Five-brane thresholds and membrane instantons in four...

-branes.leads to

ntribu-dicteddescrip-rpotentialanes andbriefly

ldrys-In the

Nuclear Physics B 736 (2006) 1–33

Five-brane thresholds and membrane instantonsin four-dimensional heterotic M-theory

Luca Carlevaro, Jean-Pierre Derendinger∗

Physics Institute, Neuchâtel University, A.-L. Breguet 1, CH-2000 Neuchâtel, Switzerland

Received 3 March 2005; received in revised form 31 October 2005; accepted 28 November 2005

Available online 15 December 2005

Abstract

The effective four-dimensional supergravity of M-theory compactified on the orbifoldS1/Z2 and aCalabi–Yau threefold includes in general moduli supermultiplets describing massless modes of fiveFor each brane, one of these fields corresponds to fluctuations along the interval. The five-brane alsomodifications of the anomaly-cancelling terms in the eleven-dimensional theory, including gauge cotions located on their world-volumes. We obtain the interactions of the brane “interval modulus” preby these five-brane-induced anomaly-cancelling terms and we construct their effective supergravitytion. In the condensed phase, these interaction terms generate an effective non-perturbative supewhich can also be interpreted as instanton effects of open membranes stretching between five-brtheS1/Z2 fixed hyperplanes. Aspects of the vacuum structure of the effective supergravity are alsodiscussed. 2005 Elsevier B.V. All rights reserved.

1. Introduction

HeteroticE8 ×E8 strings compactified to four dimensions on a six-dimensional spaceK6 arealso described by M-theory compactified onK7 ≡ S1/Z2 × K6 [1,2]. In particular, it is straight-forward to verify[3–6] that the effectiveN4 = 1 supergravity found in Calabi–Yau or orbifocompactifications of perturbative heterotic strings[7,8] is reproduced by brane-free M-theoconfigurations with compact spaceK7. A novelty of the M-theory approach lies in the posibility to concretely analyse physical effects of non-perturbative brane configurations.

* Corresponding author.E-mail addresses: [email protected](L. Carlevaro),[email protected](J.-P. Derendinger).

0550-3213/$ – see front matter 2005 Elsevier B.V. All rights reserved.doi:10.1016/j.nuclphysb.2005.11.019

mailto:[email protected]


http://dx.doi.org/10.1016/j.nuclphysb.2005.11.019

2 L. Carlevaro, J.-P. Derendinger / Nuclear Physics B 736 (2006) 1–33

mem-ons

thecon-

s).orphicairs oflfour-

modesg fluc-

chiralsu-.eens beenthe pos-

-nd

ive su-ectionsrplanese-branee then

typesalpecif-eresult-niversalssible

ton

tions

e the

low-energy effective supergravity approximation, configurations with five-branes and/orbranes (two-branes)[1,9,10]of compactified M-theory can be studied from simple modificatiof the field equations predicted by eleven-dimensional supergravity[11].

An obvious distinction in the nature of five-brane and membrane effects follows fromalignement conditions applying to their respective world-volumes if one requires that thefiguration admits (exact or spontaneously broken)N4 = 1 supersymmetry (four superchargeEach five-brane world-volume is the product of four-dimensional space–time and a holomtwo-cycle in the Calabi–Yau threefold and conditions apply on the respective cycles of pworld-volumes[1,10]. Five-brane massless excitations[12], which belong to six-dimensionachiral supersymmetry multiplets expanded in modes of the two-cycle, lead then to newdimensional fields to be included in the effective supergravity description. Some of thesedo not depend on the detail of the Calabi–Yau geometry: the five-brane modulus describintuations along theS1/Z2 direction, the two-index antisymmetric tensorBµν with self-dual fieldstrength and their fermionicN4 = 1 partner. These states can be assembled either in asupermultiplet which we will callS or, in a dual version, in a linear multiplet. The effectivepergravity for this “universal five-brane modulus” supermultiplet has been studied in Ref[13](see also Ref.[14]).1 Firstly, the Kähler potential of the theory with this new superfield has bobtained and the absence of direct contributions to the (perturbative) superpotential hademonstrated. Secondly, on the basis of the four-dimensional superspace structure only,sible appearance of new threshold corrections has been emphasized.

In contrast, open membrane euclidean world-volumes include theS1/Z2 direction and a cycle in K6 [10,17]. They stretch between theS1/Z2 fixed planes, or between a fixed plane aa five-brane, or between pairs of five-branes. Their effects in the four-dimensional effectpergravity are then localized in space–time, they can be viewed as instanton-like corrto the interaction Lagrangian. While open membrane stretching between the fixed hypecorrespond in the string approach to world-sheet instantons, membranes ending on a fivdescribe forces acting on this brane. Their contributions to the effective supergravity arexpected to lead to new (non-derivative since the world-volume includesS1/Z2) interactionsinvolving the five-brane modulus.

The corrections to the effective four-dimensional supergravity induced by the variousof membranes have been studied in Refs.[14,18,19]. They were found to contribute to the chirF -density part of the Lagrangian density, in the form of a non-perturbative superpotential. Sically, an interaction bilinear in the five-brane fermion in superfieldS has been computed in thfour-dimensional background with the five-brane and open membranes ending on it. Theing non-perturbative superpotential shows an exponential dependence on the five-brane umodulus typical of instanton calculus. To isolate the membrane contributions from other ponon-perturbative sources, a specific regime is chosen.2 As a consequence, even if the instancalculation clearly establishes the existence of an exponential dependence onS, it does not al-low to infer how this exponential term would combine with other non-perturbative contribuwhich, like gauge instantons, are expected as well.3

1 And, as a function of a non-trivial background value of the five-brane modulus, Refs.[15,16].2 For instance, Moore, Peradze and Saulina[14] select a regime where “open membrane instanton effects ar

leading source of non-perturbative effects”.3 Writing the complete non-perturbative superpotential as a sum of contributions, as for instance in Ref.[14], is an

assumption which needs to be justified.

L. Carlevaro, J.-P. Derendinger / Nuclear Physics B 736 (2006) 1–33 3

tronglynd openpersym-e

n-oduliple sum

l theoryns in-ng theensedstretch-to the

s notion onts.minglyis a con-perfieldning

l terme mul-ctive

d struc-tential

micro-braneuge

fectiveuper-to ourm” of

he

The relevance to physics of the M-theory system with five-branes and membranes srelies upon the structure of superpotentials generated by fluxes, gaugino condensates amembrane instantons. Phenomenological questions addressed in the literature include sumetry breaking and gaugino condensation[16,19], five-brane stabilization (stabilization of thmodulusS) [14,19], stabilization of all moduli[20], the existence of stable de Sitter vacua[20,21], inflationaly phases and potentials[21] and cosmic strings[22]. These analyses use in geeral simplifying assumptions, in the Kähler metric which shows a severe mixing of all mwhen five-brane fields are present, or in the superpotential which is assumed to be a simof non-perturbative contributions.

In the present paper, we use the anomaly-cancelling terms of the eleven-dimensionaon the orbifoldS1/Z2, as modified when five-branes are present, to derive new interactiovolving the five-brane universal modulus supermultiplet which describes fluctuations aloS1/Z2 orbifold direction. These new interactions are then shown to induce, in the condphase, the effective non-perturbative instanton superpotential expected from membranesing between a fixed hyperplane and a five-brane. This superpotential correctly reducesresults of Refs.[14,18,19]in the regime considered in these articles, but its derivation doerequire choosing a particular limiting regime. This approach provides then direct informatthe non-perturbative superpotential with combined gauge and membrane instantons effec

The fact that these four-dimensional interactions can be obtained by considering seeunrelated arguments (membrane instanton calculus or gauge anomaly-cancelling terms)sequence of the superfield structure of the four-dimensional theory. We use an effective suformulation[13,23]which fully respects the symmetry and supersymmetry ingredients defithe microscopic system: the modifications of the Bianchi identities and of the topologicainduced by fixed planes and five-branes. It also respects the symmetries of the five-brantiplet, with its self-dual three-form field. Thus, a given superspace contribution in the effeLagrangian describes various aspects of the microscopic theory, related by the superfielture of the effective theory. This method has been applied to the derivation of the Kähler po[13], including non-linear couplings of the five-brane superfieldS, and we will see in Section4.1that these kinetic terms can be derived from (at least) two quite distinct sectors of thescopic theory. They can certainly be derived from the Calabi–Yau reduction of the five-Born–Infeld Lagrangian[13]. But they can also be derived from a universal correction to gakinetic terms, quadratic inS, induced byS1/Z2 anomaly-cancellation.

Schematically, our argument goes as follows. Since we confine ourselves to the effour-dimensional supergravity with up to two derivatives, for which a (superconformal) sspace formulation exists, counterterms cancelling Lorentz anomalies will be irrelevantdiscussion.4 Gauge anomaly-cancelling terms are then entirely due to the “topological tereleven-dimensional supergravity[11]

− 1

24κ2

∫C3 ∧ G4 ∧ G4.

Sources for the Bianchi identity verified byG4 are provided by the two fixed hyperplanes of tS1/Z2 orbifold and by the aligned five-branes, so that

G4 = dC3 + G4,planes+ G4,branes.

4 They would however lead to similar phenomena.


both

e form

ofg

euni-

sholdiversal. The

ion. Itsstanton

m M-anizing

brane-hesntial,Conclu-

nd

s-malousz termcisely

n the

se thanrivelues of

The contributionG4,planes depends on the gauge curvatures living on the planes, andcorrections explicitly depend on theS1/Z2 coordinate and respect theZ2 symmetry used todefine the orbifold projection. The topological term leads then to a gauge interaction of th

− 1

12κ2

∫C3 ∧ G4,planes∧ G4,branes.

This term gives rise in particular to a gauge interaction of the massless orbifold modesC3located on the five-brane world-volumes and depending explicitly on their position alonS1.And, after integration over the Calabi–Yau space, it produces a coupling toF ∧ F of the ax-ionic partner ImT of the Calabi–Yau volume modulus5 ReT which depends on the five-branlocations alongS1. The superfield structure developed for the effective supergravity of theversal five-brane modulus[13] can then be used to understand this interaction as a threcorrection with a calculable dependence on the five-brane modulus. In particular, the unpart of these contributions can be derived from the Dirac–Born–Infeld kinetic Lagrangiannon-perturbative effective superpotential follows then from standard gaugino condensatdependence on the five-brane modulus is precisely the one expected from membrane incalculations, as performed for instance in Ref.[14].

Along similar lines, a description of some new charged matter contributions arising frotheory anomaly-cancellation can be given. An interesting feature is that the structure orgfive-brane threshold corrections is carried over to these matter interactions.

The paper is organised as follows. The anomaly counterterm at the origin of the five-related gauge interactions is derived in Section2. The reduction to four dimensions and tidentification of the obtained terms as superfield densities are then discussed in Section3 and4. Then, in Section5, condensation is performed to derive the non-perturbative superpotecompare it with membrane instanton results and discuss some physical consequences.sions and comments are added in Section6 andAppendix Acollects conventions, notations atechnical details.

2. The anomaly counterterm

Ten-dimensional hyperplanes located at theZ2 fixed points alongS1 and five-branes act asources of the Bianchi identity verified by the four-form fieldG4 of eleven-dimensional supergravity. As a consequence of these contributions, the topological term acquires anovariations under local symmetries. Together with the variations of the Green–Schwar[1,2,9,24], of order four in the Riemann curvature, these anomalous variations are prethose required to cancel perturbative gauge and Lorentz anomalies generated by theZ2 orb-ifold projection of the eleven-dimensional theory and by the chiral gauge multiplets living ohyperplanes.

The modification of the Bianchi identity and of its solutionG4 of course leads to modificationof the effective action. All modifications generated by the topological term would have mortwo derivatives in the heterotic, ten-dimensional, smallS1 radius limit. But compactifying furtheto four dimensions on the Calabi–Yau spaceK6 also generates modifications of the effectaction at the level of two-derivative gauge terms, because of the non-trivial background va

5 For the bulk moduliT andS, we use the terminology familiar from string compactifications in whichT is the volumemodulus andS the dilaton or string coupling modulus. The terminology commonly used in the context ofM-theory, asfor instance in Refs.[14,15], is unfortunately different. Our conventions are precisely stated inAppendix A.


whichscriptionourowever

ciatedworld-rem

cn

t

cussedn

inated

nes at

〈trR2〉 and〈trF 2〉. The purpose of this section is to precisely derive some of these termsarise whenever five-branes are present. We then begin by recalling some aspects of the deof M-theory on the orbifoldS1/Z2. It should be noted that some ambiguities remain inunderstanding of this description. The gauge sector relevant to our problem escapes hthese ambiguities.

The explicit formulation of the modified Bianchi identity uses two types of sources, assowith hyperplanes supporting Yang–Mills ten-dimensional supermultiplets and five branesvolumes. On both hyperplanes (labelled byi = 1,2) live gauge supermultiplets with curvatutwo-formsFi . The quantity appearing in the Bianchi identity is the gauge invariant four-for

(2.1)I4,i = 1

(4π)2

[1

30TrF 2

i − 1

2trR2

], dI4,i = 0 (i = 1,2),

whereR is the Lorentz curvature two-form. Five-branes compatible withN4 = 1 (four super-charges) supersymmetry wrap space–timeM4 and a two-cycle inK6. The transverse Diradistributionδ

(5)

W6,Ifor five-brane numberI with world-volumeW6,I is the five-brane source i

the Bianchi identity, which then reads6

(2.2)dG4 = 2(4πκ2)1/3

[1

2

∑I

δ(5)(W6,I ) −∑

i

δiI4,i

].

The one-forms

(2.3)δ1 = δ(y) dy, δ2 = δ(y − π)dy

localize the gauge sources on theZ2-fixed hyperplanes.7 In order to respect theZ2 symmetryused in the orbifold projection, we actually label with indexI pairs of five-branes located a±yI .8

The procedure to resolve ambiguities in the solution of the Bianchi identity has been disin detail in the literature.9 The general solution of Eq.(2.2) includes several arbitrary integratioconstants which are constrained by consistency conditions: the four-form fieldG4 should begauge-invariant and globally well-defined, its action should be well-defined, the fields elimby theZ2 truncation should be gauge-invariant, as well as the massive modes of theS1 expansionof the surviving states. These conditions should be verified for any number of five-braarbitrary locations. Writing

(2.4)G4 = dC3 + G4,branes+ G4,planes,

the consistency constraints point to a unique solution for the hyperplane contribution10:

G4,planes= −(4πκ2)1/3 ∑

i

(εiI4,i − dy

π∧ ω3,i

)

(2.5)= −(4πκ2)1/3 ∑

i

(d[εiω3,i] − 2δi ∧ ω3,i

),

6 Supersymmetry forbids that both five-branes and anti-five-branes couple to theS1/Z2 orbifold.7 Appendix Acollects our conventions.8 And one may then choose 0 yI π .9 For instance in Refs.[25] and[26], and references therein.

10 This is the solution with “b = 1” in Refs.[25,26].


,

edborater in the

where, as discussed in detail inAppendix A,

(2.6)εi(y) = sgn(y − yi) − y − yi

π, y1 = 0, y2 = π, y ∈ [−π,π]

and the Chern–Simons three-forms are defined by

(2.7)dω3,i = I4,i , i = 1,2.

A similar discussion can be made for the five-brane contribution toG4. As already mentionedthe five-branes are space–time filling and wrap a holomorphic two-cycle inK6. This implies thatone can certainly write

(2.8)δ(5)(W6,I ) = [δ(y − yI ) + δ(y + yI )

]dy ∧ δ(4)(W6,I ),

whereδ(4)(W6,I ) is now a four-form Dirac distribution inK6 such that∫M4×K6

I6 ∧ δ(4)(W6,I ) =∫

W6,I

I6

for any six-formI6. The natural solution of the Bianchi identity is then to include inG4 the branecontribution

G4,branes=(4πκ2)1/3 ∑

I

[1

2εyI

(y)δ(4)(W6,I ) − 1

πdy ∧ θ(3)(W6,I )

]+ dC3

= −(4πκ2)1/3 ∑

I

δ(y − yI ) + δ(y + yI )

dy ∧ θ(3)(W6,I )

(2.9)+ d

C3 + 1

2

(4πκ2)1/3 ∑

I

εyI(y)θ(3)(W6,I )

,

wheredθ(3)(W6,I ) = δ(4)(W6,I ) and

εyI(y) = sgn(y − yI ) + sgn(y + yI ) − 2y

π, 0 yI π,

(2.10)dεyI(y) = 2

[δ(y − yI ) + δ(y + yI ) − 1

π

]dy.

Notice thatε1(y) = 12 ε0(y) andε2(y) = 1

2 επ (y). The addition in Eq.(2.9)of the termdC3 allowsfor the introduction of brane modes contributions into the topological term, if necessary[26]: it isknown[27,28] that a gauging byC3 of the three-form field of five-brane fluctuations is inducby consistent coupling of five-branes to eleven-dimensional supergravity. We will not elaon this point here since in four dimensions the needed terms would automatically appeasupersymmetrization of the effective theory.11

To summarize, one can write

G4 = dC3 + 2(4πκ2)1/3 ∑

i

δi ∧ ω3,i

− (4πκ2)1/3 ∑

I

δ(y − yI ) + δ(y + yI )

dy ∧ θ(3)(W6,I ),

11 See Section4.


ariant.

ly, is

f thely,touge-

s the

–

lumeich also

c-

(2.11)C3 = C3 − (4πκ2)1/3

[∑i

εiω3,i − 1

2

∑I

εyI(y)θ(3)(W6,I )

]+ C3,

andG4 anddC3 only differ at locations where hyperplanes or five-branes sit.The gauge transformation ofC3 is completely fixed by consistency of theZ2 orbifold pro-

jection. The topological term indicates thatC3 is intrinsically odd. The componentsCABC ,A,B,C = 0, . . . ,9, are then projected out and should then for consistency be gauge invThis condition implies

δC3 = (4πκ2)1/3 1

πdy ∧

∑i

ω12,i ,

(2.12)δC3 = (4πκ2)1/3 ∑

i

[2δi ∧ ω1

2,i − dεiω

12,i

],

whereδω3,i = dω12,i . The correct modified topological term, which cancels anomalies local

then[26]

(2.13)− 1

24κ2

∫C3 ∧ G4 ∧ G4.

Our goal is to infer from this modified topological term four-dimensional interactions omasslessS1/Z2×K6 modes. The substitution of Eqs.(2.11)leads to two classes of terms. Firstcontributions involving the massless modeCABy (A,B = 0, . . . ,9). This mode correspondsthe antisymmetric tensorBAB of ten-dimensional sixteen-supercharge supergravity. From gatransformation(2.12), one deduces that the appropriate definition withδB = (4π)2 ∑

i ω12,i is12

(2.14)BAB = (4πκ2)−1/3

(4π)2πRS1C(0)ABy, C

(0)ABy = 1

2π

π∫−π

dy CABy,

whereRS1 is theS1 radius. In terms of this massless field, the topological term producefollowing four-dimensional interactions

(2.15)− 1

32π2

∫

S1×K6

B2 ∧[∑

i,j

εiεj I4,i ∧ I4,j −∑i,I

εi εyII4,i ∧ δ(4)(W6,I )

],

with∫S1 = ∫ π

−πdy. The first contribution, when integrated overS1 only, generates the Green

Schwarz gauge anomaly-cancelling terms expected for theE8 × E8 heterotic string[24]. Itsconsequences for Calabi–Yau compactifications have been studied long ago[8,29]. When inte-grated overS1×K6, it leads to gauge threshold corrections depending on the Calabi–Yau vomodulus as well as dilaton-dependent charged-matter terms. The second contribution, whdepends on theS1 position of the five-branes, is of interest to us.

The modified topological term also produces the followingC3-independent gauge interations:

12 Taking into account the factor(4π)−2 in the definitions ofI4,i andω3,i . This is the definition of the two-form field

commonly used in ten dimensions, with dimension (mass)2.


ella-itions

on,

t the

−1

3

∫

S1×K6

[−

∑i,j,k

εiεjω3,i ∧ I4,j ∧ ω3,k + 1

2

∑i,J,k

εi εyJω3,i ∧ δ(4)(W6,J ) ∧ ω3,k

(2.16)+ 1

2

∑i,J,k

εi εyJI4,i ∧ θ(3)(W6,J ) ∧ ω3,k +

∑i,j,K

εiεjω3,i ∧ I4,j ∧ θ(3)(W6,K)

].

After integration overy, the first contribution is a local counterterm allowed by anomaly canction [26].13 The next three terms are non-trivial brane contributions depending on their posalongS1.

The overlap integrals overS1 give an interesting result. First introduce the numbers

(2.17)dij =π∫

−π

dy εiεj = π

3(3δij − 1)

for the first integrals in contributions(2.15) and (2.16). For those depending on the brane positidefine the variables

(2.18)∆I,1 = yI

π, ∆I,2 = 1− yI

π(0< yI < π; 0 < ∆I,j < 1),

the distances from braneI to the fixed planes, with normalization∆I,1 + ∆I,2 = 1. Then,

1

2π

π∫−π

dy ε1(y)εyI(y) = (∆I,2)

2 − 1

3= y2

I

π2− 2

yI

π+ 2

3,

(2.19)1

2π

π∫−π

dy ε2(y)εyI(y) = (∆I,1)

2 − 1

3= y2

I

π2− 1

3.

It will be important for the supersymmetrization of the four-dimensional interactions thaterms quadratic inyI are identical in both integrals. With these results, contributions(2.15)leadto

− 1

16π

∫K6

B2 ∧[

1

3

(I24,1 + I2

4,2 − I4,1I4,2)

(2.20)−∑I

δ(4)(W6,I ) ∧(

∆2I,2 − 1

3

)I4,1 +

(∆2

I,1 − 1

3

)I4,2

],

while expressions(2.16)give

π

3

∫K6

ω3,1 ∧ ω3,2 ∧ (I4,1 − I4,2) + 3

2

∑I

(∆I,1 − ∆I,2)δ(4)(W6,I ) ∧ ω3,1 ∧ ω3,2

−∑I

θ (3)(W6,I ) ∧[(∆I,2ω3,1 − ∆I,1ω3,2) ∧ (∆I,2I4,1 − ∆I,1I4,2)

(2.21)− ω3,1 ∧ I4,1 − ω3,2 ∧ I4,2 + 1

2ω3,1 ∧ I4,2 + 1

2ω3,2 ∧ I4,1

],

13 The anomaly twelve-form obtained from descent equations vanishes.


lanemilarly,g of the

udesxpres-

ated to

the

e

gravity

heir

after some partial integrations.The anomalous gauge variation of contributions(2.20) and (2.21)can be written as

−π

3

(ω1

2,1I24,1 + ω1

2,2I24,2

)

(2.22)+ 2π

3

∑I

δ(4)(W6,I ) ∧ [∆I,2ω

12,1 − ∆I,1ω

12,2

] ∧ [∆I,2I4,1 − ∆I,1I4,2].

Applying descent equations to these variations leads to the formal anomaly twelve-form

I12 = −π

3

[I4,1 + I4,2 −

∑I

δ(4)(W6,I )

]

∧[I24,1 + I2

4,2 − I4,1I4,2 +∑I

δ(4)(W6,I ) ∧ (1− 3∆2

I,2

)I4,1 + (

1− 3∆2I,1

)I4,2

]

(2.23)= I12,het. +∑I

δ(4)(W6,I )I8,I ,

since for four-dimensional space–time-filling five-branes,δ(4)(W6,I )∧ δ(4)(W6,J ) = 0. The con-tribution of each five-brane is encoded in the eight-form

(2.24)I8,I = π[∆I,2I4,1 − ∆I,1I4,2]2,while the heterotic contribution is as usualI12,het. = −π

3 [I34,1 + I3

4,2].The form ofI8,I , Eq.(2.24), clearly shows that the distance from the brane to the first p

acts as the (inverse squared) coupling of the gauge fields living on the second plane. Sithe distance from the brane to the second plane acts as the (inverse squared) couplingauge fields living on the first plane.

At this point, the conclusion is that the effective, four-dimensional supergravity inclgauge contributions due to five-branes which arise from the Calabi–Yau reduction of esions(2.20) and (2.21), as derived from the modified topological term(2.13). In the next twosections, we perform this reduction keeping only the “universal” massless modes unrelgeometrical details ofK6, but including the five-brane modulus along theS1 direction, and wewrite the effective four-dimensional supergravity using superconformal tensor calculus.

3. Reduction to four dimensions

In the reduction to four dimensions, we use the freedom to rescale moduli fields to setS1

circumference 2πRS1 and the Calabi–Yau volumeV6 to unity.14

As usual, the massless modes of the metric tensor expanded onM4 × K7 includegµν , thescalar fieldgyy and massless modes of the internal metricgik . Among these, we only keep thuniversal, Kähler-metric volume modulus. Similarly, the antisymmetric tensorCABy leads to amasslessBµν and we only keep the universal massless mode of the internal tensorBik . With thesebosonic modes and their fermionic partners, the reduction of eleven-dimensional super

14 The four-dimensional gravitational constant is thenκ24 = κ2, but we nevertheless use different symbols since t

mass dimensions differ.


ng

t.

.

can be described by two chiral multipletsS andT with the familiar Kähler potential[7,8]15

(3.1)K = − ln(S + S) − 3 ln(T + T ).

Following Eq.(2.14), we define

(3.2)Bµν = (4π)2

2

(4πκ2)−1/3

Cµνy, Bij = i

κ24

ImT δij

andBµν is dual to ImS.With five-branes, vector fields on the two fixed hyperplanes gauge an algebraG1 × G2 which

is further broken by the Calabi–Yau compactification. Embedding theSU(3) holonomy intoG1×G2 defines the four-dimensional gauge groupG(4) as the stabilizer of thisSU(3) in G1 × G2.

Calabi–Yau reduction of the ten-dimensional gauge fields16Aα(i)B leads then to the correspondi

gauge fieldsAa(i)µ . It also produces a set ofSU(3)-singlet complex scalar fieldsAm(i) in some

representation ofG(4).With up to two derivatives, Riemann curvature contributions in counterterms(2.20)–(2.21)

can be omitted. The Calabi–Yau reduction ofI4,i delivers then:

Ii,µνρσ = 3!(4π)2

∑α

Fα(i)[µν F

α(i)ρσ ] ,

Ii,µνkl = − 4

(4π)2

∑m

(D[µAm(i)

)(Dν]Am(i)

)δkl

= 2i

(4π)2

∑m

∂[µ(Am(i)Dν]Am(i) − Am(i)Dν]Am(i)

)iδkl ,

(3.3)Ii,µjkl = 2

(4π)2∂µ

(λi,mnpAm(i)An(i)Ap(i)

)εjkl .

In the last expression,λi,mnp is the symmetric tensor invariant underG(4) that may arise fromthe internal Chern–Simons termωi,jkl . We will use the notations

(3.4)λA3 =∑

i

λA3i , λA3

i ≡ λi,mnpAm(i)An(i)Ap(i)

to denote this cubic holomorphic couplings which also appear in the superpotential

(3.5)W = c + λA3.

Finally, I4,i has a non-trivial background value〈I4,i〉ij kl .With these results, the reduction to four dimensions of the first line in expression(2.20), which

depends onB2 and exists without five-brane can be written

15 In general, we use the same notation for a chiral supermultiplet and for its lowest complex scalar componen16 We find useful to keep track of the plane indexi = 1,2 andα(i) is then an index in the adjoint representation ofGi .Similarly, a(i) will be used for the adjoint ofG(4) andm(i) for the representation spanned by complex scalar fields


ulusibered in

-

tencemterms

d up

ce the

Lplane= 1

2(4π)4

∑i,j

dij

1

κ24

e4〈Ij 〉 ImT Fα(i)µν F α(i)µν

− i〈Ij 〉εµνρσ (∂µBνρ)∑m

[Am(i)

(Dσ Am(i)

) − Am(i)(DσAm(i)

)]

(3.6)− i

(4π)2εµνρσ (∂µBνρ)

(λA3

i ∂σ

(λA3

j

) − ∂σ

(λA3

i

)λA3

j

).

The background value ofI4,i is encoded in the integral over the Calabi–Yau manifold

(3.7)〈Ii〉 = V −16

∫K6

〈Ii〉klklδj j εjklεj kl .

In expression(3.6), the first term is a threshold correction depending on the volume modalready well known from the heterotic strings[8,29]. The second and third contributions descrinteractions of matter scalars with the string coupling multiplet. They have been considedetail in Refs.[3,5,23].

The reduction of the five-brane contribution in expression(2.20) leads to the following La-grangian terms:

Lbrane= 1

8(4π)3

∑I,i

aI

(∆2

I,i − 1

3

)[1

κ24

e4 ImT Fα(i)µν F α(i)µν

(3.8)− iεµνρσ ∂µBνρ

∑m

[Am(i)

(Dσ Am(i)


)]].

The area of the Calabi–Yau two-cycle (with coordinatez) wrapped by the five-brane worldvolume can be written

(3.9)aI =∫C2,I

dz dz∂zm

∂z

∂zn

∂zδmn.

The first term in(3.8) is the five-brane contribution to gauge threshold corrections. Its exishas been demonstrated in an explicit background calculation by Lukas, Ovrut and Waldra[15,16]. The second term is again a matter interaction with the string coupling multiplet. Bothdepend on the positionsyI of the five-branes. Hence, they depend on theS1/Z2 five-brane mod-ulus.

The terms collected in expression(2.21)are somewhat ambiguous since they are defineto contributions which, like the first one or any counterterm of the formθ(3)(W6,I ) ∧ I7, do notcontribute to the gauge-invariant anomaly twelve-form. To reduce the first term, introdufour-dimensional Chern–Simons forms

(3.10)∂[µωi,νρσ ] = 1

4Fi,[µνFi,ρσ ], ∂[µωi,ν]j k = 1

2Ii,µνj k, ∂µωi,jkl = Ii,µjkl .

The first term then generates couplings of charged matter scalars to gauge fields:

Lplane= i

3(4π)5εµνρσ ω1,µνρ

[λA3

2∂σ

(λA3

2

) − λA32∂σ

(λA3

2

) + λA31∂σ

(λA3

2

)

− λA31∂σ

(λA3

2

) + 2[λA3

2∂σ

(λA3

1

) − λA32∂σ

(λA3

1

)]] + (1 ↔ 2)


tensionoweverf these

the

artic-gnes is

tionntisym-ipletur-

l mod-

of

se-

cnirectly

m

− i

12(4π)3εµνρσ ω1,µνρ

∑m

(Am(2)Dσ Am(2) − Am(2)DσAm(2)

)(〈I1〉 − 〈I2〉)

(3.11)+ (1 ↔ 2).

As we will see in the next section, these terms do not have a natural supersymmetric exin general, a fact which may have some relation to their ambiguous character. Notice hthat in the minimal embedding of the Calabi–Yau background into one plane only, most omixing terms are absent and a natural supersymmetrization exists.

Likewise, the second term in Eq.(2.21) yields gauge-matter interactions depending onfive-brane positions alongS1:

Lbrane= i

8(4π)3

∑I

aI (∆I,1 − ∆I,2)εµνρσ ω1,µνρ

(3.12)×∑m

[Am(2)

(Dσ Am(2)

) − Am(2)(DσAm(2)

)] + (1↔ 2).

In the next section, we will derive the effective four-dimensional supergravity in the pular case of a single five-brane. To simplify, we will omit the indexI and the correspondinsums. We will however use a formulation in which restoring contributions of several brastraightforward.

4. The effective supergravity

The universalS1/Z2 five-brane modulus describing fluctuations along the interval direchas a supersymmetric bosonic partner arising from the mode expansion of the five-brane ametric tensorBmn. Six-dimensional (world-volume) supersymmetry of the five-brane multrequires that the three-form curvatureHmnp of this tensor is self-dual. For the massless fodimensional universal mode, self-duality is the condition17

(4.1)Hµνρ ≡ 3∂[µBνρ] = e4εµνρσ ∂σ Bij ≡ e4εµνρ

σ Hσ ij .

Then, clearly, the four-dimensional supersymmetric description of the five-brane universaulus uses either a linear multipletL with the tensorHµνρ and a real scalarC for the modulus, or achiral multiplet with complex scalarS and ImS related toBij . The supersymmetric extension

condition(4.1) is chiral-linear duality, the duality transformation exchanging superfieldsS andL [30].

The fact that the chiral multipletS is dual to a linear multiplet has three important con

quences for its supergravity couplings. Firstly, the Kähler potential is a function ofS + ¯S only.

Secondly, the holomorphic gauge kinetic function can only depend linearly onS. These twoconsequences follow from the intrinsic gauge invariance ofBµν , which translates into axionishift symmetry ofS in the chiral formulation. Thirdly, the superpotential does not depend oS.In supergravity, in contrast to global supersymmetry, this statement is ambiguous and dlinked to the first consequence above. The superpotentialW and the Kähler potentialK are notindependent: the entire theory depends only onG = K+ ln |W |2. Terms can then be moved fro

17 Omitting fermion and covariantization contributions.


state-

n

ten-

nsation,

led tos a com-oupling

three

e mul-e thenhi, the

three

rms ofis thatliversuli andan be

or into K provided they are harmonic functions of the complex chiral fields. The correct

ment is then thatG may only depend onS + ¯S. Moving terms fromK to the superpotential ca

artificially generate a dependence onS of the form

(4.2)Wnew= ebSW(zi

),

whereb is a real constant andzi denotes all other chiral multiplets, and a new Kähler po

tial Knew(S + ¯S, zi, zi ) such that the functionG remains unchanged,K + ln |W |2 = Knew +

ln |Wnew|2. Notice that adding aS-independent term to the superpotential(4.2) is not allowed.Non-perturbative exponential superpotentials generated, for instance, by gaugino condeandadded to a perturbative superpotential are then incompatible with chiral-linear duality.18

The effective four-dimensional supergravity depends on three moduli multiplets coupsupergravity, gauge and charged matter superfields. Each of the three moduli scalars haponent of an antisymmetric tensor as supersymmetry partner. More precisely, the string cmodulus is in the multiplet describingGµνρy , theK6 volume modulus is paired withGµiky and

the five-braneS1/Z2 modulus is the partner of the componentsHµνρ or Hµik of the self-dualantisymmetric tensor living on the brane world-volume. We find, as explained in Ref.[23], moreefficient to formulate the theory using superconformal tensor calculus and to introducemoduli vector superfields to describe these moduli multiplets19:

V (w = 2, n = 0): Gµνρy, string coupling modulus, . . . ,

VT (w = n = 0): Gµiky, Calabi–Yau volume modulus, . . . ,

V (w = n = 0): Hµνρ, five-braneS1/Z2 modulus, . . . .

The components of the antisymmetric tensors are identified with the vector fields in thestiplets, the moduli scalars with their real lowest components. These vector multiplets arsubmitted to Bianchi identities obtained from theK7 reduction of the eleven-dimensional Biancidentity forG4, Eq.(2.2), and the self-duality condition of the five-brane tensor. In each casesupersymmetrized Bianchi identity also reduces the number of off-shell states from 8B + 8F in avector multiplet to 4B + 4F . These Bianchi identities are imposed as the field equations ofLagrange-multiplier superfields:

S (w = n = 0): a chiral multiplet for the Bianchi identity verified byV,

LT (w = 2, n = 0): a linear multiplet for the Bianchi identity verified byVT ,

S (w = n = 0): a chiral multiplet for the self-duality condition of the

brane tensor, applied toV .

Eliminating these Lagrange multiplier superfields defines the three vector multiplets in tethe physical fields solving Bianchi identities. The important advantage of this proceduresupersymmetrizing the theory before eliminating Lagrange multipliers automatically dethe correct non-linear couplings of source terms (brane and plane contributions) to modthen the correct Kähler potential. Alternatively, equivalent (dual) versions of the theory cobtained by eliminating some vector multiplets instead of the Lagrange multipliers.

18 See however Ref.[31] for an analysis.19 The Weyl weight isw andn is the chiral weight.


ving one gaugetsbyral

etry

well-

d in,of

d

-

uge-

DBI)

beene

These six multiplets describing bulk and brane states are supplemented by states lithe fixed hyperplanes. In the notation defined in the previous section, these states includchiral superfieldsWα(i) (i = 1,2 as usual,w = n = 3/2) and charged matter chiral multiple(w = n = 0) in some representation of the gauge group. They will be collectively denotedM

and they contain the complex scalar componentsAm(i). Finally, we need the compensating chimultiplet S0 (w = n = 1) to gauge fix the superconformal theory to super-Poincaré symmonly.

With this set of superfields, the Lagrangian nicely splits in a sum of five terms withdefined higher-dimensional interpretations:

(4.3)L= Lbulk +LBianchi+Lkinetic +Lthresholds+Lsuperpotential.

The bulk Lagrangian[32]

(4.4)Lbulk = −[(S0S0VT )3/2(2V )−1/2]

D

can be directly obtained from theK7 reduction of eleven-dimensional supergravity, expresseterms ofG4. It depends onV (string coupling multiplet) andVT (K6 volume modulus multiplet)and of the compensatorS0. In Eq. (4.4), [· · ·]D denotes the invariant real density formulasuperconformal calculus, as reviewed and developed in for instance Ref.[33]. Similarly, [· · ·]Fwill below denote the chiral density formula.20

The coupling of plane and brane fields (Wα(i), M andV ) to bulk multiplets is entirely encodein LBianchi, which reads[13]

(4.5)

LBianchi=[−(S + S)(V + 2Ω1 + 2Ω2) + LT

(VT + 2Me2AM

) + 1

2τ(S + ¯

S)V V

]D

,

where Ω1 and Ω2 are the Chern–Simons multiplets (w = 2, n = 0) for the gauge algebra arising on each hyperplane, defined by21 ∑

α Wα(i)Wα(i) = 16Σ(Ωi), and Me2AM ≡∑m,i M

m(i)[e2AM]m(i) is the Wess–Zumino Lagrangian superfield. This contribution is gainvariant since[(S + S)(δΩ1 + δΩ2)]D is a derivative. The dimensionless numberτ is propor-tional to the five-brane tensionT5. In our units, it reads

(4.6)τ = 2

(4π)3a,

wherea is the area of the two-cycle wrapped by the brane inK6, as defined in Eq.(3.9). Noticeshift symmetriesδS = ic, δS = id (c, d real).

The kinetic terms of the five-brane fields arise from reduction of the Dirac–Born–Infeld (Lagrangian:

(4.7)Lkinetic = −τ[V VT V 2]

D.

They are quadratic inV , a consequence of the form of the DBI action, and the prefactorV VT

is the coupling to the supergravity background. Notice that since this term is linear inV , it willnaturally assemble with the contribution inS + S in Eq.(4.5).

At this point, the contributions from bulk, plane and five-brane kinetic Lagrangians haveconsidered, with tensor fields inV , VT and V verifying Bianchi identities modified by plan

20 In global supersymmetry,[· · ·]D and[· · ·]F would be∫

d4θ [· · ·] and∫

d2θ [· · ·] + h.c.21 The operationΣ(· · ·) is the superconformal analog of1DD in global superspace.
8


l term

to

odulusg the

olog-ctions

ent

lannotthen

oduliliminat-

and brane sources. But we still have to consider further contributions from the topologicawith modifiedG4, as obtained in the previous section. These terms will be collected inLthreshold.The symmetries of expressions(4.4)–(4.7)allow the introduction of the following correctionsgauge kinetic terms[13] (threshold corrections):

Lthresholds=[−2

∑i

βiΩi

(VT + 2Me2AM

)]D

(4.8)

+ τ

4

[S

∑i

βiWiWi

]F

+[V

ε∣∣αM3

∣∣2 − 2τ∑

i

gi(V )Mie2AMi

]D

.

The first contribution corresponds to threshold corrections depending on the volume m[8,29,34]. Gauge invariance of the full Lagrangian with this term is obtained by postulatinappropriate variation of the linear multipletLT in LBianchi:

δLT = 2∑

i

βiδΩi.

The second and third contributions are threshold corrections depending on theS1/Z2 locationof the five-brane and/or, for the last one, on matter multiplets. The coefficientsβi , βi , ε and thefunctionsgi(V ) can be obtained, as explained below, from Calabi–Yau reduction of the topical term with anomaly-cancelling modifications. Symmetries of the theory leave these fununconstrained but the terms considered here only require linear functions,gi(V ) = γi V + δi . Fi-nally, the quantityαM3 in expression(4.8)denotes the holomorphic cubic invariant also presin the matter superpotential[7,8].

