−D24(l, q,p)−D0(l, q,p),

D311(p, q, l)= −D27(l, q,p)−D313(l, q,p),

D312(p, q, l)= −D27(l, q,p)−D312(l, q,p),

(B.6)D313(p, q, l)= −D27(l, q,p)−D311(l, q,p).

Appendix C. Tensor and color structure of triangles

The two generic three-point functions depicted in Fig. 11 have the following expressions

Tµ1µ21 (q1, q2)

(C.1)= 1

4mt

∫d4k

iπ2Tr

(1

/k −mt

γ µ11

/k + /q1 −mt

γ µ21

/k + /q1 + /q2 −mt

),

Tµ1µ22 (q1, q2)

(C.2)= 1

4mt

∫d4k

iπ2Tr

(1

/k −mt

γ µ21

/k + /q2 −mt

γ µ11

/k + /q1 + /q2 −mt

),

whereq1 andq2 are outgoing momenta and where we put an overall factor 1/(4mt) infront to delete a corresponding term coming from the trace over the top-quark. Using thecharge-conjugation matrixC

(C.3)CγµC−1 = −γ T

µ ,

we can derive (Furry’s theorem)

(C.4)Tµ1µ21 (q1, q2)= T

µ1µ22 (q1, q2)≡ T µ1µ2(q1, q2).

In addition,

(C.5)qµ11 Tµ1µ2(q1, q2)= q

µ22 Tµ1µ2(q1, q2)= 0,

expresses the gauge invariance of the triangle graphs. The generic tensor structuresatisfying Eq. (C.5) is then

T µ1µ2(q1, q2)= FT(q2

1, q22, (q1 + q2)

2)T µ1µ2T (q1, q2)

+ FL(q2

1, q22, (q1 + q2)

2)T µ1µ2L (q1, q2)

(C.6)= FT Tµ1µ2T + FLT

µ1µ2L ,

where (dropping the dependence on external momentaq1 andq2, for ease of notation)

(C.7)Tµ1µ2T = q1 · q2g

µ1µ2 − qµ21 q

µ12 ,

(C.8)Tµ1µ2L = q2

1q22g

µ1µ2 − q21q

µ12 q

µ22 − q2

2qµ11 q

µ21 + q1 · q2q

µ11 q

µ22 ,


Fig. 11. Two three-point functions connected by charge conjugation.

and

FL(q2

1, q22,Q

2)

(C.9)

= − 1

2 detQ2

[2− 3q2

1q2 ·QdetQ2

](B0(q1)−B0(Q)

)

+[2− 3q2

2q1 ·QdetQ2

](B0(q2)−B0(Q)

)

−[4m2

t + q21 + q2

2 +Q2 − 3q21q

22Q

2

detQ2

]C0(q1, q2)+R

,

FT(q2

1, q22,Q

2)

(C.10)

= − 1

2 detQ2

Q2[B0(q1)+B0(q2)− 2B0(Q)− 2q1 · q2C0(q1, q2)

]+ (

q21 − q2

2

)(B0(q1)−B0(q2)

) − q1 · q2FL.

Here, the negative ofQ = q1 + q2 denotes the four-momentum of the Higgs boson,detQ2 = q2

1q22 − (q1 · q2)

2 is the Gram determinant, and the terms proportional toR = −2are pole residues inD = 4 dimensions, originating from contributions proportional to(D − 4)C24(q1, q2) in the tensor reduction procedure. Please note that even though theB0 functions are divergent inε = (4−D)/2, the form factorsFL andFT are finite.

If one of the external momenta, for example,q1, is light-like (real photon or gluon), withpolarization vectorε1, then

(C.11)Tµ1µ2L = q

µ11

(q1 · q2q

µ22 − q2

2qµ21

)if q2

1 = 0,

and as a consequence of the orthogonalityq1 · ε1 = 0 we have

(C.12)ε1µ1Tµ1µ2L = 0,

i.e., the form factorFL does not contribute when an on-shell gluon or photon is attachedto the triangle graph.

The color structure of the sum of the two diagrams in Fig. 11 is straightforward. In fact,both the diagrams have the same color structure and we can write (see Eq. (C.4))

(C.13)Tr(ta1ta2

)Tµ1µ21 (q1, q2)+ Tr

(ta2ta1

)Tµ1µ22 (q1, q2)= δa1a2T µ1µ2(q1, q2).


Appendix D. Tensor and color structure of boxes

The two generic four-point functions connected by charge conjugation, depicted inFig. 12, have the following expressions

Bµ1µ2µ31 (q1, q2, q3)

= 1

4mt

∫d4k

iπ2 Tr

(1

/k −mt

γ µ11

/k + /q1 −mt

γ µ21

/k + /q12 −mt

γ µ31

/k + /q123−mt

),

Bµ1µ2µ32 (q1, q2, q3)

(D.1)

= 1

4mt

∫d4k

iπ2 Tr

(1

/k −mt

γ µ31

/k + /q3 −mt

γ µ21

/k + /q23 −mt

γ µ11

/k + /q123−mt

),

whereq1, q2 andq3 are the outgoing momenta,qij = qi + qj andqijk = qi + qj + qk. Theoverall factor 1/(4mt) cancels a corresponding term coming from the trace over the topquark. From charge conjugation we have

(D.2)Bµ1µ2µ31 (q1, q2, q3)= −Bµ1µ2µ3

2 (q1, q2, q3)≡ Bµ1µ2µ3(q1, q2, q3).

The color structure of the sum of the two diagrams depicted in Fig. 12 is

Tr(ta1ta2ta3

)Bµ1µ2µ31 (q1, q2, q3)+ Tr

(ta3ta2ta1

)Bµ1µ2µ32 (q1, q2, q3)

= [Tr

(ta1ta2ta3

) − Tr(ta3ta2ta1

)]Bµ1µ2µ3(q1, q2, q3)

(D.3)= i

2f a1a2a3 Bµ1µ2µ3(q1, q2, q3),

where we have used the identity

(D.4)Tr(ta1ta2ta3

) = 1

4

(da1a2a3 + if a1a2a3

),

and the anti-symmetry of the structure constantf a1a2a3, together with the symmetry ofda1a2a3.

The sum over the six-gluon permutations of boxes is then proportional to the single colorfactorf a1a2a3, and because of Bose symmetry of the gluons, the kinematic box factor

Bµ1µ2µ3(q1, q2, q3)

(D.5)= Bµ1µ2µ3(q1, q2, q3)+ Bµ2µ3µ1(q2, q3, q1)+ Bµ3µ1µ2(q3, q1, q2),

is totally anti-symmetric in the gluon indices(qi,µi), i = 1,2,3.

Fig. 12. Two four-point functions connected by charge conjugation.


D.1. qg → qgH and gg → ggH

The general structure of Eq. (D.5) can be further restricted for the processes we areinvestigating:qg → qgH andgg → ggH . In fact, in both processes, two gluons of thebox are on-shell, and the amplitude is contracted by the two corresponding polarizationvectors εi , while the third gluon is contracted with the conserved currentJ

µ321 (see

Eq. (3.5)), in theqg → qgH process, and with the conserved current of the two on-shellgluons in the three-gluon vertex, in thegg → ggH case.

This gives rise to a few simplification in the structure of Eq. (D.5). In fact, a parityeven, three-index tensor which depends on three independent momenta (here taken as thethree outgoing gluon momentaqi , i = 1,2,3) can be written in terms of 36 independenttensor structures, 9 of typegµ1µ2q

µ3i and permutations, plus 27 tensors of typeq

µ1i q

µ2j q

µ3k .

However, any terms proportional toqµ11 , qµ2

2 , orqµ33 vanish by virtue of the transversity of

the gluon polarization vectorsεµi

i and because the current on the vertexµ3 is conserved.This leaves us with 14 possible tensor structures, that can be further reduced to three oncewe impose the total anti-symmetry in the gluon indices(qi,µi)

Bµ1µ2µ3 = gµ1µ2qµ31 Ba(q1, q2, q3)+ gµ2µ3q

µ12 Ba(q2, q3, q1)

+ gµ3µ1qµ23 Ba(q3, q1, q2)− gµ2µ1q

µ32 Ba(q2, q1, q3)

− gµ1µ3qµ21 Ba(q1, q3, q2)− gµ3µ2q

µ13 Ba(q3, q2, q1)

+ qµ13 q

µ23 q

µ31 Bb(q1, q2, q3)+ q

µ12 q

µ21 q

µ31 Bb(q2, q3, q1)

+ qµ12 q

µ23 q

µ32 Bb(q3, q1, q2)− q

µ13 q

µ23 q

µ32 Bb(q2, q1, q3)

− qµ12 q

µ21 q

µ32 Bb(q1, q3, q2)− q

µ13 q

µ21 q

µ31 Bb(q3, q2, q1)

(D.6)+ qµ12 q

µ23 q

µ31 Bc(q1, q2, q3)− q

µ13 q

µ21 q

µ32 Bc(q2, q1, q3).

Note that Bose symmetry implies thatBc must be invariant under cyclic permutations ofits arguments

(D.7)Bc(q1, q2, q3)= Bc(q2, q3, q1)= Bc(q3, q1, q2).

However, a convenient choice of gauge will remove theBc terms altogether, as shown inSection 3.2, using the polarization vectors defined in Eqs. (3.27) and (3.28).

The scalar functions appearing in Eq. (D.6) are given by

Ba(q1, q2, q3)

= 1

2q2 · q3

[D0(q1, q2, q3)+D0(q2, q3, q1)+D0(q3, q1, q2)

]− q1 · q2

[D13(q2, q3, q1)+D12(q3, q1, q2)−D13(q3, q2, q1)

]−C0(q1, q2 + q3)

(D.8)− 4[D313(q2, q3, q1)+D312(q3, q1, q2)−D313(q3, q2, q1)

],

Bb(q1, q2, q3)

= D13(q1, q2, q3)+D12(q2, q3, q1)−D13(q2, q1, q3)


(D.9)

+ 4[D37(q1, q2, q3)+D23(q1, q2, q3)+D38(q2, q3, q1)+D26(q2, q3, q1)

−D39(q2, q1, q3)−D23(q2, q1, q3)],

Bc(q1, q2, q3)

= −1

2

[D0(q1, q2, q3)+D0(q2, q3, q1)+D0(q3, q1, q2)

]

(D.10)

+ 4[D26(q1, q2, q3)+D26(q2, q3, q1)+D26(q3, q1, q2)+D310(q1, q2, q3)

+D310(q2, q3, q1)+D310(q3, q1, q2)].

Further simplifications appear in the evaluation ofqg → qgH matrix elements. Infact, since the polarization vectors for the two on-shell gluons are either proportionalto xµ or yµ (see Eqs. (3.27) and (3.28)), we only need the contractionsBxxµ3 =xµ1xµ2B

µ1µ2µ3(q1, q2, q3), Bxyµ3 = xµ1yµ2B

µ1µ2µ3, etc. Theµ3 index will be contracted

with the fermion currentJµ321 (see Eq. (3.20)). Sinceq1, q2, q3 andy span Minkowski

space,J21 can be expanded as

(D.11)Jµ21 = 1

detQ3

(q1 · J21u

µ + q2 · J21vµ + y · J21y

µ),

where

(D.12)uµ = q2 · q3xµ + detQ3

q1 · q2qµ2 ,

(D.13)vµ = q1 · q3xµ + detQ3

q1 · q2qµ1 ,

and they satisfy the orthogonality relations

u · q1 = detQ3, u · q2 = 0, u · q3 = 0, u · y = 0,

(D.14)v · q1 = 0, v · q2 = detQ3, v · q3 = 0, v · y = 0,

by virtue of the two on-shell conditionsq21 = 0 andq2

2 = 0. Note that there is noq3

contraction in Eq. (D.11), sinceq3 · J21 = 0 by current conservation.The orthogonality ofyµ to all gluon momentaqi , i = 1,2,3, implies that all contractions

of Bµ1µ2µ3 with an odd number ofy will vanish. This leaves us with six non-zerocontractions of the tensor box integrals,Byyu, Byyv , Byxy , Bxyy , Bxxu andBxxv. Via theBose symmetry of the tensor integral in Eq. (D.6), the first four and the last two are relatedby a permutation of gluon momenta:

(D.15)Byyu = (detQ3)2Ba(q1, q2, q3),

(D.16)Byyv = −(detQ3)2Ba(q2, q1, q3),

(D.17)Byxy = − (detQ3)2

q1 · q2Ba(q3, q1, q2),

(D.18)Bxyy = (detQ3)2

q1 · q2Ba(q3, q2, q1),


Bxxu = (detQ3)2Ba(q1, q2, q3)− q2 · q3

q1 · q2

[Ba(q3, q1, q2)−Ba(q3, q2, q1)

]+ detQ3

(q1 · q2)2Bb(q1, q2, q3)

= − (detQ3)2

q1 · q2

×(q1 · q2)

2[D13(q2, q3, q1)+D12(q3, q1, q2)−D13(q3, q2, q1)]

− 1

2q1 · q2q2 · q3

[D0(q1, q2, q3)+D0(q3, q1, q2)

]− q2 · q3q2 ·

(q3 + q1

2

)D0(q2, q3, q1)

+ 4q1 · q2[D313(q2, q3, q1)+D312(q3, q1, q2)−D313(q3, q2, q1)

]+ [

q2 · q3q3 · (q1 − q2)− q23q1 · q2

]× [

D13(q1, q2, q3)+D12(q2, q3, q1)−D13(q2, q1, q3)]

− 4q2 · q3[2(D313(q1, q2, q3)+D312(q2, q3, q1)−D313(q2, q1, q3)

)+D27(q2, q3, q1)

]− 4

detQ3

q1 · q2

[D37(q1, q2, q3)+D23(q1, q2, q3)+D38(q2, q3, q1)

+D26(q2, q3, q1)−D39(q2, q1, q3)−D23(q2, q1, q3)](D.19)+ q1 · q2C0(q1, q2 + q3)

,

Bxxv = −(detQ3)2Ba(q2, q1, q3)+ q1 · q3

q1 · q2

[Ba(q3, q1, q2)−Ba(q3, q2, q1)

](D.20)+ detQ3

(q1 · q2)2Bb(q2, q1, q3)

= −Bxxu(q1 ↔ q2).

Note that in the replacementBxxv = −Bxxu(q1 ↔ q2), the Gram determinant, detQ3, is tobe treated as totally symmetric under interchange of gluon momentaq1, q2 andq3.

Appendix E. Tensor and color structure of pentagons

The two generic five-point functions connected by charge conjugation depicted inFig. 13 have the following expression

Pµ1µ2µ3µ41 (q1, q2, q3, q4)

= 1

4mt

∫d4k

iπ2Tr

(1

/k −mt

γ µ41

/k + /q4 −mt

γ µ11

/k + /q14 −mt

γ µ2

× 1

/k + /q124−mt

γ µ31

/k + /q1234−mt

),


Fig. 13. Two five-point functions connected by charge conjugation.

(E.1)

Pµ1µ2µ3µ42 (q1, q2, q3, q4)

= 1

4mt

∫d4k

iπ2 Tr

(1

/k −mt

γ µ31

/k + /q3 −mt

γ µ21

/k + /q23 −mt

γ µ1

× 1

/k + /q123−mt

γ µ41

/k + /q1234−mt

),

whereq1, q2, q3 andq4 are the outgoing momenta (qij = qi + qj and similar ones forqijlandqijln) and where we put an overall factor 1/(4mt) in front to delete a correspondingterm coming from the trace over the top-quark. From charge conjugation we have

(E.2)

Pµ1µ2µ3µ41 (q1, q2, q3, q4)= P

µ1µ2µ3µ42 (q1, q2, q3, q4)≡ Pµ1µ2µ3µ4(q1, q2, q3, q4).

Finally, the color structure of the sum of the two diagrams depicted in Fig. 13 is

Tr(ta1ta2ta3ta4

)Pµ1µ2µ3µ41 (q1, q2, q3, q4)

+ Tr(ta4ta3ta2ta1

)Pµ1µ2µ3µ42 (q1, q2, q3, q4)

(E.3)= [Tr

(ta1ta2ta3ta4

) + Tr(ta1ta4ta3ta2

)]Pµ1µ2µ3µ4(q1, q2, q3, q4).

Further details about the color structure of pentagons are given in Section 3.3.

References

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[6] S. Catani, D. de Florian, M. Grazzini, JHEP 0105 (2001) 025, hep-ph/0102227;R. Harlander, W. Kilgore, Phys. Rev. D 64 (2001) 013015, hep-ph/0102241.

http://lepewwg.web.cern.ch/LEPEWWG/


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R.P. Kauffman, S.V. Desai, D. Risal, Phys. Rev. D 55 (1997) 4005;R.P. Kauffman, S.V. Desai, D. Risal, Phys. Rev. D 58 (1998) 119901, Erratum, hep-ph/9610541.

[13] H.L. Lai et al., Phys. Rev. D 55 (1997) 1280, hep-ph/9606399.[14] D. Rainwater, R. Szalapski, D. Zeppenfeld, Phys. Rev. D 54 (1996) 6680;

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Nuclear Physics B 616 [PM] (2001) 403–418www.elsevier.com/locate/npe

Nonlinear supersymmetry on the plane inmagnetic field and quasi-exactly solvable systems

Sergey M. Klishevicha,b, Mikhail S. Plyushchaya,ba Departamento de Física, Universidad de Santiago de Chile, Casilla 307, Santiago 2, Chile

b Institute for High Energy Physics, Protvino, Russia

Received 21 May 2001; accepted 6 August 2001

Abstract

The nonlinearn-supersymmetry with holomorphic supercharges is investigated for the 2D systemdescribing the motion of a charged spin-1/2 particle in an external magnetic field. The universalalgebraic structure underlying the holomorphicn-supersymmetry is found. It is shown that theessential difference of the 2D realization of the holomorphicn-supersymmetry from the 1D caserecently analysed by us consists in appearance of the central charge entering non-trivially intothe superalgebra. The relation of the 2D holomorphicn-supersymmetry to the 1D quasi-exactlysolvable (QES) problems is demonstrated by means of the reduction of the systems with hyperbolicor trigonometric form of the magnetic field. The reduction of then-supersymmetric system withthe polynomial magnetic field results in the family of the one-dimensional QES systems with thesextic potential. Unlike the original 2D holomorphic supersymmetry, the reduced 1D supersymmetryassociated withx6 + · · · family is characterized by the non-holomorphic supercharges of the specialform found by Aoyama et al. 2001 Elsevier Science B.V. All rights reserved.

PACS: 03.65.-w; 11.10.Lm; 11.30.Pb

1. Introduction

The nonlinear supersymmetry is one of the new developments of quantum mechanics[1–11] revealing itself variously in such different systems as the parabosonic [3] andparafermionic [4] oscillator models, the fermion-monopole system [8,9], and theP,T -invariant systems of planar fermions [10] and Chern–Simons fields [11]. Being a naturalgeneralization of the usual supersymmetry [12–15], it is characterized by the polynomialsuperalgebra resembling the nonlinear finiteW -algebras [16].

A simple universal algebraic structure with oscillator-like bosonic and oscillator fermi-onic variables underlies the usual (linear) supersymmetry in classical and quantum me-chanics [12–15]. The classical nonlinear supersymmetry with holomorphic supercharges

E-mail addresses: [email protected] (S.M. Klishevich), [email protected] (M.S. Plyushchay).



404 S.M. Klishevich, M.S. Plyushchay / Nuclear Physics B 616 [PM] (2001) 403–418

[3] has also the transparent algebraic structure related to then-fold mapping of the complexplane associated with the oscillator-like bosonic variables [6]. In what follows we will referto the nonlinear supersymmetry generated by the holomorphic supercharges with Poissonbracket (anticommutator) being proportional to thenth order polynomial in the Hamil-tonian as to theholomorphic n-supersymmetry. However, the attempt to quantize the non-linear supersymmetry immediately faces the problem of the quantum anomaly. The quan-tization of the one-dimensional systems was investigated by us in detail in Ref. [6] wherewe showed that the anomaly-free quantum systems with holomorphicn-supersymmetryturn out to be closely related to the quasi-exactly solvable (QES) systems [17–20].

This paper is devoted to generalization of the analysis of Ref. [6] for the case of two-dimensional systems. It will allow us not only to find a universal algebraic structureunderlying the holomorphicn-supersymmetry at the quantum level, but also to demonstratea nontrivial relation of the holomorphicn-supersymmetry to the non-holomorphicnonlinear supersymmetry of Aoyama et al. [7], and to establish the relationship of the2D holomorphicn-supersymmetry with the family of QES systems with sextic potential[17,18,20,21] not comprised by the 1D holomorphicn-supersymmetry. Nowadays, thisspecial class of QES systems attracts attention in the context of theP,T -invariant quantummechanics [22–28].

The paper is organized as follows. In Section 2 we consider the holomorphicn-supersymmetry realized in the classical 2D system describing the motion of a chargedspin-1/2 particle in an external magnetic field. Section 3 illustrates the simplest anomaly-free quantum realization of the holomorphicn-supersymmetry in the case of the constantmagnetic field. Section 4 is devoted to investigation of the general aspects of theanomaly-free quantization of the holomorphicn-supersymmetry. We show that thequantum mechanicaln-supersymmetry can be realized only for magnetic field of specialconfigurations of the exponential and quadratic form. Here we find the universal algebraicstructure underlying the holomorphicn-supersymmetry. The nonlinear superalgebra withthe central charge is discussed in Section 5, where we consider also the reduction of the2D systems with the exponential magnetic field to the 1D systems with the holomorphicn-supersymmetry. In Section 6 we show that the spectral problem of the 2D system withthe quadratic magnetic field is equivalent to that of the 1D QES systems with the sexticpotential, and observe the relation of the 2D holomorphicn-supersymmetry to the non-holomorphicN -fold supersymmetry [7]. In Section 7 the brief summary of the obtainedresults is presented and some open problems to be interesting for further investigation arediscussed.

2. Classical n-supersymmetry

The classical Hamiltonian of a charged spin-1/2 particle (−e = m = 1) with gyromag-netic ratiog moving in a plane and subjected to a magnetic fieldB(x) is given by

(2.1)H = 1

2P2 + gB(x)θ+θ−,

S.M. Klishevich, M.S. Plyushchay / Nuclear Physics B 616 [PM] (2001) 403–418 405

whereP = p+A(x), A(x) is a 2D gauge potential,B(x) = ∂1A2−∂2A1. The variablesxi ,pi , i = 1,2, and complex Grassman variablesθ±, (θ+)∗ = θ−, are canonically conjugatewith respect to the Poisson brackets,xi,pj PB = δij , θ−, θ+PB = −i. For even valuesof the gyromagnetic ratiog = 2n, n ∈ N, the system (2.1) is endowed with the nonlinearn-supersymmetry. In this case the Hamiltonian (2.1) takes the form

(2.2)Hn = 1

2Z+Z− + i

2nZ−,Z+

PBθ+θ−,

(2.3)Z± =P2 ∓ iP1,

which admits the existence of the odd integrals of motion

(2.4)Q±n = 2− n

2(Z∓)

nθ±

generating the nonlinearn-superalgebra [3]

(2.5)Q−

n ,Q+n

PB = −i(Hn)

n,Q±

n ,Hn

PB = 0.

This n-superalgebra does not depend on the explicit form of the even complex conjugatevariablesZ±. Therefore, in principle,Z± in generators (2.2) and (2.4) can be arbitraryfunctions of the bosonic dynamical variables of the system.

The nilpotent quantityN = θ+θ− is another obvious even integral of motion. When thegauge potentialA(x) is a 2D vector, the system (2.2) possesses the additional even integralof motionL= εij xipj . The integralsN andL generate theU(1) rotations of the odd,θ±,and even,Z±, variables, respectively. Their linear combination

(2.6)Jn = L+ nN

is in involution with the supercharges,Jn,Q±n PB = 0, and plays the role of the central

charge of the classicaln-superalgebra. As we shall see, at the quantum level the form of thenonlinearn-superalgebra (2.5) is modified generically by the appearance of the nontrivialcentral charge in the anticommutator of the supercharges.

3. Quantum n-supersymmetry: constant magnetic field

We start our investigation of the quantum two-dimensional nonlinear supersymmetrywith considering the simplest case of theconstant magnetic field. The quantumn-supersymmetric Hamiltonian for such a system is

(3.1)Hn = 1

4

Z+,Z− + n

2Bσ3,

whereZ± are the quantum analogues of the variables (2.3) withPi = −i∂i − 12εij xjB,

[P1,P2] = −iB, corresponding to the choice of the symmetric gauge. Here and inwhat follows we puth = 1. The quantum Hamiltonian (3.1) is related to the classicalanalogue (2.2) via the quantization prescriptionθ± = 1

2(σ1 ± iσ2), Z+Z− → 12Z+,Z−,

θ+θ− → 12[θ+, θ−] = 1

2σ3.


The system (3.1) has the integrals of motionPi = −i∂i + 12εij xjB, which are in

involution with Pi and satisfy the relation[P1, P2] = iB. In terms of the creation–annihilation operators

a± = 1√2|B| (P1 ± iεP2), b± = 1√

2|B|(P1 ∓ iεP2

),

[a−, a+] = 1, [b−, b+] = 1, ε = signB, the Hamiltonian (3.1) is represented in the form

Hn = |B|(a+a− + 1

2+ n

2εσ3

).

Since the Hamiltonian does not depend onb±, the energy levels of the system are infinitelydegenerate. Then-supersymmetry of the system (3.1) is generated by the supercharges

(3.2)Q±n = |B| n2

(a∓)nθ± for B > 0,(a±)nθ± for B < 0,

Q−

n ,Q+n

=(Hn + n− 1

2B

)(Hn + n− 3

2B

)· · ·

(Hn − n− 3

2B

)(Hn − n− 1

2B

),

[Q±

n ,Hn

] = 0.

Therefore, the present 2D system corresponds to then-supersymmetric 1D oscillator [3].Due to the axial symmetry of the system, the operator

(3.3)Jn = 1

BHn − εb+b−

is (up to an inessential additive constant) the quantum analogue of the classicalintegral (2.6). However, here the quantum central chargeJn of the n-superalgebra playsa secondary role since it is represented in terms ofHn and integralsb±.

As in the case of the one-dimensional theory [6], the attempt to generalize then-supersymmetry of the system (3.1) to the case of the magnetic field of general formfaces the problem of quantum anomaly. In the next section we show that the generalizationis nevertheless possible for the magnetic field of special form. Such a phenomenon has analgebraic foundation and is similar to that taking place in the 1D theory.

4. Quantum n-supersymmetry: general magnetic field

Here we investigate the general case of then-supersymmetry withholomorphicsupercharges for the 2D charged particle in an external magnetic field.

A priori, the quantization prescription that respects the classicaln-supersymmetry is notknown. We take the quantum Hamiltonian of the general form

(4.1)H = 1

2P2 + V (x)+L(x)N

with Pi = −i∂i +Ai(x), i = 1,2,N = θ+θ− = 12(σ3 + 1), and fix the unknown functions

from the condition of existence of then-supersymmetry. To begin with, we analyse the


n = 2 supersymmetry, and then will generalize the construction for arbitrary naturaln. Byanalogy with the one-dimensionaln = 2 supersymmetry, we consider the second-order oddholomorphic operators [6]

(4.2)Q±2 = 1

2

((Z∓)2 − q

)θ±,

whereq ∈ C andZ± are the quantum analogues of (2.3). These odd operators commutewith the Hamiltonian whenL(x)= 2B(x), V (x) = −B(x), and magnetic field obeys theequations

(4.3)(∂1∂2 + 2 Imq)B(x)= 0,(∂2

1 − ∂22 − 4 Req

)B(x) = 0.

Let us note that the expression (4.2) is the most general form of the second-orderholomorphic supercharges since the term linear inZ± is excluded from them by thecondition[H2,Q

±2 ] = 0.

The potentialV (x) has the pure quantum nature being proportional toh. Since theresulting Hamiltonian has the form

(4.4)H2 = 1

2P2 +B(x)σ3,

one can treatV (x) as a quantum correction term providing the same quantizationprescription as in the case of the constant magnetic field.

In the complex variables

(4.5)z = 1

2(x1 + ix2), z = 1

2(x1 − ix2),

Eqs. (4.3) can be rewritten equivalently as

(4.6)(∂2 −ω2)B(z, z)= 0,

whereB∗ = B, ω2 = 4q , and the notation∂ = ∂z (∂ = ∂z) is introduced. Below we shallsee that the holomorphic nonlinearn = 2 supersymmetry given by the supercharges (4.2)and Hamiltonian (4.4) with magnetic fieldB defined by Eq. (4.6) admits the generalizationfor the case of arbitraryn ∈ N with magnetic fieldB of the same structure.

The general solution to Eq. (4.6) is

(4.7)B(z, z)=w+eωz+ωz +w−e−(ωz+ωz) +weωz−ωz + we−(ωz−ωz),

wherew± ∈ R, w ∈ C, w = w∗. On the other hand, forω = 0 the solution to Eq. (4.6) isthe polynomial,

(4.8)B(x) = c((x1 − x10)

2 + (x2 − x20)2) + c0,

with c, c0, x10, x20 being some real constants.Though the latter solution can be obtained formally from (4.7) in the limitω → 0

by rescaling appropriately the parametersw±, w, the corresponding limit procedure issingular and the cases (4.7) and (4.8) have to be treated separately.


Fig. 1. Magnetic field with signBu(−∞) = signBu(+∞).

Rewriting the magnetic field (4.7) in terms of the real variablesx1,2, we have

B(x) =w+ exp(xω)+w− exp(−xω)+w exp(ix × ω)+ w exp(−ix × ω),

wherex × ω = εij xiωj , and we have introduced the constant two-dimensional vectorω =(Reω,− Imω). The vectorω defines the preferable coordinate system,

(4.9)u= 1

|ω| (ωz+ ωz)= xω

|ω| , v = − i

|ω| (ωz− ωz) = x × ω

|ω| ,

related to the initial one by the rotation. In the new coordinates the magnetic field isrepresented in the form

(4.10)B(u, v) = Bu +Bv,

where

Bu =w+e|ω|u +w−e−|ω|u, Bv =wei|ω|v + we−i|ω|v.

Thus, the magnetic field is hyperbolic in theu direction and periodic in thev direction.A typical example of the magnetic field with signBu(−∞)= signBu(+∞) is depicted onFig. 1.

For analysing the nonlinearn-supersymmetry for arbitraryn ∈ N, it is convenient tointroduce the complex oscillator-like operators

(4.11)Z = ∂ +W(z, z), Z = −∂ + W (z, z),

where the complex superpotential is defined by ReW = A2(x), ImW = A1(x). TheoperatorsZ, Z obey the relation

(4.12)[Z, Z

] = 2B(z, z).

Then-supersymmetric Hamiltonian has the form

(4.13)Hn = 1

4

Z,Z

+ n

4

[Z, Z

]σ3


generalizing (4.4) to the case of arbitraryn. Eq. (4.6) can be rewritten as the algebraicrelation

(4.14)[Z,

[Z,

[Z, Z

]]] = ω2[Z, Z],

which can be treated as an “integrability condition” of the nonlinear holomorphicsupersymmetry. Eqs. (4.13) and (4.14) allow us to prove algebraically by the mathematicalinduction that the supercharges defined by the relations

(4.15)Q+n+2 = 1

2

(Z2 −

(n+ 1

2

)2

ω2)Q+

n , Q+0 = θ+, Q+

1 = 2− 12Zθ+,

are preserved. This recurrent relation reproduces correctly the superchargesQ±2 con-

structed above.Since the conservation of the supercharges is proved algebraically, the operatorsZ, Z

can have any nature (the action ofZ, Z is supposed to be associative). For example, theycan have a matrix structure. With this observation the nonlinear supersymmetry can beapplied to the case of matrix Hamiltonians [20,29–32].

Thus, the introduction of the operatorsZ, Z allows us to reduce the two-dimensionalnonlinearn-supersymmetry to the pure algebraic construction.

5. Superalgebra for ω = 0

Unlike the case of linear supersymmetry, the form of nonlinear superalgebra generatedby the operatorsQ±

n andHn defined via relations (4.13)–(4.15) depends essentially on theconcrete representation of the operatorsZ, Z satisfying the relation (4.14). We will useonly the representation (4.11). In this case then-supersymmetric system (4.13) withω = 0has the central charge

(5.1)Jn = −1

4

(ω2Z2 + ω2Z2) + ∂BZ + ∂BZ −B2 + n

2∂∂Bσ3.

The anticommutator of the supercharges contains it for anyn > 1. For example, then =2,3,4 nonlinear superalgebras are

Q−

2 ,Q+2

=H 22 + 1

4J2 + |ω|4

64,

Q−

3 ,Q+3

=H 33 +H3J3 + 1

4|ω|4H3 + 2|ω|2(|w|2 −w+w−

),

(5.2)

Q−

4 ,Q+4

=H 44 + 5

2H 2

4J4 + 41

25|ω|4H 2

4 + 9

24J 2

4 + 12|ω|2(|w|2 −w+w−)H4

+ 45

27 |ω|4J4 + 33

212|ω|8 − 9

2|ω|4(|w|2 +w+w−

).

Let us discuss the eigenvalue problem for the Hamiltonian (4.13) with complexsuperpotentialW(z, z) (see the definition (4.11)). The superpotential corresponding to the


magnetic field (4.7) is

W(z, z)= 1

ω

(w+eωz+ωz −w−e−(ωz+ωz) −weωz−ωz + we−(ωz−ωz)

)

+ f (z)+ i

z∫F(z, ζ ) dζ ,

whereF(z, ζ )∗ = F(ζ, z) andf (z) is a holomorphic function. These arbitrary functionsare associated with the gauge freedom of the system.

In general, the zero modes of the superchargeQ+n can be found. For the sake of

simplicity we consider the casen = 2 with the following zero modes ofQ+2 :

(5.3)ψ =(c+(z)e

12ωz + c−(z)e− 1

2ωz)e∫ z

f (ζ ) dζ−∫ zW(ζ,z) dζ ,

wheref (z)= f (z)∗. Now, let us look for the eigenfunctions of the Hamiltonian associatedwith the zero modes. Substitution of (5.3) into the corresponding stationary Schrödingerequation gives the following coupled equations forc±(z):

4(w−e−ωz + weωz

)c+(z)+ 4Ec−(z)−ωc−′(z) = 0,

4(w+eωz +we−ωz

)c−(z)+ 4Ec+(z)+ωc+′(z) = 0.

This system of differential equations can be reduced to the Riccati equation for the functiony(z)= c+(z)/c−(z):

(5.4)ωy ′ + 4(e−ωzw− + eωzw

)y2 + 8Ey + 4

(eωzw+ + e−ωzw

) = 0.

Hence, the holomorphic supersymmetry allows ones to reduce the two-dimensionalspectral problem associated with the zero modes to the one-dimensional differentialequation of the first order. Unfortunately, we have not succeeded in finding of the generalsolution to Eq. (5.4). Therefore, in what follows we consider some special cases of theexponential magnetic field.

In the preferable coordinate system (4.9) the Hamiltonian takes the form

(5.5)Hn = 1

2P2u + 1

2P2v + n

2B(u, v)σ3,

wherePu = − i∂u +Au(u, v), Pv = − i∂v +Av(u, v). For the magnetic field (4.10) thegauge potential can be chosen in the form

(5.6)Au(u, v) = 1

|ω|2B′v, Av(u, v) = 1

|ω|2B′u.

Let us consider the system (5.5) with thereduced magnetic field:Bv = 0 (w = 0). Thenthe gauge (5.6) is asymmetric and the central charge takes the form

Jn = −1

4|ω|2p2

v + |ω|2Hn − 4w+w−,

wherepv = −i∂v. Therefore, in this case instead ofJn, the integralpv can be consideredas independent central charge. It generates translations in thev-direction. Then, e.g., in the


casen = 2 the superalgebra is reduced to the form

Q−2 ,Q

+2 =

(H2 + |ω|2

8

)2

− 1

16|ω|2p2

v −w+w−.

In the gauge (5.6) the Hamiltonian can be written as

(5.7)Hn = −1

2∂2u + 1

2

(|ω|−2B ′u − i∂v

)2 + n

2Buσ3.

The coordinatev is cyclic. Representing the wave functions in the factorised formψ(u,v) = eivpvψ(u), we reduce the Hamiltonian (5.7) to the one-dimensional QESHamiltonian acting on the functionsψ(u):

(5.8)Hn = −1

2∂2u + 1

2W(u)2 + n

2W ′(u)σ3,

where

(5.9)W(u) = 1

|ω|(w+e|ω|u −w−e−|ω|u) + pv.

Hence, the 2Dn-supersymmetric system with thereduced magnetic field is equivalentto the 1Dn-supersymmetric system. In particular, using the results of Ref. [6] on theQES nature of then-supersymmetric system (5.8) with the superpotential (5.9), one cancalculate explicitlyn “Landau levels” and find the corresponding wave functions in thedescribed reduced case. One has also to note that, on the other hand, for some choiceof the parameters of the superpotential (5.9), the well-known exactly solvable systemwith the Morse potential can be reproduced [6]. Hence, in this case one can find all thecorresponding eigenstates and eigenvalues. The term “Landau levels” is justified hereby the analogy with the case of the constant magnetic field in which the Hamiltonianeigenstates are bounded only in one of two directions corresponding to the continuousvariables.

The reduced magnetic field withBu = 0 (w± = 0) can be considered exactly in thesame way. In this case the resulting 1Dn-supersymmetric system is characterized bythe trigonometric superpotential. Ifv ∈ R, the correspondingn wave functions whichcan be found algebraically are not normalizable. The normalizability can be achieved byconsideringv ∈ [0, 2π

|ω|k], k ∈ N, i.e. by identifying the initial configuration space as acylinder. However, the detailed consideration of such a problem lies out of the scope of thepresent paper.

6. Polynomial magnetic field and x6 + ··· family of quasi-exactly solvable potentials

Let us turn now to the case of the polynomial magnetic field. The operator

(6.1)Jn = 1

4c

(∂B(z, z)Z + ∂B(z, z)Z −B2(z, z)+ n

2∂∂B(z, z)σ3

)

is the integral of motion of the system (4.13) with the magnetic field (4.8). It canbe obtained from the operatorJn (5.1) in the limit ω → 0 via the same rescaling of


the parameters of the exponential magnetic field which transforms (4.7) into (4.8). Theessential feature of this integral is its linearity in derivatives.

The polynomial magnetic field (4.8) is invariant under rotations about the point(x10, x20). Therefore, one can expect that the operator (6.1) should be related to a generatorof the axial symmetry. To use the benefit of this symmetry, we pass over to the polarcoordinate system with the center at the point(x10, x20). Then the magnetic field is radial,

(6.2)B(r) = cr2 + c0,

and the Hamiltonian reads as

(6.3)Hn = −1

2

(D2

r + r−2D2ϕ + r−1Dr

) + n

2B(r)σ3.

HereDr = ∂r + iAr(r, ϕ), Dϕ = ∂ϕ + iAϕ(r,ϕ), the magnetic field is given by

(6.4)B = r−1(∂rAϕ − ∂ϕAr),

and the supercharges have the simple structure (cf. with Eq. (3.2)):

(6.5)Q+n = 2− n

2Znθ+, Q−n = 2− n

2 Znθ−.

As in the caseω = 0, the anticommutator of the supercharges (6.5) is a polynomial of thenth degree inHn, Q−

n ,Q+n = Hn

n + P(Hn,Jn), whereP(Hn,Jn) denotes a polynomialof the(n− 1)th degree. For example, forn = 2,3,4 we have

Q−2 ,Q

+2

=H 22 + cJ2,

Q−3 ,Q

+3

=H 33 + 4cH3J3 − 2c0c,

(6.6)Q−

4 ,Q+4

=H 44 + 10cH 2

4J4 + 9c2J 24 − 12c0cH4 − 9c2.

These expressions can be obtained from (5.2) via the limiting procedure discussed above.For the radial magnetic field it is convenient to use the asymmetric gauge

(6.7)Aϕ = 1

4cr4 + 1

2c0r

2, Ar = 0.

One could add an additive constant toAϕ since this does not affect the magnetic field (6.4).But such a constant would lead to a singular gauge potential in the Cartesian coordinatesowing to the singular at the coordinate origin nature of the polar system. By the same reasonthe constant cannot be removed by a gauge transformation. Therefore it has to vanish.

In the gauge (6.7), the Hamiltonian (6.3) is simplified:

(6.8)Hn = −1

2

(∂2r + r−1∂r − r−2(A2

ϕ(r)− 2iAϕ(r)∂ϕ − ∂2ϕ

)) + n

2B(r)σ3.

The angle variableϕ is cyclic and the eigenfunctions of (6.8) can be represented as

(6.9)Ψ (r,ϕ)=(eim

′ϕχm′(r)eimϕψm(r)

), m, m′ ∈ N.

In the gauge (6.7) the integralJn takes the form

Jn = −i∂ϕ − c20

4c+ n

2σ3.


Up to a constant, this integral is equal to (3.3). Moreover, the system with the constantmagnetic field is recovered in the limitc → 0. In this casecJn → −c2

0/4 that means thatJn disappears from (6.6) recovering the corresponding superalgebra for the system withthe constant field.

The simultaneous eigenstates of the operatorsHn andJn have the structure

(6.10)Ψm(r,ϕ)=(ei(m−n)ϕχm(r)

eimϕψm(r)

)

and satisfy the equation

(6.11)JnΨm(r,ϕ)=(m− n

2− c2

0

4c

)Ψm(r,ϕ).

Thus, the integralJn is associated with the axial symmetry of the system underconsideration and is (up to an additive constant) the exact quantum analogue of (2.6).

Since the angle variableϕ is cyclic, the 2D Hamiltonian (6.8) can be reduced to the1D Hamiltonian. The kinetic term of the Hamiltonian (6.8) is Hermitian with respect tothe scalar product with the measuredµ = r dr dϕ. In order to obtain a one-dimensionalsystem with the usual scalar product defined by the measuredµ= dr, one has to performthe similarity transformation

(6.12)Hn → UHnU−1, Ψ →UΨ, U = √

r.

Since the system obtained after such a transformation is originated from the two-dimensional system, one should always keep in mind that the variabler belongs to thehalf-line,r ∈ [0,∞).

In what follows we refer to the Hamiltonian acting on the lower component of thestate (6.10) as the bosonic Hamiltonian and to that acting on the upper component as thefermionic one. They form then-supersymmetric system.

After transformation (6.12), the reduced bosonic one-dimensional Hamiltonian is

H(2)n = −1

2

d2

dr2 + c2

32r6 + c0c

8r4 + 1

8

(c2

0 − 2c(2n−m))r2

(6.13)+ m2 − 14

2r2 − 1

2(n−m)c0.

This Hamiltonian gives (forc > 0) the well-known family of the quasi-exactly solvablesystems [17,18,20,21]. According to the general theory of 1D QES systems, they arecharacterized by the weightj that defines the corresponding finite-dimensional non-unitaryrepresentation of the algebrasl(2,R). In the case (6.13) the integer parametern is relatedto the corresponding weightj asn= 2j + 1.

The superpartnerH(1)n can be obtained fromH(2)

n by the substitutionn → −n, m →m − n. The supersymmetric pair of the 2D Hamiltonians is related to the correspondingpair of 1D Hamiltonians as

(6.14)e−i(m−n)ϕUH(1)n U−1ei(m−n)ϕ =H(1)

n , e−imϕUH(2)n U−1eimϕ =H(2)

n .

Here we imply that the operator∂ϕ on l.h.s. acts according to the rule∂ϕeikϕ = eikϕik.


The supercharges (6.5) are non-diagonal inϕ since in the gauge (6.7) the operatorsZ,Z have the form

Z = e−iϕ

(∂r + 1

r(Aϕ(r)− i∂ϕ)

), Z = eiϕ

(−∂r + 1

r(Aϕ(r)− i∂ϕ)

).

The operatorZ decreases the angular momentum of the state in 1 whileZ increases it. Dueto this property the supercharges (6.5) perform the proper mixing of the upper and lowerstates (6.10). The reduction of the corresponding 2D differential operators to 1D looks like

(6.15)e−i(m−n)ϕUZnU−1eimϕ = Zn, e−imϕUZnU−1ei(m−n)ϕ =Z†n,

with Zn given by

Zn =(A− n− 1

r

)(A− n− 2

r

)· · ·A,

whereA = ddr

+W(r) and the superpotential is

(6.16)W(r) = 1

4cr3 + 1

2c0r + m− 1

2

r.

Using the relations (6.14), (6.15) and[Q±n ,Hn] = 0, one can obtain the one-dimensional

intertwining relations:

ZnH(2)n =H(1)

n Zn, Z†nH(1)

n =H(2)n Z†

n.

Hence, the one-dimensional odd operators

(6.17)Q+n = 2− n

2Znθ+, Q−

n = (Q+

n

)†

are the true supercharges of the 1Dn-supersymmetric system,

(6.18)[Q±

n ,Hn

] = 0, whereHn =(H(1)

n 00 H(2)

n

).

The form of the anticommutator of the supercharges can be obtained from the corre-

sponding two-dimensional case via the formal substitutionJn → m− n2 − c2

04c . In the one-

dimensionaln-supersymmetric system (6.13), (6.17) the integer parameterm can be for-mally extended to the whole real line (m ∈ R). A similar prescription is used when thetwo-particle Calogero model is treated as that appearing from the reduction of the 3D os-cillator [15]. Note that here the Calogero model can also be obtained from (6.13) in thecasec = 0 corresponding toB = const. From the point of view of the 1D quasi-exactlysolvable systems, the Hamiltonian (6.13) withc > 0 andm< 1

2 hasn bound states whichcan be found algebraically. But from the viewpoint of the nonlinear supersymmetry, thesen algebraic states are the zero modes of the odd operatorQ+

n . The factorised form of thesupercharges allows us to find the explicit form of the zero modes:

(6.19)ψm(r)= Pn−1(r2)r

12−m exp

(− c

16r4 − c0

4r2

),

wherePn is a polynomial of thenth degree non-vanishing at zero. The same form for thealgebraic states is given by thesl(2,R) partial algebraization scheme [17,18]. Substituting


the combination (6.19) into the corresponding stationary Schrödinger equation, thealgebraic system of equations for energy and coefficients inPn−1(r

2) can be obtained.On the other hand, the energies of then levels are the roots of the polynomial that theanticommutator of the supercharges is proportional to.

Form 12 the superchargeQ+

n has no zero modes, and hence, the 1Dn-supersymmetryis spontaneously broken.

It is necessary to stress that form2 = 14 the Hamiltonian (6.13) has the non-singular

sextic potential and hence, in principle the system can be treated on the whole line,r ∈ R. The approach based on the finite-dimensional representations of the algebrasl(2,R)

allows ones to find exactlyn even bound states form = 12 and n odd states form =

−12. But this approach gives no explanation why the intermediate states of the opposite

parity are omitted. Having in mind the tight relationship between the Lee-algebraicapproach and the holomorphicn-supersymmetry (see also [6] for the details), one cansay that the explanation lies in originating the system with the sextic potential from thetwo-dimensionaln-supersymmetric system. Here it is necessary to note that even forthe Hamiltonian (6.13) with the non-singular potential (atm2 = 1

4) the correspondingsupercharges are singular at zero.

It is worth emphasizing that in the initial 2D supersymmetric system the superchargesare holomorphic whereas in the reduced 1D system they have a non-holomorphic form.The reduced 1D supersymmetric system belongs to the so-called generalizedN -foldsupersymmetry of type A [7], being a generalization of the one-dimensional holomorphicsupersymmetry [6].

We have obtained an interesting picture for the case of the polynomial magnetic field.The two-dimensional system with the nonlinearn-supersymmetry described in terms of theholomorphic supercharges (6.5) corresponds to an infinite number of the one-dimensionalsupersymmetric systems (6.18) with the non-holomorphic supercharges: for every (integer)m satisfying the relationm < 1, the 1Dn-supersymmetry is exact, while form 1 it isspontaneously broken. On the other hand, the 2D holomorphicn-supersymmetry is exact.It is characterized by the infinite-dimensional subspace of zero modes. However, there isessential difference between the casesn = 1 andn 2. For the linear supersymmetry thebasic relationQ−

1 ,Q+1 = H1 means that all the zero modes have zero energy, and so,

the ground state of such 2D system is infinitely degenerate inm. On the contrary, for thenonlinear supersymmetry withn 2 the corresponding zero modes are non-degenerate inthe energy by virtue of the nontrivial presence of the central chargeJn in the superalgebra.

7. Discussion and outlook

To conclude, let us summarize and discuss the obtained results and indicate someproblems that deserve further attention.

• The holomorphicn-supersymmetry of the 2D system describing the motion of acharged spin-1/2 particle in an external magnetic field provides the gyromagneticratiog = 2n both at the classical and quantum levels.


This is a natural generalization of the well-known restriction on the value of gyromagneticratio (g = 2) related to the linear supersymmetry [15]. Here it is necessary to havein mind that saying about the spin-1/2 particle with gyromagnetic ratiog = 2n, weproceed from the structure of then-supersymmetric 2D Hamiltonian given by Eqs. (4.13),(4.12). However, the complete (3D) spin structure does not appear in the construction,and hence, the 2D system (4.13), (4.12) could also be interpreted, e.g., as the spin-n/2,g = 2 particle with separated polarizationssz = ±n/2. Such alternative interpretationis in correspondence with the relationship between the nonlinear supersymmetry andparasupersymmetry discussed in Ref. [3].

• The algebraic foundation of the holomorphicn-supersymmetry is ascertained.

The Hamiltonian (4.13) and the supercharges (4.15) of any holomorphic supersymmetricsystem are defined in terms of the operatorsZ andZ only. The “integrability condition”(4.14) arises at the quantum level and guarantees the conservation of the supercharges ina pure algebraic way. Thus, the formulation of the nonlinear holomorphic supersymmetrydoes not depend on representation of the operatorsZ andZ. In this sense the holomorphicn-supersymmetry can be treated as a direct algebraic extension of the usual linearsupersymmetry. The one-dimensional representation of the operatorsZ, Z was explored inRef. [6] while here we have investigated the realization of the holomorphic supersymmetryin the systems on the plane. We have found that there is an essential difference of the two-dimensional realization from the one-dimensional case.

• In the 2D systems the additional integral of motionJn has been found (seeexpressions (5.1) and (6.1)). This integral is a central charge that enters non-triviallyinto the nonlinear superalgebra (see Eqs. (5.2) and (6.6)).

Technically, the holomorphicn-supersymmetry facilitates finding the form of thisadditional integral. This is the important point since, in general, such a problem is ratherlaborious.

For the 2D systems the “integrability condition” (4.14) is represented as the differentialequation (4.6) for the magnetic field. The general solution has the exponential form (4.7)for ω = 0, or the polynomial form (4.8) forω = 0. The latter solution can be formallyobtained from the former in the limitω → 0. In the exponential case the magnetic field hasthe orthogonal hyperbolic and trigonometric directions. The oscillating behaviour of thefield in one direction means that the eigenstates of the Hamiltonian are not normalizable.This situation is similar to the case of the constant magnetic field. However, for suchexponential configuration of the magnetic field we have not succeeded in finding the energylevels of the Hamiltonian associated with the zero modes of the corresponding supercharge.

• The systems with the magnetic field of the pure hyperbolic or pure trigonometric formhave been reduced to the one-dimensional problems with the nonlinear holomorphicsupersymmetry [6].

In Ref. [15] a similar reduction of the 2D system with linear supersymmetry wasconsidered in the context of application of the shape-invariant potentials to the 2D spectralproblem.


• The n-supersymmetric system with the polynomial magnetic field (4.8) has beenreduced to the well-known family of one-dimensional QES systems with the sexticpotential.

This reduction confirms the intimate relationship between the nonlinear holomorphicsupersymmetry and the QES systems observed in Ref. [6]. Moreover, it reveals thenontrivial relation of the holomorphicn-supersymmetry to the non-holomorphicN -foldsupersymmetry discussed by Aoyama et al. [7].1 It is interesting to note that fromthe point of view of the reduction, the two-dimensional holomorphic supersymmetricsystem contains the infinite set of one-dimensional systems with the non-holomorphicsupersymmetry in the exact and spontaneously broken phases. Having in mind the observedrelationship between the 2D holomorphic and 1D non-holomorphic supersymmetries, itwould be reasonable to clarify the following question:Is it possible to treat the 1Dnon-holomorphic N -fold supersymmetry of the general form [7] as a reduction of someholomorphic n-supersymmetry realized in the 2D Riemannian geometry?

The underlying algebraic structure of the nonlinear holomorphic supersymmetry allowsones to apply it for investigation of the wide class of quantum mechanical systemsincluding the models described by the matrix Hamiltonians and the models on a non-commutative space. In Refs. [20,29–32] the matrix Hamiltonians were considered in thecontext of the QES systems. Therefore it would be interesting to investigate the possiblerelation of the matrix realization of the holomorphic supersymmetry to such systems. It isworth noting that the matrix extension of the two-dimensional system (4.1) correspondsto the non-relativistic particle in an external non-Abelian gauge field. In the case of themodels on the non-commutative space [33,34], the action of quantum mechanical operatorsis associative that, in principle, is enough for realizing the holomorphicn-supersymmetry(4.13)–(4.15).

At present time, the great attention is attracted by the so-calledPT -invariant systems[22–28] described by the non-Hermitian Hamiltonians with a real spectrum. In Refs. [35,36], an extension of the notion of usual supersymmetry was proposed for such systems.The QES systems have also found an application to this subject. Since the discussedholomorphicn-supersymmetry inherits the properties of the supersymmetric and QESsystems, it would be interesting to extend the construction to the case of thePT -invariantsystems.

Acknowledgements

The work was supported by the grants 1010073 and 3000006 from FONDECYT (Chile)and by DYCIT (USACH).

1 The relation of the non-holomorphicN -fold supersymmetry [7] with the superpotential (6.16) to the familyof QES systems with the sextic potential was also noted in Ref. [25].


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Fractional helicity, Lorentz symmetry breaking,compactification and anyons

S.M. Klishevicha,b, M.S. Plyushchaya,b, M. Rausch de Traubenbergc,d

a Departamento de Física, Universidad de Santiago de Chile, Casilla 307, Santiago 2, Chileb Institute for High Energy Physics, Protvino, Russia

c Laboratoire de Physique Mathématique, Université Montpellier II, Place E. Bataillon,34095 Montpellier, France

d Laboratoire de Physique Théorique, 3 rue de l’Université, 67084 Strasbourg, France

Received 25 June 2001; accepted 31 August 2001

Abstract

We construct the covariant, spinor sets of relativistic wave equations for a massless field onthe basis of the two copies of the R-deformed Heisenberg algebra. For the finite-dimensionalrepresentations of the algebra they give a universal description of the states with integer and half-integer helicity. The infinite-dimensional representations correspond formally to the massless stateswith fractional (real) helicity. The solutions of the latter type, however, break down the(3+ 1)DPoincaré invariance to the(2+ 1)D Poincaré invariance, and via a compactification on a circle aconsistent theory for massive anyons ind = 2+ 1 is produced. A general analysis of the “helicityequation” shows that the(3+ 1)D Poincaré group has no massless irreducible representations withthe trivial non-compact part of the little group constructed on the basis of the infinite-dimensionalrepresentations ofsl(2,C). This result is in contrast with the massive case where integer and half-integer spin states can be described on the basis of such representations, and means, in particular,that the(3+ 1)D Dirac positive energy covariant equations have no massless limit. 2001 ElsevierScience B.V. All rights reserved.