The superpotential arises from the componentsGijky of G4. They also verify a non-triviaBianchi identity which is not modified by five-branes since three holomorphic directions cbe transverse to their world-volumes. The superpotential contribution to the Lagrangian is

(4.9)Lsuperpotential=[S3

0W]F, with W = c + αM3.

The constantc being the ‘flux’ of the heterotic three-form in directionεijk [7,35].This formulation of the effective supergravity, with six superfields to describe three m

supermultiplets leads to several equivalent forms, depending on the choice made when eing the three superfluous multiplets. The Lagrange multipliersS, LT andS imply the followingconstraints on the vector multiplets:

S: V = L − 2∑

i

Ωi, L linear(w = 2, n = 0),

LT : VT = T + T − 2Me2AM, T chiral (w = n = 0),

(4.10)S: V = V −1(

L + 4∑

i

βiΩi

), L linear(w = 2, n = 0).

EliminatingS, LT andS leads then to a formulation where moduli are described byL, T andL,two linear and one chiral multiplets:


percon-

condileixingsed

Iteei–Yauext

vac-point

L=[− 1√

2

[S0S0

(T + T − 2Me2AM

)]3/2(

L − 2∑

i

Ωi

)−1/2

− τ(T + T − 2Me2AM

)(L − 2

∑i

Ωi

)−1(L + 4

∑j

βjΩj

)2

+(

L − 2∑

i

Ωi

)ε∣∣αM3

∣∣2 − 2τ∑j

δj Mj e2AMj

− 2τ

(L + 4

∑i

βiΩi

)∑j

γj Mj e2AMj

]D

(4.11)+[S3

0W + 1

4

∑i

βiTWiWi

]F

.

Component expressions for this apparently complicated theory can be obtained using suformal tensor calculus[33]. Notice that plane contributions (superfieldsΩi , Wi andMi ) nowappear in the bulk Lagrangian (first line) and also in the five-brane DBI contribution (seline). Five-brane contributions (multipletL) appear in gauge kinetic terms (a “plane term”) whthreshold corrections (third and fourth lines) involve plane and five-brane fields. These mare induced by the modified Bianchi identities, Eqs.(4.10), and by threshold corrections requirby anomaly cancellation.

The kinetic term quadratic in the five-brane modulus superfieldL appears in the second line.can clearly be derived from the DBI Lagrangian, as done in Ref.[13]. But the superfield structurimplied by the modified Bianchi identity leading to the first Eq.(4.10)also implies that the samkinetic term can be obtained from gauge threshold corrections which follow from Calabreduction of the (modified) topological term. This point will be explicitly verified in the nparagraph.

This unfamiliar supergravity theory is particularly useful to study its scalar potential anduum structure since linear multiplets do not have auxiliary fields. We will come back to thislater on and especially when studying condensation.

It is however more common to formulate the supergravity theory with chiral moduliS andT ,and then to eliminateV andLT using their field equations. One obtains:

(4.12)L= −3

2

[S0S0e

−K/3]D

+[S3

0W + 1

4

∑i

(S + βiT + τ βi S)W iW i

]F

,

with the Kähler potential

(4.13)K = − ln(S + S − ∆) − 3 ln(T + T − 2Me2AM

)and

(4.14)

∆ = −τ(T + T − 2Me2AM

)V 2 + 1

2τ(S + ¯

S)V + ε∣∣αM3

∣∣2 − 2τ∑

i

gi(V )Mie2AMi.

The field equation of the vector multipletV implies then

(4.15)V = (4VT )−1(

S + ¯S − 4

∑γiM

ie2AMi

),

i


is

hirale the

cation

ts.

-e

with VT as in the second Eq.(4.10). The fully chiral formulation of the effective supergravitythen defined by Kähler potential(4.13)with now

∆ = τ

16

(T + T − 2Me2AM

)−1(

S + ¯S − 4

∑i

γiMie2AMi

)2

(4.16)+ ε∣∣αM3

∣∣2 − 2τ∑

i

δiMie2AMi,

gauge kinetic functions

(4.17)f i = S + βiT + τ βi S,

and superpotential(4.9). The presence of the five-brane then introduces mixing of the three cmultipletsS, T andS and the off-diagonal elements of the Kähler metric severely complicatanalysis of the theory.

Before returning to the analysis of the effective supergravity, we need a precise identifiof the supergravity fields in terms of massless modes of theK7 compactification.

The notation we use for component fields is as follows. Vector multipletsV , V andVT haverespectively vector fieldsVµ, Vµ andTµ and (lowest) real scalarC, C andCT . And we use thesame notation for chiral multipletsS, T andS and for their complex scalar lowest componenEqs.(4.10)indicate then that

CT = 2(ReT − MM), Tµ = −2∂µ ImT − 2iM(DµM) + 2i(DµM)M,

that the lowest scalar component of the string coupling linear multipletL is alsoC and that thefive-brane linear multipletL has a real scalar = CC. Relation(4.15)also implies that

ReS = 4C(ReT − MM) + 2∑

i

γiMiMi,

∂µ Im S = −4Vµ(ReT − MM) + 4C(∂µ ImT + iMDµM − iMDµM)

(4.18)− 2i∑

i

γi

(MiDµMi − MiDµMi

).

The scalar fieldC has background value proportional toy, the five-brane location alongS1. Inour units with 2πRS1 = 1= V6, the four-dimensional gravitational constant isκ4 = κ and

(4.19)〈C〉 = 1

κ4yRS1 = y

2πκ.

In order to derive the identification of the matter scalarsAm(i) of Eq. (3.3)and the superconformal multipletsMm(i), we note that the componentGµijy of the four-form is related to thvector component of the real multipletVT , which is

(VT )µ = −2[∂µ ImT + iM(DµM) − i(DµM)M

].

On the other hand, using Eqs.(3.2) and (3.3), we also find

Gµijy = ∂µCijy − (4πκ2)1/3

π

∑k

ωk,µij

= 1 (4πκ2)−2/3

(∂µ ImT − i

κ2 [A(i)DµA(i) − A(i)DµA(i)

])δij .
2π 2π


in thenon-

ined inf the

yauge

. The

)

g linear

By comparison, one obtains

(4.20)A(i) =√

2π

κM(i),

with an irrelevant sign choice.Finally, the gauge fields inΩi or Wa(i) are the massless modesAa(i)

µ .

4.1. Gauge coupling constants

The effective supergravity Lagrangian predicts a very specific moduli dependencesuper-Yang–Mills sector relevant to the determination of the effective superpotential withperturbative configurations. The field-dependent gauge couplings can of course be obtaany formulation of the theory. But the closest relation to the higher-dimensional origin oeffective supergravity is realized with supermultipletsL (for the string dilaton multiplet),T(Calabi–Yau volume modulus) andL (for the five-braneS1/Z2 modulus). This is the theordefined by Eq.(4.11) in which tensor calculus leads to the following (inverse squared) gcouplings:

1

g2i

= 1

2

(z0z0(T + T − 2MM)

2C

)3/2

+ τ

2(T + T − 2MM)

[C2 + 4βi C

]

(4.21)+ βi ReT + 1

2

[ε∣∣αM3

∣∣2 − 2τ∑j

(δj − βi γj )MjMj

].

The complex scalarz0 is the lowest component of the compensating multipletS0. In the Poincarétheory, it is a function of the physical scalars chosen to obtain a specific “gravity frame”Einstein frame where the gravity Lagrangian is− 1

2κ24e4R4 corresponds to22

(4.22)2κ24C =

(z0z0(T + T − 2MM)

2C

)−3/2

.

Without branes or threshold corrections the dimensionless field 4κ24C is then the (universal

gauge couplingg2i .

On the other hand, the chiral version of the theory has gauge kinetic functions(4.17)and then:

(4.23)1

g2i

= Re(S + βiT + τ βi S).

The equality of these two expressions is encoded in the duality transformations exchanginmultipletsL andL with S andS:

1

2(S + ¯

S) = 2(T + T − 2Me2AM

)V + 2

∑i

γiMie2AMi,

22 See for instance[13,36].


tions

,

oup-

turningfor

and

ane

tion ofrm in

1

2(S + S) = 1

2

(S0S0(T + T − 2Me2AM)

2V

)3/2

+ 1

2

[ε∣∣αM3

∣∣2 − 2τ∑

i

δiMie2AMi

]

(4.24)+ τ

2

(T + T − 2Me2AM

)V 2,

with V andV as in Eqs.(4.10). The lowest scalar components of these two superfield equashow the equality of(4.21) and (4.23).

The “natural” definition of the dilatonϕ with kinetic Lagrangian(∂µ lnϕ)2/4 is to identify

− ln(S + S − ∆) ←→ − ln(2ϕ)

in the Kähler potential(4.13), i.e.

ϕ = 1

2

(z0z0(T + T − 2MM)

2C

)3/2

(4.25)= ReS − τ

32

[S + ¯S − 4

∑i γiM

iMi]2T + T − 2MM

+ τ∑

i

δiMiMi − 1

2ε∣∣αM3

∣∣2.From now on, we omit charged matter terms, as we expect〈M〉 = 0. In terms of the dilaton

the gauge couplings read then

1

g2i

= ϕ + ReT[τ C2 + 4τ βiC + βi

]

(4.26)= ϕ + ReT

[τ

16

(ReS

ReT

)2

+ τ βi

ReS

ReT+ βi

].

They display a universal23 correction quadratic in the five-brane location, as well as gauge grdependent corrections linear inC or constant. The chiral version has only terms linear inS, T andS: the universal quadratic correction has been absorbed in the non-harmonic redefinitionthe dilatonϕ into ReS. And of course the quadratic term reappears in the Kähler potentialS

[see Eqs.(4.13) and (4.16)].We now restore the summation over several five-branes and split the coefficientsβi according

to βi = β(pl.)i + β

(br.)i

∑I τI since they receive in general contributions from both planes

five-branes.24 Using the identification(4.19), one obtains

(4.27)1

g2i

= ϕ + β(pl.)i ReT + ReT

∑I

τI

[β

(br.)i + 4

κβi

(yI

2π

)+ 1

κ2

(yI

2π

)2].

Notice that since the exchange 1↔ 2 of the plane indices is equivalent to moving the five-brfrom yI to π − yI , we expect

β1 + β2 = − 1

4κ, β

(br.)2 = β

(br.)1 + 2

κβ1 + 1

4κ2.

The next step is to compare these results with the terms obtained from the reducthe topological term and especially with the brane contributions described by the first te

23 I.e. identical for all group factors, all values of indexi.24 The constantsβ(pl.)

i, β

(br.)i

andβi should not depend onI .


rav-ine of

audulusd

of the

ge

Eq.(3.8):

e−14 ∆Lbrane= 1

4

∑I

τI

κ2ImT

[(yI

2π

)2

− yI

2π+ 1

6

]TrF (1)F (1)

(4.28)+[(

yI

2π

)2

− 1

12

]TrF (2)F (2)

.

The terms of order(yI )2 have identical coefficient for both planes. If our effective superg

ity is correct, this contribution should appear in the DBI term appearing in the second lLagrangian(4.11). The vector component of the Chern–Simons multipletΩ is

[Ω]vector≡ Ωµ = 1

8e4εµνρσ ωνρσ + · · ·

and the component expansion of(4.11)includes then

−4e4τI C2I (∂µ ImT )

∑i

Ωµi + · · · .

Integrating by part for constant valuesCI = 〈CI 〉 = yI /(2πκ), we obtain

(4.29)1

4e4

τI

κ2

(yI

2π

)2

ImT∑

i

TrF (i)µν F (i)µν + · · · ,

which fits correctly the quadratic term in(4.28). It is then not needed to perform the Calabi–Yreduction of the five-brane DBI Lagrangian to find the kinetic terms of the five-brane mosuperfield: knowledge of the superfield structure, Eq.(4.11), and of the gauge terms implieby the topological term is sufficient. Similarly, the terms of orderyI and(yI )

0 in the effectivesupergravity and in the reduction of the topological term can be used to find the valuescoefficientsβ(2)

i andβi .The second line of Lagrangian(4.11)indicates that the DBI contribution also includes gau

terms of orderC, which after partial integration read

(4.30)1

κImT

∑I

τI

yI

2π

∑i

βi TrF (i)µν F (i)µν.

By comparison with(4.28), we find

(4.31)β1 = − 1

4κ, β2 = 0.

Finally, comparison of theyI -independent terms in Eq.(4.28)with the first term of(4.8), whichincludes

(4.32)−2∑

i

[βiΩiVT ]D = 1

4ImT

∑i

βi TrF (i)µν F (i)µν + · · · ,

indicates that

(4.33)β(br.)1 = 1

6κ2, β

(br.)2 = − 1

12κ2.

As expected, exchanging planes 1↔ 2 is equivalent toyI ↔ π − yI .


d from

the

o

l, both

r in

Finally, as usual, the coefficientsβ(pl.)i can be read directly from the first line of Eq.(3.6),

which includes contributions to the topological terms arising from the hyperplanes only:

(4.34)β(pl.)i = 2

(4π)4κ2

∑j

dij 〈Ij 〉,

or

(4.35)β(pl.)1 = 1

6(4π)3κ2

(2〈I1〉 − 〈I2〉

), β

(pl.)2 = 1

6(4π)3κ2

(2〈I2〉 − 〈I1〉

).

Notice that

β(pl.)1 + β

(pl.)2 = 1

6(4π)3κ2

(〈I1〉 + 〈I2〉)

vanishes in the minimal embedding without five-brane[8,29].To summarize, in terms of the dilaton, the gauge couplings on both planes, as calculate

the modified topological term, read:

1

g21

= ϕ + β(pl.)1 ReT + 1

κ2ReT

∑I

τI

[(yI

2π

)2

−(

yI

2π

)+ 1

6

],

(4.36)1

g22

= ϕ + β(pl.)2 ReT + 1

κ2ReT

∑I

τI

[(yI

2π

)2

− 1

12

].

A nicer expression reminiscent of Eqs.(2.20)or (2.23)uses the distance from the brane toplanes:

1

g21

= ϕ + β(pl.)1 ReT + 1

4κ2ReT

∑I

τI

[(∆I,2)

2 − 1

3

],

(4.37)1

g22

= ϕ + β(pl.)2 ReT + 1

4κ2ReT

∑I

τI

[(∆I,1)

2 − 1

3

],

where∆I,1 = yI /π , and∆I,2 = 1 − yI /π , as in Eq.(2.18). The contribution of a five-brane tthe gauge couplings on one hyperplane decreases quadratically from a maximum value

1

6κ2τI ReT

when the brane lies on the plane, to a minimal value

− 1

12κ2τI ReT

when the five-brane lies on the opposite plane. For a five-brane in the middle of the intervagauge couplings receive the correction

− 1

48κ2τI ReT .

With however(∆I,i)2 = −∆I,1∆I,2 + ∆I,i (i = 1,2), the term quadratic iny is necessarily

universal and the two (inverse squared) gauge couplings differ only by a contribution lineay:

(4.38)1

g2− 1

g2= ReT

2(4π)3κ2

[〈I1〉 − 〈I2〉 +

∑aI

(1− 2yI

π

)],

1 2 I


e

lethe

logicalntitiesmulti-may

l term

with ∆I,2 − ∆I,1 = 1 − 2yI /π and in terms of the areaaI of the two-cycle wrapped by thfive-brane inK6 [see Eq.(4.6)].

The normalization of the four-formsI4,1 andI4,2 is such that their integrals over a four-cycin K6 are half-integers. SimilaryaI is an intersection number of the two-cycle wrapped bybrane with the four-cycle, in units of the Calabi–Yau volume.25 These statements follow fromthe integrated Bianchi identity verified byG4 and from rewriting Eqs.(3.7) and (3.9)in the form

aI = −i

∫K6

δ(4)(W6,I ) ∧ dz ∧ dz,

〈Ii〉 = −i

∫K6

〈Ii〉 ∧ dzdz.

Eq. (4.38)matches then nicely the idea that a five-brane moved to the hyperplane aty = 0 de-creases the instanton number on this plane, or on the second plane when moved toy = π .

In the chiral version of the theory, the gauge kinetic functions are

f2 = S + β(pl.)2 T − 1

12κ2

∑I

τI T ,

(4.39)f1 = f2 + [β

(pl.)1 − β

(pl.)2

]T + 1

4κ2

∑I

τI [T − κS].

Since

〈ReT 〉 − κ〈ReS〉 = 〈ReT 〉(

1− 2y

π

),

the difference is again Eq.(4.38).

4.2. Discussion of some matter terms

We have seen that the gauge part of the five-brane-induced contributions to the topoterms are due, in the effective supergravity, either to the effect of the modified Bianchi ideon the Dirac–Born–Infeld Lagrangian, or to threshold corrections. Since charged matterpletsM arise from the fields living on the fixed hyperplanes, as do gauge multiplets, weexpect that some or all matter contributions obtained from the reduction of the topologicacan also be derived from the DBI effective Lagrangian.

As an illustration, we will establish that the charged matter term in expression(3.8) arisesfor its part quadratic inyI from the DBI Lagrangian, while the terms linear and constant inyI

originate from threshold corrections. Since the vector component ofV is related to the stringantisymmetric tensorBAB by

(4.40)vµ = − 2π

8e4εµνρσ ∂νBρσ ,

the string-matter coupling term in Eq.(3.8) takes the form

25 We have chosenV6 = 1.


BI

the

erac-

f mat-

e littleed tonder-

g in

− i

8(4π)3

∑I,i

aI

(∆2

I,i − 1

3

)εµνρσ ∂µBνρ

∑m

[Am(i)

(Dσ Am(i)


)]

= − 2i

κ2e4

∑I

τI vµ

(yI

2π

)2[M(DµM) − M(DµM)

]

(4.41)+∑

i

gi (yI )[Mi

(DµMi

) − Mi(DµMi

)].

The functionsgi (yI ) are at most linear inyI :

(4.42)g1(yI ) = − yI

2π+ 1

6, g2(yI ) = − 1

12.

The first term in the r.h.s. of Eq.(4.41) is universal and can clearly be retrieved from the DLagrangian in the second line of theory(4.11)by selecting the matter contribution insideVT ,

(4.43)−2ie4τI C2I vµ

[M(DµM) − M(DµM)

] + · · · ,and identifying as usualC2 with y2

I (2πκ)−2.The second term in the r.h.s. of Eq. (4.41) originates from matter threshold corrections in

supergravity Lagrangian(4.11). The relevant term is:

(4.44)−2ie4τI vµ

∑i

gi(C)[Mi

(DµMi

) − Mi(DµMi

)] + · · · .

Comparison with Eq.(4.41)indicates that

(4.45)γ1 = − 1

κ, γ2 = 0, δ1 = 1

6κ2, δ2 = − 1

12κ2.

Interestingly enough,δi = β(br.)i andγi = 4βi .

Finally, we briefly return to the issue of tracing back the supersymmetric origin of inttions such as expressions(3.11) and (3.12). As already mentioned in Section3, Eq.(3.11)seemsin general hard to cast in a supersymmetric form because of the complicated mixing oter contributions from both hyperplanes. This feature is also present in Eq.(3.12), forbiddingby the same token its supersymmetrization for a generic background. We however havinformation on the nature of four-dimensional matter counter-terms which could be addanomaly-cancelling corrections and could radically change the picture. At this level of ustanding, this discussion cannot be conclusive.

Nevertheless, Eq.(3.11)allows a supersymmetric formulation for the standard embeddinthe gauge group in which chiral matter multiplets only appear on one plane, sayA ≡ A2. Then,Eq.(3.11)reduces to

Lplane= iεµνρσ ω1,µνρ

1

3(4π)5

[λA3∂σ

(λA3) − λA3∂σ

(λA3)]

(4.46)− 1

12(4π)3

∑m

(AmDσ Am − AmDσAm

)(〈I1〉 − 〈I2〉)

,

which extends to the supersymmetric density

(4.47)1

2 6

[Ω1

∣∣αM3∣∣2]

D− 1

2 2

(〈I1〉 − 〈I2〉)[

Ω1Me2AM]D

.
3(4π) κ 3(4π) κ


eshold

np proce-solving

nciplethe

ld. It isg are

matter

den-

te,y then,ions

he

The second term could in principle correspond to the first contribution appearing in the thrcorrection(4.8).

5. Condensation, the non-perturbative superpotential and membrane instantons

The non-perturbative superpotential arises from theF -density in the supergravity Lagrangia(4.3)when some or all gauge fields condense. It can be evaluated using a standard two-stedure: first obtain the effective action for condensates and then eliminate the condensate by(approximately in general) its field equation. Computing the effective action amounts in prito couple the superfieldWW to an external source, integrate the gauge fields and performLegendre transformation exchanging the source field with the (classical) condensate fiewell known that the symmetry content of super-Yang–Mills theory and anomaly-matchinsufficiently restrictive to accurately describe condensation[37].

As usual, we assume that the gauge multiplet which first condenses does not couple tomultipletsM . We then split the gauge group intoG0 × ∏

a Ga , where the simple groupG0 con-denses and matter multiplets only transform under

∏a Ga . The terms involvingG0 gauge fields

in the Lagrangian reduce then to

(5.1)1

4

[(S + β0T + τ β0S)W0W0

]F.

Following Ref.[37], these contributions are simply replaced in the effective action for consates by

(5.2)

1

4

[(S + β0T + τ β0S)U + b0

24π2

U ln

(U

µ3

)− U

]F

+ [S0S0K

(US−3

0 , U S−30

)]D

,

whereU is the (classical) chiral superfield (w = n = 3) describing the gaugino condensa〈U 〉 = 〈λλ〉. The coefficient of the Veneziano–Yankielowicz superpotential is dictated banomaly of the superconformal chiralU(1). It is proportional to the one-loop beta functiob0 = 3C(G0), and the scale parameterµ is the energy at which gauge couplings in express(4.37)are defined. Finally, the leading contribution to the Kähler potentialK is controlled by thescaling dimension (Weyl weight) ofU .

The effective Lagrangian with condensateU can be obtained by collecting all terms in t“microscopic” Lagrangian(4.3), with contributions(5.1) replaced by the effective terms(5.2):

Leff. =[−

S0S0(T + T − 2Me2AM

)3/2(2V )−1/2 − (S + S)V

− τV V 2(T + T − 2Me2AM) + τ

2(S + ¯

S)V V

+ V

ε∣∣αM3

∣∣2 − 2τ∑

i

gi(V )Mie2AMi

+ S0S0K

(US−3

0 , U S−30

)]D

+ 1

4

[∑a

(S + βaT + τ βaS)WaWa

]F

(5.3)

+[S3

0

(c + αM3) + 1

4(S + β0T + τ β0S)U + C(G0)

32π2

U ln

(U/µ3) − U

]F

.


ul-the

ith

e

tion of

-

on-

sev-

sen

As before, vector multipletsV and V are constrained by the field equations of Lagrange mtipliers S andS, which impose modified Bianchi identities. Rewriting their contributions inform

(5.4)

[2S

Σ(V ) + 1

8U + 1

8

∑a

WaWa

− τ S

Σ(V V ) − 1

4β0U − 1

4

∑a

βaWaWa

]F

,

multiplier S implies

(5.5)U = −8Σ(V0), V = V0 − 2∑a

Ωa,

with a real vector multipletV0 (8 bosons+ 8 fermions). Then, multiplierS requires

(5.6)V V = L0 − 2β0V0 + 4∑a

βaΩa,

with a linear multipletL0 andV as given in Eqs.(5.5). These solutions can be compared wEqs.(4.10), which apply before condensation ofW0W0. Clearly, the real vector multipletV0describesL− 2Ω0|cond., i.e. it includes the string coupling linear multipletL and the condensatfield [31]. Similarly, the linear multipletL0 replacesL + 2β0L.

To obtain the non-perturbative superpotential however, one first chooses the formulathe theory with chiral multiplets only. The elimination of vector multipletsV andV is as in theprevious section. Omitting from here on gauge fieldsWa and charged matter fieldsM , the chiralformulation of the effective Lagrangian is

Leff. =[−3

2S0S0e

−K/3 + S0S0K

]D

(5.7)+[S3

0

(c + αM3) + 1

4(S + β0T + τ β0S)U + C(G0)

32π2

U ln

(U/µ3) − U

]F

.

The Kähler potentialK is as in Eqs.(4.13) and (4.16), with full mixing of multipletsS, T andS. To derive the non-perturbative effective potential, neglectK .26 The field equation of the condensate fieldU implies then

(5.8)U = µ3 exp

(− 8π2

C(G0)[S + β0T + τ β0S]

)≡ U,

and the effective superpotential becomes

(5.9)Wnp = c + αM3 − C(G0)

32π2US−3

0 .

This superpotential is thesum of the ‘microscopic’ superpotential and the non-perturbative ctribution of the gauge condensate. The non-perturbative contribution is theexponential of thesum of the string coupling, Calabi–Yau volume and five-brane moduli contributions. Witheral condensates, the non-perturbative piece would be replaced by asum of similar terms over allgauge simple factors which condense.

26 Disregarding the Kähler potentialK is the same as consideringU as a constant background field with value choto extremize the action.


n-

nesm theutions.ot a sumstructurea non-

uctureions..

of

kineticn when-

com-ctablethenitted

To get a qualitative picture of the effect of the five-brane, use Eqs.(4.39) and (4.38)to rewritethe condensate as a function of the five-brane position alongS1, assuming first that the condesate arises on the hyperplane aty = 0:

|U |1st plane= |µ|3 exp

(− 8π2

C(G0)

[ReS + 1

(4π)3κ2ReT

1

3〈I1〉 − 1

6〈I2〉 + 1

3a − y

πa

])

(5.10)= |µ|3 exp

(− 8π2

C(G0)g21

).

If the condensate arises on the hyperplane aty = π :

|U |2nd plane= |µ|3 exp

(− 8π2

C(G0)g22

)

(5.11)= |U |1st plane× exp

(4π2

C(G0)

ReT

(4π)3κ2

[〈I1〉 − 〈I2〉 + a

(1− 2y

π

)]).

The non-perturbative superpotential(5.9) and the condensates(5.10) and (5.11)display thedependence on the five-brane location onS1 expected from explicit calculations of membrainstanton corrections in the four-dimensional effective theory[14,18]. We have obtained thidependence from the analysis of the fundamental Bianchi identity of M-theory and fro(modified) topological term, showing in this way that open membrane instanton contribfind their higher-dimensional origin in anomaly-cancellation in the presence of five-branes

This observation has a second consequence. The non-perturbative superpotential is nof exponential terms generated by gaugino condensates and membrane instantons, awhich is not in any case expected to appear in the effective supergravity. Instead, we findperturbative term which is the exponential of a sum of terms linear in the chiral fields, a strcharacteristic of threshold corrections induced by anomaly-cancellation in higher dimens

In our reduction scheme, the “microscopic” superpotentialc + αM3 is moduli-independentIt is however known that T-duality induces a holomorphic dependence onT compatible with oursupermultiplet structure as described in Lagrangian(5.3). The existence of dual descriptionsmoduliS andS in terms of constrained vector multipletsV andV or in terms of linear multipletsL andL implies that the “microscopic” superpotential cannot depend onS or S.

5.1. The scalar potential

Because of the mixing of the three moduli multipletsS, T and S in the Kähler metric, thescalar potential present in the component expansion of the effective Lagrangian(5.3) is not pos-itive and analysing its vacuum structure is a severe problem. This mixing is due to theterms of the five-brane massless modes, it is unavoidable whenever five-branes, and theeverS, are present.

We may however gain insight by deriving the scalar potential directly in terms of theponents of the constrained vector multiplets. This version of the theory is indeed more trathan the chiral one since the mixing of moduli fields is simpler. The relevant multiplets areT , V , V for the moduli andU for the condensate. Charged matter terms are as before omsince we are interested in vacua where they vanish. In the Einstein frame(4.22), the relationbetween the dilaton(4.25)and the lowest scalar componentC of V is

(5.12)ϕ = 1

4κ2C
4


ined in

he

theoryaray

ing

y

and the scalar potential is eventually expressed in terms of the physical dilaton. As explaEqs.(5.5), eliminatingS with Ωa omitted generates the modified Bianchi identityU = −8Σ(V ),whereV is the vector field describingL − 2Ω0|cond.. To derive the potential, we only need tscalar components ofV andU ,

V = (C,0,H,K,0,0, d), U = (u,0, fU ),

(5.13)u = −4(H − iK), fU = 4d.

Since a non-zero condensateu also switches on the fieldH − iK of the dilaton multipletV ,the gaugino condensate clearly breaks supersymmetry in this sector, as expected in awhere the dilaton couples to gauge fields. EliminatingS defines the five-brane (effective) linemultiplet V V = L0 − 2β0V and, since linear multiplets do not have auxiliary fields, we msimply write

L0 = (CC + 2β0C,0,0,0,0,0,0),

in terms of the lowest scalar componentC of V , when deriving the scalar potential. The resultscalar potential is then a function of the physical scalarsC (the dilaton, see Eq.(5.12)), theS1/Z2modulusC = y/(2πκ), the Calabi–Yau volume modulusT and the gaugino condensateu. It isalso a function of the auxiliary fieldsd , fT andf0 (in the compensating multipletS0) which canbe easily eliminated.

The Kähler potentialK generates a term quadratic ind . We will write the scalar potential brestrictingK to its leading term[37]

(5.14)K(US−3

0 , U S−30

)S0S0 = A(UU)1/3,

with an arbitrary normalisation constantA 0. The scalar potential as a function ofd , C, C, T

andu reads then:

Veff. = −32

9A(uu)−2/3d2 − Bd

+ uu

16C

[3

2

1

4κ24C

+ τ(C + 2β0)2(T + T )

]

+ 2κ24C

κ44(T + T )3

−2

∣∣∣∣W + 1

8κ3

4u

(T + T

2κ24C

)3/2∣∣∣∣2

(5.15)

+ (T + T )2

3

∣∣∣∣WT − 3

T + TW + 1

4κ3

4u(T + T )3/2

(2κ24C)1/2

(β0 + τ

[C2 + 4β0C

])∣∣∣∣2

.

The first two terms arise respectively from the condensate Kähler potential term[S0S0K]D =A[(UU)1/3]D and from the condensateF -density

[Wcond.]F ≡[

1

4(S + β0T + τ β0S)U + b0

96π2

U ln

(U/µ3) − U

]F

.

The coefficientB relates the gaugino condensate fieldu and the gauge couplingg20, as defined

in Eq.(4.26):

(5.16)B = 2

g2+ b0

24π2ln

(uu

µ6

)= 8 Re

∂

∂uWcond..

0


d byial

uplings

g-rbativeeli

ily leadn

e

d

The standard field-theory value of the condensate,

(5.17)|u| = µ3 exp

(−24π2

b0g20

),

is obtained ifB = 0 is part of the scalar potential vacuum equations.In Eq.(5.15), the second line is proportional to|H − iK|2 and the fourth line to|fT |2, and we

have included the possibility of aT -dependent perturbative superpotential, as often implieT -duality, even if our reduction scheme predictsWT = ∂W

∂T= 0. The dependence of the potent

on the five-brane positionC = y/(2πκ) is best understood by defining thedistance ∆c from thebrane to the condensate:

For a condensate on plane 1:∆c = y/π = 2κC;(5.18)For a condensate on plane 2:∆c = 1− y/π = 1− 2κC.

Using then the values of the threshold coefficients found in Eqs.(4.31) and (4.33), we find inboth cases:

β0 + τ[C2 + 4β0C

] = β(pl.)0 + τ

4κ2

[(1− ∆c)

2 − 1

3

],

(5.19)(C + 2β0)2 = 1

4κ2(1− ∆c)

2.

These results agree with the dependence on the five-brane location found in gauge co(4.37).

At this stage, we have two options. We may neglect the Kähler potentialK and assumeA = 0.Then, the auxiliary fieldd imposes the field equationB = 0 and the correct value of the gauino condensate. This procedure is equivalent to the derivation of the effective non-pertusuperpotential(5.9). The field equationB = 0 allows to eliminateu and to express the effectivpotential, which does not include the first line in expression(5.15), as a function of the moduscalarsC, C andT only.

Instead, with a non-zero Kähler potentialK (i.e. withA > 0), solving for the auxiliaryd turnsthe first two terms of the scalar potential(5.15)into

(5.20)9

128

B2

A(uu)2/3,

and a generic (non-supersymmetric) stationary point of the potential does not necessarto B = 0 and to the standard gaugino condensate(5.17). But sinceB appears quadratically ithe potential, the same stationary points withB = 0 would exist in both casesA = 0 andA > 0.Notice that the condensate term(5.20)can also be written

1

2(Kuu)

−1(

Re∂

∂uWcond.

)2

, Kuu = ∂2

∂u∂uA(uu)1/3.

This is the potential term due to the auxiliary fieldfu of the condensate chiral superfieldU , withnon-standard Weyl weightw = 3. The imaginary part of∂

∂uWcond. does not contribute to th

potential because of the constraintU = −8Σ(V ) [31].A complete analysis of the stationary values of the scalar potential(5.15)cannot be performe

analytically. In the absence of five-branes, the potential can be written in the form


. For

ne

ot

eorysed aodulusellingheorythesewith ac-d ontoold cor-ential

can beo-cyclerplaneantonutionslculusing of

V = 1

2(Kuu)

−1(

Re∂

∂uWcond.

)2

+ 1

κ24

[(2κ2

4C)2|fS |2 + 3(T + T )−2|fT |2] − 3

κ44

eK|W |2,

in terms of the Kähler potentialK = − ln(S + S)−3 ln(T + T ) with diagonal metric. A relativelysimple study of the stationary points of the potential with for instanceB = 0 can be performedas a function of the auxiliary fieldsfS andfT of the chiral dilatonS and the volume modulusT ,respectively.

But the introduction of the five-brane mode leads to mixings of the chiral superfieldsinstance, according to the second superfield Eq.(4.24), the auxiliary field in the chiral dilatonmultipletS reads

fS = κ−14

(2κ2

4C)−1/2

(T + T )−3/2[W − 1

4

(T + T

2κ24C

)3/2

κ34 u

]

(5.21)+ τ

[C2fT − 1

2(T + T )

(C2 + 2β0C

)(2κ2

4C)−1

κ24 u

].

The second term is due to the five-brane and it involves the auxiliary fieldfT , which is propor-tional to the last line in the potential(5.15). Similarly, the auxiliary component of the five-bramultiplet S is

(5.22)fS

= 4CfT − (T + T )(C + 2β0)(2κ2

4C)−1

κ24 u.

Mixings of the auxiliary fields then arise wheneverC = 0, i.e. whenever the five-brane does nlie on the fixed hyperplane aty = 0.

6. Conclusion

In this paper, we have studied the Calabi–Yau reduction of the low-energy limit of M-thon the intervalS1/Z2, with five-branes aligned to preserve four supercharges. We have ufully consistent, four-dimensional supergravity and superfield setup and included the mfield describing five-brane fluctuations in the interval direction. The gauge anomaly-canctopological term is modified as a consequence of the five-brane contributions to M-tBianchi identities. We have derived the new four-dimensional interactions induced byfive-brane modifications and shown that they lead to new gauge threshold correctionscalculable dependence on the five-brane position alongS1. In particular, these threshold corretions fit nicely the change in the instanton number expected when a five-brane is moveone of the fixed hyperplanes. Of course, when gauge condensation occurs, these threshrections explicitly appear in the effective non-perturbative superpotential, with an expondependence of the five-brane location.

The same five-brane-dependent contributions to the non-perturbative superpotentialobtained from a different perspective. It is expected that open membranes wrapping a twin the Calabi–Yau threefold and extending from a five-brane to a ten-dimensional hypegenerate, in the four-dimensional effective field theory, instanton-like contributions. Instcalculus allows to explicitly compute these instanton corrections and the resulting contribto the non-perturbative effective superpotential. Strictly speaking however, instanton caonly applies in specific limits, which in the case under scrutiny restricts the understand


. It isds im-ginaterbative

sup-ropean

duliing

e

s in the

i–Yau

the global structure of the superpotential and of the interplay of the various moduli fieldsprecisely here that our effective supergravity Lagrangian, as derived from M-theory, adportant new information. In particular, since membrane instanton corrections actually orifrom threshold corrections related to ten-dimensional anomaly cancellation, the non-pertusuperpotential is the exponential of a sum of terms linear in moduli chiral fields.

Acknowledgements

We thank Adel Bilal and Claudio Scrucca for helpful discussions. This work has beenported by the Swiss National Science Foundation and by the Commission of the EuCommunities under contract MRTN-CT-2004-005104.

Appendix A. Conventions and notations

Our conventions are as in Refs.[26] and[25]. We use theupstairs pictureM4 × K6 × S1/Z2,where theS1 coordinate isx10 = yR, with a (2π)-periodic angular variabley. We use−π <

y π when explicit values are needed because of the natural action ofZ2 in this interval. Ourindices convention for theM4 × K6 × S1/Z2 reduction is

xM = (xA,yR

) = (xµ, zi, zk, yR

), M = 0, . . . ,10, A = 0, . . . ,9, i, k = 1,2,3.

For bulk moduli, we use the terminology familiar from string compactifications: the mos = ReS andt = ReT with Kähler potential(3.1)are respectively related to the dilaton (or strcoupling) and to the Calabi–Yau volume. This convention follows from the metric

(A.1)ds211 = e−2φ/3g

(10)AB dxA dxB + e4φ/3 dy2,

which defines the string frame and the string couplinge−2φ with R = e2φ/3, together with

(A.2)g(10)AB dxA dxB = gµν dxµ dxν + V 1/3δij dzi dzj

which defines the Calabi–Yau volume. Rescalinggµν to the four-dimensional Einstein framleads to

(A.3)ds211 = e4φ/3[V −1gµν dxµ dxν + dy2] + e−2φ/3V 1/3δij dzi dzj .

Comparison with the standard eleven-dimensional metric used to diagonalize kinetic termfour-dimensionalN = 1 supergravity Lagrangian,

(A.4)ds211 = s−2/3[t−1gµν dxµ dxν + t2 dy2] + s1/3δij dzi dzj ,

leads to the identifications

(A.5)(ReT )3 = V, ReS = V e−2φ.

Hence,t is the volume modulus whiles is the dilaton or string coupling modulus.The terminology often used in the context of M-theory defines instead another Calab

volumeV in units specified by the metric(A.3), with then

V ≡ V e−2φ = ReS, (ReT )3 = V e2φ = V R3.

It seemingly exchanges the respective roles of the bulk moduli.


ald

some

ti-

In order to avoid duplication of contributions due toZ2 periodicity, our eleven-dimensionsupergravity action and Green–Schwarz terms are multiplied by 1/2 with respect to standarconventions in use forM11:

(A.6)LC.J.S. +LG.S. = 1

4κ2

[eR − 1

2G4 ∧ ∗G4 − 1

6C3 ∧ G4 ∧ G4

]− T2

4πG4 ∧ X7,

with membrane tensionT2 = 2π(4πκ2)−1/3, and

(A.7)dX7 = X8 = 1

(2π)34![

1

8trR4 − 1

32

(trR2)2

].

To respectZ2 symmetry, we assume that a five-brane with world-volume located aty0 has aZ2-mirror at −y0. The Dirac distribution transverse to its world-volumeM4 ⊂ W6 ⊂M4 × K6 is then defined by the condition

(A.8)∫

M4×K6×S1/Z2

I6 ∧ δ(5)(W6) = 2∫W6

I6

for any 6-formI6, since it takes both copies into account.The membrane and five-brane tensionsT2 andT5 are related by the Dirac–Zwanziger quan

zation condition

2κ2T2T5 = 2π

and also by[38]

(T2)2 = 2πT5.

We then express all constants in terms ofκ2, with

(A.9)(4πκ2)1/3 = 2π

T2= 2κ2T5.

With these conventions, the Bianchi identity is

(A.10)dG4 = 4π

T2

(1

2

∑I

qI δ(5)(W6,I ) −

∑i

δiI4,i

),

where the indexI labels theZ2-symmetric pairs of five-branes and the chargeqI is +1 for afive-brane,−1 for an anti-five-brane.

One subtlety when integrating the Bianchi identity is that one cannot find aZ2-odd functionε(y) such thatdε = δ(y − y0) dy. As in Ref.[25], we then usey, y0 ∈]−π,π] and

εy0(y) = sgn(y − y0) − y − y0

π, dεy0(y) =

(2δ(y − y0) − 1

π

)dy,

ε1(y) = ε0(y) = sgn(y) − y

π, dε1(y) = 2δ1 − 1

πdy,

(A.11)ε1(y) = επ (y) = sgn(y − π) − y − π

π, dε2(y) = 2δ2 − 1

πdy.

The sign function is

(A.12)sgn(r) = rif r = 0, sgn(0) = 0, sgn(r) = −sgn(−r).
|r|


brane

With this definition,ε1 andε2 are odd functions whileεy0(−y) = −ε−y0(y). The function

(A.13)εy0(y) + ε−y0(y) = sgn(y − y0) + sgn(y + y0) − 2y

π

(0< y0 < π ) is then odd with

(A.14)d[εy0(y) + ε−y0(y)

] = 2(δ(y − y0) + δ(y + y0)

)dy − 2dy

π.

This function is useful to insert five-brane sources in the Bianchi identity. Since the five-world-volumesW6,I are of the formM4 × C2 (C2 a holomorphic cycle inK6), W6,I is locatedaty = yI with a “Z2-mirror five-brane” at−yI . We then use

δ(5)(W6,I ) = (δ(y − yI ) + δ(y + yI )

)dy ∧ δ(4)(W6,I ),

(A.15)d([

εyI(y) + ε−yI

(y)]δ(4)(W6,I )

) = 2δ(5)(W6,I ) − 2

πdy ∧ δ(4)(W6,I ),

to integrate five-brane contributions to the Bianchi identity.

References

[1] E. Witten, Nucl. Phys. B 471 (1996) 135, hep-th/9602070.[2] P. Horava, E. Witten, Nucl. Phys. B 460 (1996) 506, hep-th/9510209;

P. Horava, E. Witten, Nucl. Phys. B 475 (1996) 94, hep-th/9603142.[3] H.P. Nilles, M. Olechowski, M. Yamaguchi, Phys. Lett. B 415 (1997) 24, hep-th/9707143;

H.P. Nilles, M. Olechowski, M. Yamaguchi, Nucl. Phys. B 530 (1998) 43, hep-th/9801030.[4] I. Antoniadis, M. Quirós, Phys. Lett. B 416 (1998) 327, hep-th/9707208.[5] A. Lukas, B.A. Ovrut, D. Waldram, Nucl. Phys. B 532 (1998) 43, hep-th/9710208;

A. Lukas, B.A. Ovrut, D. Waldram, Phys. Rev. D 57 (1998) 7529, hep-th/9711197.[6] P. Horava, Phys. Rev. D 54 (1996) 7561, hep-th/9608019;

J.R. Ellis, Z. Lalak, S. Pokorski, W. Pokorski, Nucl. Phys. B 540 (1999) 149, hep-ph/9805377;Z. Lalak, S. Pokorski, S. Thomas, Nucl. Phys. B 549 (1999) 63, hep-ph/9807503;J.R. Ellis, Z. Lalak, W. Pokorski, Nucl. Phys. B 559 (1999) 71, hep-th/9811133.

[7] E. Witten, Phys. Lett. B 155 (1985) 151.[8] J.-P. Derendinger, L.E. Ibáñez, H.P. Nilles, Nucl. Phys. B 267 (1986) 365.[9] M.J. Duff, J.T. Liu, R. Minasian, Nucl. Phys. B 452 (1995) 261, hep-th/9506126.

[10] K. Becker, M. Becker, A. Strominger, Nucl. Phys. B 456 (1995) 130, hep-th/9507158;M. Bershadsky, C. Vafa, V. Sadov, Nucl. Phys. B 463 (1996) 420, hep-th/9511222.

[11] E. Cremmer, B. Julia, J. Scherk, Phys. Lett. B 76 (1978) 409.[12] G.W. Gibbons, P.K. Townsend, Phys. Rev. Lett. 71 (1993) 3754, hep-th/9307049;

D.M. Kaplan, J. Michelson, Phys. Rev. D 53 (1996) 3474, hep-th/9510053.[13] J.-P. Derendinger, R. Sauser, Nucl. Phys. B 598 (2001) 87, hep-th/0009054.[14] G.W. Moore, G. Peradze, N. Saulina, Nucl. Phys. B 607 (2001) 117, hep-th/0012104.[15] A. Lukas, B.A. Ovrut, D. Waldram, Phys. Rev. D 59 (1999) 106005, hep-th/9808101.[16] A. Lukas, B.A. Ovrut, D. Waldram, JHEP 9904 (1999) 009, hep-th/9901017.[17] A. Strominger, Phys. Lett. B 383 (1996) 44, hep-th/9512059;

P.K. Townsend, Nucl. Phys. B (Proc. Suppl.) 58 (1997) 163, hep-th/9609217;P. Brax, J. Mourad, Phys. Lett. B 416 (1998) 295, hep-th/9707246;Z. Lalak, A. Lukas, B.A. Ovrut, Phys. Lett. B 425 (1998) 59, hep-th/9709214.

[18] E. Lima, B.A. Ovrut, J. Park, R. Reinbacher, Nucl. Phys. B 614 (2001) 117, hep-th/0101049;E. Lima, B.A. Ovrut, J. Park, Nucl. Phys. B 626 (2002) 113, hep-th/0102046.

[19] G. Curio, A. Krause, Nucl. Phys. B 643 (2002) 131, hep-th/0108220.[20] E.I. Buchbinder, B.A. Ovrut, Phys. Rev. D 69 (2004) 086010, hep-th/0310112;

E.I. Buchbinder, Phys. Rev. D 70 (2004) 066008, hep-th/0406101.


[21] E.I. Buchbinder, Nucl. Phys. B 711 (2005) 314, hep-th/0411062;K. Becker, M. Becker, A. Krause, Nucl. Phys. B 715 (2005) 349, hep-th/0501130.

[22] E.I. Buchbinder, Nucl. Phys. B 728 (2005) 207, hep-th/0507164.[23] J.-P. Derendinger, R. Sauser, Nucl. Phys. B 582 (2000) 231, hep-th/0003078.[24] M.B. Green, J.H. Schwarz, Phys. Lett. B 149 (1984) 117.[25] A. Bilal, J.-P. Derendinger, R. Sauser, Nucl. Phys. B 576 (2000) 347, hep-th/9912150.[26] A. Bilal, S. Metzger, Nucl. Phys. B 675 (2003) 416, hep-th/0307152.[27] P.K. Townsend, Phys. Lett. B 373 (1996) 68, hep-th/9512062;

O. Aharony, Nucl. Phys. B 476 (1996) 470, hep-th/9604103;E. Bergshoeff, M. de Roo, T. Ortin, Phys. Lett. B 386 (1996) 85, hep-th/9606118.

[28] E. Witten, Nucl. Phys. B 463 (1996) 383, hep-th/9512219;E. Witten, J. Geom. Phys. 22 (1997) 103, hep-th/9610234.

[29] L.E. Ibáñez, H.P. Nilles, Phys. Lett. B 169 (1986) 354.[30] S. Ferrara, J. Wess, B. Zumino, Phys. Lett. B 51 (1974) 239;

W. Siegel, Phys. Lett. B 85 (1979) 333.[31] C.P. Burgess, J.-P. Derendinger, F. Quevedo, M. Quirós, Phys. Lett. B 348 (1995) 428, hep-th/9501065;

C.P. Burgess, J.-P. Derendinger, F. Quevedo, M. Quirós, Ann. Phys. 250 (1996) 193, hep-th/9505171.[32] S. Cecotti, S. Ferrara, M. Villasante, Int. J. Mod. Phys. A 2 (1987) 1839.[33] T. Kugo, S. Uehara, Nucl. Phys. B 226 (1983) 49;

T. Kugo, S. Uehara, Nucl. Phys. B 222 (1983) 125.[34] J.-P. Derendinger, S. Ferrara, C. Kounnas, F. Zwirner, Nucl. Phys. B 372 (1992) 145.[35] J.-P. Derendinger, L.E. Ibáñez, H.P. Nilles, Phys. Lett. B 155 (1985) 65;

M. Dine, R. Rohm, N. Seiberg, E. Witten, Phys. Lett. B 156 (1985) 55.[36] J.P. Derendinger, F. Quevedo, M. Quirós, Nucl. Phys. B 428 (1994) 282, hep-th/9402007.[37] G. Veneziano, S. Yankielowicz, Phys. Lett. B 113 (1982) 231.[38] S.P. de Alwis, Phys. Lett. B 392 (1997) 332, hep-th/9609211.

llinear

orrec-, but then of theactoriza-de theessed

a-f semi-izationgs from


Heavy-to-lightB meson form factors at large recoilenergy—spectator-scattering corrections

M. Benekea,∗, D. Yangb

a Institut für Theoretische Physik E, RWTH Aachen, D-52056 Aachen, Germanyb Department of Physics, Nagoya University, Nagoya 464-8602, Japan

Received 6 September 2005; accepted 29 November 2005


Abstract

We complete the investigation of loop corrections to hard spectator-scattering in exclusiveB meson tolight meson transitions by computing the short-distance coefficient (jet-function) from the hard-coscale. Adding together the two coefficients from matching QCD→ SCETI → SCETII , we investigate thesize of loop effects on the ratios of heavy-to-light meson form factors at large recoil. We find the ctions from the hard and hard-collinear scales to be of approximately the same size, and significantperturbative expansions appear to be well-behaved. Our calculation provides a non-trivial verificatiofactorization arguments. We observe considerable differences between the predictions based on ftion in the heavy-quark limit and current QCD sum rule calculations of the form factors. We also incluhard-collinear correction in theB → ππ tree amplitudes, and find an enhancement of the colour-suppramplitude relative to the colour-allowed amplitude. 2005 Elsevier B.V. All rights reserved.

1. Introduction

The matrix elements of flavour-changing currentsqΓib are important strong interaction prameters in low-energy weak-interaction processes. The strong interaction dynamics oleptonic B decays is encoded in these form factors. They are also inputs to the factorformulae for hadronic two-bodyB decays[1] and radiative decays[2]. A better understandinof such quantities improves the accuracy of the extraction of the CKM matrix parameter

* Corresponding author.E-mail address:[email protected](M. Beneke).




M. Beneke, D. Yang / Nuclear Physics B 736 (2006) 34–81 35

Thus ef-attice

is well

onsidermen-ation-ulated

umma-

ofsgmorersn,notse the

n

ce, thetions

e

eansionrrent

ction tousthe

experimental data, and of searches for new phenomena in flavour-changing processes.forts are being made to compute the form factors with different methods including QCD lsimulations[3], light-cone QCD sum rules[4] and quark models[5].

It is also interesting to investigate these form factors in the heavy-quark expansion. Itknown that allB → D(∗) form factors reduce to a single (Isgur–Wise) function[6] up to cal-culable short-distance corrections at leading order in this expansion. In this paper we ctransitions ofB mesons to light mesons in the large-recoil regime, where the light meson motum is parametrically of order of the heavy-quark mass. In this regime a similar simplificapplies to heavy-to-light form factors[7]: the three (seven) independentB → pseudoscalar (vector) meson form factors reduce to one (two) function(s) up to corrections that can be calcin the hard-scattering formalism at leading order in the heavy-quark expansion[8]. The differentform factors can therefore be related in a systematic way. The factorization formula that srizes these statements reads[8]

(1)FB→Mi (E) = Ci(E)ξa(E) +

∞∫0

dω

ω

1∫0

dv Ti(E; lnω,v)φB+(ω)φM(v)

with E the energy of the light mesonM , ξa(E) the single non-perturbative form factor (onethe two form factors whenM is a vector meson), andφX the light-cone distribution amplitudeof theB meson and the light meson. The short-distance coefficientsCi and the hard-scatterinkernel Ti can be calculated in perturbation theory. The heavy-to-light form factors arecomplicated than both, theB → D(∗) form factors and light–light meson transition form factoat large momentum transfer. Contrary to the case ofB → D(∗), a spectator-scattering correctiothe second term on the right-hand side of(1), appears. On the other hand, the form factor canbe expressed in terms of a convolution of light-cone distribution amplitudes alone, becaucorresponding convolution integrals are dominated by endpoint singularities[9]. In (1) thesecontributions are factored into the functionξa(E).1 The factorization formula(1) has been showto be valid to all orders in perturbation theory[12] (see also[13,14]) in the framework of soft-collinear effective theory (SCET)[15–17]. In particular, since the two relevant short-distanscalesmb and(mbΛ)1/2 (Λ is the characteristic scale of QCD) can be separated in SCETshort-distance coefficientsTi pertaining to spectator-scattering are represented as convoluC

(B1)i J with the two factors associated with the two different scales.In the limit that not only power corrections inΛ/mb but also radiative corrections in th

strong couplingαs are neglected, the second term on the right-hand side of(1) is absent, andparameter-free relations between ratios of form factors follow[7]. Theαs contributions to(1)have been computed in[8], and the spectator-scattering termTi has been found to dominate thcorrection. This motivates an investigation of the subsequent term in the perturbative expof Ti . Since the leadingαs term is due to a tree diagram with gluon exchange between the cuquarks and the spectator antiquark, this amounts to the computation of the 1-loop correspectator-scattering. SinceTi = C

(B1)i Ja , the calculation splits into two parts. In a previo

paper[18] (see also[19]) we reported the first part of the calculation which consisted of1-loop correction to the coefficientsC(B1)

i originating from the hard scalemb. In this paper

1 The statement that the endpoint contributions are not calculable is challenged in the PQCD approach[10], whichassumes that Sudakov resummation renders them perturbative. This point is critically examined in[11]. We also notehere that our notationξa(E) does not show the dependence of the form factor on the nature of the mesonM .

36 M. Beneke, D. Yang / Nuclear Physics B 736 (2006) 34–81

Hills quiteethe

ationetail

aleT

nts ofform

s)ctionermsf-

withto

ee jet-in

nfic toperties

e

ined inma-

riving

ring is

on the

we complete the calculation with the 1-loop computation of the “jet-functions”Ja originatingfrom the “hard-collinear” scale(mbΛ)1/2. The jet-functions have also been computed byet al.[19,20]. Nevertheless, an independent calculation is useful, since the computation iinvolved and a comparison showed that the result of[20] was originally not given in a schemconsistent with theMS definition of the light-meson light-cone distribution amplitude (seediscussion in[19,20]). Furthermore, the numerical impact of these calculations on the relbetween form factors and other observables inB decays has not yet been discussed in any din the literature.

The organization of the calculation of the short-distance coefficientsCi andTi follows closelythe derivation of the factorization formula in[12]. In a first step, the effects from the hard scmb are computed and QCD is matched to an intermediate effective theory, called SCEI. InSCETI the termξa and the hard-scattering term are naturally defined by the matrix elemetwo distinct operator structures, the so-called A- and B-type operators. At this step, thefactors can be represented as

(2)FB→Mi (E) = Ci(E)ξa(E) +

∫dτ C

(B1)i (E, τ)Ξa(τ,E).

The point to note here is that the three (seven) form factors of aB → P (B → V ) transition canbe expressed in terms of one (two) form factor(s)ξa(E) and one (two) non-local form factor(Ξa(τ,E). A number of relations between form factors emerge already at this stage. In Se2we define the SCETI operator basis, and express the QCD heavy-to-light form factors in tof the SCETI hadronic matrix elements, which leads to(2). All the required short-distance coeficients of the SCETI operators can be inferred from[18,19].

Eq. (2) is useful only to a limited extent, because it introduces the form factorsΞa(τ,E),which depend on two variables. However, it has been shown that, contrary to theξa(E), theΞa(τ,E) can be factorized further into a convolution of light-cone distribution amplitudesa hard-scattering kernel (jet-function)[12]. This amounts to performing a second matchingSCETII , in which the effects at the hard-collinear scale(mbΛ)1/2 are computed. This is donin Section3. Here we discuss in detail the 1-loop calculation and renormalization of thfunctionsJa that follow from representing the SCETI matrix element of the B-type operatorsthe form

(3)Ξa(τ,E) = 1

4

∞∫0

dω

1∫0

dv Ja(τ ;v,ω)fBφB+(ω)fMφM(v).

Combining this with(2) we obtain the spectator-scattering term in(1). The calculation is done idimensional regularization which requires dealing with evanescent Dirac structures specid

dimensions. As will be discussed, a subtlety arises due to the fact that the factorization proof SCETII require a specific choice of reduction scheme. Together withJa we also determine thanomalous dimensions of the B-type operators confirming the results of[20].

The detailed numerical analysis of the corrections from the two matching steps is contaSections4 and 5. In addition to the next-to-leading order correction we also include the sumtion of formally large logarithms from the ratio of the hard and hard-collinear scale by dea renormalization group improved expression for the coefficient functionsC

(B1)i . From the size

of the 1-loop correction we conclude that the perturbative calculation of spectator-scatteunder reasonable control despite the comparatively low scale of order(mbΛ)1/2 ∼ 1.5 GeV.The combined hard and hard-collinear 1-loop correction is about (50–70)% depending


ronico-bodyformnsider

oweakays toressednching

malism

y,howsheavy-torsreasonfficient

ts

observable. This is also of interest in the context of QCD factorization calculations of hadB decays, since the same jet-function enters the spectator-scattering contributions to twdecays[21]. Section5 is devoted to a discussion of the symmetry-breaking effects on thefactor ratios and a comparison of these ratios to QCD sum rule calculations. We then cothe tensor-to-(axial-)vector form factor ratios that appear in electromagnetic and electrpenguin decays, and the numerical impact of our jet-function calculation on hadronic dectwo pions. Here we find that the new contribution increases the ratio of the colour-suppto the colour-allowed tree decay amplitude, which leads to a better description of the brafraction data. We conclude in Section6.

In Appendix A we summarize the short-distance coefficientsC(B1)i (E, τ) relevant to(2).

Some of the convolution integrals of the jet-functions and the coefficientsC(B1)i (E, τ) needed

for the numerical analysis of the spectator-scattering term are collected inAppendix B.

2. Heavy-to-light form factors in SCETI

Our first task is to express the QCD form factors in terms of matrix elements of SCETI currentsand the corresponding short-distance coefficients. We use the position-space SCET forand the notation of[12,16,18]to which we refer for further details. The “collinear” fieldsξandAc that appear in this section describe both, hard-collinear (virtualitymbΛ) and collinear(virtuality Λ2) modes. The reference vectorsv, n∓ are defined such thatv2 = 1, n2− = n2+ = 0,n−n+ = 2. Except for Section2.1we adopt a frame of reference wheren−v = 1 andv = (n− +n+)/2. In scalar products ofn−, n+ with other vectors we omit the scalar-product “dot”.

2.1. Operator basis

The relevant terms in the SCETI expansion of a heavy-to-light currentψΓiQ read[12,14,18]

(ψΓiQ)(0) =∫

ds∑j

C(A0)ij (s)O

(A0)j (s;0) +

∫ds

∑j

C(A1)ijµ (s)O

(A1)µj (s;0)

(4)+∫

ds1 ds2

∑j

C(B1)ijµ (s1, s2)O

(B1)µj (s1, s2;0) + · · · ,

where

O(A0)j (s;x) ≡ (ξWc)(x + sn+)Γ ′

j hv(x−) ≡ (ξWc)sΓ′j hv,

O(A1)jµ (s;x) ≡ (

ξ i←−D⊥cµ(in−vn+

←−Dc)

−1Wc

)sΓ ′

j hv,

(5)O(B1)jµ (s1, s2;x) ≡ 1

mb

(ξWc)s1

(W†

c iD⊥cµWc

)s2

Γ ′j hv,

and si ≡ simb/n−v. Since the collinear fieldsξ andAc describe modes of different virtualitno simpleΛ/mb-scaling rules apply to these fields. The power-counting argument that sthat the three types of operators contribute to the form factors at leading power in thequark expansion has been given in[12]. The main difference between the two types of operais their dependence on position arguments. The B-type operators are trilocal, and for thisare sometimes also referred to as “three-body” operators. The 1-loop corrections to the coefunctions of the A-type currents have been calculated in[15,18], to those of the B-type currenin [18,19].


nts intld insuchg

ading

The basis(5) is motivated by the simple expressions of the tree-level matching coefficiethis basis[16]. However, the analysis of[12] shows thatO(A1)

jµ andO(B1)jµ are operators relevan

at leading power in the 1/mb-expansion only because of the transverse collinear gluon fiethe covariant derivativeD⊥c. It is therefore advantageous to perform a basis redefinitionthat the transverse collinear gluon field appears only inO

(B1)jµ . This can be done by replacin

O(A1)jµ by

(6)(ξWc)si←−∂ ⊥µ(in−vn+

←−∂ )−1Γ ′

j hv,

which is the choice that has been adopted in[19,20]. The redefinition involves the identity(ξ i

←−D⊥cµ(in−vn+

←−Dc)

−1Wc

)sΓ ′

j hv = (ξ i←−D⊥cµWc)s

1

in−vn+←−∂

Γ ′j hv

= (ξWc)si←−∂ ⊥µ

in−vn+←−∂

Γ ′j hv − (ξWc)s

(W†

c iD⊥cµWc

)s

1

in−vn+←−∂

Γ ′j hv

(7)= (ξWc)si←−∂ ⊥µ

in−vn+←−∂

Γ ′j hv − i

∞∫−∞

drθ(r − s)

mb

(ξWc)r(W†

c iD⊥cµWc

)rΓ ′

j hv.

The second term modifiesC(B1)ijµ (s1, s2) in the new basis by an amount proportional toC

(A1)ijµ (s).2

In the new basis only the A0- and B-type operators contribute to the form factors at lepower in the 1/mb-expansion. Our new basis of operators for a given Dirac structureΓi is:

• Scalar currentJ = ψQ:

J (A0) = (ξWc)

(1− i

←−/∂ ⊥

in+←−∂

/n+2

)hv,

(8)J (B1) = 1

mb

(ξWc)[W†

c i/D⊥cWc

]hv.

• Vector currentJµ = ψγµQ:

J (A0)1–2µ = (ξWc)

(1− i

←−/∂ ⊥

in+←−∂

/n+2

)γµ, vµhv,

J (A0)3µ = (ξWc)

(1− i

←−/∂ ⊥

in+←−∂

/n+2

)n−µ

n−vhv + 2

n−v(ξWc)

i←−∂ µ⊥

in+←−∂

hv,

J (B1)1–3µ = 1

mb

(ξWc)[W†

c i/D⊥cWc

]vµ,

n−µ

n−v, γµ⊥

hv,

(9)J (B1)4µ = 1

mb

(ξWc)γµ⊥[W†

c i/D⊥cWc

]hv.

• Tensor currentJµν = ψiσµνQ:

J (A0)1–2µν = (ξWc)

(1− i

←−/∂ ⊥

in+←−∂

/n+2

)γ[µγν], v[µγν]hv,

2 In momentum space the generic modification isC(B1)new (E, τ) = C

(B1)old (E, τ) + mb/(2E)C

(A1)old (E), see alsoAppen-

dix A.


ept

n

ns areo not

r in

(vector)

r currenteof

ctors.

J (A0)3–4µν = (ξWc)

(1− i

←−/∂ ⊥

in+←−∂

/n+2

)n−[µγν]

n−v,n−[µvν]

n−v

hv

+ 2

n−v(ξWc)

i←−∂ [µ⊥

in+←−∂

γν], vν]hv,

J (B1)1–2µν = 1

mb

(ξWc)[W†

c i/D⊥cWc

]v[µγν⊥],

n−[µγν⊥]n−v

hv,

J (B1)3–4µν = 1

mb

(ξWc)

v[µγν⊥],

n−[µγν⊥]n−v

[W†

c i/D⊥cWc

]hv,

J (B1)5–6µν = 1

mb

(ξWc)[W†

c i/D⊥cWc

]n−[µvν]n−v

, γ[µ⊥γν⊥]hv,

(10)J (B1)7µν = 1

mb

(ξWc)γ[µ⊥γν⊥][W†

c i/D⊥cWc

]hv + J (B1)6

µν .

Herea[µbν] = aµbν − aνbµ. The operatorJ (B1)7µν vanishes in four dimensions, but must be k

since we regularize dimensionally.

Here we dropped the position indicess1,2 which should be clear from(5). We also droppedthe operators involving explicit factors of positionxµ, which come from the multipole expansio(see[16]), since we can always work with the QCD currents atx = 0. The choice of the A0operators is identical to that of[20], but the basis of B-operators is slightly different. As in[20]we combined the A1-operators with the A0-operators using that their coefficient functiorelated[18,22]. Since the A1-operators without the transverse hard-collinear gluon field dcontribute to the form factors at leading power[12], these extra terms to theJ (A0) will not beconsidered in the following. The SCET representation of the QCD currentJX(0) is then

(11)JX(0) =∑

i

C(A0)iX J

(A0)iX +

∑k

C(B1)kX J

(B1)kX + · · · ,

which defines the coefficient functions for the scalar (X = S), pseudoscalar(P ), vector (V ),axial-vector(A) and tensor(T ) currents. The star-product of coefficient function and operatoposition space is a convolution over the argumentssi as in(4).

The basis of the pseudoscalar (axial-vector) operators can be inferred from the scalarbasis by the replacement(ξWc) → (ξWc)γ5. In a renormalization with anticommutingγ5 (asadopted in[18] and in this paper), the short-distance coefficientsC

(A0)iX , C

(B1)kX of the scalar

and pseudoscalar current are then equal, as are those of the vector and the axial-vectoJµ5 = ψγ5γµQ (note the order ofγ5 andγµ). In Appendix Awe give the transformation of thmomentum-space coefficient functions calculated in[18] to the new basis. For the remainderthe paper we adopt the frame wheren−v = 1 andv = (n− + n+)/2.

2.2. Definition of the QCD form factors

The matrix elements of the QCD currents are decomposed into Lorentz-invariant form faFollowing the conventions of[8] the independent form factors are

⟨P(p′)

∣∣qγ µb∣∣B(p)

⟩ = f+(q2)[pµ + p′µ − m2

B − m2P

2qµ

]+ f0

(q2)m2

B − m2P

2qµ,

q q


i-

te theandof

s-space

men-

(12)⟨P(p′)

∣∣qσµνqνb∣∣B(p)

⟩ = ifT (q2)

mB + mP

[q2(pµ + p′µ) − (

m2B − m2

P

)qµ

]for pseudoscalar mesons, and

⟨V

(p′, ε∗)∣∣qγ µb

∣∣B(p)⟩ = 2iV (q2)

mB + mV

εµνρσ ε∗νp′

ρpσ ,⟨V

(p′, ε∗)∣∣qγ µγ5b

∣∣B(p)⟩

= 2mV A0(q2)ε∗ · q

q2qµ + (mB + mV )A1

(q2)[ε∗µ − ε∗ · q

q2qµ

]

− A2(q2) ε∗ · q

mB + mV

[pµ + p′µ − m2

B − m2V

q2qµ

],

⟨V

(p′, ε∗)∣∣qiσµνqνb

∣∣B(p)⟩ = 2iT1

(q2)εµνρσ ε∗

νp′ρpσ ,⟨

V(p′, ε∗)∣∣qiσµνγ5qνb

∣∣B(p)⟩

= T2(q2)[(m2

B − m2V

)ε∗µ − (

ε∗ · q)(pµ + p′µ)]

(13)+ T3(q2)(ε∗ · q)[

qµ − q2

m2B − m2

V

(pµ + p′µ)]

for vector mesons. We defineq = p − p′ and use the conventionε0123= −1. From now on weneglect terms quadratic in the light meson massesmP,V , but keep linear terms. In this approxmation we can putp′ = En− with E = n+p′/2= (m2

B − q2)/(2mB), andε∗ · n− = 0.

2.3. Definition of the SCETI form factors

Taking into account the quantum numbers of the mesons, it is straightforward to relamatrix elements of the operators defined in(8) to (10) and the corresponding pseudoscalaraxial-vector operators to a few non-vanishing SCETI matrix elements. We first note that onethe si -integrations in(4) can be done explicitly using collinear momentum conservation[12].This allows us to focus on matrix elements of A0-operators withs = 0 and of B-type operatorwith s1 = 0. To see this for the case of the B-type operators, we represent the positioncoefficient functions in terms of

(14)C(B1)ijµ (s1, s2) =

∫dx1

2π

dx2

2πe−i(x1s1+x2s2)C

(B1)ijµ (x1, x2),

where the argumentsxi of the momentum-space coefficient functions correspond to the motum fractionsxi = n+p′

i/mb of the collinear building blocks(ξWc)s1 and(W†c iD

µ⊥cWc)s2 of the

current operator. Then with(4) and (5)we obtain

⟨M(p′)

∣∣ ∫ ds1 ds2 C(B1)ijµ (s1, s2)O

(B1)µj (s1, s2;0)|Bv〉

= 1

mb

∫dτ C

(B1)ijµ

(2Eτ

mb

,2Eτ

mb

)

(15)× (2E)

∫dr

2πe−i2Eτr

⟨M(p′)

∣∣(ξWc)(0)(W†

c iDµ⊥cWc

)(rn+)Γ ′

j hv(0)|Bv〉


erseinear

on is

ns

tion is

lar

tor

with τ = 1 − τ . Abusing notation we will write the coefficient functions simply asC(B1)ijµ (E, τ)

in the following. Next, theJ (B1)-operators (except for the tensor operators with two transvindices, which we do not need in the following) are written as (Fourier transforms of) lcombinations of

(16)Jk(τ ) = 2E

∫dr

2πe−i2Eτr (ξWc)(0)

(W†

c iDµ⊥cWc

)(rn+)Γkhv(0),

where fork = 1,2,3 the Dirac matrixΓk can take one of the three expressions

(17)Γk = (γ5)γ

µ⊥ , (γ5)γν⊥γ µ⊥ , (γ5)γµ⊥γν⊥

,

and r = rmb. Here theγ5 in brackets means that this factor may be added. This notaticonvenient, because many of the results below do not depend on the extra factor ofγ5. In thefollowing definitions we leave out the position argument of a field, when it isx = 0. Eq.(15)suggests defining the B-type form factors as the matrix elements of the operatorsJk(τ ).

We therefore define the two leading-power SCETI form factors for pseudoscalar mesothrough⟨

P(p′)∣∣(ξWc)hv|Bv〉 = 2EξP (E),

(18)⟨P(p′)

∣∣(ξWc)(W†

c i/Dc⊥Wc

)(rn+)hv|Bv〉 = 2mbE

∫dτ ei2EτrΞP (τ,E).

Here|Bv〉 denotes theB meson state in the static limit (see the Lagrangian(32)) normalized to2mB (rather than 1 as is conventional in heavy quark effective theory). The second definisuch that

(19)ΞP (τ,E) = (2mbE)−1⟨P(p′)∣∣J1(τ )|Bv〉

(noγ5 in J1(τ )). Similarly, for vector mesons⟨V (p′)

∣∣(ξWc)γ5hv|Bv〉 = −2Eε∗ · vξ‖(E),⟨V (p′)

∣∣(ξWc)γ5γµ⊥hv|Bv〉 = −2E(εµ − ε · vn−µ)ξ⊥(E),⟨V (p′)

∣∣(ξWc)γ5(W†

c i/Dc⊥Wc

)(rn+)hv|Bv〉 = −2mbEε∗ · v

∫dτ ei2EτrΞ‖(τ,E),⟨

V (p′)∣∣(ξWc)γ5γµ⊥

(W†

c i/Dc⊥Wc

)(rn+)hv|Bv〉

= −2mbE(ε∗µ − ε∗ · vn−µ

)∫dτ ei2EτrΞ⊥(τ,E),⟨

V (p′)∣∣(ξWc)γ5

(W†

c i/Dc⊥Wc

)(rn+)γµ⊥hv|Bv〉

(20)= −2mbE(ε∗µ − ε∗ · vn−µ

)∫dτ ei2EτrΞ⊥(τ,E).

The tensor operatorsJ (B1)6µν , J

(B1)7µν have vanishing matrix elements between a pseudoscaB

meson and a pseudoscalar or vector meson, so the set of B-type operatorsJ1–3(τ ) (now includingγ5) is complete. We shall find in Section3 that the matrix element that definesΞ⊥(τ,E) (corre-sponding toJ3(τ )) vanishes at leading order in the 1/mb-expansion, hence we setΞ⊥(τ,E) = 0in the remainder of this section. The dependence on the polarization vectorε∗ shows that theform factors with subscripta =‖ (a =⊥) refer to longitudinally (transversely) polarized vecmesons.


ndcts at

0- andlinearof the

alar

in-

2.4. Form factor expressions

Taking the matrix element of(11), using(15), and inserting the definitions of the QCD athe SCETI form factors we derive expressions for the QCD form factors, in which the effethe scalemb are explicitly factorized. For the pseudoscalar meson form factors, we find

f+(E) = C(A0)f+ (E)ξP (E) +

∫dτ C

(B1)f+ (E, τ)ΞP (τ,E),

mB

2Ef0(E) = C

(A0)f0

(E)ξP (E) +∫

dτ C(B1)f0

(E, τ)ΞP (τ,E),

(21)mB

mB + mP

fT (E) = C(A0)fT

(E)ξP (E) +∫

dτ C(B1)fT

(E, τ)ΞP (τ,E).

Similarly, for the form factors of vector mesons

mB

mB + mV

V (E) = C(A0)V (E)ξ⊥(E) +

∫dτ C

(B1)V (E, τ)Ξ⊥(τ,E),

mV

EA0(E) = C

(A0)f0

(E)ξ‖(E) +∫

dτ C(B1)f0

(E, τ)Ξ‖(τ,E),

mB + mV

2EA1(E) = C

(A0)V (E)ξ⊥(E) +

∫dτ C

(B1)V (E, τ)Ξ⊥(τ,E),

mB + mV

2EA1(E) − mB − mV

mB

A2(E) = C(A0)f+ (E)ξ‖(E) +

∫dτ C

(B1)f+ (E, τ)Ξ‖(τ,E),

T1(E) = C(A0)T1

(E)ξ⊥(E) +∫

dτ C(B1)T1

(E, τ)Ξ⊥(τ,E),

mB

2ET2(E) = C

(A0)T1

(E)ξ⊥(E) +∫

dτ C(B1)T1

(E, τ)Ξ⊥(τ,E),

(22)mB

2ET2(E) − T3(E) = C

(A0)fT

(E)ξ‖(E) +∫

dτ C(B1)fT

(E, τ)Ξ‖(τ,E).

We recall thatE denotes the energy of the light meson. The coefficient functionsC(A0)F and

C(B1)F are defined as linear combinations of momentum-space coefficients functions of A

B-type operators. Remarkably, the ten different form factors involve only five independentcombinations of the A0- and B-type coefficients as a consequence of helicity conservationstrong interactions at leading power in the 1/mb-expansion. This implies

(23)mB

mB + mV

V (E) = mB + mV

2EA1(E), T1(E) = mB

2ET2(E)

up to power corrections[23], and the equality of the coefficients pertaining to pseudoscmesons and longitudinally polarized vector mesons. The five pairs(C

(A0)F ,C

(B1)F ) constitute the

main result of the first matching step from QCD to SCETI. The 1-loop expressions can beferred from[18,19]. They are collected inAppendix A.


ance

-

formf

2.5. Physical form factor scheme

Since the SCETI form factorsξa(E) are not known, it has been suggested in[8] to define themin terms of three QCD (or “physical”) form factors. Let

(24)ξFFP ≡ f+, ξFF⊥ ≡ mB

mB + mV

V, ξFF‖ ≡ mB + mV

2EA1 − mB − mV

mB

A2,

which corresponds to[8] except for the longitudinal form factorξFF‖ . The above definition, whichhas already been adopted in[24], is preferred, since it preserves the equality of the short-distcoefficients for the pseudoscalar meson and longitudinal vector meson form factors.

In the physical form factor scheme we have again the two identities(23), and the five remaining form factors read

mB

2Ef0 = R0ξ

FFP + (

C(B1)f0

− C(B1)f+ R0

) ΞP ,

mB

mB + mP

fT = RT ξFFP + (

C(B1)fT

− C(B1)f+ RT

) ΞP ,

T1 = R⊥ξFF⊥ + (C

(B1)T1

− C(B1)V R⊥

) Ξ⊥,

mV

EA0 = R0ξ

FF‖ + (C

(B1)f0

− C(B1)f+ R0

) Ξ‖,

(25)mB

2ET2 − T3 = RT ξFF‖ + (

C(B1)fT

− C(B1)f+ RT

) Ξ‖.

Here the asterisk is a shorthand for the convolution integral overτ . The ratiosR of A0-coefficients take much simpler expressions than the individual coefficients given inAppendix A.Up to one loop[8]

R0 ≡ C(A0)f0

C(A0)f+

= 1+ αsCF

4π[2− 2],

RT ≡ C(A0)fT

C(A0)f+

= 1+ αsCF

4π

[ln

m2b

ν2+ 2

],

(26)R⊥ ≡ C(A0)T1

C(A0)V

= 1+ αsCF

4π

[ln

m2b

ν2−

]

with

(27) ≡ − 2E

mb − 2Eln

2E

mb

,

CF = 4/3, andν the renormalization scale of the QCD tensor current. In the physicalfactor scheme there are only three non-trivial ratiosR and three non-trivial combinations oB-coefficients.

3. Jet-functions

In this section we turn to the main part of this paper, the calculation of the SCETI form fac-torsΞa(τ,E). Technically, this amounts to matching the B-type SCETI currents,Jk(τ ), whosematrix elements define theΞa(τ,E), to four-fermion operators in SCETII . These four-fermion


e de-atrmiontching

etween

o the

ombe

-

vour,initial

ght-hande set tos.

d inhed, all

operators factorize into a product of two light-cone distribution amplitudes resulting in thsired expression(3). That this can be done follows from[12], where it has been shown that leading power in the heavy quark expansion the B-type currents match only to four-feoperators with convergent convolution integrals. In terms of operators, we derive the marelation

Jk(τ ) = 2E

∫dr

2πe−i2Eτr (ξWc)(0)

(W†

c iD⊥cµWc

)(rn+)Γkhv(0)

=∫

dωdv Jk(τ ;v,ω)

[(ξWc)(sn+)

/n+2

Γ ck

(W†

c ξ)(0)

]FT

(28)×[(qsYs)(tn−)

/n−2

γ5(Y †

s hv

)(0)

]FT

+ · · ·

to the 1-loop order, where the ellipses denote terms that have vanishing matrix elements bB mesons and pseudoscalar or vector mesons, or are power-suppressed in 1/mb, Γ c

k will bedefined after(57), and the subscript “FT” denotes a Fourier transformation with respect tlight-cone variabless, t that will be made more precise later (see(45)). The functionsJk are theshort-distance coefficients of the SCETII operators, which contain the hard-collinear effects frthe scale(mbΛ)1/2 integrated out in passing to SCETII . These short-distance coefficients willcalled “jet-functions”.

3.1. Set-up of the calculation

The jet-functionsJk(τ ;v,ω) are extracted by computing both sides of(28) between appropriate quark states. We therefore consider the four-quark matrix element of the operatorsJk(τ ),

(29)Ak(τ ;v,ω) = ⟨q(p′

1)q(p′2)

∣∣Jk(τ )∣∣q(l)b(mbv)

⟩,

wherek = 1,2,3, and the momental, p′i (i = 1,2) are soft and collinear, respectively.q denotes a

light quark,b theb-quark. The quark and anti-quark in the final state may be of different flabut the flavours of the initial and final state anti-quark are identical. The quark–antiquarkand final states are assumed to be in a colour-singlet state. Since the operators on the riside of(28)do not contain derivatives, the transverse momenta of the collinear states can bzero, and we can definep′

1 = vp′ = vEn−, p′2 = vp′ with v ≡ 1− v. Likewise for the soft state

the momenta can be taken to bembv for the heavy quark andl = ωn+/2 for the light antiquarkThe four functionsΞa(τ,E) defined in(18), (20)correspond to settingΓk = (γ5)γ

µ⊥ (ΞP ,Ξ‖),Γk = γ5γν⊥γ µ⊥ (Ξ⊥), andΓk = γ5γ

µ⊥γν⊥ (Ξ⊥).The operatorsJk(τ ) generate momentum-space vertices with an arbitrary number ofn+Ac

gluons due to the Wilson linesWc. Of these only the one- and two-gluon vertices are needethe 1-loop calculation. The corresponding Feynman rules read (collinear quark lines dasgluon momenta are out-going)

(30)gsδ

(τ − n+k

n+p′

)(g

µρ⊥ − n

ρ+k

µ⊥

n+k

)T AΓk,


gi-

CET

rk.

m

e

g2s

(δ

(τ − n+k2

n+p′

)− δ

(τ − n+(k1 + k2)

n+p′

))

× nρ+

n+k1

(g

µσ⊥ − nσ+k

µ2⊥

n+k2

)+ δ

(τ − n+(k1 + k2)

n+p′

)

× nσ+n+k2

(g

µρ⊥ − n

ρ+(k1 + k2)

µ⊥

n+(k1 + k2)

)T AT BΓk

(31)+ (k1 ↔ k2,A ↔ B,ρ ↔ σ).