PACS:03.65.Pm; 11.30.Cp; 11.10.Kk; 11.30.Qc

1. Introduction

It is generally accepted that the(2 + 1)-dimensional space–time reveals specificcharacteristics which are no more valid in higher dimensions. For instance, only in(2+1)Dthe states of arbitrary spinλ ∈ R and of corresponding anyonic statistics [1,2] different

E-mail addresses:[email protected] (S.M. Klishevich), [email protected] (M.S. Plyushchay),[email protected] (M. Rausch de Traubenberg).



420 S.M. Klishevich et al. / Nuclear Physics B 616 [PM] (2001) 419–436

from the bosonic and fermionic ones do exist.1 There are several ways to introduceanyons in(2 + 1)D including the Chern–Simons [8–10] and the group-theoretical [11–15] approaches. In the latter, noticing that theSO(2,1) group is infinitely connected,the massive anyons of spinλ ∈ R are realized on the infinite-dimensional half-boundedrepresentations ofSL(2,R) [16], the universal covering group ofSO(2,1). Within thatapproach, in particular, the anyons can be described by the covariant vector [17] andspinor [18,19] sets of linear differential equations for infinite-component fields.

The little group for the(2+ 1)D massive anyon isSO(2) and it carries the same numberof physical degrees of freedom as a(2+1)D massive scalar field [20]. This specific featureis also valid for the massless states in(3+ 1)D. Indeed, in(3+ 1)D the little group inthe massless case isE(2), the group of rotations and translations in the 2D Euclideanspace. Representing its non-compact part by zero, we reduceE(2) to SO(2). With thisobservation at hands, it was shown that some(2+ 1)D models for anyons can be obtainedfrom the(3+ 1)D models of massless particles via the appropriate formal reduction [21].Further, since, unlike the massive(3+ 1)D case, the algebra of the little group gives noquantization restrictions, it seems that there is no strong reason for excluding the possibilityto consider massless states with fractional (arbitrary real) helicity [22]. Following this lineof reasoning, some time ago it was claimed that, in fact, a massless analogue of the originalDirac spinor set of equations describing a massive spin-0 field in(3+ 1)D [23,24] givesrise to the(3+ 1)D massless states of helicity±1/4 called “quartions” [25,26]. Besides,it was noted that the non-covariant formal quantization of the massless superparticlepreserving its classicalP -invariance should result in the supermultiplet with helicitystructure (−1/4,+1/4) [27]. On the other hand, there exist the topological arguments(see, e.g., [28,29]) related to the two-connectedness ofSO(3,1) which restrict the helicityof massless representations to integer and half-integer values. But then,appealing to thewell-known relations betweenD-dimensional massive and(D + 1)-dimensional masslessrepresentations, one can ask what corresponds to the(2+ 1)D massive representationswith fractional spin. Formally, it could be the(3+1)D massless irreducible representationswith fractional helicity constructed on the basis of infinite-dimensional representations ofsl(2,C). But since, due to the topological reasons, they cannot exist, what kind of defectshave to appear in such a theory and how the(2+ 1)D massive anyons emerge from thecorresponding(3+ 1)D massless theory?

In this paper we address in detail the problem of the description of fractional helicitymassless fields in(3+ 1)D on the basis of the infinite-dimensional representations of thesl(2,C) Lie algebra realized in terms of the two copies of theR-deformed Heisenbergalgebra (RDHA) [7,30,31]. The RDHA was first introduced by Yang [32] in the contextof Wigner generalized quantization schemes [33] underlying the concept of parafields andparastatistics [7] (in this context, see also Ref. [34]). This algebra and its generalizationsis nowadays exploited extensively in the mathematical and physical literature in different

1 Strictly speaking, other generalizations of statistics called parafermions and parabosons exist in any space–time dimension [3–7], but via the so-called Green anzatz they can be represented in terms of ordinary bosons andfermions.

S.M. Klishevich et al. / Nuclear Physics B 616 [PM] (2001) 419–436 421

aspects (see Refs. [18,19,35–37]). It should be emphasized that the infinite-dimensionalrepresentations ofsl(2,C) we consider have nothing to do with those representationscorresponding to the little groupE(2) with a non-trivial non-compact part, to which areusually referred as to representations with “continuous” [38] or “infinite” [28] spin.

We observe here that the corresponding irreducible infinite-dimensional representationsof sl(2,C) cannot be “exponentiated” to representations of theSL(2,C) Lie group in themassless case. In other words, the fractional helicity representation of the little groupSO(2)cannot be promoted to a representation of the(3+ 1)D Lorentz group being a subgroup ofthe corresponding massless representations of the(3 + 1)-dimensional Poincaré group.This is reflected in breaking of the Lorentz invariance at the level of solutions of thecovariant spinor set of equations for fractional helicity massless fields. The symmetrybreaking corresponds to the violation of the invariance with respect to the rotations in twodirections and to the boosts in one direction. Consequently, the Lorentz groupSL(2,C) isbroken down toSL(2,R), and via a compactification and subsequent dimensional reductionfrom (3+ 1)D to (2+ 1)D a consistent theory for massive relativistic anyons ind = 2+ 1is produced. At the same time, we show that the(3+ 1)D Poincaré group has no masslessirreducible representations characterized by the trivial non-compact part of the little groupand which would be constructed on the basis of the infinite-dimensional representations ofsl(2,C). This results in the same restriction for helicity but not of a topological origin.

The paper is organized as follows. In Section 2 we realize representation ofsl(2,C)

in terms of the two copies of the R-deformed Heisenberg algebra. The spinor set ofrelativistic equations based on this representation is considered in Section 3, where weshow that for the finite-dimensional representations of RDHA such equations universallydescribe massless states with any integer and half-integer helicity. The infinite-dimensionalrepresentations correspond formally to the states with fractional helicities. We demonstratethat such solutions, however, break down the Poincaré invariance. In Section 4 we showthat the compactification of the initial massless equations and subsequent reduction to(2+1)-dimensional space result in the consistent theory of massive anyons. The absence ofthe massless infinite-dimensional representations of the(3+1)D Poincaré group is provedin Section 5 in a generic case, independently on the concrete form of equations. Section 6is devoted to a brief discussion of the obtained results. Appendix A reviews briefly theDirac equations for massive spinless positive energy states.

2. RDHA and sl(2,C): generalization of the Schwinger construction

The first (3+ 1)D relativistic equation, due to which the infinite-dimensional unitaryrepresentations ofSL(2,C) were discovered, is the Majorana equation [39]. Its solutions,however, describereducible representations ofISO(3,1) characterized by the positiveenergy in the massive sectorp2 < 0. At the beginning of 70s Dirac [23,24] (see also [40])proposed a covariant spinor set of equations (see Appendix A) from which the Majoranaand Klein–Gordon equations appear in the form of integrability conditions. As a result, theDirac spinor set of equations possesses a massive spin-0 positive energy solutions, whereas


its vector modification considered by Staunton [40] describes massive spin-1/2 states.2 Weare interested in analysing the possibility of constructing relativistic wave equations for amassless field carrying fractional helicity. For the purpose, the fields related to the infinite-dimensional representations ofsl(2,C) will be considered.

2.1. Fock representations of the R-deformed algebra

Having in mind the analogy with the(2+ 1)D case of anyons, it is convenient to usethe infinite-dimensional representations ofsl(2,C) realized by means of the two copies ofRDHA [19,30–32] with mutually commuting generators:3[

a−, a+]=Π + νR,

R,a±

= 0, R2 =Π,

(2.1)[a−, a+

]= Π + νR,R, a±

= 0, R 2 = Π.

The operatorsa± will represent internal (spin) degrees of freedom. They generalize thetwo oscillator degrees of freedom (withν = 0) used by Dirac [23,24].

In the case of a direct sum of representations of the algebras with which we begin ouranalysis,Π and Π are the projectors on the corresponding subspaces that in a matrixrealization is reflected by the relationsΠ = 1

2(1 + σ3), Π = 12(1 − σ3). The operators

R, R have the sense of reflection operators for the internal variables andν ∈ R is thedeformation parameter. As it was mentioned above, the RDHA and its representationswere studied extensively in the literature. Here we will mainly refer to [31], where auniversality of the RDHA was observed: whenν =−(2k + 1), k ∈ N, its representationsare finite-dimensional (parafermion-like), and are infinite-dimensional if not (being unitaryparaboson-like forν >−1 [30]). The choiceν = 0 with a direct product of representationsof the two algebras was used in the Dirac [23,24] and Staunton [40] sets of equations.

For the sake of clarity and self-contained presentation, we recall briefly the constructionof representations of the algebra. The infinite-dimensional representations are built fromthe primitive vectors|0〉 and|0〉 (vacua), annihilated bya− anda−, respectively,

(2.2)Rν =|n〉 =

(a+

)n√|[n]ν !| |0〉, n ∈ N

, Rν =

|n〉 =

(a+

)n√|[n]ν !| |0〉, n ∈ N

,

with [n]ν ! = [n]ν[n − 1]ν · · · [1]ν , [0]ν ! = 1, [n]ν = n + 12(1 − (−1)n)ν. The finite-

dimensional representations of RDHA withν = −(2k + 1), k ∈ N, are built in the samevein (2.2) but in this case there is another primitive vector|2k〉 annihilated bya+, a+|2k〉 =0 [31].

Let us consider the quadratic operators

J± = 1

2

(a±

)2, J0 = 1

4

a+, a−

, J± = 1

2

(a±

)2, J0 = 1

4

a+, a−

,

2 More details on infinite component relativistic equations and correspondingSL(2,C) representations can befound in Ref. [41,42].

3 The necessary infinite-dimensional half-bounded representations ofsl(2,C) [43,44] can be realizedalternatively in terms of homogeneous monomials [45] but the RDHA construction is more convenient for ourpurposes.


forming the two copies ofsl(2,R) algebra,[J0,J±

]=±J±,[J−,J+

]= 2J0,

whereJ = J or J , and[J, J ] = 0. As a result, irreducible representationsRν and Rν

(2.2) are decomposed into the direct sums of the irreducible representations ofsl(2,R)

bounded from below as [31]

(2.3)Rν =D+1+ν

4⊕D+

3+ν4, Rν = D+

1+ν4

⊕ D+3+ν

4,

wherej0 = κ + l, l = 0,1, . . . , are the eigenvalues ofJ0 with κ = 1+ν4 and κ = 3+ν

4 ,respectively. The “left”- and “right”-handed parts of the generators of the(3+1)D Lorentzalgebrasl(2,C), Ki andKi , i = 1,2,3, obeying the relations[

Ki ,Kj

]= iεijkKk, [K, K ] = 0,

with K=K (K), are defined in terms of theso(3,1) generatorsJµν as follows:

Ki = 1

2εijkJjk + iJ0i, Ki = 1

2εijkJjk − iJ0i .

Then thesl(2,C) generatorsKi can be identified with thesl(2,R) generatorsJ0, J± =J1 ± iJ2 by means of the relations

(2.4)J0 =−K2, J1 =−iK1, J2 =−iK3.

Such identification corresponds to the concrete choice of theγ -matrices in covariant spinorformalism (see below). Any other possible identifications betweenJ andK, e.g., thoseobtained from (2.4) by a cyclic permutations ofKi , lead to other realizations of theγ -matrices related by unitary transformations.

Note here that the Fock spaces of the usual oscillators corresponding toν = 0 aredecomposed into the spin-1/4 and spin-3/4 representations ofsl(2,R) [46]. Anotherrealization ofsl(2,R) generators,J± = 1

2(a∓)2, J0 = −1

4a+, a−, results in the directsum of bounded from above infinite-dimensional representations,Rν =D−

1+ν4

⊕D−3+ν

4.

Thus, we have generalized the Schwinger construction ofsu(2) to the case of thesl(2,C)

algebra.

2.2. Covariant spinor formalism

Introducing sl(2,C) notations of dotted and undotted indices for two-dimensionalspinors, all can be rewritten in covariant notations. The spinor conventions to raise/lowerindices are as follow [47]:ψα = εαβψ

β , ψα = εαβψβ , ψα = εαβ ψβ , ψα = εαβ ψβ with

(ψα)∗ = ψα , ε12= ε12 =−1,ε12= ε12 = 1. Defining the two spinor operatorsLα andLα ,

L1 = 1√2

(a+ + a−

), L2 = i√

2

(a+ − a−

),

(2.5)L1 =1√2

(a+ + a−

), L2 =

i√2

(a+ − a−

),


a direct calculation shows that they generate theosp(4|1) superalgebra. Its bosonic part issp(4,R)∼ so(3,2)= AdS4 with generators

(2.6)Lαβ = 1

4Lα,Lβ , Lαβ = 1

4Lα,Lβ, Mαα = 1

4Lα,Lα,

but we shall be interested only in itsso(3,1) part. Let us introduce the 4D Dirac matricesin the Weyl representation,

(2.7)γµ =(

0 σµ

σµ 0

),

with

(2.8)σµαα = (1, σi), σµαα = (1,−σi).

Then theso(3,1) spinor generators

γµν = i

4[γµ, γν] = diag

(iσµν α

β, iσµναβ

)allow us to present theso(3,1) generators in the form

(2.9)Jµν = 1

2

(Lασµν α

βLβ − Lα σµναβ L

β).

The Pauli–Lubanski pseudo-vector is given by

(2.10)Wµ = 1

2εµνρσPνJρσ

with Pµ being the generator of space–time translations.

2.3. Invariant scalar product

For any physically admissible representation ofISO(3,1), the generators of the Lorentzgroup should be Hermitian. This means that for any such a representation we have toconstruct an invariant scalar product with the necessary properties. In what follows wewill discuss mainly the representationD+

λ ⊕ D+λ . Therefore, we consider in detail the

construction of the invariant scalar product for this representation only. Though for freemassless states the left- and right-handed sectors are uncoupled, the both are needed forthe construction of the invariant scalar product [48]. We consider the vectors living on thereducible representation space,Ψ ∈ Rν ⊕ Rν , i.e., Ψ = |ψ〉 + |χ〉 with |ψ〉 ∈ Rν and|χ〉 ∈ Rν .

The representations of (2.1) possess the natural involution(a±

)+ = a∓, |0〉+ = 〈0|, (a±

)+ = a∓, |0〉+ = 〈0|.This involution is not a covariant operation since it does not mix the left- and right-handedsectors. As a consequence, the state〈ψ∗| = |ψ〉+ is not contravariant while the originalstate|ψ〉 is a covariant vector of the representation space. The Lorentz generators are notHermitian with respect to such an involution. In order to obtain the covariant (Hermitian)


conjugation, we introduce the intertwining operatorΥ which permutes the left- and right-handed sectors,

Υ a± = a±Υ, Υ R = RΥ,

Υ |0〉 = |0〉, Υ |0〉 = |0〉, Υ 2 = 1.

For the finite-dimensional representation(12,0) ⊕ (0, 1

2), the matrix elements of thisoperator correspond to the usualγ 0-matrix [48]. In general this operator can be representedas

Υ =∑n

(|n〉〈n| + |n〉〈n|).The state〈ψ | ≡ 〈ψ∗|Υ is a contravariant vector of the representation. Therefore, the

sl(2,C) invariant scalar product isΨ †Ψ , where the Dirac-like conjugation is defined byΨ † = Ψ+Υ . One can verify that the Lorentz operators are Hermitian with respect to thisscalar product,

J †µν = Υ J+

µνΥ = Jµν.

For this conjugation we also have(Lα)† = Lα andR† = R.

The quantum mechanical (positively defined) probability density〈ψ∗|ψ〉 = 〈ψ |Υ |ψ〉(the left-handed part here) is not a covariant object. For example, for the above mentionedfinite-dimensional representation it corresponds to the zero component of the vectorcurrent:ψγ 0ψ , whereψ is the usual Dirac spinor.

3. Relativistic spinor equations for massless states

In this section we propose and investigate relativistic spinor equations of covariant formbased on the representation ofsl(2,C) realised in terms of RDHA. It turns out that from thealgebraic point of view they correspond to massless states of arbitrary real helicity. For thefinite-dimensional representations of RDHA the equations universally describe states withany integer and half-integer helicity. However, the application of the infinite-dimensionalrepresentations of RDHA inevitably leads to a Lorentz symmetry breaking.

3.1. Algebraic aspect of equations

Using definitions and conventions of the previous section, let us introduce the fields|ψ〉 ∈Rν , |ψ〉 ∈ Rν . By analogy with the(2+1)D case [18,19] we postulate the relativisticwave equations

(3.1)PµσµααLα|ψ〉 = 0, PµσµααL

α|ψ〉 = 0.

Let us analyse the physical content of Eqs. (3.1) from the algebraic point of view. Usingthe identity

(3.2)(Pµσµ

ααLα

)(Pνσν

ββLβ

)εαβ = iPµPµ(1+ νR),


and similar one for the right-handed part, we arrive at the equations

(3.3)PµPµ|ψ〉 = 0, PµPµ|ψ〉 = 0,

and, as a consequence, the proposed relativistic equations describe relativistic masslessfields. Moreover, it is easy to see that the solutions to Eqs. (3.1) obey also the relations

(3.4)(Wµ + 1+ν

4 Pµ)|ψ〉 = 0,

(Wµ − 1+ν

4 Pµ)|ψ〉 = 0.

This means that they have the fixed helicityλ=−1+ν4 for the left-handed sector andλ=

1+ν4 for the right-handed one. So, from the viewpoint of the Poincaré algebra Eqs. (3.1)

describe massless states with arbitrary helicity.In what follows we mainly concentrate on the left-handed part since all the correspond-

ing results for the right-handed sector can be reproduced straightforwardly. In componentsthe first equations in (3.1) read(

a+(P 0 + iP1 − P2 + P3

)− a−(P 0 − iP1 + P2 + P3

))|ψ〉 = 0,

(3.5)(a+

(P 0 − iP1 − P2 − P3

)+ a−(P 0 + iP1 + P2 − P3

))|ψ〉 = 0.

Taking the sum and the difference of these equations, we get(a+

(P 0 − P2

)+ a−(iP1 − P3

))|ψ〉 = 0,

(3.6)(a−

(P 0 + P2

)− a+(iP1 + P3

))|ψ〉 = 0.

3.2. Finite-dimensional representations: integer and half-integer helicities

Let us first consider a solution to Eqs. (3.6) in the case of the finite-dimensionalrepresentations of RDHA [31] withν = −(2k + 1), k ∈ N. As was noted above, thefinite-dimensional representations are built similarly to (2.2) and are characterised by theadditional primitive vector|2k〉 annihilated bya+, a+|2k〉 = 0. Such representations ofRDHA are also reducible with respect to thesl(2,R) algebra,Rν =D k

2⊕D k−1

2.

It is worth noting that formally the solutions to Eqs. (3.6) have the structure similar tothat of the Weyl equationpµσµψ=0 written in components as

(p0 +p3)ψ1 + (p1 − ip2)ψ2 = 0,

(p1 + ip2)ψ1 + (p0 −p3)ψ2 = 0.

Indeed, the coneΓ = pµ :pµpµ = 0, p0 = 0 can be covered by the two chartsU± =

pµ :p0 ± p3 = 0. In each chart the solution can be represented in the regular form

(3.7)ψ(p)∣∣U+ =

(ω(p)

1

)ϕ+(p)δ(p2), ψ(p)

∣∣U− =

(1

ω(p)

)ϕ−(p)δ(p2),

where the functions

ω(p)= ip2 − p1

p0 + p3, ω(p)= p1 + ip2

p3 − p0

obey on the cone the identityω(p)ω(p)= 1. On the intersectionU+ ∩U−, the functionsϕ± are related asϕ−(p)= ω(p)ϕ+(p).


The solution to Eqs. (3.6) can be considered in the same way but with the covering chartsU± = pµ :p0 ±p2 = 0. The peculiarity of thep2-direction is associated with the chosenrepresentation of theσ -matrices and can be changed to any other direction by a unitarytransformation. The solution to the first equation from (3.1) is

|ψ〉∣∣U+ = δ(p2)ϕ+(p)

k∑n=0

CnΩn(p)|2n〉,

|ψ〉∣∣U− = δ(p2)ϕ−(p)

k∑n=0

CnΩk−n(p)|2n〉,

with the functionsϕ± related onU+ ∩ U− asϕ−(p) = Ωk(p)ϕ+(p). Hereϕ± are thefunctions regular onU±, and

(3.8)Ω(p)= p3 + ip1

p0 + p2, Ω(p)= p3 − ip1

p0 − p2, Cn =

√|[2n]ν!|2nn! ,

with the identityΩ(p)Ω(p) = 1 to be valid on the cone. From the explicit form of thesolution one can see that the equations of motion (3.1) contain effectively the projector onthe even invariant subspaceD k

2(or D k

2for the right-handed sector). Here we imply that

the parity is defined with respect to the action of the operatorR (R). The obtained solutiondescribes a free left-handed massless particle with helicityk

2. For example, in the case ofhelicity 1

2 the corresponding solution is given by∣∣λ= 12

⟩∣∣∣U+

= δ(p2)ϕ+(p)(|0〉 +Ω(p)|2〉),

∣∣λ= 12

⟩∣∣∣U−

= δ(p2)ϕ−(p)(|0〉 + Ω(p)|2〉).

This solution is in the exact correspondence with the solution (3.7) to the (unitarilytransformed) Weyl equation. The solution to the second equation from (3.1) can beconsidered analogously.

Thus,in the case of the finite-dimensional representation, Eqs.(3.1)provide a consistentuniversal description of the free states with arbitrary integer and half-integer helicities.The equations have the same form for all such helicities and the information about thevalues ofλ is encoded in the parameterν.

3.3. Infinite-dimensional representations: Lorentz symmetry breaking

Eqs. (3.1) formally have covariant form and, as we have seen, give the consistentdescription of the massless finite-dimensional representations of the Poincaré group.But the situation for the infinite-dimensional representations turns out to be essentiallydifferent. A simple analysis of Eqs. (3.1) reveals some contradictory properties of thesolutions: they exist in some frames and do not exist in others. Indeed, the solution inthe frame wherepµ = (E,0,E,0) is given byψ ∝ |0〉, with |0〉 the vacuum state ofthe RDHA. However, no normalized solutions can be found in the frames wherepµ =


(E,0,−E,0), pµ = (E,±E,0,0) or pµ = (E,0,0,±E). This means thatthe solutionsare not invariant under some transformations of the Lorentz group, and so, the Lorentzinvariance is broken.Note, that if another choice of the Dirac matrices would have beendone, the Lorentz invariance would be broken in other directions. We would like to note thatthis situation with the Lorentz breaking has a formal analogy with the usual spontaneousbreaking of a global symmetry in field theory models, where the equations of motions arecovariant with respect to the corresponding symmetry group and the breaking occurs onthe level of vacuum solutions.

To make the statement on the breaking of the Lorentz invariance more precise, weobserve that for the coveringU+,U− of the cone the formal solution to the first equationfrom (3.1) exists onU+ only,

(3.9)|ψ〉 = δ(p2)ϕ(p)

∞∑n=0

CnΩn(p)|2n〉 = δ(p2)ϕ(p)exp

(12Ω(p)(a+)2

)|0〉,whereϕ(p) is a regular function, andCn andΩ(p) were defined in (3.8). Some care has tobe taken with infinite-dimensional representations since, generally, an infinite-dimensionalrepresentation of the Poincaré algebra is not obligatory a representation of the Poincarégroup. Therefore, the solution (3.9) is proper one if its norm with respect to the internalspace scalar product,

〈ψ∗|ψ〉 =∞∑n=0

C2n

∣∣Ω(p)∣∣2n,

is finite. The radius of convergence of the series is equal to

limn→∞

Cn+1

Cn

= 1.

Therefore, we arrive at the strict inequality|Ω(p)|2 < 1, which can be rewritten as

(3.10)p2(p0 + p2

)> 0.

All the formulas corresponding to the right-handed sector can be reproduced via the formalsubstitutiona± → a±, p2 → −p2. Therefore, the corresponding convergence conditionfor that sector is

(3.11)p2(p0 − p2

)< 0.

Consequently, although Eqs. (3.1) look likeSL(2,C)-invariant equations, the maximalinvariance group of the solution is the one preserving the inequality (3.10) (or (3.11) for theright-handed sector). This means that the equations are invariant underSL(2,R) subgroupof SL(2,C) (the group generated by the rotations in the plane (1–3) and by the boosts inthe directions 1 and 3, which do not violate the relations (3.10), (3.11)). This is a directconsequence of the fact that the used infinite-dimensional representation is ill defined atthe level of the Poincaré group, even if it is perfectly defined at the level of the Lie algebra.

The solution (3.9) illustrates an unusual type of Lorentz symmetry breaking. The mass-less equations are formally covariant but in the case of infinite-dimensional representations


the Lorentz invariance is strongly broken on the level of the solutions. Formally, this break-ing is associated not with the infinite-dimensional representation of theSL(2,C) group it-self but rather with the attempt to enclose it into the corresponding representation of thePoincaré group (cf. with massive case, see Appendix A). Indeed, as follows from the in-equalities (3.10), (3.11), the Lorentz violation is provoked by the transformations of themomentum, which is naturally associated with the Poincaré group.

Through the reductionsl(2,C) → sl(2,R), the infinite-dimensional representationsRν and Rν are identified. In this case the representationD±

λ can be exponentiated toa representation of the Lie groupISO(2,1). So, the Poincaré invariance in(3 + 1)D,ISO(3,1), is broken toISO(2,1), the Poincaré invariance in(2+ 1)D. As we shall seein the next section, this means that the dimensional reduction to the directionp2 gives riseto a consistent(2+1)D theory describing a massive anyon of a massm and spinλ=±1+ν

4 .One has to stress once again that for the case of the finite-dimensional representations,

when 2λ is an integer number, the problem encountered with infinite-dimensionalrepresentations is not present and, as we have seen, the consistent equations for the helicity±λ fields are obtained.

4. Compactification and reduction to (2 + 1)D anyons

We have seen that in the case of infinite-dimensional representations the theory providedby Eqs. (3.1) does not have a nontrivial content on the whole(3+ 1)D Minkowski spaceM4 since the formal solution (3.9) breaks the Lorentz invariance. But let us show that, ina sense, this problem can be “cured” by compactifying the singled out (in this casex2)direction on a circle,M4 →M3 × S1.

Fixing the space geometry in the form of the three-dimensional Minkowski space timesa circle of radiusm−1 and denoting the compactified coordinate asθ , 0 θ 2π , the state|ψ〉 in coordinate representation can be expanded as

|ψ〉 =∞∑

n=−∞einθ |ψn〉, with |ψn〉 =

∞∑k=0

ψnk(ya)|k〉,

depending only on the 3-dimensional coordinatesya, a = 0,1,2, with y0 = x0, y1 = x1

andy2 = x3.So, starting with a(3+ 1)D massless momentumPµ, it is reduced to(P 0,P 1, nm,P 3)

for each state|ψn〉. Denoting the three-dimensional momentum bypa = (P 0,P 1,P 3), themass-shell condition in(2+ 1)D reads(

papa + n2m2)|ψn〉 = 0,

i.e., the initial massless system is formally reduced to an infinite tower of massive statesplus a massless one. Forsl(2,R) the left- and right-handed representations are equivalent,and so, the dotted and undotted indices are identical. The both equations in (3.1), throughthe compactification process, lead to the equations

(4.1)(pa(γa)α

β + nmδαβ)Lβ |ψn〉 = 0.


These equations can be obtained from (3.1) by the formal substitutionP2 → nm andmultiplication byσ2 that, in fact, corresponds to lowering the free index. Theγ -matricesare given in the Majorana representation:

(γ0)αβ =−(σ2)α

β, (γ1)αβ =−i(σ3)α

β, (γ2)αβ = i(σ1)α

β.

Up to the spatial rotationy1 → −y2, y2 → y1, Eqs. (4.1) actually coincide with theequations for anyons [18,19] and, hence, forn = 0 theyconsistentlydescribe relativistic(2+1)D anyons of the spinλ=−sign(n)1+ν

4 . By analogy with (3.9), the formal solutionsto these equations in the momentum representation are given by

(4.2)|ψn〉 = δ(p2 + n2m2)ϕn(p)exp

( 12Ωn(p)(a

+)2)|0〉,

with Ωn(p) = p2+ip1p0+nm

. The condition of normalizability (3.10) is transformed into theinequality

(4.3)n(p0 + nm

)> 0.

This means that the solutions (4.2) are normalizable only in the case of energy withdefinite sign: sign(p0) = sign(n). In other words, all the states|ψn〉 with n > 0 havethe positive energy, those withn < 0 have negative energy while the state|ψ0〉 does notbelong to the spectrum of the theory. Such properties reveal the difference of the consideredcompactification from the usual procedure.

Let us discuss this issue from viewpoint of the symmetries usually associated with sucha compactification on a circle. Following Ref. [49], one can infer that in the usual casean affine Kac–Moody algebrag is always associated with a compactifiedM3 × S1 theory.This infinite-dimensional algebra consists of the loop extension ofiso(2,1),

iso(2,1)= Pan = einθP a, J a

n = einθ J a,

wherePa andJ a are the translation and Lorentz generators ofiso(2,1), respectively, andof the additional set of operatorsQn = ieinθ∂θ with the commutation relations[

Qn,Qm

]= (n−m)Qn+m,[Qn,P

am

] =−mPan+m,

[Qn,J

am

]=−mJan+m.

The affine Kac–Moody algebrag has the following natural triangular decomposition

g= g+ ⊕ g0 ⊕ g−,

where the subalgebrasg+ andg− are the sets of all the operators withn > 0 andn < 0,respectively, whileg0 = iso(2,1)⊕ u(1) with Q0 being the generator of theu(1) algebra.

In our case the symmetry corresponding to the affine algebrag is always partially broken.Indeed, the action ofg− is ill defined on the part of the spectrum with positive energy(n > 0) while g+ is ill defined on the part of the spectrum with negative energy (n < 0).Hence, the symmetry is always broken tog+ ⊕ g0 or to g0 ⊕ g−. The only symmetryof the whole spectrum isg0. Nevertheless, one can conclude that in spite of the partialbreaking of the infinite-parametric symmetry, the compactified theory has the non-trivialcontent.


The considered compactification can be treated, in principle, as that “induced” by thebreaking of the Lorentz invariance of Eqs. (3.1) for the infinite-dimensional representationscase. The “induction” is understood here in the sense that the compactified theory, unlikethe initial one, is consistent and the direction of the compactification is defined by theLorentz breaking.

The dimensional reduction emerges after choosing one level, sayn= 1 orn=−1, anddiscarding all the others. Evidently, the reduction gives rise to a consistent(2+1)D theorydescribing a massive anyon of a massm and spinλ=±1+ν

4 . Correspondingly, the formalISO(3,1) invariance is reduced to the trueISO(2,1) invariance.

Eqs. (4.1) providing solutions with fixed sign energy have a close analogy with themassive Dirac(3 + 1)D positive energy spinor equations [23,24] (see Appendix A).4

The construction of the positive energy massive Dirac equations is based, in fact, on therepresentation of the typeRν=0 ⊗ Rν=0, and in the massless limit they are reduced to theequations of the form (3.1) to be imposed on the same state. However, note that forp0 > 0,the inequalities (3.10), (3.11) are incompatible. This means that there is no referenceframe in which the normalizable solutions could exist in the left- and right-handed sectorssimultaneously. Using this observation, one can assume that the Dirac equations [23,24]have no proper solutions in the massless limit. In Appendix A we demonstrate that this isindeed the case.

5. No-go theorem for massless infinite-dimensional representations of ISO(3, 1)

We have seen that the proposed spinor sets of equations cannot be used to describe amassless field with fractional helicity in(3+ 1)D because the maximal invariant group ofthe solution is broken down to the(2+ 1)D Poincaré group. Moreover, the same problemsarise under attempt to describe the massless field of integer or half-integer helicity bymeans of infinite-dimensional representations ofsl(2,C) associated with RDHA. So, wemay wonder if this feature is general or it is specific to Eqs. (3.1) we have considered. Inother words, formally we may address the general problemif fractional helicity states in(3+ 1)D might exist.At this point, the natural question we should ask concerns the othertype of equations that could be proposed. Following Ref. [40] and the results obtained in(2+ 1)D case [17], a vector set of equations can be considered,

(5.1)(αPµ + iJ µνPν

)Ψ = 0,

whereα is some constant that defines the helicity while the representation ofΨ is not fixedhere. According to Refs. [48,50], the equations of such a form describe all irreduciblemassless finite-dimensional representations of the Poincaré group. But we are going toconsider these equations from the viewpoint of infinite-dimensional representations. Inthis sense the equations (5.1) are analogous to the massless limit of the equations proposedby Staunton in Ref. [40]. Contraction withPµ shows that we have a massless field. Then,

4 The change of the sign before mass in the Dirac equations leads to solutions with negative energy.


choosing the frame wherePµ = E(1,0, ε,0), we reduce the system of equations to thesystem

(α − ε(J0 − J0)

)Ψ = 0,(

(1+ ε)(J− + J+)+ (1− ε)(J+ + J−))Ψ = 0,

(5.2)((1+ ε)(J− − J+)− (1− ε)(J+ − J−)

)Ψ = 0.

For ε =+1, the solution has to obey the relations

(5.3)J−Ψ = J+Ψ = 0,

whereas forε =−1 they are changed for

(5.4)J+Ψ = J−Ψ = 0,

that is possible only for the finite-dimensional representations ofsl(2,C). In other words,we arrive at the same problems as before and Eqs. (5.1) are not consistent for the statescarrying fractional helicity or, generally, with all the infinite-dimensional representations.On the other hand, one can show that the dimensional reduction of (5.1) leads to the(2+ 1)D vector set of equations that, as well as the spinor equations (4.1),consistentlydescribes anyons [17].

On a general ground, any set of equations which can be proposed to describe a masslessfield of helicityλ (fractional or not) has to give rise to the “helicity equation”

(5.5)(Wµ − λPµ

)Ψ = 0,

with Wµ the Pauli–Lubanski vector [22]. The representation ofΨ is also not specifiedproviding the universality of the analysis.

Eqs. (5.5), like Eqs. (3.1) and (5.1), are not compatible for infinite-dimensionalrepresentations. Indeed, e.g., in a frame wherePµ = E(1,0, ε,0), the equations aresimplified for

(λ− ε(J0 + J0)

)Ψ = 0,(

(1+ ε)(J− − J+)+ (1− ε)(J+ − J−))Ψ = 0,

(5.6)((1+ ε)(J− + J+)− (1− ε)(J+ + J−)

)Ψ = 0.

One can see that these equations can be obtained from (5.2) by the substitutionJ0 →−J0,J± →−J± andα → λ. The reason is that Eqs. (5.5) can be reproduced from (5.1) by theformal substitutionJµν → iJ µν , α → λ, whereJ µν = 1

2εµνρσ Jρσ . For ε = 1 or ε =−1

we arrive, correspondingly, at Eqs. (5.3) or (5.4). This means that Eqs. (5.5) are compatiblefor finite-dimensional representations (with integer or half-integer helicities) only. In otherwords,the (3+ 1)D Poincaré group has no massless irreducible representations of any(integer, half-integer or fractional) helicity with the trivial non-compact part of the littlegroup constructed on the basis of infinite-dimensional representations of sl(2,C).


6. Discussion and outlook

The specific properties of the(2+ 1)-dimensional space–time admit the existence ofanyons. The little group of massive states in(2+ 1)D coincides with the compact part(which is the infinitely connected groupSO(2)) of the little group of massless statesin (3 + 1)D. Consequently, the “charge” of its universal covering groupSO(2) = R isnot quantized. The group theory justification of anyons is related to the fact that therepresentations of the fractional charge describing relativistic anyons can be extended torepresentations of the whole(2+ 1)-dimensional Lorentz group which is also infinitelyconnected. For the purpose of describing the massless states with fractional (λ ∈ R) helicitywe have constructed representations ofsl(2,C) in terms of two copies of theR-deformedHeisenberg algebra. In terms of such representations of the Poincaré algebra,

• The universal relativistic equations(3.1) describing massless states of any integerand half-integer helicity have been proposed.

This possibility is related to the existence of the finite-dimensional representations ofRDHA for ν = − (2k+1), k ∈ N. Forν >−1, the RDHA has infinite-dimensional unitaryrepresentations. In this case Eqs. (3.1) formally describe the massless states with fractionalhelicity. Analysing the solutions to the relativistic wave equations, we have traced outexplicitly that the corresponding infinite-dimensional representations of the Lie algebrasl(2,C) cannot be exponentiated to representations of the Lie groupSL(2,C) for suchmassless states. In other words, the(3 + 1)D Lorentz invariance is broken on the levelof solutions. This resembles the Lorentz symmetry breaking in supersymmetric Yang–Mills theory considered in Ref. [51], where the corresponding action is Lorentz invariantwhile the symmetry is broken at the field equation level. In spite of the Lorentz symmetrybreaking we have shown that

• The dimensional reduction of the massless equations(3.1) leads to the consistent(2+ 1)D theory of massive anyons with spinλ=±1+ν

4 .

The dimensional reduction emerges from the compactificationM4 →M3 × S1. We treatthis compactification as that induced by the Lorentz symmetry breaking in Eqs. (3.1) onthe solution level. It would be interesting to find an example of such a compactification(induced by the Lorentz symmetry breaking on solution level) in other theories. Wehope that the observed unusual symmetry breaking could be helpful in the context of theconsiderable activity looking for different mechanisms of the Lorentz symmetry violation[51–58] and compactification [59–62].

Concluding our investigation on existence of massless states with fractional helicityin the 4-dimensional space, we have analysed the fundamental equation (5.5) definingmassless irreducible representations ofISO(3,1), and have shown that

• The Poincaré group ISO(3,1) has no massless irreducible representations with thetrivial non-compact part of the little group constructed on the basis of the infinite-dimensional representations of sl(2,C).

This means, in particular, thatthe massless(3+ 1)-dimensional fractional helicity statescannot be described in a consistent way and that the integer and half-integer helicity


massless particles can be described only in terms of finite-dimensional representationsof sl(2,C). It is worth also emphasizing that the obtained restriction on values of helicity isnot of a topological origin. The topological arguments [28,29] do not forbid irreduciblerepresentations with integer and half-integer helicity constructed on the basis of theinfinite-dimensional representations ofsl(2,C). Therefore,the topology provides onlythe necessary condition for massless representations to be the true representations ofISO(3,1), which, as we see, is not the sufficient condition.This is quite an unexpected resultsince the massive irreducible representations ofISO(3,1) with integer and half-integer spincan be constructed on the basis of the infinite-dimensional representations ofsl(2,C) [23,24,40].

The situation with Eqs. (3.1) for the infinite-dimensional representation can be comparedwith the result obtained earlier in Ref. [63] in a different context. Here the proposedequations look invariant under the(3 + 1)D Poincaré transformations but relativisticinvariance is broken at the level of the solutions. In the paper [63] two of us have obtainedsimilar results, which were related, however, to the fact that the considered there Lorentzautomorphism of the underlying algebra is outer and not obligatory inner. Only when theouter automorphism becomes inner, a covariant equation can be obtained. All this means,in particular, that it is not enough to have the equation (or the set of equations) which looksinvariant (or covariant) to obtain a covariant theory.

Acknowledgements

M.P. thanks L. Avarez-Gaumé, A.I. Oksak, V.I. Tkach, G.G. Volkov and A.A. Zheltukhinfor early communications stimulated the present research and acknowledges the usefulcommunications with L.N. Lipatov and L. Ryder. M.R.T. acknowledges gratefully theuseful discussions with G. Mennessier and M.J. Slupinski. One of us (M.R.T.) wouldlike to thank USACH for its hospitality, where the part of this work was realized. Thework was supported in part by the grants 1980619, 7980044, 1010073 and 3000006 fromFONDECYT (Chile) and by DICYT (USACH).

Appendix A. Dirac positive-energy relativistic equations

The Dirac equations [23,24] corresponding to massive spinless particle with positiveenergy can be represented as

(A.1)(Pµγµ −m1

)QΨ = 0, with Q=

q1

q2

η1

η2

,

where q1 and q2 are commuting dynamical quantities andη1 and η2 denote thecorresponding conjugate momenta,[qi, ηj ] = iδij , i, j = 1,2. The matricesγµ are relatedto those in the Weyl representation (2.7) by the unitary transformationγµ = UγµU

† withthe matrix


U = 1√2

0 0 1 i

1 −i 0 0i −1 0 00 0 −i −1

.

Dirac showed that Eqs. (A.1) consistently describe massive spin-0 positive-energy states[23,24]. In the “coordinate” representation, with the operatorsqi to be diagonal, thesolution to Eqs. (A.1) has the form [23,24]

(A.2)

Ψ ∝ δ(p2 +m2)exp

(− 1

2(p0 − p3)

(m

(q2

1 + q22

)− ip1(q2

1 − q22

)+ 2ip2q1q2))

.

One can verify that this solution is normalizable in “internal” variables form = 0 only.Moreover, in the massless limit the solution (A.2) is singular on the conep2 = 0 due to thepresence of the factor(p0 − p3)

−1. This means that Eqs. (A.1) have no proper solution inthe massless limit.

It is worth noting that the formal changem → −m leads to the normalized solutionswith negative energy.

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Algebraic renormalization of twistedN = 2supersymmetry withZ = 2 central extension

B. Geyera, D. Mülschb

a Universität Leipzig, Naturwissenschaftlich-Theoretisches Zentrum and Institut für Theoretische Physik,D-04109 Leipzig, Germany

b Wissenschaftszentrum Leipzig e.V., D-04103 Leipzig, Germany

Received 26 April 2001; accepted 4 September 2001

Abstract

We study the renormalizability of (massive) topological QCD based on the algebraic BRSTtechnique by adopting a noncovariant Landau type gauge and making use of the full topologicalsuperalgebra. The most general local counter terms are determined and it is shown that in the presenceof central charges the BRST cohomology remains trivial. By imposing an additional set of stabilityconstraints it is proven that the matter action of topological QCD is perturbatively finite. 2001Elsevier Science B.V. All rights reserved.

PACS:12.60.J; 03.70; 11.10.GKeywords:Topological QCD;N = 2 twisted supersymmetry; Dimensional reduction

1. Introduction

Topological quantum field theories (TQFT) are characterized by observables dependingonly on the topology of the manifold on which these theories are defined [1]. The classicalexample is topological Yang–Mills theory (TYM) on a four-manifold as proposed byWitten [2] whose correlation functions, if computed in the weakly coupled ultraviolet limit,turn out to be related to the Donaldson invariants [3]. On the other hand, the Donaldsoninvariants of smooth four-manifolds are closely related toN = 2 supersymmetric Yang–Mills theories (SYM). In fact, by restricting to flat Euclidean space–time TYM can bereformulated through a twisting ofN = 2 SYM in Wess–Zumino gauge [2]. The resultingtheory has a fermionic symmetry being identified with the topological shift operatorQ ofthe twistedN = 2 SUSY algebra. In the cohomological formulation of Labastida–Pernici[4] and Baulieu–Singer [5] TYM is interpreted as gauge-fixed action of the Pontryagin

E-mail addresses:[email protected] (B. Geyer), [email protected] (D. Mülsch).



438 B. Geyer, D. Mülsch / Nuclear Physics B 616 [PM] (2001) 437–475

term andQ, after identifying theR-charge with the ghost number, is interpreted as theBRST operator of the equivariant part of TYM. As a result of that identification theBRST cohomology becomes completely trivial. In the derivation of ultraviolet finitenessproperties of various topological models and in the construction of their observables,besidesQ, still another fermionic symmetry, the so-called vector supersymmetry, playsan important role being generated by the vector chargeQµ of the twistedN = 2 SUSYalgebra [6–11].

Seiberg and Witten have studied also the strongly coupled infrared limit of twistedN = 2 SYM which leads to a new class of four-manifold invariants, the so-called Seiberg–Witten invariants [12]. The Seiberg–Witten theory can be considered as the twistedversion ofN = 2 supersymmetric Maxwell theory with one massless hypermultiplet [13].In a second paper, Seiberg and Witten have studied the strong coupled infrared limitof topological quantum chromodynamic (TQCD), being the twisted version ofN =2 SYM coupled to a standard hypermultiplet in the fundamental representation [14].In Ref. [15] it has been shown that by introducing theN = 2 supersymmetric baremass term to the hypermultiplet the resulting massive TQCD interpolates between theDonaldson and the Seiberg–Witten theory. Such hypermultiplets have a nonvanishingcentral charge and, therefore, an extension of theN = 2 SUSY algebra is required. Auseful approach to understand the geometry of the TQCD is based on the Mathai–Quillenformalism [16].

In the present paper we investigate the renormalization properties of twistedN = 2SUSY with two central extensions. In extending the analysis of Ref. [17] we study theultraviolet behaviour of TQCD in the framework of algebraic renormalization [18,19] byexploiting both the topological shift symmetryQ and the vector supersymmetryQµ.

The renormalization properties of twistedN = 2 SUSY without central extensions havebeen widely investigated for particular gauges [6,7,17,20–22]. In Ref. [21] it has beenshown that the cohomological nature of Witten’s topological model is totally insensitiveto quantum corrections and, in accord with a previous result [5], that gauge anomalies areabsent to all orders of perturbation theory. This analysis has been extended in Ref. [17]by choosing a noncovariant Landau type gauge. This gauge has the merit that theassociated vector supersymmetryQµ is linearly realized and, therefore, can be employedas a stability constraint in order to improve the ultraviolet finiteness properties of thatmodel.

Here, we study the renormalizability of a slightly more general extension of TQCDwhich is formed by twistingN = 2 SUSY withZ = 2 central charges. Let us recall, thatfor N = 2 the twisting procedure is unique. An undesired feature which is displayed bytwisting N = 2 SYM coupled to a standard massive hypermultiplet [14] is the fact thatit fails to be invariant underR-symmetry, the latter being broken intoZ2. In order tocircumvent this problem we consider TQCD in the presence of two central chargesZ

andZ, being complex conjugated to each other. It can be shown that this topological modelpreservesR-symmetry and, therefore, the ghost number conservation if one formallyascribes toZ andZ theR-weightsR(Z)= 2R(Q) andR(Z)= 2R(Qµ), respectively.

B. Geyer, D. Mülsch / Nuclear Physics B 616 [PM] (2001) 437–475 439

Furthermore, by adopting the noncovariant Landau type gauge it will be proven that thewhole set of stability constraints considered in Ref. [17] can be applied also in that generalcase—except for the so-called ghost for the ghost equation. Moreover, we shall be able togive an additional set of constraints which ensures the perturbative finiteness of the matterpart of TQCD, i.e., the twisted hypermultiplet is not subjected to any renormalization. It isworthwhile to emphasize here that the algebraic proof of that finiteness property extendsto all orders of perturbation theory and does not rely on the existence of any regularizationscheme.

The paper is organized as follows. Section 2 reviews the main features of TQCD. Itspurpose is, at first, to introduce our notations and to make this paper reasonably self-contained and, secondly, to bring separately into the play the various elements ofN = 2TQCD with two central charges thereby also motivating the choice of (noncovariant)Landau type gauge. In Subsection 2.1 we introduce the so-called equivariant part of theaction of TYM by fixing the topological shift symmetry in an covariantξ -gauge. Thisprocedure closely parallels the construction, in a Feynman type gauge(ξ = 1), of thetopological action in Ref. [22] but, in contrast to it, we introduce a parameterξ intothe twistedN = 2 SUSY algebra for being able also to select the Landau type gauge(ξ = 0). In Subsection 2.2 we extend the topological Yang–Mills theory by coupling itto a massive hypermultiplet (and its hermitian conjugate) thus obtaining the equivariantpart of the action of TQCD. This is achieved by a nontrivial dimensional reductionof a (D = 6)-dimensionalN = 1 SYM containing a gauge multiplet and a masslesshypermultiplet in the adjoint and some (e.g., the fundamental) representation of thegauge algebra Lie(G), respectively. The main body of that derivation is postponed toAppendix B. In Subsection 2.3 we introduce the complete action of TQCD by fixingalso the remaining gauge symmetry. It turns out that only in the Landau type gauge,ξ = 0, the complete action is invariant under both the topological shift symmetryQ

and the vector supersymmetryQµ. In Section 3 we construct the action of TQCD in anoncovariant Landau type gauge which allows to realize the vector supersymmetry linearlyand, lateron, to employ it as an additional stability constraint. In Section 4 we encloseall the symmetry operators of the theory into a single nilpotent BRST operatorsT byassociating to each generator of the topological superalgebra a global ghost. We establishthe corresponding classical Slavnov–Taylor identity and derive the gauge conditions,the antighost equations and some global constraints, being related to the noncovariantLandau type gauge. In Section 5 the problem of renormalizability is treated by standardcohomological methods. It is proven that the BRST cohomology is trivial and the mostgeneral local counterterms are determined. Appendix A contains the Euclidean spinorconventions being used thourough this paper. Appendix B gives a detailed derivation ofthe twistedN = 2 SYM coupled to a (massive) hypermultiplet with two central charges byusing the method of dimensional reduction [23].

In this paper we use the convention[T i, T j ] = f ijkT k and tr(T iT j ) = δij , theantihermitean generatorsT i being a basis of the Lie algebra Lie(G) of the gauge groupG,which we assume to be a simple compact Lie group, in some representationR, e.g.,a fundamental representation. We also adopt the matrix notationϕ = ϕiT i and δ/δϕ =


T iδ/δϕi for any field ϕi transforming according to the adjoint representation ofG.Furthermore, we use the conventionϕ for the hermitian conjugate of some fieldϕ.

2. Topological QCD with Z = 2 central extensions

2.1. Topological Yang–Mills theory: equivariant part

One of the possibilities to introduce TYM consists in twisting a set of conventional(spinorial) superchargesQA

a andQAa of N = 2 SYM in (D = 4)-dimensional Euclideanspace–time [2]. Thereby, theN = 2 gauge multiplet consists of a (antihermitean)gauge fieldAµ, the Sp(2)-doublets of chiral and of antichiral 2-spinors,λAa and λAa ,respectively, and a complex scalar fieldφ. In order to close the SUSY algebra it isnecessary to introduce a (symmetric) auxiliary fieldχab = χba . All the fields of thatoff-shell gauge multipletV = Aµ,λAa, λAa, φ, φ, χab are in the adjoint representationand take their values in the Lie algebra Lie(G) of some compact gauge groupG. Therotation group of Euclidean space–time,SO(4), is locally isomorphic toSU(2)L⊗SU(2)Rand the spinor indices in the fundamental representation ofSU(2)L andSU(2)R will bedenoted byA = 1,2 and A = 1, 2, respectively. The global internal symmetry groupof N = 2 SUSY is Sp(2) ⊗ U(1)R corresponding to symplectic rotations and chiraltransformations (R-symmetry). TheR-charges (chiral weights) ofQA

a and QAa are 1and−1, respectively. The internalSp(2) indices, labelling the differentN = 2 charges, aredenoted bya = 1,2.

In the absence of a central extension, i.e., in a theory without massive fields which willbe considered first, theN = 2 SUSY algebra in the Wess–Zumino gauge is characterizedby the eight spinorial superchargesQA

a andQAa which, together with the generatorPµ ≡i∂µ of space–time translations, obey the relations

QAa,QB

b=−4εabεABδG(φ),

QAa, QBb

=−2δab(σµ

)AB

(Pµ + iδG(Aµ)

),

(1)QAa,

QBb

= 4εabεABδG(φ),

with φ being the hermitean conjugate ofφ. (The conventions of Euclidean spinor algebraare collected in Appendix A; the on-shell version of the algebra (1) is derived inAppendix B). Since in Wess–Zumino gauge the supersymmetry is realized nonlinearly thealgebra (1) closes only modulo the field dependent gauge transformationsδG(ω), ω =Aµ,φ, φ, respectively (cf. Eqs. (B.2)).

As explained in Ref. [2], TYM is obtained fromN = 2 SYM by replacing the groupSU(2)L ⊗ Sp(2) through its diagonal subgroup or, in other words, by identifying theinternal indexa with the spinor indexA. According to this twisting procedure oneconstructs from the generatorsQA

a andQAa the twisted ones,

(2)Q= 1

2εABQAB, Qµ = 1

2i(σµ)

AB QAB, Qµν = 1

2(σµν)

ABQAB,


being a scalarQ, a vectorQµ and a self-dual tensorQµν , respectively. Since it turnsout that TYM is already completely specified by the topological shift symmetryQ andthe vector supersymmetryQµ, we actually do not take into account the self-dual tensorsupersymmetryQµν in the following. In terms ofQ andQµ, from (1) for the twisted (ortopological) superalgebra we get [8,9]

Q,Q = −2δG(φ),Q, Qµ

=−iPµ + δG(Aµ),(3)

Qµ, Qν

=−2δµνδG(φ).

From the first of these relations it follows that scalar superchargeQ is equivariantlynilpotent, i.e., it squares to gauge transformationsδG(φ) generated byφ, and, applyingthe Jacobi identity onφ, it follows that the fieldφ, from which the Donaldson invariant isconstructed, must beQ-invariant,Qφ = 0.

The second relation, being typical for a topological theory, states that, due to theexistence of the vector superchargeQµ, the space–time translationsPµ can be representedas aQ-anticommutator modulo the gauge transformationδG(Aµ). This allows, startingfrom the Donaldson invariant,O, to construct all the (global) observables of TYM [2] byapplying successively the vector superchargeQµ [22] (see also Ref. [1d]),

O ≡∫d4x trφ2, QµO, Qµ

QνO,

(4)QµQνQρO, Qµ

QνQρQσO.

(It is worthwhile to note that the same observables can also be recovered through the so-called equivariant cohomology, compare, e.g., Ref. [24]). In this manner one obtains thatpart of the action of TYM which results by fixing, in a Feynman type gauge, the topologicalshift symmetry [22]:

(5)W(ξ=1)T = 1

4

∫d4x trFµνF µν − 1

24εµνρσ Qµ

QνQρQσ

∫d4x trφ2,

where

Fµν = 1

2εµνρσ F

ρσ , Fµν = ∂µAν − ∂νAµ + [Aµ,Aν],Dµ = ∂µ + [Aµ, · ],

here,Fµν is the dual of the YM field strengthFµν andDµ is the covariant derivative (inadjoint representation). The first term in (5) is the classical action (Pontryagin term) and thesecond one is the gauge-fixing term which removes the degeneracy of the classical actionwith respect to the topological shift symmetryQ still leaving out of account the originalgauge symmetry. Usually,W(ξ=1)

T is called equivariant part of the topological action beingdefined by the equivariant cohomology of TYM (prior introduction of the gauge ghostfield). Obviously, the Wess–Zumino gauge of supersymmetry immediately leads to theFeynman type gauge of TYM which is mediated by the vector chargeQµ. Of course,other gauges may be used as well, e.g., those introduced by Eq. (6) below. For notational


simplicity we have set the gauge coupling equal to one,g = 1. Let us also remark that, inprinciple, the Pontryagin term could be multiplied byθ with some arbitrary parameterθwhich, here, is set equal to one,θ = 1.