In light-cone gaugen+Ac = 0 the variableτ corresponds to the fraction of total collinear lontudinal momentumn+p′ carried by the transverse hard-collinear gluon.

In addition the calculation requires the collinear interactions from the leading-power SLagrangian,

L= ξ

(in−D + i/D⊥c

1

in+Dc

i/D⊥c

)/n+2

ξ − 1

2tr(Fµν

c Fµνc

)(32)+ hvivDshv + qs i/Dsqs − 1

2tr(Fµν

s Fµνs

),

as well as the sub-leading interaction[16]

(33)L(1)ξq = qsW

†c i/D⊥cξ − ξ i

←−/D⊥cWcqs

that converts the soft spectator antiquark in theB meson into an energetic, collinear antiquaThe Feynman rules for the collinear interactions can be found in[15], while (33) implies thevertices (collinear quark line dashed, soft quark line solid, gluon momenta outgoing)

(34)igsTA

(γ

ρ⊥ − n

ρ+/p⊥n+p

),

(35)

−ig2s T

AT B nρ+

n+k1

(γ σ⊥ − nσ+/p⊥

n+p

)+ (k1 ↔ k2,A ↔ B,ρ ↔ σ).

We note thatn+p = n+(k1 + k2) andp⊥ = (k1 + k2)⊥, since the corresponding soft momentucomponents are neglected in collinear-soft interaction vertices (multipole expansion).

3.2. Unrenormalized matrix element

3.2.1. TreeThe tree contribution to(29) is shown inFig. 1. The gluon momentum (outgoing from th

operator vertex) is given byk = p′2 − l, and withk2 = −vn+p′n−l = −2Eωv, we obtain

(36)A(0)k (τ ;v,ω) = −g2

s CF 1δ(τ − v)Γk ⊗ γ

µ⊥ ,

Nc 2Eωv


bers are

es

earr loopon. Theusece

Fig. 1. Tree diagram.

Fig. 2. 1-loop diagrams. The first diagram summarizes all gluon propagator corrections. Diagrams without numomitted when the calculation is done by expansion of QCD diagrams.

where Γ1 ⊗ Γ2 means uc(p′1)Γ1uh(mbv) v(l)Γ2vc(p

′2). The heavy quark spinor satisfi

/vuh(mbv) = uh(mbv), and for the collinear spinors/n−vc(p′2) = uc(p

′1)/n− = 0.

3.2.2. One loopA generic 1-loop diagram in SCETI contains contributions from the hard-collinear, collin

and soft momentum region. For our external momentum configuration soft and collineaintegrals are scaleless, so the diagram computation extracts the hard-collinear contributiSCETI diagrams are shown inFig. 2, omitting diagrams that are obviously scaleless. Wedimensional regularization (d = 4 − 2ε) for both ultraviolet and infrared singularities, henscaleless integrals vanish.


uted thender

ollinears”aicallyrated bynns

n-

hn. The

dent

nt

here).f

tertermlly, the

We computed the 1-loop diagrams in two different ways. First, we used the SCETI Feynmanrules as given above and computed the diagrams as shown in the figure. Second, we compmatrix element(29) with full QCD Feynman rules, but expanded the Feynman integrand uthe assumption that the loop momentum is hard-collinear, and the external momenta cand soft, respectively. This method, also known as the “strategy of expanding by region[25,26], gives precisely the same result as the first computation, but it turns out to be algebrsimpler, because it avoids having to use the more complicated vertex Feynman rules genethe SCET Lagrangian. There are also fewer diagrams to compute. There are no two-gluoqqgg

vertices in full QCD, and the unnumbered diagrams inFig. 2 are absent. In both computatiowe write

(37)ddk

(2π)d= 1

8π2dn+ k dn− k

dd−2k⊥(2π)d−2

,

and first perform then−k-integral by contour integration. Thek⊥-integral reduces to a convetional Feynman integral; then+k-integration is eliminated by theδ-function in(30)or (31)withthe exception of diagrams such as(1), (5) or (6).

It is convenient to perform the calculation without specifying the Dirac matrixΓk of theSCETI operator. The unrenormalized matrix element(29)can be written as

(38)A(ur)

k = [A(0) + A(1)

]Γk ⊗ γ µ⊥ + B(1)γ ρ⊥γ µ⊥Γk ⊗ γρ⊥ + C(1)γ ρ⊥γ λ⊥Γk ⊗ γ µ⊥γλ⊥γρ⊥

in terms of Dirac structures that cannot be reduced further ind dimensions. The notation is sucthat for any quantity the superscript (0) denotes the tree and (1) the 1-loop contributiocoefficients of all three structures are infrared divergent, but onlyA(1) andB(1) are ultravioletdivergent. It follows that all B-type currents can be renormalized with only two indepenrenormalization constants. Specifically, inserting the three Dirac structures(17), we obtain

A(ur)1 = [

A + (d − 2)B](γ5)γµ⊥ ⊗ γ µ⊥ + C(γ5)γ

ρ⊥γ λ⊥γ µ⊥ ⊗ γµ⊥γλ⊥γρ⊥ ,

A(ur)2 = A(γ5)γν⊥γµ⊥ ⊗ γ µ⊥ + (4− d)B(γ5)γµ⊥γν⊥ ⊗ γ µ⊥

+ C(γ5)γρ⊥γ λ⊥γν⊥γ µ⊥ ⊗ γµ⊥γλ⊥γρ⊥ ,

(39)A(ur)3 = [

A + (d − 2)B](γ5)γµ⊥γν⊥ ⊗ γ µ⊥ + C(γ5)γ

ρ⊥γ λ⊥γ µ⊥γν⊥ ⊗ γµ⊥γλ⊥γρ⊥ .

Here and in the following we use anti-commutingγ5, and the bracket aroundγ5 refers to thetwo cases, whereγ5 is or is not included inΓk . From this it can be seen that the UV divergeparts have the same Dirac structure as the original operator. Hence, the operatorsJk(τ ) andall the B-type current operators do not mix under renormalization (in the basis adoptedThe renormalization constant for the operatorJk(τ ) with Γ1,3 is related to the divergent part oA(1) + 2B(1), the one forJk(τ ) with Γ2 to A(1).

3.3. Ultraviolet counterterms

The counterterm diagrams are obtained from the tree diagram by insertions of a counvertex into the gluon propagator, the quark–gluon vertex and the operator vertex. Finaon-shell matrix element must be multiplied by the propagator residue factorsR

1/2hv

R1/2qs

Rξ . Theyare related to the field renormalization constants by

(40)RX = ZOSX ,

ZX


n

note thatted to

matrix

r-sionalr,

d-nitiestion

heET

whereZOSX is the renormalization constant of fieldX in the on-shell scheme (by definitio

ROSX = 1) andZX is the field renormalization constant in theMS scheme.The Lagrangian counterterms are standard. As regards the operator counterterms, we

the three operatorsJk(τ ) do not mix (see above), hence the renormalized operator is relathe bare operator (expressed in terms of bare fields) by

J1,3(τ ) =∫

dτ ′ Z‖(τ, τ ′)J bare1,3 (τ ′),

(41)J2(τ ) =∫

dτ ′ Z⊥(τ, τ ′)J bare2 (τ ′),

which defines the operator renormalization kernels. Here we used thatJ1(τ ) andJ3(τ ) renor-malize identically.

Putting together all the renormalization constants, the on-shell ultraviolet-renormalizedelements ofJ1 andJ3 follow from (39)by the replacement

A + (d − 2)B → A + (d − 2)B + Z(1)‖ A(0)

(42)+(

2Z(1)g + 1

2

(Z

OS(1)hv

+ ZOS(1)qs

+ 2ZOS(1)ξ

))A(0),

while the renormalized matrix element ofJ2 is the second line of(39)with

(43)A → A + Z(1)⊥ A(0) +

(2Z(1)

g + 1

2

(Z

OS(1)hv

+ ZOS(1)qs

+ 2ZOS(1)ξ

))A(0).

The asterisk denotes convolution in the variableτ ′ as in the definition of the operator renomalization kernels. The on-shell field renormalization factors are all equal to 1 in dimenregularization, soZOS(1)

X = 0. Zg is the standardMS strong coupling renormalization factogbare

s = Zggs with (CA = 3, Tf = 1/2)

(44)Zg = 1− αsβ0

8πε, β0 = 11CA

3− 4

3nf Tf .

The matrix elements(39) with the substitutions(42), (43)are now ultraviolet finite, but infraredivergent. The infrared divergences are reproduced by the SCETII computation of the matrix element, resulting in a finite jet-function. This can be used todeterminethe operator renormalizatiokernelsZ‖ andZ⊥ alternative to the direct computation of the operators’ ultraviolet singularperformed in[20]. That is, after extracting the jet-function, we shall obtain the renormalizafactors by requiring that they render the result finite.

3.4. Matching to SCETII and extraction of the jet function

3.4.1. SCETII matrix elementTo obtain the jet-function, we need the SCETII matrix elements on the right-hand side of t

matching equation(28)to the 1-loop order. In the absence of collinear-soft interactions in SCIIthe four-quark operator factorizes into a collinear and a soft bilinear. We define

Q[Γ c

k

](v) =

[(ξWc)(sn+)

/n+2

Γ ck

(W†

c ξ)(0)

]FT

= n+p′ ∫ds e−isvn+p′

(ξWc)(sn+)/n+

Γ ck

(W†

c ξ)(0),

2π 2


rac

tion

reoryce allcluding

rator

spond tolements

P(ω) =[(qsYs)(tn−)

/n−2

γ5(Y †

s hv

)(0)

]FT

(45)= 1

2π

∫dt eitω(qsYs)(tn−)

/n−2

γ5(Y †

s hv

)(0),

and the quark matrix elements

Φqq(v′, v) = ⟨q(p′

1)q(p′2)

∣∣Q[Γ c

k

](v′)|0〉,

(46)Φbq(ω′,ω) = 〈0|P(ω′)∣∣q(l)bv(0)

⟩.

The Dirac matricesΓ ck will be determined later by the Fierz transformation of the first Di

matrix product on the right-hand sides of(39). Note that the hadronic matrix elements ofQ

andP are precisely the meson light-cone distribution amplitudes. Collinear-soft factoriza3

in SCETII means that the matrix element ofQ[Γ ck ](v)P (ω), which appears in(28), factorizes

into

(47)⟨M(p′)

∣∣Q[Γ c

k

](v)P (ω)|Bv〉 = ⟨

M(p′)∣∣Q[

Γ ck

](v)|0〉〈0|P(ω)|Bv〉,

which is a product of light-cone distribution amplitudes.With these definitions the tree quark matrix element of the SCETII four-quark operator is

given by

(48)⟨q(p′

1)q(p′2)

∣∣Q[Γ c

k

](v′)P (ω′)

∣∣q(l)b(mbv)⟩(0) = δ(v − v′)δ(ω − ω′)/n+

2Γ c

k ⊗ /n−2

γ5,

where nowΓ1 ⊗ Γ2 meansuc(p′1)Γ1vc(p

′2)v(l)Γ2uh(mbv). The “tensor products”⊗ and⊗ are

related by a Fierz transformation as discussed below.The SCETII computation of the matrix elements(46) is very simple, since in the collinea

sector SCETII is equivalent to full QCD, and in the soft sector to heavy quark effective th[12]. The external momenta do not allow to form a non-vanishing kinematic invariant, henloop integrals are scaleless and vanish. The matrix elements are given by tree diagrams intree diagrams with counterterm insertions. We can therefore write

Φqq(v′, v) = ZQ(v′, v)/n+2

Γ ck ,

(49)Φbq(ω′,ω) = ZP (ω′,ω)/n−2

γ5,

with ZQ(v′, v) andZP (ω′,ω) the renormalization kernels that relate the renormalized opeto the bare operator expressed in terms of the bare fields,

Q[Γ c

k

](v) =

1∫0

dwZQ(v,w)Q[Γ c

k

]bare(w),

(50)P(ω) =1∫

0

dω′ ZP (ω,ω′)P bare(ω′).

3 We recall here that in general the apparent factorization of collinear and soft degrees of freedom in SCETII is notvalid [12,27]due to the non-existence of a regulator that preserves factorization. The unregulated integrals correendpoint-divergent convolution integrals. However, it has been shown that for the particular case of the matrix eof the B-type currents, the convolution integrals must be convergent[12], so collinear-soft factorizationis valid here.


or the

sonion

opraredever a

The required 1-loop renormalization factors have been computed in other contexts. Fcollinear operatorZQ(v,w) is the Brodsky–Lepage kernel[28]

ZQ(v,w) = δ(v − w) + Z(1)Q (v,w) + · · · ,

Z(1)Q (v,w) = −αsCF

2πε

1

ww

[vw

θ(w − v)

w − v+ vw

θ(v − w)

v − w

]+

(51)− 1

2δ(v − w) + ∆

(v

wθ(w − v) + v

wθ(v − w)

),

where∆ = 1 applies toΓ ck = (γ5)1 (pseudoscalar meson, longitudinally polarized vector me

distribution amplitudes) and∆ = 0 toΓ ck = γ ν⊥ (transversely polarized vector meson distribut

amplitude), and the plus-distribution is defined for symmetric kernelsf as

(52)∫

dw[f (v,w)

]+g(w) =

∫dw f (v,w)

(g(w) − g(v)

).

Similarly, the renormalization ofP(ω) has been worked out in[29] with the result

ZP (ω,ω′) = δ(ω − ω′) + Z(1)P (ω,ω′) + · · · ,

Z(1)P (ω,ω′) = αsCF

4π

[(1

ε2+ 2

εln

µ

ω− 5

2ε

)δ(ω − ω′)

(53)− 2ω

ε

(1

ω′θ(ω′ − ω)

ω′ − ω+ 1

ω

θ(ω − ω′)ω − ω′

)+

].

3.4.2. Extraction of the jet-functionThe jet-function is extracted from the quark matrix element of(28),

(54)Ak(τ ;v,ω) =∞∫

0

dω′1∫

0

dv′ Jk(τ ;ω′, v′)Φbq (ω′,ω)Φqq(v′, v).

This gives

(55)J(0)k (τ ;ω,v)

/n+2

Γ ck ⊗ /n−

2γ5 = A(0)(τ ;v,ω),

at tree level, and

J(1)k (τ ;ω,v)

/n+2

Γ ck ⊗ /n−

2γ5

= A(1)k (τ ;v,ω) −

[ ∞∫0

dω′ Z(1)P (ω′,ω)J

(0)k (τ ;ω′, v)

(56)+1∫

0

dv′ Z(1)Q (v′, v)J

(0)k (τ ;ω,v′)

]/n+2

Γ ck ⊗ /n−

2γ5

at the 1-loop order.A(1)k (τ ;v,ω) is the ultraviolet-renormalized but infrared divergent 1-lo

matrix element ofJk(τ ). The subtraction on the right-hand side precisely cancels the infdivergences, so that the short-distance jet-function is finite as it should be. There is how


ed

sogether

n

struc-decay

er) signer-

f

the

nsider

iple

Fierz

difficulty in executing the subtraction in the form of(56), sinceA(1)k (τ ;v,ω) is expressed in

terms of the spinor product⊗, corresponding to SCETII operators with spinor indices contractin [ξhv][qsξ ], while the left-hand side and the subtraction term involve the product⊗, corre-sponding to operators with spinor indices contracted in[ξ ξ ][qshv]. The standard Fierz identitiethat relate these structures are valid only in four dimensions. We shall discuss this issue twith the reduction of evanescent Dirac structures appearing in(39) in the following subsection.

None of this applies to the tree-level equation(55), where all quantities are finite. We catherefore apply the four-dimensional Fierz identities to the tree-level terms in(39), so that for

k = 1: (γ5)γµ⊥ ⊗ γ µ⊥ = (±1)/n+2

γ5(γ5) ⊗ /n−2

γ5 + . . . ,

k = 2: (γ5)γν⊥γµ⊥ ⊗ γ µ⊥ = −/n+2

γν⊥γ5(γ5) ⊗ /n−2

γ5 + . . . ,

(57)k = 3: (γ5)γµ⊥γν⊥ ⊗ γ µ⊥ = . . . .

The Fierz transformation produces a number of 4-fermion operators with different Diractures. In the following we discuss only those operators (jet-functions) that contribute to theof a pseudoscalarB meson. The ellipses denote terms involving/n−/2 and/n−/2γν⊥(γ5) to theright of ⊗, which vanish when the matrix element〈0|qs[· · ·]hv|B〉 with the pseudoscalarB me-son state is evaluated. Hence, these terms will not be considered further. The upper (lowrefers to the operators on the left-hand side without (with) theγ5 factor. Here and below thFierz identities are given for four-fermionoperatorsand the extra minus sign from the field pemutation relative to the identities for matrix products is already included. Comparison o(57)with the definition(45) of Q[Γ c

k ] determines the Dirac matrixΓ ck in the collinear SCETII op-

erator. Fork = 1, we haveΓk = (γ5)γµ⊥ and the correspondingΓ ck = (±1)γ5(γ5); for k = 2,

Γk = (γ5)γν⊥γµ⊥ and Γ ck = −γν⊥γ5(γ5); for k = 3, Γk = (γ5)γµ⊥γν⊥ and Γ c

k = 0, i.e., thereis no contribution for pseudoscalarB mesons. Hence, comparing(55), (57)to (36) leads to thetree-level jet function

(58)J(0)k (τ ;v,ω) = −g2

s CF

Nc

1

2Eωvδ(τ − v)

for k = 1,2, and 0 fork = 3.Using the 1-loop expressions for the SCETII renormalization kernels, we can evaluate

subtraction term in square brackets in(56)with the result

(59)−g2s CF

Nc

1

2Eωv

αsCF

4πδ(τ − v)

(1

ε2+ 2

εln

µ

ω− 5

2ε

)+ v

τZ

(1)Q (τ , v)

.

3.4.3. Evanescent operators and Fierz transformationWe now discuss the reduction of the Dirac structure and the Fierz transformation. We co

in detail the caseΓk = γ µ⊥ . According to the first equation of(39) the UV renormalized matrixelement of the corresponding SCETI currentJ1(τ ) is

(60)A1(τ ;v,ω) = Aγµ⊥ ⊗ γ µ⊥ + Cγ ρ⊥γ λ⊥γ µ⊥ ⊗ γµ⊥γλ⊥γρ⊥ ,

whereA stands for the right-hand side of(42). The second Dirac structure reduces to a multof the first one in four dimensions, and the first one satisfies the Fierz relation(57) in four dimen-sions. In the following we discuss separately the reduction of the Dirac structure and the


ingle

ra-

jet-matrix

rared-relationan takeen-e

-

itraryer-.

transformation ind dimensions, although it is possible to perform both reductions in a sstep.

Indicating only the field content and Dirac structure we define the operatorsO1 = (ξγµ⊥hv) ×(qsγ

µ⊥ξ) andO ′1 = (ξγ ρ⊥γ λ⊥γ µ⊥hv)(qsγµ⊥γλ⊥γρ⊥ξ), and rewrite the previous equation as

(61)A1(τ ;v,ω) = [A(0) + A(1)

]〈O1〉(0) + C(1)〈O ′1〉(0),

where the tree-level matrix elements ofO(′)1 just reproduce the Dirac structures, i.e.,〈O1〉(0) =

uc(p′1)γ

µ⊥uh(mbv)v(l)γµ⊥vc(p′2), etc. Ind = 4, O ′

1 = 4O1, so we define the evanescent opetor E = O ′

1 − f (ε)O1, wheref (0) = 4, but otherwisef (ε) is arbitrary. Hence,

(62)A1(τ ;v,ω) = [A(0) + A(1) + f (ε)C(1)

]〈O1〉(0) + C(1)〈E〉(0).

SinceC(1) has a 1/ε infrared divergence, the coefficient of the “physical” operatorO1 dependson the up to now arbitrary prescriptionf (ε) in the definition of the evanescent operator. Thefunctions are obtained by expressing this equation in terms of the renormalized operatorelements. To the 1-loop order it is sufficient to use

(63)〈O1〉 = (1+ M

(1)O1

)〈O1〉(0) + M(1)O1E

〈E〉(0), 〈E〉 = ⟨E(0)

⟩,

which results in

A1(τ ;v,ω) = [A(0) + A(1) + f (ε)C(1) − A(0)M

(1)O1

]〈O1〉 + [C(1) − A(0)M

(1)O1E

]〈E〉(64)

d→4→ [J

(0)O1

+ J(1)O1

]〈O1〉.The coefficients of the operator matrix elements in the first line of this equation are now inffinite short-distance quantities, and the equation is interpreted as an operator matchingvalid also when the matrix elements are taken between hadronic final states. Hence we cthe limit d → 4 in which 〈E〉 = 0, since the tree-level matrix element vanishes in four dimsions.4 The 1-loop jet-functionJ (1)

O1can be read off from(64). It depends on the choice of th

evanescent operator, i.e., the order-ε term off (ε), but so does〈O1〉 as can be seen from(63),and the physical amplitude is scheme-independent.

The jet-functionJO1 is not the desired result, because instead ofO1 we must use the Fierztransformed operator

(65)P1 =(

ξ/n+2

γ5ξ

)(qs

/n−2

γ5hv

)+ . . . ,

where the ellipses denote terms that are irrelevant for us, as in(57). This Fierz-ordering isuniquely singled out by collinear-soft factorization in SCETII . Radiative corrections toP1 occuronly within the collinear factor or within the soft factor, hence the Dirac structures in arbloop diagrams can be reduced to the tree structureP1. It follows that whatever evanescent opator one may write down in this Fierz-ordering decouples fromP1, and can simply be ignoredHence, even ind dimensions we have

(66)A1(τ ;v,ω) = [J

(0)P1

+ J(1)P1

]〈P1〉,

4 In higher orders one usually uses a non-minimal subtraction scheme to ensure that〈E〉 = 0, but this is not relevant tothe present calculation.


se,

trixht-ngual

chosen

e

o thel

ction,ut, be-

ace.

. Theal

oret opera-

a-

with J(1)P1

the desired 1-loop jet-function. Comparing this equation to(64) we haveJP1 = JO1,if the renormalized matrix elements〈P1〉 and 〈O1〉 are equal. In general this is not the cabecause the Fierz identity that relatesO1 andP1 in four dimensions is not valid ford = 4. Now,the renormalization scheme forP1 is fixed by the requirement that the collinear and soft maelements coincide with the standardMS definition of the light meson and heavy meson ligcone distribution amplitudes, respectively, so〈P1〉 is completely defined. However, by choosif (ε), we can adjust the definition ofO1, such that the infrared-finite matrix elements are eqin the limit d → 4.

Since the jet-functions are independent of the infrared regularization, any one can befor the calculation, and it is convenient to compare the matrix elements ofO1 andP1 by assumingan off-shell infrared regularization. The ultraviolet-renormalized 1-loop matrix elements ar

〈O1〉 = (1+ Moff

O1

)〈O1〉(0) + MoffO1E

〈E〉(0),

(67)〈P1〉 = (1+ Moff

P1

)〈P1〉(0).

There are no 1/ε poles in these equations due to the use of off-shell IR regularization, sevanescent term drops out ford → 4. Furthermore,〈O1〉(0) = 〈P1〉(0) by the four-dimensionaFierz-equivalence ofO1 andP1, hence we need to defineO1 such that the differenceMoff

O1−

MoffP1

= 0. At the 1-loop order one finds that the self-energy corrections and the vertex correwhere the gluon is exchanged between the soft light quark and the heavy quark, drop ocause the coupling to the heavy quark is proportional tovµ and does not introduce a new Dirstructure. Only the gluon exchange between the collinearξ -fields can give a non-zero differencIn the case ofO1 the part of this diagram that does not cancel in the differenceMoff

O1− Moff

P1in-

volvesγ ρ⊥γ λ⊥γ µ⊥ ⊗ γµ⊥γλ⊥γρ⊥ , and its contribution toMO1 is therefore proportional tof (ε).In the case ofP1, the Dirac structure is

(68)γ α⊥γ β⊥ /n+2

γβ⊥γα⊥ ⊗ /n−2

= (d − 2)2〈P1〉(0).

Including the overall coefficient, we find

(69)MoffO1

− MoffP1

= −[C(1)

]div

(f (ε) − (d − 2)2)⟨P (0)

1

⟩,

where[C(1)]div is the divergent part ofC(1). In order for this to vanish, we must choosef (ε) =(d − 2)2. In other words, we must define

(70)γ ρ⊥γ λ⊥γ µ⊥ ⊗ γµ⊥γλ⊥γρ⊥ = (d − 2)2γ µ⊥ ⊗ γµ⊥ .

With thisJP1 = JO1, so returning to(64)we have

(71)J(1)P1

= limd→4

(A(1) + (d − 2)2C(1) − A(0)M

(1)P1

).

The termA(0)M(1)P1

is nothing but the subtraction term(59) that appears in(56), while A(1) and

C(1) can be read off from the 1-loop calculation that leads to(38) and (42), so the previousequation gives the final result for the correctly renormalized and subtracted jet-functionabove argument can be repeated forΓk = γ5γ

µ⊥ , and one finds that the jet-function is identicto the one forΓk = γ µ⊥ as was expected.

The case of B-type currentsJ2,3(τ ) with one uncontracted transverse index is slightly mcomplicated than the scalar case, because there are two physical and two evanescentors. Corresponding toΓk = (γ5)γ

ν⊥γ µ⊥ and (γ5)γµ⊥γ ν⊥ we define the two physical oper

torsO2 = (ξ (γ5)γν⊥γµ⊥hv)(qsγµ⊥ξ) andO3 = (ξ (γ5)γµ⊥γν⊥hv)(qsγ

µ⊥ξ). Requiring that the


ond-the

i-

at

ns wesh

ance

renormalized matrix elements ofO2,3 equal the renormalized matrix elements of the corresping operatorsP2,3 in the other Fierz-ordering fixes uniquely the prescription for reducingDirac structures multiplyingC in (39) and ensures that the SCETII operators are correctly minmally subtracted. A short calculation analogous to the one discussed above gives

(γ5)γρ⊥γ λ⊥γν⊥γ µ⊥ ⊗ γµ⊥γλ⊥γρ⊥ = (d − 4)2 (γ5)γν⊥γµ⊥ ⊗ γ µ⊥,

(72)(γ5)γρ⊥γ λ⊥γ µ⊥γν⊥ ⊗ γµ⊥γλ⊥γρ⊥ = (d − 2)2 (γ5)γµ⊥γν⊥ ⊗ γ µ⊥ .

Referring to(57) we see that the matrix element ofO3 vanishes for a pseudoscalarB meson.Hence the B-type operatorJ3(τ ) with Γk = (γ5)γµ⊥γν⊥ has a vanishing jet-function just astree level. On the other hand, inserting the first equation of(72) into the second of(39), we findthat the B-type operatorJ2(τ ) matches toJP2P2 with

(73)J(1)P2

= limd→4

(A(1) + (d − 4)2C(1) − A(0)M

(1)P2

).

HereA(1) andC(1) are defined by(39) and (43)and the subtraction term is given by(56)exceptthat now∆ = 0 must be used in the Brodsky–Lepage kernel. SinceC(1) has only a single 1/εpole, the term(d − 4)2C(1) does not contribute to the final result for the jet-function.

3.5. Final results for the jet-functions and B-type current renormalization kernels

We shall now denote the jet-functionJP1 (JP2) that arises in the matching ofJ1(τ ) (J2(τ ))asJ‖ (J⊥). To the 1-loop order we write

(74)Ja = −g2s CF

Nc

1

2Eωv

(δ(τ − v) + αs

4πja(τ ;v,ω)

)

with a =‖,⊥. Applying the subtraction procedure described in the previous subsectioobtain the ultraviolet and infrared finite 1-loop correctionsja(τ ;v,ω). The resulting expressionstill depend on the yet undetermined renormalization factorsZ

(1)a of the B-type currents throug

(42), (43). They can now be determined by the condition thatja(τ ;v,ω) must not contain 1/εpoles. We choose theMS scheme to be consistent with the definition of the B-type short-distcoefficients in[18].

3.5.1. Renormalization kernelsWe expand theZ-factors in(41)as

(75)Za(τ, τ′) = δ(τ − τ ′) + αs

4πz(1)a (τ, τ ′),

and obtain

z(1)‖ (τ, τ ′) = (−CF )

(1

ε2+ 2

εln

µ

2E

)δ(τ − τ ′) − 1

ε

[z1(τ, τ

′) + z2(τ, τ′)],

(76)z(1)⊥ (τ, τ ′) = (−CF )

(1

ε2+ 2

εln

µ

2E

)δ(τ − τ ′) − z1(τ, τ

′)ε

,

with

z1(τ, τ′) = δ(τ − τ ′)

CF

[−2 ln τ + 5

]+ CA ln

τ

2 τ


kernels-of the

+ CA

[θ(τ − τ ′)τ − τ ′ + θ(τ ′ − τ)

τ ′ − τ

]+

+(

CF − CA

2

)2τ

τ τ ′ θ(τ − τ ′)

− CA

(1

τ τ ′ θ(τ − τ ′) + 1

τ ′ θ(τ ′ − τ)

),

z2(τ, τ′) = (−2)

(CF − CA

2

)(ττ ′

τ ′ θ(τ ′ − τ) + τ

(1+ 1

τ+ 1

τ ′

)θ(τ − τ ′)

)

(77)+ CA

(τ τ ′

τ ′

(1+ 1

τ

)θ(τ − τ ′) + τ

(1+ 1

τ ′

)θ(τ ′ − τ)

).

The anomalous dimensions of the B-type operators are derived from the renormalizationin the standard way (see(99)). Our result is in complete agreement with[20], where the anomalous dimension has been obtained by extracting directly the ultraviolet divergent partsSCETI diagrams.5

3.5.2. Jet-functionsThe 1-loop jet-functions read

j‖(τ ;v,ω) = Aδ(τ − v) +(

CF − CA

2

)[2B]+

+ CF

[θ(v − τ)

2τ

vv

(L + ln

(v − τ)τ

v+ v(v − τ)

τ τ

)

− 2vτ

v

(L + ln τ τ + vτ

vτ

)]

−(

CF − CA

2

)[θ(τ − v)

2(v − τ)2

vvτ

(L + ln

(τ − v)τ

v− vτ

(v − τ)2

)

+ θ(v − τ)2(v − τ)

vv

(L + ln(v − τ) + τ τ

(v − τ)(v − τ)ln

v

τ− v

v − τ

)

(78)+ θ(τ − v)2

τ

(L + ln(τ − v) + v

τ − vln

v

τ− τ

v

)],

j⊥(τ ;v,ω) = Aδ(τ − v) +(

CF − CA

2

)[2B]+

+ CF

[θ(v − τ)

2

v

(L + ln

(v − τ)τ

v− 1

)− θ(τ − v)

2τ

vτ

]

−(

CF − CA

2

)[θ(τ − v)

2τ

vτ

(L + ln

(τ − v)τ

v− v − τ

vτ

)

+ θ(v − τ)2

v

(L + ln(v − τ) + τ

v − τln

v

τ− 1

v

)

(79)+ θ(τ − v)2

vτ

(L + ln(τ − v) + vτ

τ − vln

v

τ− 1

)]

5 The variableu in [20] corresponds to ourτ = 1− τ , theirv to our τ ′.


h

nt of

ans-is

n 1)

f

with

A = CF

[(L + ln v)2 − 13

3(L + ln v) − π2

6+ 80

9

]

+(

CF − CA

2

)[(L + lnv)2 − (L + ln v)2 + 22

3(L + ln v) + 2π2

3− 152

9

]

+ nf Tf

[4

3(L + ln v) − 20

9

],

(80)B = θ(τ − v)

τ − v

(L + ln(τ − v)

) + θ(v − τ)

v − τ

(L + ln(v − τ)

)andL = ln(n+p′n−l/µ2) = ln(2Eω/µ2). The SU(3) group factors areCF = 4/3,CA = 3,Tf =1/2, andnf denotes the number of light quark flavours. Once again we find agreement wit[19,20].

3.5.3. Hard-scattering form factorsHere we express the hard-scattering (SCETI) form factorsΞ(τ,E) defined in(18), (20)as

convolutions of the above jet-functions and light-cone distribution amplitudes.The light-cone distribution amplitudes of the light mesons follow from the matrix eleme

Q[Γ ck ](v) defined in(45),

(81)⟨M(p′)

∣∣Q[Γ c

k

](v)|0〉 =

−ifP EφP (v), Γ ck = /n+

2 γ5,

−ifV ‖E mV ε∗·vE

φV ‖(v), Γ ck = /n+

2 ,

−ifV ⊥E(ε∗α − ε∗ · vnα−)φV ⊥(v), Γ ck = /n+

2 γ α⊥.

The three cases correspond toM being a pseudoscalar meson, longitudinally polarized or trversely polarized vector meson, respectively. Similarly, theB meson distribution amplituderelated to the matrix element ofP(ω) in (45)such that

(82)〈0|P(ω)|Bv〉 = ifBmB

2φB+(ω)

with fB the HQETB meson decay constant (but defined such that is has mass dimensio6,that is

(83)fB = K(µ)fB(µ) = K(µ)Fstat(µ)√mB

with Fstat(µ) themB -independent decay constant in the static limit (HQET) and

(84)K(µ) = 1+ αsCF

4π

(3 ln

mb

µ− 2

)

a short-distance coefficient[30].

6 With the conventions of heavy-quark effective theory our|Bv〉 corresponds to√

mB |Bv〉, and the right-hand side o(82) is mB -independent when expressed in terms ofFstat.


tresult

eavy-merical

egral

c-

We can now evaluate

⟨P(p′)

∣∣J1(τ )|Bv〉 = 2E

∫dr

2πe−i2Eτr

⟨P(p′)

∣∣(ξWc)(0)(W†

c i/D⊥cWc

)(rn+)hv(0)|Bv〉

=∫

dωdv J‖(τ ;v,ω)⟨P(p′)

∣∣Q[/n+2

γ5

](v)P (ω)|Bv〉

(85)= mBE

2

∫dωdv J‖(τ ;v,ω)fBφB+(ω)fP φP (v),

where we have used(28) and (47). Comparing this to the definition(18) and (19)gives theexpression forΞP . Proceeding in the same way for the other form factors we obtain

ΞP (τ,E) = mB

4mb

fBφB+ fP φP J‖,

E

mV

Ξ‖(τ,E) = mB

4mb

fBφB+ fV ‖φV ‖ J‖,

Ξ⊥(τ,E) = mB

4mb

fBφB+ fV ⊥φV ⊥ J⊥,

(86)Ξ⊥(τ,E) = 0.

The asterisk stands for the convolutions inω andv as in(85). This together with the expliciexpressions for the 1-loop jet-functions is the main technical result of this paper. Using thisin (25) allows us to investigate numerically the corrections to the symmetry relations for hto-light form factors at the 1-loop order. The subsequent sections are devoted to this nuinvestigation.

4. Numerical analysis

4.1. Jet-functions

It will be seen below that the jet-function appears in the form factors in the form of the int

(87)Ia ≡ λB

〈v−1〉M

1∫0

dv

vφM(v)

∞∫0

dω

ωφB+(ω)

1∫0

dτ

(δ(τ − v) + αs

4πja(τ ;v,ω)

),

which is normalized to 1 in the absence of theαs -correction. We now evaluate the 1-loop corretion.

4.1.1. Light-cone distribution amplitudesThe light-cone distribution amplitude (LCDA) of the light meson,φM(v), is conventionally

expanded into the eigenfunctions of the 1-loop renormalization kernel,

(88)φM(v) = 6vv

[1+

∞∑n=1

aMn C

(3/2)n (2v − 1)

],


define

at theenolog-

scale-d mini-

to the

of the

r atnddopt a

whereaMn andC

(3/2)n (v) are the Gegenbauer moments and polynomials, respectively. We

the quantity

(89)⟨v−1⟩

M≡

1∫0

dv

vφM(v) = 3

(1+

∞∑n=1

aMn

).

In practice the expansion will be truncated after the second term, since it is believed thhigher Gegenbauer moments are negligible, or accounted for approximately by phenomical determinations of the first two moments. The LCDAs and Gegenbauer moments areand scheme-dependent. Our computation of the jet-function corresponds to the modifiemal subtraction (MS) scheme and the renormalization scaleµ of the LCDA (not indicated by itsarguments) is equal to the scaleµ that appears in the expressions(78), (79)for the jet-functions.In particular the scale-dependence of〈v−1〉M is given by

(90)µd

dµ

⟨v−1⟩

M= αsCF

π

1∫0

dv

vφM(v)

4− ∆

2+ ln v

v[1− v∆]

,

which follows from(51).7 We recall that∆ = 0 for transversely polarized vector mesonsM , and∆ = 1 for pseudoscalar mesons or longitudinally polarized vector mesons.

The first inverse moment of the LCDA of theB meson,

(91)1

λB

≡∞∫

0

dω

ωφB+(ω)

is a key quantity in exclusiveB decays[1]. We define the averages

(92)⟨f (ω)

⟩ ≡ λB

∞∫0

dω

ωφB+(ω)f (ω).

The LCDA andλB are scale-dependent. Our computation of the jet-function correspondsmodified minimal subtraction scheme and the renormalization scaleµ is equal to the scaleµ thatappears in the expressions for the jet-functions. The scale-dependence of 1/λB is given by

(93)µd

dµ

(fB

λB

)= αsCF

π

fB

λB

3

4+ 1

2−

⟨ln

µ

ω

⟩which follows from(53). The first term in the bracket comes from the scale-dependencestatic decay constantfB .

Sinceω is of orderΛ, only logarithmic modifications of the first inverse moment appealeading order in the 1/mb-expansion. It can be seen from(78) and (79)that the 1-loop calculatioinvolves the two logarithmic moments〈L〉, 〈L2〉 with L = ln(2Eω/µ2). The entire energy anscale-dependence of the 1-loop jet-functions is contained in these two quantities. We asimple one-parameter model for the shape of the distribution amplitude[31],

7 Note that(51) describes the scale dependence offMφM(v), and

µd

dµfM = αsCF

π

∆ − 1

2fM.


erom-

.ion of

ons

le

,

tive cor-gn thatsupportis an

approxi-ent wouldculty if

(94)φB+(ω) = ω

λ2B

e−ω/λB ,

which relates the two logarithmic moments to the parameterλB ,

(95)〈L〉 = ln2EλBe−γE

µ2,

⟨L2⟩ = ln2 2EλBe−γE

µ2+ π2

6

with γE = 0.577216. . . .The functional form(94) and the Gegenbauer momentsaM

n are assumed to be given at somreference scaleµ0 of order(mbΛ)1/2. This avoids having to evolve the meson parameters fthe hadronic scaleΛ to the hard-collinear scale with the SCETII renormalization group equations.8

4.1.2. Integrated jet-functionsWe proceed to the evaluation ofIa . The integrals

∫ 10 dτ ja(τ ;v,ω) are given in analytic form

in Appendix B. Integration overω introduces the logarithmic moments〈L〉, 〈L2〉 defined aboveThe finalv-integration can be done numerically, or term by term in the Gegenbauer expansthe light meson light-cone distribution amplitude. Up to the second moment, we obtain

I‖ = 1+ αs(µ)

4π

3

〈v−1〉M(

4

3

[1+ aM

1 + aM2

]⟨L2⟩ − [

5.24+ 8.93aM1 + 10.86aM

2

]〈L〉

+ [3.99+ 8.67aM

1 + 13.47aM2

]),

I⊥ = 1+ αs(µ)

4π

3

〈v−1〉M(

4

3

[1+ aM

1 + aM2

]⟨L2⟩ − [

4.90+ 8.93aM1 + 10.81aM

2

]〈L〉

(96)+ [0.73+ 6.78aM

1 + 11.48aM2

]),

with9 nf = 4, Tf = 1/2, CF = 4/3 andCA = 3. The analytic expressions of these convolutiare also given inAppendix B.

These results do not contain large logarithms, whenµ is of order of the hard-collinear sca(mbΛ)1/2 ≈ 1.5 GeV, i.e.,〈L〉, 〈L2〉 are of order 1 forE ∼ mb and µ ∼ (mbΛ)1/2. Sinceαs(1.5 GeV)/(4π) is approximately 0.029,〈L2〉 ≈ 2.5 and〈L〉 ≈ −1 (for typical parameters)the perturbative corrections to the jet-functions are about (20–50)%, depending ona =‖,⊥ andthe precise values of the Gegenbauer moments. We may therefore conclude that perturbarections to hard spectator-scattering are non-negligible. At the same time there is no sithe series expansion is not well-behaved despite the comparatively low scale, lendingto the possibility of performing perturbative factorization at the hard-collinear scale. Thisimportant result, already mentioned in[20,32], since theoretical calculations of exclusiveB de-cays in general rely on this possibility. The present calculation and the one in[19,20]are the firstcomputations of quantum corrections to spectator-scattering.

8 The corresponding anomalous dimensions are given by(51), (53).9 We assume four massless quark flavours throughout this numerical analysis for simplicity. This is not a good

mation for the charm quark, whose mass is of the same order as the hard-collinear scale. A more precise treatmkeep the charm quark mass in the fermion loop correction to the jet-function. This could be done without diffisuch precision were required.


n

erithmicply as

nd hases-

lso

g

rithmicup im-ator. Theor which

4.2. Renormalization group improvement of theC(B1) coefficients

The complete hard-scattering term involves∫ 1

0 dτ [CB1 J ], so we now turn to the evaluatioof the CB1. With µ of order(mbΛ)1/2, the jet-function is free from large logarithms, butCB1

involves up to two powers of logarithms ln(mb/Λ) per loop from the ratio of the hard to thhard-collinear scale. In the following we derive an expression that sums the leading-logaand double-logarithmic terms to all orders in perturbation theory. We shall refer to this simthe leading-logarithmic approximation (LL).10

4.2.1. Solution to the renormalization group equationThe renormalization-group formalism that accomplishes this summation is standard a

been applied to the B-type SCETI currents in[20]. We shall recapitulate the relevant exprsions to define the notation. In the following we drop the superscript “B1” onC(B1) and denoteby C(E, τ ;µ) a generic short-distance coefficient. With the renormalization kernels(76), (77)the renormalization group equation is derived from the requirement thatC(E, τ ;µ)J (B1)(µ) isindependent of the QCD/SCETI factorization scale. This implies

(97)µd

dµC(E, τ ;µ) = −Γcusp(αs) ln

µ

2EC(E, τ ;µ) +

1∫0

dτ ′ γa(τ′, τ )C(E, τ ′;µ)

with

(98)Γcusp(αs) =∞∑

n=0

Γn

(αs

4π

)n+1

the so-called universal cusp anomalous dimension. Comparison with(76), (77)givesΓ0 = 4CF

and

(99)γa(τ, τ′) = −αs(µ)

2π

[z1(τ, τ

′) + ∆az2(τ, τ′)],

where∆a = 1 for a =‖ and∆a = 0 for a =⊥. Due to the presence of double logarithms we aneed the two-loop cusp anomalous dimension[33]

(100)Γ1 = 4CF

([67

9− π2

3

]CA − 20

9nf Tf

).

The short-distance coefficients of the B-type operators(8) to (10) and the coefficients appearinin (21), (22)evolve with the anomalous dimensionsγa(τ

′, τ ) as follows:

a =‖ C(B1)S,P , C

(B1)1−3V,A , C

(B1)1,2,5−7T , C

(B1)f+ , C

(B1)f0

, C(B1)fT

,

(101)a =⊥ C(B1)4V,A , C

(B1)3,4T , C

(B1)V , C

(B1)T1

.

10 In the literature on Sudakov resummation the analogous approximation is usually called “next-to-leading-logaapproximation”. We prefer the term “leading-logarithmic”, since, as in other applications of renormalization-groproved perturbation theory, the approximation requires only the 1-loop anomalous dimension of a generic opercomplication from double-logarithms is reflected by the presence of the so-called cusp anomalous dimension, fthe 2-loop coefficient is needed already in the leading-logarithmic approximation.


le

res-

The general solution to the renormalization group equation(97) reads

(102)C(E, τ ;µ) = e−S(E;µh,µ)

1∫0

dτ ′ Ua(τ, τ′;µh,µ)C(E, τ ′;µh),

where

S(E;µh,µ) =αs(µ)∫

αs(µh)

dαs

Γcusp(αs)

β(αs)

αs∫αs(2E)

dα′s

β(α′s)

(103)=αs(µ)∫

αs(µh)

dαs

Γcusp(αs)

β(αs)

αs∫αs(µh)

dα′s

β(α′s)

+ lnµh

2E

αs(µ)∫αs(µh)

dαs

Γcusp(αs)

β(αs)

with the QCDβ-function

β(αs) = µdαs

dµ= −2αs

∞∑n=0

βn

(αs

4π

)n+1

,

(104)β0 = 11

3CA − 4

3nf Tf , β1 = 34

3C2

A −(

20

3CA + 4CF

)nf Tf .

The evolution kernelUa(τ, τ′;µh,µ) satisfies the integro-differential equation

(105)µd

dµUa(τ, τ

′;µh,µ) =1∫

0

dτ ′′ γa(τ′′, τ )Ua(τ

′′, τ ′;µh,µ),

with initial conditionUa(τ, τ′;µh,µh) = δ(τ − τ ′). To sum the large logarithms the initial sca

µh should be of ordermb, and the evolution ends atµ of order(mbΛ)1/2.Several simplifications occur in the leading-logarithmic approximation. Eq.(103)can be in-

tegrated to

S(E;µh,µ) = − Γ0

2β0ln r ln

µh

2E+ Γ0

4β20

(4π

αs(µh)

[ln r − 1+ 1

r

]− β1

2β0ln2 r

(106)+(

Γ1

Γ0− β1

β0

)[r − 1− ln r]

),

with r = αs(µ)/αs(µh) > 1. Furthermore, the initial condition is given by the tree-level expsion forC(E, τ ;µh), which is independent ofτ (andµh), hence theτ ′ integration in(102)canbe done. We then have

(107)C(LL)(E, τ ;µ) = e−S(E;µh,µ)Ua(τ ;µh,µ)C(0)(E),

whereC(0)(E) is the tree coefficient, andUa(τ ;µh,µ) = ∫ 10 dτ ′ Ua(τ, τ

′;µh,µ) satisfies

(108)µd

dµUa(τ ;µh,µ) =

1∫0

dτ ′ γa(τ′, τ )Ua(τ

′;µh,µ),


ays

tion

e usesate

cientshmic

rms areg order

to

Fig. 3.Ua(τ,µh,µ) for µh = mb = 4.8 GeV andµ = 1.5 GeV. The upper curves refer toU‖, the lower ones toU⊥.Solid lines: exact numerical integration. Dashed lines: approximate solutions.

with initial conditionUa(τ ;µh,µh) = 1. This equation must be solved numerically. We alwuse two-loop running ofαs(µ), and putµh = mb = 4.8 GeV. As input we takeαs(4.8 GeV) =0.215, which givesαs(1.5 GeV) = 0.359 (four massless flavours). The result of this integrais shown inFig. 3for µ = 1.5 GeV. We have found that the solution to

(109)µd

dµU

appa (τ,µh,µ) =

[ 1∫0

dτ ′ γa(τ′, τ )

]U

appa (τ,µh,µ),

given by

(110)Uappa (τ,µh,µ) =

(αs(µ)

αs(µh)

)−γa(τ)/(2β0)

with αs

4πγa(τ ) = ∫ 1

0 dτ ′γa(τ′, τ ) and

γ‖(τ ) = −CF + 4

(CF − CA

2

)ln τ

τ,

(111)γ⊥(τ ) = −CF

(4τ ln τ

τ+ 1

)+ 4

(CF − CA

2

)(1+ τ

τln τ + τ ln τ

τ

)

provides a very good approximation (better than 1%) to the exact solution, provided on1-loop running forαs with αs(4.8 GeV) = 0.215 in the approximate solution. The approximexpressions are also shown inFig. 3.

4.2.2. NLO+ LL approximationWe are now in the position to give expressions for the B-type short-distance coeffi

C(E, τ ;µ), which include the complete 1-loop correction as well as the leading-logaritterms. The formula is

C(E, τ ;µ) = C(0)(E) + C(1)(E, τ ;µ) − C(0)(E)[e−SUa

]αs

(E, τ ;µ)

(112)+ C(0)(E)[e−SUa − 1

](E, τ ;µ).

The meaning of the four terms on the right-hand side is as follows: the first and second tethe tree and 1-loop coefficients, respectively. Together they constitute the next-to-leadin(NLO) approximation toC(E, τ ;µ). The fourth term is the sum of leading-logarithmic terms


luded

,tted);g or-

ms at

evolve

order),ersen

light-edIt ise

s is a

all orders minus the tree. Finally, the third term subtracts the logarithmic terms already incin the full 1-loop correctionC(1)(E, τ ;µ). The subtraction is given by

[e−SUa

]αs

(E, τ ;µ)

(113)= αs(µ)

4π

−Γ0

2

[ln2

(µ

µh

)+ 2 ln

(µh

2E

)ln

(µ

µh

)]+ ln

(µ

µh

)γa(τ )

.

To analyze the structure of the correction, we display inFig. 4 the following approximationsall normalized to the tree coefficient: (i) tree plus the logarithmic terms at 1-loop (dash-do(ii) previous approximation plus the non-logarithmic term, i.e., the complete next-to-leadinder result (dashed); (iii) previous approximation plus the sum of leading-logarithmic terorder α2

s and beyond (solid). In the numerical implementation we setµh = mb = 4.8 GeV,E = mbx/2, and regard the coefficients as functions of energy fractionx and the convolutionvariableτ . SinceE must be of ordermb, x cannot be chosen too small. We takex = 1 andx = 0.6as representative examples of large (maximal) and small energy of the light meson, andto µ = 1.5 GeV. We also fix the scale of the QCD tensor current operator toν = mb = 4.8 GeV.Fig. 4shows four of the five combinations,C

(B1)X , X ∈ f+, f0, fT , T1,V , which appear in the

hard-scattering contribution to the vector and tensor current form factors(21), (22). Only C(B1)V

is not shown, because its tree coefficient vanishes, hence onlyC(1)(E, τ ;µ) in (112)is non-zero.The following observations can be made from the figure: (a) the logarithmic term at

αs (dash-dotted lines) is a very poor approximation to the fullαs coefficient (dashed linesespecially forC(B1)

T1, which is the only one of the four coefficients shown involving the transv

anomalous dimensionγ⊥. (b) Except near the endpointsτ = 0,1, where the relative correctiodiverges, the typical next-to-leading order correction from the hard scalemb is of order 30%. Theendpoint singularities are logarithmic and disappear when the correction is folded with themeson distribution amplitude (integration overτ ). (c) The effect of the logarithmically enhancterms beyond the orderαs is negligible (difference between the solid and dashed lines).largest forC(B1)

T1towards largerτ since hereU⊥(τ ;µh,µ) is significantly different from 1, se

Fig. 3.

4.3. Spectator-scattering correction

According to(21), (22), (25) and (86)the form factorsFX(E) are given by

(114)FX(E) = C(A0)X ξa(E) + HX(E)

with the spectator-scattering term

(115)HX(E) = mB

4mb

fBfM

1∫0

dv φM(v)

∞∫0

dωφB+(ω)

1∫0

dτ C(B1)X (E, τ)Ja(τ ;v,ω).

Herea =‖ (or P in case ofξa) for X = f+, f0, fT anda =⊥ for X = V,T1. We have nowassembled all the pieces required for the evaluation ofHX(E) at orderα2

s (1-loop). From nowon we setmB/mb in (115)to 1, since the difference between the meson and the quark maspower correction beyond the accuracy of the present calculation.


esrms

Fig. 4. The short-distance coefficientsC(B1)X

(E, τ,µ) relevant to the form factorsX at two representative energy valu(x = 1 (left) andx = 0.6 (right)) atµ = 1.5 GeV normalized to the tree approximation. Dash-dotted: logarithmic teat orderαs ; dashed: full NLO approximation; solid: NLO plus logarithmic summation.


(here-

results

n, we

nctiontion

ssionsn witheandallyt

in the

the

At the leading order we insert the tree expressions for the B-type coefficient functionafter we again drop the superscript “B1”) and the jet-function and obtain

(116)H(0)X (E) = −παs(µ)CF

Nc

fBfM 〈v−1〉M2EλB

C(0)X (E).

Here as before the superscript “(0)” refers to the tree approximation. This agrees with theof [8].

To obtain the next-to-leading order result including the renormalization-group summatioinsert(112) and the jet-function into(115), and neglect cross terms of orderα3

s in the productCX Ja . The result is

HX(E) = H(0)X (E) ×

1+ 1

〈v−1〉M∫

dv

vφM(v)

C(1)(E, v)

C(0)(E)+ [Ia − 1]

− 1

〈v−1〉M∫

dv

vφM(v)

[e−SUa

]αs

(E, v)

(117)+ 1

〈v−1〉M∫

dv

vφM(v)

[e−SUa − 1

](E, v)

.

The second term in the bracket is the 1-loop hard correction; the third comes from the jet-fuand is defined in(96); the fourth and fifth are related to the renormalization group summaas in (112). The integration of the second term can be done analytically, but the expreare lengthy. They are given for selected short-distance coefficients and for the integratiothe asymptotic distribution amplitude inAppendix B. It is as straightforward to perform thintegration overφM(v) numerically. The integration of the subtraction term is elementarygiven by(113)together with(111). The integration of the last term can only be done numericusing the numerical solution of the integro-differential equation forUa . SinceS is independenof v, one needs the integrals

1∫0

dv

vφM(v,µ)U‖(v,µh,µ) = 3.037+ 3.058aM

1 + 3.051aM2 + · · · ,

(118)

1∫0

dv

vφM(v,µ)U⊥(v,µh,µ) = 2.795+ 2.980aM

1 + 3.003aM2 + · · · .

The numerical values are given forµh = 4.8 GeV andµ = 1.5 GeV.To illustrate these results we consider the three coefficients relevant to the form factors

physical form factor scheme defined in(25). Let

C(B1)0+ (τ,E) = C

(B1)f0

(τ,E) − C(B1)f+ (τ,E)R0(E),

C(B1)T + (τ,E) = C

(B1)fT

(τ,E) − C(B1)f+ (τ,E)RT (E),

(119)C(B1)T1V

(τ,E) = C(B1)T1

(τ,E) − C(B1)V (τ,E)R⊥(E).

ChoosingλB = 0.35 GeV, x = 0.85 (corresponding to light meson energyE = xmB/2 =2.24 GeV or momentum transferq2 = 4.18 GeV), and asymptotic distribution amplitudes,


orderows the

themaller.ectator-

lized to

sto

ionsplitudereog order

Fig. 5. HX(E)/H(0)X

(E) for X = 0+, T +, T1V . The two upper curves represent the complete next-to-leadingresult with (solid) and without (dashed) the renormalization group summation. The middle (dash-dotted) line shjet-function (hard-collinear) correction alone, the lower (dashed) line the hard correction alone.

curly bracket in(117)evaluates to

X = 0+: 1+ 0.283[C] + 0.371[jet] + 0.059[log, αs] − 0.073[all logs],X = T +: 1+ 0.213[C] + 0.371[jet] + 0.059[log, αs] − 0.073[all logs],

(120)X = T1V : 1+ 0.209[C] + 0.268[jet] + 0.169[log, αs] − 0.147[all logs],where the five terms correspond to the five terms on the right-hand side of(117). We observethat the hard correction [C] and the jet-function correction [jet] are of similar size, whilesum of higher-order logarithms (the sum of the last two terms) is at least a factor of 10 sThe total correction to the tree result amounts to an enhancement of (50–70)% of the spscattering effect. These features are independent of the value ofE as can be seen fromFig. 5,which displays the weak energy-dependence of the spectator-scattering correction normathe tree result.

The dependence of these results on the hadronic input parametersλB , aM1 , aM

2 is roughlyas follows.λB enters the relative correction through the moments(95) and therefore affectthe jet-function terms only. ChoosingλB = 0.5 GeV (0.25 GeV) changes the number 0.3710.295 (0.452), and 0.268 to 0.195 (0.346), an uncertainty characteristic for all energy fractx.Furthermore, there is an uncertainty due to the model for the shape of the distribution amthat correlates the logarithmic moments withλB , which we do not attempt to quantify. Theis a larger dependence of the tree resultH

(0)X (E) on λB , since it is inversely proportional t

λB . Positive Gegenbauer moments increase the tree result and the relative next-to-leadincorrection. This can be seen from(96) for the jet-function correction. ForaM

2 = 0.2, the totalrelative next-to-leading order correction increases from 64% forX = +0 (50% forX = T1V )


metersnic

s

lu-lutionntities

tants

n-th,-

cal-ughnfon andings

dleading-onsfrom

to 75% (62%). Finally, there is a dependence on the renormalization scaleµ, which we fixedto 1.5 GeV. In order to estimate this dependence, one must fix the hadronic input paraat someµ0 and evolve them toµ using(90), (93). Since the scale-dependence of the hadroparameters is within their uncertainty, we do not perform this estimate here.

5. B decay phenomenology

In this section we discuss three applications of our results toB decays. We restrict ourselveto decays to pions orρ mesons, since the results for kaons are qualitatively very similar.

We use the following parameters: theb-quark pole massmb = 4.8 GeV; the renormalizationscale of the QCD tensor currentν = mb; the initial scale for the renormalization group evotion µh = mb; the renormalization scale and final scale of the renormalization group evoµ = 1.5 GeV. This is also the (hard-collinear) scale at which all other scale-dependent quasuch as meson light-cone distribution amplitudes and the scale-dependent decays consfB ,fM⊥ are evaluated. The strong coupling is obtained fromαs(mb) = 0.215 by employing 2-loop

running (Λ(nf =4)

MS= 323.6 MeV), which givesαs(1.5 GeV) = 0.359. The pion andρ meson

parameters arefπ = 130.7 MeV, fρ‖ = 209 MeV, fρ⊥ = 150 MeV, and the second Gegebauer moment is assumed to beaM

2 = 0.1 for the pion and the distribution amplitudes of bothe longitudinal and transverseρ meson. TheB meson mass ismB = 5.28 GeV and the decay constantfB = fB/K(1.5 GeV) = 200 MeV. We assume the model(94) for the B mesondistribution amplitude andλB = 0.35 GeV. This is somewhat smaller than the value 0.46 GeVsuggested by QCD sum rule calculations[34]. Allowing λB to vary from 0.25 GeV to 0.5 GeVimplies that the value ofλB is the single most important uncertainty in the final numericalculation. The SCETI form factorsξa(E) are defined in the physical form factor scheme throfull QCD form factors according to(24). The full QCD form factors needed for this definitioare taken from the light-cone QCD sum rule calculations[35] including the parameterization otheir q2 dependence. On the basis of this input we can compute the remaining seven piρ meson form factors using(25). We relate hadronic to partonic variables by first eliminatE throughE = xmb/2 in the coefficient functions. The energy fractionx is then interpreted ax = 1 − q2/m2

B = 2E/mB , when we plot hadronic form factors as functions ofq2 or hadronicenergyE.

5.1. Symmetry-breaking corrections to form factor ratios

In the absence of radiative and power corrections, the factorization formula(1) impliesparameter-free relations between form factors[7], since onlyξa(E) appears on the right-hanside, which cancels in ratios. These relations receive corrections, which are calculable atpower in the 1/mb expansion given the above-mentioned input parameters[8]. The seven relations between the total of ten pion andρ meson form factors are obtained from the two relati(23), which do not receive any perturbative corrections, and the five relations that follow(25)by dividing through the appropriateξFF

a . For instance, the second and third equations of(25)imply

R1(E) ≡ mB

mB + mP

fT (E)

f+(E)= RT (E) +

1∫dτ C

(B1)T + (τ,E)

ΞP (τ,E)

f+(E),

0


ls the

(and

s

isplayowingluatingcorre-

an the

ouslyr-ction

% (re-rule

tios

cal-e in theym-

e basisre alsolreadybasedthetio

ns be-ations.an theclusion

(121)R2(E) ≡ mB + mV

mB

T1(E)

V (E)= R⊥(E) + mB + mV

mB

1∫0

dτ C(B1)T1V

(τ,E)Ξ⊥(τ,E)

V (E),

with C(B1)T + (τ,E), C(B1)

T1V(τ,E) the combinations of coefficient functions defined in(119). Similar

relations follow for the other form factors. The second term on the right-hand side equahard spectator-scattering term(115) divided by the appropriateξFF

a (E). Putting together(26),(116) and (117)we obtain the form factor ratios including the new next-to-leading orderresummed) correction to the spectator-scattering term.

The result of this computation is shown inFig. 6for the various form factor ratios. The ratioare normalized such that in absence of any radiative corrections they equal 1 for allq2. Our finalresult, which includesR to orderαs and the spectator-scattering term to orderα2

s as well as thesummation of leading logarithms to all orders is shown as the solid (black) curves. To dthe size of the various contributions to the complete result, we also show the result follfrom neglecting the spectator-scattering term (dash-dotted (red) curves), and from evathe spectator-scattering contribution in leading-order (long-dashed (blue) curves), whichsponds to the previous results[8]. As has already been discussed in Section4 the new NLOcorrection always enhances the symmetry-breaking effect. The correction fromR(E) in (121)is always smaller than the spectator-scattering contribution. In fact, it is even smaller thNLO spectator-scattering term, despite the fact that the latter is formally of orderα2

s . Overall, thedeviations from the symmetry-limit range up to 40%, which is significant but not anomallarge given that the typical scales involved are in the range of 1.5 GeV. The theoretical uncetainties in the relative NLO spectator-scattering term have been discussed before in Se4.The more important unknown factor resides in the normalization of the tree contribution(116),which involves the product

(122)fBfM〈v−1〉M

λBξa(E)

of hadronic parameters. We estimate the theoretical errors of the factors to be around 15fB ),15% (fρ⊥), 10% (〈v−1〉M ), 30% (λB ) and 15% (ξa(E)), so it is clear that the curves in the figuare affected by a significant normalization uncertainty. In particular, adopting the QCD sumresultλB = 0.46 GeV rather than 0.35 GeV decreases the deviations of the form factor rafrom unity by about 30%.

It is instructive to compare this result for the form factor ratios with the QCD sum ruleculations. The corresponding sum rule ratios are shown as dashed (black; black and blulower right panel) curves inFig. 6. One notices that the sum rule calculation satisfies the smetry relations remarkably well—the ratios are in general closer to 1 than predicted on thof the heavy-quark limit corrected by radiative and spectator-scattering effects. There asignificant differences concerning the sign of the correction similar to those observed ain [8]. It is unclear whether the differences between the sum rule calculations and thoseon the heavy-quark limit are due to 1/mb power corrections or ununderstood systematics ofsum rule calculation (see the discussion in[36]). For instance, the sum-rule result for the rainvolving mB/(2E)T2 −T3 has changed from about 0.7 to almost 1.2 with the update[35] of theform factor calculations. This may not be surprising, since the ratio involves cancellatiotween form factors and may be particularly sensitive to the uncertainties of sum rule calculIn such cases the SCET calculation of form factor relations is probably more reliable thQCD sum rule method. In general, the comparison of the two methods leads to the con


cectiond):the twoe shown

sider-better

rest to

Fig. 6. Corrections to theB → π andB → ρ form factor ratios as a function ofq2. The ratios equal 1 in the absenof radiative corrections. Solid (black) line: full result, including NLO and resummed leading-logarithmic correto spectator-scattering; Long-dashed (blue): without NLO+ LL correction to spectator-scattering; Dash-dotted (rewithout any spectator-scattering term. Dashed (black): QCD sum rule calculation. The lower right panel showsform factor ratios that equal 1 at leading power. For comparison, the QCD sum rule results for these two ratios ar(upper (blue) line refers toA1/V , lower (black) line toT2/T1).

that the theoretical calculations of form factors with QCD sum rules are affected by conable uncertainties until the systematics of and discrepancies with the heavy-quark limit areunderstood.

5.2. Radiative vs. semi-leptonic decay

Factorization calculations of radiative and hadronic two-bodyB decays involving only lightmesons (and leptons) make use of the form factors at maximal recoil. It is therefore of inteinvestigate the short-distance corrections atx = 1, i.e.,E = mB/2 or q2 = 0. In addition to the


ely

ericalatios

ements outors

s

exact relations(23), the first and fourth relations of(25)also degenerate to

(123)mB

2Ef0(E) = ξFF

P ,mV

EA0(E) = ξFF‖

as a consequence of the equations of motion. This leaves only two interesting ratios, namR1andR2 defined in(121). The last ratio in(25) involving T2 andT3 can be obtained fromR1replacingξFF

P = f+ by ξFF‖ .At x = 1 we obtain the analytic expressions (assuming the asymptotic formφM(v) = 6vv for

the light-meson distribution amplitude)

R1 = 1+ αs

4π

[−8

3ln

ν

mb

+ 8

3

]− 8παs(µ)

3

fBfM

MBλBξFFP

1+ αs(µ)

4π

×[−8

3ln

ν

mb

− 8

3ln2 mb

µ+ 4

3

⟨ln2 mbω

µ2

⟩+

(8

3− 2π2

9

)ln

mb

µ

+(

−9+ π2

9+ 2nf

3

)⟨ln

mbω

µ2

⟩+ 103

6+ 5π2

9− 19nf

9+ δ

‖log

]

= 1+ 0.046(RT ) − 0.1651+ 0.540(NLO spec.) − 0.01

(δ‖log

)= 0.794,

R2 = 1+ αs

4π

[−8

3ln

ν

mb

− 4

3

]+ 4παs(µ)

3

fBfM⊥MBλBξFF⊥

1+ αs(µ)

4π

×[−8

3ln

ν

mb

− 8

3ln2 mb

µ+ 4

3

⟨ln2 mbω

µ2

⟩+

(−2

3− 2π2

9

)ln

mb

µ

+(

−26

3+ π2

9+ 2nf

3

)⟨ln

mbω

µ2

⟩+ 27

2+ π2 − 2ζ(3)

3− 19nf

9+ δ⊥

log

]= 1− 0.023(RT ) + 0.086

1+ 0.418(NLO spec.) + 0.03

(δ⊥

log

)(124)= 1.102,

whereδalog denotes the small effect form the renormalization-group summation. The num

results refer to the pion (R1) andρ meson (R2) with the parameters as specified above. The rR1 with ξFF

P = f+ replaced byξFF‖ and pion parameters replaced byρ meson parameters give0.707 instead of 0.794. For comparison the QCD sum rule calculation[35] givesR1 = 0.96 (1.02for ρ and the relation involvingT2, T3) andR2 = 0.95.

The factorization approach allows us to make predictions for the exclusive radiativeB de-caysB → Mγ [2,37] andB → Ml+l− [38]. The decaysB → ργ together withB → K∗γ areparticularly interesting, because they may lead to a determination of the CKM matrix el|Vtd | or constrain flavour non-universality in penguin transitions. The main limitation turnto be the poor knowledge of SU(3) flavour symmetry breaking in the ratio of tensor form factT

ρ1 (0)/T K∗

1 (0) [39,40]. In [39] it was therefore suggested to take the ratio of theB → ργ tothe differential semi-leptonicB → ρlν branching fraction, which avoids the problem of SU(3)-breaking, but introduces the ratioT1/V of ρ-meson form factors atq2 = 0. The method relieon normalizing theB → ργ rate to the differential decay rate

(125)d2Γ (B → ρlν)

dq2 d cosθ∝ (1+ cosθ)2H 2− + (1− cosθ)2(H 2+ + 2H 2

0

)


tiveting

ds isizationd

thele-

e rel-

lies an

ectionbys not

near cosθ = 1 (with θ the angle between the neutrino momentum and theB meson momentumin the lν center-of-mass frame) andq2 = 0. The angle cut has the effect of isolating the negahelicity form factorH−, which has a simple expression in the heavy-quark limit. Neglecquadratic effects in the light meson mass,

H−(q2) =

√q2

m2B

(mB + mρ

mB

A1(q2) + m2

B − q2

mB(mB + mρ)V

(q2))

(126)= 2

√q2

m2B

(1− q2

m2B

)mB

mB + mρ

V(q2),

hence the ratio of branching fractions involves

(127)H 2−T 2

1

= 4q2

m2B

(1− q2

m2B

)2 H 2−ξFF⊥

2

q2→0→ 4q2

m2B

1

R22

.

Assigning a 60% uncertainty to the spectator-scattering contribution toR2 we obtain 1/R22 =

0.82±0.12 to be compared with the QCD sum rule value 1/R22 = 1.11±0.22, where we assigne

a 10% uncertainty to the calculation of[35]. The disagreement between the two numberunfortunate and should be resolved. Assuming the result of the calculation in the factorapproach, we obtain a 10% uncertainty on|Vtd/Vub| from the form factor ratio in the methoproposed in[39]. This does not include an uncertainty from power-suppressed effects.

The tensor-to-vector ratioR2(q2) also appears in the forward–backward asymmetry in

electroweak penguin decaysB → Ml+l−. The complete calculation of the decay matrix ement divides into “factorizable” and “non-factorizable” contributions[8,38]. In this terminology,“factorizable” contributions are related to the heavy-to-light form factors, and hence are thevant ones here. Inserting

(128)(mB + mV )T1

V= (mB − mV )

T2

A1= mBR2

into Eq. (75) of[38], we obtain the differential forward–backward asymmetry[8]

(129)dAFB

dq2∝ Re

[C9 + Y

(q2) + 2mbMB

q2Ceff

7 R2(q2) + non-fact. terms

].

Since the dependence onq2 is mainly throughR2(q2)/q2 it follows that the increase ofR2(q

2)

by several percent due to the next-to-leading order spectator-scattering correction impincrease in the position of the asymmetry zero in approximately the same proportion.

5.3. Hadronic decays

The jet-function computed in this paper also appears in the next-to-leading order corrto spectator-scattering in hadronic two-bodyB decays to light mesons. We outline this effectthe example ofB → ππ decays, keeping the discussion short, since the NLO correction iyet completely available.

The factorization formula reads[1]

(130)〈ππ |Qi |B〉 = f+(0)T Ii ∗ fπφπ + T II

i ∗ fBφB+ ∗ fπφπ ∗ fπφπ ,


up toefficient

f apick uphard-ictrongrk

ssed”m. Thetoeectator-st

iongressed

tion toeters

eic two-

ur-and 1.57er in

jet-d

de small,

whereQi denotes an operator from the effective weak Hamiltonian, and the formula holdspower corrections. The second term describes spectator-scattering. Its short-distance cois a convolutionT II

i = CIIi J‖, whereCII

i is the coefficient function of a generalization oB-type operator that takes into account the second pion. Since the pion that does notthe spectator anti-quark from theB meson decouples at the hard-scale, the physics at thecollinear scale is exactly the same as in theB → π transition. Hence the jet-function in hadrondecays equalsJ‖ [21], which has been computed above. Note that this implies that the srescattering phases are all generated at the hard scalemb (at leading order in the heavy-quaexpansion), since the jet-function is real.

Spectator-scattering is particularly important for decay amplitudes of the “colour-supprefinal stateπ0π0, because the colour-suppression is absent for the spectator-scattering tersituation is opposite for the colour-allowed final stateπ−π+. Both amplitudes are relevantπ−π0. In the following we shall therefore focus on the coefficientα2(ππ) that describes thcolour-suppressed tree amplitude. We emphasize that a complete NLO calculation of spscattering requires the calculation of the hard coefficientCII

i as well. The remarks below mutherefore be understood as preliminary.

Following the notation of[41] (Eqs.(35) and (47)) we write

α2(ππ) = C2 + C1

Nc

+ C1

Nc

αs(µ)CF

4πV2(π)

(131)+ C1

Nc

παs(µh)CF

Nc

[H tw2

2 (ππ)I‖ + H tw32 (ππ)

],

where nowµ should be chosen of ordermb andµh is a hard-collinear scale assumed to beµh =√Λhµ with Λh = 0.5 GeV. TheCi are Wilson coefficients from the effective weak-interact

Hamiltonian,V2(π) is a vertex correction, andH tw22 (ππ) + H tw3

2 (ππ) the spectator-scatterinterm at tree level, which we separated into a leading-power (“tw2”) and a power-supp“chirally enhanced” (“tw3”) term. The new ingredient in this formula is the factorI‖, whichequals 1 in the absence of the NLO correction to the jet-function, and is given by(96) includingthe correction. Exactly the same modification applies to the spectator-scattering contribuα1(ππ) and the leading-power pieces of the penguin amplitudes. Numerically, with paramdefined in[41], we obtain

α2(ππ) = 0.17− [0.17+ 0.08i]V2 + [0.11· 1.37]H tw2

2 ·I‖ + [0.07]H tw32

(default),

[0.29· 1.57]H tw22 ·I‖ + [0.17]H tw3

2(S4)

(132)=

0.22(0.18) − 0.08i (default),

0.64(0.47) − 0.08i (S4).

The various terms and factors correspond to those in(131) and we show the numbers for thdefault parameter set and the set S4 that provides a better overall description of hadronbody modes. Due to the near cancellation11 of the tree term with the vertex correction the colosuppressed tree amplitude comes essentially from spectator-scattering. The factors 1.37show the effect from the NLO correction to the jet-function. In the final line the numbbrackets gives the result from[41], the unbracketed number corresponds to including the newfunction term. To illustrate the implications of these results, we show inTable 1the CP-average

11 The size of the loop correction is due to the absence of colour-suppression, which makes the tree amplituand is therefore not an indication of failure of the perturbative expansion.


t

ithouttiont tominaryhave aoretical

ctttering

madech theInffectshardtion ofs thefpects ofher et

ctator-m thed. We

s fromrder ofh-input. In ad-

Table 1Tree amplitude coefficientsα1 andα2, and the CP-averagedππ branching ratios in units of 10−6 in the default and S4scenario of[41] showing the effect of the NLO jet function correction

Scenario Default, LO jet Default, NLO jet S4, LO jet S4, NLO je

α1(ππ) 0.99+ 0.02i 0.98+ 0.02i 0.88+ 0.02i 0.81+ 0.02iα2(ππ) 0.18− 0.08i 0.22− 0.08i 0.47− 0.08i 0.64− 0.08iBr(B0 → π+π−) 8.86 8.62 5.17 4.58Br(B0 → π0π0) 0.35 0.40 0.70 1.13Br(B− → π−π0) 6.03 6.28 5.07 5.87

B → ππ branching fractions corresponding to the four cases (default vs. S4, with and wNLO jet-function correction). For simplicity, we only consider the NLO jet function correcto the tree amplitudesα1(ππ) andα2(ππ) (colour-allowed and colour-suppressed), but nothe penguin amplitudes, since this gives the dominant effect (and the results are prelianyway, see above). It is clearly seen that the NLO correction to spectator-scattering cansignificant effect. The enhancement of the colour-suppressed tree amplitude brings the thecomputation in better agreement with data, since the largeπ0π0 rate and the smallπ+π− toπ−π0 ratio favour a large colour-suppressed tree amplitude[41]. We do not discuss the direCP asymmetries, since we expect the still missing NLO hard correction to spectator-sca(which includes a new source of rescattering phases) to be the more important factor.

6. Conclusion

Spectator-scattering plays an important role in the theory of exclusiveB decays. It is alsorather complicated, because several scales,mb (hard),

√mbΛ (hard-collinear), andΛ (hadronic)

are involved. The development of QCD factorization and soft-collinear effective theory hasit possible to formulate the calculation in terms of two separate matching steps, in whieffects from the short-distance scalesmb and

√mbΛ are calculated in perturbation theory.

previous work[18] we began the calculation of 1-loop corrections to spectator-scattering ein heavy-to-light meson form factors in the large-recoil region with the computation of thecoefficient functions. In this paper we have completed the second step with the computathe hard-collinear coefficient function, also called jet-function. Since the calculation involvedefinition of various renormalized operators in QCD, SCETI, and SCETII , and the treatment oevanescent operators in dimensional regularization, we have described the technical asthis work in some length. Our results provide a check of similar results obtained by Becal. [19,20]. The jet-function computed here is relevant to many differentB decays, includingradiative and hadronicB decays in the QCD factorization approach.

The results may be summarized as follows: we find significant enhancements of spescattering at next-to-leading order, which increase the deviation of form factor ratios froasymptotic heavy quark limit, in which perturbative and power corrections are neglectehave also included the summation of formally large logarithms lnmb/Λ, but found this effect tobe negligible compared to the full 1-loop correction. Despite the small scale of order 1.5 GeVinvolved, there is no sign that a perturbative treatment is not applicable. The 1-loop effectthe hard scale and the hard-collinear scale are about equally important, being on the o(20–40)% (depending on parameters), at least in theMS factorization scheme adopted througout this work. It follows that the dominant theoretical uncertainties are related to hadronicparameters such as moments of light-cone distribution amplitudes and decay constants


diativee NLOcementdes.sincetchingcase

u-

cor-

thankmer ofander-.B. is

e Theo-

e

dition to the symmetry-breaking corrections to form factor ratios, we also discussed raand hadronic two-body decays. Although the jet-function constitutes only one aspect of thcorrection to spectator-scattering in hadronic decays, we have seen that the NLO enhanhas interesting implications for final states with significant colour-suppressed tree amplitu

We would also like to emphasize the theoretical conclusions from this calculation,the form factors are up to now the only observables, for which a complete two-step main soft-collinear effective theory has been explicitly carried out to the 1-loop level in awith spectator-scattering. The factorization arguments that lead to the formula(1) rely on thedemonstration that the B-type SCETI currents can be matched to SCETII without encounteringendpoint-divergent convolution integrals, which would violate naive SCETII factorization[12,13]. The calculations performed here and in[18] provide an explicit verification of these argments at the 1-loop level.

Note added

We have been informed of related work by G. Kirilin, in which he computes the 1-looprection to the jet-functionJ‖, and to the coefficient functionC(B1)

f+ [42].

Acknowledgements

We are grateful to S. Jäger for careful reading of the manuscript. M.B. would like tothe INT, Seattle and KITP, Santa Barbara for their generous hospitality during the sum2004, when most of this work was being done. D.Y. acknowledges support from the Alexvon-Humboldt Stiftung and the Japan Society for the Promotion of Science. The work of Msupported in part by the DFG Sonderforschungsbereich/Transregio 9 “Computergestütztretische Teilchenphysik”.

Appendix A. Short-distance coefficients

A.1. Change of basis

The coefficient functions of the operators defined in(8) to (10) are given in terms of thosdefined and calculated in[18] (denoted with subscript “old”) as follows:scalar:

C(A0)S = C

(A0)Sold ,

(A.1)C(B1)S = C

(B1)Sold − C

(A0)Sold

x;

vector:

C(A0)1V = C

(A0)1V old , C

(A0)2V = C

(A0)3V old ,

C(A0)3V = C

(A0)2V old − C

(A0)1V old ,

C(B1)1V = C

(B1)3V old − 1

x

(2C

(A0)1V old + C

(A0)3V old

),

C(B1)2V = −C

(B1)2V old + 1(

2C(A0)1V old − C

(A0)2V old

),

x


l-vector

-

C(B1)3V = C

(B1)1V old + 1

2C

(B1)4V old + C

(A0)2V old

x,

(A.2)C(B1)4V = 1

2C

(B1)4V old − 1

x

(C

(A0)1V old − C

(A0)2V old

);tensor:

C(A0)1T = −1

2C

(A0)1T old , C

(A0)2T = C

(A0)3T old ,

C(A0)3T = C

(A0)2T old − C

(A0)1T old , C

(A0)4T = C

(A0)1T old + C

(A0)3T old − C

(A0)4T old ,

C(B1)1T = C

(B1)2T old − 1

2C

(B1)6T old + 1

x

(C

(A0)1T old + C

(A0)4T old

),

C(B1)2T = C

(B1)1T old − C

(B1)5T old − C

(A0)1T old

x,

C(B1)3T = −1

2C

(B1)6T old − 1

x

(C

(A0)1T old + C

(A0)3T old − C

(A0)4T old

),

C(B1)4T = −C

(B1)5T old + 1

x

(C

(A0)1T old − C

(A0)2T old

),

C(B1)5T = C

(B1)3T old − 1

x

(2C

(A0)2T old + 2C

(A0)3T old − C

(A0)4T old

),

C(B1)6T = 1

2C

(B1)4T old − 1

2x

(C

(A0)1T old − 2C

(A0)2T old

),

(A.3)C(B1)7T = 1

4

(C

(B1)7T old − C

(B1)4T old

) + 1

2x

(C

(A0)1T old − C

(A0)2T old

).