From the third relation in (3) it follows that also the operatorQµ is equivariantlynilpotent, but now modulo the gauge transformationδG(φ), and, for the same reason asbefore, that the fieldφ must beQµ-invariant, Qµφ = 0. Let us stress that the secondterm in (5) is different from zero if and only ifQµ does not anticommute with itself,showing the importance of the presence of the gauge transformation on the right-hand sideof Qµ, Qν = −2δµνδG(φ). On the other hand, that gauge-fixing in (5) is incomplete.In order to obtain the complete action of TYM one still has to add a further gauge-fixingterm removing the degeneracy of the classical action with respect to the remaining gaugesymmetry (which will be done in Subsection 2.3). However, that additional term turnsout to be invariant under the vector supersymmetryQµ if and only if Qµ, Qν = 0, i.e.,if Qµ is strictly nilpotent. Hence, by choosing a Feynman type gauge the vector chargeQµ cannot be really a symmetry operator of the complete gauge-fixed action of TYM.However, such a situation can be circumvented by choosing a Landau type gauge, i.e., bymodifying the second term of the action (5) appropriately.

In order to prepare the frame for a more general covariant gauge let us deform thetopological superalgebra (3) without changing its topological character:

Q,Q = −2δG(φ),Q, Q(ξ)

µ

=−iPµ + δG(Aµ),(6)

Q(ξ)µ , Q(ξ)

ν

=−2ξδµνδG(φ),

thus makingQµ alsoξ -dependent. Here,ξ is the gauge parameter interpolating betweenFeynman (ξ = 1) and Landau (ξ = 0) type gauge. In turn, the topological action (5)changes into

(7)W(ξ)T =

1

4ξ

∫d4x trFµνF

µν − 1

24ξεµνρσ Q(ξ)

µQ(ξ)νQ(ξ)ρQ(ξ)σ

∫d4x trφ2,

where both terms are rescaled by 1/ξ in order to ensure that the action is well-defined alsofor ξ = 0 (see Eq. (10) below). From (7) it follows that, by construction,W

(ξ)T will be left

invariant by the twisted operatorsQ andQ(ξ)µ ,

QW(ξ)T = 0, Q(ξ)

µ W(ξ)T = 0.

The symmetry transformations corresponding toQ and Q(ξ)µ can be obtained by

applying the previous twisting procedure to the transformation laws generated byQAa and

QAa (see Eqs. (B.6) for the on-shell transformations in the Feynman type gauge). Beforeshowing their explicit form let us briefly introduce the twisted gauge multipletVT of TYMby its relation to the gauge multipletV = Aµ,λAa, λAa, φ, φ, χab of N = 2 SYM.

The antichiral spinorλAa , being the hermitean conjugate ofλAa , is related to aGrassmann-odd vector fieldψµ through

ψµ =−1

2(σµ)

AB λAB,


Table 1

Aµ ψµ χµν η λµν φ φ Q QµGhost number 0 1 −1 −1 0 2 −2 1 −1Mass dimension 1 3/2 3/2 3/2 2 1 1 1/2 1/2Scale dimension 1 1 2 2 2 0 2 0 1Z-, Z-charge 0 0 0 0 0 0 0 0 0Grassmann parity even odd odd odd even even even odd odd

which is the topological ghost;φ is the ghost of the topological ghost andφ is thecorresponding antighost. The chiral spinorλAa is associated with both a Grassmann-odd antisymmetric self-dual tensor fieldχµν ≡ χ+µν and a Grassmann-odd scalar fieldηaccording to

χµν =−1

2i(σµν)

ABλAB, η=−1

2iεABλAB,

whereη plays the role of an auxiliary field. Finally, the symmetric auxiliary fieldχab isrelated to a Grassmann-even antisymmetric self-dual tensor fieldλµν ≡ λ+µν through

λµν = 1

2(σµν)

ABχAB,

which, again, ensures the closure of the twisted superalgebra (6). Hence, the field content ofTYM is given by the following twisted gauge multipletVT = Aµ,ψµ,χµν, η,φ, φ, λµνwhose properties, together with those of the topological charges, are displayed in Table 1.

Let us now give the transformation law for the twisted gauge multiplet:(i) The topological shift symmetryQ takes the form

QAµ =ψµ, Qψµ =Dµφ, Qφ = 0,

(8)Qφ = η, Qη= [φ, φ], Qχµν = λµν, Qλµν = [χµν,φ],it is independent of the choice ofξ . Indeed, after rescaling byξ the fields φ, η andχµν , λµν—which belong to the nonminimal part (in the sense of BRST symmetry) ofthe multipletVT—these transformation rules obviously remain unchanged.

(ii) The transformations rules for the vector supersymmetryQ(ξ)µ , are given by

Q(ξ)µ Aν = ξ(δµνη+ χµν), Q(ξ)

µ ψν = Fµν − ξ(δµν[φ, φ] + λµν

),

Q(ξ)µ φ =ψµ, Q(ξ)

µ φ = 0, Q(ξ)µ η=Dµφ,

Q(ξ)µ χρσ = δµρDσ φ − δµσDρφ + εµνρσDνφ,

Q(ξ)µ λρσ =Dµχρσ + δµρ

([φ,ψσ ] −Dση)− δµσ ([φ,ψρ ] −Dρη)(9)+ εµνρσ

([φ,ψν] −Dνη).

These transformations sensitively depend on the choice of the gauge parameterξ . Notice,that for ξ = 0 the operatorQ(ξ)

µ leaves the gauge fieldAµ inert and, therefore, doesnot change the classical action. Thus, the vector supersymmetry represents a nontrivialsymmetry only with respect the gauge-fixing terms of TYM.


Now, we are in a position to determine the second term in the topological action (7)explicitely. As a result one gets (remind thatλµν andχµν are self-dual)

(10)

W(ξ)T =

∫d4x tr

λµνFµν − 2χµνDµψν + 2ηDµψµ + 2φψµ,ψµ + 2φD2φ

− ξ( 12λ

µνλµν − 12χ

µν[χµν,φ] + 2[φ, φ][φ, φ] − 2η[η,φ]).The relative factor in front of both terms ofW(ξ)

T in Eq. (7) (as well as (5)) has been choosenin such a way that the Pontryagin term is cancelled by the contributions of the gauge-fixingterm and that the remainder can be written as an exactQ-cocycle,

(11)W(ξ)T =QΨ (ξ)

T with Q(ξ)µ Ψ

(ξ)T = 0,

whereΨ (ξ)T is the gauge fermion,

(12)Ψ(ξ)T =

∫d4x tr

χµνFµν + 2φDµψµ − ξ

( 12χ

µνλµν + 2η[φ, φ]).This is that crucial property which allows to interpret TYM as a cohomological theory [2].Furthermore, let us stress that by imposing the vector supersymmetryQ(ξ)

µ W(ξ)T = 0, which

implies Q(ξ)µ Ψ

(ξ)T = 0, the gauge fermionΨ (ξ)

T is determined uniquely.

2.2. Massive topological QCD: equivariant part

So far we have implicitely assumed that the complex fieldφ does not induce a centralextension of the SUSY algebra (1), or equivalently, that its vacuum expectation value iszero [25]. Now we remove that restriction and considerN = 2 SYM coupled to a massivehypermultiplet which, after twisting, leads to topological QCD.

In this more general case we are faced with aN = 2 SUSY algebra with two centralchargesZ andZ,

QAa,QB

b=−4εabεAB

(Z+ δG(φ)

),

QAa, QBb

=−2δab(σµ

)AB

(Pµ + iδG(Aµ)

),

(13)QAa,

QBb

= 4εabεAB(Z+ δG(φ)),

Z being the complex conjugate ofZ. Thereby,Z andZ, together withPµ, satisfy thecondition 4ZZ = PµPµ. The eigenvalues ofZ andZ are±m and±m, respectively, wherethe positive (negative) sign relates to the hypermultipletY (Y ). Here, we have introducedtwo central charges in order to ensure that, lateron, theR-symmetry remains unbroken ifwe formally assign toZ andZ theR-charges 2 and−2, respectively. (By coupling thegauge multiplet to the standard massive hypermultiplet [26], i.e., for only one (real) centralchargeZ = Z, theR-symmetry is broken intoZ2, cf. Ref. [4].)

In order to construct the topological action with central chargesZ and Z we mayconsider in a(D = 6)-dimensional space–time [23] aN = 1 SYM whose gauge multipletAM,λa is coupled to a (massless) hypermultipletψ,ζa in some representationR ofthe gauge groupG (for a detailed presentation, see, Appendix B). Then, one carries out a


Table 2

αA βA αA βA ζA ζA χA χA

Ghost number 1 1 −1 −1 0 0 0 0Mass dimension 3/2 3/2 3/2 3/2 1 1 2 2Scale dimension 1 1 2 2 1 1 2 2Z-, Z-charge 1 −1 −1 1 1 −1 1 −1Grassmann parity even even even even odd odd odd odd

nontrivial dimensional reduction onto a torus with respect to the extra spatial dimensionsx5 andx6 [27], thereby, introducing two (real) masses,m5=m+ m andm6= i(m−m),which are the inverse periods of the fields of the hypermultiplet with respect tox5 andx6.The extra componentsP5 andP6 of the generator of space–time translation are relatedto the central charges according toP5 = Z + Z andP6 = i(Z − Z), respectively. In thesame way, the extra componentsA5 andA6 of the gauge field, which are assumed to beindependent onx5 andx6, are identified with the complex scalar field according toA5=−i(φ+ φ) andA6= φ − φ.

After compactification, assuming that only the first nonzero modes are excited, thehypermultiplet Y (and its hermitean conjugateY ) becomes massive. The resultinghypermultipletY consists of two Weyl spinors being the components of the corresponding(Dirac) spinor,q = (αA, βA)T , aSp(2)-doublet of complex scalar fieldsζ a and, in order toclose the superalgebra (13) also off-shell, aSp(2)-doublet of complex auxiliary fieldsχa .After the twisting procedure the scalar fieldsζ a become the components of a bispinorfield ζA. The appearance of bispinors after twisting is a new feature of TQCD. Similarly,the auxiliary fieldχa should be replaced by a bispinor fieldχA. But, a further and somehowsurprising feature of the twisting procedure is that, in the presence of central charges,twisting of the superalgebra (13) does not lead to a topological superalgebra which closesoff-shell [28]—despite having started with a closed algebra!

However, one can proceed like in the case of twistingN = 2 conformal supergrav-ity [29]. Namely, replacingχa instead byχA through another bispinorχA the resultingtopological superalgebra closes off-shell [16]. Thus, the multiplets from which TQCD isconstructed consist of the (massless) twisted gauge multipletVT = Aµ,ψµ,χµν, η,φ,φ, λµν in the adjoint representation ofG and two twisted massive hypermultipletsYT =αA, βA, ζA, χA andYT = βA, αA, ζA,χA in the representationR. Their properties aredisplayed in Table 2.

The twisted action for the matter fields, including the terms for the new auxiliarly fieldsχA andχA, is

W(ξ)M =

∫d4x

iαA

(σµ

)AB

−→Dµα

B + iβA←−Dµ

(σµ

)ABβB − 2mβAαA

+ iαA(σµ

)ABψµζ

B − iζA(σµ

)ABψµβB − 2αA(φ +m)βA

− iχA(σµ

)AB

−→Dµζ

B + iζA←−Dµ

(σµ

)ABχB − 2χAχA + 2mζA(φ +m)ζA


− ξ(

14βA

(σµν

)ABχµνζB + 1

4ζA

(σµν

)ABλµνζB − 1

4 ζA(σµν

)ABχµναB

(14)

+ ζAηαA − ζAφ(φ +m)ζA+ 2βAφαA + βAηζA − ζA(φ +m)φζA

),

where the covariant derivatives in theR-representation are given by

−→Dµ = −→∂ µ +Aµ, ←−

Dµ =←−∂ µ −Aµ.In the Feynman type gauge (ξ = 1) this part of the action is symmetric inm and m.Furthermore, let us recall that in the twisted theory theR-symmetry is identified with theghost number conservation. Hence, by inspection of (14), we observe that the mass termspreserve the ghost number if we formally assign tom andm the ghost number 2 and−2,respectively.

Let us now consider the transformation law of the twisted hypermultiplets.(i) The topological shift symmetryQ takes the form

QζA = αA, QαA =−(φ +m)ζA, QβA = χA, QχA =−(φ +m)βA,

(15)

QζA = βA, QβA = ζA(φ +m), QαA = χA, QχA = αA(φ +m),

again, it is independent of the choice of the gauge parameterξ . But now, the anticommuta-tor of the operatorQ includes, besides the gauge transformation−2δG(φ), form = 0 alsothe central charge transformationZ. Together with those ofZ they are given by

ZVT = 0, ZYT =mYT , ZYT =−mYT ,(16)ZVT = 0, ZYT = mYT , ZYT =−mYT .

(ii) The transformation rules for the vector supersymmetryQ(ξ)µ are given by

Q(ξ)µ ζA = i(σµ)ABβB , Q(ξ)

µ βA = i(σµ)AB(ξφ + m)ζB,Q(ξ)µ αA =−i(σµ)ABχB +

−→Dµζ

A,

Q(ξ)µ χA =−i(σµ)AB(ξφ + m)αB − ξi(σµ)ABηζB +

−→DµβA,

Q(ξ)µ ζA =−i(σµ)ABαB , Q(ξ)

µ αA = i(σµ)AB ζB(ξφ + m),Q(ξ)µ βA = i(σµ)ABχB + ζA

←−Dµ,

(17)Q(ξ)µ χA =−i(σµ)ABβB(ξφ + m)− ξi(σµ)AB ζBη+ αA←−Dµ.

As before, the anticommutatorQ(ξ)

µ , Q(ξ)ν

does no longer coincide with theξ -dependent

gauge transformation−2ξδµνδG(φ) alone but in addition it includes the (complexconjugate) central charge transformationZ.

By a rather lengthy calculation it can be proven that the transformations (15)–(17)together with (8) and (9) satisfy the following topological superalgebra:

Q,Q = −2(Z + δG(φ)

),


Q, Q(ξ)

µ

=−iPµ + δG(Aµ),(18)

Q(ξ)µ , Q(ξ)

ν

=−2δµν(Z+ ξδG(φ)).

By making use of theQ-transformations (15) it is easy to show that the twisted actionW(ξ)M can be written as an exactQ-cocycle,

(19)W(ξ)M =QΨ (ξ)

M with Q(ξ)µ Ψ

(ξ)M = 0, ZΨ

(ξ)M = 0, ZΨ (ξ)

M = 0,

where them-dependent gauge fermionΨ (ξ)M is given by

(20)

Ψ(ξ)M =−

∫d4x

iαA

(σµ

)AB

−→Dµζ

B − iζA←−Dµ

(σµ

)ABβB + m

(ζAα

A − βAζA)

+ αAχA + χAβA + ξ(ζAφα

A − βAφζA + 14 ζA(σ

µν)ABχµνζB).

Let us stress that the gauge fermionΨ (ξ)M is completely specified by the topological shift

symmetryQ and the vector supersymmetryQ(ξ)µ . Furthermore, identifyingZ = Z and,

therefore,m= m, we obtain for the matter action, Eq. (14), the result of [15].Hence, the twistedN = 2 SYM coupled to the massive hypermultipletsYT andYT is

described by the action

(21)W(ξ) =W(ξ)T +W(ξ)

M =Q(Ψ(ξ)T +Ψ (ξ)

M

).

This defines completely the equivariant part of TQCD. It is invariant under the topologicalsuperalgebra (18) and possesses the crucial property of being an exactQ-cocycle.

2.3. Massive topological QCD: complete action in Landau gauge

Next, we are faced with the problem to construct the complete action of TQCD. Thisis achieved by adding to the equivariant part a supplementary gauge-fixing term whichremoves the residual gauge degeneracy of the classical action thereby preserving theinvariance under the topological superalgebra (18). The last requirement turns out to bevery restrictive and can be fulfilled only when in (21) the Landau type gaugeξ = 0 hasbeen chosen.

To begin with, let us introduce the following prolonged BRST operator

(22)sQ = s +Q,which, according to the first of Eq. (18), is required to be nilpotent modulo the centralchargeZ, i.e., sQ, sQ = −2Z. In order to identifys with the (usual) nilpotent BRSToperator, let us introduce the gauge ghostC by assuming, as usual, that the topologicalghostφ is the exactQ-cocycle,φ = QC. Then, applyingQ,Q = −2(Z + δG(φ)) onC it follows that s is defined bys = δG(C) when acting on the gauge multipletV andsC = C2 when acting on the gauge ghost. The transformation law for the antighostCand the auxiliary fieldB will be defined bysC = B + C,C andsB = [C,φ] + [C,B].Notice, thatC, C andB take their values in Lie(G) and have vanishing central charges;their properties are summarized in Table 3.


Table 3

C C B

Ghost number 1 −1 0Mass dimension 1/2 3/2 2Scale dimension 0 2 2Z-, Z-charge 0 0 0Grassmann parity odd odd even

Together with the action of the topological shift symmetryQ, Eq. (8), we get the BRSTtransformation:

sQAµ =ψµ −DµC, sQψµ =Dµφ + C,ψµ,sQC = φ +C2, sQφ = [C,φ],sQφ = η+ [C, φ], sQη= [φ, φ] + C,η,sQχµν = λµν + C,χµν, sQλµν = [χµν,φ] + [C,λµν ],

(23)sQC = B + C,C, sQB = [C,φ] + [C,B].Now, we are in a position to define the complete action of TYM by adding to the

topological action (11) and (12) the remaining gauge fixing partWYM ,

(24)W(ξ)TYM =W(ξ)

T +WYM with W(ξ)T = sQΨ (ξ)

T , WYM = sQΨYM ,

whereΨYM is the YM gauge fermion in the Landau gauge,

(25)ΨYM = 2∫d4x tr

C∂µAµ.

(In principle, also here we could have choosen a gauge fermionΨ(ξ ′)YM in an intermediate

gauge with another parameterξ ′ which, for the Landau gauge, is set equal to zero.) Notice,thatW(ξ)

T can be rewritten in the formW(ξ)T = sQΨ (ξ)

T , sinceΨ (ξ)T is gauge invariant, i.e.,

sΨ(ξ)T = 0. Hence, by construction, the action (24) is invariant under the modified BRST

transformations, Eqs. (23). For the irreducible part of TYM we obtain

(26)WYM = sQΨYM = 2∫d4x tr

B∂µAµ −C∂µDµC − C∂µψµ

.

Let us now study the invariance properties ofWYM under theQ(ξ)µ -transformations.

By applying Q, Q(ξ)µ = −iPµ + δG(Aµ) on C and usingQC = φ we get Q(ξ)

µ C =−Aµ. This allows to cast the previous relation into the formsQ, Q(ξ)

µ = −iPµ. Then,by making use of this new relation, from the requirementssQC = B + C,C andsQB =[C,φ] + [C,B] we getQ(ξ)

µC = 0 andQ(ξ)

µ B =DµC. Now, recalling thatQ(ξ)µ leavesAµ

inert if ξ = 0, see Eq. (9), by applyingQ(ξ)

µ , Q(ξ)ν

=−2δµν(Z+ξδG(φ)) onC and usingQ(ξ)µ C = −Aµ we conclude that the irreducible partWYM cannot be invariant under the

vector supersymmetryQ(ξ)µ as long asξ = 0.


For that reason, from now on we shall choose the Landau type gauge,ξ = 0, for thetopological action. In that gauge the transformations of the complete gauge multiplet underthe vector supersymmetryQ(0)

µ read (cf. Eq. (9)):

Q(0)µ Aν = 0, Q(0)

µ ψν = Fµν, Q(0)µ C =−Aµ, Q(0)

µ φ =ψµ,Q(0)µ B =DµC, Q(0)

µC = 0, Q(0)

µ φ = 0,

Q(0)µ η=Dµφ, Q(0)

µ χρσ = δµρDσ φ − δµσDρφ + εµνρσDνφ,

Q(0)µ λρσ =Dµχρσ + δµρ

([φ,ψσ ] −Dση)− δµσ ([φ,ψρ ] −Dρη)(27)+ εµνρσ

([φ,ψν] −Dνη)

and for the massive hypermultiplets we obtain (cf. Eq. (17))

Q(0)µ ζA = i(σµ)ABβB , Q(0)

µ βA = im(σµ)ABζB,Q(0)µ αA =−i(σµ)ABχB +

−→Dµζ

A, Q(0)µ χA =−im(σµ)ABαB +

−→DµβA,

Q(0)µ ζA =−i(σµ)ABαB , Q(0)

µ αA = im(σµ)ABζB,(28)Q(0)

µ βA = i(σµ)ABχB + ζA←−Dµ, Q(0)

µ χA =−im(σµ)ABβB + αA←−Dµ.

The corresponding BRST transformations of the hypermultiplet read (cf. Eq. (15))

sQζA = αA +CζA, sQα

A =−(φ +m)ζA +CαA,sQβA = χA +CβA, sQχA =−(φ +m)βA +CχA,sQζA = βA − ζAC, sQβA = ζA(φ +m)+ βAC,

(29)sQαA = χA + αAC, sQχ

A = αA(φ +m)− χAC.By a tedious but straightforward calculation it can be proven that the transformations

(16), (23) and (27) together with (28) and (29) satisfy the following superalgebra:

(30)sQ, sQ = −2Z,sQ, Q(0)

µ

=−iPµ, Q(0)µ , Q(0)

ν

=−2δµνZ,which is obtained from (18) after substitutingQ by sQ and puttingξ equal to zero.

From (10) for the topological action in the Landau type gauge we obtain

(31)

W(0)T =

∫d4x tr

λµνFµν − 2χµνDµψν + 2ηDµψµ + 2φD2φ + 2φ

ψµ,ψµ

,

and from (14) for the matter action we immediately get

(32)

W(0)M =

∫d4x

iαA

(σµ

)AB

−→Dµα

B + iβA←−Dµ

(σµ

)ABβB − 2mβAα

A

+ iαA(σµ

)ABψµζ

B − iζA(σµ

)ABψµβB + 2mζA(φ +m)ζA

− iχA(σµ

)AB

−→Dµζ

B + iζA←−Dµ

(σµ

)ABχB − 2χAχA

− 2αA(φ +m)βA.


Here,W(0)M can be rewritten in the formW(0)

M = sQΨ (0)M , sinceΨ (0)

M is gauge invariant, i.e.,

sΨ(0)M = 0.After all, putting anything together, we arrive at the complete action of TQCD (in

Landau gauge) we are looking for,

(33)W(0)TQCD=W(0)

TYM +W(0)M = sQ

(Ψ(0)T +Ψ (0)

YM

)+ sQΨ (0)M ,

with

(34)Ψ(0)T =

∫d4x tr

χµνFµν + 2φDµψµ

,

(35)

Ψ(0)M =−

∫d4x

iαA

(σµ

)AB

−→Dµζ

B − iζA←−Dµ

(σµ

)ABβB

+ m(ζAα

A − βAζA)+ αAχA + χAβA

andΨYM given by Eq. (25). It is invariant under the topological superalgebra (30) andpossesses the crucial property of being an exactsQ-cocycle.

3. Noncovariant Landau type gauge

In Ref. [17] it has been shown that if instead of the gauge covariant choice,Dµψµ = 0,

cf. Eq. (34), the noncovariant Landau type gauge,∂µψµ = 0, is imposed then the action of

TYM obeys a larger set of global constraints than in the former case. In particular, it turnsout that the associated vector supersymmetry, in the following denoted simply byQµ,is linearly realized. Therefore, it can be employed as an additional stability constraint inorder to improve the ultraviolet behaviour of TYM, restraining the number of independentinvariant counterterms. Motivated by that result, it will be shown that the same gauge canbe imposed also in the case of TQCD and that, except for the so-called ghost for the ghostequation [17], there is an additional set of stability constraints which improve the finitenessproperties displayed by this model.

To begin with, let us first express the BRST transformations (23) by redefining thefieldsη, λµν andB, being only auxiliary ones, according to

η→ η− [C, φ], λµν→ λµν − C,χµν

, B→ B − C,C,

in such a way that all the fields belonging to the nonminimal sector occure as trivial BRST-doublets,

sQAµ =ψµ −DµC, sQψµ =Dµφ + C,ψµ, sQC = φ +C2,

sQφ = [C,φ], sQφ = η, sQη= 0, sQχµν = λµν, sQλµν = 0,

(36)sQC = B, sQB = 0.

Next, let us adopt the preceding construction of TQCD in the covariant Landau typegauge as a guiding principle for the gauge-fixed action of TYM in the noncovariant Landau


type gauge [17], namely, requiring

(37)WTYM = sQΨTYM , QµΨTYM = 0,

with the following modification of the gauge fermion (cf. Eqs. (25) and (34))

(38)ΨTYM =∫d4x tr

χµνFµν + 2φ∂µψµ + 2C∂µAµ

,

which consists in choosing a linear gauge condition not only for the ordinary gaugesymmetry but also for the topological shift symmetry. Then, by making use of (36), onegets

(39)

WTYM =∫d4x tr

(λµν −

C,χµν)Fµν − 2χµνDµψν + 2η∂µψµ

+ 2φ∂µ(Dµφ + C,ψµ

)+ 2B∂µAµ − 2C∂µ(ψµ −DµC).

Moreover, it is easily seen that this action exhibits, besides the BRST symmetry (36),invariance under the following linear vector supersymmetry [17]:

QµAν = 0, Qµψν = ∂µAν, QµC = 0, Qµφ = ∂µC,Qµφ = 0, Qµη= ∂µφ, Qµχρσ = 0,

(40)Qµλρσ = ∂µχρσ , QµC = ∂µφ, QµB = ∂µC − ∂µη.

These transformations are determined by the second of the Eq. (37) and the requirement,together with (36), to obey the topological superalgebra (30) without central extensions.

Now, we are faced with the problem to construct an action of TQCD which, on theone hand, bears in mind the noncovariant gauge choice∂µψ

µ = 0 and, on the other hand,satisfies the topological superalgebra (30) with the central chargesZ andZ. This problemamounts to look for a linear vector supersymmetryQµ of the matter fields, too, whichtogether with (29), (36) and (40) obeys the superalgebra (30). It is not difficult to convinceoneself that this requirement is indeed fulfilled by defining the action ofQµ on the matterfields as

QµζA = i(σµ)AB βB , QµβA = im(σµ)ABζB,

QµαA =−i(σµ)AB χB + ∂µζA, QµχA =−im(σµ)ABαB + ∂µβA,

QµζA =−i(σµ)AB αB , QµαA = im(σµ)AB ζB,

(41)QµβA = i(σµ)ABχB + ∂µζA, QµχA =−im(σµ)ABβB + ∂µαA.

The transformation rules (41) are obtained from (28) by simply replacing the gaugecovariant derivative through the ordinary partial derivative.

Since Qµ leaves the gauge fieldAµ inert, from (41) one infers that now the abovereplacement procedure simply can be repeated in order to get the action of TQCD:

(42)WTQCD=WTYM +WM, WM = sQΨM, QµΨM = 0,


with the following linearized gauge fermion (see Eq. (20))

(43)

ΨM =−∫d4x

iαA

(σµ

)AB∂µζ

B − i(∂µζA)(σµ)ABβB+ m(

ζAαA − βAζA

)+ αAχA + χAβA,

and, by making use of (29), the action corresponding to the matter fields reads:

(44)

WM =∫d4x

iαA

(σµ

)AB∂µα

B + i(∂µβA)(σµ

)ABβB − 2mβAα

A

+ iαA(σµ

)AB(∂µC)ζ

B − iζA(σµ

)AB(∂µC)βB + 2mζA(φ +m)ζA

− iχA(σµ

)AB∂µζ

B + i(∂µζA)(σµ

)ABχB − 2χAχA

− 2αA(φ +m)βA.

Finally, let us notice that the ghostC enters into the action (39) only as derivative∂µCas well as through the combinationsη − [C, φ], λµν − C,χµν andB − C,C. Theseglobal constraints can be expressed by the so-called ghost equation [30], usually valid inthe Landau type gauge. There exists still another set of global constraints, the so-calledghost for the ghost equation [17], being related to the noncovariant Landau type gauge. Itexpresses the fact that the ghost for the ghostφ enters into the action (39) only as derivative∂µφ as well as through the combinationB−[φ, φ]. On the other hand, by inspection of theaction (44) one observes that only the first of the above-mentioned ghost equation can beimposed as global constraint, whereas the second one leads to a nonlinear breaking term.

4. Slavnov–Taylor identity and stability constraints

In the previous section we have constructed the complete gauge-fixed action (42) ofTQCD being invariant under the whole set of symmetry operatorssQ, Qµ, Pµ, Z andZ obeying the topological superalgebra (30). Our aim is now to collect all the symmetryproperties of that action into auniqueWard identity—the Slavnov–Taylor identity. Thiscould be achieved in the conventional Batalin–Vilkovisky (BV) approach [31] where forevery field a corresponding antifield with opposite Grassmann parity is introduced andthen the symmetry properties are compactly formulated by some master equation. Here,however, we adopt the completely equivalent method of Ref. [32], where the antifields(sources) are introduced only for the fields belonging to the minimal sector, and wherethe symmetry operatorssQ, Qµ, Pµ, Z andZ are collected into a unique nilpotent BRSToperatorsT by associating to each of the generatorsQµ, Pµ, Z andZ a global ghostρµ,ξµ, ξ andξ , respectively.

In this context let us recall that identifying theR-charge with the ghost number hasthe meaning of setting the global ghostρ associated to the generatorQ equal to one, i.e.,ρ = 1. This already has been done by defining the operatorsQ according tosQ = s+Q andregardingQ as a BRST-like operator. Then, the aforementioned Slavnov–Taylor identitycan be derived in the usual manner by coupling the antifields to the nonlinear parts of field


Table 4

ρµ ξµ ξ ξ

Ghost number 2 1 −1 3Mass dimension 0 −1/2 −1/2 −1/2Grassmann parity even odd odd odd

transformations generated bysQ. In addition, the action (42) obeys a set of gauge-fixingand antighost conditions as well as global constraints being related to the noncovariantLandau type gauge.

4.1. Introduction of antifields

Let us begin by introducing a set of global ghostsρµ, ξµ, ξ and ξ , associated,respectively, to the generatorsQµ, Pµ, Z andZ, and defining, in this way, the total BRSToperator

sT = sQ + ρµQµ − iξµPµ + ξZ+ ξZ, with sT ρµ = 0,

(45)sT ξµ =−ρµ, sT ξ = 1, sT ξ =−ρµρµ.

Here, the transformation rules of the global ghosts are chosen in such a way thatsT isstrictly nilpotent. Then, it holds

sT WTQCD= 0, sT , sT = 0,

where all the relevant features of the superalgebra (30) are now encoded in the nilpotency ofthe unique BRST operatorsT . The properties of the global ghosts are displayed in Table 4.

Let us now introduce the antifields of the minimal sector,V ∗T = A∗µ,ψ∗µ,C∗, φ∗,transforming according to

(46)sQψ∗µ =A∗µ, sQA

∗µ = 0, sQφ

∗ = C∗, sQC∗ = 0

and

(47)Qµψ∗ν = 0, QµA

∗ν = ∂µψ∗ν , Qµφ

∗ = 0, QµC∗ = ∂µφ∗,

so that the superalgebra (30) is satisfied. In particular, from Eq. (46) it is obvious that theantifields are grouped in BRST-doublets.

Next, we extend the action (39) by adding a pure BRST invariant termsQΥT,

(48)STYM =WTYM + sQΥT,

with

ΥT =−∫d4x tr

A∗µAµ +ψ∗µψµ −C∗C + φ∗φ

(49)

sQΥT =∫d4x tr

A∗µ

(sQA

µ −ψµ)−ψ∗µ(sQψ

µ)+C∗(sQC − φ)+ φ∗(sQφ).


Table 5

A∗µ ψ∗µ C∗ φ∗

Ghost number −1 −2 −2 −3Mass dimension 3 5/2 5/2 3Scale dimension 3 3 4 4Z-, Z-charge 0 0 0 0Grassmann parity odd even even odd

Notice, thatΥT, by virtue of (40), does not violate the vector supersymmetryQµ, i.e., itholdsQµΥT = 0. Then, by making use of (36), for the antifield dependent terms we obtain

(50)sQΥT =∫d4x tr

−A∗µDµC −ψ∗µ

(Dµφ +

C,ψµ)+C∗C2+ φ∗[C,φ]

.

Here, it is worthwhile do draw the attention to a particular feature of the operatorsQ.First, since the antifields in the transformation law (46) appear in BRST-doublets they donot contribute to the BRST cohomology [33]. Second, the antifields in (48) do not coupleto the linear partsψµ andφ of the field transformationssQAµ andsQC (cf. Eq. (36)).

Furthermore, as a useful hint we remark that also the fields of the minimal sector couldbe cast into BRST-doublets as well, namely,

sQAµ =ψµ, sQψµ = 0, sQC = φ, sQφ = 0.

This might be achieved by redefiningψµ andφ according to the replacements

ψµ→ ψµ +DµC and φ→ φ −C2.

For that reason, one expects that the cohomology of the operatorsQ is completely trivial[5,24]. However, needless to say, such redefinitions can be performed only at the lowestorder of perturbation theory.

Finally, let us display the properties of the antifields of the gauge multiplet in Table 5.Proceeding in the same manner as before, let us introduce for each matter field a corre-

sponding antifield,Y ∗T =α∗A, βA∗, ζ ∗A, χ A∗

andY ∗T =

βA∗, α∗

A, ζ A∗, χ ∗

A

, transforming

according to

sQα∗A = ζ ∗A, sQζ

∗A =−mα∗A, sQχ

A∗ = βA∗, sQβA∗ = −mχA∗,

(51)sQβA∗ = ζ A∗, sQζ

A∗ =mβA∗, sQχ∗A= α∗

A, sQα

∗A=mχ ∗

A,

and

QµχA∗ = i(σµ)ABα∗B, Qµα

∗A =−im(σµ)ABχ B∗,

QµβA∗ = −i(σµ)ABζ ∗B + ∂µχA∗, Qµζ

∗A = im(σµ)AB βB∗ + ∂µα∗A,

Qµχ∗A= i(σµ)ABβB∗, Qµβ

A∗ = im(σµ)ABχ ∗B,(52)Qµα

∗A=−i(σµ)AB ζ B∗ + ∂µχ ∗A, Qµζ

A∗ = −im(σµ)ABα∗B + ∂µβA∗,


which together with the central charge transformations

ZV ∗T = 0, ZY ∗T =−mY ∗T , ZY ∗T =mY ∗T ,(53)ZV ∗T = 0, ZY ∗T =−mY ∗T , ZY ∗T = mY ∗T ,

obey the superalgebra (30).Next, let us add also to the matter action (44) a pure BRST invariant termsQΥM,

(54)SM =WM + sQΥM,

with

ΥM =−∫d4x

α∗AαA + ζ ∗AζA + χ A∗χA − βA∗βA+ βA∗βA + ζ A∗ζA + χ ∗AχA − α∗AαA

(55)

sQΥM =∫d4x

− α∗A

(sQα

A −mζA)+ ζ ∗A(sQζ

A − αA)+ χ A∗(sQχA −mβA)+ βA∗(sQβA − χA)− βA∗(sQβA +mζA)+ ζ A∗(sQζA − βA)+ χ ∗

A

(sQχ

A +mαA)+ α∗A

(sQα

A − χA).

As before, by virtue of (41), it can be verified thatΥM does not spoil the vectorsupersymmetryQµ, i.e., it holdsQµΥM = 0. Thus, by making use of (29), for the antifielddependent terms we get

(56)

sQΥM =∫d4x

α∗A

(φζA −CαA)+ ζ ∗ACζA − χ A∗(φβA −CχA)

+ βA∗CβA − βA∗(ζAφ + βAC)− ζ A∗ζAC + χ ∗A(αAφ − χAC)

+ α∗AαAC

.

Again, here one observes that when carrying out in (29) the replacements

αA→ αA −CζA, βA→ βA + ζAC,χA→ χA −CβA, χA→ χA − αAC,

a doublet-like structure of the matter fields would follow,

sQζA = αA, sQα

A =mζA, sQβA = χA, sQχA =mβA,sQζA = βA, sQβA =−mζA, sQα

A = χA, sQχA =−mαA,

showing that the cohomology of the operatorsQ, when acting on the space of integratedlocal polynomials of the fields and antifields with vanishing central charge, would be trivial,too.

The properties of the matter-antifields are displayed in Table 6.Finally, putting together (48) and (54) we get the extended antifield-dependent action of

TQCD,

(57)STQCD= STYM + SM , sT STQCD= 0,

where any of its symmetry properties is encoded in the single operatorsT .


Table 6

α∗A

βA∗ α∗A

βA∗ ζ ∗A

ζA∗ χ ∗A

χA∗

Ghost number −2 −2 −2 −2 −1 −1 −1 −1Mass dimension 5/2 5/2 5/2 5/2 3 3 2 2Scale dimension 3 3 2 2 3 3 1 1Z-, Z-charge 1 −1 −1 1 1 −1 −1 1Grassmann parity odd odd odd odd even even even even

4.2. Slavnov–Taylor identity

We translate now the symmetry property (57) into the Slavnov–Taylor identity,

(58)ST (S)= 0,

of the classical actionSTQCD which, for notational simplicity, has been denoted byS. Here,ST (S) displays the following expansion with respect to the global ghosts,

(59)

ST (S)≡ SQ(S)+ ρµQµS − iξµPµS + ξZS + ξZS − ρµ ∂S∂ξµ+ ∂S∂ξ− ρµρµ ∂S

∂ξ,

whereQµ, Pµ, Z andZ denote the Ward operators of vector supersymmetry transforma-tions, space–time translations and central charge transformations in the space of fields andantifields, respectively. Furthermore, we represent the nilpotent BRST operatorsT by alinear nilpotent functional differential operator:

(60)ST = SQ + ρµQµ − iξµPµ + ξZ+ ξZ− ρµ ∂

∂ξµ+ ∂

∂ξ− ρµρµ ∂

∂ξ,

where Qµ, Pµ, Z and Z, together with SQ, are required to obey the topologicalsuperalgebra

(61)SQ,SQ = −2Z,SQ,Qµ

=−iPµ, Qµ,Qν

=−2δµνZ.From Eq. (61) it follows thatSQ becomes a nilpotent operator when acting on the space ofintegrated local polynomials with vanishing central charge (and being independent of theglobal ghosts). Finally, we obtainST ,ST = 0.

In (59) the operatorSQ(S) lumps together both the linear and nonlinear parts of theBRST transformations, the latter ones being expressed by derivatives with respect to theantifields (see Eqs. (36), (46), (49), (51) and (55)). It takes the form

SQ(S)=∫d4x tr

(δS

δA∗µ+ψµ

)δS

δAµ+

(δS

δC∗+ φ

)δS

δC−

(δS

δψµ−A∗µ

)δS

δψ∗µ

+(δS

δφ+C∗

)δS

δφ∗+B δS

δC + ηδS

δφ+ 1

2λµν

δS

δχµν


(62)

+∫d4x

(δS

δζ ∗A+ αA

)δS

δζA+mζA δS

δαA+

(δS

δβA∗+ χA

)δS

δβA

+mβAδS

δχA+

(δS

δαA+ ζ ∗A

)δS

δα∗A−mα∗A

δS

δζ ∗A

−(δS

δχA− βA∗

)δS

δχ A∗−mχA∗ δS

δβA∗+

(δS

δζ A∗+ βA

)δS

δζA

−mζA δSδβA+

(δS

δα∗A

+ χA)δS

δαA−mαA δS

δχA

+(δS

δβA+ ζ A∗

)δS

δβA∗+mβA∗ δS

δζ A∗−

(δS

δχA− α∗

A

)δS

δχ ∗A

+mχ ∗A

δS

δα∗A

.

FromSQ(S) one reads off thelinearizedSlavnov–Taylor operator

SQ =∫d4x tr

(δS

δA∗µ+ψµ

)δ

δAµ+ δS

δAµ

δ

δA∗µ+

(δS

δC∗+ φ

)δ

δC+ δS

δC

δ

δC∗

−(δS

δψµ−A∗µ

)δ

δψ∗µ− δS

δψ∗µδ

δψµ+

(δS

δφ+C∗

)δ

δφ∗+ δS

δφ∗δ

δφ

+B δ

δC + ηδ

δφ+ 1

2λµν

δ

δχµν

+∫d4x

(δS

δζ ∗A+ αA

)δ

δζA− δS

δζA

δ

δζ ∗A+

(δS

δβA∗+ χA

)δ

δβA

− δS

δβA

δ

δβA∗+

(δS

δαA+ ζ ∗A

)δ

δα∗A− δS

δα∗Aδ

δαA

−(δS

δχA− βA∗

)δ

δχ A∗+ δS

δχ A∗δ

δχA+mζA δ

δαA+mβA

δ

δχA

−mα∗Aδ

δζ ∗A−mχA∗ δ

δβA∗+

(δS

δζ A∗+ βA

)δ

δζA− δS

δζA

δ

δζ A∗

+(δS

δα∗A

+ χA)

δ

δαA− δS

δαA

δ

δα∗A

+(δS

δβA+ ζ A∗

)δ

δβA∗

− δS

δβA∗δ

δβA−

(δS

δχA− α∗

A

)δ

δχ ∗A

+ δS

δχ ∗A

δ

δχA−mζA δ

δβA

(63)−mαA δ

δχA+mβA∗ δ

δζ A∗+mχ ∗

A

δ

δα∗A

.

If the action S is a solution ofSQ(S) = 0 then, by a tedious but straightforwardcalculation, one can show thatSQ obeys the relationSQ,SQ = −2Z, where the centralcharge operatorsZ andZ are given by (cf. Eqs. (16) and (53))


Z/m=Z/m=

∫d4x

− αA δ

δαA− βA

δ

δβA− ζA δ

δζA− χA

δ

δχA+ βA δ

δβA+ αA δ

δαA

+ ζA δ

δζA+ χA δ

δχA+ α∗A

δ

δα∗A+ βA∗ δ

δβA∗+ ζ ∗A

δ

δζ ∗A

+ χ A∗ δ

δχ A∗− βA∗ δ

δβA∗− α∗

A

δ

δα∗A

− ζ A∗ δ

δζ A∗− χ ∗

A

δ

δχ ∗A

.

Furthermore, ifS is also a solution ofQµS = 0, whereQµ is given by (cf. Eqs. (40), (41),(47) and (52))

Qµ =∫d4x tr

∂µAν

δ

δψν+ ∂µψ∗ν

δ

δA∗ν+ ∂µC δ

δφ+ ∂µφ∗ δ

δC∗

+ ∂µφ δ

δC + (∂µC − ∂µη) δ

δB+ ∂µφ δ

δη+ 1

2∂µχρσ

δ

δλρσ

+∫d4x

(∂µζA + i(σµ)ABχB

) δ

δβA+ (

∂µζA − i(σµ)ABχB

) δ

δαA

+ (∂µβ

A∗ − im(σµ)ABα∗B) δ

δζ A∗+ (

∂µα∗A + im(σµ)AB βB∗

) δ

δζ ∗A

+ i(σµ)AB βBδ

δζA+ im(σµ)ABζB

δ

δβA+ (

∂µαA − im(σµ)ABβB

) δ

δχA

− i(σµ)AB αBδ

δζA+ im(σµ)AB ζB δ

δαA+ (

∂µβA − im(σµ)ABαB) δ

δχA

+ i(σµ)ABβB∗δ

δχ ∗A

+ im(σµ)ABχ ∗Bδ

δβA∗ +(∂µχ

∗A− i(σµ)AB ζ B∗

) δ

δα∗A

+ i(σµ)ABα∗Bδ

δχ A∗− im(σµ)AB χ B∗

δ

δα∗A+ (

∂µχA∗ − i(σµ)ABζ ∗B

) δ

δβA∗

,

then one can verify that the vector supersymmetryQµ allows to decompose the translationoperatorPµ as Qµ,SQ = −iPµ. Moreover, one can establish, in accordance with thesuperalgebra (61), that it holdsQµ,Qν = −2δµνZ.

4.3. Equations of motion and global constraints

Besides obeying the Slavnov–Taylor identity (58), the complete actionS turns out tobe characterized by further constraints, namely, both Landau gauge-fixing conditions (seeEqs. (39) and (50))

(64)δS

δB= 2∂µAµ,

δS

δη= 2∂µψµ,

and the corresponding antighost equations of motion

(65)δS

δC + 2∂µδS

δA∗µ=−2∂µψ

µ,δS

δφ+ 2∂µ

δS

δψ∗µ= 0,


where the terms on the right-hand side, being linear in the quantum fields, are classicalbreakings, i.e., they are not subjected to any specific renormalization.

As already emphasized in Section 3 the essential reason for imposing the noncovariantLandau type gauge relies on the fact that in such a case the actionS exhibits much largersymmetries than in a covariant one. In particular, it can be easily verified that in this gaugethe dependence ofS on the whole set of matter fields is completely fixed by the followinglinearly broken Ward identities (see Eqs. (44) and (56)),

δS

δχA=−i(σµ)

AB∂µζ

B − 2χA−Cχ ∗A,

δS

δαA= i(σµ)

AB∂µ

(δS

δζ ∗B+ αB

)+ 2

(δS

δχ A∗+mβA

)

+C(δS

δχA− α∗

A

)+

(δS

δC∗+ φ

)χ ∗A,

δS

δχA=−i(σµ)AB

∂µζB + 2χA − χ A∗C,

δS

δβA= i(σµ)AB

∂µ

(δS

δζ B∗+ βB

)− 2

(δS

δχ ∗A

−mαA)

−(δS

δχA− βA∗

)C − χ A∗

(δS

δC∗+ φ

),

δS

δαA=−i(σµ)

AB∂µα

B − 2mβA − α∗AC,

δS

δζ A= − i(σµ)

AB∂µ

(δS

δα∗B

+ χ B)+ 2m

(δS

δβA∗+mζA

)

−(δS

δαA+ ζ ∗A

)C − α∗A

(δS

δC∗+ φ

),

δS

δβA=−i(σµ)AB

∂µβB − 2mαA +CβA∗,

(66)

δS

δζA= − i(σµ)AB

∂µ

(δS

δβB∗+ χB

)+ 2m

(δS

δα∗A−mζA

)

−C(δS

δβA+ ζ A∗

)+

(δS

δC∗+ φ

)βA∗.

The stability constraints (64)–(66), which drastically reduce the number of independentinvariant counterterms, can be established to all orders of perturbation theory by using therenormalized quantum action principles [19].

Moreover, there exists a set of global constraints, usually valid in the Landau type gauge.The first one is the so-called ghost equation [30], which in the present case reads


(67)

GS =∫d4x

[A∗µ,Aµ

]+ [ψ∗µ,ψµ

]− [C∗,C

]+ [φ∗, φ

]− T i(ζ ∗AT iζA + α∗AT iαA− βA∗T iβA + χ A∗T iχA − ζ A∗ζAT i − βA∗βAT i

+ α∗AαAT i − χ ∗

AχAT i

),

with

G=∫d4x

δ

δC+

[C, δδB

]+

[φ,

δ

δη

]+ 1

2

[χµν,

δ

δλµν

],

where the terms on the right-hand side of that equation are linear classical breakings, i.e.,they will not get radiative corrections.

As usual, commuting the ghost equation (67) with the ST identity (58) one gets a furtherglobal constraint fulfilled by the actionS, namely, the Ward identity for the rigid gaugeinvariance,

(68)RS = 0,

whereR= ST ,G denotes the Ward operator for rigid gauge transformations in the spaceof fields and antifields, expressing the fact that all the (anti)fields belong to either theadjoint or theR-representation of the gauge groupG.

It is known [19] that the structure of the invariant counterterms entirely will be governedby a set of classical stability constraints, provided they can be extended to all orders ofperturbation theory. The proof, that the constraints (67) and (68) are extendable to anyorder of perturbation theory, can be appreciably simplified by adopting again the strategyof Ref. [32]. One associates to each operatorG andR a global ghost,γ andτ , respectively,which take their values in Lie(G), and introduces the following operator:

(69)OT = ST + tr

γG+ τR+ [τ, γ ] ∂

∂γ− γ ∂

∂τ+ τ2 ∂

∂τ

.

One easily verifies thatOT is nilpotent,

OT ,OT = 0.

Furthermore, if it can be proven that the integrated cohomology ofOT turns out tobe empty, then the integrated cohomology ofST is empty as well and, besides theST identity (58), the constraints (66) and (67) can be employed to single out invariantcounterterms. (The proof, that the integrated cohomology ofOT is indeed empty, willbe given in Section 5.) The properties of these additional global ghosts are displayed inTable 7.

5. BRST cohomology: anomalies and invariant counterterms

Let us discuss now the renormalizability of topological QCD in the framework ofthe algebraic BRST technique [19] which allows for a systematic study of the quantumextension of the BRST symmetry. In that framework the proof of renormalizability is


Table 7

γ τ

Ghost number 2 1Mass dimension 0 0Parity even odd

related to the characterization of some cohomology classes of the linearized ST-operatorST , Eq. (60), which turns out to be essential for the (possible absence of) anomalies and theconstruction of the invariant counterterms. Let us recall that both the anomalies∆A and theinvariant counterterms∆C of the (quantum) actionS are integrated local polynomials inthe (anti)fields with (mass) dimension four and ghost number, respectively, one and zero.In addition, they are constrained by the following consistency conditions

(70)ST ∆A = 0, ∆A = ST ∆A, gh(∆A)= 1,

and

SQ∆C= 0,∂∆C

∂ρµ= 0,

∂∆C

∂ξµ= 0,

∂∆C

∂ξ= 0,

(71)∂∆C

∂ξ= 0, gh(∆C)= 0.

Therefore, the cohomological relevant solutions of Eqs. (70) and (71) are the nontrivialcocycles of the integrated cohomology ofST andSQ, respectively. Let us mention thatboth,∆A and∆C, by virtue of [Z,ST ] = 0, must have vanishing central charge, i.e.,Z∆A = 0 andZ∆C= 0; analogously forZ.

In order to characterize the integrated cohomology ofST we introduce the filtration

FT = 2ρµ∂

∂ρµ+ ξµ ∂

∂ξµ− ξ ∂

∂ξ+ ξ ∂

∂ξ+ m ∂

∂m+m ∂

∂m,

which obviously, by virtue of[m∂/∂m,Z] = Z, induces a separation ofST according to

ST =∑n=0

S(n)T ,[FT ,S

(n)T

]= nS(n)T ,

S(0)T ≡ S(m=0)Q being them- (andm-) independent part ofSQ, Eq. (63). Since form = 0

andm= 0 the central charges vanish the operatorS(0)T is strictly nilpotent,S(0)T ,S(0)T

= 0.

According to the general results on cohomology [34], the integrated cohomology ofST isisomorphic to a subspace of the integrated cohomology ofS(0)T .

In order to characterize the integrated cohomology ofS(0)T we introduce the filtration

FQ =∫d4x tr

Aµ

δ

δAµ+ψµ δ

δψµ+ 1

2χµν

δ

δχµν+ 1

2λµν

δ

δλµν

+ C δ

δC +Bδ

δB+ φ δ

δφ+ η δ

δη+ 2C

δ

δC+ 2φ

δ

δφ


+∫d4x

αA

δ

δαA+ βA

δ

δβA+ ζ A δ

δζ A+ χA

δ

δχA

+ βA δ

δβA+ αA δ

δαA+ ζA δ

δζA+ χ A δ

δχ A

,

which has the structure of a counting operator. Therefore,FQ possesses the property of

decomposingS(0)T as follows:

S(0)T =∑n=0

S(n,0)T ,[FQ,S

(n,0)T

]= nS(n,0)T ,

whereS(0,0)T is just that linear,m- andm-independent part of the operatorSQ which alsodoes not depend onS:

S(0,0)T =∫d4x tr

ψµ

δ

δAµ+A∗µ

δ

δψ∗µ+ φ δ

δC+C∗ δ

δφ∗+B δ

δC+ η δ

δφ+ 1

2λµν

δ

δχµν

+∫d4x

αA

δ

δζ A+ ζ ∗A

δ

δα∗A+ βA δ

δζA+ ζ A∗ δ

δβA∗

+ χAδ

δβA+ βA∗ δ

δχ A∗+ χ A δ

δαA+ α∗

A

δ

δχ ∗A

.

Since in S(0,0)T all the (anti)fields appear in BRST-doublets, one concludes that the

integrated cohomology ofS(0,0)T is empty [33,34]. Thus, the integrated cohomology of theoperatorST is empty as well, due to the fact that it is, in turn, isomorphic to a subspaceof the integrated cohomology ofS(0,0)T . This result implies that the general solutions of theconsistency conditions (70) and (71) are given by

∆A = 0,

and

(72)∆C= SQ∆C, gh(∆C)=−1,

where ∆C is the most general integrated local polynomial in the (anti)fields, withdimension 7/2 and ghost number minus one. Hence, the ST identity (58) is anomaly freeand the invariant counterterms are trivial cocycles of the integrated cohomology ofSQ.

The absence of anomalies of the ST identity,SQ(S)= 0, as well as of the Ward identitiesQµS = 0, PµS = 0, ZS = 0 andZS = 0, with S being independent ofρµ, ξµ, ξ andξ ,implies that∆C, besides (71), is subjected also to the following consistency conditions,

(73)Qµ∆C= 0, Pµ∆C= 0, Z∆C= 0, Z∆C= 0.

Furthermore, from the ghost equation (67) and the Ward identity (68), which can beimposed to any order of perturbation theory, it follows that∆C is required to obey the


following constraints, too,

(74)G∆C= 0, R∆C= 0.

In order to prove that the stability constraints (67) and (68) can be extended to thequantum level it is sufficient to verify that the integrated cohomology of the nilpotentoperatorOT , Eq. (69), is empty. For that purpose, let us introduce the filtration

NT = FT + tr

2γ

∂

∂γ+ 2τ

∂

∂τ

,

which clearly induces a separation ofOT , namely,

(75)OT =∑n=0

O(n)T ,

[NT ,O

(n)T

]= nO(n)T ,

with

(76)O(0)T = S(0)T − tr

γ∂

∂τ

.

Since the integrated cohomology ofSQ(0) is empty and because the extra term in (76)

appears as BRST-doublet the integrated cohomology ofO(0)T is empty as well and,

therefore, with the same reasoning as before, the integrated cohomology ofOT is empty,too.

From the stability constraints (64)–(66) for∆C we get a further set of restrictions:

δ∆C

δB= 0,

δ∆C

δC + 2∂µδ∆C

δA∗µ= 0,

(77)δ∆C

δη= 0,

δ∆C

δφ+ 2∂µ

δ∆C

δψ∗µ= 0

and

δ∆C

δχA= 0,

δ∆C

δαA− i(σµ)

AB∂µδ∆C

δζ ∗B− 2

δ∆C

δχ A∗−C δ∆C

δχA− δ∆C

δC∗χ ∗A= 0,

δ∆C

δχA= 0,

δ∆C

δβA− i(σµ)AB

∂µδ∆C

δζ B∗+ 2

δ∆C

δχ ∗A

+ δ∆C

δχAC + χ A∗ δ∆C

δC∗= 0,

δ∆C

δαA= 0,

δ∆C

δζA+ i(σµ)

AB∂µδ∆C

δα∗B

− 2mδ∆C

δβA∗+ δ∆C

δαAC + α∗A

δ∆C

δC∗= 0,

(78)

δ∆C

δβA= 0,

δ∆C

δζA+ i(σµ)AB

∂µδ∆C

δβB∗− 2m

δ∆C

δα∗A+C δ∆C

δβA− δ∆C

δC∗βA∗ = 0.