In the new basis the pseudoscalar coefficients equal the scalar coefficients, and the axiacoefficients equal the vector coefficients. Furthermorex = n−vn+p′/mb = 2E/mb with E theenergy of the light meson.

A.2. Coefficients appearing in the form factors

The five independent A0-coefficients appearing in the SCETI representation of the form factors(21), (22)are given by

C(A0)f+ = C

(A0)1V + x

2C

(A0)2V + C

(A0)3V

= 1+ αsCF

4π

[−2 ln2

(x

µ

)+ 5 ln

(x

µ

)− 2 Li2(1− x) − π2

12− 3 lnx − 6

],

C(A0)f0

= C(A0)1V +

(1− x

2

)C

(A0)2V + C

(A0)3V

= 1+ αsCF

4π

[−2 ln2

(x

µ

)+ 5 ln

(x

µ

)− 2 Li2(1− x) − π2

12− 3− 5x

1− xlnx − 4

],


Dre-ed

C(A0)fT

= −2C(A0)1T + C

(A0)2T − C

(A0)4T

= 1+ αsCF

4π

[−2 ln ν − 2 ln2

(x

µ

)+ 5 ln

(x

µ

)− 2 Li2(1− x) − π2

12

− 3− x

1− xlnx − 6

],

C(A0)V = C

(A0)1V

= 1+ αsCF

4π

[−2 ln2

(x

µ

)+ 5 ln

(x

µ

)− 2 Li2(1− x) − π2

12− 3− 2x

1− xlnx − 6

],

C(A0)T1

= −2C(A0)1T +

(1− x

2

)C

(A0)2T + C

(A0)3T

(A.4)

= 1+ αsCF

4π

[−2 ln ν − 2 ln2

(x

µ

)+ 5 ln

(x

µ

)− 2 Li2(1− x) − π2

12− 3 lnx − 6

].

The variableE used in(21), (22)is related tox throughx = n−v n+p′/mb = 2E/mb. We alsodefineαs = αs(µ), µ = µ/mb andν = ν/mb, whereν is the renormalization scale of the QCtensor current, andµ is the SCETI renormalization scale. Theµ dependence cancels the corsponding dependence of the SCETI form factorsξa(E). The heavy quark mass is renormalizin the pole scheme. The five independent B-coefficients are given by

C(B1)f+ = x

2C

(B1)1V + C

(B1)2V

=(

−2+ 1

x

)1+ αsCF

4π

[−2 ln2

(x

µ

)+ ln

(x

µ

)− 3

1− 2xlnx

− 2 Li2(1− x) − π2

12− 2(1− x)

1− 2x+ x

(1− 2x)(1− xξ)− 4(1− x)

(1− 2x)ξln ξ

+ x(2− xξ)

(1− 2x)(1− xξ)2ln(xξ)

]

+ αs

4π

(CF − CA

2

)[4

ξln ξ ln µ + 2

ξF (x, xξ ) + 2

x(1− 2x)ξ ξG

+ 2

1− 2x

(2(1− x)

ξln ξ + 3− 2x

ξln ξ − x

1− xξln(xξ)

)],

C(B1)f0

=(

1− x

2

)C

(B1)1V + C

(B1)2V

= −1

x

1+ αsCF

4π

[−2 ln2

(x

µ

)+ ln

(x

µ

)− 3 lnx − 2 Li2(1− x) − π2

12

+ 2

ξ

((2− x) lnx

1− x− (2− xξ) ln(xξ)

1− xξ

)+ x(2− xξ)

(1− xξ)2ln(xξ) + x

1− xξ

]

+ αs

4π

(CF − CA

2

)[4

ξln ξ ln µ + 2

ξln ξ − 2x

1− xξln(xξ) + 2

ξF (x, xξ )

− 2¯ G

],

xξξ


d

s given

C(B1)fT

= −C(B1)5T

= 1

x

1+ αsCF

4π

[−2 ln ν − 2 ln2

(x

µ

)+ ln

(x

µ

)− 3 lnx − 2 Li2(1− x) − π2

12− 2

+ 2

ξ

((2− x) lnx

1− x− (2− xξ) ln(xξ)

1− xξ

)− x(2− xξ) ln(xξ)

(1− xξ)2− x

1− xξ

]

+ αs

4π

(CF − CA

2

)[4

ξln ξ ln µ + 2

ξln ξ + 2x

1− xξln(xξ) + 2

ξF (x, xξ )

− 2

xξ ξG

],

C(B1)V = C

(B1)4V

= αsCF

4π

1

1− xξ

[x lnx

1− x− ln(xξ)

1− xξ− ln ξ

ξ− 1

]

+ αs

4π

(CF − CA

2

)[− 2

xξln ξ − 2

xξln ξ + 2 ln(xξ )

1− xξ+ 2 ln(xξ)

1− xξ− 2

x2ξ ξG

],

C(B1)T1

=(

1− x

2

)C

(B1)3T + C

(B1)4T

= (−1)

1+ αsCF

4π

[−2 ln ν − 2 ln2

(x

µ

)+ ln

(x

µ

)+ lnx − 2 Li2(1− x)

− π2

12− 1− 4ξ

ξln ξ ln µ − 2

ξln ξ − 2ξ

ξF (x, xξ)

]

+ αs

4π

(CF − CA

2

)[(4(1+ ξ)

ξln ξ + 4ξ

ξln ξ

)ln µ − 2 ln ξ + 2 lnξ

− 2F(xξ , xξ) + 2

ξF (x, xξ ) + 2

ξF (x, xξ) − 2

xξG

(A.5)− 2

xξ

(Li2(1− x) − Li2(1− xξ)

) − 2

].

The variablesE andτ used in(21), (22)are related tox andξ throughx = 2E/mb andξ = τ .Diagrammaticallyξ corresponds ton+p′

2/n+p′, the fractional longitudinal momentum carrieby the transverse collinear gluon in the B-type current operator. We also useξ ≡ 1 − ξ , andintroduced the two abbreviations

F(y, z) ≡ ln2 y − lny + Li2(1− y) − ln2 z + ln z − Li2(1− z),

(A.6)G ≡ Li2(1− x) − Li2(1− xξ ) − Li2(1− xξ) + π2

6.

These results are obtained by taking the appropriate linear combinations of the coefficientin [18]. The variables(x1, x2) used there are related to(x, ξ) by x1 = xξ andx2 = xξ (ξ ∈ [0,1]).


Appendix B. Integrals of coefficient functions

B.1. Integration of the jet-function

The integrals∫ 1

0 dτ ja(τ ;v,ω) of the jet-functions(78), (79)are given by

1∫0

dτ j‖(τ ;v,ω)

= CF L2 +(

CF

[−7

3+ 2 ln v

]+

(CF − CA

2

)[22

3+ 2 lnv

v

]+ 4

3nf Tf

)L

+ CF

[53

9− π2

6+

(−4

3+ 1

v

)ln v + ln2 v

]

+(

CF − CA

2

)[−170

9+

(22

3− 2

v

)ln v + 1

v

(π2

3− 2 lnv + ln2 v

+ 2(1− 2v)(Li2(v) − Li2(v)

))]+

(−5

3+ ln v

)4

3nf Tf ,

1∫0

dτ j⊥(τ ;v,ω)

= CF L2 +(

CF

[−7

3+ 2 ln v

]+

(CF − CA

2

)[22

3+ 2v

vln v + 2(2− v) lnv

v

]

+ 4

3nf Tf

)L

+ CF

[44

9− π2

6+

(−7

3+ 2

v

)ln v + ln2 v

]

+(

CF − CA

2

)[−152

9+

(22

3− 2

v

)ln v + v

vln2 v + 1

vv

(π2

3− 2v lnv

(B.1)+ v(2− v) ln2 v + 2(1− 2v)(Li2(v) − Li2(v)

))]+

(−5

3+ ln v

)4

3nf Tf .

The analytic expressions of the integrals enteringIa (see(87)) read (settingnf = 4, Tf = 1/2andCF = 4/3 andCA = 3)

λB

3

1∫0

dv

vφM(v)

∞∫0

dω

ωφB+(ω)

1∫0

dτ j‖(τ ;v,ω)

= 4

3

⟨L2⟩ + (

−19

3+ π2

9

)〈L〉 + 169

18− 2π2

9− 8

3ζ(3)

+ aM1

[4

3

⟨L2⟩ + (

−110

9+ π2

3

)〈L〉 + 464

27+ π2

9− 8ζ(3)

]

+ aM2

[4⟨

L2⟩ + (−157+ 2π2)

〈L〉 + 646+ 8π2

− 16ζ(3)

],

3 9 3 27 9


e three

λB

3

1∫0

dv

vφM(v)

∞∫0

dω

ωφB+(ω)

1∫0

dτ j⊥(τ ;v,ω)

= 4

3

⟨L2⟩ + (

−6+ π2

9

)〈L〉 + 65

9− π2

3− 8

3ζ(3)

+ aM1

[4

3

⟨L2⟩ + (

−110

9+ π2

3

)〈L〉 + 413

27+ π2

9− 8ζ(3)

]

(B.2)+ aM2

[4

3

⟨L2⟩ + (

−313

18+ 2π2

3

)〈L〉 + 4975

216+ 7π2

9− 16ζ(3)

].

B.2. Convolution ofC(B1)X with the light-cone distribution amplitude

Because the expressions are lengthy, we only list the results for the convolution of thcombinations of coefficients functions in the physical form factor scheme as defined in(25), andassume that the light-cone distribution amplitude are given by their asymptotic formsφM(v) =6vv. The convolution integrals read

1∫0

dv

vφM(v)

(C

(B1)f0

− C(B1)f+ R0

)(x, v)

= −6(1− x)

x

1+ αsCF

4π

[−2 ln2

(x

µ

)+ ln

(x

µ

)−

(2− 1

(1− x)2

)lnx

−(

2+ 2

x

)Li2(1− x) − π2

12

x − 4

x+ 6+ x

1− x

]

+ αs

4π

(CF − CA

2

)[(4π2

3− 8

)ln

(x

µ

)− 4x lnx

1− x+ 4 lnx Li2(x)

− 4 Li3(1− x) − 8 Li3(x) + 8− 4π2

3− 4ζ(3)

],

1∫0

dv

vφM(v)

(C

(B1)fT

− C(B1)f+ RT

)(x, v)

= 6

1+ αsCF

4π

[−2 ln ν − 2 ln2

(x

µ

)+ ln

(x

µ

)+

(2+ 2

x+ 1

1− x

)lnx

−(

2+ 2

x+ 2

x2

)Li2(1− x) − π2

12

x2 − 4x − 4

x2+ 3− 2

x

]

+ αs

4π

(CF − CA

2

)[(4π2

3− 8

)ln

(x

µ

)+ 4+ 4x

xlnx + 4 lnx Li2(x)

− 4

x2Li2(1− x) −

(4− 4

x2

)Li3(1− x) − 8 Li3(x) + 8− 4

x

− 2π2(2− 1 − 1

2

)−

(4+ 4

2

)ζ(3)

],

3 x x x


hep-

58.

1∫0

dv

vφM(v)

(C

(B1)T1

− C(B1)V R⊥

)(x, v)

= −3

1+ αsCF

4π

[−2 ln ν − 2 ln2

(x

µ

)− ln

(x

µ

)+

(2+ 4

x+ 2

1− x

)lnx

−(

2+ 2

x+ 4

x2

)Li2(1− x) − π2

12

x2 − 4x − 8

x2+ 9

2− 4

x

]

+ αs

4π

(CF − CA

2

)[(4π2

3− 4

)ln

(x

µ

)+ 2 lnx − 4(1− x)

xlnx Li2(x)

+ 4

xLi2(1− x) −

(4− 4

x− 4

x2

)Li3(1− x) + 8(1− x)

xLi3(x)

(B.3)+ 1− 2π2

3− 4

(1+ 1

x+ 1

x2

)ζ(3)

]

with x = 2E/mb, µ = µ/mb, ν = ν/mb, andαs = αs(µ).

References

[1] M. Beneke, G. Buchalla, M. Neubert, C.T. Sachrajda, Phys. Rev. Lett. 83 (1999) 1914, hep-ph/9905312;M. Beneke, G. Buchalla, M. Neubert, C.T. Sachrajda, Nucl. Phys. B 591 (2000) 313, hep-ph/0006124.

[2] M. Beneke, Th. Feldmann, D. Seidel, Nucl. Phys. B 612 (2001) 25, hep-ph/0106067;S.W. Bosch, G. Buchalla, Nucl. Phys. B 621 (2002) 459, hep-ph/0106081.

[3] A. Abada, et al., Nucl. Phys. B 416 (1994) 675, hep-lat/9308007;C.W. Bernard, P. Hsieh, A. Soni, Phys. Rev. Lett. 72 (1994) 1402, hep-lat/9311010;A. Abada, et al., Nucl. Phys. B (Proc. Suppl.) 119 (2003) 641, hep-lat/0209092;UKQCD Collaboration, K.C. Bowler, J.F. Gill, C.M. Maynard, J.M. Flynn, JHEP 0405 (2004) 035,lat/0402023;J. Shigemitsu, et al., hep-lat/0408019;M. Okamoto, et al., Nucl. Phys. B (Proc. Suppl.) 140 (2005) 461, hep-lat/0409116.

[4] V.L. Chernyak, I.R. Zhitnitsky, Nucl. Phys. B 345 (1990) 137;V.M. Belyaev, A. Khodjamirian, R. Rückl, Z. Phys. C 60 (1993) 349, hep-ph/9305348;A. Khodjamirian, R. Rückl, S. Weinzierl, O.I. Yakovlev, Phys. Lett. B 410 (1997) 275, hep-ph/9706303;E. Bagan, P. Ball, V.M. Braun, Phys. Lett. B 417 (1998) 154, hep-ph/9709243;P. Ball, V.M. Braun, Phys. Rev. D 58 (1998) 094016, hep-ph/9805422;P. Ball, R. Zwicky, Phys. Rev. D 71 (2005) 014015, hep-ph/0406232;P. Ball, R. Zwicky, Phys. Rev. D 71 (2005) 014029, hep-ph/0412079.

[5] M. Wirbel, B. Stech, M. Bauer, Z. Phys. C 29 (1985) 637;N. Isgur, D. Scora, B. Grinstein, M.B. Wise, Phys. Rev. D 39 (1989) 799;D. Melikhov, B. Stech, Phys. Rev. D 62 (2000) 014006, hep-ph/0001113.

[6] N. Isgur, M.B. Wise, Phys. Lett. B 237 (1990) 527.[7] J. Charles, A. Le Yaouanc, L. Oliver, O. Pène, J.C. Raynal, Phys. Rev. D 60 (1999) 014001, hep-ph/98123[8] M. Beneke, Th. Feldmann, Nucl. Phys. B 592 (2001) 3, hep-ph/0008255.[9] A. Szczepaniak, E.M. Henley, S.J. Brodsky, Phys. Lett. B 243 (1990) 287.

[10] Y.Y. Keum, H.N. Li, A.I. Sanda, Phys. Rev. D 63 (2001) 054008, hep-ph/0004173;C.D. Lü, K. Ukai, M.Z. Yang, Phys. Rev. D 63 (2001) 074009, hep-ph/0004213.

[11] S. Descotes-Genon, C.T. Sachrajda, Nucl. Phys. B 625 (2002) 239, hep-ph/0109260.[12] M. Beneke, Th. Feldmann, Nucl. Phys. B 685 (2004) 249, hep-ph/0311335;

M. Beneke, Th. Feldmann, Eur. Phys. J. C 33 (2004) S241, hep-ph/0308303.[13] B.O. Lange, M. Neubert, Nucl. Phys. B 690 (2004) 249, hep-ph/0311345.[14] C.W. Bauer, D. Pirjol, I.W. Stewart, Phys. Rev. D 67 (2003) 071502, hep-ph/0211069.


12529;

[15] C.W. Bauer, S. Fleming, D. Pirjol, I.W. Stewart, Phys. Rev. D 63 (2001) 114020, hep-ph/0011336;C.W. Bauer, D. Pirjol, I.W. Stewart, Phys. Rev. D 65 (2002) 054022, hep-ph/0109045.

[16] M. Beneke, A.P. Chapovsky, M. Diehl, Th. Feldmann, Nucl. Phys. B 643 (2002) 431, hep-ph/0206152;M. Beneke, Th. Feldmann, Phys. Lett. B 553 (2003) 267, hep-ph/0211358.

[17] R.J. Hill, M. Neubert, Nucl. Phys. B 657 (2003) 229, hep-ph/0211018.[18] M. Beneke, Y. Kiyo, D.S. Yang, Nucl. Phys. B 692 (2004) 232, hep-ph/0402241.[19] T. Becher, R.J. Hill, JHEP 0410 (2004) 055, hep-ph/0408344.[20] R.J. Hill, T. Becher, S.J. Lee, M. Neubert, JHEP 0407 (2004) 081, hep-ph/0404217.[21] C.W. Bauer, D. Pirjol, I.Z. Rothstein, I.W. Stewart, Phys. Rev. D 70 (2004) 054015, hep-ph/0401188.[22] D. Pirjol, I.W. Stewart, Phys. Rev. D 67 (2003) 094005, hep-ph/0211251;

D. Pirjol, I.W. Stewart, Phys. Rev. D 69 (2004) 019903, Erratum.[23] G. Burdman, G. Hiller, Phys. Rev. D 63 (2001) 113008, hep-ph/0011266.[24] M. Beneke, Th. Feldmann, D. Seidel, Eur. Phys. J. C 41 (2005) 173, hep-ph/0412400.[25] M. Beneke, V.A. Smirnov, Nucl. Phys. B 522 (1998) 321, hep-ph/9711391.[26] V.A. Smirnov, E.R. Rakhmetov, Theor. Math. Phys. 120 (1999) 870, Teor. Mat. Fiz. 120 (1999) 64, hep-ph/98

V.A. Smirnov, Applied Asymptotic Expansions in Momenta and Masses, Springer-Verlag, Berlin, 2002.[27] T. Becher, R.J. Hill, M. Neubert, Phys. Rev. D 69 (2004) 054017, hep-ph/0308122.[28] G.P. Lepage, S.J. Brodsky, Phys. Rev. D 22 (1980) 2157;

A.V. Efremov, A.V. Radyushkin, Phys. Lett. B 94 (1980) 245.[29] B.O. Lange, M. Neubert, Phys. Rev. Lett. 91 (2003) 102001, hep-ph/0303082.[30] X.D. Ji, M.J. Musolf, Phys. Lett. B 257 (1991) 409.[31] A.G. Grozin, M. Neubert, Phys. Rev. D 55 (1997) 272, hep-ph/9607366.[32] M. Beneke, G. Buchalla, M. Neubert, C.T. Sachrajda, hep-ph/0411171.[33] J. Kodaira, L. Trentadue, Phys. Lett. B 112 (1982) 66.[34] V.M. Braun, D.Y. Ivanov, G.P. Korchemsky, Phys. Rev. D 69 (2004) 034014, hep-ph/0309330;

A. Khodjamirian, T. Mannel, N. Offen, hep-ph/0504091.[35] P. Ball, R. Zwicky, Ref.[4].[36] F. De Fazio, Th. Feldmann, T. Hurth, hep-ph/0504088.[37] J.G. Chay, C. Kim, Phys. Rev. D 68 (2003) 034013, hep-ph/0305033;

S. Descotes-Genon, C.T. Sachrajda, Nucl. Phys. B 693 (2004) 103, hep-ph/0403277;T. Becher, R.J. Hill, M. Neubert, hep-ph/0503263.

[38] M. Beneke, Th. Feldmann, D. Seidel, Ref.[2].[39] S.W. Bosch, G. Buchalla, JHEP 0501 (2005) 035, hep-ph/0408231.[40] A. Ali, E. Lunghi, A.Y. Parkhomenko, Phys. Lett. B 595 (2004) 323, hep-ph/0405075;

M. Beneke, T. Feldmann, D. Seidel, Ref.[24].[41] M. Beneke, M. Neubert, Nucl. Phys. B 675 (2003) 333, hep-ph/0308039;

M. Beneke, G. Buchalla, M. Neubert, C.T. Sachrajda, Nucl. Phys. B 606 (2001) 245, hep-ph/0104110.[42] G.G. Kirilin, hep-ph/0508235.

s. Theusing

iant andacena–mpute thee scalarderateility ofthe way

beliant strongtr gaugeonfine-difficult


Bulk dynamics in confining gauge theories

Marcus Berga,∗, Michael Haacka, Wolfgang Mückb

a Kavli Institute for Theoretical Physics, University of California Santa Barbara, CA 93106-4030, USAb Dipartimento di Scienze Fisiche, Università degli Studi di Napoli “Federico II”, via Cintia, 80126 Napoli, Italy

Received 2 September 2005; accepted 29 November 2005


Abstract

We consider gauge/string duality (in the supergravity approximation) for confining gauge theoriesystem under scrutiny is a 5-dimensional consistent truncation of type IIB supergravity obtainedthe Papadopoulos–Tseytlin ansatz with boundary momentum added. We develop a gauge-invarsigma-model-covariant approach to the dynamics of 5-dimensional bulk fluctuations. For the MaldNuñez subsystem, we study glueball mass spectra. For the Klebanov–Strassler subsystem, we colinearized equations of motion for the 7-scalar system, and show that a 3-scalar sector containing thdual to the gluino bilinear decouples in the UV. We solve the fluctuation equations exactly in the “moUV” approximation and check this approximation numerically. Our results demonstrate the feasibanalyzing the generally coupled equations for scalar bulk fluctuations, and constitute a step ontowards computing correlators in confining gauge theories. 2005 Published by Elsevier B.V.

1. Introduction

Gauge/string duality offers an alternative approach to aspects of supersymmetric non-Agauge theories that are hard to describe with conventional techniques. For example, acoupling many non-Abelian gauge theories exhibitconfinement, the familiar yet still somewhamysterious phenomenon that the only finite-energy states are singlets under the cologroup: at colliders, we never see quarks directly, only colorless hadrons. The details of cment, and of other non-perturbative phenomena such as chiral symmetry breaking, are

* Corresponding author.E-mail addresses:[email protected](M. Berg),[email protected](M. Haack),[email protected]

(W. Mück).

0550-3213/$ – see front matter 2005 Published by Elsevier B.V.doi:10.1016/j.nuclphysb.2005.11.029





M. Berg et al. / Nuclear Physics B 736 (2006) 82–132 83

rbativeg on aical

the-

ce then,enwhosewith thisS. De-ies hasmean

is thenfielde of the

Z flowuctionndaryrated)tes theo cure

osingstingputedlators

spon-cpaceo theviourary. Ifsimpleiour in

elatorsuch a

assler

n thecontext,

to capture with conventional gauge theory methods. In the dual picture, the non-pertugauge theory regime is typically described by weakly coupled closed strings propagatinspace of higher dimensionality (thebulk), and their dynamics can be approximated by classsupergravity.

One of the most powerful applications of gauge/string duality is the calculation of fieldory correlation functions from the dual bulk dynamics. This idea was developed in[1–3] forsuperconformal gauge theories, whose gravity duals are anti-de Sitter (AdS) spaces. Sinin a program known asholographic renormalization([4–9] and references therein), it has besystematically generalized to gauge theories that are conformal in the ultraviolet (UV),gravity duals are asymptotically AdS spaces. There are several reasons to push aheadline of research. First, confining gauge theories have duals that are not asymptotically Adspite some progress, the holographic calculation of correlators in confining gauge theornot yet been carried out in any controlled approximation (we shall specify later what weby that). Second, interesting new supergravity solutions have been found recently[10–13]. Theyare regular and thus qualify as dual configurations of ground states in gauge theories. Itonly natural to investigate the possibility of calculating correlation functions for their dualtheories. In this paper, we report on progress towards this general goal and sharpen somremaining challenges.

Some of the asymptotically AdS backgrounds studied in the literature (such as the GPP[14]) were originally envisaged as toy-model duals of confining gauge theories. The obstrto being full-fledged duals is a naked curvature singularity at finite distance from the bouinto the bulk. One would have liked to interpret this distance as the (dynamically genescale of onset of confinement in the dual field theory, but unbounded curvature invalidause of the supergravity approximation to string theory. Although string theory appears tthese curvature singularities by the enhancing mechanism[15] or the Myers effect[16], correla-tors are always computed in the supergravity approximation. In practice, this involves impregularity conditions on the bulk fluctuations at the curvature singularity. It would be intereto quantify the precise effect of the string theory resolution on the explicit correlators comin singular supergravity backgrounds, but it would be simpler if one could compute corredirectly from regular duals.

An even simpler approach to computing correlators is thehard-wall approximation, which hasbeen used in the effort to connect gauge/string duality to real QCD. This “AdS/QCD corredence” studies problems like meson–hadron coupling universality[17–19] and deep inelastiscattering[20]. In the hard-wall approximation, one replaces the regular solution by AdS scut off at a minimal radiusrIR, the idea being that some of the physics should be insensitive tdetails of the geometry in the deep infrared (IR) region, while retaining conformal UV beha(seeFig. 1). Then, the issue arises which boundary conditions to impose at the IR boundone were able to compute correlators directly in the regular solution at least for somecases, a qualitative picture of which hard-wall boundary conditions best mimic the behavthe regular case could be pieced together.1

Thus, we are interested in the question to what extent it is feasible to compute corrdirectly from regular supergravity duals of confining gauge theories. The first example of sbulk configuration was the warped deformed conifold solution found by Klebanov and Str

1 Incidentally, in [21], cut-off AdS was used to model the dynamics of D-brane inflationary cosmology oKlebanov–Strassler background. It is not unreasonable to hope that our methods will also prove useful in thatfor the same reasons as for AdS/QCD.

84 M. Berg et al. / Nuclear Physics B 736 (2006) 82–132

ings insingular

ade ofrcan beacena–

sted

nfigu--AdS

holo-ce of aThus,soniane field

Sat theplingsewhatsomes can

phicm su-tainedlin)t wass, hadharony

Fig. 1. Bulk toy models used in the literature. In the hard-wall approximation, the bulk is exactly AdS, so couplthe gauge theory do not run (represented by straight sides in the figure). In singular approximations, like theconifold, there is logarithmic running, but also a curvature singularity (represented by the black dot).

(KS) [22]. TheN = 1 supersymmetric gauge theory dual to this solution undergoes a cascSeiberg dualities, as recently explained in greater detail in the lecture notes by Strassle[23].2

Importantly, the curvature remains small everywhere, so the supergravity approximationused at all energies. Another example is the wrapped D5-brane, also known as the MaldNuñez (MN) solution[25]. In the infrared, it shares many properties withN = 1 SYM theory[26–29], but it becomes six-dimensional little string theory in the UV. We are mostly interein the KS solution, since there the supergravity approximation is under full control.

Even before addressing the implementation of gauge/string duality in such bulk corations, it is worth noting that the dual field theory interpretation of non-asymptoticallysupergravity configurations poses a conceptual problem (which we do not resolve): ingraphic renormalization, the asymptotically AdS bulk region corresponds to the presenWilsonian renormalization group (RG) UV fixed point in the 4-dimensional gauge theory.in its absence, one might wonder whether the dual gauge theory is well-defined in the Wilsense. Several viewpoints on this are possible. One may defer the UV completion of ththeory to string theory, as in the MN solution. Alternatively, one can attempt todefinethe fieldtheory by its holographic dual, as advocated in[30]. Another hope may be to embed the Ksolution into a more complicated configuration with an asymptotically AdS region, so thdual field theory is UV-conformal, but there is an intermediate energy range where coudo run logarithmically as in the KS solution. Here, we adopt a pragmatic approach, somlike [30]. We extrapolate from AdS/CFT that the bulk dynamics encodes information aboutdual field theory, which might only be an effective theory, and try to see which of its featurebe extracted by existing holographic renormalization technology.

Optimistically, then, we would like to investigate whether techniques similar to holograrenormalization can be used to calculate (effective) field theory correlation functions fropergravity duals in non-asymptotically AdS setups. First results in this direction were obby Krasnitz[31–33] for certain 2-point functions in the singular conifold (Klebanov–Tseytbackground[34], and in the limit of very large energy. No counterterms were obtained, but iargued that the particular correlators studied would only have received minor correctioncounterterms been included. Counterterms were recently studied in a tour de force by Aet al.[30], who obtained renormalized one-point functions of the stress-energy tensor.

2 See also[24] for a nice review of the KS solution.


interre-

rators

ite co-blem);wheree “fluc-

heolution

ionalcribed

ted in.il-

lutions,y gener-s of theansatz,egra-ationsigmaystemro thisr haveto).

flowpriately

d solu-variantlismholo-ive”

five-ationsoupledntain-

all three

To address the problem of computing correlators systematically, one must face threelated issues:

(1) define precisely the duality relations between supergravity fields and field theory ope(the “dictionary” problem);

(2) renormalize the bulk prescription for correlation functions, that is, compute the requisvariant counterterms and show the absence of divergences (the “renormalization” pro

(3) solve for the dynamics of supergravity fluctuations about the background of interest,the fluctuations must be allowed to vary along the external space–time coordinates (thtuation” problem).

In this paper, we will mostly address the last issue.3 We focus on gauge theories dual to tregular supergravity solutions discussed above: the MN solution and especially the KS sof type IIB supergravity in ten dimensions.

In holographic renormalization, the bulk dynamics is 5-dimensional (for a 4-dimensgauge theory). Thus, we need to find a sector of type IIB supergravity which can be desby a 5-dimensional system, while allowing for the background solutions we are interesPapadopoulos and Tseytlin (PT)[35] found an effective 1-dimensional action (subject to a Hamtonian constraint) that is general enough to describe both the MN and KS background sowhere the fields only depend on the radial coordinate. This suggests that one can suitablalize their ansatz to allow the parameterizing scalar fields to depend also on the coordinate4 dimensions of the gauge theory. In other words, we add boundary momentum to the PTwhich leads to a 5-dimensional effective theory. We will show that, after imposing an intbility constraint that is automatically satisfied for the MN and KS systems, this generalizconstitutes a consistent truncation of type IIB supergravity and gives rise to a non-linearmodel of scalars coupled to 5-dimensional gravity. Moreover, the resulting 5-dimensional sfalls into a general class of actions dubbed “fake supergravity” actions in[36], since the scalapotential is determined by a function resembling a superpotential. We will mostly stick tterminology (i.e., “fake supergravity”), even though the background solutions we considebeen shown to preserve some supersymmety[37,38], and one might expect the full systembe embeddable in a supersymmetric system (see Section3 for some further comments on thisThese fake supergravity actions are formally similar to those governing holographic RGbackgrounds in standard AdS/CFT, which suggests that they can be studied using approgeneralized AdS/CFT techniques.

Thus, we need to study the dynamics of fluctuations about the (MN and KS) backgrountions in the effective five-dimensional bulk system. To this end, we generalize the gauge-informalism developed in[39] to generic multi-scalar systems. The gauge-invariant formaovercomes technical difficulties encountered in early work on correlation functions ingraphic RG flows[40–42]. These difficulties arose from the fact that the fluctuations of “actscalars (those with a non-trivial radial background profile) couple to the fluctuations of thedimensional metric already at the linear level, making it inconsistent to set the metric fluctuto zero when studying the scalar fluctuations, or vice versa. Consistent treatment of the csystem typically involved, even in the simplest cases, third-order differential equations co

3 Although it may seem that the first and second issues should be resolved first, this point is moot: ultimatelyquestions have to be addressed, and as we shall see, the solution to one may help with the others.


Happily,m, andal-

re were

ystem,tackleults ofuationsutionsfuture,l

elatorsed forreover,

on thexample,des andof statesione

roblemwever,ent in-uplingsystem.l tosolu-fnsider

lly byontentns in

ell, soll KS

rly 1980sl

geome-variabless wasalism

ing spurious gauge redundancies that needed to be painstakingly factored out by hand.fluctuations are manifestly disentangled at the linear level in the gauge-invariant formalistheir equations of motion are second order.4 The formalism was applied to the holographic cculation of three-point functions and scattering amplitudes in[49] (see also[50] for earlier workon three-point functions), and the main ideas for the generalization that we undertake hepresented in[51].

These two ingredients—consistent truncation to a five-dimensional fake supergravity sand a general gauge-invariant formalism to describe its fluctuations—put us in a position tothe “fluctuation problem” in the list above, and now we proceed to summarize the new resour work. We first note that in most cases, one can only expect to solve the fluctuation eqnumerically, but there are notable exceptions and simplifying limits where analytical solare possible. Still, with the hope that issues 1 and 2 in the above list will be solved in thewe wish to emphasize that numerical integration ofclassicalgauge-invariant ordinary differentiaequations is a vastly simpler problem than numerically computing the corresponding corrby lattice methods directly in the gauge theory, so the formulation of equations well-suitnumerical analysis should be important even in the absence of analytical solutions. Moeven without solving issues 1 and 2, there are physical quantities that should not dependcounterterms and which can, therefore, be addressed directly using our methods. For eglueball masses in the gauge theory correspond to the existence of normalizable bulk modo not depend on renormalization details. As an example, we calculate the mass spectrafor the N = 1 gauge theory dual of the MN solution, up to a caveat discussed in Sect5.We note that mass spectra obtained in the literature[52,53] disagree with ours, which will bdiscussed more thoroughly in Sections5 and 7.

For the KS background, one can only hope to obtain numerical results, so we pose the pin terms of gauge-invariant variables and leave numerical evaluation to future work. Howe cananalytically study the scalar fluctuations of the KS system (i.e., the 7 scalars presthe KS ansatz) in the singular Klebanov–Tseytlin (KT) background[34], which is a sensible approximation to the ultraviolet region of the KS background. In this case, we observe decobetween the 4-scalar KT system and the 3 additional scalars that are present in the KSWe will refer to this group of 3 scalars as thegluino sector, because it contains the scalar duathe gluino bilinear trλλ. The remaining equations are simple enough to allow for analyticaltions in the “moderate UV” regime considered by Krasnitz[31–33], in terms of combinations oBessel functions and logarithms. For the ultraviolet physics of the KS gauge theory, we cothe Krasnitz approximation to be controlled, since we will be able to check it numericacomputing the same solutions in KS. We leave a thorough check for future work and courselves with comparing our analytical results to numerical solutions of the full equatiothe KT background. The result is that the Krasnitz approximation seems to work very wwe expect our analytical solutions to be useful as guidance in numerical work in the fubackground.

4 Gauge-invariant variables for linearized scalar-gravity systems have been studied in cosmology since the ea[43,44]. Those variables are similar to the ones used in holographic renormalization in[4,45,46]; typical cosmologicabackgrounds are themselves very similar to the Poincaré-sliced AdS domain walls used in the simplest RG flowtries. The connection between the linearized cosmology variables and linearized holographic-renormalizationwas studied in[47]. Also, holography of finite-temperature field theories using linearized gauge-invariant variableinitiated in[48]. Although the applications in this paper are worked out at the linear level, our gauge-invariant formis defined non-linearly.


n-ationsys-

IBiven in

tionss andianceupled)s aboutof thelained

ively.rationgrationrity. Itble to

he MNto theingulars are

xtractd.ts.

wingraphics foro non-s wasch to

del of

Now, let us outline the rest of the paper. In Section2, we start by reviewing briefly the essetials of holographic renormalization in AdS/CFT, in particular the dictionary and renormalizproblems. On the way, we will introduce generic fake supergravity, which is the typical bulktem in holographic RG flows. Then, in Section3, we perform a consistent truncation of type Isupergravity to a 5-dimensional fake supergravity system. Details of the calculation are gAppendix A.

Section4 is dedicated to the generalization of the gauge-invariant analysis of bulk fluctua[39] to a generic “fake supergravity” system, allowing for an arbitrary number of scalaran arbitrary (but invertible) sigma-model metric. The principle of reparametrization invaris the beacon that guides us to the main result of this section: a system of (generally cosecond-order differential equations, which describes the dynamics of the scalar fluctuationPoincaré-sliced domain walls in a manifestly gauge-invariant fashion. The presentationgauge-invariant method is intended to be pedagogical: the principal line of argument is expin the main text, while details are included in the appendices.

In Sections5 and 6we use our techniques to study the MN and KS systems, respectFor both, we shall first derive the most general background solutions including all integconstants. Although the regular bulk configurations correspond to a unique choice of inteconstants, we find it useful to keep the constant governing the resolution of the singuladetermines the vacuum expectation value of the gluino bilinear, and by tuning it one is aconsider regimes where analytic solutions to the fluctuation equations are possible. In tsystem, we discover a number of normalizable (subleading only) modes, which we linkmass values of glueball states. In the KS system, we perform the calculation in the sKT background and, in addition, apply the Krasnitz approximation. The resulting solutionvery similar to the ones Krasnitz found in simpler cases, but we refrain from trying to ecorrelators given that the “dictionary” and “renormalization” problems have yet to be solve

Finally, Section7 contains conclusions and a discussion of possible further developmen

2. Review: correlation functions from AdS/CFT

In this section, we briefly review some essentials of holographic renormalization, follothe three-pronged list of problems discussed in the introduction. We review how hologrenormalization systematically resolves the “dictionary” and “renormalization” problemasymptotically AdS bulk geometries. These two steps must ultimately be generalized tasymptotically AdS setups. We leave their general resolution to future work (initial progresmade in[30]), but we will comment on some of the specific challenges. A general approathe third issue in the list, the fluctuation problem, will be described in detail in Section4.

Let us start by introducing a generic “fake supergravity” system, a non-linear sigma moscalar fields with a particular potential, coupled to gravity ind +1 dimensions (typically,d = 4).Its action is given by5

(2.1)S =∫

dd+1x√

g

[−1

4R + 1

2Gab(φ)∂µφa∂µφb + V (φ)

],

5 We follow the curvature conventions of MTW and Wald[54,55], i.e., the signature is mostly “+”, and Rijkl =

∂kΓ ij l + Γ i

kmΓ mjl − (k ↔ l). This has the opposite sign of the convention used in[39].


.

s havepe IIBs

theed

veralupled

ey aregh thegiven

ient

xed-th

en

arianthee dualf pri-

latorsNCs)

n

where the potential,V (φ), follows from a superpotential,W(φ), by

(2.2)V (φ) = 1

2GabWaWb − d

d − 1W2.

The matrixGab(φ) is the inverse of the sigma model metricGab(φ). Our notation is as in[39],i.e., derivatives ofW with respect to fields are indicated as subscripts, as inWa = ∂W/∂φa .Moreover, the sigma model metric and its inverse are used to lower and raise field indices

Actions of the form(2.1) arise in a variety of cases, such as the familiar truncation ofN =8, d = 5 gauged supergravity, where several holographic RG flow background solutionbeen found. As we shall see in the next subsection, other consistent truncations of tysupergravity can also give rise to effective actions of the form(2.1). This richness in applicationis our main motivation for considering the generic case in detail.

The existence of the superpotentialW ensures that the equations of motion arising fromaction(2.1)admit a particular class of solutions withd-dimensional Poincaré invariance, callPoincaré-sliced domain walls6 or holographic RG flow backgrounds:

(2.3)ds2 = dr2 + e2A(r)ηij dxi dxj , φa = φa(r).

That is, the radial domain wall in the metric is supported by a radial profile of one or sescalars (the “active” scalars). The background fields are determined by the following cofirst-order equations:

(2.4)∂rA(r) = − 2

d − 1W(φ), ∂r φ

a(r) = GabWb.

These relations do not specify the background uniquely (integration constants!), but thsufficient for the general analysis carried out in this section. We also note that, althouvarious backgrounds we study in this paper are “logarithmically warped” and not usuallyin the form(2.3), one can always reach this form by a change of radial variable.