The constraints (73) imply that∆C is required to be invariant under vector supersymme-try transformations, space–time translations and central charge transformations. From (74)one infers that the ghostC enters into∆C either as derivative∂µC or through the combi-nations

B − C,C, η− [C, φ] and λµν − C,χµν

,

respectively, and that∆C can be taken to be rigid gauge invariant.


Concerning the constraints (77) it follows that the auxiliary fieldsB andη cannot appearin ∆C and that the antighostsC andφ can enter only through the combinations

A∗µ − 2∂µC and ψ∗µ − 2∂µφ,

respectively. Furthermore, from (78) it follows that the matter fieldsχ A, χA, αA andβAcannot appear in∆C and thatαA, βA, ζ A and ζA can enter only through the followingcombinations:

ζ ∗A − i(σµ

)AB∂µα

B , χ A∗ + 2αA, C∗ + χAαA,ζ A∗ − i(σµ)AB

∂µβB, χ ∗A− 2βA, C∗ − βAχ A∗,

α∗A+ i(σµ)

AB∂µζ

B, βA∗ + 2mζA, C∗ − ζAα∗A,βA∗ + i(σµ)AB

∂µζB, α∗A + 2mζA, C∗ − βA∗ζA.Let us now turn to the computation of∆C. First of all we point out that the whole set

of constraints is stable under the action ofSQ and, therefore,∆C = SQ∆C satisfies theseconstraints if∆C will do. Although, generally,∆C does not have to obey them it may bechosen to do so, except for the constraintG∆C = 0 [17]. Since the constraints above arevery restrictive they will give rise to a rather small set of independent counterterms. Indeed,the only possible invariant counterterms which are compatible with the constraints (73),(74) and (77) are obtained from∆C = SQ∆C by choosing for∆C the following sevencombinations of fields and antifields,

∆C=∫d4x tr

z1

(A∗µ − 2∂µC

)Aµ + z1

(ψ∗µ − 2∂µφ

)ψµ

+ z2(ψ∗µ − 2∂µφ

)∂µC + z3χ

µν∂µAν + z4χµν[Aµ,Aν]

+

∫d4x

(z5χ

A∗ + z72αA)χA −

(z5β

A∗ + z7i(σµ

)AB∂µζB

)βA

+ (z6χ∗A− z72βA

)χA − (

z6α∗A+ z7i

(σµ

)AB∂µζ

B)αA

+ (z5α∗A + z72mζA

)αA + (

z5ζ∗A − z7i

(σµ

)AB∂µα

B)ζA

+ (z6β

A∗ + z72mζA)βA +

(z6ζ

A∗ − z7i(σµ

)AB∂µβB

)ζA

,

wherez1, z2, . . . , z7, are arbitrary coefficients. This set of independent counterterms isreduced further by applying the last constraints (78), leading to some relations among therenormalization factors, namelyz5= z7, z6= z7 andz7= 0, respectively. Hence, one endsup with only four possible invariant counterterms,

(79)

∆C=∫d4x tr

z1

(A∗µ − 2∂µC

)Aµ + z1

(ψ∗µ − 2∂µφ

)ψµ

+ z2(ψ∗µ − 2∂µφ

)∂µC + z3χ

µν∂µAν + z4χµν[Aµ,Aν]

.

Here, some remarks are in order. First, since all the quantum corrections to the actionS

are trivial SQ-cocycles there appears no physical coupling parameter in TQCD and,therefore, the coefficientsz1, z2, z3 andz4 are anomalous dimensions redefinig the fields


and antifields. Second, an unexpected feature of the noncovariant Landau type gaugeadopted here is the fact that the independent counterterms of TQCD agree with those ofTYM. This means, on the one hand, that the matter action of TQCD is finite, i.e., it doesnot receive any radiative corrections from higher loops, and, on the other hand, that therequirement of the ghost for the ghost equation is not actually necessary. Third, as alreadypointed out in Refs. [17,21], as a consequence of the fact that the Landau type gauge,ξ = 0,is stable under renormalization (i.e., that only in this gaugeξ receives no renormalization),the counterterm

∆C∼∫d4x tr

F+µνFµν+

,

disappears. This result is essential in preserving the topological nature of the model.Furthermore, the absence of that counterterm guarantees that theβ-function vanishes. Thisis in accordance with an one-loop computation carried out in Refs. [1,7].

6. Concluding remarks

In this paper we have studied the renormalizability of twistedN = 2 supersymmetrywith Z = 2 central charges. By coupling the gauge multiplet to the standard massivehypermultiplet, i.e., with only one central charge, theR-symmetry is broken intoZ2. Insuch a case we are faced with the situation that the ghost number of the gauge-fixed actionand, in consequence of this, the cohomology classes of the counter terms and anomaliesare not uniquely characterized. Here, it has been shown that this problem can be avoidedby introducing two central chargesZ andZ, being complex conjugate to each other, andformally ascribing to them, as well as to their eigenvalues±m and±m, theR-weights(ghost numbers)R(Z)=R(m)= 2R(Q) andR(Z)=R(m)= 2R(Qµ).

By adopting the noncovariant Landau type gauge and by making use of both thetopological shift symmetryQ and the vector supersymmetryQ it has been proven that thetwisted hypermultiplets are not subjected to any renormalization, i.e., the matter action ofTQCD is perturbatively finite. Thus, in that particular gauge, which should be as acceptableas any other gauge choice, TQCD is renormalizable with the same counter terms as TYM.Since the invariant counterterms are trivial BRST-cocycles no physical parameters appearin that model.

In this paper we have not analyzed the question whether the choice of a noncovariantgauge, which significantly differs from the original one of TYM, may eventually changethe construction of the topological observables of that model.

Furthermore, no attention has been paid to a possible nontrivialθ term in the topologicalaction. The question whether this term, without loss of generality, can be dropped at thebeginning or whether theθ angle does receive radiation corrections from higher loops willbe studied elsewhere.


Appendix A. Two-spinor notations in Euclidean space–time

The Weyl 2-spinor conventions in Euclidean space–time adopted in this paper are thoseof Appendix E in Ref. [35]. The matrices(σµ)AB and(σµ)AB , being invariant numericaltensors ofSL(2,C) if µ transforms according to the vector representation ofSO(4), aredefined by

(σµ)AB = (σα, iI2)

AB, (σµ)AB = (σα,−iI2)AB,

with σα (α = 1,2,3) being the Pauli matrices.Another set of invariant tensors are the antisymmetric matricesεAB andεAB ,

εAB =(

0 1−1 0

)= εAB, εAB =

(0 −11 0

)= εAB ,

with εAB andεAB being defined byεACεCB =−δAB andεACεCB =−δAB . These tensorsraise and lower the spinor indices according toψA = ψBεBA, ψA = εABψB andψA =ψBεBA, ψA = εABψB , respectively.

The matrices(σµ)AB and(σµ)AB satisfy the Clifford algebra

(σµ)AC(σν)CB + (σν)AC(σµ)CB = 2δµνδAB,

(σµ)AC(σν)CB + (σν)AC(σµ)CB = 2δµνδ

AB ,

and, in addition, the completeness relations

(σµ)AB(σν

)BA = 2δνµ, (σµ)AB(σµ

)CD= 2εACεBD,

(σµ)AB(σµ

)CD = 2δDAδCB , (σµ)

AB(σµ

)CD = 2εACεBD.

Since(σµ)AB is the hermitean conjugate of(σµ)AB , it holds(σµ)AB ≡ εACεBD(σµ)DC =(σµ)

BA and, lowering its indices,(σµ)AB = (σµ)BA. Hence,(σµ)AB and (σµ)AB aresymmetric in their spinor indices.

The self-dual and antiself-dualSO(4) generators(σµν)AB and (σµν)AB , respectively,being antisymmetric in their vector indices and symmetric in their spinor indices, aredefined by

(σµν)AB = (σµ)AC(σν)CB + δµνεAB, (σµν)

AB =+(σµν)AB(σµν)AB = (σµ)AC(σν)CB − δµνεAB , (σµν)AB =−(σµν)AB .

A vectorVµ and an antisymmetric tensorTµν are represented by

Vµ =−1

2(σµ)

ABVAB, VAB =(σµ

)ABVµ

and

Tµν = 1

4(σµ)

AB(σν)CDTABCD = T +µν + T −µν, TABCD = εBDT +AC + εACT −BD,

respectively, whereT ±µν =±(σµν)ABT ±AB is the (anti)self-dual part ofTµν .


Finally, some often used identities are

(σρ)AC(σµν)C

B = δρµ(σν)AB − δρν(σµ)AB − εµνρσ(σσ

)AB,

(σρ)AC(σµν)CB = δρµ(σν)AB − δρν(σµ)AB + εµνρσ

(σσ

)AB,

(σµν)AC (σρσ )C

B = 2(δµρδνσ − δνρδµσ − εµνρσ )εAB − δµρ(σνσ )AB+ δνρ(σµσ )AB − δνσ (σµρ)AB + δµσ (σνρ)AB ,

(σµν)AC(σρσ )CB = 2(δµρδνσ − δνρδµσ + εµνρσ )εAB − δµρ(σνσ )AB+ δνρ(σµσ )AB − δνσ (σµρ)AB + δµσ (σνρ)AB

and

(σµν)AB (σρσ )AB = 2(δµρδνσ − δνρδµσ − εµνρσ ),(

σµν)AB

(σµν)CD = 8εACεBD − 4εABεCD,

(σµν)AB(σρσ )AB = 2(δµρδνσ − δνρδµσ + εµνρσ ),(

σµν)AB(σµν)CD = 8εACεBD − 4εABεCD,

here, the antisymmetric tensorεµνρσ is normalized according toε1234= 1.

Appendix B. Twisting of N = 2 supersymmetric theories with two central charges Z

and Z

In this appendix we use the method of dimensional reduction in order to include centralcharges in the superalgebra ofN = 2 SYM coupled to two (massive) hypermultiplets (inthe fundamental and its conjugate representation ofSL(2,C)).

As is well known there exists a close relationship between extendedN = 2 SYMin D = 4 dimensions and simpleN = 1 SYM with gauge multiplet(AM,λ),M =1, . . . ,6, in D = 6 dimensions. The transition from the latter to the former is achievedby a trivial dimensional reduction, namely, by demanding that the gauge potentialAM

and the chiral spinorsλ, λ are independent of the extra dimensionsx5 and x6. Afterthat dimensional reduction the extra components ofAM simply become complex scalarfields,A5=−i(φ+ φ) andA6 = φ − φ, and the rotation group in(D = 6)-dimensionalEuclidean space–time is broken down according toSO(6) ⊃ SO(4) ⊗ SO(2). Afterreduction the chiral fields transform under the spinor representation of the universalcovering groupSL(2,C)⊗UR(1). (Sinceλ andλ are not subjected to a symplectic realitycondition theSp(2) internal symmetry of theN = 2 SYM is not accounted for in Ref. [23].This problem can be circumvented by reformulatingN = 1 SYM in such manner that theSp(2) symmetry will be manifest (see, below).)

There exist alsonontrivial dimensional reductions which allow to generate also centralcharges in both the massive matter multiplets [36] and the massive ghost excitations[37]. The central chargeZ in the standard massive hypermultiplet [26] occurs by onlycompactifying the sixth dimensionx6 into a circle [27] and reducing the fifth dimension


trivially. In order to get two central chargesZ andZ, being complex conjugate to eachother, one has to compactify the extra dimensionsx5 andx6 into a torus and to assume thatthe complex scalar fieldsζ , ζ and the antichiral Dirac spinorψ of which the (originallymassless) hypermultiplet consists are periodic inx5 andx6 with the (inverse) periodsm5=m+ m andm6 = i(m−m), respectively [27]. Thereby, the central charges are identifiedwith the extra components of the generator of space–time translations according toP5 =Z+ Z andP6= i(Z−Z), respectively.

In order to implement the central chargesZ and Z into the matter multiplet let usstart fromN = 1 SYM coupled to a (massless) hypermultiplet in(D = 6)-dimensionalEuclidean space–time in Wess–Zumino gauge:

(B.1)W(D=6) =W(N=1)SYM +W(Z=0)

M ,

which is built from an antihermitean vector fieldAM (M = 1, . . . ,6) and aSp(2)-doubletof chiral (symplectic) Majorana spinors [38]λa (a = 1,2) in the adjoint representation ofthe gauge group,

W(N=1)SYM =

∫d6x tr

14F

MNFMN − 12iλ

aΓ MDMλa,

and from a Sp(2)-doublet of complex scalar fieldsζa , ζ a ≡ ζ†a , and an antichiral

Dirac spinorψ in some representationR (with generatorsT i ), e.g., the fundamentalrepresentation, of the gauge group,

W(Z=0)M =

∫d6x

iψΓ MDMψ − iψλaζ a + iζaλaψ − 1

2

(ζaD

2Mζ

a)

− 1

4

(ζaT

iζ b)(ζbT

iζ a)

with

DM = ∂M +AiMT i.Here, the 8-dimensional Dirac matricesΓM andΓ7 are represented as follows:

Γµ = I2⊗ γµ, γµ =(

0 −(σµ)AB(σµ)AB 0

), µ= 1,2,3,4,

Γ4+α = σα ⊗ γ5, γ5=−i(δAB 0

0 −δAB), α = 1,2,3,

whereγµ andγ5 are the (usual) 4-dimensional Dirac matrices andσα are the Pauli matrices.They obey the relations

ΓM,ΓN = −2δMN I8, ΣMN =−1

2[ΓM,ΓN ],

with

Σµν = I2⊗ σµν, σµν =((σµν)

AB 0

0 (σµν)AB

), µ, ν = 1,2,3,4,

Σµ,4+α =−σα ⊗ γµγ5, Σ4+α,4+β = 1

2[σα,σβ ] ⊗ I4, α,β = 1,2,3.


In order to ensure that the action (B.1) is manifestly invariant under the internalsymmetry groupSp(2,R) ∼= SU(2) the chiral 8-spinorsλa and λa are required to obeyboth the Weyl conditionλa = iΓ7λa (chirality condition) and theSp(2) covariant Majoranacondition λa = −CΓ5λ

Ta (symplectic reality condition) [38], whereC is the charge

conjugation matrix. These conditions on the 8-spinorsλa and λa restrict them to be ofthe form

λa =

iλAa

00λAa

, λa = (

0,−iλAa, λAa,0), C = I2⊗

(εAB 0

0 εAB

),

with the chiral and antichiral 2-spinorsλAa and λAa = (λAa)†, respectively. TheSp(2)indexa is raised and lowered as follows,λAa = λAbεba andλAa = εabλAb whereεab isthe invariant tensor ofSp(2), εacεcb =−δab (analogously forλAa).

The antichiral 8-spinorsψ with ψ =−iΓ7ψ andψ = iψ†Γ4 are of the form

ψ =

0iβAαA

0

, ψ = (−iβA,0,0, αA)

,

where the Weyl 2-spinorsαA, βA andαA, βA transform according to the fundamental andits hermitean conjugate representation ofSL(2,C), respectively.

The action (B.1) is invariant under the gauge transformationsδG(ω) with ω≡ ωiT i ,δG(ω)Aµ =−Dµω, δG(ω)λa = [ω,λa], δG(ω)λ

a = [ω, λa],

(B.2)

δG(ω)ψ = ωψ, δG(ω)ψ =−ψω, δG(ω)ζa = ωζa, δG(ω)ζa =−ζaω,

and the following rigid on-shell supersymmetry transformationsδQ = ρAa QAa−ρAaQAa

with the constant chiral symplectic Majorana spinorsρa andρa ,

δQAµ = 1

2ρaΓ µλa − 1

2λaΓµρa, δQλa =−1

2iΣµνFµνρa + iT i

(ζaT

iζ b)ρb,

δQλa = 1

2iρaΣµνFµν − iρb

(ζbT

iζ a)T i, δQζ

a = 2ρaψ,

δQψ = iΓ µ−→Dµζaρa, δQζa = 2ψρa, δQψ =−iρaζa←−DµΓµ.

The corresponding 8-component spinorial supercharges

Qa =

iQAa

00QA

a

, Qa =

(0,−iQA

a,QAa,0),

obey, together with the generatorsPM (M = 1, . . . ,6) of space–time translations, theN = 2 supersymmetry algebra,

(B.3)Qa ⊗ Qb + Qb ⊗Qa .=−δab( I8+ iΓ7)ΓM

(PM + iδG(AM)

),


where the symbol.= means that the algebra is satisfied only on-shell, i.e., by taking into

account the equations of motion. This algebra can also be closed off-shell at the expenseof introducing two sets of auxiliary fields, namely theSp(2)-vector fieldχab = χba and thetwo conjugateSp(2) 2-spinor fieldsχa andχa ≡ χ†

a .Let us now compactify the fifth and sixth dimension by a nontrivial dimensional

reduction, demanding thatAM andλa are independent onx5 andx6 whereasζa andψare periodic inx5 andx6 with the (inverse) periodsm5 andm6, respectively,

∂5,6AM = 0, ∂5,6λa = 0, ∂5,6ζa = im5,6ζ

a, ∂5,6ψ = im5,6ψ.

(Here it has been assumed that the higher modes are not stimulated.) We further define

A5=−i(φ+ φ), A6= φ − φ, m5=m+ m, m6= i(m−m),where the independence of the gauge transformations uponx5 andx6 have madeA5 andA6 into a complex scalar fieldφ, φ ≡ φ†.

After this procedure the dimensional reduced action in four dimensions becomes

W(D=4) =W(N=2)SYM +W(Z=2)

M ,

with

(B.4)

W(N=2)SYM =

∫d4x tr

1

4FµνFµν − 2

(Dµφ

)(Dµφ)− 2[φ, φ][φ, φ]

− iλAa(σµ

)ABDµλBa + λAa[φ,λAa] + λAa

[φ, λAa

]and

W(Z=2)M =

∫d4x

iαA

(σµ

)AB

−→Dµα

B − 2αA(φ +m)βA − i(αAλAa + βAλAa

)ζ a

+ iβA←−Dµ

(σµ

)ABβB − 2βA(φ + m)αA + iζa

(λA

aαA + λAaβA)

(B.5)

+ 1

2

(Dµζa

)(Dµζ

a)+ ζaφ +m, φ + mζ a − 1

4

(ζaT

iζ b)(ζbT

iζ a),

which is obviously invariant under the internal symmetry groupSp(2) ⊗ UR(1) if we(formally) ascribe tom and m the sameR-charges as toφ and φ. The correspondingon-shell supersymmetry transformations in the presence of the central charges are

δQAµ = ρAa(σµ)ABλBa − λBa(σµ)ABρAa, δQφ = iρAaλAa,δQφ =−iλAaρAa,δQλA

a = − 1

2i(σµν

)ABρBaFµν + 2i[φ, φ]ρAa + iT i

(ζbT

iζ a)ρAb

+ 2ρBa(σµ

)ABDµφ,

(B.6)

δQλAa =1

2iρB a

(σµν

)ABFµν − 2iρAa[φ, φ]

− iρAb(ζaT

iζ b)T i + 2

(σµ

)ABρBaDµφ,


and

δQζa = 2ρAaβA + 2ρAaαA,

δQαA =−iρBa

(σµ

)AB−→Dµζ

a + 2ρAa(φ +m)ζ a,δQβA = iρBa

(σµ

)AB

−→Dµζ

a + 2ρAa(φ + m)ζ a,δQζa = 2βAρ

Aa + 2αAρAa,

δQβA =−iζa←−Dµ

(σµ

)ABρBa + 2ζa(φ +m)ρAa,

(B.7)δQαA = iζa←−Dµ

(σµ

)ABρB

a + 2ζa(φ + m)ρAa.Identifying the central charges with certain combinations of the space–time translations onthe torus, namelyP5 = Z + Z andP6 = i(Z−Z), and reverting to a two-spinor notationthe supersymmetry algebra (B.3) can be recast into the form

QAa,QB

b .=−4εabεAB

(Z+ δG(φ)

),

QAa, QBb

.=−2δab(σµ

)AB

(Pµ + iδG(Aµ)

),

(B.8)QAa,

QBb

.= 4εabεAB(Z+ δG(φ)),

where the central charge transformations are given by

ZV = 0, ZV = 0

for the on-shell gauge multipletV = Aµ,λA

a, λAa, φ, φ

and

ZY =mY, ZY =−mY , ZY = mY, Z Y =−mYfor the (massive) on-shell hypermultipletsY =

αA, βA, ζa

andY = αA, βA, ζa

being

hermitean conjugate to each other.In order to derive the twisted actions (10) and (14) we identify in (B.4) and (B.5) the

internal indexa with the spinor indexA. It is precisely that identification which defines thetwisting procedure [2]. In addition, we introduce another set of auxiliary fields,χAB = χBAandχA, χA, in order to get an off-shell realization of the twistedN = 2 superalgebra. Thisgives

(B.9)

WTSYM =∫d4x tr

14F

µνFµν − 2(Dµφ

)(Dµφ)− 2[φ, φ][φ, φ] + χABχBA

− iλAC(σµ

)ABDµλBC + λAB [φ,λAB] + λAB

[φ, λAB

]and

(B.10)

WM =∫d4x

iαA

(σµ

)AB

−→Dµα

B − 2αA(φ +m)βA − i(αAλAB + iβAλAB

)ζB

+ iβA←−Dµ

(σµ

)ABβB − 2βA(φ + m)αA + iζB

(λA

BαA + λAB βA)

+ 12D

µζADµζB + ζAφ +m, φ + mζA+ ζAχABζB − 2χAχA

,

with the off-shell hypermultipletsYT =αA, βA, ζ

A; χA

andYT =αA, βA, ζA;χA

.


Now we are able to construct the complete set ofN = 2 twisted generators,

Q= 1

2εABQAB, Qµ = 1

2i(σµ)

AB QAB, Qµν = 1

2(σµν)

ABQAB,

being, respectively, a scalarQ, a vectorQµ and a self-dual tensorQµν , by substituting

in (B.6) and (B.7) forρAa andρAa the following expressions,

ρAB = 1

2ρεAB, ρAB = 1

2iρµ(σµ)

AB, ρAB = 1

2ρµν(σµν)

AB,

whereρ, ρµ andρµν are some new global symmetry parameters associated toQ, Qµ andQµν , respectively. Then, again disregarding the generatorQµν , the twisted actions (B.9)and (B.10) will be separately invariant under the following twisted supersymmetrytransformationsδT = ρQ+ ρµQµ,

δT Aµ = ρAC(σµ)ABλBC − λBC(σµ)ABρAC,δT φ = iρAB λAB, δT φ =−iλABρAB,δT λAC =−1

2i(σµν)ABρ

BCFµν + 2i[φ, φ]ρAC − 2iχCBρAB

+ 2ρBC(σµ

)ABDµφ,

δT λAC = 1

2iρBC

(σµν

)ABFµν − 2iρA

C [φ, φ] + 2iρABχB

C + 2(σµ

)ABρBCDµφ,

(B.11)

δT χAB =−1

2DµλCA

(σµ

)CDρDB + i

[φ,λCA

]ρCB

− 1

2ρCA

(σµ

)CDDµλDB + iρCA

[λCB, φ

]+ (A↔ B)

and

δT ζB = 2ρAB βA + 2ρA

BαA,

δT αA =−iρBC(σµ)AB

−→Dµζ

C + 2ρAB(φ +m)ζB − 2ρBAχB ,

δT βA = iρBC(σµ)AB−→Dµζ

C + 2ρAB(φ + m)ζB − ρBBχA,

(B.12)

δT χA =−1

2iβB←−Dµ(σ

µ)ABρCC + αA(φ +m)ρCC − 1

2iζBλ

ABρCC

+ iαB←−Dµ(σµ)BCρ

AC + 2βB(φ + m)ρAB − iζCλBCρAB,

δT ζB = 2βAρAB + 2αAρAB,

δT αA = iζC←−Dµ

(σµ

)ABρB

C + 2ζB(φ + m)ρAB + χAρBB,δT βA =−iζC←−Dµ

(σµ

)ABρBC + 2ζB(φ +m)ρAB + 2χBρBA,

(B.13)

δT χA =1

2iρC

C(σµ

)AB

−→Dµα

B − ρCC(φ +m)βA −1

2iρC

CλABζB

− iρAC(σµ

)CB−→DµβB − 2ρAB(φ + m)αB − iρABλBCζC.


In order to show that the action (B.9) and (B.10) is equivalent to the topological actionas given by (7) and (13) we carry out the following replacements:

χAB→ χAB − 1

4

(σµν

)ABFµν, χA→ χA − 1

2iζB←−Dµ

(σµ

)AB,

χA→ χA +1

2i(σµ

)AB

−→Dµζ

B,

and we revert the spinor notation to the more familar vector notation by introducing thefollowing set of fields:

ψµ =−1

2(σµ)AB λ

AB, η=−1

2iεABλ

AB,

χµν =−1

2i(σµν)ABλ

AB, λµν = 1

2(σµν)ABχ

AB.

Then, one easily verifies that the resulting action is just the topological action (7), (13) forξ = 1 with the Pontryagin term subtracted, i.e., it is determined by the invariant polynomialtrφ2 through the very attractive form [22],

WTSYM =WT − 1

4

∫d4x trFµνFµν

=− 1

24εµνρσ Qµ

QνQρQσ

∫d4x trφ2,

thus also giving a suggestive idea of the usefulness of the vector operatorQµ. Performingthe same replacements in (B.10) we arrive at the matter action (14) forξ = 1.

Finally, in order to establish the relationship between the twistedN = 2 SYM and TYMone identifies theR-charge with the ghost number, i.e., one goes over from a conventionalQFT to a cohomological one [4,5]. This is achieved by simply setting in (B.11)–(B.13)the ghostρ associated to the singlet operatorQ equal to one, i.e.,ρ = 1. Thereby, theghost numbers of the remaining fields have to be redefined. In this way one recoversthe topological shift symmetry and the vector supersymmetry introduced in (8), (9), (15)and (17) forξ = 1.

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M.F. Sohnius, Nucl. Phys. B 138 (1978) 109.[27] M.F. Sohnius, Phys. Rep. 128 (1985) 40.[28] M. Alvarez, J.M.F. Labastida, Phys. Lett. B 315 (1993) 251;

M. Alvarez, J.M.F. Labastida, Nucl. Phys. B 437 (1995) 356.[29] A. Karlhede, M. Rocek, Phys. Lett. B 212 (1988) 51.[30] A. Blasi, O. Piguet, S.P. Sorella, Nucl. Phys. B 356 (1991) 154.[31] I.A. Batalin, G.A. Vilkovisky, Phys. Lett. B 102 (1981) 27;

I.A. Batalin, G.A. Vilkovisky, Phys. Rev. D 28 (1983) 2567.[32] C. Becchi, A. Blasi, G. Bonneau, R. Collina, D. Delduc, Commun. Math. Phys. 120 (1988) 121.[33] F. Brandt, N. Dragon, M. Kreuzer, Phys. Lett. B 231 (1989) 263;

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NT = 4 equivariant extension of the 3D topologicalmodel of Blau and Thompson

B. Geyera, D. Mülschb

a Universität Leipzig, Naturwissenschaftlich-Theoretisches Zentrum and Institut für Theoretische Physik,D-04109 Leipzig, Germany

b Wissenschaftszentrum Leipzig e.V., D-04103 Leipzig, Germany


Abstract

The Blau–ThompsonNT = 2, D = 3 non-equivariant topological model, obtained through theso-called ‘novel’ twist ofN = 4, D = 3 super-Yang–Mills theory, is extended to aNT = 4, D = 3topological theory. The latter, formally, may be regarded as a topological non-trivial deformationof the NT = 2, D = 4 Yamron–Vafa–Witten theory after dimensional reduction toD = 3. Forcompleteness also the dimensional reduction of the half-twistedNT = 2, D = 4 Yamron model isexplicitly constructed. 2001 Elsevier Science B.V. All rights reserved.

PACS: 11.15.E; 03.70; 02.40.PKeywords: Topological Yang–Mills theory; Dimensional reduction; ‘Novel’ twist

1. Introduction

Topological quantum field theory (TQFT) has become an interesting link betweenphysics and mathematics. It has connected diverse areas and many of the advanced ideasin QFT and string theory with the ones involved in topology (see, e.g., Refs. [1–13]).TQFTs with simple,NT = 1, topological symmetry have been widely studied in differentspace–time dimensions, e.g., the topological sigma models inD = 2 [14], the Chern–Simons gauge theory inD = 3 [4] and the Donaldson–Witten theory inD = 4 [3],namely, from both the perturbative and the non-perturbative point of view. TQFTs withextended,NT > 1, topological symmetry have also been considered, e.g., as effective worldvolume theories of D3-branes [15], D2-branes [16] and M5-branes [17] in string theorywrapping supersymmetric cycles of higher-dimensional compactification manifolds. Theyprovide a promising arena for testing key ideas asS-duality [18,19], large-N dynamics ofsupersymmetric gauge theories and, eventually, the AdS/CFT conjecture [20].

E-mail addresses: [email protected] (B. Geyer), [email protected] (D. Mülsch).




Usually, the topological supersymmetry is realized equivariantly, i.e., prior to theintroduction of gauge ghosts the cohomology closes only modulo equations of motions.However, by introducing the ‘novel’ topological twist of theN = 4, D = 3 super-Yang–Mills theory (SYM) Blau and Thompson [16] obtained aNT = 2, D = 3 topologicalmodel whose topological shift symmetry is strictly nilpotent even prior to the introductionof the gauge ghosts. Such theories are intrinsic forD = 3 and, obviously, quite special.After carrying out a dimensional reduction toD = 2 one obtains aNT = 4 Hodge-type cohomological theory [21]. In such theories, completely analogous to de Rhamcohomology, there exists besides the topological shift operator also a co-shift operatorboth of which are nilpotent and interrelated by a discrete Hodge-type duality.

Motivated by this interesting possibility we looked for an equivariantNT = 4 extensionof the Blau–Thompson model which might provide, after dimensional reduction, anotherNT = 8 Hodge-type cohomological theory. If such an equivariant extension would existit had to be equivalent to one of the various non-equivalentNT 2, D = 3 topologicaltheories.

A complete group theoretical classification of all the topological twists ofN = 4 andN = 8 super-Yang–Mills theory (SYM) inD = 3 has been given in [16]. According to thatanalysis in both cases there exist exactly two possible topological twists (see Diagram 1).

In the caseN = 4,D = 3 SYM the ‘standard’NT = 2 topological twist gives the super-BF model whereas the so-called ‘novel’NT = 2 topological twist by construction leadsto a model, henceforth called Blau–Thompson (BT) model, which precisely enjoys theabove mentioned property, i.e., which has no bosonic scalar fields and hence no underlyingequivariant cohomology.

Also in Ref. [16] it has been shown, without constructing the corresponding modelsexplicitly, that there exist only two (partial) topological twists ofN = 8, D = 3 SYM,provided one excludes theories involving higher spin fields. These models having anunderlyingNT = 4 andNT = 2 topological symmetry describe world-volume theoriesof D2-brane instantons wrapping supersymmetric three-cycles of Calabi–Yau three-foldsandG2-holonomy Joyce manifolds.

Moreover, in [16] it has been shown that theNT = 4, D = 3 model is just thedimensional reduction of either of the twoNT = 2, D = 4 models toD = 3, namely,the so-calledA-model constructed by Yamron [22] and Vafa–Witten [18] and theB-model constructed by Marcus [23], whereas theNT = 2, D = 3 model arises from thedimensional reduction of the ‘half-twisted’NT = 1,D = 4 theory [22].

In this paper we construct explicitly both theNT = 2 and theNT = 4 model inD =3 dimensions. Thereby, we restrict ourselves to Euclidean space–time. In searching ofcohomological Hodge-type theories this restriction is convenient since, after performinga dimensional reduction toD = 2, the topological co-shift symmetry is partially encodedin the vector supersymmetry. Under that restriction both theories can be characterizeduniquely by imposing besides the topological shift symmetryQa (and possiblyQa) alsothe vector supersymmetryQa

α (and possiblyQaα). Furthermore, the latter model coincides

with theNT = 4 equivariant extension of the Blau–Thompson model which we consider atfirst instance. The equivalence of both models obtains by deforming explicitly the Yamron–


Diagram 1. Different twisting ofN = 2,N = 4 andN = 8 SYM theories and interrelations of varioustopological theories inD = 3 andD = 4.

Vafa–Witten theory, i.e., theA-model, after dimensional reduction toD = 3. In addition,we remark that we have not been able to find an appropriate equivariant extension of thenovel topological model preserving the number of topological superchargesNT = 2. Onthe other hand, due to the completeness of the classification [16], such an extension shouldreally not be possible.


The paper is organized as follows. Following Ref. [16], in Section 2 we recall thestructure of the two possible topological twists ofN = 4, D = 3 SYM, leading to theNT = 2, D = 3 super-BF model [24–26] and the novelNT = 2, D = 3 Blau–Thompsonmodel. In Section 3 we construct aNT = 4 equivariant extension of the Blau–Thompsonmodel. In Section 4 it is shown that this extension coincides with the Yamron–Vafa–Witten theory after carrying out a dimensional reduction toD = 3, i.e., with theNT = 4topological twist ofN = 8,D = 3 SYM. In Section 5 we construct theNT = 2 topologicalmodel ofN = 8,D = 3 SYM which is the extension of theNT = 2,D = 3 super-BF modelby a spinorial hypermultiplet.

2. The two topological twists of N = 4, D = 3 SYM theory:super-BF and Blau–Thompson model

In this section we briefly recall the two possible topological twists ofN = 4 SYM inD =3 dimensional Euclidean space–time which obtains by dimensional reduction ofN = 1,D = 6 SYM, either directly or viaN = 2, D = 4 SYM, toD = 3 (cf., upper half of theDiagram 1). As pointed out in Ref. [27], this theory has a global(SU(2)R ⊗ SU(2)N )⊗SU(2)E symmetry, where theSU(2)R group primarily results from the symmetry of thefermions ofN = 1 SYM inD = 6.

After dimensional reduction the gauge multiplet ofN = 4 SYM inD = 3 contains threescalar fields which transform in the vector representation under the groupSU(2)N , theinternal Euclidean symmetry group arising from the decompositionSpin(6)→ SU(2)N ⊗SU(2)E . The symmetry groupSU(2)E is the Euclidean rotation group inD = 3.

There are only two essentially different possibilities to construct topological models withNT = 2 scalar topological supercharges, arising from twistingN = 4 SYM inD = 3 [16].

2.1. The super-BF model

The standard twist consists in replacingSU(2)E ⊗ SU(2)R through its diagonalsubgroup. This leads to the universal gauge multipletAα,ψ

aα ,φ

ab, ηa, a = 1,2; α =1,2,3, of theNT = 2, D = 3 super-BF model [24–26], which is built up from the gaugefield Aα , a SU(2)N doublet of Grassmann-odd topological ghost-antighost vector fieldsψaα = ψα,χα, a SU(2)N triplet of Grassmann-even ghost-for-ghost scalar fieldsφab =

φ, τ, φ, whereτ plays the role of a Higgs field, and aSU(2)N doublet of Grassmann-oddscalar fieldsηa = λ,η, respectively. (Let us recall, thatφab is symmetric,φab = φba .) Inorder to close the topological superalgebra (see Eq. (4) below) it is necessary to introducethe bosonic auxiliary vector fieldBα . All the fields are in the adjoint representation andtake their values in the Lie algebraLie(G) of some compact gauge groupG.

The twisted action of thisD = 3 super-BF (Casson or Euler character) model [24] withaNT = 2 off-shell equivariantly nilpotent topological supersymmetryQa is given by


(1)

SBF =∫

d3x trεαβγ Bγ Fαβ + εαβγ εabψ

aγDαψ

bβ − 2εabηaDαψb

α

− 2ηa[φab, η

b] − 2ψαa

[φab,ψ

bα

] + φabD2φab

− [φab,φcd ][φab,φcd

] − 2BαBα

,

whereFαβ = ∂αAβ − ∂βAα + [Aα,Aβ ] andDα = ∂α + [Aα, · ] is the YM field strengthand the covariant derivative in the adjoint representation, respectively;εαβγ is the totallyantisymmetric Levi-Civita tensor inD = 3, ε123= 1, andεab is the invariant tensor of thegroupSU(2)N , ε12 = 1. The internal indexa, which labels the differentNT = 2 charges,is raised and lowered as follows:ϕa = ϕbεba andϕa = εabϕb with εacεcb = −δab .

The action (1) can be cast into theQa -exact form

SBF = 12εabQ

aQbXBF,

where

XBF = SCS+∫d3x tr

εabψ

αaψbα + εabη

aηb,

with

SCS =∫

d3x trεαβγ

(Aα∂βAγ + 2

3AαAβAγ

),

being the Chern–Simons action. Let us notice, that the gauge bosonXBF is not uniquelyspecified, namely, substituting forεabηaηb the expression2

3εabφcd [φac,φbd ] gives thesame action. Since the BF-termεαβγ Bγ Fαβ in (1) has an on-shell first-stage reduciblegauge symmetry,δG(ω)Bγ = −Dγω, the gauge-fixing terms can be derived also by theBatalin–Vilkovisky procedure [13,28]. Let us notice, that the gauge-fixed action of theD = 3 super-BF theory can also be obtained by a dimensional reduction [29] of theNT =1,D = 4 Donaldson–Witten theory [3] which results by twisting theN = 2,D = 4 SYM(see Diagram 1).

The off-shell equivariantly nilpotent topological shift symmetryQa takes the form

QaAα =ψaα,

(2)

Qaφbc = 12ε

abηc + 12ε

acηb, Qaηb = −εcd[φac,φbd

],

Qaψbα =Dαφ

ab + εabBα, QaBα = −12Dαη

a − εcd[φac,ψd

α

],

it agrees with that of Refs. [18,30] (up to the ubiquitous factor of12 in front of ηa) and [16].

In addition, by restricting to flat Euclidean space–time the action (1) is invariant also underthe following vector supersymmetryQa

α ,

QaαAβ = δαβη

a + εαβγ ψγa,

Qaαφ

bc = −12ε

abψcα − 1

2εacψb

α,

Qaαη

b =Dαφab + εabBα,

Qaαψ

bβ = −εabFαβ + εabεαβγ B

γ − εαβγDγ φab + δαβεcd

[φac,φbd

],

(3)QaαBβ =Dαψ

aβ − 1

2Dβψaα + εαβγ D

γ ηa + εcd[φac, εαβγ ψ

γd − δαβηd].


By a straightforward calculation it can be verified that the four superchargesQa andQaα , together with the generatorPα = i∂α of space–time translations, obey the following

topological superalgebra:Qa,Qb

= −2δG(φab

),

Qa, Qbα

= εab(−iPα + δG(Aα)

),

(4)Qa

α,Qbβ

.= −2δαβδG(φab

),

where the symbol.= means that the corresponding relation is satisfied only on-shell, i.e.,

by taking into account the equation of motions. Since both the supersymmetriesQa andQaµ are realized non-linearly, the superalgebra (4) closes only modulo the field-dependent

gauge transformationsδG(ω), ω = Aα,φab, which are defined byδG(ω)Aα = −Dαω

and δG(ω)ϕ = [ω,ϕ], ϕ = φab, ηa,ψaα ,Bα. Let us emphasize that the form of the

action (1) is not completely specified by the topological supersymmetryQa , i.e., it is notthe most general action compatible with the gauge and theQa -invariance. Nevertheless,it turns out to be uniquely characterized by imposing the vector supersymmetryQa

α . TheconditionsQaSBF = Qa

αSBF = 0 fix all the relative numerical coefficients of the action (1),allowing, in particular, for a single coupling constant.

2.2. The Blau–Thompson model

The second twist ofN = 4 SYM in D = 3 consists in replacingSU(2)E ⊗ SU(2)Nthrough its diagonal subgroup. This yields the novelNT = 2 topological twist introducedby Blau and Thompson [16]. The gauge multipletAα,Vα,ψ

aα , η

a of this topologicalmodel is built up from the gauge fieldAα , a bosonic vector fieldVα , a SU(2)R doublet ofGrassmann-odd topological ghost-antighost vector fieldsψa

α = ψα,χα and aSU(2)Rdoublet of Grassmann-odd scalar fieldsηa = λ, η. In order to close the topologicalsuperalgebra (see Eq. (8) below) it is necessary to introduce a further set of bosonicauxiliary fields, namely two vector fieldsBα , Bα and a scalar fieldY , respectively.

The twisted action of this topological model is given by [16]

(5)

SBT = 12

∫d3x tr

− iεαβγBγ Fαβ(A+ iV )

− iεαβγ εabψaγDα(A+ iV )ψb

β − 4BαBα − 4Y 2

+ iεαβγ Bγ Fαβ(A− iV )− 2εabηaDα(A− iV )ψbα

− 4YDα(A)Vα,

and can be rewritten as sum of a BF-like topological term and aQa-exact term,

SBT = 12

∫d3x tr

iεαβγ Bγ Fαβ(A− iV )

+ 12εabQ

aQbXBT,

with the gauge boson

XBT = −14iSCS(A+ iV )−

∫d3x tr

iBαVα + 1

2εabηaηb

.


Here, the Chern–Simons actionSCS(A + iV ) is formed by the complexified gauge fieldAα + iVα . A striking, but somewhat unusual feature of this model is that there areno bosonic scalar fields and hence no underlying equivariantQa -cohomology (afterdimensional reduction the three scalar fields are combined to form the vector fieldVα).Another special feature is thatAα − iVα is Qa -invariant. Thus, as pointed out in [16],any gauge invariant functional ofAα − iVα , constrained byFαβ(A − iV ) = 0, is a goodobservable (e.g., bosonic Wilson loops). Moreover, since this twisted model differs fromtheD = 3 super-BF model by an exchange ofSU(2)R andSU(2)N , in Ref. [16] it has beenspeculated, that it can be regarded as providing a mirror description of the Casson model.

Let us now give the transformation laws which leave the action (5) invariant. The off-shell nilpotent topological supersymmetryQa takes the form [16]

QaAα =ψaα, QaVα = −iψa

α ,

Qaψbα = 2εabBα, Qaηb = −2iεabY,

QaBα = 0, QaBα = −Dα(A− iV )ηa,

(6)QaY = 0,

i.e., prior to the introduction of gauge ghosts, theNT = 2 topological supersymmetryQa

is not equivariant, but rather strictly nilpotent. In addition, by restricting to flat Euclideanspace–time, the action (5) is left invariant under the following vector supersymmetryQa

α ,

QaαAβ = δαβη

a − iεαβγ ψγa,

Qaαη

b = 2εabBα,

QaαBβ = −iεαβγDγ (A+ iV )ηa,

QaαVβ = −iδαβ ηa + εαβγ ψ

γa,

Qaαψ

bβ = −2εabFαβ(A)− 2iεabDα(A)Vβ + 2iεabεαβγ Bγ − 2iδαβε

abY,

QaαBβ = 2Dα(A)ψ

aβ −Dβ(A+ iV )ψa

α + iεαβγDγ (A− iV )ηa,

(7)QaαY = iDα(A− iV )ηa.

The scalar and the vector supercharges,Qa andQaα , together with the generatorPα of

space–time translations, satisfy the following topological superalgebra:Qa,Qb

= 0,Qa,Qb

α

= εab(−iPα + δG(Aα − iVα)

),

(8)Qaα,Q

bβ

.= 2iεabεαβγ(−iP γ + δG

(Aγ − iV γ

)).

As before, all relative numerical factors of the action (5), except for an overall uniquecoupling constant, are fixed by imposing the requirementsQaSBT =Qa

αSBT = 0.

3. NT = 4 equivariant extension of Blau–Thompson model

After having characterized the two possible topological twists ofN = 4,D = 3 SYM letus turn to the question whether the topological model constructed by Blau and Thompson


can be regarded as deformation of another one with underlyingNT 2 equivariantcohomology. Under the requirement of preserving the number of topological supercharges,NT = 2, we have not been able to find an appropriate equivariant extension of this model.However, if the conditionNT = 2 is relaxed, it is not difficult to show that this model,formally, can be recovered by deforming a cohomological theory with an extendedNT = 4topological supersymmetry. Such a theory has an additional global symmetry group, whichwill be denoted bySU(2)R. In Section 4 it will be shown that this cohomological theorycoincides with theNT = 4 topological twist ofN = 8 SYM inD = 3.

The construction of theNT = 4 equivariant extension of the Blau–Thompson topologi-cal model is governed by the following strategy.

First, after eliminating, by the use of the equations of motion, in the action (5) the auxil-iary fieldsBα , Bα andY the bosonic part of the resulting action involves the complexifiedgauge fieldsAα ± iVα. This bears a strong resemblance to the so-called topologicalB-twist of N = 4, D = 4 SYM studied by Markus [23] (recalling thatAα , Vα andY areanti-Hermitean). Hence,Bα should be regarded as the anti-Hermitean conjugate ofBα .

Second, we introduce aSU(2)R doublet of Grassmann-odd topological ghost-antighostvector fieldsψa

α = ψα, χα and aSU(2)R doublet of Grassmann-odd scalar fieldsηa =λ,η, which should be regarded as the Hermitean conjugate ofψa

α and ηa , respectively.Then, we construct anSU(2)R ⊗ SU(2)R invariant action by adding to (5) appropriateψa

α -andηa -dependent terms.

Third, we introduce aSU(2)R ⊗ SU(2)R quartet of Grassmann-even ghost-for-ghostscalar fieldsζ ab = φ, τ + iρ, τ − iρ, φ and ζ ab ≡ ζ ba = φ, τ − iρ, τ + iρ, φ, whereτ ± iρ plays the role of a complexified Higgs field.

Finally, we complete the action by adding suitableζ ab- and ζ ab-dependent termsanalogous to theφab-dependent terms in the action (1) of the super-BF model.

Proceeding in that way one gets the followingNT = 4 equivariant extension of theaction (5),

(9)

S(NT =4) = 12

∫d3x tr

− iεαβγBγ Fαβ(A+ iV )− iεαβγ εabψ

aγ Dα(A+ iV )ψb

β

− 2εabηaDα(A− iV )ψbα + 2ζab

ηa, ηb

+ 2ζabψαa,ψb

α

+ ζabD

2(A+ iV )ζ ab − [ζab, ζcd ][ζ ab, ζ cd

] − 4BαBα

+ iεαβγ Bγ Fαβ(A− iV )+ iεαβγ εabψaγ Dα(A− iV )ψb

β

− 2εabηaDα(A+ iV )ψbα + 2ζab

ηa, ηb

+ 2ζabψαa, ψb

α

+ ζabD

2(A− iV )ζ ab − [ζab, ζcd

][ζ ab, ζ cd

]− 4YDα(A)Vα − 4Y 2

,

which, by construction, is manifestly invariant under Hermitean conjugation or, equiva-lently, under the discrete symmetry

(Aα,Vα,Bα,Bα,Y

) → (Aα,−Vα,−Bα,−Bα,−Y

),

(10)(ψaα, ψ

aα , η

a, ηa, ζ ab, ζ ab) → (

iψaα ,−iψa

α , iηa,−iηa, ζ ab, ζ ab),


exchanging the scalar and the vector supercharges with their conjugate ones. Thus, theunderlying equivariant cohomology should be aNT = 4 supersymmetry. This is indeedthe case. By an explicit calculation one establishes that the action (9) is invariant under thefollowing off-shell equivariantly nilpotent topological supersymmetryQa ,

QaAα =ψaα, QaVα = −iψa

α ,

Qaζ bc = εacηb, Qaζ bc = εabηc,

Qaψbα = 2εabBα, Qaψb

α = 2Dα(A− iV )ζ ab,

Qaηb = 0, Qaηb = −2iεabY − 2εcd[ζ ac, ζ bd

],

QaBα = 0, QaBα = −Dα(A− iV )ηa − 2εcd[ζ ac, ψd

α

],

(11)QaY = iεcd[ζ ac, ηd

],

which, formally, may be regarded as deformation of the topological supersymmetrydisplayed in (6). This deformation is, of course, topological non-trivial since some of thefields, namelyηa , ψa

α , ζ ab andζ ab, must be deformed equal to zero. In addition, applyingthe discrete symmetry (10) onQa , which mapsQa to iQa

α , one gets a further one, namelythe conjugate topological supersymmetryQa ,

QaAα = ψaα ,

QaVα = iψaα ,

Qaζ bc = εacηb, Qaζ bc = εabηc,

Qaψbα = 2εabBα, Qaψb

α = 2Dα(A+ iV )ζ ab,

Qaηb = 0, Qaηb = 2iεabY − 2εcd[ζ ac, ζ bd

],

QaBα = 0, QaBα = −Dα(A+ iV )ηa − 2εcd[ζ ac,ψd

α

],

(12)QaY = −iεcd[ζ ac, ηd

],

proving, as promised, that the action (9) actually possesses aNT = 4 topologicalsupersymmetry.

By restricting to flat Euclidean space–time one can convince oneself by a rather lengthycalculation that the action (9) is also invariant under the following vector supersymmetryQaα :

QaαAβ = δαβη

a + iεαβγ ψγ a,

Qaαζ

bc = −εabψcα,

Qaαψ

bβ = −2iεαβγDγ (A− iV )ζ ab,

Qaαη

b = 2εabBα,

QaαBβ = iεαβγD

γ (A− iV )ηa,

QaαVβ = iδαβη

a + εαβγ ψγ a,

Qaαζ

bc = −εacψbα,

Qaαψ

bβ = − 2εabFαβ(A)+ 2iεabDα(A)Vβ − 2iεabεαβγ B

γ

+ 2δαβεcd[ζ ac, ζ bd

] + 2iδαβεabY,


Qaαη

b = 2Dα(A+ iV )ζ ab,

QaαBβ = 2Dα(A)ψ

aβ −Dβ(A− iV )ψa

α − iεαβγDγ (A+ iV )ηa

− 2εcd[ζ ac, δαβη

d + iεαβγ ψγd

],

(13)QaαY = −iDα(A+ iV )ηa − iεcd

[ζ ac,ψd

α

].

Combining Qaα with the discrete symmetry (10), which mapsQa

α into iQaα , gives rise to

the conjugate vector supersymmetryQaα ,

QaαAβ = δαβη

a − iεαβγ ψγa,

Qaαζ

bc = −εabψcα,

Qaαψ

bβ = 2iεαβγDγ (A+ iV )ζ ab,

Qaαη

b = 2εabBα,

QaαBβ = −iεαβγDγ (A+ iV )ηa,

QaαVβ = −iδαβ ηa + εαβγ ψ

γa,

Qaαζ

bc = −εacψbα,

Qaαψ

bβ = − 2εabFαβ(A)− 2iεabDα(A)Vβ + 2iεabεαβγ Bγ

+ 2δαβεcd[ζ ac, ζ bd

] − 2iδαβεabY,

Qaαη

b = 2Dα(A− iV )ζ ab,

QaαBβ = 2Dα(A)ψ

aβ −Dβ(A+ iV )ψa

α + iεαβγDγ (A− iV )ηa

− 2εcd[ζ ac, δαβη

d − iεαβγ ψγ d

],

(14)QaαY = iDα(A− iV )ηa + iεcd

[ζ ac, ψd

α

],

which, formally, may be regarded as deformation of the vector supersymmetry given in (7).The eight superchargesQa , Qa , Qa

α andQaα , together with the generatorPα of space–time

translations, obey the following topological superalgebra,Qa,Qb

= 0,Qa, Qb

= −4δG(ζ ab

),Qa, Qb

= 0,Qa,Qb

= −4δG(ζ ab

),

Qa, Qbα

= 0,Qa,Qb

α

= 2εab(−iPα + δG(Aα − iVα)

),Qa,Qb

α

= 0,Qa, Qb

α

= 2εab(−iPα + δG(Aα + iVα)

),

Qaα,

Qbβ

.= −4εabδαβδG(ζ ab

),

Qaα,Q

bβ

.= 2iεabεαβγ

(−iP γ + δG(Aγ − iV γ

)),

(15)

Qaα,Q

bβ

.= −4εabδαβδG(ζ ab

),

Qaα,

Qbβ

.= −2iεabεαβγ

(−iP γ + δG(Aγ + iV γ

)),

which is just theNT = 4 equivariant extension of the topological superalgebra (8).In summary, so far we have seen that the novelNT = 2 topological twist ofN = 4,

D = 3 SYM can be regarded, formally, as restriction of aNT = 4 topological gauge theory.


4. Dimensional reduction of NT = 2, D = 4 topological YM theory to D = 3

Now, our aim is to show that theNT = 4 equivariant extension of the Blau–Thompsonmodel introduced in Section 3 is precisely one of the two essentially different topologicaltwists ofN = 8,D = 4 SYM which, on the other hand, is just the dimensional reductionof either of the twoNT = 2,D = 4 models toD = 3, namely, either theA-model [18,22]or theB-model [23]. However, before proceeding, let us briefly recall the two possibletopological twists ofN = 8,D = 3 SYM and its relation to the three essentially differenttopological twists of theN = 4,D = 4 SYM.

TheN = 8, D = 3 SYM obtains by dimensional reduction ofN = 1, D = 10 SYM,either directly or viaN = 4, D = 4 SYM, toD = 3 (cf., lower half of the Diagram 1).The global symmetry group ofN = 8,D = 3 SYM is SU(2)E ⊗ Spin(7). In decomposingSpin(7), with respect to the twist procedure, some restrictions have to be required [16]:

First, the twisted theory should contain at least one scalar topological supercharge.Second, among the spinor representations ofSpin(7) no ones with spin 2 shouldappear.Third, for a full topological twist only half-integral spins should appear among thespinor representations.

Under these restrictionsSpin(7) decomposes asSpin(7) → SU(2)R ⊗ SU(2)R ⊗SU(2)N , so that the maximal residual global symmetry group isSU(2)R ⊗ SU(2)R.

Now, choosing the diagonal subgroup ofSU(2)E ⊗ SU(2)N one gets a twisted theorywith an underlyingNT = 4 equivariant cohomology. Below, it will be shown that the actionof this model is precisely the one given in (9).

The other topological twist is obtained by taking the diagonal subgroup ofSU(2)E ⊗SU(2)R and gives rise to aNT = 2 theory with global symmetry groupSU(2)N ⊗ SU(2)R .The action of that theory will be explicitly constructed in Section 5.

Furthermore, there are three topological twists ofN = 4, D = 4 SYM, namely theA-model, which is theNT = 2 equivariant extension of theNT = 1, D = 4 Donaldson–Witten model [3], theB-model, which formally can be regarded as a deformation [16] oftheNT = 2,D = 4 super-BF model [28], and the half-twistedNT = 1,D = 4 model [22],the Donaldson–Witten theory coupled to a spinorial hypermultiplet. Therefore, one mighthave expected that there are at least three topological twists ofN = 8,D = 3 SYM. But, aspointed out in [16], the dimensional reduction of either of the twoNT = 2,D = 4 theories,i.e., theA- and theB-model, give rise to equivalentD = 3 topological gauge theories, sothat, under the above-mentioned restrictions, there are only two different twists.

Now we want to show that the dimensional reduction of theNT = 2, D = 4 Yamron–Vafa–Witten theory with global symmetry groupSU(2)R, i.e., theA-model, leads preciselyto the action given in (9). The gauge multiplet of this theory consists of the gauge fieldAµ, a Grassmann-even self-dual tensor fieldMµν , a SU(2)R doublet of Grassmann-oddself-dual ghost-antighost tensor fieldsχa

µν = ψµν,χµν, aSU(2)R doublet of Grassmann-odd ghost-antighost vector fieldsψa

µ = ψµ,χµ, a SU(2)R doublet of Grassmann-oddscalar fieldsηa = λ,η, and aSU(2)R triplet of Grassmann-even ghost-for-ghost complex


scalarsφab = φ, τ, φ. For the closure of the topological superalgebra (see Eq. (19) below)it is necessary to introduce a set of bosonic auxiliary fields, namely the self-dual tensor fieldGµν and the vector fieldHµ. All the fields are in the adjoint representation and take theirvalues in the Lie algebraLie(G) of some compact gauge groupG.