For the system(2.4) to admit an asymptotically AdS solution, it is necessary and sufficthat the superpotentialW possess a local extremum with a non-zero value, i.e.,Wa(φ0) = 0 forall a. Then,φ0 is called a fixed point. Without loss of generality, we can assume that the fipoint value ofW is negative7: W(φ0) = −(d −1)/(2L), whereL is the characteristic AdS lengscale which is often set toL = 1.

Let us now briefly review how the issues discussed in Section1 are addressed in AdS/CFT. Wstart with issue 1, the dictionary between gauge theory operators and bulk fields. The actio(2.1)is manifestly invariant under field redefinitions—this is indeed the point of the gauge-invformalism that we develop in Section4—but this invariance is given up when formulating tone-to-one correspondence between bulk fields and primary conformal operators of thgauge theory. As is well known, conformal invariance imposes that two-point functions omary conformal operators of different weights vanish:

(2.5)⟨O∆(x1)O∆′(x2)

⟩ = 0 for ∆ = ∆′.

In AdS/CFT, this orthogonality property is achieved for the holographically calculated correby the following choice of field variables. Let us consider Riemann normal coordinates (R

6 As opposed to, for example, the AdS-sliced domain walls studied in[9,56], where thed-dimensional boundary cabe AdS instead of flat space.

7 Note that an overall sign change ofW can be absorbed by changing the sign of the coordinater .


uch

the

tof

ionsn-e samens un-fromwn

ticallynt pa-llyound-to thecorre-otionseriesl to antic

ion ofon, theintctions,adraticd dif-ms are

g

[55,57] in field space around the fixed pointφ0. This means that we choose field variables sthatφ0 = 0, and that the sigma model connections (defined later in(4.1)) vanish atφ0. This stillleaves us the freedom to imposeGab(φ0) = δab and, by means of a rotation, to diagonalizesymmetric matrix of second derivatives ofW at φ0. With this choice of parametrization,W hasthe following expansion around the fixed point,

(2.6)W = − (d − 1)

2L− 1

2

∑a

λa

(φa

)2 + · · · ,

where the ellipsis stands for terms that are at least cubic inφ. Using the AdS/CFT dictionary, iis now a simple matter to establish that the fieldsφa are dual to primary conformal operatorsdimensions8

(2.7)∆a = d

2±

∣∣∣∣d2 − λa

∣∣∣∣.For pure AdS,(2.6) ensures that the matrix of holographically calculated two-point functis diagonal, that is, Eq.(2.5) follows. In general,(2.6) is not enough to unambiguously idetify a map between supergravity modes and field theory operators. For operators with thdimension, one can usually distinguish them by other quantum numbers like transformatioderR-symmetry groups. (When even that fails, one can try to use additional informationthe correlators[7].) It is fair to say that the dictionary question is well understood in knoasymptotically AdS examples.

The second issue, renormalization, is solved in general for bulk systems with asymptoAdS bulk geometries by holographic renormalization. The reader is referred to the relevapers[4–9,60,61]and lecture notes[62] for details. Holographic renormalization systematicaremoves the divergences by first formulating the bulk theory on a bulk space with cut-off bary located well in the asymptotic UV region. Covariant local counterterms are addedaction so that removing the cutoff yields a finite generating functional, and therefore finitelation functions. The result of this procedure is most compactly described in terms of the nof sourcesandresponses, which are the coefficients in front of the leading and subleading sin the asymptotic expansion of the bulk fields, respectively. That is, a bulk scalar that is duaoperator of dimension∆a (with + sign in(2.7)) displays asymptotic behavior of the schemaform

(2.8)φa(x, r) ≈ e−(d−∆a)r[φa(x) + · · ·] + e−∆ar

[φa(x) + · · ·],

where φ and φ denote the source and response functions, respectively. Up to the additscheme-dependent local terms, which arise from adding finite counterterms to the actiresponse function represents theexact one-point functionof the dual operator, i.e., the one-pofunction in the presence of sources. Thus, in order to calculate higher correlation funone needs to solve the dynamics of bulk fluctuations up to the required order (e.g., qufor 3-point functions), extract the response function from their asymptotic behavior, anferentiate with respect to the sources. It is important to note that although the local terscheme-dependent, in general they cannot just be dropped. As was stressed in[4,5], correlation

8 Usually the upper sign applies. The lower sign can be chosen if|d/2 − λa | < 1, and is accompanied by imposinirregular boundary conditions on the bulk fields[7,58,59].


enti-

flow,int

AdS

uldof this

involve

enor-

asymp-not be aater inuld be

s,gsible toe the-rategyal limit.

e bulkd of thens thatnt localeneral

Sec-

G flowrator.

functions computed in conflicting schemes will in general fail to fulfill the requisite Ward idties. The most efficient renormalization method to date is that presented in[9], that homes in onthe minimal calculation needed for each correlator.

Here is an example of a correlation function calculated in this fashion: take the GPPZwhich is N = 4 SYM deformed by a∆ = 3 operator insertion. The result for the two-pofunction of this operator for arbitrary boundary momentump is [4,41,42]9

(2.9)

⟨Oφ(p)Oφ(−p)

⟩ = N2

2π2

p2

2

[ψ

(3

2+ 1

2

√1− p2

)+ ψ

(3

2− 1

2

√1− p2

)− 2ψ(1)

],

whereψ(z) = Γ ′(z)/Γ (z). Note that the only scale in this expression is the asymptoticlength scaleL, which has been set to unity and is easily restored replacingp → pL. The ultravi-olet (p2 → ∞) asymptotics is that of the limiting conformal theory, namely〈Oφ(p)Oφ(−p)〉 →p2∆−4 logp. The infrared regime (smallp2) encodes the spectrum in a series of poles. It wobe very interesting to understand the connection, if any, between AdS/CFT correlatorstype and high-energy correlators computed by integrability in QCD (see e.g.[63]), summinglarge numbers of certain classes of diagrams. It is intriguing that those correlators alsotheψ function.

To end this section, let us outline how we imagine approaching the “dictionary” and “rmalization” problems in the non-asymptotically AdS case. The absence of a fixed point ofW , asin the KS and MN solutions, invalidates some of the strategies discussed above. First, thetotic behavior must probably be studied on a case-by-case basis: in general, there maybasis in which the scalars decouple asymptotically (we will encounter examples of this lthe MN and KS systems). This means that the bulk field/boundary operator dictionary shoreformulated as finding suitable source functions for the boundary operators.10 One possibilityis to generalize the AdS/CFT definition of the source function as follows: A system ofn cou-pled second-order differential equations forn scalars has 2n independent asymptotic solutionof whichn can be regarded as “leading” andn as “subleading”. Then coefficients of the leadinsolutions can be defined to be sources of dual gauge theory operators. It might be posexploit the coupling between the bulk fields to describe operator mixing in the dual gaugory, but we leave this interesting question for the future. For the KS case, we follow the stadopted so far in the literature: to consider fields that are mass eigenstates in the conformThis does not, of course, resolve the dictionary issue in the non-conformal case.

Second, in AdS/CFT, asymptotically AdS behavior implies that the divergent terms of thon-shell action can be ordered into a double expansion in powers of the scalar fields annumber of boundary derivatives, with higher order terms being less divergent. This meathe number of divergent terms is finite, and that they can be cancelled by adding covariacounterterms at a cutoff boundary. At present, we have no equivalent prescription for gbulk systems, although the results of[30] are very promising.

A general approach to solving the “fluctuation problem” will be described in detail intion 4. But first, we need to derive the system in which we will study fluctuations.

9 The correlators given in these papers differ by a scheme-dependent local term.10 Note that this is so even in generic AdS/CFT, where the bulk fields may not decouple in holographic Rbackgrounds in some cases, making it ambiguous to speak of the dual bulk field of a specific gauge theory ope


sotion

is sug-lars thatll fivey the-tion is

s giventhemain

tinctionction isand

epend

3. Adding boundary momentum to the PT ansatz

The Papadopoulos–Tseytlin (PT) ansatz for type IIB supergravity solutions with fluxe[35]reduces the problem of finding these particular flux solutions to solving the equations of mderiving from an effective one-dimensional action subject to a zero-energy constraint. Thgests that it should be possible to generalize the PT ansatz in such a way that the scaparametrize the 10-dimensional solution depend not only on a “radial” variable, but on a“external” variables. This corresponds to allowing for non-zero momentum in the boundarory, as required for computing correlators as functions of momentum. Such a generalizaindeed possible, and we shall present the result in this section, with the technical detailin Appendix A. In order not to unnecessarily overload the notation, we deviate slightly fromconvention used in the appendix by dropping tildes from the 5-dimensional objects. In thetext, the meaning of the symbols should be clear from the context, whereas a clearer disis needed for the detailed calculations in the appendices. The resulting five-dimensional aof the form(2.1). It will be important for us that in many cases of interest, including the KSMN systems, a superpotentialW generating the potentialV via (2.2) is known[35].

The equations of motion of type IIB supergravity in the Einstein frame are

RMN = 1

2∂MΦ∂NΦ + 1

2e2Φ∂MC∂NC + 1

96g2

s FMPQRSFPQRSN

+ gs

4

(e−ΦHMPQH

PQN + eΦFMPQF

PQN

)(3.1)− 1

48gMN

(e−ΦHPQRHPQR + eΦFPQRF PQR

),

(3.2)d dΦ = e2ΦdC ∧ dC − gs

2e−ΦH3 ∧ H3 + gs

2eΦF3 ∧ F3,

(3.3)d(e2Φ dC

) = −gseΦH3 ∧ F3,

(3.4)d(eΦ F3

) = gsF5 ∧ H3,

(3.5)d(e−ΦH3 − CeΦF3

) = −gsF5 ∧ F3,

(3.6)F5 = F5,

where we have used the notation

F3 = dC2, H3 = dB2, F5 = dC4, F3 = F3 − CH3, F5 = F5 + B2 ∧ F3.

From the last definition follows the Bianchi identity

(3.7)dF5 = H3 ∧ F3.

In the following we setgs = 1 andα′ = 1.Our ansatz for a consistent truncation follows PT closely, but allows the scalar fields to d

on all five external coordinates. Thus, we take

ds210 = e2p−x ds2

5 + (ex+g + a2ex−g

)(e2

1 + e22

)+ ex−g

[e2

3 + e24 − 2a(e1e3 + e2e4)

] + e−6p−xe25,


r

xIIB

n

by PT,onrability

eapped

d MN

ved

ds25 = gµν dyµ dyν,

H3 = h2 e5 ∧ (e4 ∧ e2 + e3 ∧ e1) + dyµ ∧ [∂µh1(e4 ∧ e3 + e2 ∧ e1)

+ ∂µh2(e4 ∧ e1 − e3 ∧ e2) + ∂µχ(−e4 ∧ e3 + e2 ∧ e1)],

F3 = Pe5 ∧ [

e4 ∧ e3 + e2 ∧ e1 − b(e4 ∧ e1 − e3 ∧ e2)]

+ dyµ ∧ [∂µb(e4 ∧ e2 + e3 ∧ e1)

],

Φ = Φ(y), C = 0,

(3.8)F5 = F5 + F5, F5 = Ke1 ∧ e2 ∧ e3 ∧ e4 ∧ e5,

wherep, x, g, a, b, h1, h2,K andχ are functions of the external coordinatesyµ, andP is aconstant measuring the units of 3-form flux across the 3-cycle ofT 1,1 in the UV. For readersfamiliar with the KS background, it may be useful to note thatΦ = χ = 0 in KS, and the othefields have backgrounds as given later in Section6.1.

We are using the KS convention for the forms,11 i.e.,

e1 = −sinθ1 dφ1, e2 = dθ1, e3 = cosψ sinθ2 dφ2 − sinψ dθ2,

(3.9)e4 = sinψ sinθ2 dφ2 + cosψ dθ2, e5 = dψ + cosθ1 dφ1 + cosθ2 dφ2.

We note that the first term in the ansatz forF3 is essentiallyω3 = g5 ∧ ω2 in KS notation, andas it will turn out thatb → 0 in the UV, we see that the ansatz forF3 indeed describes a flupiercing the 3-cycle ofT 1,1 in the UV. Thus, we have parametrized the 10d fields of typesupergravity by a 5d metric,gµν , and a set of ten scalars,Φ, p, x, g, a, b, h1, h2, K andχ . As in[35], one finds (again, details are relegated toAppendix A) that some of the equations of motio(3.1)–(3.7)impose constraints on this system of fields, namely

(3.10)K = Q + 2P(h1 + bh2),

for a constantQ that sets the AdS scale whenP = 0, and

(3.11)∂µχ = (e2g + 2a2 + e−2ga4 − e−2g)∂µh1 + 2a(1− e−2g + a2e−2g)∂µh2

e2g + (1− a2)2e−2g + 2a2.

Although this latter constraint is a 5d generalization of the analogous constraint foundunlike in their case it does not only eliminateχ from the action, but also imposes restrictionsthe possible sets of independent fields. These restrictions arise from the demand of integ(∂ν∂µχ = ∂µ∂νχ ) of the five first-order partial differential equations(3.11). Considering thefour special cases given in[35], one finds that(3.11) is satisfied for the singular conifold (thKT solution, a special case of the KS system), the deformed conifold (KS), and the wrD5-brane (MN), but not in general for fluctuations about the resolved conifold[64]. Furthercomments on this appear below. Thus, we shall, in the following, consider only the KS ansystems.

Imposing the constraints(3.10) and (3.11), the remaining equations of motion can be derifrom the 5-dimensional action

(3.12)S5 =∫

d5y√

g

[−1

4R + 1

2Gab(φ)∂µφa∂µφb + V (φ)

],

11 The relation to the PT and MN conventions can be found in footnote 7 of[35].


e su-tion in

f tenriteion oflars inal in aever,

or

calars.ple,tial isealon thest in

assler

m the

with sigma model metric

Gab(φ)∂µφa∂µφb

= ∂µx∂µx + 1

2∂µg∂µg + 6∂µp∂µp + 1

2e−2g∂µa∂µa + 1

4∂µΦ∂µΦ

+ 1

2P 2eΦ−2x∂µb∂µb + e−Φ−2x

e2g + 2a2 + e−2g(1− a2)2

(1+ 2e−2ga2)∂µh1∂

µh1

(3.13)

+ 1

2

[e2g + 2a2 + e−2g

(1+ a2)2]

∂µh2∂µh2 + 2a

[e−2g

(a2 + 1

) + 1]∂µh1∂

µh2

,

and potential

V (φ) = −1

2e2p−2x

[eg + (

1+ a2)e−g] + 1

8e−4p−4x

[e2g + (

a2 − 1)2e−2g + 2a2]

+ 1

4a2e−2g+8p + 1

8P 2eΦ−2x+8p

[e2g + e−2g

(a2 − 2ab + 1

)2 + 2(a − b)2](3.14)+ 1

4e−Φ−2x+8ph2

2 + 1

8e8p−4x

[Q + 2P(h1 + bh2)

]2.

As emphasized above, we must remember that integrability of(3.11)effectively restricts us tothe KS and MN systems. With this restriction, the system(3.12)with kinetic terms(3.13)andpotential(3.14) represents a consistent truncation of type IIB supergravity. Moreover, thperpotential exists and is known in both cases. We show the consistency of the truncaAppendix A.

It is an interesting and (as far as we know) open question how the truncation of(3.13) and(3.14)to the KS system can be made manifestly supersymmetric. As explained in[35] (and aswe review in Section6.1), this truncation introduces one more constraint on the system oscalars, cf.(6.1). Together with(3.10) and (3.11)this leaves seven independent scalars. To wdown a manifestly supersymmetric effective action for them might require a generalizatthe ansatz(3.8). To some readers it may seem discouraging that the number of real scathe KS system is odd, as four-dimensional intuition would indicate that the superpotentisupersymmetric theory ought to be a holomorphic function in complex field variables. Howthis intuition does not apply in odd dimensions. InN = 2 theories in five dimensions, the vectmultiplet only contains a real scalar, so it is conceivable that a potential of the form(2.2)could beappropriate for a supersymmetric theory, even if the derivatives are with respect to real sA similar situation arises inN = 2 supersymmetric theories in three dimensions (for examthose obtained from Calabi–Yau fourfold compactifications of M-theory). There, the potengiven by an expression similar to(2.2) but involving two functions, one depending on the rscalars of the vector multiplets and the other being a holomorphic function dependingremaining scalars[65–68]. It would be interesting to see whether it is also possible (at leacertain cases) to rewrite the general form of the potential in a five-dimensional gaugedN = 2supergravity, given in[69], in a form that resembles(2.2).

We would also like to connect the discussion above to the work on the Klebanov–StrGoldstone mode found in[70,71] (this mode was predicted already in[72]).12 Since we ar-gued that the analysis of fluctuations about the resolved conifold does not extend fro

12 See also[73].


eneral-of theill not

theuationf theed

rice ofm. Oneuch anetweenalll ones.

el

odelf-ratic

rds,ariant

nts

one-dimensional to the five-dimensional truncation in any obvious way, one needs to gize the ansatz to satisfy the integrability constraint if one wants to study the dynamicsGoldstone mode multiplet. We have no reason to doubt that this is possible, but we wpursue it further here.

4. Real fluctuations in fake supergravity

4.1. The sigma-model covariant field expansion

It is our aim to study the dynamics of the fake supergravity system(2.1), (2.2)on someknown backgrounds of the form(2.3), (2.4). In this section, we shall expand the fields aroundbackground, exploiting the geometric nature of the physical variables to formulate the fluctdynamics gauge-invariantly. Our arguments will closely follow the original development ogauge-invariant method for a single scalar in[39], but important new ingredients will be needin order to account for the general sigma model.

As is well known in gravity, reparametrization invariance of space–time comes at the pdragging along redundant metric variables together with the physical degrees of freedoattempts to reduce redundancy by gauge fixing, but as mentioned in the introduction, sapproach causes problems for fluctuations in holographic RG flows, due to the coupling bmetric and scalar fluctuations. Thus, following[39], we shall start from a clean slate keepingmetric degrees of freedom and describe in the next subsection how to isolate the physica

The geometry of the sigma-model target space is characterized by the metricGab(φ), whichwe assume to be invertible, the inverse being denotedGab(φ). One can define the sigma-modconnection

(4.1)Gabc = 1

2Gad(∂cGdb + ∂bGdc − ∂dGbc),

and its curvature tensor

(4.2)Rabcd = ∂cGa

bd − ∂dGabc + Ga

ceGebd − Ga

deGebc.

We also define the covariant field derivative as usual, e.g.,

(4.3)DaAb ≡ Aa|b ≡ ∂bAa − GcabAc.

All indices after a bar “|” are intended as covariant field derivatives according to(4.3). Moreover,field indices are lowered and raised withGab andGab, respectively.

Armed with this notation, it is straightforward to expand the scalar fields in a sigma-mcovariant fashion. The naive ansatzφa = φa + ϕa , introducingϕa simply as the coordinate diference between the pointsφ andφ in field space, leads to non-covariant expressions at quadand higher orders, because theseϕa do not form a vector in (tangent) field space. In other wothe coordinate difference is not a geometric object. However, it is well known that a covexpansion is provided by theexponential map[55,57],

(4.4)φa = expφ (ϕ)a ≡ φa + ϕa − 1

2Ga

bcϕbϕc + · · · ,

where the higher order terms have been omitted, and the connectionGabc is evaluated atφ.

Geometrically,ϕ represents the tangent vector atφ of the geodesic curve connecting the poiφ andφ, and its length is equal to the geodesic distance betweenφ andφ; seeFig. 2.


i-yback-y the

onnt.l

the

a-

),

Details

r fromulk and

Fig. 2. Illustration of the exponential map.

It is also a standard result that the componentsϕa coincide with the Riemann normal coordnates (RNCs) (with origin atφ) of the pointφ (see e.g.[57]). This fact can be used to simplifthe task of writing equations in a manifestly sigma-model covariant form. Namely, given aground pointφ, we can use RNCs to describe some neighborhood of it and then emplofollowing properties at the origin of the RNC system,

(4.5)Gabc = 0, Ra

bcd = ∂cGabd − ∂dGa

bc,

in order to express everything in terms of tensors. Because the background fields dependr , wemust be careful to use(4.5)only outsider-derivatives, but the simplifications are still significa

Finally, let us also define a “background-covariant” derivativeDr , which acts on sigma-modetensors as, e.g.,

(4.6)Drϕa = ∂rϕ

a + GabcW

bϕc.

If a tensorAa depends onr only implicitly through its background dependence, then we findidentity

(4.7)DrAa(φ) = Wb(φ)DbAa(φ).

The background-covariant derivativeDr will be important in our presentation of the field equtions in Section4.4.

4.2. Gauge transformations and invariants

The form of the background solution(2.3) lends itself well to the ADM (or time-slicingformalism for parametrizing the metric degrees of freedom[54,55]. Instead of slicing in timewe shall write a general bulk metric in the radially-sliced form

(4.8)ds2 = (n2 + nin

i)dr2 + 2ni dr dxi + gij dxi dxj

wheregij is the induced metric on the hypersurfaces of constantr , andn andni are the lapsefunction and shift vector, respectively. It will be important to us that the objectsn, ni andgij

transform properly under coordinate transformations of the radial-slice hypersurfaces.concerning the geometry of hypersurfaces are reviewed inAppendix B. Again, we will not puttildes on the bulk quantities in the main text, as the meaning of the symbols should be cleathe context. In contrast, tildes are used in the appendices in order to clearly distinguish bhypersurface quantities.

We can now expand the radially-sliced metric around the background configuration:

(4.9)gij = e2A(r)(ηij + hij ), ni = νi, n = 1+ ν,


t theat

e setally, i.e.,

a form

oupleds startomor-

in-

e writ-For

-ation

ns,

wherehij , νi andν denote small fluctuations. Henceforth, we shall adopt the notation thaindices of metric fluctuations, as well as of derivatives∂i , are raised and lowered using the fl(Minkowski/Euclidean) metric,ηij .

Now let us turn to the question of isolating the physical degrees of freedom from thof fluctuationshi

j , νi, ν,ϕa introduced so far. In the earlier AdS/CFT literature one usu

removed the redundancy following from diffeomorphism invariance by partial gauge fixingby placing conditions on certain components of the metric, such asn ≡ 1, ni ≡ 0. And indeed,it is always possible to perform a change of coordinates which transforms the metric intothat satisfies the gauge conditions.

Alas, as discussed in the introduction, partial gauge fixing can create problems in csystems. Instead, we will obtain the equations of motion in gauge-invariant form. Let uby considering the effect of diffeomorphisms on the fluctuation fields. We consider a diffephism of the form

(4.10)xµ = expx′[ξ(x′)

]µ = x′µ + ξµ(x′) − 1

2Γ µ

νρ(x′)ξν(x′)ξρ(x′) + · · · ,whereξ is infinitesimal. Notice that we found it convenient to apply the diffeomorphismversely, i.e., we have expressed the old coordinatesxµ in terms of the new coordinatesx′µ. Theuse of the exponential map implies that also the transformation laws for the fields can bten covariantly (the functionsξµ(x′) are thought of as the components of a vector field).example, a scalar field transforms as

(4.11)δφ = ξµ∂µφ + 1

2ξµξν∇µ∂νφ + · · · ,

whereas a covariant tensor of rank two transforms as

δEµν = ξλ∇λEµν + (∇µξλ)(

Eλν + ξρ∇ρEλν

) + (∇νξλ)(

Eµλ + ξρ∇ρEµλ

)+ (∇µξλ

)(∇νξρ)Eλρ + 1

2ξρξλ

(∇ρ∇λEµν − RσλµρEσν − Rσ

λνρEµσ

)(4.12)+ · · · .

For the metric tensorgµν , (4.12)simplifies to

(4.13)δgµν = ∇µξν + ∇νξµ + (∇µξλ)(∇νξλ) − Rµλνρξλξρ + · · · .

Eqs.(4.11) and (4.12)are most easily derived using RNCs aroundx′ and using(4.5). The secondorder terms inξ have been included here in order to illustrate the covariance of the transformlaws. For our purposes, the linear terms will suffice.

Splitting the fake supergravity fields into background and fluctuations, as defined in(4.9) and(4.4), the transformations(4.11) and (4.13)become gauge transformations for the fluctuatioto lowest order:

δϕa = Waξr +O(f ), δν = ∂rξr +O(f ),

δνi = ∂iξ r + e2A∂rξi +O(f ),

(4.14)δhij = ∂j ξ

i + ∂i(ηjkξ

k) − 4

d − 1Wδi

j ξr +O(f ).


r-

fields

s-oid

owestnalysisometricnt of the

e shallsome

n

nvariant

By O(f n) we mean terms of ordern in the fluctuationsϕa,hij , νi, ν. Furthermore, let usdecomposehi

j as follows,

(4.15)hij = hT T i

j + ∂iεj + ∂j εi + ∂i∂j H + 1

d − 1δij h,

wherehT T ij denotes the traceless transverse part, andεi is a transverse vector. It is straightfo

ward to obtain from(4.14)

δhT T ij = O(f ), δεi = Πi

j ξj +O(f ),

(4.16)δH = 2∂iξi +O(f ), δh = −4Wξr +O(f ).

The symbolΠij denotes the transverse projector,

(4.17)Πij = δi

j − 1

∂i∂j .

The main idea of our approach is to construct gauge-invariant combinations from thehT T i

j , εi , h,H,ν, νi, ϕa. Using the transformation laws(4.14) and (4.16), this is straightfor-

ward, and to lowest order, one finds the gauge-invariant fields13

(4.18)aa = ϕa + Wa h

4W+O

(f 2),

(4.19)b = ν + ∂r

(h

4W

)+O

(f 2),

(4.20)c = e−2A∂iνi + e−2A h

4W− 1

2∂rH +O

(f 2),

(4.21)di = e−2AΠij ν

j − ∂rεi +O

(f 2),

(4.22)eij = hT T i

j +O(f 2).

The variablesc anddi both arise fromδνi , which has been split into its longitudinal and tranverse parts. We chose theFraktur typeface for the gauge invariant variables in order to avconfusion with the field indices, and still keep notational similarity with[39]. Notice thatc anddi have been rescaled with respect to[39] for later convenience.

Although we have carried out the construction of gauge-invariant variables only to lorder, and this is all we will need here, it is necessary for consistency that the preceding acan be extended to higher orders, in principle. In this context it becomes clear that the genature of the field expansions, as expressed by the exponential map, is a crucial ingrediemethod.

Finally, let us prepare the ground for the arguments of the next subsection, where wanalyze the implications of gauge-invariance on the equations of motion. Let us introducemore compact notation. Consider the set of gauge-invariant fields,I = aa,b, c,di , ei

j . Fromthe definitions(4.18)–(4.22)we see that there is a one-to-one correspondence betweenI and asubset of the fluctuation fields,Y = ϕa, ν, νi, hT T i

j . We also collect the remaining fluctuatio

13 The choice of gauge-invariant variables is not unique, of course, as any combination of them will be gauge-ias well.


o

ne

y

theng thate

This

stablish

scalarch

ns are

m

variables into a set,X = h,H, εi. Henceforth, the symbolsI , X andY shall be used also tdenote members of the corresponding sets.

One can better understand the correspondence betweenI andY by noting that(4.18)–(4.22)can be rewritten as

(4.23)Y = I + y(X) +O(f 2),

wherey is a linear functional of the fieldsX. Going to quadratic order in the fluctuations, owould find

(4.24)Y = I + y(X) + α(X,X) + β(X, I) +O(f 3),

whereα andβ are bi-linear in their arguments. Terms of the formγ (I, I ) do not appear, as thecan be absorbed intoI .

We interpret the gauge-invariant variablesI as the physical degrees of freedom, whereas(d + 1) variablesX represent the redundant metric variables. This can be seen by observione can solve the transformation laws(4.16)for the generatorsξµ, which yields equations of thform

(4.25)ξµ = zµ(δX) +O(f 2) = δzµ(X) +O

(f 2),

with zµ being a linear functional.

4.3. Einstein’s equations and gauge invariance

It is our aim to cast the equations of motion into an explicitly gauge-invariant form.means that the final equations should contain only the variablesI and make no reference toXandY . Reparametrization invariance suggests that this should be possible, and we shall ethe precise details in this subsection.

Let us consider Einstein’s equations, symbolically written as

(4.26)Eµν = 0,

but it is clear that the arguments given below hold also for the equations of motion for thefields. To start, let us expand the left-hand side of(4.26)around the background solution, whiyields, symbolically,

(4.27)Eµν = E(1)1

µν (X) + E(1)2µν (Y ) + E(2)1

µν (X,X) + E(2)2µν (X,Y ) + E(2)3

µν (Y,Y ) +O(f 3).

Here,E(1) andE(2) denote linear and bilinear terms, respectively. The background equatiosatisfied identically. SubstitutingI for Y using(4.24)yields

(4.28)Eµν = E(1)1µν (X) + E(1)2

µν (I ) + E(2)1µν (X,X) + E(2)2

µν (X, I) + E(2)3µν (I, I ) +O

(f 3).

Notice that the functionalsE(1)2 andE(2)3 are unchanged (Y is just replaced byI ), whereas theothers are modified by theX-dependent terms of(4.24), which we indicate by adorning thewith a tilde. For example,E(2)2 receives contributions fromE(2)2, E(2)3 andE(1)2.

In order to simplify(4.28), we consider its transformation under the diffeomorphism(4.10).On the one hand, from the general transformation law of tensors(4.12)we find, using also(4.25),that it should transform as

(4.29)δEµν = [∂µδzλ(X)

]Eλν + [

∂νδzλ(X)

]Eµλ + δzλ(X)∂λEµν +O

(f 3).


ght-

second-consis-

endentof mo-

ge

otion

On the other hand, the variation of(4.28)is

(4.30)δEµν = E(1)1µν (δX) + 2E(2)1

µν (δX,X) + E(2)2µν (δX, I) +O

(f 3).

Let us compare(4.29) and (4.30)order-by-order. The absence of first-order terms on the rihand side of(4.29)implies that

(4.31)E(1)1µν (X) = 0.

It can easily be checked that this is indeed the case. Then, substitutingEµν = E(1)2µν (I ) +O(f 2)

into the right-hand side of(4.29)yields

(4.32)δEµν = δ

[∂µzλ(X)

]E

(1)2λν (I ) + [

∂νzλ(X)

]E

(1)2µλ (I ) + zλ(X)∂λE

(1)2µν (I )

+O(f 3).

Comparing(4.32)with the second-order terms of(4.30), we obtain

E(2)1µν (X,X) = 0,

(4.33)E(2)2µν (X, I) = [

∂µzλ(X)]E

(1)2λν (I ) + [

∂νzλ(X)

]E

(1)2µλ (I ) + zλ(X)∂λE

(1)2µν (I ).

Hence, we find that a simple expansion of Einstein’s equations yields gauge-dependentorder terms, but they contain the (gauge-independent) first-order equation, and so cantently be dropped. Happily, we arrive at the following equation, which involves onlyI :

(4.34)E(1)2µν (I ) + E(2)3

µν (I, I ) +O(f 3) = 0.

The argument generalizes recursively to higher orders. One will find that the gauge-depterms of any given order can be consistently dropped, because they contain the equationtion of lower orders.

Eq.(4.34)and its higher-order generalizations are obtained using the following recipe:

Expand the equations of motion to the desired order dropping the fieldsX

and replacing every fieldY by its gauge-invariant counterpartI .

This rule is summarized by the following substitutions,

(4.35)ϕa → aa, ν → b, e−2Aνi → di + ∂i

c, hij → ei

j .

Sinceeij is traceless and transverse, the calculational simplifications arising from(4.35)are con-

siderable. For the reader’s reference, the expressions that result from(4.35)for some geometricobjects are listed at the end ofAppendix C.

Let us conclude with the remark that, although the rules(4.35)can be interpreted as the gauchoiceX = 0, the equations we found are truly gauge invariant.

4.4. Equations of motion

In this section, we shall put the above preliminaries into practice. The equations of mthat follow from the action(2.1)are

(4.36)∇2φa + Gabcg

µν(∂µφb

)(∂νφ

c) − V a = 0


o

tion inelations

order

lar

ure.

easily

for the scalar fields, and Einstein’s equations

(4.37)Eµν = −Rµν + 2Gab

(∂µφa

)(∂νφ

b) + 4

d − 1gµνV = 0.

Notice that we use the opposite sign convention for the curvature with respect to[39,51].We are interested in the physical, gauge-invariant content of(4.36) and (4.37)to quadratic

order in the fluctuations around an RG flow background of the form(2.3), (2.4). As we sawin the last section, the physical content is obtained by expanding the fields according t(4.9)and (4.4)and then applying the substitution rules(4.35). Since we defined the expansion(4.4)geometrically, we will obtain sigma-model covariant expressions. To carry out this calculapractice, it is easiest to use RNCs at a given point in field space, so that one can use the r(4.5)outsider-derivatives.

In the following, we shall present the linearized equations of motion, and indicate higherterms as sources, the relevant quadratic terms of which are listed inAppendix D. For intermediatesteps we refer the reader toAppendix C. Let us start with the equation of motion for the scafields(4.36), which gives rise to the following fluctuation equation,[

D2r − 2d

d − 1WDr + e−2A

]aa − (

V a |c −RabcdWbWd

)ac − Wa(c + ∂rb) − 2V ab

(4.38)= J a.

Note the appearance of the field-space curvature tensor in the potential term.Second, the normal component of Einstein’s equations14 gives rise to

(4.39)−4W c + 4Wa

(Dra

a) − 4Vaa

a − 8V b = J.

Third, the mixed components of(4.37)yield

(4.40)−1

2di − 2W∂ib − 2Wa∂ia

a = Ji.

The appearance of the fieldsaa , b, c anddi on the left-hand sides of(4.38)–(4.40)seems toindicate the coupling between the fluctuations of active scalars (non-zeroWa) to those of themetric, which is familiar from the AdS/CFT calculation of two-point functions in the literatHowever, the gauge-invariant formalism resolves this issue, because(4.39) and (4.40)can besolved algebraically (in momentum space) for the metric fluctuationsb, c anddi , so that thecoupling of metric and scalar fluctuations at linear order is completely disentangled. Oneobtains

(4.41)b = − 1

WWaa

a − 1

2W

∂iJ i,

(4.42)c = Wa

W

(δabDr − Wa |b + WaWb

W

)ab − 1

4WJ + 1

2

(WaW

a

W2− 2d

d − 1

)∂i

Ji,

(4.43)di = − 2

Πji Jj .

We proceed by substituting(4.41) and (4.42)into (4.38), using also the identities

V a = Wa|bWb − 2d

d − 1WWa,

14 More precisely, it is the equation obtained by multiplying(4.37)by NµNν − gij Xµi

Xνj.


uationsder dif-ractions

d

se theand

e fluc-t

perator,of the

e that ae shall

irr

ssless

(4.44)V a |c = DrWa |c +Ra

bcdWbWd + Wa|bWb|c − 2d

d − 1

(WaWc + WWa |c

),

which follow from(2.2) and (2.4), and we end up with the second-order differential equation

[(δabDr + Wa |b − WaWb

W− 2d

d − 1Wδa

b

)(δbc Dr − Wb|c + WbWc

W

)+ δa

c e−2A]ac

(4.45)= J a,

where the source termJ a is related to the sourcesJ a , J andJi by

(4.46)J a = J a − Wa

4WJ − 1

2

(δabDr + Wa |b − WaWb

W− 2d

d − 1Wδa

b

)(Wb

W

∂i

Ji

).

Eq. (4.43) implies that we can dropdi in the source terms (to quadratic order). Eq.(4.45) isthe main result of the gauge-invariant approach and governs the dynamics of scalar fluctaround generic Poincaré-sliced domain wall backgrounds. Being a system of second-orferential equations, one can use the standard Green’s function method to treat the inteperturbatively.

A feature that is evident from the linearized version of(4.45)is the existence of a backgrounmode in the fluctuations. It is independent of the boundary variablesxi , and is simply given by

(4.47)aa = αWa

W,

whereα is an infinitesimal constant. In standard holographic renormalization, one can ubackground mode(4.47) to establish the existence of finite sources (CFT deformations)vacuum expectation values in the dual field theory. Asymptotically each component of thtuation vector is dual to a conformal primary operator (as explained in Section2); a componenof Wa/W that behaves asymptotically as the leading term of the general solution of(4.45)is in-terpreted as a background source deforming the CFT action by the corresponding dual owhile a background mode component that behaves asymptotically as the subleading termgeneral solution represents a vacuum expectation value of the dual operator. We believstatement of this kind can be made also in the general non-asymptotically AdS case, and wpresent an example for the MN system in Section5.

Let us also consider the tangential components of(4.37). Because of the Bianchi identity, thetrace and divergence are implied by(4.38), (4.39) and (4.40), which is easily checked at lineaorder. Thus, we can use the traceless transverse projector,

(4.48)Πikjl = 1

2

(ΠikΠjl + Πi

l Πkj

) − 1

d − 1Πi

jΠkl ,

in order to obtain the independent components. This yields

(4.49)

(∂2r − 2d

d − 1W∂r + e−2A

)eij = J i

j .

As expected, the physical fluctuations of the metric satisfy the equation of motion of a mascalar field.


ve 5d

n in

educes

olutionsa-

e origin

e-r

5. The Maldacena–Nuñez system

5.1. Review of the background solution

The MN system is obtained by imposing the following relations on the general effectiaction obtained in Section315:

Q = 0, h1 = h2 = 0, b = a,

(5.1)Φ = −6p − g − 2 lnP, x = 1

2g − 3p.

Together with(5.1), the constraints(3.10) and (3.11)imply alsoK = 0 andχ = 0. (Notice thata constant inχ is irrelevant.) It is straightforward to check from the equations of motioAppendix Athat this truncation is consistent, i.e., the equations of motion forb, h1, h2, Φ andx are satisfied or implied by those fora, p andg. Notice that, having absorbed the constantP 2

into eΦ , it has disappeared from the equations of motion. Hence, the effective 5d action rto the form(2.1), with three scalar fields (g, a,p), the sigma model metric

(5.2)Gab∂µφa∂µφb = ∂µg∂µg + e−2g∂µa∂µa + 24∂µp∂µp,

and the superpotential16

(5.3)W = −1

2e4p

[(a2 − 1

)2e−4g + 2(a2 + 1

)e−2g + 1

]1/2.

Let us briefly summarize the most general Poincaré-sliced domain wall background s(2.3) for this system. It is obtained by solving(2.4) and coincides with the family of solutionfound in[74]. In the following,g, a andp will denote the background fields, while the fluctutions are described by the gauge-invariant variablesaa . Introducing a new radial coordinate,ρ,by

(5.4)∂ρ = 2e−4p∂r ,

one can show from the equations forg anda that

(5.5)[(

a2 − 1)2 + 2

(a2 + 1

)e2g + e4g

]1/2 = 4ρ.

The integration constant arising here has no physical meaning and has been used to fix thof ρ. Then, one easily obtains

(5.6)a = 2ρ

sinh(2ρ + c), e2g = 4ρ coth(2ρ + c) − (

a2 + 1),

wherec is an integration constant with allowed values 0 c ∞. We shall discuss the interprtation ofc in the next subsection. The MN solution corresponds toc = 0 and is the only regulasolution. All others suffer from a naked curvature singularity.

It is also easy to show from(2.4) that

(5.7)e−2Ae−8p = C2,

15 We correct formula (5.25) of[35].16 We have adjusted the overall factor of the superpotential of[35] to our conventions.


s fortion

d”erravity

ur’stion.tates,s an ap-se the

ergrav-nes are-

eter-,er-

ter

e

le

he

where the integration constantC determines the 4d reference scale. We shall setC2 = 1/4 forlater convenience. The explicit solution forp can be found by pluggingΦ from the literatureinto (5.1), but it will not be needed here.

5.2. The role ofc

The family of background solutions of the MN system suffers from naked singularitieall c except for the casec = 0, which is regular. Hence, on the supergravity side the integraconstantc governs the resolution of the singularity. However, the scalara(ρ) is the dual ofthe gluino bilinearλ2 [26], so c, which entersa(ρ) in (5.6), also determines the “measurevalue of the gluino condensate,〈λ2〉, which is of non-perturbative field theory origin. In othwords,c identifies the “amount” of non-perturbative physics that is captured by the supergsolution.