The action of the Yamron–Vafa–Witten model, with anNT = 2 off-shell equivariantlynilpotent topological shift symmetryQa , is given by [18,22] (see also [31])

(16)

SYVW =∫d4x tr

GµνFµν − 1

4Gµν

[Mµ

λ,Mνλ

] + 2εabχµνaDµψbν

− 14εabχ

µνa[Mµ

λ,χbνλ

] − εabψµa

[Mµν,ψ

νb] − 2εabηaDµψb

µ

− 12εabη

a[Mµν,χb

µν

] − 12G

µνGµν

+ 2φabηa, ηb

+ 2φabψµa,ψb

µ

+ 12φab

χµνa,χb

µν

+ φabD

2φab − 14

[φab,M

µν][φab,Mµν

]− [φab,φcd ]

[φab,φcd

] + 2HµDνMµν − 2HµHµ

,

and can be cast into theQa -exact form

SYVW = 12εabQ

aQbXYVW ,


XYVW =∫

d4x trMµνFµν − 1

12Mµν

[Mµ

λ,Mνλ

]

− 12G

µνMµν + εabψµaψb

µ + εabηaηb

.

The complete set of symmetry transformations which fix all the relative numerical factorsin (16), except for an overall coupling constant, is given by the topological supersymmetryQa ,

QaAµ =ψaµ, QaMµν = χa

µν,

Qaφbc = 12ε

abηc + 12ε


],

Qaψbµ =Dµφ

ab + εabHµ, QaHµ = −12Dµη

a − εcd[φac,ψd

µ

],

Qaχbµν = [

Mµν,φab

] + εabGµν, QaGµν = −12

[Mµν,η

a] − εcd

[φac,χd

µν

],

(17)

and by the vector supersymmetryQaµ,

QaµAν = δµνη

a + χaµν,

QaµMρσ = −δµ[ρψa

σ ] − εµνρσψνa,

Qaµφ

bc = −12ε

abψcµ − 1

2εacψb

µ,

Qaµη

b =Dµφab + εabHµ,

Qaµψ

bν = −εabFµν + δµνεcd

[φac,φbd

] + εabGµν − [Mµν,φ

ab],

QaµHν =Dµψ

aν − 1

2Dνψaµ + εcd

[φac,χd

µν − δµνηd] + [

Mµν,ηa],


Qaµχ

bρσ = δµ[ρDσ ]φab + εµνρσD

νφab − εabδµ[ρHσ ]− εabεµνρσH

ν − εabDµMρσ ,

(18)

QaµGρσ =Dµχ

aρσ − δµ[ρDσ ]ηa − εµνρσD

νηa − εcd[φac, δµ[ρψ

dσ ] + εµνρσψ

νd]

+ 12

[ψaµ,Mρσ

],

which, together with the space–time translationsPµ, obey the topological superalgebraQa,Qb

= −2δG(φab

),

Qa, Qbµ

= εab(−iPµ + δG(Aµ)

),

(19)Qa

µ,Qbν

.= −2δµνδG(φab

) − εabδG(Mµν).

Our next goal is to show how the action (9) is obtained from (16) by a dimensionalreduction. To this end we perform a(3+ 1)-decomposition of the action (16), i.e., we splitthe space–time coordinates intoxµ = xα, x4, α = 1,2,3, wherexα andx4 denote thespatial and the temporal part, respectively. As a next step, we assume that no field dependsonx4, i.e.,∂4 = 0, so that the integration overx4 factors out and, therefore, can be ignored.Furthermore, we rename the temporal part ofAµ, ψa

µ andHµ, according to

A4 = 2ρ, ψa4 = ηa, H4 = Y,

reserving the notationAα andψaα for the spatial part ofAµ andψa

µ, and identifyMµν ,χµν , Gµν and the spatial part ofHµ with

Mαβ = εαβγ Vγ , Mα4 = Vα, χa

αβ = εαβγ ψγ a, χa

α4 = ψaα ,

Gαβ = εαβγ B γ + εαβγDγ ρ, Gα4 = Bα +Dαρ, Hα = Bα − [Vα,ρ],

respectively. Here, theρ-dependent shifts inGαβ , Gα4 andHα ensure that after carryingout the dimensional reduction the Hermitean conjugateQa of the scalar superchargeQa

coincides with the temporal partQa4 of the vector superchargeQa

µ, i.e., Qa = Qa4. Then,

after squeezing (16) toD = 3, we arrive at the following reduced action

(20)

S(NT =4) =∫d3x tr

εαβγ BγFαβ − εαβγ Bγ [Vα,Vβ ] − 2BαBα

+ 2εabεαβγ ψaγDαψ

bβ + εabε

αβγ Vγ(ψaα ,ψ

bβ

− ψaα , ψ

bβ

)+ 2εαβγBγDαVβ − 2BαBα − 2εab

(ηaDαψb

α + ηaDαψbα

)+ 2εabV α

(ηa, ψb

α

− ηa,ψb

α

)+ 2φab

(ηa, ηb

+ ψαa,ψb

α

)+ 4ρεab

(ηa, ηb

− ψαa, ψb

α

)+ 2φab

(ηa, ηb

+ ψαa, ψb

α

) + φabD2φab

+ 2ρD2ρ − [V α,φab

][Vα,φ

ab]

− 2[V α,ρ

][Vα,ρ] − [φab,φcd ][φab,φcd

]− 4[ρ,φab]

[ρ,φab

] − 2YDαVα − 2Y 2,


which is manifestly invariant under the discrete symmetry(Aα,Vα,Bα,Bα,Y

) → (Aα,−Vα,−Bα,Bα,−Y

),

(21)(ψaα, ψ

aα , η

a, ηa, φab, ρ) → (

ψaα ,ψ

aα , η

a, ηa,φab,−ρ),

exhibiting that the global symmetry group is actuallySU(2)R ⊗ SU(2)R .Next, after squeezing (17) and (18) toD = 3, for the transformation laws generated by

the scalar and vector supercharges,Qa and Qaα , we get

QaAα =ψaα,

Qaφbc = 12ε

abηc + 12ε

acηb,

Qaηb = −εcd[φac,φbd

],

Qaψbα =Dαφ

ab − εab[Vα,ρ] + εabBα,

QaBα = −12Dαη

a + 12

[Vα, η

a] − εcd

[φac,ψd

α

] − [ρ, ψa

α

],

QaVα = ψaα ,

Qaρ = 12η

a,

Qaηb = 2[ρ,φab

] + εabY,

Qaψbα = [

Vα,φab

] + εabDαρ + εabBα,

QaBα = −12Dαη

a − 12

[Vα,η

a] − εcd

[φac, ψd

α

] + [ρ,ψa

α

],

(22)QaY = −[ρ,ηa

] − εcd[φac, ηd

],

and

QaαAβ = δαβη

a + εαβγ ψγ a,

Qaαφ

bc = −12ε

abψcα − 1

2εacψb

α,

Qaαη

b =Dαφab − εab[Vα,ρ] + εabBα,

Qaαψ

bβ = − εabFαβ + εabεαβγ Bγ + δαβεcd

[φac,φbd

] + εabεαβγDγ ρ

− εαβγ[V γ ,φab

],

QaαBβ =Dαψ

aβ − 1

2Dβψaα + 1

2

[Vβ, ψ

aα

] − εcd[φac, δαβη

d − εαβγ ψγ d

]+ [

ρ, δαβηa − εαβγ ψ

γa] + εαβγ

[V γ ,ηa

],

QaαVβ = −δαβηa + εαβγ ψ

γa,

Qaαρ = 1

2ψaα ,

Qaαη

b = −εabDαρ − [Vα,φ

ab] + εabBα,

Qaαψ

bβ = − εabDαVβ + εabεαβγ B

γ + 2δαβ[ρ,φab

] − εαβγ Dγ φab

− εabεαβγ[V γ ,ρ

] − εabδαβY,

QaαBβ =Dαψ

aβ − 1

2Dβψaα − 1

2

[Vβ,ψ

aα

] − εcd[φac, δαβη

d − εαβγ ψγd

]− [

ρ, δαβηa − εαβγ ψ

γ a] + εαβγ D

γ ηa,

(23)QaαY =Dαη

a + [Vα,η

a] + εcd

[φac, ψd

α

] − [ρ,ψa

α

],


respectively. The transformation laws generated by the conjugate supercharges,Qa andQaα , are obtained from (22) and (23) by carrying out the replacements (21), mappingQa

to Qa and Qaα toQa

α , respectively.To make contact with the action and the transformations given in (9), (11) and (13)

we express (20), (22) and (23) in terms of the complexified fieldsAα ± iVα , Bα ± iBα ,ψaα ± iψa

α , ηa ± iηa and in terms of the complexified superchargesQa ± iQa , Qaα ± iQa

α ,respectively. In addition, we combineφab andρ to form the complex scalar fields

ζ ab = φab + iεabρ and ζ ab = φab − iεabρ,

recalling thatφab is symmetric,φab = φba . Then, after carrying out the redefinitions

Bα + iBα → Bα, Bα − iBα → Bα,

ηa + iηa → ηa, ηa − iηa → ηa,

ψaα + iψa

α →ψaα , ψa

α − iψaα → ψa

α ,

and

Qa + iQa →Qa, Qa − iQa → Qa,

Qaα + iQa

α → Qaα,

Qaα − iQa

α →Qaα,

it is easily seen that the resulting action and transformations are precisely the ones given inSection 4. Hence, it is proven that by a dimensional reduction of either of the twoNT = 2,D = 4 theories one recovers theNT = 4 equivariant extension of the Blau–Thompsonmodel proposed in Section 4.

5. Dimensional reduction of half-twisted NT = 1, D = 4 Yamron model

After having described theNT = 4 topological twist ofN = 8, D = 3 SYM, arisingfrom the dimensional reduction of either of the twoNT = 2,D = 4 theories, now, for thesake of completeness, we will also explicitly construct the otherNT = 2 topological modelarising either by partially twistingN = 8,D = 3 SYM or by dimensional reduction of thehalf-twistedNT = 1,D = 4 theory [22] (see Diagram 1).

The global symmetry group of this theory isSU(2)N ⊗ SU(2)R . The action of thispartially twisted theory can be described as the coupling of theNT = 2, D = 3 super-BF theory to a spinorial hypermultipletλabA , ζ aA. This hypermultiplet is built up froma SU(2)N ⊗ SU(2)R quartet of Grassmann-odd spinor fieldsλabA and aSU(2)R doubletof Grassmann-even spinor fieldsζ aA. In order to close the topological superalgebra (seeEq. (29) below), we introduce aSU(2)R doublet of bosonic auxiliary spinor fieldsY a

A.The spinor indices are denoted byA = 1,2. All the spinor fields are taken in the adjointrepresentation of the gauge groupG.

Omitting any details, after dimensional reduction of the half-twistedNT = 1, D = 4theory the reduced action, with an underlyingNT = 2 off-shell equivariantly nilpotent


topological shift symmetryQa , splits up in twoSU(2)N ⊗ SU(2)R invariant parts

(24)S(NT =2) = SBF + SM,

where the first part is just the action of theNT = 2, D = 3 super-BF theory given in (1)and the second one is the matter action

(25)

SM =∫

d3x tr

− iλAab(σα

)ABDαλ

abB − 2iλAab

(σα

)AB[ψaα, ζ

bB

]

+ 2λAab[ηa, ζAb

] + 2εcdλacA[φab,λ

Abd]

+ [φab, ζAc][φab, ζAc

] − iεabζaA

(σα

)AB[Bα, ζ

bB

]− 2iεabY a

A

(σα

)ABDαζ

bB + 2YAaYAa

,

here,Dα is the covariant derivative (in the adjoint representation) andσα are the Paulimatrices,

(σα)AC(σβ)CB = δαβεAB + iεαβγ

(σγ

)AB

, α = 1,2,3.

The spinor indexA is raised and lowered as follows,ϕA = ϕBεBA andϕA = εABϕB withεACεCB = −δAB .

The action (25) can be cast in theQa -exact form

SM = 12εabQ

aQbXM,


XM = −12

∫d3x tr

λAabλ

Aab + iεabζaA

(σα

)ABDαζ

bB

.

By a straightforward calculation in can be verified that both parts in (24) areseparately invariant under the off-shell equivariantly nilpotent topological supersymmetryQa (cf., Eq. (2)),

QaAα =ψaα, Qaζ bA = λabA ,

Qaφbc = 12ε

abηc + 12ε


],

Qaψbα =Dαφ

ab + εabBα, QaBα = −12Dαη

a − εcd[φac,ψd

α

],

(26)QaλbcA = −[φab, ζ cA

] + εabY cA, QaY b

A = 12

[ηa, ζ bA

] − εcd[φac, λdbA

],

and under the vector supersymmetryQaα (cf., Eq. (3)),

QaαAβ = δαβη

a + εαβγ ψγa,

Qaαζ

bA = i(σα)ABλ

Bab,

Qaαφ

bc = −12ε

abψcα − 1

2εacψb

α,

Qaαη

b =Dαφab + εabBα,

Qaαψ

bβ = −εabFαβ + εabεαβγ B

γ − εαβγDγ φab + δµνεcd

[φac,φbd

],

QaαBβ =Dαψ

aβ − 1

2Dβψaα + εαβγ D

γ ηa + εcd[φac, εαβγ ψ

γd − δαβηd],


Qaαλ

bcA = −εabDαζ

cA + i(σα)AB

([φab, ζBc

] + εabYBc),

(27)QaαY

bA =Dαλ

abA + 1

2

[ψaα , ζ

bA

] − i(σµ)AB([ηa, ζBb

] − εcd[φac, λBdb

]).

Furthermore, it can be proven that the sum of the two parts in (24) is invariant under thetransformations generated by the spinorial superchargesQab

A ,

QabA Aα = i(σα)ABλ

Bab,

QabA ζ cB = εABε

bcηa − i(σα

)AB

εbcψaα ,

QabA φcd = 1

2εacλdbA + 1

2εadλcbA ,

QabA ηc = [

φac, ζ bA] − εacY b

A,

QabA ψc

α = −εacDαζbA + i(σα)AB

([φac, ζBb

] + εacYBb),

QabA Bα = 1

2DαλabA + [

ψaα, ζ

bA

] − i(σα)AB([ηa, ζBb

] − εcd[φac, λBdb

]),

QabA λcdB = εac

[ζ bA, ζ

dB

] + εAB[φac,φbd

] + i(σα

)AB

εbd(Dαφ

ac − εacBα

),

(28)

QabA Y c

B = − 12

[λabA , ζ cB

] + [λacB , ζ bA

] − εAB[ηa,φbc

]− i

(σα

)AB

εbc(Dαη

a + εef[φae,ψf

α

]).

By the symmetry requirementsQaSBF = QaαSBF = 0 andQaSM = Qa

αSM = 0, togetherwith Qab

A (SBF +SM)= 0, all the relative numerical coefficients of the actionSBF +SM areuniquely fixed, except for a single overall coupling constant.

The eight superchargesQa , Qaα andQab

A , together with the generatorPα of space–timetranslations, satisfy the topological superalgebra

Qa,Qb

= −2δG(φab

),

Qa, Qbα

= εab(−iPα + δG(Aα)

),

Qa,QbcA

= −εabδG(ζ cA

),Qa

α,Qbβ

.= −2δαβδG(φab

),Qa

α,QbcA

.= −iεab(σα)ABδG(ζBc

),

(29)QacA ,Qbd

B

.= 2εcdεABδG(φab

) − iεabεcd(σα

)AB

(−iPα + δG(Aα)).

Finally, let us notice that the dimensional reduction of the half-twistedNT = 1, D = 4theory can also be described by decomposingSpin(7) → G2 → SU(2)N ⊗ SU(2)R [16].In that description, however, only the diagonal subgroup of the global symmetry groupSU(2)N ⊗ SU(2)R is manifest. The action of this theory coincides with the dimensionalreduction of theNT = 1, D = 4 Donaldson–Witten theory [3] coupled to the standardhypermultiplet [32] toD = 3.

6. Conclusions and remarks

In the present paper we proposed aNT = 4 equivariant extension of the Blau–ThompsonNT = 2 non-equivariant topological model inD = 3 Euclidean space–time. Furthermore,we showed, by proving the equivalence with the dimensional reduction toD = 3 of


the Yamron–Vafa–WittenNT = 2, D = 4 theory, that this extended topological modelcoincides with one of the two inequivalent topological models which, according to theclassification of [16], may be constructed by twisting theN = 8, D = 3 super-Yang–Mills theory. In addition, we constructed explicitly also the other topological model whichmay by obtained, namely, theNT = 2, D = 3 super-BF theory coupled to a spinorialhypermultiplet.

All these models, including theNT = 2,D = 3 super-BF model, are considered in theflat Euclidean space–time where, besides the topological shift symmetry also the vectorsupersymmetry can be constructed. All these symmetries have been given explicitly. Theactions of the corresponding theories are shown to be uniquely determined, up to an overallcoupling constant, when they are required to be invariant not only with respect to thetopological shift symmetry but also with respect to the vector supersymmetry. Furthermore,it is shown that these symmetry operations, together with the generators of space–timetranslations, fulfill corresponding topological superalgebras.

As already mentioned in the introduction, the Blau–ThompsonNT = 2 non-equivarianttopological model and itsNT = 4 equivariant extension provide not only the link betweenthe various topological theories arising from twistingN = 1, D = 6 or N = 1, D = 10super-Yang–Mills theory, but also are pre-candidates, after carrying out a dimensionalreduction toD = 2, for Hodge-type cohomological theories inD = 2. Of course, insearching for all of the Hodge-type cohomological theories inD = 2 a complete grouptheoretical classification of the topological models inD = 2 and their explicit constructionwould be necessary.

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[17] J.A. Minahan, D. Nemeschansky, C. Vafa, N.P. Warner, Nucl. Phys. B 527 (1998) 581.[18] C. Vafa, E. Witten, Nucl. Phys. B 431 (1994) 3.[19] M. Bershadsky, A. Johansen, V. Sadov, C. Vafa, Nucl. Phys. B 448 (1995) 116.[20] J. Maldacena, Adv. Theor. Math. Phys. 2 (1998) 231.[21] B. Geyer, D. Mülsch, Phys. Lett. B 518 (2001) 181.[22] J. Yamron, Phys. Lett. B 213 (1988) 325.[23] N. Marcus, Nucl. Phys. B 452 (1995) 331.[24] E. Witten, Nucl. Phys. B 323 (1989) 113.[25] D. Birmingham, M. Blau, G. Thompson, Int. J. Mod. Phys. A 5 (1990) 4721.[26] M. Blau, G. Thompson, Commun. Math. Phys. 152 (1993) 41.[27] N. Seiberg, E. Witten, Gauge dynamics and compactifications to three dimensions, in: The

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Branched coverings and interacting matrix stringsin two dimensions

M. Billó a, A. D’Addaa, P. Proveroa,ba Dipartimento di Fisica Teorica dell’Università di Torino and Istituto Nazionale di Fisica Nucleare,

Sezione di Torino, via P. Giuria 1, I-10125 Torino, Italyb Dipartimento di Scienze e Tecnologie Avanzate, Università del Piemonte Orientale, I-15100 Alessandria, Italy

Received 9 April 2001; accepted 2 August 2001

Abstract

We construct the lattice gauge theory of the groupGN , the semidirect product of the permutationgroup SN with U(1)N , on an arbitrary Riemann surface. This theory describes the branchedcoverings of a two-dimensional target surface by strings carrying aU(1) gauge field on the world-sheet. These are the nonsupersymmetric matrix strings that arise in the unitary gauge quantization ofa generalized two-dimensional Yang–Mills theory. By classifying the irreducible representations ofGN , we give the most general formulation of the lattice gauge theory ofGN , which includes arbitrarybranching points on the world-sheet and describes the splitting and joining of strings. 2001 ElsevierScience B.V. All rights reserved.

PACS:11.15.-q; 11.15.Ha; 11.25.-w

1. Introduction

The relation between two-dimensional Yang–Mills theories in the largeNc limit andtwo-dimensional string theories was established by Gross and Taylor in a series ofpapers [1–3]. They showed that the coefficients of the 1/Nc expansion of the YM partitionfunction, with gauge groupSU(Nc) on a Riemann surfaceM of areaA and arbitrary genusG count the number of string configurations without folds, namely, the number of branchedcoverings of the surface. As a matter of fact two-dimensional YM does not give exactly apure theory of coverings for two reasons: the presence of two distinct chiral sectors, whichare weakly coupled by pointlike tubes and correspond to the two possible orientations ofthe world-sheet, and the presence, forG> 1, of the so-calledΩ−1 points, whose geometricmeaning was eventually clarified in [4,5].

E-mail addresses:[email protected] (M. Billó), [email protected] (A. D’Adda), [email protected](P. Provero).



496 M. Billó et al. / Nuclear Physics B 616 [PM] (2001) 495–516

A lattice gauge theory of complexNc ×Nc matrices that describes, in the largeNc limit,a pure theory of branched coverings with just one chiral sector and withoutΩ−1 pointswas later discovered in [6,7]. In fact, when expanded in powers ofq = e−A, the partitionfunction of this complex matrix model can be regarded as the generating function of thenumber of branched coveringsZG,N that wrap aroundM N times. Actually for finiteNc

only the firstNc coefficient of the expansion correctly reproduce the correspondingZG,N

and the generating function is only reproduced in the largeNc limit. A similar propertyholds for two-dimensional YM theories [8].

For fixedN , the number of branched coveringsZG,N is itself the partition function ofa lattice gauge theory [2,6,7] whose gauge group is the symmetric groupSN and whoseplaquette action is the standard heat-kernel action forSN :

(1)e−Spl(P ,Apl) = 1

(N !)2∑r

drchr (P )eApl gr ,

whereP ∈ SN andr labels the representations ofSN of characters chr and dimensionsdr ;Apl is the area of the plaquette and the arbitrary coefficientsgr carry all the informationabout the density of all different types of branch points. The relation between the latticematrix theory of [7] and theSN gauge theory is a direct consequence of Frobenius formula,as shown in [6,7].

A completely different way of obtaining a string theory from generalized two-dimensional YM theories was proposed in [9,10]. The mechanism is analogue to the onethrough which string configurations emerge in “Matrix string theory” [11] and it is basedon the quantization of a generalized1 YM theory in a unitary gauge. More precisely, onestarts from the generalized action

(2)Sgen=∫M

dµV (B)− i tr∫M

(dA− iA∧A)B,

where the gauge group isU(N) andV (B) is an arbitrary gauge invariant potential ofthe N × N Hermitian matrix2 B. One chooses the unitary gauge in which the matrixB is diagonal. This gauge choice is incomplete, as it leaves a residualU(1)N gaugeinvariance. It is also affected by Gribov ambiguities as at each point of the space–timemanifold the gauge is determined up to an arbitrary permutation of the eigenvalues ofB.A global smooth diagonalization is then in general not possible because, as one goes rounda loop noncontractible to a point, the eigenvalues ofB may be subjected to a permutation.This gives origin to twisted sectors, which are in one-to-one correspondence with thehomomorphismsΠ1(x | M) → SN of the homotopy group ofM onto the symmetricgroupSN , namely, with the unbranchedN -coverings ofM.

In four space–time dimensions unitary gauges lead to divergences when two eigenvaluescoincide, and the theory is apparently nonrenormalizable [24] in these gauges. In two di-

1 Generalized YM theories have been discussed in [12,13].2 We denote here the number of colours byN as in this case it coincides with the number of times the world-

sheet of the string wraps round the target space. This marks an important difference between this approach andthe one of [2] or [6].

M. Billó et al. / Nuclear Physics B 616 [PM] (2001) 495–516 497

mensions these divergences still occur, but they are exactly cancelled in flat space–time. Infact there is an exact cancellation, due to a fermionic symmetry, between the contributionsof the nondiagonal components of the gauge field and the ghost–antighost system [9]. Thisfermionic symmetry is, however, anomalous in presence of space–time curvature, and itappears that the only consistent way to eliminate the resulting divergences is to make useof the arbitrariness of the potentialV (B) in (2) and introduce in it a term, which dependson the curvature, that cancels exactly the anomaly. With this prescription the theory be-comes completely Abelian, consisting ofN U(1) gauge theories, whose field strengths arethe eigenvalues ofB, and which are only coupled, in each twisted sector, by the bound-ary conditions induced by the homomorphismΠ1(x |M) → SN . The original generalizedYM theory, with U(N) gauge group, is then described by a string theory (unbranchedcoverings) that coversM N times, and has aU(1) gauge theory on its world-sheet.

As shown in [10], this theory can be formulated as a lattice gauge theory whose gaugegroupGN is the semi-direct product ofSN andU(1)N defined by the multiplication rule

(3)(P,ϕ)(Q, θ) = (PQ,ϕ + Pθ),

where(P,ϕ) can be represented as theN ×N matrix

(4)(P,ϕ)ij = eiϕi δiP (j)

andP is an element ofSN . The twisted sectors considered in [10] did not include thepossibility of branch points. Correspondingly, the plaquette action in theGN lattice gaugetheory had the form

(5)e−Spl = δ(Ppl)∑ni

exp

N∑i=1

(iniϕ

(pl)i −Aplv(ni)

),

wherev(ni) is the remaining part ofV (B) after the anomaly cancellation. The plaquetteaction for theSN subgroup in (5) is just a delta functionδ(Ppl); this denotes the absenceof branch points.

The problem of introducing twisted sectors that allow branch points, so that the stringscarrying theU(1) field can join and split, will be addressed in this paper. From the pointof view of the lattice gauge theory ofGN this is equivalent to writing the most generalheat kernel action forGN (as it is done in Eq. (1) for the groupSN ). This entails findingan explicit expression for the characters of the irreducible representions ofGN in terms ofthe characters ofSN andU(1)N . From the point of view of the (generalized) Yang–Millstheory, the string interactions should originate from nonperturbative effects, in analogy towhat happens in the supersymmetric case ofN = 8 super Yang–Mills theory. In this casethe string theory that arises in the diagonal gauge3 is the matrix string of [11], which isbelieved to be equivalent to the Type IIB string theory in the light-cone frame. The stringinteractions [14–18] arise precisely because there are classical instantonic configurations[19–21] which lead to a structure of eigenvalues describing a branched covering of the

3 Here the field which are diagonalized in the Yang–Mills strong coupling limit are the eight scalarsuperpartners of the gauge field.


target space–time manifold. The string coupling constantgs is, consistently with thispicture, proportional to 1/g2

YM .In our case it is not yet clear how to explicitely derive analogous instantonic configura-

tions for the generalized Yang–Mills theory; nevertheless we can investigate the interactingtheory, i.e., the theory of branched coverings, irrespectively of its dynamical origin.

2. N -coverings as SN gauge theory

In this section we will show that the partition function ofN -coverings of a generalRiemann surface can be expressed as the partition function of a lattice gauge theory, definedon a cell decomposition of the surface, with the symmetric groupSN as the gauge group.

A covering of atargetRiemann surfaceΣT is essentially a smooth map

(6)f : ΣW → ΣT

from acoveringsurfaceΣW (the “world-sheet”) toΣT (the “target”) such that each pointof ΣT has exactlyN counterimages inΣW . Each of these counterimages is said to belongto a differentsheetof the covering surface. Consider now a pathγ in ΣT , from a pointP1

to a pointP2. It is natural to define theN liftings γi of γ as the paths onΣW such that

(7)f γi = γ.

If the liftings of all closed, homotopically trivial paths are closed, the covering is saidunbranched. Otherwise it has branch points: these are points inΣT such that a closedloop around them, when lifted, goes from one sheet of the coverings to a different one.Therefore, each branch point is associated to a nontrivial permutation of theN sheets ofthe coverings.

Consider now a cell decomposition of the target surfaceΣT , made ofN0 sites,N1 linksandN2 plaquettes, so that the Euler formula gives the genusG as

(8)N0 −N1 +N2 = 2− 2G.

To construct a branchedN -covering of this discretized target surface, considerN copies ofeach plaquettep of ΣT : these will be the plaquettes ofΣW . Consider now two neighboringplaquettesp1 andp2 in ΣT , let l be their common link, and glue each of theN copies ofp1 to one of theN copies ofp2 alongl according to a permutationP ∈ SN . Repeat theprocedure for all links to construct the discretized version of the covering surfaceΣW .

In this way one associates a permutationP ∈ SN to each link ofΣT . It is easier tovisualizeP as associated to an oriented link of thedual lattice, that is the one in whichsites and plaquettes are exchanged, as shown in Fig. 1. A dual link goes from a plaquettep1 to a plaquettep2 and its associated permutation dictates how to glue copies ofp1 tocopies ofp2.

A closed path around a sites of ΣT will be represented in the discretized version as aplaquette of the dual lattice. If the ordered product of the permutations around the plaquetteis not the identical permutation, thens is a branch point of our covering: the liftings of the


Fig. 1. Dual lattice: the solid lines are the links of the original lattice and the dashed lines the links ofthe dual lattice. The plaquettesp1 andp2 are joined by a dual link to which a permutationQ ∈ SNis associated:Q dictates how to glue together copies ofp1 to copies ofp2 to construct the coveringsurface.

closed path toΣW are not all closed, as some of them start on a sheet and end on a differentone. Therefore, in our discretization the branch points of the coverings are localized on thelattice sites.

Now we want to construct a model in which all possible branchedN -coverings of agiven (discretized) target surface are counted with Boltzmann weights depending on theirbranch point structure. The previous discussion suggests that this model can be written asa lattice gauge theory withSN as the gauge group.

The possible types of branch points are in one-to-one correspondence with the conjugacyclasses of permutations, that is with the partitions ofN . To each conjugacy class wewant to assign an arbitrary, positive Boltzmann weight. Therefore, we must construct alattice gauge theory of the symmetric group, defined on the discretized target surface: theBoltzmann weight of a configuration will be

(9)∏s

ws(Ps),

where the product is extended to all the sites,Ps ∈ SN is the ordered product of thepermutations on the dual links around the sites, that is around the dual plaquettecorresponding to the sites and the weightsws(Ps) vary in general from site to site anddepend on the conjugacy class ofPs only, namely:

(10)ws(P ) = ws

(QPQ−1) ∀ P,Q ∈ SN .

The partition function of our model is then

(11)ZN =(

1

N !)N2∑

P

∏s

ws(Ps),


where the sum is extended to all the configurations, that is to all ways of assigning apermutation to each link, and the product to all sites. The normalization takes into accountthe arbitrary relabelings of the sheets on top of each plaquette.

This partition function can be computed by first expanding the class functionws(P ) incharacters of the symmetric group:

(12)ws(P ) = 1

N !∑r

drws(r)chr (P ),

(13)ws(r) =∑P∈SN

chr (P )

drws(P ).

We then follow the standard method used for solving two-dimensional lattice gauge theory,first introduced in Refs. [22,23]. That is, we use the orthogonality and completenessproperties of the characters (already implicitly used in deriving (12) and (13)), which inthe case of the symmetric group read:

(14)∑r

chr (Q)chr (P ) =∑R

δ(PRQ−1R−1),

(15)1

N !∑P∈SN

chr (P1P)chr ′(P−1P2

)= δrr ′chr (P1P2)

dr,

(16)1

N !∑P∈SN

chr(PP1P

−1P2)= chr (P1)chr (P2)

dr.

These properties can be used to integrate over all the internal dual links of the discretizedsurface. Suppose, for example, we want to integrate overQ as shown in Fig. 2 (right).Using the character expansion Eq. (12) and the orthogonality property Eq. (15), we find

(17)∑Q

ws2

(P1P2Q

−1)ws1(QP3P4) = 1

N !∑r

drws1(r)ws2(r)chr (P1P2P3P4).

In this way one can integrate over all the internal links, and end up with an effective one-site model, whose form depends on the topology of the target surface. Let us consider first

Fig. 2. Left: one-plaquette discretization of the torus. The solid lines are the original lattice linksexiting from the sites. The dashed lines are the dual links, to which the permutationsP andQ areassociated. A closed path around the sites gives rise to the permutationPQP−1Q−1, so that if thisis not the identical permutation thens is a branch point. Right: integration over internal (dual) links.


the partition function on the disk: ifP is the permutation around the boundary we get

(18)ZN,disk(P ) = (N !)−2∑r

drchr (P )

N0∏s=1

ws(r).

From this expression one can calculate the partition function for a surface withoutboundaries and arbitrary genusG. This is done by representing the surface as a polygonwith sides suitably identified:

(19)ZN,G =∑

P1,Q1,...,PG,QG

ZN,disk(P1Q1P

−11 Q−1

1 · · ·PGQGP−1G Q−1

G

).

At this point one can use Eqs. (15), (16) to perform the sum overP1,Q1, . . . ,PG,QG andobtain the final form of the partition function:

(20)ZN,G = (N !)2G−2∑r

d2−2Gr

N0∏s=1

ws(r).

This expression4 allows one to calculate the number of coverings with any prescribedbranch point number and structure. As an example, let us consider the limiting case ofunbranchedN -coverings: this is obtained by choosing

(21)ws(P ) = δ(P ) ∀s.In this case Eq. (13) gives simply

(22)ws(r) = 1

so that

(23)ZN,G =∑r

(dr

N !)2−2G

and in particular for the torus we obtain the well-known result

(24)ZN,1 =∑r

1= p(N),

wherep(N) is the number of partitions ofN .The previous example can be generalized to coverings with any branch point structure

by associating to each sites a permutationQs and choosing the Boltzmann weightws(P )

of the form:

(25)ws(P ) = δ(P,Qs) = 1

N !∑R∈SN

δ(PRQsR

−1),4 This partition function has been studied already in the literature, for instance in connection with the

interpretation of two-dimensional Yang–Mills theories as string theories [2].


which means thatws(P ) is different from zero only ifP is in the same conjugacy classasQs . From (25) and the orthogonality of the characters we get:

(26)ws(r) = chr (Qs)

dr.

By inserting (26) into (18) and (20) we obtain:

(27)ZN,disk(P, Qs

)= 1

N !2∑r

drchr (P )∏s

(chr (Qs)

dr

)and

(28)ZN,G

(Qs)=

∑r

(dr

N !)2−2G∏

s

(chr (Qs)

dr

).

The r.h.s. of both Eqs. (27) and (28) depend only on the equivalence classes ofQs andultimately on how many branch point of each equivalence class are present on the surface.Let us then denote byQ the equivalence classes, and bypQ the number of branch pointsin the equivalence classQ; then Eqs. (27) and (28) can be rewritten as:

(29)ZN,disk(P,pQ) = 1

N !2∑r

drchr (P )∏Q =1

(chr (Q)

dr

)pQ

,

(30)ZN,G(pQ) = N !2G−2∑r

d2−2G−pr

∏Q=1

(chr (Q)

)pQ ,

wherep = ∑Q=1 pQ is the total number of branch points. In Eqs. (29) and (30) the

numberspQ of branched points of a given type are kept fixed. Suppose now to considerthe numberspQ as additional degrees of freedom, and to let them vary. This leads us toconsider a partition function, that in a broad sense we can call grand-canonical, given by

(31)ZN,disk(P,A) =∑pQ

Ap∏Q =1

(σQgQ)pQ

pQ! ZN,disk(P,pQ),

where A is the area of the disk,σQ the number of permutations in the conjugacyclassQ, andgQ a Boltzmann weight, referred to the unit area, attached to a branch pointcharacterized by a permutationQ ∈ Q. The factorpQ! accounts for the fact that branchpoints in the same conjugacy class are indistinguishable. The sum is over all thepQ with

Q = 1. It can be done explicitely and gives

(32)ZN,disk(P,A) = 1

N !2∑r

drchr (P )eAgr ,

with

(33)gr =∑Q =1

gQchr (Q)

dr,

where of coursegQ = gQ if Q ∈ Q. In Eq. (33) the sum over the conjugacy classesQ

has been replaced by the sum over the group elements, using the fact that according to the


definition ofσQ we have∑

Q σQ ≡∑Q. The quantitygr in (32) can be thought of as an

arbitrary function of the representationr.The partition functionZN,disk(P,A) can be used as the building block of a different lat-

tice gauge theory of the groupSN . Consider once again a cell decomposition of a Riemannsurface of genusG, and associate an element ofSN to each link. Let us now associateto each plaquetteα of areaAα in our cell decomposition a Boltzmann weight given byZN,disk(Pα,Aα), wherePα is the ordered product of the link elements aroundα. Unlikethe original theory defined in (11), this theory has no branch points on its sites, but a densedistribution of branch points (characterized bygr or equivalently bygQ) on each plaquette.The integration over the link variables can be performed again by means of the orthogo-nality properties of the characters, and the result for a Riemann surface of genusG is

(34)ZN,G(A) =∑r

(dr

N !)2−2G

eAgr .

We can regard the theory introduced at the beginning of this section, with the branch pointsat the sites of the cell decomposition as a “microscopic” theory, and the one defined by thepartition function (34) as a “continuum” limit. The theory characterized by the plaquetteaction (32) has the same structure as a generalized two-dimensional YM theory, but withSN as gauge group. As a special case we can consider the one where all branch points arequadratic. This is obtained from (33) by choosing

(35)gr = g ξr2 ≡ gN(N − 1)

2

ch(21)

dr,

where21 donotes the conjugacy class consisting of just one exchange.ξr2 can be expressedin terms of the lengthsmα andnβ of the rows and the columns of the Young tableau la-beling the representationr asξr2 = 1/2

(∑α m

2α −∑

β n2β

). It is related to the quadratic

Casimir of the representation of a unitary group corresponding to the same Young tableau.The statistics of branched coverings with a finite numberp of branched points, as

summarized in Eqs. (29) and (30), can also be viewed in a slightly different way. Considera disk withp holes, and let the holonomies on the internal and external boundaries be,respectively,Qs andP . A cell decomposition of this surface is shown in Fig. 3 and anelement ofSN can be associated to each link of the cell decomposition. Assuming thatthere is no branch point on the surface we associate to each plaquetteα the Boltzmannweight

(36)wα = δ(Pα),

wherePα is the ordered product of the elements ofSN associated to the links around theplaquetteα. There is a second type of plaquettes in the cell decomposition that we areconsidering, namely, the plaquettes that correspond to the boundaries of the holes. We canchoose the holonomies on these boundaries to be fixed and given byQs as in Eq. (27). Thisis equivalent to associate the Boltzmann weightws(Ps) = δ(Ps,Qs) introduced in Eq. (25)to the corresponding plaquettes. We have in this way constructed anSN lattice gaugetheory with two types of plaquettes, the ones obtained by the cell decomposition of the


Fig. 3. Cell decomposition of a disk withp holes (p = 4 in this case).

surface, and the ones corresponding to the holes, which are in fact nothing else but blown-up branched points. Clearly, the partition function of such lattice theory is again givenby (29). Instead, if we associate to all boundary plaquettes the same Boltzmann weightws(Ps) =∑

r drAgrchr (Ps) and we sum over the number of boundaries we reproduce the“continuum” partition function (34).

This construction is not obviously limited to the gauge groupSN . Consider for instanceYM theory on a surface of genusG with p boundaries, with a cell decompositionanalogous to the one of the disk in Fig. 3. We associate to the internal plaquettesα aBoltzmann weight

(37)wα(gα) =∑R

dRχR(gα),

wheregα is the ordered product of the link variables around the plaquette. This defines onthe surface a topological (BF) theory. To each of thep plaquettes forming the boundarieswe associate instead a Boltzmann weight

(38)ws(gs) =∑R

dRCRAχR(gs),

3. Matrix strings of generalized YM2, coverings, and the lattice gauge theory of GN

As already discussed in the Introduction, the quantization in the unitary gauge of ageneralizedU(N) Yang–Mills theory is described [9,10] by a theory of coverings of thespace–time manifold, i.e., a string theory. This theory is, however, not the pure theoryof coverings that we described in the previous section as a lattice gauge theory of thepermutation groupSN . In fact aU(1) gauge theory is defined on the world-sheet of thestring. This is the result of the residual gauge freedom after choosing the unitary gauge inwhich the auxiliary fieldB is diagonal.


We have already shown in [10] that in the case ofunbranchedcoverings a string theorywith U(1) gauge fields on the world-sheet can be described as a lattice gauge theory inwhich the gauge group is the semi-direct productGN of SN andU(1)N . The elements ofthis group, which we denote by(P,φ), can be represented byU(N) matrices of the specialform

(39)(P,φ)ij = eiφi δi,P (j),

whereP ∈ SN .In order to understand how the lattice gauge theory of the groupGN originates, let

us go back to Fig. 1, at the beginning of Section 2, and consider againN copies ofeach plaquettep. Although we are interested here in introducing on the world-sheet agauge groupU(1), we consider the case of an arbitrary gauge groupG to make theconstruction completely general. Let us introduce on each plaquettep a matter fieldΨ α

i (p)

that transform under a given representation ofG. In Ψ αi (p) the indexi = 1,2, . . . ,N ,

labels the copies of the plaquette andα is the index of the representation ofG. A gaugetransformation consists in a relabeling of the sheets induced by a permutationP and, oneach sheeti, in the gauge transformation induced bygi ∈G, namely:

(40)Ψ αi (p) −→ (P,g)Ψ α

i (p) = Dαβ(gi)Ψ

β

P−1(i)(p),

whereDαβ(gi) denote the matrix elements ofgi in the given representation. From Eq. (40)

the composition rule of two elements of the gauge group can be easily derived and foundto define the semi-direct product ofSN andGN :

(41)(P,g)(Q,h) = (PQ,g · Ph) with (g · Ph)i = gihP−1(i).

From (41) one also obtains:

(42)(P,g)−1 = (P−1,P−1g−1).

Consider now the dual lattice. The matter fields sit on the sitess of the dual lattice, while onthe links we have to define gauge connections, given by group elements, which are neededto define covariant differences, the discrete analogue of covariant derivatives. Nontrivialholonomies are associated to the plaquettes of the dual lattice, namely, to the sites of theoriginal lattice, and are given by the ordered product along the plaquette of the groupelements on the links. Let(Ps,g(s)) be such product relative to a plaquettes. A gaugetransformation on the plaquette variable(Ps,g(s)) is given by:

(43)(Ps,g(s)

)−→ (Q,h)(Ps,g(s)

)(Q,h)−1 = (

QPsQ−1,h ·Qg ·QPQ−1h−1).

For theSN part, this gauge transformation amounts, as in the model of the previous section,to a relabeling of the sheets and leaves unchanged the decomposition into cycles ofPs ,which describes the branching structure of the points. As for theG gauge transformations,they can be derived from (43) by settingQ = 1, and read

(44)gi(s) −→ hi · gi(s) · h−1P−1(i)

.

Clearly, unlessP−1(i) = i, the transformation given in Eq. (44) is not a gaugetransformation ongi(s). This is due to the fact that the lifting of a closed loop around


the points on the target spaceΣT is in general not closed. The closed loops on the world-sheetΣW are given by the cycles ofPs , and correspondingly the gauge covariant loopvariables are the productsgi1(s)gi2(s) · · ·gik (s) where(i1, i2, . . . , ik) is a cycle ofPs .

We have shown that the theory of branchedN -coverings with aG-gauge theory of on theworld-sheet is equivalent to a lattice gauge theory where the gauge group is the semi-directproduct ofSN andGN . In order to write the partition function of this theory, we need to findthe irreducible representations of such group. Although this can be done for an arbitrarygroupG, we shall restrict ourselves in the rest of this section to the caseG = U(1). Theextension to arbitrary groups, although cumbersome from the point of view of notations, israther straightforward.

3.1. Representation theory ofGN

Consider now the groupGN , the semi-direct product ofSN andU(1)N , whose elementswe denote by(P,φ) whereP is an element ofSN andφ stands for the set of invariantanglesφi , i = 1,2, . . . ,N , that characterize an element ofU(1)N . The product of twogeneric elements of the group can be obtained from (39) or directly from (41) by puttinggi = eiφi :

(45)(P,φ)(Q, θ) = (PQ,φ + Pθ),

where(Pθ)i = θP−1(i). With this product law, the inverse of an element is

(46)(P,φ)−1 = (P−1,−P−1φ

)and the expression of a conjugated element is

(47)(Q, θ)(P,φ)(Q, θ)−1 = (QPQ−1, θ +Qφ −QPQ−1θ

).

The structure of the groupGN is similar to that of the Poincaré group, theU(1)N

elements playing the role of the translations and the permutations the role of the Lorentzrotations. To describe the irreducible representations ofGN we can thus follow Wigner’smethod of induced representations usually employed for the Poincaré group.

We begin by choosing an irreducible representationn ≡ ni of U(1)N , with i =1, . . . ,N andni ∈ Z. This is nothing but the assignment of quantized momenta in all of theN compact directions: theU(1)N elementφ is represented by the phase exp

(i∑

i niφi).

The chosenunorderedN -ple of momenta is invariant under the action of the permutationgroup; however, this is not the case for the specific ordering of them which defines therepresentationn. We use again the notation

(48)Pn = nP

−1(i)

to denote a permutation of the momenta. In the analogy with the Poincaré group, theunorderedN -ple of theni ’s is the invariant that plays the role of the squared mass.

Given the representationn, Wigner’s idea is to construct the representations of the semi-direct product group in terms of irreducible representations of the so-calledlittle groupLn ⊂ SN , which contains those permutations that preserven. We can split any permutation


P ∈ SN as

(49)P = π p :

p ∈Ln,

π ∈ SN/Ln.

While pn = n, the elementsπ in the cosetSN/Ln act nontrivially, mappingn into πn,with

(50)(πn)i = nπ−1(i),

in accordance with Eq. (48). Notice that the coset classπ is of course defined only up tolittle group transformations. We can single out a specific representativeπ in this class byrequiring, for instance, that

(51)π(i) < π(j) if i < j, with ni = nj .

This removes any ambiguity from the decomposition Eq. (49); where not otherwiseindicated, we assume such a choice in what follows.

The little group is determined, up to isomorphisms, only by thestructureof the setnidefining the representation. The construction is the following: let us denote byNn with∑

a Na = N the number of times a given momentumn appears in the setni. In otherwords there areNn values ofi for whichni = n. Then the little group consists of the directproduct

(52)⊗n

SNn

of the symmetric groupsSNn acting on the subsets of indicesi for which ni = n. In thePoincaré group there are only two different little group structures corresponding to the masssquared being bigger than or equal to zero. In our case we have many more possibilities,specified by the possible degeneracies amongst the set ofni ’s.

The explicit expressionLn in Eq. (49) of the little group Eq. (52) as a subgroup ofSN

depends on the specific ordering of theni ’s that defines the representationn. If we modifythe orderingn to πn, we have an isomorphic expression

(53)Lπn = πLn π−1.

As it follows from Eq. (52), an irreducible representationr of the little groupLn isa tensor product of irreducible representationsra of the symmetric groupsSNa , for a =1, . . . ,M. The irreducible representationsra , which are in one-to-one correspondence withthe Young tableaux ofNa boxes, and their characters chra have been discussed in theprevious section. We denote the matrix elements ofp ∈Ln in the representationr by

(54)[Dr(p)

]αβ =

M⊗a=1

[Dra (pa)

]αa

βa ,

wherepa is the component ofp in theath factor,SNa , of the little group. Correspondingly,the characters Chr(p) in this representation are given by

(55)Chr(p) =M∏a=1

chra (pa).


Consider now the action of a permutationP on a state with momentan and whichtransforms in the irreducible representationr of the little group. Denoting such a state as|α〉(n), we have

(56)P |α〉(n) = [Dr(p)

]αβ |β〉(πn),

having used the decomposition Eq. (49) with a specific choice of coset representative,e.g., the one in Eq. (51). We see that the states at fixedn do not span a representation bythemselves, and we are forced to form a single representation including all of the possiblereorderings of theni ’s, parametrized by the classes ofSN/Ln. This is analogous to theobvious fact that a Lorentz rotationΛ on a state of momentumpµ produces a state ofrotated momentumΛµ

νpν .

Eq. (56) suggests that we can represent, for each fixedU(1)N representationn, the entiregroupGN on a finite-dimensional space spanned by state vectors|σ ;α〉, with σ ∈ SN/Ln

andα in the carrier space of an irreducible representationr of the little groupLn, as above.The matrix representing an element(P,φ) ∈ GN can then be written, using the compositeindexA = (σ ;α), as

(57)[Dn,r(P,φ)

]AB = ei

∑i (σn)

iφi[Dn(P )

]στ[Dr(σ−1P τ

)]αβ,

where

(58)[Dn(P )

]στ = δ

(P−1σn, τn

).

The [Dn(P )]σ τ factor simply states that we have nonzero matrix elements in the spaceof σ, τ indices, i.e., inSN/Ln, σn (momenta of the state we are acting on) toτn. Thisbeing the case, we see thatσ−1Pτn = n, i.e., for any choice of representatives of the cosetclasses,σ−1Pτ is an element of the little group and can rightly appear in the last factorin Eq. (57). However, which specific elementσ−1P τ one gets depends on the choice ofrepresentatives, so to fully describe the representation matrices we have to make a choice,e.g., the one of Eq. (51), as indicated in Eq. (57). Different choices of coset representativeslead to equivalent representations.

It is easy to verify that the matricesDn,r defined as in Eq. (57) provide indeed arepresentation of the groupGN , i.e., that

(59)[Dn,r(P,φ)

]AB[Dn,r(Q, θ)

]BC = [

Dn,r(PQ,φ +Pθ)]AC.

One can furthermore show that the representationsDn,r areirreducible, by employingSchur’s lemma: if one assumes that a matrixFA

B commutes with all the elementsDn,r(P,φ), one finds thatF has to be proportional to the identity matrix.

The representationsDµn,r, with µ ∈ SN/Ln, obtained by considering a differentreordering of theni ’s, can be shown to be equivalent to the representationsDn,r.

Summarizing, the set of inequivalent irreducible representations of the groupGN

is described by the possible unorderedN -ples of momentani and the irreduciblerepresentations of the corresponding little groupLn.


The characters in these representations we denote by CHn,r and are given, as it followsfrom Eq. (57), by

(60)

CHn,r(P,φ) =∑A

[Dn,r(P,φ)

]AA =

∑σ

ei∑

i (σn)iφi δ(P−1σn, σn

)Chr(σ

−1Pσ).

The dimension of the representationDn,r, which we denote bydn,r, is given byCHn,r(1,0), i.e., by

(61)dn,r =∑σ

dr = |SN/Ln|dr = N !∏n Nn!dr,

wheredr is the dimension of the chosen irreducible representationr of the little group.An extreme case is the one in which all theni ’s are different; in this case the little group

is trivial and has only the trivial representation (r = 0); all representationsDn,r=0 havethen dimensionN !, and are given by:

(62)[Dn,r=0(P,φ)

]RQ

= ei∑

i (Qn)iφi δ(P−1Q,R

),

whereQ andR are indices ofSN/Ln which coincides in this case withSN . 5 At the otherend we have the case where all theni ’s are equal, the little group coincides withSN , and theonly coset class is the identity one. All representationsDn,r have the same dimensionalitydr as the chosen representation of the little group, and coincide, up to theU(1)N phasefactor, with the representations ofSN .

The characters given in Eq. (60) satisfy the usual orthogonality and completenessrelations, which we will need to construct and solve the lattice theory. One has the “fusion”rule ∫

Dφ1

N !∑P∈SN

CHn,r((Q1, θ1)(P,φ)−1)CHn′,r′

((P,φ)(Q2, θ2)

)(63)= δn,n′ δr,r′

CHn,r((Q1, θ1)(Q2, θ2))

dn,r,

which in the particular caseQ1 = Q2 = 1 and θ1 = θ2 = 0 becomes the orthogonalityrelation

(64)∫

Dφ1

N !∑P∈SN

CHn,r((P,φ)−1)CHn′,r′(P,φ) = δn,n′δr,r′ .

In both equations the measure onU(1)N is defined asDφ ≡∏i dφi/2π . Besides we have

the following “fission” property:∫Dφ

1

N !∑P∈SN

CHn,r((P,φ)(Q1, θ1)(P,φ)−1(Q2, θ2)

)(65)= 1

dn,rCHn,r(Q1, θ1)CHn,r(Q2, θ2)

5 Notice that if one omits in (62) theU(1)N phase factors, one obtains the regular representation ofSN which,however, is reducible.


and the completeness relation

(66)1

N !∑

n

∑r

CHn,r(Q, θ)CHn,r(P,φ) = δGN

[(P,φ), (Q, θ)

].

TheGN -invariant delta-function appearing in the l.h.s above is explicitly given by

δGN

[(P,φ), (Q, θ)

]= 1

N !∑R∈SN

∫Dψδ

((P,φ)(R,ψ)(Q, θ)(R,ψ)−1)

(67)=∑R

δ(PRQR−1) 1

(2π)N

N∏l=1

rl∏A=1

2πδ

(l−1∑α=0

φl,A,α +l−1∑α=0

(Rθ)l,A,α

),

where we decomposedP into rl cycles of lengthl (l = 1, . . . ,N ) and replaced the indexi = 1, . . . ,N with the multi-index

(68)(l,A,α), l = 1, . . . ,N, A = 1, . . . , rl, α = 0, . . . , l − 1.

Notice that theGN -invariant delta-function depends only on the sums of the anglesφ

belonging to the same cycles ofP and on the sums of the anglesθ belonging to the samecycles ofR−1PR, that is ofQ. A particular case of (66) is obtained by puttingQ= 1 andθi = 0, and reads:

(69)1

N !∑

n

∑r

dn,rCHn,r(P,φ) = δ(P )(2π)Nδ(φ1) · · · δ(φN).

All these formulas can be checked by direct although somewhat cumbersome calculations.

3.2. Branched coverings endowed with aU(1)N flux and the lattice gauge theory ofGN

We are now ready to generalize the results of Section 2, and study a theory of branchedN -coverings of a Riemann surfaceΣT , endowed with aU(1) gauge theory on the world-sheetΣW . According to the discussion at the beginning of the present section this willbe described by a lattice gauge theory ofGN , namely, by a theory where an element(P,φ) ∈ GN is associated to each link in the dual lattice. A specific covering is determinednow by its branch-point structureandby the values of theU(1)N fluxes. To each plaquettes of the dual lattice (that is, to each site of the original lattice) we associate a weightWs(Ps,φs), where(Ps,φs) is the ordered product of the elements ofGN associated to thelinks around the plaquettes. Such weights must be class functions, that is they depend onthe conjugacy class of(Ps,φs) only. The partition function is obtained summing over allconfigurations:

(70)ZG,N =(

1

N !)N2 ∑

(P ,φ)

∏s

Ws(Ps,φs)


(the normalization is as in Eq. (11)). We can expand the weightWs(P,φ) into charactersof GN ,

(71)WS(P,φ) = 1

N !∑n,r

dn,rWs(n, r)CHn,r(P,φ),

where, according to Eq. (64), we have

(72)Ws(n, r) =∑P

∫Dφ

CHn,r((P,φ)−1)

dn,rWs(P,φ).

Using the character expansion, it is possible to integrate over all internal links of thedual lattice to obtain finally, just as in Section 2, the partition function on a disk:

(73)Zdisk,N(P,φ) = (N !)−2∑n,r

dn,r

N0∏s=1

Ws(n, r)CHn,r(P,φ),

where(P,φ) is the group element associated to the boundary of the disk, and from this thepartition function for a closed surface of genusG:

(74)ZG,N =∑n,r

(dn,r

N !)2−2G N0∏

s=1

Ws(n, r).

In [10] we considered the case of unbranched coverings, and we showed that itcorresponds to the following choice of weights:

(75)∀s: Ws(P,φ) = δ(P )∑ni

e∑N

i=1(iniφi−As

∑i v(ni)

),

whereAs is the area of the dual plaquettes. The lattice gauge theory corresponding to thisplaquette action was shown to be equivalent to the generalizedU(N) YM theory based ona potentialV (F) if the finite potential

∑i v(ni) is obtained fromV (ni) by subtracting

the logarithmic divergences that arise when the genusG of the target manifold is differentfrom 1. Theni ’s correspond to the quantized eigenvalues of the diagonal components ofF .With this choice, we get from Eq. (72)

(76)Ws(n, r) = e−As

∑i v(ni),

that is, theW ’s have no dependence on the little group representationr. The partitionfunction for a closed surface is then simply

(77)ZG,N =∑n,r

(dn,r

N !)2−2G

e−A∑i v(ni),

whereA = ∑s As is the total area of the target surface. In [10], the grand-canonical

partition functionZG(q) ≡ ∑N ZG,Nq

N was investigated by directly enumerating thecoverings and associating to each connected component a generalizedU(1) partitionfunction

∑n∈Z

e−Av(n). A closed expression was given in the case of the torus, and alist of the first few terms (lowest values ofN ) in its q expansion was given in Appendix


of [10] also for other values ofG. A closed expression for the grand canonical partitionfunctionZG(q), generalizing the one given in [10] for the torus, can be obtained from (77)by using Eq. (61) and the relations

∑i v(ni ) =∑

n Nnv(n) andN =∑n Nn. We find that

the infinite sums factorize in a product overn, namely:

(78)ZG(q) ≡∑N

ZG,NqN =

∏n

ZG

(e−Av(n)q

),

whereZG(q) in the grand-canonical partition function for the unbranched coverings, thatis:

(79)ZG(q)=∑N

ZN,GqN,

andZN,G is given in (23). In Appendix A, the torus grand-canonical functionZG(q) willbe described in more detail to appreciate the relation between the treatment of [10] and themore general one given here.

Let us go back to the general case of an arbitrary branching structure. We follow thesame pattern as in Section 2 and associate to each dual plaquettes an element(Qs, θs)

of GN . This is done by assigning to it the weight

(80)Ws(P,φ) = δGN

((Qs, θs), (P,φ)

),

where theGN -invariant delta-function is the one defined in (67). The permutationQs givesthe branching structure at the sites of the original lattice, while the invariant anglesθs,idetermine theU(1) holonomies. Notice, however, that closed loops on the world-sheetaround the sites are in one-to-one correspondence to the cycles of the permutationQs .The correspondingU(1) gauge invariant holonomies are given by the anglesθs,l,A =∑

i∈l,A θs,i, wherel,A denotes theAth cycle of lengthl in Qs . These are the onlyangles that appear, according to (67), in the definition of the covariant delta-function andare the only angles which are left invariant, according to the general discussion at thebeginning of the present section, under conjugacy transformations ofGN . From Eq. (80)and the orthogonality of characters we obtain

(81)Ws(n, r) = CHn,r(Qs, θs)

dn,r.

The partition function on a disk and on a closed surface of genusG can be obtained byinserting (81) in (73) and (74) leading to

(82)

Zdisk,N((P,φ),

(Qs, θs)

)= (N !)−2∑n,r

dn,r

N0∏s=1

(CHn,r(Qs, θs)

dn,r

)CHn,r(P,φ)

and

(83)ZG,N

((Qs, θs)

)= (N !)2G−2∑n,r

d2−2Gn,r

N0∏s=1

(CHn,r(Qs, θs)

dn,r

).

The continuum limit can be taken on Eqs. (82) and (83) following the same track as inSection 2. Let us assume that each branch point with holonomy(Qs, θs) appears with


a couplingAg[(Qs, θs)], whereA is the area of the surface andg[(Qs, θs)] is a classfunction ofGN . Let us then definegn,r as the coefficients of the expansion ofg[(Qs, θs)]in characters ogGN :

(84)g[(Qs, θs)

]= 1

N !∑n,r

dn,rgn,rCHn,r(Qs, θs).

If we multiply the partition functions in (82) and (83) by∏

s Ag[(Qs, θs)], integrate over(Qs, θs) and sum over the numberN0 of branch points (with a 1/N0! factor) we obtain

(85)Zdisk,N((P,φ),A

)= (N !)−2∑n,r

dn,rCHn,r(P,φ)eAgn,r

and

(86)ZG,N(A) = (N !)2G−2∑n,r

d2−2Gn,r eAgn,r .

The partition function given in (86) is very general, as it corresponds to arbitrary weightsfor the different types of branch points, and to arbitraryU(1) holonomies associated toeach type of branch point. Consider now a more specific case, where in all sitess thereis either no branch point or one corresponding to a single exchange. This means that thecouplingg[(Q, θ)] consists of two terms:

(87)g[(Q, θ)

]= g0[(Q, θ)

]+ λg1[(Q, θ)

],

whereλ is a free parameter. The functionsg0[(Q, θ)] andg1[(Q, θ)] are the most generalclass functions ofGN with support, respectively, inQ = 1 and inQ consisting of a singleexchange. Their explicit expression is:

(88)g0[(Q, θ)

]= δ(Q)∑ni

e∑

i iniθi v(n1, n2, . . . , nN),

(89)g1[(Q, θ)

]=∑R

δ(RP12R

−1Q)∑

ni δn1,n2f (n1;n3, . . . , nN)ei

∑i ni θR(i) ,

whereP12 is a permutation consisting of the exchange of the labels 1 and 2,v(n1, n2,

. . . , nN) andf (n1;n3, . . . , nN) are arbitrary functions, which are symmetric, respectively,under permutations of all theni ’s and of theni ’s with i = 3,4, . . . ,N . We consider nowthe case where thef (n1;n3, . . . , nN) = 1, andv(n1, n2, . . . , nN) =∑

i v(ni). The formercondition implies that there is noU(1) holonomy attached to the quadratic branch points or,in other words, that theU(1) electromagnetic field is not localized on the branch points, butdistributed on the world-sheet. The conditionv(n1, n2, . . . , nN) =∑

i v(ni) is equivalentto the statement that theU(1) gauge action is local on the world-sheet, namely, that whathappens on one sheet has no effect on the others. With this choice the coefficientsgn,r canbe easily calculated, leading to:

(90)gn,r =∑n

(Nnv(n) + λξ

rn2

),

where as beforeNn is the number of times the integern appears inn, rn is therepresentation ofSNn in r, andξr2 is defined in (35). We can now insert (90) into (86)


and consider the grand-canonical partition function, defined as in the case without branchpoints (see Eq. (78)). It is not difficult to verify that with the choice (90) forgn,r, thegrand-canonical partition function factorizes in an infinite product overn, namely,

(91)ZG(A, q) ≡∑N

ZG,N(A)qN =∏n

ZG

(e−Av(n)q, λA

).

Here ZG

(q,A

)is the grand canonical partition function for coverings with quadratic

branch points, namely:

(92)ZG(q,A) ≡∑N

ZG,N(A)qN =∑N

∑r |SN

(dr

N !)2−2G

eAξ r2qN,

where the second sum at the r.h.s. is over the representationsr of SN .

Acknowledgements

We thank M. Caselle for many useful discussions.

Appendix A. Unbranched coverings

In this appendix, we make more explicit some points of the discussion, given in the maintext after Eq. (77), of the grand-canonical partition functions describing the particular caseof unbranched covers.

The partition functions for coverings with aU(1) on their world-sheet were alreadydiscussed, in the unbranched case, in [10]. In that paper a procedure was given to constructthe grand-canonical partition functionZG(q) ≡∑

N ZG,NqN . One starts from the grand-

canonical partition functionZG(q) for the pure coverings, and considers the correspondingfree energyFG(q). Each connected world-sheet coveringk times the target is thenweighed by a factor ofz(kA), wherez(A) ≡∑

n∈Zexp(−Av(n)) is the generalizedU(1)

partition function. This produces the free energyFG(q) for our theory, which can then beexponentiated to obtain the partition functionZG(q).

In the case of the torus,G = 1, the free energy counting connected coverings is givenby F1(q) = ∑

p

∑m|p(1/m)qp (herem | p means “m dividesp”), so that an explicit

expression forZ1(q) is

(A.1)Z1(q) = exp

(∑p

∑m|p

1

mqpz(pA)

).

The partition functionsZ1,N at fixedN are then straight-forwardly obtained by expandingEq. (A.1) to the desired order (see Appendix of [10] for the first few values ofN ). Forinstance, forN = 2 andN = 3 one gets

(A.2)Z1,2 = 1

2

(z2(A)+ 3z(2A)

),


(A.3)Z1,3 = 1

6

(z3(A)+ 9z(2A)+ 8z(3A)

).

On the other hand, it was noticed in [9,10] that the partition function Eq. (A.1) can bere-expressed as an infinite product:

(A.4)Z1(q) =∏n∈Z

∞∏k=1

1

1− qke−kAv(n)=∏n∈Z

( ∞∑N=0

p(s)qse−sAv(n)

),

p(s) being the number of partitions ofs. This coincides with our general formula Eq. (78),which forG = 1 reduces to

(A.5)Z1(q) =∏n

(∑s

Zs,1qse−sAv(n)

),

since for the pure coverings of the torus one hasZs,1 = p(s), as given in Eq. (24).The equality between the expressions Eqs. (A.1) and (A.4) of the grand-canonical

partition function summarizes the rearrangements by which, for any fixedN , onereconstructs from the sum over the momentan appearing in Eq. (77) the independent sumsover integers which appear in the expansion of Eq. (A.1). Such an expansion containsin fact products ofU(1) partition functionsz(kA). For instance, let us consider the caseN = 2. The inequivalent set of momentan are: (i)n1 < n2, and (ii)n1 = n2. In the case (i),the little group is trivial and so it only has one representation; in case (ii) the little groupis S2, which admits two irreducible representations. We have, therefore,

Z1,2 = 1

2

∑n1 =n2

e−A(v(n1)+v(n2)) + 2∑n1

e−2Av(n1)

=∑n1,n2

e−A(v(n1)+v(n2)) +(

2− 1

2

)∑n1

e−2Av(n1) = 1

2z2(A)+ 3

2z(2A),

in agreement with Eq. (A.2). One can easily repeat the same check forN = 3 and (withincreasing effort) higherN ’s.

When the target space has genusG> 1, in [10] the expressions ofZG,N could be workedout case by case, but a closed expression could not be exhibited. The reason is that noclosed form is in fact known for the free energyFG(q) of connected coverings in this case.Here, attacking the problem by the point of view of theGN lattice gauge theory, we haveobtained in Eq. (77) such a closed expression.

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On the level-dependence of Wess–Zumino–Wittenthree-point functions

Jørgen Rasmussen1, Mark A. Walton2

Physics Department, University of Lethbridge, Lethbridge, AB T1K 3M4 Canada

Received 31 May 2001; accepted 5 July 2001

Abstract

Three-point functions of Wess–Zumino–Witten models are investigated. In particular, we studythe level-dependence of three-point functions in the models based on algebrassu(3) andsu(4). Wefind a correspondence with Berenstein–Zelevinsky triangles. Using previous work connecting thosetriangles to the fusion multiplicities, and the Gepner–Witten depth rule, we explain how to constructthe full three-point functions. We show how their level-dependence is similar to that of the relatedfusion multiplicity. For example, the concept of threshold level plays a prominent role, as it does forfusion. 2001 Elsevier Science B.V. All rights reserved.

PACS: 11.25.HfKeywords: Conformal field theory; WZW model; Correlation functions; Affine fusion

1. Introduction

A general study of Wess–Zumino–Witten (WZW) three-point functions was initiatedin [1]. At sufficiently high levelk, three-point functions for so-called generating-functionprimary fields were written as

(1.1)〈φλ(z1, x1)φµ(z2, x2)φν(z3, x3)〉 = ∆(z1, z2, z3)F(k)λ,µ,ν(x1, x2, x3).

A (generating-function) primary fieldφλ(z, x) is labelled by a dominant weightλ in

(1.2)Pk =

σ =

r∑j=1

σjΛj

∣∣∣∣∣σj ∈ Z,∀j = 1, . . . , r;r∑

j=1

σja∨j k

,

so that its dependence on the levelk is implicit. HereΛj is thej th fundamental weightof a simple Lie algebraXr , of rankr, anda∨j is the corresponding comark. That is, the

E-mail addresses: [email protected] (J. Rasmussen), [email protected] (M.A. Walton).1 Supported in part by a PIMS Postdoctoral Fellowship and by NSERC.2 Supported in part by NSERC.



518 J. Rasmussen, M.A. Walton / Nuclear Physics B 616 [PM] (2001) 517–536

highest root ofXr has corootθ∨ =∑rj=1 a∨jα∨

j , whereα∨j is coroot to thej th simple

rootαj . Henceforth, we will normalise the long rootsα2long = 2, so thatθ∨ = θ .

In φλ(z, x), z is the holomorphic worldsheet coordinate, andx represents the flagvariables — there is an independent variablexα for every positive rootα ∈ R> of thesimple Lie algebraXr .

In (1.1),

(1.3)

∆(z1, z2, z3) := (z1 − z2)−∆1−∆2+∆3(z2 − z3)

∆1−∆2−∆3(z3 − z1)−∆1+∆2−∆3,

where∆j, j = 1,2,3, denote the conformal weights of the three primary fields. That is,the primary fieldφλ(z1, x1) obeys the following operator product expansions (OPEs) withthe energy–momentum tensorT (z)

(1.4)T (z)φλ(z1, x1) ∼ ∆1

(z − z1)2φλ(z1, x1) + 1

z − z1

∂φλ(z1, x1)

∂z1,

and the currentsJa(z)

(1.5)Ja(z)φλ(z1, x1) ∼ −1

z − z1ω(Ja)(x1, ∂, λ)φλ(z1, x1).

Similar OPEs are obeyed byφµ(z2, x2) andφν(z3, x3). Ja(x1, ∂, λ) denotes a differentialoperator (inx1) that realises the corresponding finite-dimensional Lie algebra generatorJa — see [1,2] for more details. In (1.5),ω(Ja) is the generator ofXr obtained fromJa bythe Chevalley involutionω. Since the conformal weights depend on the level, the function∆(z1, z2, z3) has an implicit and well-known dependence on the level.

Here we investigate the extension of (1.1) from high level to all levelsk ∈ Z. We willtreat different levels together, investigating the so-called level-dependence of the three-point functions. The implicit level-dependence of the factor∆(z1, z2, z3) is well-known,so the object of study will beF (k)

λ,µ,ν(x1, x2, x3).Section 2 is a general discussion of three-point functions. It also contains a review

of threshold levels in fusion and indicates how they enter consideration of three-pointfunctions. Sections 3 and 4 treat the casessu(3) andsu(4), respectively. (The case ofsu(2)is covered in Appendix A.) Section 5 is a concluding discussion.

2. Three-point functions

It is important here that the Ward identities can be written in terms of the differentialoperatorsJa(x, ∂, λ). For example, we have

(2.1)0 =(

3∑j=1

Ja(xj , ∂,Λ(j))

)F

(k)Λ(1),Λ(2),Λ(3)

(x1, x2, x3).

Here we usedΛ(1) := λ, Λ(2) := µ, Λ(3) := ν, for convenience of notation.Ward identities encoding the level-dependence, however, are not imposed by (2.1).

Those were used by Gepner and Witten in the derivation of their depth rule [3]. LetEθ

J. Rasmussen, M.A. Walton / Nuclear Physics B 616 [PM] (2001) 517–536 519

denote the generator ofXr that raises the weight by the highest rootθ of Xr . In the formuseful to us, a symmetrical version of the Gepner–Witten constraint is

(2.2)0 =3∏

j=1

(Eθ(xj , ∂,Λ(j))

)j F (k)Λ(1),Λ(2),Λ(3)

(x1, x2, x3),

∀(1, 2, 3) ∈ (Z)⊗3, such that1 + 2 + 3 > k.The differential operator expression forEθ is particularly simple:

(2.3)Eθ(x, ∂,λ) = ∂

∂xθ.

As a consequence, the differential operator realisation is ideally suited to the implementa-tion of the Gepner–Witten depth rule: (2.2) takes a simple, useful form,

(2.4)0 =3∏

j=1

(∂

∂xθj

)j

F(k)Λ(1),Λ(2),Λ(3)

(x1, x2, x3),

∀(1, 2, 3) ∈ (Z)⊗3, such that1 + 2 + 3 > k. This says that the levelk must be

greater than or equal to the highest power ofxθ in F(k)λ,µ,ν , usingxθ

j = xθ for all j = 1,2,3.Now Ja |0〉〉 = 0, for all generatorsJa of the simple Lie algebra, where|0〉〉 represents the

scalar state for the diagonal subalgebra ofXr ⊕ Xr ⊕ Xr . From this, we will see shortlythat

(2.5)F(k)λ,µ,ν(x1, x2, x3) =

(3⊗

j=1

〈Λ(j)|G+(xj )P

)|0〉〉

satisfies the Ward identities (2.1). Here we follow the notation of [1], so that

(2.6)G+(x) := exp

( ∑α∈R>

xαEα

),

with Eα being the raising operator ofXr associated toα ∈ R>. P is the projector from thealgebraXr ⊕ Xr ⊕ Xr to its diagonal subalgebra. Because

(2.7)0 =3⊗

j=1

〈Λ(j)|G+(xj )P (Ja |0〉〉),

we get

0=((

3⊗j=1

〈Λ(j)|G+(xj )

)∆(Ja)

)|0〉〉

(2.8)=3∑

=1

Ja(x, ∂,Λ())

3⊗j=1

〈Λ(j)|G+(xj ) |0〉〉,

usingPJa = ∆(Ja)P with ∆(Ja) = (Ja ⊗ I ⊗ I)⊕ (I ⊗ Ja ⊗ I)⊕ (I ⊗ I ⊗ Ja). So (2.5)obeys (2.1), but the condition (2.2) must still be imposed.


Let L(λ) denote the representation ofXr of highest weightλ. We can decompose thepolynomial (2.5) as

F(k)λ,µ,ν(x1, x2, x3) =

(∑a

W[a]λ,µ,ν(x1, x2, x3) 〈〈0|a

)|0〉〉

(2.9)=∑a

W[a]λ,µ,ν(x1, x2, x3),

where 〈〈0|a denotes one of the (normalised) singlet states in the triple tensor productL(λ) ⊗ L(µ) ⊗ L(ν). The k-dependence will enter in the summation range (see (2.15)below).

Incidentally, we note thatW(a)λ,µ,ν(x1, x2, x3) is a generating function for the Clebsch–

Gordan coefficients of a couplingL(λ) ⊗ L(µ) ⊗ L(ν) ⊃ L(0). We can write

(2.10)|0〉〉a =∑

|u〉∈L(λ)

∑|v〉∈L(µ)

∑|w〉∈L(ν)

C[a]u,v,w|u〉 ⊗ |v〉 ⊗ |w〉,

whereC[a]u,v,w denotes the Clebsch–Gordan coefficient appropriate to the coupling indicated

by a. Define

(2.11)Kλu (x) := 〈λ|G+(x)|u〉,

for |u〉 ∈ L(λ). Then (2.5) and (2.9) yield

W(a)λ,µ,ν(x1, x2, x3)

(2.12)=∑

|u〉∈L(λ)

∑|v〉∈L(µ)

∑|w〉∈L(ν)

C[a]u,v,wKλ

u(x1)Kµv (x2)K

νw(x3).

A key step in [1] was to write the polynomialsF (k)λ,µ,ν(x1, x2, x3) as sums of products of

elementary polynomials:

(2.13)F(k)λ,µ,ν(x) =

∑a

∏E∈E

[RE(x)

]pa(E).

HereE denotes the set of elementary (three-point) couplings of the algebraXr , RE(x) thecorresponding polynomial, and(x) is short for(x1, x2, x3). One necessary constraint onthe products appearing in the decomposition (2.13) is determined by the weightλ,µ, ν.If we define wt(E) to be the weight of an elementary couplingE, then

(2.14)∑E∈E

p(E)wt(E) = λ,µ, ν

must hold. The setsE must be found for each algebra (see [4], for example). Thereare also certain algebraic relations among the elementary couplings (and so among theelementary polynomials) that must be taken into account. These relations are sometimescalled syzygies. They can be implemented by excluding certain products as redundant fromthe sum in (2.13).

Once the syzygies are implemented, each summand in (2.13) counts a coupling betweenthe primary fieldsφλ(z1, x1), φµ(z2, x2) andφν(z3, x3). That is, each summand contributes


1 to the fusion multiplicityT (k)λ,µ,ν that corresponds to the three-point function (1.1). We

therefore rewrite (2.9) as

(2.15)F(k)λ,µ,ν(x) =

T(k)λ,µ,ν∑a=1

W[a]λ,µ,ν(x),

with

(2.16)W[a]λ,µ,ν(x) =

∏E∈E

[RE(x)

]pa(E).

Since each factorRE(x) is itself a valid polynomial, theXr -symmetry is respectedautomatically. Incidentally, here one also sees the usefulness of thex-dependence ofthe generating function primary fields: it makes the different summandsW

[a]λ,µ,ν(x)

independent.Thek-dependence of a fixedW [a]

λ,µ,ν(x) is straightforward, since by (2.4) we must have

(2.17)0 =3∏

j=1

(∂

∂xθj

)j

W[a]Λ(1),Λ(2),Λ(3)

(x1, x2, x3),

∀(1, 2, 3) ∈ (Z)⊗3, such that1 + 2 + 3 > k.

2.1. Threshold levels

Let ta denote the maximum power ofxθ in W[a]λ,µ,ν(x) with non-zero coefficient:

(2.18)

ta = min

t ∈ Z

∣∣∣∣∣3∏

j=1

(∂

∂xθj

)j

W[a]λ,µ,ν(x) = 0,

∀(1, 2, 3) ∈ (Z)⊗3,

3∑j=1

j = t + 1

.

ThenW[a]λ,µ,ν(x) will not contribute to the sum (2.15) ifta is greater than the levelk, but

does contribute ifta k. That is,ta is athreshold level for W[a]λ,µ,ν(x).

Since (2.18) only involvesxαj for α = θ , it appears to violate the symmetry of the

horizontal algebraXr . It therefore must be interpreted with care — one could just dropthe higher powers ofxθ , for example, to obtain incorrect results. The condition should beused to restrict theXr -invariant summands in (2.15), as we will discuss below.

The concept of threshold level has been useful in WZW fusion, when the fusionis viewed as a truncation of the tensor product of simple Lie algebras [5–8]. Thedecomposition of a tensor productL(λ) ⊗ L(µ) of such representations can be writtenas

(2.19)L(λ) ⊗ L(µ) =⊕ν∈P

T νλ,µL(ν),


where

(2.20)P =σ =

r∑j=1

σjΛj

∣∣∣∣∣σj ∈ Z, ∀j = 1, . . . , r

(compare to (1.2)). More important for three-point functions is

(2.21)L(λ) ⊗ L(µ) ⊗ L(ν) ⊃ Tλ,µ,νL(0),

where the triple product multiplicitiesTλ,µ,ν are related to the conventional tensor productmultiplicitiesT ν

λ,µ by

(2.22)Tλ,µ,ν = T ν+λ,µ ,

ν+ being the weight conjugate toν. Fusion products⊗k can be written in a similar way:

(2.23)L(λ) ⊗k L(µ) =⊕ν∈Pk

T(k) νλ,µ L(ν)

and

(2.24)L(λ) ⊗k L(µ) ⊗k L(ν) ⊃ T(k)λ,µ,νL(0),

where

(2.25)T(k)λ,µ,ν = T

(k)ν+λ,µ .

That fusion is a truncated tensor product,

(2.26)T(k)λ,µ,ν T

(k+1)λ,µ,ν , lim

k→∞T(k)λ,µ,ν = Tλ,µ,ν,

follows from the Gepner–Witten depth rule [3,6].One can encode all this in a simple fashion using the threshold level. The triple product

multiplicities can be written as a sum over terms of specific threshold level:

(2.27)Tλ,µ,ν =∞∑t=0

n[t ]λ,µ,ν,

where the threshold multiplicityn[t ]λ,µ,ν is the number of couplings of threshold levelt . We

can write the fusion triple product multiplicity in a similar way:

(2.28)T(k)λ,µ,ν =

k∑t=0

n[t ]λ,µ,ν.

To compare this with a similar result for three-point functions, we defineW[a]λ,µ,ν(x; t) :=

W[a]λ,µ,ν(x), if ta = t . The goal is to write

(2.29)F(k)λ,µ,ν(x) =

k∑t=0

n[t]λ,µ,ν∑a=1

W[a]λ,µ,ν(x; t),

where the values of the indexa have been redefined appropriately.


We emphasise here that a choice must be made to write (2.29). Due to the syzygies,different representations ofW [a] will exist, in general with different threshold levelsta .

By (2.18), one sees that the threshold level of a product of polynomials is just the sumof their individual threshold levels. We define the threshold levelt (E) of an elementarycouplingE ∈ E by

(2.30)

t (E) := min

t ∈ Z

∣∣∣∣∣3∏

j=1

(∂

∂xθj

)j

RE(x1, x2, x3) = 0,∀3∑

j=1

j = t + 1

.

Then when (2.16) gets updated to

(2.31)W[a]λ,µ,ν(x; ta) =

∏E∈E

[RE(x)

]pa(E),

we must have

(2.32)ta =∑E∈E

pa(E)t (E).

ForXr = Ar∼= su(N) (so thatN = r + 1), the couplings counted inTλ,µ,ν are in one-

to-one correspondence with Berenstein–Zelevinsky (BZ) triangles [9] of weightλ,µ, ν.The BZ triangles were used to studysu(3) fusion in [6,7];su(4) fusion was studied in [8],which also alluded tosu(N) generalisation. Forsu(3) andsu(4), it was found that everyBZ triangle could be assigned a threshold level. We will now “lift” those results to thecorresponding polynomials, by associating a BZ triangle to each polynomialW [a](x; t).Not only will this exercise promote the previous results to something containing moreinformation, it will also provide the justification for some of them.

3. su(3) three-point functions, triangles and fusion

An su(3) BZ triangle of weightλ,µ, ν looks like:

(3.1)

m13

n12 l23

m23 m12

n13 l12 n23 l13

Its entrieslij ,mij , nij ∈ Z determine the Dynkin labels ofλ,µ, ν ∈ P:

m13 + n12 = λ1, n13 + l12 = µ1, l13 + m12= ν1,

(3.2)m23 + n13 = λ2, n23 + l13 = µ2, l23 + m13= ν2.

We call these the outer constraints on the BZ entries.For generalsu(N), the BZ entries determine three elements ofZR>:

λ + µ − ν+ =∑α∈R>

nαα =: nR>,


ν + λ − µ+ =∑α∈R>

mαα =: m · R>,

(3.3)µ + ν − λ+ =∑α∈R>

lαα =: l · R>.

Using an orthonormal basisej | j = 1, . . . ,N; ei · ej = δi,j of RN , the positive roots can

be written as

(3.4)R> = ei − ej | 1 i < j N.This explains the indicesij on the BZ entries.

If the su(3) version of (3.3) is written, by applying the conjugation operation+, one canderive the consistency conditions

n12 + m23 = n23 + m12,

l12 + m23 = l23 + m12,

(3.5)l12 + n23 = l23 + n12.

These are the hexagon constraints, so named because of the geometry of (3.1).The triple tensor product multiplicityTλ,µ,ν equals the number of BZ triangles with non-

negative integer entries that satisfy both the outer and hexagon constraints. Now a sum ofBZ triangles is also a BZ triangle, but with a new weight equalling the sum of the weights ofthe added triangles. Compare this to (2.14); a product of “three-point polynomials” (RE ’s,and so by extensionW [a]

λ,µ,ν ’s) is also a three-point polynomial, but with the weights added.Thus, adding triangles corresponds to multiplying polynomials. Clearly then, we shouldfind the triangles associated to the elementary polynomials, the elementary triangles. Thenthe triangle associated with an arbitrary polynomialW

[a]λ,µ,ν(x) can be found simply from

its factorisation into powers of elementary polynomials.The elementary polynomials forsu(3) and su(4) were worked out completely in [1].

Recall thatαb denotes thebth simple root. Forsu(3), R> = α1, θ = α1 + α2, α2. Wewrite x

αb

j =: xbj for the flag coordinates related to the simple roots. The variablexθ only

appears in the combinations

(3.6)x±j := 1

2x1j x

2j ± xθ

j .

This can be seen as follows. LetFα denote the lowering operator ofXr associated withα ∈ R>. In the differential operator realisation of the algebraXr , one finds [2]

(3.7)Fα(x, ∂,λ) =∑β∈R>

Vβ−α(x)

∂

∂xβ+

r∑j=1

Pj−α(x)λj .

HereV β−α(x) andPj

−α(x) denote polynomials, explicitly written in [2], and only involvingxθ in the combinations (3.6). This property is carried over to the elementary polynomials,by a straightforward analysis.

Although for generalXr it does not hold, forsu(3) andsu(4), an elementary couplingE ∈ E is uniquely specified by its weight. We will label the corresponding polynomials


with those weights.su(3) has 8 elementary polynomials. Six are related to

(3.8)R1,2,0(x) := RΛ1,Λ2,0(x1, x2, x3) = x+1 + x−

2 − x11x

22;

they can be obtained from (3.8) by permuting the subscripts ofx1, x2, x3 and the threeweightsΛ1,Λ2,0 in the identical way. These six elementary polynomials are actuallyelementary polynomials for two-point couplings — see [10]. The other two elementarypolynomials are

R1,1,1(x) := RΛ1,Λ1,Λ1(x1, x2, x3)

(3.9)= x12x

+3 − x+

2 x13 + x1

1

(x+

2 − x+3

)+ x+1

(x1

3 − x12

),

andR2,2,2(x) := RΛ2,Λ2,Λ2(x1, x2, x3), obtained fromR1,1,1(x) by x+

j → x−j andx1

j →x2j .The elementarysu(3) BZ triangles were written in [6]. They are

(3.10)

01 0

0 00 0 1 0

00 0

0 01 0 0 0

01 0

0 10 1 0 0

and those obtained by applying the triangle symmetries that generate the dihedral groupD3 ∼= S3. TheS3 orbits of the first two triangles of (3.10) are 3-dimensional, and that ofthe last is 2-dimensional, giving the correct number of 8 elementary triangles.

It is simple to assign these elementary triangles to elementary polynomials. For example,those of (3.10) correspond toR1,2,0, R2,1,0 andR1,1,1, respectively. The threshold levelsare also easily found. Becausexθ only appears in the combinationsx± of (3.6), thethreshold level is simply the maximum power ofx±. By (3.8) and (3.9), we see that all8 elementary polynomials (and so elementary couplingsE) have threshold levelt (E) = 1.

If we write λ = λ1Λ1 + λ2Λ

2 = (λ1, λ2), then consider the example of the three-point function with weight(1,1), (1,1), (1,1). The tensor product multiplicity is 2, withthreshold levels 2 and 3 for the two couplings. We write

(3.11)W[1](1,1),(1,1),(1,1)(x;2)= R1,1,1(x)R2,2,2(x)

and

(3.12)W[2](1,1),(1,1),(1,1)(x;3)= R2,1,0(x)R0,2,1(x)R1,0,2(x),

for the polynomials of threshold levels 2 and 3, respectively.But another productR1,2,0R0,1,2R2,0,1 has weight(1,1), (1,1), (1,1) — why is it not

included? Forsu(3), there is a single syzygy

(3.13)R1,1,1R2,2,2 = R1,2,0R0,1,2R2,0,1 + R2,1,0R0,2,1R1,0,2

to be taken into account. Only two linear combinations of the threeR-monomials in (3.13)can be included, since three are not independent. Here the triangle correspondencemay guide us, since there are no relations on the BZ triangles themselves — they are


simply counted to get the tensor product multiplicity. Associating a BZ triangle to eachelementary polynomial implements the syzygies automatically: bothR1,1,1R2,2,2 andR1,2,0R0,1,2R2,0,1 correspond to the same BZ triangle. In order to produce the thresholdlevels2,3, it is the productR1,2,0R0,1,2R2,0,1 that should be excluded.

That means that if

W[a]λ,µ,ν = (

R1,2,0)p1,2,0(R0,1,2)p0,1,2

(R2,0,1)p2,0,1

(R2,1,0)p2,1,0

(3.14)× (R0,2,1)p0,2,1

(R1,0,2)p1,0,2

(R1,1,1)p1,1,1

(R2,2,2)p2,2,2,

then at least one ofp1,2,0,p0,1,2,p2,0,1 must vanish. Now, (3.14) corresponds to thetriangle

(3.15)

p1,0,2

p1,2,0 +p1,1,1 p0,1,2 + p2,2,2

p2,0,1 + p2,2,2 p2,0,1 +p1,1,1

p2,1,0 p0,1,2 +p1,1,1 p1,2,0 + p2,2,2 p0,2,1

Let’s assume thatp1,2,0 = 0. Becauset (E) = 1 for all elementary couplings, we see thatthe threshold level of (3.14) is just the sum of all the powerspi,j,k , or t = n13 + m13 +l23 + m12 + l13 = n13 + ν1 + ν2. Notice thatp1,2,0 = 0 ensures thatn13 + ν1 + ν2 maxm13 +µ1 +µ2, l13 + λ1 + λ2. By considering the other possibilities,p0,1,2 = 0 andp2,0,1 = 0, we find the cyclically symmetric formula

(3.16)t = maxl13 + λ1 + λ2,m13 + µ1 + µ2, n13 + ν1 + ν2

,

confirming the result of [6].Of course, our results constitute much more than a verification of the results of [6] on

fusion multiplicities. We have shown how to construct the full three-point functions (3.14).In addition, we have demonstrated that the fusion results “lift” in a straightforward mannerto those for three-point functions.

Furthermore, in [6] certain obstacles were found in a direct application of the Gepner–Witten depth rule. Here we have shown how to implement the depth rule in a very simpleway, using a differential operator realisation ofXr . We will have more to say on this subjectbelow.

The triangles helped us implement the syzygy (3.13), by choosing a product to forbid,R1,2,0R0,1,2R2,0,1. But a different choice could certainly have been made, with no effecton the results at large level. Because the syzygy is inhomogeneous in the threshold level,however, the choice becomes important at smaller levels. For example, if we had forbiddenthe productR1,1,1R2,2,2, then the additivity (2.32) of threshold levels would be lost.Another possibility is to forbidR2,1,0R0,2,1R1,0,2. This is equivalent to choosing theorientation around the BZ triangle that is opposite to that used in (3.1). Of course, onecould also forbid a linear combination of theR-monomials in (3.13), but such a choice isnot convenient for us.

It is often preferable to work directly with the polynomialsW [a]λ,µ,ν(x), rather than to

first express them as products of the elementary polynomialsRE(x). But we can find thetriangle of a polynomialW [a]

λ,µ,ν(x) directly. For our purposes, rewrite (3.3) as


λ +(µ −

∑α∈R>

nαα

)+ (−ν+)= 0,

ν +(λ −

∑α∈R>

mαα

)+ (−µ+)= 0,

(3.17)µ +(ν −

∑α∈R>

lαα

)+ (−λ+)= 0.

The first of these expressions suggests that we look at the contribution|λ〉⊗|µ′〉⊗ |−ν+〉.Here|λ〉 is the highest weight state ofL(λ), | −ν+〉 is the lowest weight state ofL(ν), and|µ′〉 denotes a state ofL(µ), of weightµ′ := µ − n · R>.

Let x[β]3 denote any linear combination of products

(3.18)∏

α∈R>

(xα

3

)bαsuch that

∑bα α = β . By construction,x[0]

1 = 1 isolates the highest weight state|λ〉,while the lowest weight state|−ν+〉 corresponds tox[ν+ν+]

3 (see (2.11), for exam-

ple). Now, a fixedW[a]λ,µ,ν should include a unique term (with non-vanishing coeffi-

cient) x[0]1 x

[µ−µ′]2 x

[ν+ν+]3 , or x

[0]1 x

[n·R>]2 x

[ν+ν+]3 . Let that term be denotedW [a]

λ,µ,ν(x) |x

[0]1 x

[n·R>]2 x

[ν+ν+]3 . We find that its factorx[n·R>]

2 suffices to determine thenα, i.e., onethird of the BZ entries. The other two thirds are obtained in a similar way, using the othertwo equations of (3.17).

Simply put, we find

(3.19)x[n·R>]2 = (

x12

)n12(x−

2

)n13(x2

2

)n23.

This result and its two siblings can be encoded in the form of a BZ triangle:

(3.20)

x−1

x12 x2

3

x21 x1

1

x−2 x1

3 x22 x−

3

This diagram can be used to find the BZ triangle corresponding to a given polynomialW

[a]λ,µ,ν(x). We should emphasise, however, that it does not allow us to construct the full

polynomial from a prescribed BZ triangle, only a few terms in it.A refined version of the Gepner–Witten depth rule [3] was conjectured in [6,11] (see

also [12]). It is less symmetric than (2.2), but is a natural generalisation of a well-knownformula in the theory of simple Lie algebras. It will prove to be more convenient in somecomputations. The realisation-independent form of the formula is

(3.21)

T(k)λ,µ,ν = dim

v ∈ L

(µ; ν+ − λ

) ∣∣Fαi

ν+i +1v = 0,

∀i ∈ 1,2, . . . , r; Eθk−ν·θ+1v = 0

.


HereL(µ; ν+ − λ) is the subspace ofL(µ) of weightν+ − λ, andν+i is theith Dynkin

label ofν+. The well-known classical formula for tensor product multiplicities (see [13],for example) is obtained in the limit of large levelk → ∞. In this case then the constraintEθ

k−ν·θ+1 v = 0 is always satisfied, and can be dropped.If written similar to (2.17), it is

(3.22)0 =(

∂

∂xθ2

)

W[a]Λ(1),Λ(2),Λ(3)

(x)|x[0]1 x

[n·R>]2 x

[ν+ν+]3 ,

∀ ∈ Z, such that > k − ν · θ . Consequently, we find

(3.23)ta = min

t ∈ Z

∣∣∣∣(

∂

∂xθ2

)t+1

W[a]λ,µ,ν(x)|x[0]

1 x[n·R>]2 x

[ν+ν+]3 = 0

+ ν · θ.

Clearly, the last two formulas treat the 3 weights in asymmetric manner. Consequently,5 other formulas of this type can be written, based on the 5 other permutations ofλ,µ, ν.

Using these formulas, we easily verify the threshold levels computed using thesymmetric version of the depth rule. This provides evidence for the refinement of thedepth rule (3.21), conjectured in [6,11]. This asymmetric form has the advantage that iteffectively reduces consideration from the coupling of three states to a single state, onethat necessarily couples to a simple highest-weight state and a simple lowest-weight state.Consequently, to use (3.22), we need only a small part of the polynomialW

[a]λ,µ,ν ; for (2.17),

the full polynomial is required.

4. su(4) three-point functions, triangles and fusion

The results forsu(4) are technically more complicated than those forsu(3). They alsoinclude an interesting new feature, discussed below.

The 18 elementary couplings may be specified by their weights,λ,µ, ν, or Λ(1),Λ(2),

Λ(3) [1]. Permuting 1,2,3 in Λ(1),Λ(2),Λ(3) andx1, x2, x3 produces other elementarycouplings. The polynomial-triangle correspondence breaks the permutationS3 symmetryto the cyclicZ3 symmetry (see (3.17), e.g.). So only cyclic permutations of 1,2,3 willproduce other elementary couplings with related BZ triangles.

Of the 18 elementary couplings, 9 are of the two-point type [10]. They separate intothree cyclic-permutation orbits, with representative polynomials

(4.1)R1,3,0 := RΛ1,Λ3,0, R3,1,0 := RΛ3,Λ1,0 and R2,2,0 := RΛ2,Λ2,0.

The 9 elementary polynomials of three-point type form another three orbits, withrepresentatives

(4.2)R1,1,2 := RΛ1,Λ1,Λ2, R2,3,3 := RΛ2,Λ3,Λ3

, and R13,2,2 := RΛ1+Λ3,Λ2,Λ2.

By (2.18) or (3.23), we find that all these couplings have threshold level 1, exceptR13,2,2,R2,13,2,R2,2,13, which havet = 2.


For su(4) the BZ triangle is

(4.3)

m14

n12 l34

m24 m13

n13 l23 n23 l24

m34 m23 m12

n14 l12 n24 l13 n34 l14

with corresponding Dynkin labels given by the outer constraints

m14 + n12 = λ1, n14 + l12 = µ1, l14 + m12 = ν1,

m24 + n13 = λ2, n24 + l13 = µ2, l24 + m13 = ν2,

(4.4)m34 + n14 = λ3, n34 + l14 = µ3, l34 + m14 = ν3.

The su(4) BZ triangle contains three hexagons, and there are inner constraints (hexagonidentities) related to each of them in the obvious way.

The elementary polynomials of (4.1) and (4.2) can be related easily to elementary BZtriangles. The non-zero BZ entries corresponding to (4.1) are

(4.5)n12 = n23 = n34 = 1, n14 = 1 and n13 = n24 = 1,and those related to (4.2) are

n12 = l12 = m13 = l23 = 1, m24 = l34 = n23 = n34 = 1 and

(4.6)n12 = m34 = l23 = n24 = m13 = 1.Thesu(4) polynomial→ triangle correspondence cannot be given as simply as forsu(3),

in (3.19), (3.20). Instead, we write

K1,0,0(x2) → ,K−1,1,0(x2) → n12 = 1,K0,−1,1(x2) → n13 = 1,

(4.7)K0,0,−1(x2) → n14 = 1,

K0,1,0(x2) → ,K1,−1,1(x2) → n23 = 1,K−1,0,1(x2) → n12 = n23 = 1,K1,0,−1(x2) → n24 = 1,

K−1,1,−1(x2) → n12 = n24 = 1,(4.8)K0,−1,0(x2) → n13 = n24 = 1,

K0,0,1(x2) → ,K0,1,−1(x2) → n34 = 1,K1,−1,0(x2) → n34 = n23 = 1,

(4.9)K−1,0,0(x2) → n34 = n12 = n23 = 1,


(4.10)−L1(x2) + 2L2(x2) − L3(x2) → n13 = n34 = 1.The K- andL-polynomials were defined in [1]. Those in (4.7)–(4.9) are in one-to-onecorrespondence with the states inL(Λ1), L(Λ2), L(Λ3), respectively. Eq. (4.10) picksout one state in the three-dimensional subspace of weight 0 in the adjoint representationL(Λ1 + Λ3).

The rules (4.7)–(4.9) and (4.10), can be used to find the triangle that corresponds to agiven polynomial directly, without first decomposing the polynomial into its elementaryfactors.

For su(4), there are 15 syzygies. They can be obtained from the following 5, bypermuting the superscripts cyclically in all polynomials:

(4.11)SR13,2,2R2,13,2,R2,2,0R1,1,2R3,3,2,R0,2,2R2,0,2R3,1,0R1,3,0,

(4.12)SR13,2,2R2,3,3,R1,0,3R2,2,0R3,3,2,R1,3,0R2,0,2R3,2,3,

(4.13)SR2,2,13R1,1,2,R1,0,3R0,2,2R2,1,1,R0,1,3R2,0,2R1,2,1,

(4.14)SR13,2,2R0,1,3,R1,1,2R3,2,3,R1,0,3R3,1,0R0,2,2,

(4.15)SR2,13,2R1,0,3,R1,1,2R2,3,3,R0,1,3R1,3,0R2,0,2.

HereSA,B,C indicates there exists one independent vanishing linear combinationαA+βB + γC = 0, with α,β, γ = 0.3

We will implement each syzygy by choosing a forbidden product from among its“component”R-monomials. That is, we will not consider eliminating linear combinationsof the componentR-monomials as redundant, although that is certainly possible. Evenwith this limitation, however, the choice is not unique. As we did forsu(3), we will use thetriangle correspondence, and specific fusion multiplicities to guide us. We will preserve theadditivity of the threshold level, and by choosing a symmetric set of forbidden couplings,the cyclicZ3 symmetry of the triangles.

The fusions relevant to (4.11)–(4.15) are

(4.16)L(1,1,1)⊗ L(1,1,1)⊗ L(0,2,0) ⊃ L(0,0,0)3,4,

(4.17)L(1,1,1)⊗ L(0,1,1)⊗ L(0,1,1) ⊃ L(0,0,0)3,3,

(4.18)L(1,1,0)⊗ (1,1,0)⊗ L(1,1,1) ⊃ L(0,0,0)3,3,

(4.19)L(1,0,1)⊗ L(1,1,0)⊗ L(0,1,1) ⊃ L(0,0,0)2,3,

(4.20)L(1,1,0)⊗ L(1,0,1)⊗ L(0,1,1) ⊃ L(0,0,0)2,3,

respectively. Here the notation isL(λ1, λ2, λ3) = L(λ), with λ = ∑3i=1λiΛ

i . Also,in (4.16), e.g.,L(0,0,0)3,4 indicates thatT(1,1,1),(1,1,1),(0,2,0) = 2, and that the correspond-ing set of threshold levels is3,4. Compare these threshold levels with those of (4.11):4,3,4. EitherR13,2,2R2,13,2 or R0,2,2R2,0,2R3,1,0R1,3,0 must be forbidden (recall thatwe do not consider the linear combinations thereof as possible forbidden terms). Since

3 The explicit linear combinations are given in Ref. [1]. We take the opportunity to correct a sign error there:the parameteru5 = −2.


R13,2,2R2,13,2 andR2,2,0R1,1,2R3,3,2 correspond to the same BZ triangle, however, weforbid R13,2,2R2,13,2.

Analysing the syzygies (4.11)–(4.14), yields the forbidden products

(4.21)R13,2,2R2,13,2,

(4.22)R13,2,2R2,3,3 or R1,3,0R2,0,2R3,2,3,

(4.23)R2,2,13R1,1,2 or R0,1,3R2,0,2R1,2,1,

(4.24)R13,2,2R0,1,3,

(4.25)R0,1,3R1,3,0R2,0,2,

respectively. The consequences of the other 10 syzygies are obtained by cyclicallypermuting the superscripts in (4.21)–(4.25). We will specify a choice of forbidden productslater.

First we treat a new feature that arises in thesu(4) case. Whenλ = µ = ν = Λ1 + Λ3,the highest weight of the adjoint representation, the tensor product multiplicity is 2, andthe threshold levels are 2 and 3. But the possibleR-monomials of this weight are

(4.26)R1,3,0R0,1,3R3,0,1 and R3,1,0R0,3,1R1,0,3,

both with threshold level 3. This discrepancy was noticed first in [8], where considerationswere based entirely on BZ triangles, and was verified in [14]. Here we can explain why ithappens: the terms in theR-monomials (4.26) of maximum degree inxθ are cancelled intheir sum. That is,

(4.27)R13,13,13 := R1,3,0R0,1,3R3,0,1 + R3,1,0R0,3,1R1,0,3

has threshold level 2, as found by (2.18).One can persist in using the original set of elementary couplings obtained from (4.1),

(4.2). When considering three-point functions of arbitrary weightλ,µ, ν, then linearcombinations such as (4.27) may have to be found, in order to be consistent with the fusionmultiplicities. In general, however, this is a tedious task.

Alternatively, as suggested by (4.27), we can continue to work in the same framework byintroducingR13,13,13 as a new elementary polynomial. This follows the procedure of [8]and [14]. The price to be paid is the introduction of extra syzygies involving the newelementary coupling, so that a single coupling is not counted more than once. In particular,we must implement

(4.28)SR13,13,13,R1,3,0R0,1,3R3,0,1,R3,1,0R0,3,1R1,0,3

as a new syzygy, by forbidding eitherR1,3,0R0,1,3R3,0,1 or R3,1,0R0,3,1R1,0,3 (forexample). SinceR3,1,0R0,3,1R1,0,3 corresponds to the simpler triangle (with non-vanishingentries l14 = m14 = n14 = 1) we choose to forbid it, so that the new coupling thencorresponds to this simple triangle.

This introduction of a new elementary coupling is perhaps not too surprising, if onelooks back at thesu(3) case. Recall the single syzygy (3.13) forsu(3). As far as tensorproducts are concerned, it can be implemented by forbidding the productR1,1,1R2,2,2.


As noted above, however, two other possibilities (forbiddingR1,2,0R0,1,2R2,0,1 orforbidding R2,1,0R0,2,1R1,0,2) are more convenient for fusion, since they preserve in asimple way the additivity of the threshold level. If we insist on forbiddingR1,1,1R2,2,2,however, the situation is completely analogous to thesu(4) case just considered. TheR-monomials for the couplingλ = µ = ν = Λ1 + Λ2 are thenR1,2,0R0,1,2R2,0,1

and R2,1,0R0,2,1R1,0,2, each with threshold level 3; the correct threshold levels are2, 3, however. But a new elementary polynomialR12,12,12 := R1,2,0R0,1,2R2,0,1 +R2,1,0R0,2,1R1,0,2 can be introduced, and it has threshold level 2. Of course, a new syzygySR12,12,12,R1,2,0R0,1,2R2,0,1,R2,1,0R0,2,1R1,0,2 is also necessary.

The difference between thesu(3) and su(4) cases originates in the syzygies. Forsu(3), implementing the syzygy (3.13) makes the linear combinationR1,2,0R0,1,2R2,0,1 +R2,1,0R0,2,1R1,0,2 inaccessible. No such syzygy is present in thesu(4) case, however. ThecombinationR1,3,0R0,1,3R3,0,1 + R3,1,0R0,3,1R1,0,3 can be included, and so generates anew elementary coupling.

It is interesting to use the asymmetric depth rule, (3.21) and (3.22), to analyse thesephenomena at the level of states. A single relation amongR-monomials implies anx[m·R>]

1

relation, anx[n·R>]2 relation, and anx[l·R>]

3 relation. Thesu(3) syzygy (3.13), for example,yields

(4.29)x1x2 = x− + x+

three times, once for eachx → xi , i = 1,2,3. In terms of lowering operators, (4.29) is aconsequence of the relation

(4.30)[Fα1,Fα2] = −Fθ .

It is identical in the three cases because of the symmetry of (3.13), and encodes a relationbetween three states of weight 0 in thesu(3) representationL(1,1). Since only two suchstates are independent, a relation like (4.29) is inevitable. A basis of the two-dimensionalspace can be chosen with depths1,1, or with 1,0, giving rise to threshold levels3,3 or 3,2. Since the latter is correct forL(1,1)⊗3 ⊃ L(0), we see that choosing thecorrectR-monomialR1,1,1R2,2,2 = R1,2,0R0,1,2R2,0,1 + R2,1,0R0,2,1R2,0,1 correspondsto choosing a basis of states of the required depths.

Many of thesu(4) syzygies are not symmetric and so lead through the asymmetric depthrule to three distinctxi-relations. But the definition of the new elementary coupling (4.27),is symmetric. It yields

(4.31)x1x23 + x12x3 = −K−100(x) + K00−1(x),

in the notation of [1], where

K00−1 = xθ + 1

2x1x23 + 1

2x12x3 + 1

6x1x2x3,

(4.32)K−100= xθ − 1

2x1x23 − 1

2x12x3 + 1

6x1x2x3.

This indicates that a linear combination of two depth-1 states of weight 0 inL(1,0,1) is astate of depth 0.


In an arbitrary basis, finding such linear combinations are necessary when using thedepth rule [6,11]. It is not surprising, therefore, that the new coupling (4.27) must beintroduced; it is perhaps surprising that (4.27) is the only new coupling needed. This hasnot been proved, however. We defer to [14], where some counting arguments supportedthis assumption.

We conclude our construction ofsu(4) three-point functions by listing a choice of a setof forbidden couplings:

(4.33)

R13,2,2R2,13,2, R2,13,2R2,2,13, R2,2,13R13,2,2,

R13,2,2R2,3,3, R2,13,2R3,2,3, R2,2,13R3,3,2,

R2,2,13R1,1,2, R13,2,2R2,1,1, R2,13,2R1,2,1,

R13,2,2R0,1,3, R2,13,2R3,0,1, R2,2,13R1,3,0,

R0,1,3R1,3,0R2,0,2, R3,0,1R0,1,3R2,2,0, R1,3,0R3,0,1R0,2,2,

R3,1,0R0,3,1R1,0,3.The three-point function of weightλ,µ, ν can be constructed from the sum over allR-monomials of that weight (as in (2.15), (2.16)) that do not contain the forbiddenproducts (4.33).

To avoid confusion, we should state that our choice of forbidden products will not leadto the formula of [8,14] assigning a threshold level to eachsu(4) BZ triangle. Since it isonly the set of threshold levels for a fixed weightλ,µ, ν that is physically relevant, thisis not a problem. We have not attempted to find a different formula, resulting from thechoice (4.33), assigning a threshold level to a BZ triangle. Threshold levels can be veryeasily found from the relevantR-monomials (using (2.32)), and theseR-monomials mustbe constructed in order to write the three-point function. When the interest is just fusion,and not the three-point functions, a formula like that of [8,14] is very helpful (see [15],e.g.). Here it represents an intermediate step we can eliminate.

5. Discussion

Building on the work [1], we have shown how to construct WZW three-point functionsfor generating-function primary fields, for the algebrassu(3) andsu(4). More generally,our approach is ideally suited to the use on three-point functions of the Gepner–Wittendepth rule, and makes it clear that the level-dependence of WZW three-point functionsmimics that of the corresponding fusion multiplicities, for all algebras.

In addition, we showed that a refined version [6,11,12] of the original depth rule leadsto a correspondence between three-point functions and BZ triangles. This correspondencewas very helpful forsu(3) andsu(4). For example, it allowed us to use previous work onsu(3) andsu(4) fusion and BZ triangles [6–8]. It also helped eliminate the redundanciesencoded in the syzygies.

We will conclude with a brief discussion of possible uses and improvements of thiswork. We hope to address some of these issues in the future.


To be more precise, we found a correspondence between the polynomialsRE(x) andW

[a]λ,µ,ν(x) and the BZ triangles, for the algebrassu(3) andsu(4). (For completeness, the

su(2) case is treated in a short appendix.) That is, given such a polynomial, we showedhow to find the associated triangle. The polynomials themselves are more important thanthe triangles, however. It would be interesting to try to find an algorithm to construct theappropriate polynomial from a BZ triangle.

As described above, when respecting the additivity of the threshold level, the trianglesmay favour certain ways of imposing the syzygies. However, it is sometimes preferable toimpose them in other ways. Furthermore, BZ triangles, or their analogues, have not beenconstructed for all algebras. If one is interested inG2 three-point functions, for example,one cannot rely on BZ constructions as a guide.

In those cases, one should consider the polynomials themselves, instead of firstprojecting to the corresponding triangles, and working with them. For example, onecan write polytope volume formulas for the fusion multiplicities [17] that are analogousto those written in [18] for the tensor product multiplicities forsu(N). These multiplesum formulas generalise the expressions for fusion multiplicities in [7,15,16], derivedusing BZ techniques. In particular, they are derived using virtual three-point functions,generalisations of the virtual BZ triangles and diagrams used in [7,15,16,18].

Finally, consider the generalisation to higher rank — forsu(N) or other algebras.Our method of studying the level-dependence should work simply for any algebra, ofany rank. However, we have relied on the method of elementary couplings to constructthe polynomialsW [a]

λ,µ,ν , before the level-dependent constraints are imposed. Since thenumbers of elementary couplings and syzygies climb very rapidly, this part of theconstruction becomes impractical for higher ranks. A method of finding the polynomialsW

[a]λ,µ,ν directly, without first factoring into elementary polynomials, might make larger

ranks more tractable.

Acknowledgements

We thank Chris Cummins and Pierre Mathieu for comments on the manuscript.

Appendix A. su(2) three-point functions, triangles and fusion

Since there is only one positive root,α1 = θ , we usex1j = xθ

j = xj , j = 1,2,3. Similarly,we write

Eα1(x, ∂, λ) =: E(x, ∂,λ) = ∂

∂x,

Hα1(x, ∂, λ) =: H(x, ∂,λ) = −2x∂

∂x+ λ,

(A.1)Fα1(x, ∂, λ) =: F(x, ∂,λ) = −x2 ∂

∂x+ λx

for the differential-operator realisation ofsu(2).


There are three elementary couplings,

(A.2)R1,1,0(x) = x1 − x2, R0,1,1(x) = x2 − x3, R1,0,1(x) = x3 − x1,

with no syzygies. For a weightλ,µ, ν = λ1Λ1,µ1Λ

1, ν1Λ1, we therefore find

(A.3)Wλ,µ,ν = [R1,1,0](λ1+µ1−ν1)/2[

R0,1,1](−λ1+µ1+ν1)/2[R1,0,1](λ1−µ1+ν1)/2

.

An su(2) BZ triangle is very simple:

(A.4)m

n l

To be consistent with (3.1) and (4.3), we should writem = m12, etc., sinceα1 = e1 − e2,wheree1, e2 is an orthonormal basis ofR2. We dropped the subscripts for simplicity,however. The outer constraints are

(A.5)λ1 = m + n, µ1 = n + l, ν1 = l + m,

and there are no inner constraints, since the triangle contains no hexagons.The elementary polynomials of (A.2) are associated to the triangles

(A.6)0

1 00

0 11

0 0

respectively. Comparing this correspondence with (A.3), we see that

(λ1 + µ1 − ν1)/2 = n, (−λ1 + µ1 + ν1)/2 = l,

(A.7)(λ1 − µ1 + ν1)/2 = m,

as must be, by the outer constraints (A.5).The symmetric form (2.17) of the depth rule gives threshold levelt = 1 for the three

elementary couplings, and thresholdt = l + m + n = (λ1 + µ1 + ν1)/2 for the generalpolynomial (A.3). This is consistent with the additivity of the threshold level undermultiplication ofR-monomials.

The asymmetric form (3.22) of the depth rule is in agreement. For the generalpolynomial (A.3), we find

(A.8)x[m·R>]1 = (−x1)

m, x[n·R>]2 = (−x2)

n, x[l·R>]3 = (−x3)

l .

The second of these corresponds to a state of depthd = n, and so givest = d + ν · θ =n + (l + m). The first and third can be analysed using formulas obtained by permutingλ,µ, ν andx1, x2, x3 cyclically in (3.22). We gett = d +µ · θ = m+ (n+ l) from the first,andt = d + λ · θ = l + (m + n) from the third monomial of (A.8).

From (A.8), we see that

(A.9)x1

x2 x3

is thesu(2) analogue of thesu(3) result (3.20).


References

[1] J. Rasmussen, Int. J. Mod. Phys. A 14 (1999) 1225.[2] J. Rasmussen, Applications of free fields in 2D current algebra, Ph.D. thesis, Niels Bohr

Institute, 1996, hep-th/9601167;J.L. Petersen, J. Rasmussen, M. Yu, Nucl. Phys. B 502 (1997) 649.

[3] D. Gepner, E. Witten, Nucl. Phys. B 278 (1986) 493.[4] L. Bégin, C. Cummins, P. Mathieu, J. Math. Phys. 41 (2000) 7611.[5] C.J. Cummins, P. Mathieu, M.A. Walton, Phys. Lett. B 254 (1991) 386.[6] A.N. Kirillov, P. Mathieu, D. Sénéchal, M.A. Walton, Nucl. Phys. B 391 (1993) 651.[7] L. Bégin, P. Mathieu, M.A. Walton, Mod. Phys. Lett. A 7 (1992) 3255.[8] L. Bégin, A.N. Kirillov, P. Mathieu, M.A. Walton, Lett. Math. Phys. 28 (1993) 257.[9] A.D. Berenstein, A.V. Zelevinsky, J. Alg. Comb. 1 (1992) 7.

[10] J. Rasmussen, Mod. Phys. Lett. A 13 (1998) 1213;J. Rasmussen, Mod. Phys. Lett. A 13 (1998) 1281.

[11] A.N. Kirillov, P. Mathieu, D. Sénéchal, M.A. Walton, in: Group-Theoretical Methods inPhysics, Proc. of the XIXth International Colloquium, Salamanca, Spain, 1992, Vol. 1,CIEMAT, Madrid, 1993.

[12] M.A. Walton, Can. J. Phys. 72 (1994) 527.[13] D.P. Zhelobenko, Compact Lie Groups and their Representations, Am. Math. Soc., 1973,

Theorem 4, Section 78.[14] L. Bégin, C. Cummins, P. Mathieu, J. Math. Phys. 41 (2000) 7640.[15] J. Rasmussen, M.A. Walton, Affinesu(3) andsu(4) fusion multiplicities as polytope volumes,

hep-th/0106287.[16] J. Rasmussen, M.A. Walton, Fusion multiplicities as polytope volumes:N -point and higher-

genussu(2) fusion, hep-th/0104240.[17] J. Rasmussen, M.A. Walton, work in progress.[18] J. Rasmussen, M.A. Walton,su(N) tensor product multiplicities and virtual Berenstein–

Zelevinsky triangles, math-ph/0010051.


Hyperbolicity of partition function andquantum gravity

Kazuhiro HikamiDepartment of Physics, Graduate School of Science, University of Tokyo, Hongo 7-3-1, Bunkyo,

Tokyo 113-0033, Japan

Received 28 May 2001; accepted 17 August 2001

Abstract

We study a geometry of the partition function which is defined in terms of a solution of the five-term relation. It is shown that the 3-dimensional hyperbolic structure or the EuclideanAdS3 naturallyarises in the classical limit of this invariant. We discuss that the oriented ideal tetrahedron can beassigned to the partition function of string. 2001 Elsevier Science B.V. All rights reserved.

PACS: 04.60.-m; 04.60.Nc; 03.65.-w

1. Introduction

This article is devoted to reveal a relationship between the partition function and3-dimensional hyperbolic geometry. From the physical viewpoint, the 3-dimensionalhyperbolic spaceH3 corresponds to the Euclidean anti-de Sitter spaceAdS3, and manystudies concerning the AdS/CFT correspondence [1] including string theory on AdS andthe SL(2,R) WZNW model have been done (for a review, see Ref. [2], and referencestherein). Our motivation is based on a recent conjecture that an asymptotic behavior ofa specific value of the colored Jones polynomial gives the hyperbolic volume of knotcomplement [3–8]. This opens up a geometrical study of the quantum knot invariantssuch as the Jones polynomial, which have been defined based on the quantum group. Herewe shall give a geometrical picture of the partition function which indicates an aspect ofthe AdS/CFT correspondence; we show that the bosonic and fermionic partition functionsconstitute as oriented ideal tetrahedra inH

3. This partition function could be related witha string partition function on knot, which is a collection of torus and is on the boundaryof H3.

E-mail address: [email protected] (K. Hikami).



538 K. Hikami / Nuclear Physics B 616 [PM] (2001) 537–548

The organization of this paper is as follows. In Section 2 we briefly review a fact of thehyperbolic spaceH3. The hyperbolic volume of manifold is computed based on the idealtriangulation, and we explain how to construct the invariant of the hyperbolic manifoldfrom this triangulation. Note that almost all orientable 3-manifolds are believed to admit ahyperbolic structure. In Section 3 we study several properties of the quantum dilogarithmfunction. Key identity is the five-term relation, and we define the partition functionof manifold by assigning oriented tetrahedron to the quantum dilogarithm function. InSection 4 we study a classical limit of the partition function. We show that the orientedtetrahedron which is assigned to the quantum dilogarithm can be regarded as the idealtetrahedron inH3, and that the hyperbolicity consistency condition in gluing tetrahedraexactly coincides with a condition at the critical point of the partition function. In Section 5we consider an application in physics, and we clarify a relationship with the AdS/CFTcorrespondence. As was pointed out in Refs. [9,10], the quantum dilogarithm function usedhere is a modular double of theq-exponential function, and we claim that our quantuminvariant can be regarded as the partition function of string from cylinder. We show thatthe chemical potential and the average number of angular momentum determines the idealtetrahedron. The last section is for concluding remarks.

2. Hyperbolic space H3

The 3-dimensional hyperbolic spaceH3 (see, e.g., Refs. [11–13]) or the Euclideanversion of the anti-de Sitter spaceAdS3, is defined as the space-like surface in4-dimensional Minkowski space:

−x20 + x2

1 + x22 + x2

3 = −1, x0 > 0.

Here the metric is ds2 = −dx20 + dx2

1 + dx22 + dx2

3, and it has a constant negativecurvature−1. The hyperbolic spaceH3 is conformally mapped into the 3-dimensionallydisk D3 (the Poincaré model) via

x → (x1, x2, x3)

1+ x0,

x0 = chξ, x1 = shξ sinθ cosϕ,x2 = shξ sinθ sinϕ, x3 = shξ cosθ,

with ds2 = dξ2 + sh2 ξ(dθ2 + sin2 θ dϕ2), and a boundary∂H3 of the hyperbolic space

is homeomorphic toS2. The orientation-preserving isometry ofH3 is isomorphic toPSL(2,C), and the hyperbolic 3-manifoldM is defined byH3/Γ whereΓ is a discrete,torsion-free subgroup ofPSL(2,C). Hereafter to consider the ideal tetrahedron, we use theBeltrami half-space modelX0 > 0 with coordinate(X0,X1,X2), and the metric is givenby

ds2 = dX20 + dX2

1 + dX22

X20

.

K. Hikami / Nuclear Physics B 616 [PM] (2001) 537–548 539

Fig. 1.

The transformation is explicitly given by

X−10 = chξ + shξ cosθ, X1 = X0 shξ sinθ cosϕ, X2 = X0 shξ sinθ sinϕ.

With this metric geodesics are semi-circles orthogonal toX0 = 0 ∪ ∞.Every non-compact finite volume hyperbolic 3-manifold admits a decomposition into

a finite number of ideal polyhedra [11,14]. The ideal polyhedron means that all verticeslie on ∂H3, and all of edges are hyperbolic geodesics. Among the ideal polyhedron theideal tetrahedron is completely determined by a single complex numberz with positiveimaginary part, which we call modulus, or cross-ratio. The Euclidean triangle near anyvertex cut out from ideal tetrahedron by a horosphere section is similar to the triangle withvertices 0, 1, andz in C (see Fig. 1), and opposite edges of the tetrahedron have the samedihedral angle. For fixed edgee, the parameter which describes the dihedral angle is thengiven in terms of modulus by

zi(e) ∈zi ,

1

1− zi,1− 1

zi

.

Furthermore, with the modulusz, the hyperbolic volume of the ideal tetrahedron is givenby the Bloch–Wigner functionD(z) (see Eq. (A.3)) [15]. Depending on the orientation oftetrahedron, we can extend the modulus toz ∈ C \ 0,1.

Gluing a finite collection of ideal tetrahedra together results in a 3-manifoldM admittinga hyperbolic structure of finite volume [11]. To endow the hyperbolic structure inM, thefollowing gluing conditions must be satisfied:

1. Triangles cut out of adjacent tetrahedra to every edgeeν fill neatly around edge, i.e.,we require the consistency condition∑

j

argzj (eν) = 2π,∏j

zj (eν) = 1.

2. The developing map near the ideal vertex yields a Euclidean structure on thehorosphere.

I.e., when∂M is a collection of tori,M − ∂M have a hyperbolic structure if and onlyif for each edge the hyperbolic consistency condition is fulfilled. Closed manifold couldbe obtained topologically by Dehn filling a certain hyperbolic link complement. When a


3-manifoldM is triangulated in this way, a sum of the hyperbolic volume of the idealtetrahedron

(2.1)Vol(M) =∑i

D(zi)

is a topological invariant ofM.The volume Vol(M) of oriented hyperbolic 3-manifoldM has an analytic relationship

with the Chern–Simons invariant CS(M) [16,17]. Namely,

(2.2)VCS(M) = Vol(M)+ i CS(M)

is a natural complexification of Vol(M), and in general the formulae to compute CS(M)

give Vol(M). The invariant VCS(M) is induced from the pre-Bloch group [18] by

i VCS : P(C) −→ C,

(2.3)[zi] −→L(1− zi).

Here[z] satisfies

(2.4)[x] − [y] +[y

x

]−[

1− x−1

1− y−1

]+[

1− x

1− y

]= 0,

andL(z) is the Rogers dilogarithm function (A.2). Fact that both the hyperbolic volume(2.1) and its analytic continuation (2.2) are invariant ofM is based on that the Blochinvariant defined by

(2.5)β(M) =∑i

[zi]

is the invariant of the manifold [19].

3. Quantum dilogarithm function and partition function

We define the integralΦγ (ϕ) by

(3.1)Φγ (ϕ) = exp

( ∫R+i0

e−iϕx

4 sh(γ x)sh(πx)

dx

x

),

which was first introduced by Faddeev [9]. As we will discuss later, this function wasoriginally introduced as a dualization or modular double of theq-exponential function.The function enjoys the five-term relation [20,21],

(3.2)Φγ (p)Φγ (q) = Φγ (q)Φγ (p + q)Φγ (p),

wherep andq are the canonically conjugate operators satisfying

(3.3)[p, q] = −2iγ.

The five-term relation (3.2) is recast into a simple form,

(3.4)S2,3S1,2 = S1,2S1,3S2,3,


where theS-operator is defined by

(3.5)S1,2 = e1

2iγ q1p2 Φγ

(p1 + q2 − p2

).

Herep1 = p ⊗ 1, q2 = 1 ⊗ q, and so on. The matrix element of theS-operator over themomentum space,p|p〉 = p|p〉, is given by [7]

〈p1,p2|S∣∣p′

1,p′2

⟩= δ

(p1 + p2 − p′

1

)Φγ

(p′

2 − p2 + iπ + iγ)e

12iγ

(− π2+γ2

6 − γπ2 +p1(p

′2−p2)

),

〈p1,p2|S−1∣∣p′

1,p′2

⟩(3.6)= δ

(p1 − p′

1 − p′2

) 1

Φγ (p2 − p′2 − iπ − iγ )

e1

2iγ( π2+γ2

6 + γπ2 −p′

1(p2−p′2)).

We shall give the geometrical picture for this operator. A key is the five-term relation(3.4), and we associate the (3-dimensional) oriented tetrahedron to theS-operators; each2-simplex is assigned the momentum (Fig. 2). With this identification, we can glue two2-simplexes in pairs so that every orientation of edges matches. For instance, the pentagonrelation (3.4) can be depicted as Fig. 3.

When 3-manifoldM is decomposed into oriented ideal tetrahedra, we define thepartition functionτ (M) by

(3.7)τ (M) =∫ ∫

dp∏

〈pi,pj |S±1|pk,pl〉.

In the following we study in detail the classical limitγ → 0 of the partition functionτ (M).

〈p1,p2|S∣∣p′1,p

′2

⟩: 〈p1,p2|S−1

∣∣p′1,p

′2

⟩:

Fig. 2. TheS-operators (3.6) denote the oriented tetrahedron. Parametersp are assigned to everyface, andza denotes a dihedral angle. In the classical limitγ → 0, those become hyperbolic

ideal tetrahedra with modulus ep′2−p2, and the dihedral anglesza at the edges of the simplex are

z1 = ep′2−p2, z2 = 1− z−1

1 , andz3 = (1− z1)−1, respectively, with opposite edges having the same

angle.


Fig. 3. Pentagon identity (3.4).

4. Classical limit and hyperbolic geometry

We study the classical limitγ → 0 of matrix elements (3.6) of theS-operators. Assimple calculation leads that the Faddeev integral reduces to the Euler dilogarithm function(A.1)

(4.1)Φγ (x) = exp1

2iγ

(Li2(−ex

)+ O(γ )),

we have

〈p1,p2|S∣∣p′

1,p′2

⟩= δ

(p1 + p2 − p′

1

)exp

(− 1

2iγV(p′

2 − p2,p1)+ O

(γ 0)),

〈p1,p2|S−1∣∣p′

1,p′2

⟩(4.2)= δ

(p1 − p′

1 − p′2

)exp

(1

2iγV(p2 − p′

2,p′1

)+ O(γ 0)),

where

(4.3)V (x, y) = π2

6− Li2

(ex)− xy.

Thus the operatorsS±1, which is assigned to each oriented tetrahedron in the partitionfunction (3.7), asymptotically give the functionV (x, y) satisfying

(4.4)V (x, y) = L(1− ex

)+ 1

2

(∂V (x, y)

∂ logx+ ∂V (x, y)

∂ logy

),

(4.5)

ImV (x, y) = D(1− ex

)+ log∣∣ex∣∣ Im

(∂V (x, y)

∂x

)+ log

∣∣ey∣∣ Im

(∂V (x, y)

∂y

).

Here we have used the Rogers dilogarithm (A.2) and the Bloch–Wigner function(A.3). As the Bloch–Wigner functionD(z) denotes the hyperbolic volume of the idealtetrahedron [15], we are forced to study a relationship between theS-operator at the criticalpoint and the hyperbolic geometry.


For our purpose we reconsider the five-term relation (3.4) in the classical limit. Wesubstitute the asymptotic form (4.2) into matrix form of Eq. (3.4),

〈p1,p2,p3|S2,3S1,2∣∣p′

1,p′2,p

′3

⟩= 〈p1,p2,p3|S1,2S1,3S2,3∣∣p′

1,p′2,p

′3

⟩.

The LHS reduces to∫dx 〈p2,p3|S

∣∣x,p′3

⟩〈p1, x|S∣∣p′1,p

′2

⟩= δ

(p1 + p2 + p3 − p′

1

)(4.6)× exp

1

2iγ

(−V(p′

3 −p3,p2)− V

(p′

2 − p2 − p3,p1)+ O(γ )

),

while the RHS gives∫ ∫ ∫dy dzdw 〈p1,p2|S|y, z〉〈y,p3|S

∣∣p′1,w

⟩〈z,w|S∣∣p′2,p

′3

⟩= δ

(p1 + p2 + p3 − p′

1

)

(4.7)

×∫

dz exp1

2iγ

(− π2

2+ Li2

(ez−p2

)+ Li2(ep

′2−p3−z

)+ Li2(ep

′3−p′

2+z)

+ z(−p2 + p′

3 − p′2 + z

)− p1p2

+ (p′

2 − p3)(p1 + p2) + O(γ )

).

Above integral w.r.t.z is evaluated by use of the saddle point method, whose critical pointis given by

(4.8)(1− ew−p3

)−1 (1− ep2−z) (

1− ew−p′3)= 1, w = p′

2 − z.

The validity of the saddle point method can be checked directly. In fact, substituting asolution

e−z = e−p2 + e−p′2+p′

3 − e−p2−p3+p′3,

into the integral (4.7), and equating both hands sides (4.6), (4.7) at critical point, we get

Li2(ez−p2

)+ Li2(ep

′2−p3−z

)+ Li2(ep

′3−p′

2+z)+ (z − p2)

(z − p′

2 + p′3

)− π2

6

= Li2(ep

′3−p3

)+ Li2(ep

′2−p2−p3

),

which is nothing but Schaeffer’s five-term relation of the Euler dilogarithm function,

Li2

(1− x−1

1− y−1

)= Li2(x) − Li2(y) + Li2

(y

x

)+ Li2

(1− x

1− y

)

− π2

6+ logx log

(1− x

1− y

),

with x = ez−p2 andy = ep′2−p2−p3.

A geometrical meaning of the saddle point equation (4.8) becomes clear once we regardthe oriented tetrahedron in Fig. 2 as a hyperbolic ideal tetrahedron; the modulus is ep′

2−p2,


and the dihedral anglesza at the edges of the simplex are, respectively, given by

(4.9)z1 = ep′2−p2, z2 = 1− z−1

1 , z3 = (1− z1)−1,

with opposite edges having the same angle. Then the condition (4.8) exactly coincideswith the hyperbolic consistency condition for gluing three tetrahedra around central edgeof RHS in Fig. 3.

The correspondence between the saddle point condition and the hyperbolic consistencycondition can be checked for other forms of the five-term relations such asS2,3S1,2S

−12,3 =

S1,2S1,3, and a trivial identityS1,2S−11,2 = 1. See Ref. [7] for further discussions, where the

colored Jones type invariant was constructed based on theS-operator, and the hyperbolicstructure of the knot complement was studied in detail. As a result the oriented tetrahedron,which we have assigned to theS-operator based on the five-term relation (3.2), is identifiedwith the ideal tetrahedron in the classical limit. Correspondingly the partition function ofthe 3-manifoldM defined in Eq. (3.7) reduces to

(4.10)limγ→0

2iγ logτ (M) =∑i

L(1− ezi

),

where the modulus ezi is to be fixed from the hyperbolicity consistency gluing conditions.Looking at the Bloch invariant (2.3), this limit may give both the hyperbolic volume and theChern–Simons invariant ofM (we are not sure which solution of a set of the hyperbolicityequations is dominant in the partition functionτ (M) in the classical limit).

We note that the functionV (x, y) satisfies a one parameter deformation of Eq. (2.4):

V (logx,p1) + V

(log

(y

x

),p1 + p2

)+ V

(log

(1− x

1− y

), z

)

= V

(log

(1− x−1

1− y−1

),p2

)+ V (logy,p1),

wherex, y, andz are given above in terms ofp2, p3, p′2 andp′

3. This relation might behelpful for the extended Bloch group.

5. Physical interpretation

We have seen that the partition function (3.7) defined in terms of the Faddeev integral(3.1) is closely connected with the hyperbolic geometry. Especially we have shown thatthe classical limit of (imaginary part of ) matrix elements coincides with the hyperbolicvolume of the ideal tetrahedron, and that the saddle point equation coincides with thehyperbolic gluing condition. As the 3-dimensional hyperbolic geometry is nothing butthe EuclideanAdS3 space, we recall here the so-called AdS/CFT correspondence; gravitytheory on a 3-dimensional anti-de Sitter space is equivalent to the conformal field theory onthe boundary. This was initiated from the observation [22] that the asymptotic symmetrygroup ofAdS3 is generated by left and right Virasoro algebras. In this correspondence theCFT lives on the cylindrical boundary ofAdS3, and in our viewpoint it seems that thiscylindrical boundary could be identified with a knotK.


To clarify our observation further, it is preferable to recall properties of the Faddeevintegral [9,21]. Collecting residues in integral in Eq. (3.1), we get (we assume Imγ > 0)

(5.1)Φγ (ϕ) = Eq(eϕ)

EQ(eϕπ/γ ),

where

(5.2)q = eiγ , Q = e−iπ2/γ ,

andEq(w) is theq-exponential function defined by

(5.3)Eq(w) =∞∏n=0

(1+ q2n+1w

).

Theq-exponential function itself satisfies the five-term relation [23],

(5.4)Eq(b)Eq(a) = Eq(a)Eq

(q−1ab

)Eq(b),

where we have used the Weyl commuting operators

ab = q2ba.

With a definition of a deformation parameterq (5.2), and settinga = expq andb = expp,we can check from Eq. (3.3) that we can naively replace the Faddeev integralΦγ (w)

in Eq. (3.6) withEq(ew) as was shown in Eq. (5.1). In fact asymptotic behavior of theq-exponential is same with that ofΦγ (ϕ). We can see that, using the Euler–Maclaurinformula, the classical limitγ → 0 of Eq(ew) reduces to the Euler dilogarithm function asin Eq. (4.1) as follows:

logEq(w) =∞∑n=0

log(1+ eiγ (2n+1)w

)∼ 1

2iγ

0∫−w

log(1− s)

sds = 1

2iγLi2(−w).

A remaining part,EQ(eϕπ/γ ), also satisfies the five-term relation (5.4) replacingq by Q,and Eq. (3.2) follows from a commutativity between the vertex operator ep and a dualvertex operator eqπ/γ . In this sense the integral (3.1) is a dualization, a modular double oftheq-exponential function. This type of modular invariance,γ ↔ π2/γ , can also be seenin the CFT.

As a result, matrix element,〈p1,p2|S−1|p′1,p

′2〉, defined in Eq. (3.6) can be rewritten

in terms of a functionZB(µ,N) defined by

(5.5)ZB(µ,N) = e1

2iγ (C−µN) 1

Eq(−q−1eµ),

where we setC = π2+γ 2

6 + γπ2 . The functionZB(µ,N) is familiar in the CFT, and

coincides with the partition function of the free boson up to constant. The constant termC

cancels with a contribution from its super partner whose partition function can be written


Fig. 4. Cylinder in a limitγ → 0.

as

(5.6)ZF(µ,N) = e1

2iγ (−C+µN)Eq

(−qeµ),

which plays a role of the matrix element,〈p1,p2|S|p′1,p

′2〉 in Eq. (3.6). As a result, the

above functionsZB,F(µ,N) can be regarded as the partition functions of bosonic/fermionicstring from the cylinder, and the classical limitγ → 0 corresponds to a long strip limit(Fig. 4). By this identification,µ and N denote the chemical potential and the scaledaverage number of angular momentum, respectively.

It is now clear that, by replacingΦγ (ϕ) in Eq. (3.6) with theq-exponential function(5.3), the bosonic string corresponds to the oriented tetrahedron (say, left in Fig. 2) whileits super-partner is depicted as a mirror image (say, right in Fig. 2). It seems that thequantum five-term relation (3.2) indicates a possibility to construct the quantum invariant(3.7) by usingnoncommutative thermodynamical variablesµ andN as in Eq. (3.3). In thisformalism, string interactions should be seen as constraints amongµ andN , and in theclassical limit they describe the hyperbolic consistency condition.

To conclude the string partition functions (5.5), (5.6) can be regarded as the orientedideal tetrahedron in the 3-dimensional hyperbolic spaceH

3. This fact seems to indicate ageometrical aspect of the AdS/CFT correspondence.

6. Concluding remarks and discussions

We have studied the hyperbolic structure of the partition function which is definedby use of the quantum dilogarithm function (Faddeev integral). We have shown that theclassical limit of the Faddeev integral describes the ideal tetrahedron in the hyperbolicspaceH3 which is the space-like surface in 4-dimensional Minkowski space. Thoughwe have originally defined the partition function in terms of the Faddeev integral, wecan replace it with theq-exponential function as we have discussed in Section 5. Asthe q-exponential function denotes a partition function of free bosonic/fermionic stringsfrom cylinder, we have proposed that the hyperbolic structureH3 can naturally arise in(classical limit of ) the string partition function. It seems that it indicates a new aspect ofthe AdS/CFT correspondence. As a result, the modulus of the oriented ideal tetrahedronwhich is assigned to partition functions of free bosonic/fermionic strings, is given explicitlyin terms of the chemical potential and the average number of angular momentum, andthe string interaction should be regarded as a quantization of the hyperbolic consistencyconditions in gluing ideal tetrahedra. We note that we have assigned theq-exponentialfunction to hyperbolic ideal tetrahedron to construct the quantum invariant (3.7), while an


idea of the Regge calculus [24] is that the three dimensional gravity is a sum over the 6j

symbol, whose connection with a volume of the tetrahedron was recently discussed [25].Promising is a fact that the classical limit (4.10) of the partition functionτ (M) is

suggested to give both the hyperbolic volume and the Chern–Simons invariant (2.3)of manifold M. As it was shown [26] that the Einstein–Hilbert action with negativecosmological constantΛ = −1/02,

S = 1

16πG

∫d3x

√−g

(R + 2

02

)reduces to 2 sets of the Chern–Simons actions by taking a linear combination of the spinconnection and the vierbein as the gauge field, the partition functionτ (M) seems to be acandidate for quantum gravity.

The AdS/CFT correspondence enables us to derive the Bekenstein–Hawking entropyof black hole microscopically [27]. Therein it was shown that the entropy coincideswith the asymptotic growth of the number of states in CFT with central chargec [28].We see that the computation of the central chargec is essentially same with ourcomputation of the classical limit of the partition function (4.10), once we have identifiedthe integralΦγ (ϕ) with the partition function of free bosonic/fermionic strings writtenby theq-exponential function. As we have seen based on the “volume conjecture” thatthe asymptotic behavior determines the hyperbolic geometrical structure, it would help usto understand the thermodynamics of black hole, such as the BTZ solution [29] whichis locally isometric toAdS3. It remains for future studies to relate explicitly a particularCFT with a specific manifoldM through the partition functionτ (M). In such studies aninterpretation presented in Ref. [30] will be useful.

Appendix A. Dilogarithm function

• The Euler dilogarithm,

(A.1)Li2(z) =∞∑n=1

zn

n2 = −z∫

0

dslog(1− s)

s.

• The Rogers dilogarithm,

(A.2)L(z) = Li2(z) + 1

2log(z) log(1− z).

• The Bloch–Wigner function,

(A.3)D(z) = ImLi2(z) + arg(1− z) log|z|.

References

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[3] R.M. Kashaev, Mod. Phys. Lett. A 10 (1995) 1409.[4] R.M. Kashaev, Lett. Math. Phys. 39 (1997) 269.[5] H. Murakami, J. Murakami, Acta Math. 186 (2001) 85.[6] Y. Yokota, math.QA/0009165.[7] K. Hikami, Int. J. Mod. Phys. A 16 (2001) 3309.[8] S. Baseilhac, R. Benedetti, math.GT/0101234.[9] L.D. Faddeev, Lett. Math. Phys. 34 (1995) 249.

[10] L.D. Faddeev, in: G. Dito, D. Sternheimer (Eds.), Quantization, Deformations, and Symmetries,Conference Mosh Flato 1999, Vol. I, Kluwer, Dordrecht, 2000, pp. 149–156.

[11] W.P. Thurston, The Geometry and Topology of Three-Manifolds, Lecture Notes in PrincetonUniversity, Princeton, 1980.

[12] J.G. Ratcliffe, Foundations of Hyperbolic Manifolds, Springer-Verlag, Berlin, 1994;R. Benedetti, C. Petronio, Lectures on Hyperbolic Geometry, Springer-Verlag, Berlin, 1992.

[13] P.J. Callahan, A.W. Reid, Chaos, Solitons & Fractals 9 (1998) 705.[14] D.B.A. Epstein, R.C. Penner, J. Diff. Geom. 27 (1988) 67.[15] J. Milnor, Bull. Am. Math. Soc. 6 (1982) 9.[16] W.Z. Neumann, D. Zagier, Topology 24 (1985) 307.[17] T. Yoshida, Invent. Math. 81 (1985) 473.[18] W.D. Neumann, in: B. Apanasov, W.D. Neumann, A.W. Reid, L. Siebenmann (Eds.), Topology

’90, de Gruyter, Berlin, 1992, pp. 243–271.[19] W.D. Neumann, J. Yang, Duke Math. J. 96 (1999) 29.[20] L. Chekhov, V.V. Fock, Theor. Math. Phys. 120 (1999) 1245.[21] L.D. Faddeev, R.M. Kashaev, A.Y. Volkov, Commun. Math. Phys. 219 (2001) 199.[22] J.D. Brown, M. Henneaux, Commun. Math. Phys. 104 (1986) 207.[23] L.D. Faddeev, R.M. Kashaev, Mod. Phys. Lett. A 9 (1994) 427.[24] G. Ponzano, T. Regge, in: F. Bloch (Ed.), Spectroscopic and Group Theoretical Methods in

Physics, North-Holland, 1968, pp. 1–58.[25] J. Roberts, Geom. Topol. 3 (1999) 21.[26] E. Witten, Nucl. Phys. B 311 (1988/1989) 46.[27] A. Strominger, C. Vafa, Phys. Lett. B 379 (1996) 99.[28] J.L. Cardy, Nucl. Phys. B 270 (1986) 186.[29] M. Bañados, C. Teitelboim, J. Zanelli, Phys. Rev. Lett. 69 (1992) 1849;

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Integrability and symmetry algebra associatedwith N = 2 KP flows

Sasanka Ghosh, Debojit SarmaDepartment of Physics, Indian Institute of Technology, North Guwahati, Guwahati 781 031, India


Abstract

We show the complete integrability ofN = 2 nonstandard KP flows establishing the bi-Hamiltonian structures. One of Hamiltonian structures is shown to be isomorphic to the nonlinearN = 2 W∞ algebra with the bosonic sector havingW1+∞ ⊕ W∞ structure. A consistent free fieldrepresentation of the super conformal algebra is obtained. The bosonic generators are found to be anadmixture of free fermions and free complex bosons, unlike the linear one. The fermionic generatorsbecome exponential in free fields, in general. 2001 Elsevier Science B.V. All rights reserved.

PACS: 11.25.Hf; 11.30.Pb; 11.30.LyKeywords: Supersymmetric KP; Bi-Hamiltonian;W∞ algebras; Universal symmetry

1. Introduction

The close relationship between the conformal algebras and the rich symmetry associatedwith integrable systems is well understood. The Hamiltonian structures of the integrablehierarchies have been found to be isomorphic to the various higher spin conformal algebrasat the classical level. This was realized when it was shown that theWn algebra incorporatesin its classical limit the Hamiltonian structure of the nonlinear integrable systems, i.e.,the generalizednth KdV hierarchy [1–3]. This technique of obtaining classical conformalalgebras through the integrable hierarchies is indeed a powerful one and its importancewas recognized when the existence of a new higher spin conformal algebra was realizedat the classical level [4]. Prior to this all the known conformal algebras were obtained firstin terms of free fields via the bootstrap approach and subsequently their relation to thesymmetries of the integrable hierarchies was obtained.

Generalized KdV hierarchy, in general, is significant in its own right because of itsrich symmetry structures. But it becomes physically more relevant since the equations of

E-mail addresses: [email protected] (S. Ghosh), [email protected] (D. Sarma).



550 S. Ghosh, D. Sarma / Nuclear Physics B 616 [PM] (2001) 549–573

motion and symmetries of 2D quantum gravity [5] can be formulated in terms of KdV-likeequations.

The KP hierarchy incorporates all KdV hierarchies [6] and thus there exists the distinctpossibility that it is the bedrock of 2D quantum gravity [7]. It was believed that thelargen limit of Wn algebra, namely,W1+∞ andW∞ algebras would provide the necessaryframework in this direction. The naive approach to the largen limit, however, gives riseto linear algebras which are truly infinite dimensional symmetry algebras containing allthe conformal spins [8], but the fact that they are linear prevents them from being theright candidates for universal algebras, there being no straightforward mechanism whicheffectively truncates the spin content of these algebras and produces the nonlinear featuresof Wn. A nonlinear realization of theW algebra, namely,W∞ algebra was obtainedby the bootstrap approach and identified with the 2nd Hamiltonian structure of the KPhierarchy [9,10]. It is a universalW algebra containing allWn algebras. This was obtainedby associating the symmetry algebra of theSL(2,R)k/U(1) coset model with theW∞algebra characterized by the labelk and then by showing that the symmetry algebratruncates toWn algebra fork = −n [11].

Subsequently Manin and Radul [12] provided the supersymmetric extension of the KPhierarchy, and this was based on the odd parity super-Lax formulation. But constructionof the Hamiltonian structure for odd parity Lax operator, following the Drinfeld–Sokolovformalism, is not well understood yet [13]. Later on, an even parity Lax operator associatedwith supersymmetric KP hierarchy was obtained [14,15] and a supersymmetric extensionof the linearW algebra was realized [16]. The connection of theN = 2 super KdVhierarchies with affine Lie algebras was demonstrated by Inami and Kanno [17,18] whichis a step forward towards anN = 2 super analogue of the Drinfeld–Sokolov formulation.This indicates that there ought to be consistentN = 2 super-Lax formulation of the superKP hierarchy which should be Hamiltonian with respect to the super Gelfand–Dikii bracketof the second kind and also should reduce to the Lax operators considered by Inami andKanno under suitable reduction. Consequently, the existence of a nonlinear realization ofthe superW∞ algebra [4] may be shown through the super KP formulation. Unlike all otherknown conformal algebras, there is no bootstrap approach to find the higher spin extensionof N = 2 superconformal algebra, namely, the superW∞ algebra through the free fieldrepresentations.

While the bosonic KP hierarchy and their connection to matrix model have been studiedextensively, not much is known about the higher spin extension ofN = 2 superconformalsymmetry. Moreover, conventional formulation of the supersymmetric matrix model failedto describe nonperturbative superstring theory; it gives nothing but the ordinary matrixmodel [19]. The super KP formalism may throw some light in this direction even if nosuper symmetric extension of the matrix model can be formulated.

Renewed interest in the study ofN = 2 andN = 4 supersymmetry in the contextof quantum gravity and their connections with integrable systems, opened up a seriesof studies relating to the symmetry structures ofN = 2 and N = 4 supersymmetricintegrable models [20]. A major breakthrough in recent times is the nonperturbativesolution of N = 2 super Yang–Mills equation and their connection with integrable

S. Ghosh, D. Sarma / Nuclear Physics B 616 [PM] (2001) 549–573 551

systems [21]. Investigations have also been made in recent times to obtainτ functionsfor supersymmetric integrable hierarchies [22]. Motivated by these works, we intend tofurther explore the superW∞ algebra and particularly its free field representation, thesignificance of which cannot be overestimated since the underlying representation ofa conformal field theory is essentially a free field representation. Moreover it plays amajor role in the classification of various conformal algebras. Further, the quantizationof classical symmetry algebras becomes straightforward in terms of free fields.

In this paper, we show that nonstandard supersymmetric KP flows following theGelfand–Dikii method, are bi-Hamiltonian. We further show that one of the Hamiltonianstructures is a candidate for a higher spin extension ofN = 2 superconformal algebra andhas the required number of spin fields and the bosonic sector of the algebra has the rightstructure with two commuting sets of bosonic generators. We will also obtain the free fieldrepresentation of the generators which turns out to be nonlinear. The generators in thefermionic sector are exponential in the free fields. It will be apparent that in the bosonicsector the representation of generators are not trivial extensions of linear representation.However, unlike the linear symmetry algebras, all the generators cannot be expressed interms of the free fields in a closed form. A few of the lower order generators are explicitlywritten down and an algorithm for constructing higher order generators will be indicated.This brings us closer to establishing that superW∞ is a universal symmetry containing allfinite-dimensional bosonic andN = 2 supersymmetricW algebras.

The organization of the paper is as follows. In Section 2 we introduce theN = 2 superKP model and obtain its bi-Hamiltonian structures through the Gelfand–Dikii method. Thisestablishes the complete integrability of the system. In Section 3 we show that the secondHamiltonian structure ofN = 2 super KP hierarchy exhibits the appropriate structure of asuperW∞ algebra. In particular, the bosonic sector of this nonlinear algebra is shown topossess the requiredW1+∞ ⊕W∞ structure. We obtain a nonlinear free field representationof the bosonic and fermionic generators in a suitable basis in Section 4. Unlike the linearrepresentation of the bosonic generators which was in terms of bilinears of free fields—either fermionic or bosonic; here the consistent free field representation of these algebrascomprises an admixture of free fermions and free complex scalar fields. Section 5 is theconcluding one.

2. N = 2 Super KP hierarchy and bi-Hamiltonian structures

In an earlier work [4], it was shown that with an even parity super-Lax operator theHamiltonian structure leading to a nonlinear superW∞ algebra becomes local, thus makingit a right candidate for a universal symmetry algebra containing all finite-dimensionalbosonic as well asN = 2 supersymmetric algebras. For completeness and future referencewe mention the explicit forms of the Hamiltonian structure and the dynamical equations oftheN = 2 super KP hierarchy. The super Lax operator of the associatedN = 2 super KP


hierarchy is given by

(2.1)L =D2 +∞∑i=0

ui−1(X)D−i

where,D is the superderivative withD2 = d/dx andui−1(X) are superfields inX = (x, θ)

space,θ being Grassmann odd coordinate. The grading ofui−1(X) is |ui−1| = i so thatu2i−1 are bosonic superfields, whereasu2i are fermionic ones.

The local Poisson bracket algebra among the coefficient fieldsui−1(X) can be obtainedfollowing the method of Gelfand and Dikii [4]. This has the explicit form

uj−1(X),uk−1(Y )

2

(2.2)

=[

−j+1∑m=0

[j + 1

m

](−1)j (k+m+1)+[m/2]uj+k−mD

m

+k+1∑m=0

[k + 1

m

](−1)jm+(k+1)(m+1)Dmuj+k−m

+j−1∑m=0

k−1∑l=0

([j

m+ 1

][k

l + 1

]−

[j − 1

m

][k − 1

l

])× (−1)j (m+1)+k+l+[m/2]uj−m−2D

m+l+1uk−l−2

+k−1∑n=0

k−n−1∑l=0

[k − n− 1

l

](−1)j (n+l)+(l+1)(n+k+1)un−1D

luj+k−n−l−2

−j+k−n−l−1∑

m=0

k−1∑n=0

k−n−1∑l=0

[j − 1

m

][n+ l − 1

l

](−1)j (m+n+l+k+1)+n(l+1)+[m/2]

× uj+k−m−l−2Dm+lun−1

](X − Y ).

Notice that the symmetry algebra (2.2) possesses the following features of interest.1. The algebra is antisymmetric and satisfies the Jacobi identity.2. The lowest subalgebra contains two super fields, namely,u−1(X) and u0(X) and

becomes isomorphic to the classical analogue of theN = 2 super conformal algebra.3. The Hamiltonian structure along with conserved quantites [4] provides a set of

dynamical equations of theN = 2 KP hierarchy consistent with the nonstandard flowequation, namely,

(2.3)dL

dtn= [

Ln>0,L

],

where the super-Lax operatorL is given in (2.1). In (2.3) ‘> 0’ implies the+ve part ofLn

withoutD0 term. The significance of nonstandard flow equation becomes apparent fromthe explicit forms of the following set of dynamical equations. In fact, the nonstandardflows provide the nontrivial dynamics to the lowest superfieldu−1 which is instrumental inmaking the Poisson bracket structure (2.2) local. The evolution equations corresponding to


the lowest three time-flows of the hierarchy which follow from (2.1), (2.2) are given belowfor completeness and for future reference:

dui−1

dt1= u

[2]i−1,

dui−1

dt2= 2u[2]

i+1 + u[4]i−1 + 2u0u

[1]i−1 + 2u−1u

[2]i−1 − 2

[i + 1

1

]uiu

[1]−1

− 2(1+ (−1)i

)u0ui + 2

i−1∑m=0

[i

m+ 1

](−1)i+1+[−m/2]ui−m−1u

[m+1]0

+ 2i−1∑m=0

[i + 1

m+ 2

](−1)[m/2]ui−m−1u

[m+2]−1 ,

dui−1

dt3= 3u[2]

i+3 + 3u[4]i+1 + u

[6]i−1 + 6u−1u

[2]i+1 + 3u−1u

[4]i−1

− 3

[i + 3

1

]ui+2u

[1]−1 + 3

[i + 3

2

]ui+1u

[2]−1 + 3

[i + 3

3

]uiu

[3]−1

− 3(1+ (−1)i

)u0ui+2 + 3u0u

[1]i+1 − 3(−1)iu0u

[2]i + 3u0u

[3]i−1

+ 3

[i + 2

1

](−1)iui+1u

[1]0 − 3

[i + 2

2

](−1)iuiu

[2]0

+ 3(u1 + 2u[2]

−1 + u2−1

)u

[2]i−1 − 3

[i + 1

1

]ui

(u1 + u

[2]−1 + u2−1

)[1]

+ 3(u2 + 2u−1u0 + u

[2]0

)u

[1]i−1 − 3

(1+ (−1)i

)(u2 + 2u−1u0 + u

[2]0

)ui

− 3i−1∑m=0

[i + 3

m+ 4

](−1)[m/2]ui−m−1u

[m+4]−1

− 3i−1∑m=0

[i + 2

m+ 3

](−1)i+[−m/2]ui−m−1u

[m+3]0

+ 3i−1∑m=0

[i + 1

m+ 2

](−1)[m/2]ui−m−1

(u1 + u

[2]−1 + u2−1

)[m+2]

(2.4)+ 3i−1∑m=0

[i

m+ 1

](−1)i+[−m/2]ui−m−1

(u2 + 2u−1u0 + u

[2]0

)[m+1].

In order to show the dynamical equations associated with the nonstandard flows arecompletely integrable, we show the existence of another Hamiltonian structure makingtheN = 2 supersymmetric KP hierarchy bi-Hamiltonian. The super Gelfand–Dikii bracketof the first kind is defined by

(2.5)FP (L),FQ(L)

1 = −Tr

([P,Q]L) = −Tr([L,P ]Q)

,


whereP andQ are the auxiliary fields defined as

(2.6)P =∞∑

j=−2

Djpj , Q =∞∑

j=−2

Djqj ,

with the grading|pj | = |qj | = j so that the linear functionalFP (L) (and similarlyFQ(L))becomes

(2.7)FP (L) = Tr(LP) =∞∑i=0

∫dX (−1)i+1ui−1(X)pi−1(X).

Consequently the L.H.S. of (2.5) becomesFP (L),FQ(L)

(2.8)=

∞∑i,j=0

∫dX

∫dY (−1)i+jpi−1(X)

ui−1(X),uj−1(Y )

qj−1(Y ).

Notice that (2.8) does not involve terms likep−2 andq−2 since the superfields begin fromu−1(X) in the Lax operator (2.1). This consistency is ensured by setting the coefficient oftheD term in the commutator [L,P ] to zero. This leads to the constraint

p−1 =∞∑

j=−1

p[2j+2]2j+1 +

∞∑i,m=0

(−1)iui−1p[2m]i+2m+1

(2.9)+∞∑j=0

j−1∑m=0

[j

m

](−1)j (m+1)(pj uj−m−2)

[m].

Using the constraint (2.9) we obtain from (2.8) the following Poisson bracket among thesuperfields

ui−1(X),uj−1(Y )

1

(2.10)

=[

− δi,0Dj+1 + (−1)[i/2]δj,0Di+1

−i+j−1∑m=0

(−1)i(j+m+1)+[m/2][i − 1

m

]ui+j−m−2D

m

−i+j−1∑m=0

(−1)m(i+1)+j (m+1)[j − 1

m

]Dmui+j−m−2

](X − Y ).

It is seen, however, that the first Hamiltonian structure above does not correctly reproducethe equations of motion (2.4). This inconsistency arises because the super-Lax operatorconsidered is not a pure differential operator and was observed also in other cases involvingpseudo-differential operators [9,15]. This indicates, like previous cases, a modification ofHamilton’s equation and consequently the Hamiltonian structure is required. If we modifyHamilton’s equation of motion to the form given below,


dui−1(X)

dtn=

ui−1(X),Hn+1

1 − (−1)i∫

dYui−1(X),u−1(Y )

1sResL

n(Y )

(2.11)− (−1)i∫

dY ui−1(X),u0(Y )1sRes(LnD−1(Y )

),

where

(2.12)

sResLn(X)=∞∑j=0

δj,0δHn+1

δuj−1(X)=

∞∑j=0

j−1∑l=0

[j − 1

l + 1

]Dl−1uj−l−2(X)

δHn+1

δuj−1(X)

and

sResLnD−1(X)

=∞∑j=0

(−1)jDjX

δHn+1

δuj (X)

=∞∑j=0

j−1∑l=0

(−1)jD−2uj−l−2(X)Dl

X +[j − 1

l

](−1)l(j+1)Dl−2

X uj−l−2(X)

(2.13)× δHn+1

δuj−1(X),

it reproduces the equations of motion correctly. Substituting (2.12) and (2.13) in (2.11), theequation of motion can be rewritten in the form

(2.14)dui−1(X)

dtn=

ui−1(X),Hn+1

1

which eventually leads us to the correct form of the first Hamiltonian structure asui−1(X),uj−1(Y )

1

(2.15)

=[

−i+j−1∑m=0

[i − 1

m

](−1)i(j+m+1)+[m/2]ui+j−m−2D

m

+[j − 1

m

](−1)m(i+1)+j (m+1)Dmui+j−m−2

+i−1∑m=0

j−1∑l=0

[j − 1

l + 1

](−1)m(i+1)+j+[m/2]Dmui−m−2D

l−1uj−l−2

−[i − 1

m+ 1

](−1)[−m/2]ui−m−2D

m−1uj−l−2Dl

+[i − 1

m

][j − 1

l + 1

](−1)j+i(m+1)+[m/2]ui−m−2D

m+l−1uj−l−2

+[i − 1

m+ 1

][j − 1

l

](−1)(l+1)(j+1)+[−m/2]

× ui−m−2Dm+l−1uj−l−2

](X − Y )


which provides the dynamical equations of the hierarchy associated with the super-Laxoperator (2.1) and the nonstandard flows (2.3). It is evident that the above Hamiltonianstructure is manifestly antisymmetric and satisfies the Jacobi identity. But this Hamiltonianstructure, unlike the earlier one (2.2), is nonlocal and may not be associated with theconformal symmetry. This feature is noticed in other supersymmetric integrable hierarchiesalso that one of the two Hamiltonian structures becomes nonlocal [15]. The existence oftwo Hamiltonian structures, nonetheless, confirms the complete integrability of the evenparity super KP hierarchy.

3. Nonlinear super W∞ algebra

In this section we show that the local superalgebra (2.2) obtained in the previous sectionis a higher spin extension ofN = 2 conformal algebra containing all conformal spins. Thenonlinear nature of superalgebra endows it with rich algebraic structures.

If the super fields are expressed in component form as

(3.1)u2i−1(X) = ub2i−1(x)+ θuf

2i−1(x), u2i(X) = uf

2i (x)+ θub2i(x)

the odd bosonic fieldsub2i−1(x) have conformal weightsi + 1, whereas the even bosonicfields ub2i (x) have conformal weightsi + 2 (i = 0,1,2, . . .), with respect to the stresstensor

(3.2)T (x)= ub0(x)− 1

2∂xu

b−1(x).

On the other hand, both odd fermionic fieldsuf2i−1(x) and even fermionic fieldsuf2i (x)

have conformal weightsi + 32 (i = 0,1,2, . . .) with respect to the same stress tensor (3.2).

The stress tensorT (x) belongs to theN = 2 conformal algebra being a subalgebraof (2.2). The conformal weights of the component fields are evident from the followingrelations.

T (x),ub2i−1(y)

=[(i + 1)ub2i−1(y)∂y + (

ub2i−1(y))′ − i−2∑

m=0

(−1)m(

i

m+ 2

)ub2i−2m−3(y)∂

m+2y

− 1

2

i−1∑m=0

(−1)m(

i

m+ 1

)ub2i−2m−3(y)∂

m+2y

]δ(x − y),

T (x),ub2i(y)

=

[(i + 2)ub2i (y)∂y + (

ub2i (y))′ − i−2∑

m=0

(−1)m(

i + 1m+ 2

)ub2i−2m−2(y)∂

m+2y

+ 1

2

i−1∑m=0

(−1)m(

i + 1m+ 1

)ub2i−2m−1(y)∂

m+2y

]δ(x − y),

S. Ghosh, D. Sarma / Nuclear Physics B 616 [PM] (2001) 549–573 557T (x),u

f2i−1(y)

=

[(i + 3

2

)uf

2i−1(y)∂y + (uf

2i−1(y)) −

i−2∑m=0

(−1)m(

i

m+ 2

)uf

2i−2m−3(y)∂m+2y

+ 1

2

i−1∑m=0

(−1)m(

i

m+ 1

)uf

2i−2m−3(y)∂m+2y

]δ(x − y),

T (x),u

f

2i(y)

=[(

i + 3

2

)uf

2i (y)∂y + (uf

2i (y))′ − i−2∑

m=0

(−1)m(

i + 1m+ 2

)uf

2i−2m−2(y)∂m+2y

](3.3)× δ(x − y).

This ensures the presence of a nonlinear supersymmetric conformal algebra in the Hamil-tonian structure (2.2) of theN = 2 super KP hierarchy. The Poisson brackets among all thecomponent fields are given in Appendix A in a basis which will be defined later.

The W1+∞ ⊕ W∞ structure of the bosonic sector, however, is not apparent in ourcase from the Poisson bracket between two types of bosons,ub2i−1 and ub2i . This is incontrast to the other supersymmetric algebras. We shall establish that the bosonic sector ofthis algebra, indeed, has the requiredW1+∞ ⊕ W∞ structure [16]. This step is crucialin obtaining the free field representation of the generators. To carry out this program,a suitable basis is required. Notice that the odd bosons themselvesu2i−1 form a closedalgebra (A.5) and consequently for the odd bosons the new set of generators may beconstructed from a linear combination of the fields as considered in [14], namely,

(3.4)Wn+1 = 2−nn!(2n− 1)!!

n∑l=0

(n

l

)(n+ l

l

)ub(n−l)2l−1 .

Since the odd bosons form a closed algebra among themselves, the algebra of the new setof generators are also closed and constitute theW1+∞ algebra. Using the Poisson bracketamongst the component fields given in Appendix A, we obtain following Poisson bracketsfor the lower order odd boson generators.

W1, W1 = 0,

W2, W1 = W1∂δ(x − y),

W2, W2 = [

2W2∂ + W ′2

]δ(x − y),

W3, W1 = 2W2∂δ(x − y),

W3, W2 =

[3W3∂ + W ′

3 + 16W1∂

3]δ(x − y),

W3, W3 =

[4W4∂ + 2W ′

4 + 35W

′′2 ∂ + 2W3W1∂ − 1

3W′′1 W1∂

− 2W22∂ + 1

2W21∂ + W ′

2∂2 + 2

3W3∂3 + 2

15W′′′2

+ W ′3W1 − 1

6W′′′1 W1 − 2W ′

2W2 + W3W′1 + 1

3W′′1 W

′1

]δ(x − y),

W4, W1 =

[3W3∂ + 1

10W′′1 ∂ − 3

10W′1∂

2 + 110W1∂

3]δ(x − y),


(3.5)W4, W2

=[4W4∂ + W ′

4 + 710W2∂

3]δ(x − y).

In the even boson,ub2i sector, however, the Poisson bracket relation is complex and itappears that the generators neither form a closed algebra nor do they commute with theodd bosons. Apparently, therefore, the direct sum structure is not maintained as requiredfor the supersymmetricW algebra. This problem can be circumvented and for the evenbosons also a suitable basis with these desirable properties can be obtained. The first stepin making them commute is to redefine the even bosons as a linear combination of odd andeven bosons of equal spin as in [14],

(3.6)vb2j = ub2j+1 + ub2j

and similarly we choose a linear combination of generators for the odd fermions

(3.7)vf

2j−1 = uf

2j−1 − uf

2j .

In the linear superW∞ algebra, this is sufficient to ensure commutation between odd andeven bosons, and thereby establish theW1+∞ ⊕W∞ structure, but in this case it is observedthat the odd bosons commute with only the lowest spin even boson generator, i.e.,

(3.8)ub2j−1, v

b0

= 0

and the Poisson brackets with higher even bosonsvb2j (j = 0) are nonzero. Interestingly,nonlinear combinations of bosons and fermions exist which commute with all odd bosons.This may be achieved, as the second step, by taking the most general combinations of thefields of the appropriate conformal spin and the coefficient of the terms may be determinedby allowing them to commute with the odd bosons. This procedure can be carried out forall even boson generators thereby yielding the mutually commuting set of generators. Theexplicit expressions of a few even boson generators are given below.

W2 = vb0,

W3 = vb2 + 1

2vb

′0 + ub−1v

b0 + u

f

0 vf

−1,

W4 = vb4 + vb′

2 + 1

5vb

′′0 + 2vb2u

b−1 + u

f2 v

f−1

(3.9)+ uf

0 vf

1 + vb′

0 ub−1 + uf

0 vf ′−1 + vb0u

b2

−1 + ub−1uf

0 vf

−1

and so on. The distinguishing character of this set is that spin 3 and higher generators arenonlinear. For the spin 3 generator it is a bilinear combination of bosonic as well as spin32 generators. For the spin 4 generator, this combination is more involved, having termstrilinear in the fields. This indicates that for higher spin generators, the basis becomesmore and more nonlinear. But most importantly,these generators are such that the desiredproperty is exhibited, namely,

(3.10)ub2j−1,W2

= 0,ub2j−1,W3

= 0,ub2j−1,W4

= 0,

i.e., the new set of even boson generators commute with all the odd bosons. Moreover, theW boson generators produce the requisite form ofW∞ algebra as can be observed from


the following Poisson bracket relations.

W2,W2 = [−2W2∂ −W ′2

]δ(x − y),

W3,W2 = [−3W3∂ −W ′3

]δ(x − y),

W3,W3 =[− 4W4∂ − 2W2

2∂ − 920W

′′2 ∂ − 2W ′

4 − (W2

2

)′− 1

10W′′′2 − 3

4W′2∂

2 − 12W2∂

3]δ(x − y),

(3.11)W4,W2 =[−4W4∂ −W ′

4 − 25W2∂

3]δ(x − y).

This algebra is isomorphic to classical analogue of theW∞ algebra [11]. The procedureoutlined above, although straightforward, becomes extremely difficult to use in obtainingthe still higher spin generators and to show the closure of the algebra explicitly. Toobtain the genenerators of higher spin we employ a different strategy. TheW4 generator,for example, may be obtained straightforwardly from theW3,W3 Poisson bracketalgebra (3.11) by ensuring the closure of the algebra following the classical analogueof the W∞ algebra [11]. Importantly theW4 generator thus obtained matches with thatof (3.9), which commutes with the odd bosons (3.10) and the form is unique. Similarly,the explicit form of theW5 generator may be obtained from the Poisson bracket relation,W4,W3 by ensuring the closure of the algebra following the classical analogue ofW∞algebra. It is found that the leading term of theW4,W3 algebra becomesvb6. This, indeed,ensures the presence of theW5 generator in the algebra. TheW5 generator exhibits theexplicit form

W5 = vb6 + 3

2vb

′4 + 9

14vb

′′2 + 1

14vb

′′′0 + 3ub−1v

b4 + 3ub−1v

b′2 + 3ub

2

−1vb2 + 3

2ub

2

−1vb′0

+ ub3

−1vb0 − ub

′−1v

b′0 + ub−1v

b′′0 + 1

7

(ub−1v

b0

)′′ + uf

2 vf

1 + uf

4 vf

−1 + uf

0 vf

3

+ 3

2uf

0 vf ′1 − 1

2uf ′0 v

f

1 + 1

2uf ′2 v

f

−1 + 3

2uf

2 vf ′−1 − u

f ′0 v

f ′−1 + u

f

0 vf ′′1

+ 1

7

(uf

0 vf

−1

)′′ + 1

2ub−1u

f ′0 v

f

−1 − 1

2ub

′−1u

f

0 vf

−1 + 3

2ub−1u

f

0 vf ′−1 + ub

2

−1uf

0 vf

−1

(3.12)+ 2ub−1uf

2 vf

−1 + 2ub−1uf

0 vf

1 + ub1uf

0 vf

−1.

It is found by explicit calculation that theW5 generator commutes with the odd bosons,ub2j−1. The spin six generatorW6, in a similar way, may be obtained from the Poisson

bracket relation,W4,W4, whose leading term turns out to bevb8. Thus the first threegenerators may be obtained by the bootstrap approach and from spin four onwards all thegenerators may be found following the above procedure. This strategy of obtaining bosonichigher spin generators evidently guarantees the closure of the algebra being isomorphicto the classicalW∞ algebra. We have checked upto spin six generators explicitly. Butto obtain the explicit forms of all higher generators becomes very difficult, although thestrategy is quite clear. Moreover, this strategy also ensures that in the bosonic limit thesuperW∞ reduces to theW∞ algebra. In this way, we may establish theW1+∞ ⊕ W∞structure in the bosonic sector of the superW∞ algebra.


In the fermionic sector both the fermions,uf2i andvf2i−1 form closed algebras (A.2),

(A.4) separately. This reveals that a linear representation such as the one in [14] alwaysexists for the fermions. We shall, however, show the existence a nonlinear basis for boththe fermions in conjunction with the odd bosons. This follows from the observation thatthe even fermions,uf2i as well as the odd fermions,vf2i−1 and the odd bosons also satisfyseparately a subalgebra (see Appendix A). Nonetheless, it turns out that nonlinear basis hasan interesting consequence. Both the fermions satisfy identical algebras in the nonlinearbasis with an added advantage of generating the minimal algebra.

The new basis for the even fermion generators may be constructed as a suitable nonlinearcombination of even fermionsuf2i and odd bosonsub2i−1 in the form

J3/2 = −uf0 ,

J5/2 = −uf

2 − uf ′0 + ub−1u

f

0 ,

(3.13)J7/2 = −uf

4 − uf ′2 − 5

4uf ′′0 + ub−1u

f

2 + ub1uf

0 + ub′

−1uf

0 − ub2

−1uf

0 .

Thus the new set of generators become more and more nonlinear as in the even boson case.But it is seen that the nonlinear basis for the even fermions (3.13) can be recast in terms ofthe bilinears of the generators only as

J3/2 = −uf0 ,

J5/2 = −uf

2 + J ′3/2 − W1J3/2,

(3.14)J7/2 = −uf

4 + J ′5/2 + 1

4J ′′

3/2 − W1J5/2 + 2W1J′3/2 − W2J3/2 + 1

2W ′

1J3/2.

This demonstrates that all the even fermions, in general, can be written as bilinears inJ

andW having the form

Jn+3/2 = −uf

2n + 2−n(n− 1)!(2n− 1)!!

n−1∑l=0

(n

l

)(n+ l + 1l + 1

)J(n−l)l+3/2

(3.15)−n−1∑m=0

n−m−1∑k=0

m∑l=0

BnklC

nml(−1)k+l J

(n−m−l−1)k+3/2 Wm−l

l+1 ,

(n = 0,1,2, . . .), whereBnkl andCn

ml are thec-number coefficients. While the values ofCnml can be easily extracted from (3.4) by determiningub2i−1 in terms ofWi+1, it appears

that there is no straightforward procedure to determine the explicit expressions of theBnkl

for arbitary spins. The closed algebras among the even fermions and odd bosons stronglycorroborates the existence of a nonlinear basis for all higher spin generators and therebyensuring the coefficientsBn

kl can always be determined for all higher spin generators. Theeven fermions of higher spins may be obtained from (A.1) and (A.9) through the Poissonbrackets of the even fermionsuf2k with theW2 andW2 generators,

(3.16)W2, u

f

2k

= −uf

2k+2 −k∑

m=0

(k + 1m+ 1

)∂m+1u

f

2k−2m


and W2, u

f

2k

= −uf

2k+2 − 1

2uf

2k + 1

2uf

2k∂

(3.17)+k∑

l=0

(k

l

)uf

0 ∂lub2k−2l−1 −

k∑l=0

(k

l

)ub−1∂

luf

2k−2l .

The presence of theuf2k+2 term in (3.16), (3.17) clearly confirms that the next higher spingenerator can be generated from the preceding one. Finally, (3.16) and (3.17) togetherwith the closure of the fermion generators determine the higher spin generators explicitly.As a consequence, the closure of the algebra among the even fermions is ensured in thenonlinear basis. To show the closure property, the algebra among a few even fermiongenerators are given below.

J3/2, J3/2 = 0,

J5/2, J3/2 = 0,

J5/2, J5/2 = 2J ′

3/2J3/2δ(x − y),J7/2, J3/2

= −J ′3/2J3/2δ(x − y),

J7/2, J5/2 = [−3J5/2J

′3/2 − 3 J5/2J3/2∂ + 2J ′

3/2J3/2∂]δ(x − y),

(3.18)

J7/2, J7/2

=[6J ′

5/2J5/2 − 3J ′5/2J

′3/2 + 3J5/2J

′′3/2 + 3

4 J′′3/2J

′3/2

− 54J

′′′3/2J3/2 − 5

2 J′′3/2J3/2∂ − 5

2 J′3/2J3/2∂

2]δ(x − y).

It is interesting to note that unlike the linear algebra, the generators do not commute amongthemselves. This is a significant departure from the linear representation and compels oneto make the free field representations nonlinear as will be observed later. This differencebecomes evident from the spin 5/2 generator onwards. For example, the Poisson bracket ofthe J5/2 generator with itself becomes nonzero. In fact, the self brackets of the generators,in general, cannot be made to zero by any change of basis. Thus the nonlinear algebracannot be reduced trivially to the linear one. The algebra ofW andW bosons with evenfermions are given in Appendix B.

The arguments regarding the existence of the nonlinear basis for the even fermiongenerators stated above are also valid for the odd fermions since the odd fermionsv

f

2i−1and the odd bosonsub2i−1 form a closed algebra among themselves (A.3)–(A.5). We can,

therefore, generate a nonlinear basis for the odd fermionsvf

2i−1 as well in terms of oddfermions and the odd bosonsub2i−1. For example, a consistent nonlinear basis for a fewodd fermions may be identified as

J3/2 = vf

−1,

J5/2 = −vf

1 + ub−1vf

−1,

(3.19)J7/2 = vf3 + v

f ′1 + 5

4vf ′′−1 − ub−1v

f1 − ub1v

f−1 − ub−1v

f ′−1 + ub

2

−1vf−1.


It is straightforward to observe that the generators in (3.19) can be rewritten as bilinearcombinations ofW andJ like the even fermions. This can also be argued from the factthat odd fermions and odd bosons form a subalgebra and thus can be generalised for otherhigher spin generators. The higher spin generators, however, may be obtained from thelower ones following the Poisson bracket relations resulting from (A.3), (A.8)

(3.20)W2, v

f

2k−1

= vf

2k+1 − vf

2k−1∂ + ub−1vf

2k−1 − ub2k−1vf

−1,

W2, v

f

2k−1

= vf

2k+1 −k−1∑l=0

(k − 1l

)ub−1∂

luf

2k−2l−3

(3.21)+[

k−1∑m=0

(k

m+ 1

)+ 1

2

k∑m=0

(k

m

)]∂m+1v

f

2k−2m−1

and by demanding the closure of the algebra among the odd fermions. To demonstrate theidentical algebra among the odd fermions and the even fermions in the nonlinear basis, wegive below the algebra among a few odd fermion generators explicitly.

J3/2, J3/2 = 0,

J5/2, J3/2 = 0,

J5/2, J5/2 = 2J ′3/2J3/2δ(x − y),

J7/2, J3/2 = −J ′3/2J3/2δ(x − y),

J7/2, J5/2 = [−3J5/2J′3/2 − 3J5/2J3/2∂ + 2J ′

3/2J3/2∂]δ(x − y),

(3.22)

J7/2, J7/2 =[6J ′

5/2J5/2 − 3J ′5/2J

′3/2 + 3J5/2J

′′3/2 + 3

4J′′3/2J

′3/2

− 54J

′′′3/2J3/2 − 5

2J′′3/2J3/2∂ − 5

2J′3/2J3/2∂

2]δ(x − y).

The algebra of odd fermions with even fermions as well as with the bosons are also givenin Appendix B.

The superW∞ algebra, in its own right, deserves to be a candidate for a universalalgebra, unifying all finite-dimensional bosonicW algebras as well as supersymmetricW

algebras. The presence of classical analogue ofW∞ algebra and a direct sum basis in thebosonic sector guarantees that all finite-dimensional bosonicW algebras can be obtainedunder suitable truncation. Since the superW∞ algebra is a higher spin extension ofN = 2conformal algebra, it is expected to contain all finite-dimensionalN = 2 supersymmetricW algebras, like the bosonic universal algebra. But a systematic analysis of the truncationof superW∞ algebra through some noncompact coset model is yet to be studied.

4. Free-field representation

In this section we construct a consistent free field representation of the generatorsdiscussed in the earlier section. We will show that all the generators can be representedby the free complex bosons,φ(x, t) and φ(x, t) and free fermionsψ(x, t) andψ∗(x, t),


which satisfy the following Poisson bracket algebras

(4.1)ψ∗(x),ψ(y)

= δ(x − y)

and

(4.2)∂φ(x), ∂φ(y)

= ∂xδ(x − y).

The nontrivial Poisson bracket algebras among the even fermionsJn+3/2 (3.13)–(3.15)as well as odd fermionsJn+3/2 (3.19) ensure a significant departure of the free fieldrepresentations from the linear ones [16]. We will see that the representations of thefermion generators become not only nonlinear, and in general exponential, but also asuitable combinations of both types of bosons and fermions. This makes the free fieldrepresentation distinct and important. In the bosonic sector such a nontrivial change in thefree field representation is not apparent from the Poisson bracket algebras of the generators.But it will be observed that the nontrivial Poisson bracket algebras of the fermions becomeresponsible for a nontrivial change in the free field representation of the bosons over thelinear ones.

In order to reproduce the Poisson brackets for the even fermion generators, the free fieldrepresentation of all the generators, in general, turns out to be exponential of the bosonfields. The explicit forms of the representation a few even fermions may be given in orderto observe the change in the fermion sector.

(4.3)J3/2 =ψ∗∂φeiεφ2,

(4.4)J5/2 = [ψ∗′

∂φ +ψ∗(a∂φ + b∂φ)∂φ − bψψ∗ψ∗′]eiεφ2,

J7/2 = 5

4(ψ∗∂φeiεφ2)′′ −ψ∗′

(∂φeiεφ2)′ + bψψ∗ψ∗′(eiεφ2)′

+ψ∗′(a∂φ + b∂φ)∂φeiεφ2 −ψ∗(a∂φ + b∂φ)(∂φeiεφ2)′

+ψ∗(a∂φ + b∂φ)2∂φeiεφ2 + bψ ′ψ∗ψ∗′eiεφ2

(4.5)+ 2b2ψψ∗ψ∗′∂φeiεφ2 + 3abψψ∗ψ∗′

∂φeiεφ2,

where,a, b and ε are real parameters andφ2 = 12i (φ − φ). Notice that the significant

change of the free field representation becomes obvious fromJ5/2 onwards. This is dueto the presence of bothψ andψ∗ fields in the representations, the presence of both thefields being essential to reproduce the algebra (3.18). In fact, the presence of both kindsof fermions,ψ andψ∗ is inevitable to reproduce the nonzero algebras consistently in theeven fermion sector. An algorithmic procedure can be developed following (3.16), (3.17)to reproduce the free field representation of all even fermion generators, but it involvesexplicit representations of theW2 and W2 generators, which will be obtained later on.It is important to note that in order to reproduce the even fermion algebras (3.18), theconsistency condition demands that all the parameters are not independent, but are relatedby

(4.6)bε√

2− aε√

2− ab = 1.


Evidently, two more parameters still remain arbitrary, which cannot be fixed at the classicallevel. The quantization of the algebra may fix the arbitrary parameters through its centralcharge. The relation (4.6) dictates that all the parametersa, b andε cannot be set to be zerosimultaneously. This implies the nonlinear free field representation cannot be reduced tothe linear one [16] trivially, which is a crucial observation and will also be seen in the oddfermion sector.

The similar Poisson bracket structures of the odd fermionsJn+3/2 and the even fermionsJn+3/2, however, indicate the similar free field representations of the generators for the oddfermions. We give below the representations of the odd fermions upto spin 7/2. The otherhigher spin generators can be constructed in a similar fashion as in the even fermion case.The representations of the odd fermions may given as

(4.7)J3/2 = −ψ∂φe−iεφ2,

(4.8)J5/2 = [−ψ ′∂φ +ψ(a∂φ + b∂φ)∂φ + aψψ∗ψ ′]e−iεφ2,

J7/2 = −5

4

(ψ∂φe−iεφ2

)′′ +ψ ′(∂φe−iεφ2)′ − aψψ∗ψ ′(e−iεφ2

)′−ψ ′(a∂φ + b∂φ)∂φe−iεφ2 −ψ(a∂φ + b∂φ)

(∂φe−iεφ2

)′−ψ(a∂φ + b∂φ)2∂φe−iεφ2 − aψψ∗′

ψ ′e−iεφ2

(4.9)+ 2a2ψψ∗ψ ′∂φe−iεφ2 − 3abψψ∗ψ ′∂φe−iεφ2.

The complex nature of the representations of the odd and even fermions ensure that the oddand fermions generators cannot be represented by a pair of free fields and their complexconjugates, like the linear one. The nonzero values ofa andb make the representationsdifferent from each other. The representations of other higher spin generators may followfrom (3.20), (3.21) like the even fermion case.

In the bosonic sector, the free field representations become more involved. To be precise,in the linear representation [16], theW1+∞ algebra was realized in terms of the bilinearsof a free fermion field and its conjugate whereas theW∞ algebra was realized fromthe bilinears of a free complex scalar field and its conjugate. This had the advantage ofautomatically ensuring the direct sum basis of the generators of these algebras. But sucha simple representation cannot be considered in the present case. This, in turn, leads to aninconsistency in the fermion sectors and consequently demands for a significant change inthe free field representations in the bosonic sector.

For the odd bosons we have the following consistent representation. The linear part ofthe representation may be written in terms of the fermion bilinears. Thus the lowest one isidentical to that of the linear representation, namely,

(4.10)W1 = −ψ∗ψ.

On the other hand in spin 2, we have trilinear terms,

(4.11)W2 = −1

2

(ψ∗′

ψ −ψ∗ψ ′) −ψ∗ψ(a∂φ + b∂φ)

and this is the most general form of the spin 2 generator, but involving complex bosonfields. From spin three onwards the representation becomes complicated having more


and more nonlinear combinations of the free fields. We will, therefore, follow a differentstrategy from spin three onwards. Following a similar procedure as in the odd boson case,we can obtain the spin 2 generator of the other sector. The most general form of theW2

generator is

(4.12)W2 = −∂φ∂φ −ψ∗ψ(a∂φ + b∂φ)

which commutes with odd bosons. For both the spin 2 generators the last term turns out tobe trilinear and more so this is the only possible term that exists at the spin 2 level being amixture of bosonic and fermionic fields.

For higher spin generators, however, the representations of both types of the bosonicgenerators may be obtained from the leading order terms of the Poisson brackets,Jn+3/2, J3/2, the Jn+3/2 being given in (3.15). This will immediately follow from thePoisson bracket relation (A.7),

(4.13)

uf

2n, vf

−1

=[(

ub2n+1 − vb2n) −

n∑m=0

(n+ 1m+ 1

)(−1)mub2n−2m−1∂

m+1

]δ(x − y).

The leading order terms in (4.13) evidently will be a combination of(Wj+2 − Wj+2)

term and the suitable combinations of lower order spin terms. The representation ofWj+2

generator and consequently theWj+2 generator may be obtained explicitly by exploitingthe commuting property ofW and W generators and from the Poisson bracket relationWj+2,W2. The consistency of these representations may be checked by comparing thealgebra among the other bosonic generators. For example, the free field representationsW3

andW3 generators may be obtained as follows. The Poisson bracketJ5/2, J3/2 using thefree field representations of the fermion generatorsJ5/2 andJ3/2 (4.4), (4.7) is found to be

J5/2, J3/2

(4.14)

=[

− ∂φ∂2φ + ε√2∂φ∂φ(∂φ − ∂φ)− ∂φ∂φ(a∂φ + b∂φ)

−ψ∗′ψ ′ − 2aψ∗ψ ′∂φ − bψ∗ψ ′∂φ +ψ∗′

ψ∂φ

+ aε√2ψ∗ψ(∂φ)2 − b

ε√2ψ∗ψ(∂φ)2 − bψ∗ψ∂2φ

− (∂φ∂φ +ψ∗′

ψ + 2ψ∗ψ(a∂φ + b∂φ))∂

]δ(x − y)

comparing the same with the Poisson bracket of generatorsJ5/2 andJ3/2 (B.14) allows usto determine the free field representation ofW3 − W3 from the leading order term since thefree field representation of the nonleading order terms are already known. Explicitly,

W3 − W3 = 1

2

[∂2φ∂φ − ∂φ∂2φ

] +(

−a + ε√2

)(∂φ)2∂φ −

(b + ε√

2

)∂φ(∂φ)2

+ 1

2ψ∗ψ

(a∂2φ − b∂2φ

) −ψ∗ψ(a∂φ + b∂φ)2 + 2∂φ∂φψ∗ψ


+ ε√2ψ∗ψ

[a(∂φ)2 − b(∂φ)2

] − 1

2

(ψ∗′ψ +ψ∗ψ ′)(a∂φ − b∂φ)

+ 1

6

(ψ∗′′ψ − 4ψ∗′ψ ′ +ψ∗ψ ′′) +ψ∗ψ(a∂φ + b∂φ)2

(4.15)+ (ψ∗′ψ −ψ∗ψ ′)(a∂φ + b∂φ).

Next, we determine the Poisson bracket betweenW3 − W3 andW2. Since the odd andeven boson generators commute (3.10), this operation allows one to findW3 from (3.11)and henceW3. This procedure eventually leads to the following explicit forms of spin 3generators

W3 = −1

6

(ψ∗′′ψ − 4ψ∗′ψ ′ +ψ∗ψ ′′) −ψ∗ψ(a∂φ + b∂φ)2

(4.16)− (ψ∗′ψ −ψ∗ψ ′)(a∂φ + b∂φ),

W3 = 1

2

[∂2φ∂φ − ∂φ∂2φ

] +(

−a + ε√2

)(∂φ)2∂φ −

(b + ε√

2

)∂φ(∂φ)2

+ 1

2ψ∗ψ

(a∂2φ − b∂2φ

) −ψ∗ψ(a∂φ + b∂φ)2 + 2∂φ∂φψ∗ψ

(4.17)+ ε√2ψ∗ψ

[a(∂φ)2 − b(∂φ)2

] − 1

2

(ψ∗′ψ +ψ∗ψ ′)(a∂φ − b∂φ),

wherea andb are the same parameters, already introduced in the fermionic sectors. Noticethat the spin 3 generators acquire a complex structure and possess terms quadrilinear in freefields. In a similar manner both the spin four generators may be obtained from the PoissonbracketJ7/2, J3/2. This constitutes an algorithmic procedure by means of which freefield representations of the higher spin generators may be constructed. The relation (4.6)dictates the both the parametersa andb cannot be set to be zero simultaneously, makingthe representation essentially nonlinear. The presence of an admixture of the bosonic andfermionic terms, on the other hand, makes the representation of bosonic sector nontrivial,unlike the linear case. Importantly, even with the presence of the fermionic and bosonicfields together, we observe that the odd bosons commute with the even bosons. All thehigher spin generators may be constructed from the algebra amongst the even and oddfermionic generators (4.13) and consistency of these representations may be checked bycomparing the algebra among the bosonic generators. But the explicit forms of the higherspin generators in terms of the free fields become more and more complicated as observedfrom the spin 3 generators (4.16), (4.17). However, the strategy is quite clear.

5. Conclusion

In this paper we have shown thatN = 2 KP hierarchy associated with nonstandardflows are bi-Hamiltonian, one of the Hamiltonian structures being nonlocal. To show theexistence of bi-Hamiltonian structures is not straightforward, as it is an intricate processto obtain the correct Poisson bracket which makes theN = 2 KP flows Hamiltonian.Since one of the Hamiltonian structures is local, it becomes a candidate for a nonlinear


superW∞ algebra which is a higher spin extension ofN = 2 superconformal algebra. Thebosonic sector correctly reproduces theW1+∞ ⊕ W∞ structure with an appropriate choiceof basis. To be explicit, in the even boson sector the basis becomes highly nontrivial andnonlinear. But we have evoked a novel strategy to obtain all the generators. Consequently,theW∞ algebra becomes isomorphic to the classical analogue of the nonlinear symmetryconsidered in [11]. This ensures that the nonlinear superW∞ algebra under suitablereduction truncates and gives rise to all finite-dimensional bosonic algebras. In thefermionic sector, the odd and even fermions also form closed algebras among themselvesin a suitable basis. It turns out that the algebra among both kinds of fermions becomesdistinctly different from the linear algebra and more so they form identical algebra amongthemselves. The superW∞ algebra thus deserves to be a universal algebra unifying allfinite-dimensional bosonic as well as fermionicW algebras.

The free field representations of theN = 2 nonlinear superW∞ algebra are obtainedin terms of free complex bosons and fermions. These representations cannot be reducedto the linear one trivially. This is due the constraint condition (4.6). Moreover, therepresentation of the bosonic generators in terms of the free fields possesses a morecomplex structure having an admixture of complex bosons as well as fermions. But atthe same time the odd and the even bosonic generators mutually commute with eachother maintainingW1+∞ ⊕ W∞ structure. This is a nontrivial generalisation in contrastto the linear representation of theN = 2 superW∞ algebra. In the fermionic sector themost general representations become exponential in terms of the free fields. The free fieldrepresentations of theN = 2 nonlinear superW∞ algebra, in fact, is a major breakthroughin classifyingN = 2 super conformal algebras.

Appendix A

Poisson bracket algebra amongst the component fieldsub2i , ub2i−1, uf2i anduf2i−1.

ub2j−1(x), u

f

2k(y)

=[−

j∑m=0

(j

m

)(−1)muf2j+2k−2m∂

m

+j−1∑m=0

k∑l=0

(j − 1m

)(k

l

)(−1)muf2j−2m−2∂

m+lub2k−2l−1

−j−1∑m=0

k−1∑l=0

(j

m+ 1

)(k

l + 1

)(−1)mub2j−2m−3∂

m+l+1uf

2k−2l−2

+k−1∑n=0

k−n−1∑l=0

(k − n− 1

l

)uf

2n∂lub2j+2k−2n−2l−3


−j+k−n−l−1∑

m=0

k∑n=0

k−n∑l=0

(j − 1m

)(n+ l − 1

l

)(−1)m

(A.1)× uf2j+2k−2m−2n−2l−2∂

m+lub2n−1

]δ(x − y),

uf

2j (x), uf

2k(y)

(A.2)

=[

−j−1∑m=0

k−1∑l=0

(j

m+ 1

)(k

l + 1

)(−1)muf2j−2m−2∂

m+l+1uf

2k−2l−2

−k−1∑n=0

k−n−1∑l=0

(k − n− 1

l

)uf

2n∂lu

f

2j+2k−2n−2l−2

+j+k−n−l−1∑

m=0

k−1∑n=0

k−n−1∑l=0

(j

m

)(n+ l

l

)(−1)muf2j+2k−2m−2n−2l−2∂

m+luf

2n

]

× δ(x − y),

ub2j−1(x), v

f

2k−1(y)

(A.3)

=[

k∑m=0

(k

m

)∂mv

f

2j+2k−2m−1

+j−1∑m=0

k−1∑l=0

(j − 1m

)(k − 1l

)−

(j

m+ 1

)(k

l + 1

)

× (−1)mub2j−2m−3∂m+l+1v

f

2k−2l−3

+k−1∑n=0

k−n−1∑l=0

(k − n− 1

l

)ub2n−1∂

lvf

2j+2k−2n−2l−3

−j+k−n−l−1∑

m=0

k−1∑n=0

k−n−1∑l=0

(j − 1m

)(n+ l − 1

l

)

× (−1)mub2j+2k−2m−2n−2l−3∂m+lv

f

2n−1

]δ(x − y),

S. Ghosh, D. Sarma / Nuclear Physics B 616 [PM] (2001) 549–573 569vf2j−1(x), v

f2k−1(y)

(A.4)

=[

j−1∑m=0

k−1∑l=0

(j − 1m

)(k − 1l

)−

(j

m+ 1

)(k

l + 1

)× (−1)mvf2j−2m−3∂

m+l+1vf2k−2l−3

+k−1∑n=0

k−n−1∑l=0

(k − n− 1

l

)vf

2n−1∂lv

f

2j+2k−2n−2l−3

−j+k−n−l−1∑

m=0

k−1∑n=0

k−n−1∑l=0

( − 1m

)(n+ l − 1

l

)

× (−1)mvf2j+2k−2m−2n−2l−3∂m+lv

f

2n−1

]δ(x − y),

ub2j−1(x), u

b2k−1(y)

(A.5)

=[

−j∑

m=0

(j

m

)(−1)mub2j+2k−2m−1∂

m +k∑

m=0

(k

m

)∂mub2j+2k−2m−1

+j−1∑m=0

k−1∑l=0

(j − 1m

)(k − 1l

)−

(j

m+ 1

)(k

l + 1

)× (−1)mub2j−2m−3∂

m+l+1ub2k−2l−3

+k−1∑n=0

k−n−1∑l=0

(k − n− 1

l

)ub2n−1∂

lub2j+2k−2n−2l−3

−j+k−n−l−1∑

m=0

k−1∑n=0

k−n−1∑l=0

(j − 1m

)(n+ l − 1

l

)

× (−1)mub2j+2k−2m−2n−2l−3∂m+lub2n−1

]δ(x − y),

vb2j (x), v

b2k(y)

=

[j+1∑m=0

(j + 1m

)(−1)mvb2j+2k−2m+2∂

m −k+1∑m=0

(k + 1m

)∂mvb2j+2k−2m+2

−j−1∑m=0

k−1∑l=0

(j

m+ 1

)(k

l + 1

)(−1)mvb2j−2m−2∂

m+l+1vb2k−2l−2

−k∑

l=0

(k

l

)ub−1∂

lvb2j+2k−2l −k∑

l=0

(k

l

)uf

2j ∂lv

f

2k−2l−1


+j∑

m=0

(j

m

)(−1)mvb2j+2k−2m∂

mub−1 −j∑

m=0

(j

m

)(−1)mvf2j−2m−1∂

muf

2k

−k−1∑n=0

k−n−1∑l=0

(k − n− 1

l

)vb2n∂

lvb2j+2k−2n−2l−2

+j+k−n−l−1∑

m=0

k−1∑n=0

k−n−1∑l=0

(j

m

)(n+ l

l

)(−1)mvb2j+2k−2m−2n−2l−2∂

m+lvb2n

](A.6)× δ(x − y),

uf

2j (x), vf

2k−1(y)

=[

j+1∑m=0

(j + 1m

)(−1)mub2j+2k−2m+1∂

m −k∑

m=0

(k

m

)∂mvb2j+2k−2m

−j−1∑m=0

k−1∑l=0

(j

m+ 1

)(k

l + 1

)(−1)muf2j−2m−2∂

m+l+1vf

2k−2l−3

−j∑

m=0

(j

m

)(−1)mub2j−2m−1∂

mub2k−1 +j∑

m=0

(j

m

)(−1)mub2j+2k−2m−1∂

mub−1

−k−1∑n=0

k−n−1∑l=0

(k − n− 1

l

)uf

2n−1∂lub2j+2k−2n−2l−2

+j+k−n−l−1∑

m=0

k−1∑n=0

k−n−1∑l=0

(j

m

)(n+ l

l

)(−1)mub2j+2k−2m−2n−2l−3∂

m+lvb2n

](A.7)× δ(x − y),

vb2j (x), v

f2k−1(y)

=

[j+1∑m=0

(j + 1m

)(−1)mvf2j+2k−2m+1∂

m

−j−1∑m=0

k−1∑l=0

(j

m+ 1

)(k

l + 1

)(−1)mvb2j−2m−2∂

m+l+1vf

2k−2l−3

+j∑

m=0

(j

m

)(−1)mvf2j+2k−2m−1∂

mub−1

−j∑

m=0

(

m

)(−1)mvf2j−2m−1∂

mub2k−1


−k−1∑n=0

k−n−1∑l=0

(k − n− 1

l

)vf2n−1∂

lvb2j+2k−2n−2l−2

+j+k−n−l−1∑

m=0

k−1∑n=0

k−n−1∑l=0

(j

m

)(n+ 1l

)(−1)mvf2j+2k−2m−2n−2l−3∂

m+lvb2n

](A.8)× δ(x − y),

vb2j (x), u

f

2k(y)

=[−

k+1∑m=0

(k + 1m

)∂mu

f

2j+2k−2m+2

−j−1∑m=0

k−1∑l=0

(j

m+ 1

)(k

l + 1

)(−1)mvb2j−2m−2∂

m+l+1uf

2k−2l−2

+k∑

l=0

(k

l

)uf

2j ∂lub2k−2l−1 −

k∑l=0

(k

l

)ub−1∂

luf

2j+2k−2l

−k−1∑n=0

k−n−1∑l=0

(k − n− 1

l

)vb2n∂

luf

2j+2k−2n−2l−2

+j+k−n−l−1∑

m=0

k−1∑n=0

k−n−1∑l=0

(j

m

)(n+ l

l

)(−1)mvb2j+2k−2m−2n−2l−2∂

m+luf2n

](A.9)× δ(x − y),

ub2j−1(x), v

b2k(y)

=

[−

j−1∑m=0

k∑l=0

(j − 1m

)(k

l

)(−1)muf2j−2m−2∂

m+lvf

2k−2l−1

−j−1∑m=0

k−1∑l=0

(j

m+ 1

)(k

l + 1

)(−1)mub2j−2m−3∂

m+l+1vb2k−2l−2

−k−1∑n=0

k−n−1∑l=0

(k − n− 1

l

)uf

2n∂lv

f

2j+2k−2n−2l−3

+j+k−n−l−1∑

m=0

k∑n=0

k−n∑l=0

(j − 1m

)(n+ l − 1

l

)

(A.10)× (−1)muf2j+2k−2m−2n−2l−2∂m+lv

f

2n−1

]δ(x − y).


Appendix B

The Poisson brackets between odd bosons and even fermions are:

(B.1)W1, J3/2

= −J3/2δ(x − y),

(B.2)W2, J3/2

= [−J5/2 + 12 J

′3/2 + 1

2J3/2∂ − W1J3/2]δ(x − y),

(B.3)W1, J5/2

= [−J5/2 + J ′3/2 + J3/2∂

]δ(x − y),

W2, J5/2 = [−J7/2 + 1

2 J′5/2 + 3

4J′′3/2 − W1J5/2 + 3

2 J5/2∂ − J ′3/2∂ − 1

2 J3/2∂2]

(B.4)× δ(x − y).

The Poisson brackets between odd bosons and odd fermions are:

(B.5)W1, J3/2

= J3/2δ(x − y),

(B.6)W2, J3/2

= [−J5/2 + 12J

′3/2 + 1

2J3/2∂ + W1J3/2]δ(x − y),

(B.7)W1, J5/2

= [J5/2 − J ′

3/2 − J3/2∂]δ(x − y),

W2, J5/2 = [−J7/2 + 1

2J′5/2 + 3

4J′′3/2 + W1J5/2 + 3

2J5/2∂ − J ′3/2∂ − 1

2J3/2∂2]

(B.8)× δ(x − y).

The Poisson brackets between even bosons and even fermions are:

(B.9)W2, J3/2

= [−J5/2 − J3/2∂ − W1J3/2]δ(x − y),

(B.10)W2, J5/2

= [−J7/2 − J ′5/2 + 5

4 J′′3/2 − W1J5/2 − J5/2∂

]δ(x − y).

The Poisson brackets between even bosons and odd fermions are:

(B.11)W2, J3/2 = [−J5/2 − J3/2∂ + W1J3/2]δ(x − y),

(B.12)W2, J5/2 = [−J7/2 − J ′5/2 + 5

4J′′3/2 + W1J5/2 − J5/2∂

]δ(x − y).

The Poisson bracket between even fermions and odd fermions are:

(B.13)J3/2, J3/2

= [−W2 +W2 + 12W

′1 + W1∂

]δ(x − y),

(B.14)

J3/2, J5/2

=[W3 −W3 − W ′

2 − 12W

′2 − W2W1 + 2W2W1 + 3

2W′1W1

+ 13W

′′1 − (

W2 +W2 − 12W

′1 − W2

1

)∂]δ(x − y),

(B.15)

J5/2, J3/2

=[− W3 +W3 + 1

2W′2 + W2W1 − 2W2W1 − 1

2W′1W1

+ 16W

′′1 + (

W2 +W2 + 12W

′1 − W2

1

)∂]δ(x − y),

(B.16)

J5/2, J5/2

=[W4 −W4 − 1

2W′3 −W ′

3 − W3W1 + 3W2W1 +W2W2 + 32W

′2W1

+ 16W

′2W1 + 3

2W2W′1 + W2W

′1 − 3W2W

21 + W2W

21 − 3

10W′′2

− 110W

′′2 − W2

2 + 112W

′′1 W1 + 3

4W21 + 1

12W′′′1 − 3

2W′1W

21 − J ′

3/2J3/2

− (W3 + 2W3 − 3W2W1 − 2W2W1 − 1

6W′′1 +W ′

2 − W31

+ J3/2J3/2)∂ −W2∂

2]δ(x − y)

and so on.


References

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CUMULATIVE AUTHOR INDEX B611–B616

Abel, S.A. B611 (2001) 43ALPHA Collaboration B612 (2001) 3Amico, L. B614 (2001) 449Arndt, D. B612 (2001) 171Astier, P. B611 (2001) 3Autiero, D. B611 (2001) 3

Bais, F.A. B612 (2001) 229Bajnok, Z. B614 (2001) 405Baldisseri, A. B611 (2001) 3Baldo-Ceolin, M. B611 (2001) 3Banner, M. B611 (2001) 3Bardakci, K. B614 (2001) 71Barenboim, G. B613 (2001) 284Bartels, M. B612 (2001) 413Baseilhac, P. B612 (2001) 373Bassompierre, G. B611 (2001) 3Becher, T. B611 (2001) 367Benedetti, R. B613 (2001) 329Beneke, M. B612 (2001) 25Benslama, K. B611 (2001) 3Berkovits, N. B614 (2001) 195Bernard, D. B612 (2001) 291Besson, N. B611 (2001) 3Billó, M. B614 (2001) 254Billó, M. B616 (2001) 495Binétruy, P. B615 (2001) 219Binoth, T. B615 (2001) 385Bird, I. B611 (2001) 3Blumenfeld, B. B611 (2001) 3Bobisut, F. B611 (2001) 3Botella, F.J. B613 (2001) 284Bouchez, J. B611 (2001) 3Bowick, M. B614 (2001) 467Boyd, S. B611 (2001) 3Brandhuber, A. B611 (2001) 179Brecher, D. B613 (2001) 218Bruzzo, U. B611 (2001) 205Bueno, A. B611 (2001) 3Bunyatov, S. B611 (2001) 3Buras, A.J. B611 (2001) 488

Caffo, M. B611 (2001) 503Camilleri, L. B611 (2001) 3Cardini, A. B611 (2001) 3Cattaneo, P.W. B611 (2001) 3Catterall, S. B614 (2001) 467Cavasinni, V. B611 (2001) 3Cervera-Villanueva, A. B611 (2001) 3Chaichian, M. B611 (2001) 383Chukanov, A. B611 (2001) 3Ciafaloni, M. B613 (2001) 381Ciafaloni, P. B613 (2001) 381Collazuol, G. B611 (2001) 3Comelli, D. B613 (2001) 381Conforto, G. B611 (2001) 3Conta, C. B611 (2001) 3Contalbrigo, M. B611 (2001) 3Corianò, C. B614 (2001) 233Cousins, R. B611 (2001) 3Cvetic, M. B613 (2001) 167Cvetic, M. B615 (2001) 3Czarnecki, A. B611 (2001) 488Czyz, H. B611 (2001) 503

D’Adda, A. B616 (2001) 495Dall’Agata, G. B612 (2001) 123Daniels, D. B611 (2001) 3Deffayet, C. B615 (2001) 219Degaudenzi, H. B611 (2001) 3Degrassi, G. B611 (2001) 403Delgado, A. B613 (2001) 49Del Prete, T. B611 (2001) 3Demichev, A. B611 (2001) 383De Santo, A. B611 (2001) 3Dhar, A. B613 (2001) 105Dib, C.O. B612 (2001) 492Diehl, H.W. B612 (2001) 340Dienes, K.R. B611 (2001) 146Dignan, T. B611 (2001) 3Di Lella, L. B611 (2001) 3Di Lorenzo, A. B614 (2001) 449Do Couto e Silva, E. B611 (2001) 3Dotsenko, V.S. B613 (2001) 445

0550-3213/2001 Published by Elsevier Science B.V.PII: S0550-3213(01)00507-7

576 Nuclear Physics B 616 (2001) 575–579

Dumarchez, J. B611 (2001) 3Dürr, S. B611 (2001) 281

Ellis, M. B611 (2001) 3Enqvist, K. B614 (2001) 388Espinosa, J.R. B615 (2001) 82Espinosa, O. B612 (2001) 492

Falkowski, A. B613 (2001) 189Faraggi, A.E. B614 (2001) 233Farzan, Y. B612 (2001) 59Feldman, G.J. B611 (2001) 3Feldmann, Th. B612 (2001) 25Feng, J.L. B613 (2001) 365Ferrari, F. B612 (2001) 151Ferrari, R. B611 (2001) 3Ferrère, D. B611 (2001) 3Finkel, F. B613 (2001) 472Flaminio, V. B611 (2001) 3Foerster, A. B612 (2001) 461Fraternali, M. B611 (2001) 3Freund, M. B615 (2001) 331Fucito, F. B611 (2001) 205Fujii, A. B615 (2001) 61

Gaillard, J.-M. B611 (2001) 3Gallot, L. B614 (2001) 254Gambino, P. B611 (2001) 338Gangler, E. B611 (2001) 3Garousi, M.R. B611 (2001) 467Gehrmann, B. B612 (2001) 3Geiser, A. B611 (2001) 3Geppert, D. B611 (2001) 3Gersdorff, G.v. B613 (2001) 49Geyer, B. B616 (2001) 437Geyer, B. B616 (2001) 476Ghosh, S. B616 (2001) 549Gibin, D. B611 (2001) 3Gies, H. B613 (2001) 352Giveon, A. B615 (2001) 133Gninenko, S. B611 (2001) 3Godley, A. B611 (2001) 3Gomez-Cadenas, J.-J. B611 (2001) 3Gómez-Ullate, D. B613 (2001) 472Gomis, J. B611 (2001) 179González-López, A. B613 (2001) 472Gosset, J. B611 (2001) 3Gößling, C. B611 (2001) 3Gouanère, M. B611 (2001) 3Gould, M.D. B612 (2001) 461Grant, A. B611 (2001) 3Graziani, G. B611 (2001) 3Guadagnini, E. B613 (2001) 329Guan, X.-W. B612 (2001) 461Gubser, S.S. B611 (2001) 179Guglielmi, A. B611 (2001) 3

Guillet, J.Ph. B615 (2001) 385Gukov, S. B611 (2001) 179Gukov, S. B614 (2001) 195Gutowski, J. B615 (2001) 237

Hagner, C. B611 (2001) 3Hatsuda, M. B611 (2001) 77Hatzinikitas, A. B613 (2001) 237Hayakawa, M. B614 (2001) 171Hebecker, A. B613 (2001) 3Heinrich, G. B615 (2001) 385Hernando, J. B611 (2001) 3Herrmann, C. B612 (2001) 123Hikami, K. B616 (2001) 537Hubbard, D. B611 (2001) 3Huber, P. B615 (2001) 331Hurst, P. B611 (2001) 3Hyett, N. B611 (2001) 3

Iacopini, E. B611 (2001) 3Imaizumi, Y. B615 (2001) 61Ishibashi, N. B614 (2001) 171Ivanov, N.Ya. B615 (2001) 266

Janke, W. B614 (2001) 494Johnston, D.A. B614 (2001) 494Joseph, C. B611 (2001) 3Joshipura, A.S. B611 (2001) 227Juget, F. B611 (2001) 3

Kamimura, K. B611 (2001) 77Kazama, Y. B613 (2001) 17Keski-Vakkuri, E. B614 (2001) 388Khorsand, P. B611 (2001) 239Kim, J.E. B613 (2001) 305Kirsanov, M. B611 (2001) 3Kitazawa, Y. B613 (2001) 105Klimov, O. B611 (2001) 3Klishevich, S.M. B616 (2001) 403Klishevich, S.M. B616 (2001) 419Kogan, I.I. B615 (2001) 191Kokkonen, J. B611 (2001) 3Konechny, A. B614 (2001) 71Kovzelev, A. B611 (2001) 3Krasnoperov, A. B611 (2001) 3Kurth, S. B612 (2001) 3Kustov, D. B611 (2001) 3Kutasov, D. B615 (2001) 133Kuznetsov, V.E. B611 (2001) 3Kyae, B. B613 (2001) 305

Lacaprara, S. B611 (2001) 3Lachaud, C. B611 (2001) 3Lakic, B. B611 (2001) 3Lalak, Z. B613 (2001) 189Lambert, N.D. B615 (2001) 117

Nuclear Physics B 616 (2001) 575–579 577

Langfeld, K. B613 (2001) 352Langlois, D. B615 (2001) 219Lanza, A. B611 (2001) 3La Rotonda, L. B611 (2001) 3Laveder, M. B611 (2001) 3Lee, H.M. B613 (2001) 305Letessier-Selvon, A. B611 (2001) 3Levy, J.-M. B611 (2001) 3Liccardo, A. B614 (2001) 254Lima, E. B614 (2001) 117Lima-Santos, A. B612 (2001) 446Lindner, M. B615 (2001) 331Linssen, L. B611 (2001) 3Liu, H. B614 (2001) 279Liu, H. B614 (2001) 305Liu, H. B614 (2001) 330Liu, H. B615 (2001) 169Ljubic, A. B611 (2001) 3Long, J. B611 (2001) 3Lü, H. B613 (2001) 167Lukyanov, S. B612 (2001) 391Lunin, O. B615 (2001) 285Lupi, A. B611 (2001) 3

Ma, E. B615 (2001) 313Ma, J.P. B611 (2001) 523Mack, G. B612 (2001) 413Maggiore, N. B613 (2001) 34March-Russell, J. B613 (2001) 3Marchionni, A. B611 (2001) 3Martelli, F. B611 (2001) 3Martinelli, G. B611 (2001) 311Matchev, K.T. B613 (2001) 365Mathur, S.D. B615 (2001) 285Maxwell, C.J. B611 (2001) 423Mazumdar, A. B614 (2001) 101McGuire, S. B614 (2001) 467Méchain, X. B611 (2001) 3Mendiburu, J.-P. B611 (2001) 3Meyer, J.-P. B611 (2001) 3Mezzetto, M. B611 (2001) 3Miao, Y.-G. B612 (2001) 215Michelson, J. B614 (2001) 279Michelson, J. B614 (2001) 330Michelson, J. B615 (2001) 169Michishita, Y. B614 (2001) 26Minasian, R. B613 (2001) 87Minasian, R. B613 (2001) 127Mirjalili, A. B611 (2001) 423Mishra, S.R. B611 (2001) 3Misiak, M. B611 (2001) 338Misiak, M. B611 (2001) 488Mizoguchi, S. B611 (2001) 253Moorhead, G.F. B611 (2001) 3Mouslopoulos, S. B615 (2001) 191Müller-Kirsten, H.J.W. B612 (2001) 215

Mülsch, D. B616 (2001) 437Mülsch, D. B616 (2001) 476Muramatsu, T. B613 (2001) 17

Naumov, D. B611 (2001) 3Navarro, I. B615 (2001) 82Nédélec, P. B611 (2001) 3Nefedov, Yu. B611 (2001) 3Neubert, M. B611 (2001) 367Nguyen, X.S. B613 (2001) 445Nguyen-Mau, C. B611 (2001) 3Nishino, H. B612 (2001) 98NOMAD Collaboration B611 (2001) 3Nomura, Y. B613 (2001) 147

Ohta, N. B615 (2001) 61Olum, K.D. B611 (2001) 125Orestano, D. B611 (2001) 3Osterloh, A. B614 (2001) 449Ovrut, B. B614 (2001) 117

Palla, L. B614 (2001) 405Palma, G. B612 (2001) 413Panda, S. B614 (2001) 101Papadopoulos, G. B615 (2001) 237Papazoglou, A. B615 (2001) 191Park, D.K. B612 (2001) 215Park, J. B614 (2001) 117Paschos, E.A. B611 (2001) 227Pastore, F. B611 (2001) 3Peak, L.S. B611 (2001) 3Penati, S. B614 (2001) 367Pennacchio, E. B611 (2001) 3Peres, O.L.G. B612 (2001) 59Pérez-Lorenzana, A. B614 (2001) 101Pessard, H. B611 (2001) 3Petrov, A.A. B611 (2001) 367Petti, R. B611 (2001) 3Placci, A. B611 (2001) 3Plümacher, M. B614 (2001) 233Plyushchay, M.S. B616 (2001) 403Plyushchay, M.S. B616 (2001) 419Pokorski, S. B613 (2001) 189Polesello, G. B611 (2001) 3Pollmann, D. B611 (2001) 3Polyarush, A. B611 (2001) 3Pope, C.N. B613 (2001) 167Popov, B. B611 (2001) 3Portugal, R. B613 (2001) 237Poulsen, C. B611 (2001) 3Pradisi, G. B615 (2001) 33Prešnajder, P. B611 (2001) 383Provero, P. B616 (2001) 495

Quirós, M. B613 (2001) 49

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Raidal, M. B615 (2001) 313Rajpoot, S. B612 (2001) 98Räsänen, S. B614 (2001) 388Rasmussen, J. B616 (2001) 517Rausch de Traubenberg, M. B616 (2001) 419Read, N. B613 (2001) 409Refolli, A. B613 (2001) 64Regnault, N. B612 (2001) 291Reinbacher, R. B614 (2001) 117Remiddi, E. B611 (2001) 503Riccioni, F. B615 (2001) 33Rico, J. B611 (2001) 3Riemann, P. B611 (2001) 3Roberts, R.G. B615 (2001) 358Roda, C. B611 (2001) 3Rodejohann, W. B611 (2001) 227Roditi, I. B612 (2001) 461Rodríguez, M.A. B613 (2001) 472Rolf, J. B612 (2001) 3Romanino, A. B615 (2001) 358Ross, G.G. B615 (2001) 191Ross, G.G. B615 (2001) 358Rossi, G.C. B611 (2001) 311Rubbia, A. B611 (2001) 3Rupp, C. B612 (2001) 313Russo, J.G. B611 (2001) 93

Sachrajda, C.T. B611 (2001) 311Saffin, P.M. B613 (2001) 218Saleur, H. B613 (2001) 409Salvatore, F. B611 (2001) 3Santachiara, R. B613 (2001) 445Santambrogio, A. B613 (2001) 64Santambrogio, A. B614 (2001) 367Santiago, J. B611 (2001) 447Sarkar, U. B615 (2001) 313Sarma, D. B616 (2001) 549Schahmaneche, K. B611 (2001) 3Scharf, R. B612 (2001) 313Schmidt, B. B611 (2001) 3Schmidt, T. B611 (2001) 3Schubert, C. B615 (2001) 385Schwimmer, A. B615 (2001) 133Sconza, A. B611 (2001) 3Seidel, D. B612 (2001) 25Serban, D. B612 (2001) 291Servant, G. B611 (2001) 43Sevior, M. B611 (2001) 3Sfetsos, K. B612 (2001) 191Shadmi, Y. B613 (2001) 365Sharpe, S. B611 (2001) 311Shatashvili, S.L. B613 (2001) 87Sheikh-Jabbari, M.M. B611 (2001) 383Shiu, G. B615 (2001) 3Shpot, M. B612 (2001) 340Sibold, K. B612 (2001) 313

Siemens, X. B611 (2001) 125Sillou, D. B611 (2001) 3Slavich, P. B611 (2001) 403Slingerland, J.K. B612 (2001) 229Smirnov, A.Yu. B612 (2001) 59Smith, D. B613 (2001) 147Smye, G.E. B613 (2001) 259Soler, F.J.P. B611 (2001) 3Sozzi, G. B611 (2001) 3Stanishkov, M. B612 (2001) 373Stathakopoulos, M. B614 (2001) 494Steele, D. B611 (2001) 3Stiegler, U. B611 (2001) 3Stipc, M. B611 (2001) 3Stolarczyk, Th. B611 (2001) 3

Takács, G. B614 (2001) 405Talevi, M. B611 (2001) 311Tani, T. B611 (2001) 253Tanzini, A. B611 (2001) 205Tanzini, A. B613 (2001) 34Tareb-Reyes, M. B611 (2001) 3Taylor, G.N. B611 (2001) 3Taylor, T.R. B611 (2001) 239Tereshchenko, V. B611 (2001) 3Terzi, N. B613 (2001) 64Testa, M. B611 (2001) 311Toropin, A. B611 (2001) 3Touchard, A.-M. B611 (2001) 3Tovey, S.N. B611 (2001) 3Tran, M.-T. B611 (2001) 3Travaglini, G. B611 (2001) 205Tsesmelis, E. B611 (2001) 3Tseytlin, A.A. B611 (2001) 93Tsimpis, D. B613 (2001) 127Tsvelik, A.M. B612 (2001) 479Tureanu, A. B611 (2001) 383

Ulrichs, J. B611 (2001) 3Uranga, A.M. B615 (2001) 3Urban, J. B611 (2001) 488

Vacavant, L. B611 (2001) 3Vainshtein, A. B614 (2001) 3Valdata-Nappi, M. B611 (2001) 3Vallilo, B.C. B614 (2001) 195Valuev, V. B611 (2001) 3Vanhove, P. B613 (2001) 87Vannucci, F. B611 (2001) 3Varvell, K.E. B611 (2001) 3Velasco-Sevilla, L. B615 (2001) 358Veltri, M. B611 (2001) 3Vercesi, V. B611 (2001) 3Vidal-Sitjes, G. B611 (2001) 3Vieira, J.-M. B611 (2001) 3

Nuclear Physics B 616 (2001) 575–579 579

Vinogradova, T. B611 (2001) 3Vives, O. B613 (2001) 284

Walton, M.A. B616 (2001) 517Warner, S. B614 (2001) 467Weber, F.V. B611 (2001) 3Weiner, N. B613 (2001) 147Weisse, T. B611 (2001) 3West, P.C. B615 (2001) 117Wilson, F.F. B611 (2001) 3Winton, L.J. B611 (2001) 3Wolff, U. B612 (2001) 3

Yabsley, B.D. B611 (2001) 3Ynduráin, F.J. B611 (2001) 447Yung, A. B614 (2001) 3

Zaccone, H. B611 (2001) 3Zagermann, M. B612 (2001) 123Zanon, D. B613 (2001) 64Zhdanov, R. B613 (2001) 472Zhou, H.-Q. B612 (2001) 461Zuber, K. B611 (2001) 3Zuccon, P. B611 (2001) 3Zwirner, F. B611 (2001) 403

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