In this subsection, we will attempt to flesh out this picture qualitatively, applying Mathcoarse graining argument[75,76], before we analyze the fluctuations in the next subsecAlthough only regular solutions qualify as gravity duals of (pure) field theory quantum sthe coarse graining argument indicates that certain singular solutions have a meaning aproximation to the duals of mixed states. In this point of view, singularities appear becau“space–time foam” that is dual to the mixture of pure states cannot be resolved by supity. (We are using the terminology of[75,76] here. See also[77] for some earlier discussioof the admissibility of singular solutions.) In the case at hand, the possible pure statnaturally identified as theN equivalent vacua of SU(N) N = 1 SYM theory, which are distinguished by a phase angle in the gluino condensate.17 Let us denote theseN vacua by|n〉,wheren = 0,1,2, . . . ,N − 1. The gluino condensate in these vacua takes the values

(5.8)〈n|λ2|n〉 = Λ3e2πin/N ,

where we have absorbed theθ angle of the gauge theory in the phase ofΛ3.Now, let us form mixed states by defining the density matrix

(5.9) =N−1∑n=0

pn|n〉〈n|, withN−1∑n=0

pn = 1.

Clearly, for equal weights,pn = 1/N , we would measure〈λ2〉 = tr(λ2) = 0. For a genericmixed state, the measured value〈λ2〉 lies somewhere within theN -polygon spanned by theNpure-state values(5.8). Using standard thermodynamics arguments, it is straightforward to dmine the unique distributionpn maximizing the entropy for a given fixed〈λ2〉. Notice, howeverthat theN vacua are equivalent, and that, for largeN , which is the regime described by the supgravity approximation, the vacuum values of〈λ2〉 effectively span a circle of radius|Λ3|. Thus,up to 1/N corrections, the phase of some given〈λ2〉 is irrelevant, making the relevant paramespace for the probability distributionpn effectively one-dimensional.

Thus, from the point of view advocated in[75,76], the integration constantc can be inter-preted as a parameter that interpolates between the uniform distribution (c = ∞) and a pure stat

17 In the 10d MN solution, the location of the Dirac string for the magnetic 2-formC2 is specified by an angular variabψ that can take 2N different values for the same field theoryθ -angle, but the solution is symmetric under a shift byπ

of ψ , leavingN different configurations. Equivalently, one hasN different ways of placing probe D5-branes in tbackground, in order to obtain the same field theory action[25,27,29].


byorizon

en-gluino-

n

s,section.utlineds). Inyion,perator

t theonsserves

aboutary”agonalectrum

r.

(c = 0), with fixed phase of〈λ2〉. It would be interesting to make this interpretation preciseattempting to match the statistical entropy of a mixed state with the area of the stretched hsurrounding the dual “space–time foam”. We leave such investigations for the future.

Instead, let us confirm the role ofc in determining the measured value of the gluino condsate from the perspective of holographic renormalization. Being a one-point function, thecondensate should appear as a backgroundresponsefunction in a supergravity field (cf. the discussion in Section2). Thus, consider the background mode(4.47) of the fluctuation equatiofor an arbitrary value ofc. As noted in Section4.4, the background mode,Wa/W , is always asolution of(4.45)independent ofxi . Let us determine its asymptotic behaviour (largeρ) and seewhether it is leading or subleading. For an arbitrary value ofc, we obtain

(5.10)Wa

W∼

(− 1

2ρ,8e−cρe−2ρ,

1

6

).

The first and third components are independent ofc, i.e., universal for all background solutionand they are leading compared to the general solutions that we shall find in the next subWe have, at present, no specific interpretation of their role, although the arguments oin Section2 indicate that they should correspond to finite field theory sources (couplingcontrast, the second component is subleading and depends onc. Hence, we argue in analogwith AdS/CFT (again, we refer to Section2) that its coefficient represents a response functso it determines the vacuum expectation value of the dual operator. In this case, the dual ois the gluino bilinear. Restoring dimensions, this yields

(5.11)⟨λ2⟩ = Λ3e−c,

which fits nicely with the preceding discussion involving mixed states.

5.3. Fluctuations and mass spectra

In the following, we shall consider the equation of motion for scalar fluctuations abousingular background withc = ∞. Although we argued in the introduction that singular solutias supergravity duals should be taken with a grain of salt, doing so is quite instructive andmainly two purposes: First, this solution elegantly describes the asymptotic region (largeρ) of allbackground solutions, including the regular MN solution, so that we can learn somethingthe asymptotic behaviour of the field fluctuations, which will be important for the “dictionand “renormalization” problems. Second, the matrix equation for fluctuations becomes diand analytically solvable. Thus, we can hope to get a qualitative glimpse of the particle spof the dual field theory.

Consider the equation of motion for scalar fluctuations(4.45). In terms ofρ and going to 4dmomentum space, as well as neglecting the source terms on the right-hand side,(4.45)becomes

(5.12)[(

δab∂ρ + 2Ma

b

)(δbc ∂ρ − 2Nb

c

) − k2]ac = 0,

where we have fixed the 4d scale by the choiceC2 = 1/4, which will turn out convenient lateThe matricesMa

b andNab are given by

(5.13)Nab = e−4p

(∂bW

a − WaWb

W

), Ma

b = Nab + 2e−4p

(Ga

bcWc − Wδa

b

).


thes

ations

nc-

er ton thendition,eaturee

irst,

Notice that thep-dependence inMab andNa

b cancels out. For the casec = ∞, the matricesMab

andNab are diagonal,

Nab = diag

(− 1

2ρ,

1

2ρ− 1,0

),

(5.14)Mab = 1

4ρ − 1diag

(4ρ − 2+ 1

2ρ,1− 1

2ρ,4ρ

).

We are mostly interested in the fielda2 (the middle component), since its dual operator isgluino bilinear (+ its Hermitian conjugate). From(5.12) and (5.14), its equation of motion read

(5.15)

(∂2ρ + 4

2ρ − 1

4ρ − 1∂ρ + 4

4ρ − 1− k2

)a2 = 0.

Performing a change of variable by defining

(5.16)ρ − 1

4= αz,

with a constantα to be determined later, and using the following ansatz for the solution,

(5.17)a2 = eazzbf (z),

with constanta andb, we find that the choices

(5.18)a = −α, b = 1

4, α2(1+ k2) = 1

4lead to the equation

(5.19)

(∂2z − 1

4+ 3α

2z− 5

16z2

)f = 0.

This can be recognized as Whittaker’s equation, the solutions of which are linear combinof the two Whittaker functions

(5.20)f = M 3

2α, 34(z),M 3

2α,− 34(z)

.

Hence, using(5.17)and the relation of Whittaker’s functions to confluent hypergeometric futionsΦ andΨ [78,79], we find

(5.21)a2 ∼ e−(α+1/2)z

(αz)3/2Φ(5

4 − 32α, 5

2; z),Φ(−1

4 − 32α,−1

2; z).In standard AdS/CFT, one would impose a regularity condition in the bulk interior in ordobtain a linear combination of the two solutions, which uniquely fixes the relation betweeresponse and the source functions. Here, however, we were not able to find such a coprobably due to the curvature singularity of the background. However, there is a useful fthat can guide us in the choice of suitable solutions. From(5.16), we should demand that thsolution be invariant under a simultaneous change of sign ofz andα. Due to the identity[78]

(5.22)Φ(a,b; z) = ezΦ(b − a, b;−z),

the particular solutions(5.21)are invariant under this symmetry. This implies two things. Fwe are free to choose the solution forα in the last equation of(5.18)such that Reα > 0, which


utin

incet

hed only

ee

he

in-

allowum

f, andequentlyeball

implies also Rez > 0. Notice that the square root in the definition ofα demands a branch cin k2-space, which we place atk2 + 1 < 0. This branch cut is an indication for a continuumthe particle spectrum, form2 = −k2 > 1. (Notice that this is relative to a reference scale, swe are working in dimensionless variables. With the earlier choiceC2 = 1/4 we place the onseof the continuum conveniently at the branch pointk2 = −1.) Second, linear combinations of tsolutions should also reflect this symmetry implying that proportionality factors can depenon α2. In particular, the choice of the functionsΨ (a, b; z) instead ofΦ(a,b; z) is not allowed,cf. [78].

It is instructive to consider the asymptotic behavior of the solutions. Letα be generic andfixed, so that we can consider largez. One finds that both solutions in(5.21), and any genericlinear combination of them, behave as

(5.23)a2 ∼ e(1/2−α)zz1/4−3α/2,

but there are notable exceptions. Indeed, the confluent hypergeometric functionsΦ(a,b; z) re-duce to polynomials (Laguerre polynomials, to be precise), if the first index,a, is zero or anegative integer. In these cases, the generic leading terms(5.23) are absent. Generalizing thAdS/CFT argument[80], we interpret the corresponding values of−k2 as discrete particlmasses in the spectrum of the dual field theory.

Hence, the two solutions(5.21)give rise to two different discrete spectra

(5.24)m2n = 1− 9

(4n + 3)2, n = 0,1,2, . . . ,

and

(5.25)m2n = 1− 9

(4n + 5)2, n = 0,1,2, . . . .

Notice that there is a massless state, forn = 0 in (5.24). Moreover, both spectra approach tbranch point,−k2 = 1, for n → ∞.

Similarly, we consider the other components. The equation of motion fora3 is

(5.26)

(∂2ρ + 8ρ

4ρ − 1∂ρ − k2

)a3 = 0,

for which we obtain the solutions

(5.27)a3 ∼ e−(α+1/2)z

(αz)1/2Φ(3

4 + 12α, 3

2; z),Φ(1

4 + 12α, 1

2; z).As before,z andα are defined by(5.16) and (5.18), respectively. Hence, we find again a contuum of states for−k2 > 1. However, although the solutions(5.27)are similar to(5.21), the signin front of theα-terms in the first index of the confluent hypergeometric functions does notthem to reduce to polynomials. (Remember that Reα > 0.) Hence, there is no discrete spectrof states.

We would like to note that the solution(5.27)is very similar to (3.17) of[52]. They consideredfluctuations of the dilaton about the MN background and introduced a hard-wall cut-offound an unbounded discrete spectrum of glueball masses. This procedure was subscriticized in [53]. Due to the discussion in the previous paragraph, we do not infer glumasses from the componenta3.


metricd the

that,tes for

tra are

e havend with

ual IReouslyeachlds as weadinggular atwing,d with

regular

The treatment of componenta1 is slightly more complicated. Its equation of motion is

(5.28)

(∂2ρ + 8ρ

4ρ − 1∂ρ − 2

ρ2+ 8

4ρ − 1− k2

)a1 = 0.

The awkward double pole inρ can be removed by settinga1 = ρ−1f (ρ), which yields theequation

(5.29)

[∂2ρ +

(2+ 2

4ρ − 1− 2

ρ

)∂ρ − k2

]f = 0.

After changing variables toz by using(5.16)and making the ansatz

(5.30)f (z) = eczf (z),

we find that the choice

(5.31)c = −1

2− α,

whereα is defined as before, leads to the equation

(5.32)

4αz

[z∂2

z +(

−3

2− z

)∂z + 3

4+ 3

2α

]+

[z∂2

z +(

1

2− z

)∂z − 1

4− 1

2α

]f = 0.

The two terms in square brackets represent differential equations for confluent hypergeofunctions, which gives us a nice hint for solving the equation. Indeed, we can explicitly finsolutions, which, combined with(5.30)anda1 = ρ−1f , result in

(5.33)a1 ∼ e−(α+1/2)z

αz + 1/4

Φ(−3

4 − 32α,−3

2; z) − 4α2−13 z2Φ(5

4 − 32α, 5

2; z),(αz)1/2[Φ(−1

4 − 32α,−1

2; z) + 36α2−15 z2Φ(7

4 − 32α, 7

2; z)].Notice that both solutions respect the symmetry of simultaneously changing the signs ofα andz.The sign of theα-terms in the first index of the confluent hypergeometric functions indicatesin addition to the continuum from the branch cut, we have again a discrete spectrum of stathose values ofα, where these functions reduce to polynomials. The corresponding specgiven again by(5.24) and (5.25), but in (5.24) only valuesn = 1,2,3, . . . are allowed, whichimplies that the massless state is absent.

To conclude this section, let us discuss whether we can trust the mass spectrum wfound. This question arises since the calculation was performed in the singular backgrouc = ∞, but the true supergravity dual of a field theory vacuum is the MN solution, withc = 0.Moreover, one typically expects the boundary conditions in the interior to influence the dphysics, but we have not directly imposed any conditions except symmetry of simultanchanging the signs ofα and z. However, there are only three things that can happen toparticular mass value when the regular background withc = 0 is considered. First, there couexist a corresponding regular and subleading solution for which the mass value changego from c = ∞ to c = 0. Second, there could exist a corresponding regular and sublesolution with the same mass. Third, the corresponding subleading solution may not be reρ = 0, in which case that particular mass value would not be in the spectrum. In the followe will argue that the first of these scenarios is excluded. Remember that the backgrounc = ∞, which we have considered here, correctly describes the asymptotic region of thebackground. Hence, the asymptotic behaviour of the fluctuations we found is valid also forc = 0,


riesthatil.)

ich isgiven

rs ofing to

inationrom it.

ct a. It isanism.

not

ntions

onifolded by

e KS

ngular,

implying that the mass spectra(5.24) and (5.25)are unchanged. One can verify this by a seexpansion in e−c of the equations of motion. Also, it is a straightforward but important checkthe componenta3 decouples from the other two for any value ofc and, therefore, cannot spothe subleading behaviour. (Remember that the solutions fora3 did not give rise to mass spectra

However, it might happen that imposing a regularity condition on the fluctuations, whrequired to calculate 2-point functions, does not allow for the solution that corresponds to amass value. This mechanism can be summarized as follows: For givenk2, a1 anda2 give fourindependent solutions, two of which give rise to the mass spectrum(5.24), the other two leadingto (5.25). These solutions evolve as we go fromc = ∞ to c = 0, but their asymptotic behavioudoes not change. Forc = 0, imposing regularity conditions will select two linear combinationthese four solutions. If such a linear combination involves only the two solutions correspondthe same mass spectrum, then this spectrum will survive. If, in contrast, the linear combinvolves solutions corresponding to different mass spectra, no mass values will result fA particularly interesting case is the massless state, which belongs to the spectrum(5.24), butarises only from the componenta2, not froma1, in the analysis above. One does not expemassless glueball state to exist, and in fact, it is likely to be excluded by this mechanismless likely that only single masses, as opposed to an entire spectrum, will survive this mechThis is in contrast to the result of[53], where only a single glueball state was found. We willanswer these interesting questions in this paper, but we intend to come back to them.

6. The Klebanov–Strassler system

In this section we review thewarped deformed conifold, or the Klebanov–Strassler solutio[22]. We will be particularly interested in the “gluino sector”, the 3-scalar system of fluctuathat contains the field dual to the gluino bilinear trλλ.

6.1. Review of the background

The KS system is obtained from the general PT system by relating the fieldsa andg by therelation

(6.1)a = tanhy, e−g = coshy (KS),

whereby a new fieldy (not to be confused with the 5d coordinatesyµ used in Section3) isintroduced. This relation renders the constraint(3.11) integrable and impliesχ = 0. Moreover,one can check that the equations of motion fora andg, (A.30) and (A.31), become equivalent.

There exists an even more restricted truncation, which gives rise to the singular 10d cbackground of KT and certain fluctuations thereof. It contains four scalars and is obtainimposing

(6.2)a = b = g = h2 = 0 (KT),

which also impliesχ = 0. We shall not consider the KT system separately, but discuss thsystem in a way similar to the treatment of the MN system in Section5. That is, we will consider aclass of background solutions characterized by a parameterc, that formally interpolates betweethe KT and KS backgrounds. As in the MN case, the background solutions are typically sinexcept for the KS endpoint of the family.


aper.

the AdS

ature,nd

g

r to

Table 1Comparison of symbol and field conventions used by Apreda[81] (see also[82]), Papadopoulos and Tseytlin[35],and Klebanov and Strassler (KS)[22]. The entries n/a mean that these fields do not appear explicitly in the KS pApreda’s fields diagonalize the mass matrix in the AdS background forP = 0, Q = 2/

√27. The last two columns

contain, respectively, the mass squared of the bulk fields and the conformal dimensions of the dual operators inbackground forP = 0

Apreda PT KS m2 ∆

q 15(x − 2p) + 3

20 ln(3) + 110 ln(2) n/a 32 8

fApreda15(x + 3p) + 1

10 ln( 23) n/a 12 6

y sinh−1(ae−g) n/a −3 3

Φ Φ Φ 0 4

s −2h1 M(k + fKS) 0 4

N1 −h2 − PPT(b + 1) M2 (k − fKS) − MF 21 7

N2 −h2 + PPT(b + 1) M2 (k − fKS) + MF −3 3

PApreda −PPT ≡ −P M2 – –

For the remaining fields of the KS system, there exist a variety of conventions in the litersome of which we list for reference inTable 1.18 For the purpose of rederiving the backgrousolutions, we shall start with the PT variables(x,p, y,Φ,b,h1, h2), wherey was introduced in(6.1). The sigma-model metric(3.13)for the KS system reduces to

Gab∂µφa∂µφb = ∂µx∂µx + 6∂µp∂µp + 1

2∂µy∂µy + 1

4∂µΦ∂µΦ + P 2

2eΦ−2x∂µb∂µb

+ 1

4e−Φ−2x

[e−2y∂µ(h1 − h2)∂

µ(h1 − h2)

(6.3)+ e2y∂µ(h1 + h2)∂µ(h1 + h2)

],

and the superpotential reads[35]

(6.4)W = −1

2

(e−2p−2x + e4p coshy

) + 1

4e4p−2x(Q + 2Pbh2 + 2Ph1).

SettingP = 0, there exists an AdS fixed point. The choiceQ = 2/√

27 leads to the correspondinAdS background with unit length scale, and Apreda’s fields vanish in this background.

For the KS system, the background equations(2.3), (2.4)become

∂r(x + 3p) = 3

2e−2p−2x − e4p coshy,

∂r(x − 6p) = 2e4p coshy − 3

2e4p−2x[Q + 2Pbh2 + 2Ph1],

∂ry = −e4p sinhy, ∂rΦ = 0, ∂rb = 1

PeΦe4ph2,

∂rh1 = PeΦe4p[cosh(2y) − b sinh(2y)

],

(6.5)∂rh2 = PeΦe4p[b cosh(2y) − sinh(2y)

].

18 Note that there are typos in the first three equations of (5.24) in[35], which relate the variables used in that papethose used in[22]. The correct relations can be read off fromTable 1.


g the

d

e

We shall, in the following, rederive the background solutions of this system by followincalculations of KS[22], but adding the relevant integration constants. From(6.5)we can imme-diately read offΦ = Φ0 = const, and after introducing the KS radial coordinateτ by

(6.6)∂τ = e−4p∂r ,

we easily find

(6.7)ey = tanhτ + c

2.

For generality, we shall keep the integration constantc. In particular,c takes the values∞ and0 for the KT and KS solutions, respectively. Similar to the parameterc in the MN solutiondiscussed in Section5, it determines whether the supergravity solution is regular (c = 0) or not(c = 0). We note that(6.7)restricts the range ofτ to τ > −c.

From the equations forb, h1 andh2 one can derive the differential equation

(6.8)∂2τ b = b cosh(2y) − sinh(2y),

whose general solution is

(6.9)b = b1 cosh(τ + c) − (b1 + 1)τ + b2

sinh(τ + c).

We must setb1 = 0 in order to avoid the exponential blow-up for largeτ , andb2 can be absorbeinto a redefinition ofτ andc. Hence, we have

(6.10)b = − τ

sinh(τ + c),

from which follows immediately

(6.11)h2 = PeΦ0τ coth(τ + c) − 1

sinh(τ + c).

Then, we obtain also

(6.12)h1 = PeΦ0 coth(τ + c)[τ coth(τ + c) − 1

] + h,

whereh is an integration constant.The functionsb, h1 andh2 determine the functionK , which measures the 5-form flux in th

10d configuration(3.8).19 From(3.10), (6.10), (6.11) and (6.12)we find

(6.13)K = K0 + P 2eΦ0τ coth(τ + c) − 1

sinh2(τ + c)

[sinh(2τ + 2c) − 2τ

],

where we have abbreviated

(6.14)K0 = Q + 2P h.

Now, let us calculate the background fieldsx andp. It is convenient to use Apreda’s fieldsf

andq, the definitions of which are given inTable 1. Then, from the equation for(x + 3p) wefind

(6.15)5∂τ f = e−10f − coshy,

19 Note that this is not theK of KS.


to

-

c in

with the general solution

(6.16)e10f = coth(τ + c) − τ + f0

sinh2(τ + c),

wheref0 is again an integration constant. The remaining background equation gives rise

(6.17)

(∂τ − 4

3cothy

)e6q−8f/3 = −2 · 31/2Ke−20f/3,

whereK is given by(6.13). Isolating the homogeneous solution by the ansatz20

(6.18)e6q−8f/3 = 24/3e−4c/3 sinh4/3(τ + c)h(τ),

we obtain from(6.17)thath(τ) satisfies

∂τh = −21/331/2e4c/3[sinh(2τ + 2c) − 2τ − 2f0]−2/3

(6.19)×K0 + P 2eΦ0

τ coth(τ + c) − 1

sinh2(τ + c)

[sinh(2τ + 2c) − 2τ

].

It is instructive to consider the limitc → ∞, which describes the large-τ behaviour of allbackground solutions. In this case, we obtain explicitly

(6.20)h = 1

233/2e−4τ/3

[K0 + 2P 2eΦ0

(τ − 1

4

)]+ h0.

The choiceh0 = 0, needed in order to avoid the exponential growth in(6.18), removes the asymptotically flat region from the 10d solution.

Finally, one can show that the equation for the warp factorA in (2.3)yields

(6.21)e−2A = C2e−2x/3(2e−c)−2/3 sinh−2/3(τ + c),

where the integration constantC sets the 4d scale and will be fixed later.21

The regular KS solution is given by fixing the integration constants as follows:

(6.22)c = f0 = K0 = 0,

and imposing vanishingh for largeτ , which yields

(6.23)h = 21/331/2P 2eΦ0

∞∫τ

dϑϑ cothϑ − 1

sinh2 ϑ

[sinh(2ϑ) − 2ϑ

]1/3.

Note that our definition ofh differs from the one in[22] by a constant involving a factorε−8/3.(Although [22] fix ε to a numerical value early on, it is clear from (65) in[24] that theirh ∼ε−8/3.) Our constantC2 of (6.21), which appears in front of the external 4-dimensional metri(2.3), corresponds toε−4/3 of [22] up to numerical factors.

20 The constant factor 24/3e−4c/3 has been inserted to normalize the forefactor to unity in thec → ∞ limit.21 Note that in this formula most of the complicatedτ -dependence of e−2A is hidden in the factor e−2x/3.


e equa-

areand

n by

roundn haveed in

6.2. Fluctuation equations

We are now in a position to write down the equations of motion for fluctuations(4.45)aboutthe background solutions found in the previous subsection. Let us begin by expressing thtion of motion in terms of the KS radial coordinateτ . After multiplying (4.45)by e−8p and using(6.6)we obtain

(6.24)[(∂τ + M)(∂τ − N) + e−8p−2A]

a = 0,

where the matricesM andN are given by

(6.25)Nab = e−4p

(∂bW

a − WaWb

W

), Ma

b = Nab + 2e−4p

(Ga

bcWc + e−2p−2xδa

b

).

When we substitute the KS background in(6.25), the matrices become quite complicated andrelegated toAppendix E. We view it as an important step to have obtained them explicitly,we intend to come back to a more detailed study of them at a later date.

In the following, we shall consider fluctuations about the KT background, which is givethe choice of integration constants

(6.26)c = ∞, K0 = f0 = h0 = Φ0 = 0.

The motivation for this choice is essentially the same as for the MN system: this backgdescribes correctly the asymptotic region of the KS solution, and the equations of motioa simpler form, which can be treated analytically (with a further approximation describSection6.3).

For the background specified by the integration constants(6.26), the matricesM andN havequite a simple form. Using Apreda’s variables for the fluctuation fields,a = δ(q, f,Φ, s, y,

N1,N2), andP ≡ PPT = −PApreda, we find

(6.27)M =

4(8τ−3)3(4τ+1)

0 0 32(τ−1)

45P(16τ2−1)0 0 0

0 −23 0 − 4

15P(4τ−1)0 0 0

0 0 43 − 8

3P(4τ−1)0 0 0

20P(4τ−1)4τ+1 0 0 8(2τ+1)(4τ−3)

3(16τ2−1)0 0 0

0 0 0 0 13

83P(4τ−1)

83P(4τ−1)

0 0 0 0 0 28τ−193(4τ−1)

0

0 0 0 0 0 0 4τ−133(4τ−1)

,

(6.28)N =

16(τ−1)3(4τ+1)

0 0 49P(4τ+1)

0 0 0

0 −2 0 0 0 0 0

0 0 0 0 0 0 032P(τ−1)

4τ+1 −8P −2P 83(4τ+1)

0 0 0

0 0 0 0 −1 0 0

0 0 0 0 2P 1 0

0 0 0 0 2P 0 −1

.


ember3ck is

cause

r-

f

vious

ing to

The block-diagonal form of these matrices is a nice feature of the KT background. Remthat(6.2)defines a consistent truncation of the KS to the KT system, so that the lower left× 4off-diagonal blocks ofM andN are expected to be zero, but vanishing of the upper right bloa bonus feature. It is a particularly welcome bonus, since the gluino sectorδ(y,N1,N2) is wherewe would expect much of the interesting physics to be encoded.

We also see from(6.27) and (6.28)that the UV limitτ → ∞ and the conformal limitP → 0do not commute. One might have considered performing an expansion inP to study a “near-conformal” regime, but the order of limits would pose a problem. This is not surprising, beamong other things we have imposedK0 = 0 on the solution, which is not possible forP = 0, ascan be seen from(6.13). It is of course possible to study the conformal (Klebanov–Witten[83])system directly, but this would require changing field variables.

For the KT background, it is useful to change the radial variable by introducing22

(6.29)τ = 3 lnσ + 1

4.

Using(6.21), (6.18) and (6.20), we find that the term in(6.24)with the 4-dimensional box opeator is proportional to

(6.30)e−2A−8p ∼ C2P 2

σ 2lnσ,

where we have suppressed a numerical factor. Hence, from(6.24)and with a suitable choice othe constantC follows, in momentum space,

(6.31)

[(σ∂σ + 3M)(σ∂σ − 3N) − k2P 2

σ 2lnσ

]a = 0.

We see that fixingC indeed sets the 4-dimensional energy scale, as claimed in the presection.

We further introduce23

(6.32)v = kP

σ,

in terms of which(6.31)becomes

(6.33)

[v3∂vv

−3∂v − Yv−1∂v − Zv−2 − lnkP

v

]a = 0,

where the matricesY andZ are given by

(6.34)Y = 3(M − N) − 4, Z = 9MN + 3σ∂σ N.

In [32], fluctuations of the 4-scalar KT system were studied in a particular gauge, leadequations more complicated than, but presumably equivalent to(6.33).

22 Ourσ corresponds tor of KT up to a multiplicative factor, whereas ourr corresponds to theiru.23 Our v corresponds to Krasnitz’sy.


he

otiva-

se, not

n aswhichlizatione UV

ns,.

n

is ansly.nless

theg atld be

hus we

xima-ons

ge

ur

6.3. “Moderate UV” approximation

Despite its apparent simplicity, Eq.(6.33)has no analytic solution. A method to extract tresponse functions at leading order in the high-energy limit was developed by Krasnitz[31,32].We proceed to briefly review this method, but first we pause for a short comment on our mtion to use the method in the first place.

We are, of course, ultimately interested in all energy ranges and the confining phajust the high-energy limit. Nevertheless, we have seen that the matrices inAppendix Eare pro-hibitively complicated for analytical work, so we view the approximation in this subsectioa simple way to get a handle on the full problem in one particular regime (high energy),should provide good cross-checks for a numerical treatment. In addition, since renormais a UV problem, KT counterterms should be sufficient to renormalize KS correlators, so thregime seems a good place to start.

Here is the brief review. In[31,32], the KT solution was divided into two overlapping regiowhich we will call “moderate UV” (or “mUV”) region and “extreme UV” (or “xUV”) regionFor the purposes of this discussion, let us setP = 1; it can be restored byk → kP . In themUV region, | logv| | logk|, so we can approximate the troublesome log(k/v) in (6.33)bya constant logk. This clearly does not work forv too small, hence “moderate” UV, but wheit does work, exact solutions of the approximated equation can be found[31,32]. In the xUVregion,[31,32] treats log(k/v) as a perturbation, and expands iteratively in it. Then, thereintermediate overlap region (seeFig. 3) where both solutions should be valid simultaneouFor largek, the solutions naively appear to differ appreciably in the intermediate regime, uthere is some relation between large-k terms in the two solutions: this allows us to matchleading-order terms ink. Analytic correlators can in principle be extracted from this matchinleading order ink, but we reiterate that the dictionary and renormalization problems shoucompletely solved before any gauge theory correlators can be quoted with certainty. (Twill not perform the xUV analysis here, but we mentioned it for completeness.)

Here, we will show that the 7-scalar system is analytically solvable in the mUV approtion, generalizing the analysis of[31,32] to the present case. We will then check our solutinumerically.

The mUV regime is obtained in two steps. First, we consider the UV region, i.e., larτ ,which implies largeσ . To leading order, the matricesY andZ become

Fig. 3. Krasnitz matching for a generic fieldφ. The solution denotedφmUV is regular in the IR, and analogous to osolutions below. The solutions are matched to approximately agree in the cross-hatched overlap region.


. (The

al mo-

e

le

(6.35)Y =

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

36P 24P 6P 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 −6P 0 0

0 0 0 0 −6P 0 0

,

and

(6.36)Z =

32 0 0 0 0 0 0

0 12 0 0 0 0 0

0 0 0 0 0 0 0

336P −96P −24P 0 0 0 0

0 0 0 0 −3 0 0

0 0 0 0 42P 21 0

0 0 0 0 6P 0 −3

.

As a check, we see thatZ reproduces the masses inTable 1in the conformal limitP → 0,although we noted earlier that one would need rescaled field variables to study this limitmass does not depend on the field normalization.)

Second, as discussed earlier, the mUV region is isolated by considering large externmenta24 | logk| | logv|. (As in the discussion above, note that this limitsv from below as wellas from above.) This means that we can neglect lnv from (6.33). When this is done,k is easilyremoved from(6.33)by defining

(6.37)z = √ln(kP )v,

so that we obtain the equation

(6.38)[z3∂zz

−3∂z − Yz−1∂z − Zz−2 − 1]a = 0.

The variablez blows up in the conformal limitP → 0 (cf. the order-of-limits discussion in thprevious subsection). If needed, one can always go back to(6.31)and setP = 0 there.

With Y andZ given by the constant matrices(6.35) and (6.36), Eq. (6.38)admits analyticasolutions. We are, as usual in AdS/CFT, interested in the solutions that are regular for largz.

For the four scalars of the KT system, we obtain

a1 = z2 K6(z)

1

0

0

6P

03

− 6P

[4z2 K4(z) + 6zK1(z) + 6K0(z) − 3z2 K2(z) ln z

]

0

0

0

1

03

,

24 We recall that we use dimensionless variables.


rices

tions

ort

in the

uation

a2 = z2 K4(z)

0

1

0

0

03

− 12P

[z2 K4(z) + 2zK1(z) + 2K0(z) − z2 K2(z) ln z

]

0

0

0

1

03

,

a3 = z2 K2(z)

0

0

1

0

03

− 3P

[2zK1(z) + 2K0(z) − z2 K2(z) ln z

]

0

0

0

1

03

,

(6.39)a4 = z2 K2(z)

0

0

0

1

03

,

where the Kn are Bessel functions of ordern. For the gluino sector, we find

a5 = z2 K1(z)

04

1

0

0

− 3P

[z2 K5(z) ln z + 21zK4(z) + 7

6z2 K1(z) + 80

zK2(z)

+ 240

z2 K1(z) + 384

z3 K0(z)

]

04

0

1

0

− 3P

[z2 K1(z) ln z + zK0(z)

]

04

0

0

1

,

(6.40)a6 = z2 K5(z)

04

0

1

0

, a7 = z2 K1(z)

04

0

0

1

.

A few comments on these solutions are in order. We note that from the approximate matY

andZ, given in(6.35) and (6.36), it could have been gleaned already that the componenty is asource forN1 andN2, but not the other way around. Hence, it is not surprising that the solua6 anda7, where onlyN1 or N2 are non-zero, are significantly simpler thana5, where also they-component is turned on. Our next observation is thaty sources the other two gluino-sectfields by terms linear inP . Indeed, the matricesY andZ in (6.35), (6.36)make manifest the facthat the Apreda basis diagonalizes the mass matrix in the conformal (P = 0) limit. It makes itequally manifest that the gauge/gravity dictionary problem is significantly more pressingP = 0 case than in the conformal limit.

These analytical solutions are remarkably simple. In the face of darkShelobhorror like thefull KS matrices shown inAppendix E, these solutions may prove to be our savingEärendillight,25 provided we can convince ourselves that they actually do solve the exact KT eq

25 Seehttp://www.wikipedia.org.

http://www.wikipedia.org


r-an

in

ff.n

e ofith theuracy

ke thee

ationsds. Intagenon-

erivedwhichtantly,amics

manypoint

, onceing

Fig. 4. Moderate-UV analysis: comparison of the analytical solutions(6.40)of Eq.(6.38)with the corresponding numeical solution of(6.33)found by shooting (marked by crosses) fork = 103, P = 1. The “response functions” agree toaccuracy of 8%.

(6.33)in a suitably approximate sense. The Krasnitz approximation is valid for very largek, sowe give a representative check for moderately largek, when the approximation should just begto work.

The numerical solutions were found by shooting for approximately regular solutions of(6.33),that is, minimizing field values at the grid endpoint by tuning the derivative at a UV cuto26

Superimposing a linear combination of the solutionsa5, a6, a7 of the approximate equatio(6.38), and normalizing them to unity at the cutoff, we find good qualitative agreement inFig. 4.The derivatives at the cutoff essentially give the response functions for the given valuk(since we normalized the fields at the cutoff to unity). The numerical responses agree wanalytical solutions of the approximate equation within fairly good accuracy, and the accwill improve with energy. Above and beyond any matching procedure à la Krasnitz, we tagood agreement inFig. 4as evidence that the solutions(6.39), (6.40)may give us the crutch wneed as we embark on a numerical study of the full KS system.

7. Outlook

In this paper we have investigated aspects of the bulk dynamics of supergravity fluctuabout the duals of confining gauge theories, in particular the KS and MN backgrounour to-do list in the introduction, we called this the “fluctuation problem”. This sets the sfor addressing the problem of calculating correlators in confining gauge theories fromasymptotically AdS supergravity backgrounds. To be able to perform our analysis we da consistent truncation of type IIB supergravity to a set of scalars coupled to 5d gravity,is general enough to deal with fluctuations about the KS and MN backgrounds. Imporwe also developed a gauge-invariant and sigma-model-covariant formulation of the dynof the field fluctuations in generic, “fake-supergravity”-type systems, which should findapplications amongst the various configurations studied in the literature. Moreover, weout that the gauge-invariant formalism naturally includes higher order interactions. Hencethe “dictionary” and “renormalization” problems for holographic renormalization of confin

26 Further details will be given in future work.


e-point

dresseddity oflartest alsonce

ation

f Sec-angeenormal-

xtremet

eory.an

. Fromatedthe 10d

nte KSttemptlly.n

r of thealnd (atariablez, oncesfy then ofations

a form

amicsated itsndingsultsrwardd from

gauge theories (as introduced in the introduction) are understood, the calculation of threfunctions and scattering amplitudes (along the lines of[49]) should become straightforward.

Concerning our particular results, there are many open issues that could and will be adin the near future. For the MN system, the most interesting question is to check the valithe mass spectra(5.24) and (5.25)by numerically solving the fluctuation equation in the regubackground. As discussed in detail at the end of Section5.3, it is only the existence of the discremasses, not their particular values, which is in doubt. This question is of particular interein view of the contrasting results of[52,53]. It is an interesting point, though, that the existeof an upper bound on the masses, as is the branch point in our case, was also found in[53]. In anycase, all MN results should be considered in light of the fact that the supergravity approximis not under control in the UV region of the MN solution.

For the KS system, it should be straightforward to generalize the numerical analysis otion 6.3to the full KS background. This will not only lead to a better understanding of the rof validity of our approximate analytical solutions of Section6.3, but also pave the way for thextraction of some dual physics, once progress has been made on the dictionary and reization problems. Moreover, a more detailed analysis of the fluctuation equations in the “eUV” region, for instance, the asymptotic expansion used by Krasnitz[31,32]which we have noperformed here, might shed further light on these problems.

We would also like to comment on the question of the glueball spectrum in the KS thThis has already been studied in[84,85], where it was argued that the glueball spectrum isIR quantity. As the 3-cycle is anS3 in the IR, the fields were expanded in harmonics ofS3

in these papers, which was argued to lead to a decoupling of, for example, the dilaton(A.10), (A.7) and (A.8)it is obvious that this decoupling can never be exact. The complicdependence on the internal coordinates present in the PT ansatz simply drops out ofequation of motion for the dilaton, leaving the 5d equation given in(A.10). It might still bethat the expansions performed in[84,85] are approximately correct, but it would be importato check to what extent this is really a controllable approximation of the IR physics of thgauge theory. We are optimistic that our formalism presents a useful starting point to asuch a check by solving the linearized gauge-invariant equations for the scalars numerica

Another interesting open issue was brought up in Section3. In the PT ansatz, there is aadditional scalar field, which does not appear in the KS system: This is the superpartneGoldstone mode predicted in[72] and studied in[70,71]. Even though it seems to be an idecandidate for addressing the problem of calculating 2-point functions in the KS backgroulinear level it decouples from the other scalars as long as it depends only on the radial v[70]), it turns out that the dynamics of this mode requires a generalization of the PT ansatwe allow the field to depend on all five external coordinates, since then it does not satiintegrability constraint(3.11)in general. It would be very interesting to find a generalizatiothe PT ansatz that would lead to a 5-dimensional consistent truncation of the type IIB equof motion and include this additional mode. Attempts along those lines might also lead toof the 5-dimensional effective theory which is manifestly supersymmetric.

In all, we have found an efficient approach to (at least an important subset of) the dynof fluctuations about the supergravity duals of confining gauge theories, and demonstrapplicability in a number of examples. This is an important step towards a full understaof the “fluctuation” problem for holographic renormalization. We are hopeful that our remake also the “dictionary” and “renormalization” problems more accessible. We look foto the day when exciting new physics of confining gauge theories can be reliably extractegauge/string duality.


ata,. Za-

ations.entsby therted inges fi-sion’swasank-uriesup-racts, andM.H.

ity to

ns.e

metricand

Ricci

Acknowledgements

It is a pleasure to thank O. Aharony, D. Berenstein, M. Bianchi, A. Buchel, G. Dall’AgC. Herzog, I. Klebanov, A. Lerda, S. Mathur, P. Ouyang, H. Samtleben, A. Tseytlin and Mgermann for illuminating insights and helpful advice through discussions or email conversWe thank M. Bianchi for collaboration in an early stage of the work and C. Herzog for common a draft of the paper. M.B. was supported by the Wenner–Gren Foundations, and M.H.German Science Foundation (DFG). Moreover, the research of M.B. and M.H. was suppopart by the National Science Foundation under Grant No. PHY99-07949. W.M. acknowlednancial support from the MIUR-COFIN project 2003-023852, from the European Commis6th framework programme, project MRTN-CT-2004-005104, and from INFN. This workstarted while M.B. and M.H. were at Università di Roma 2 (‘Tor Vergata’) and they are thful for the hospitality of the string theory group. There, M.B. was supported by a Marie CFellowship, contract number HPMF-CT-2001-01311. Also, the work of M.B. and M.H. wasported in part by INFN, in part by the MIUR-COFIN contract 2003-023852, by the EU contMRTN-CT-2004-503369 and MRTN-CT-2004-512194, by the INTAS contract 03-51-6346by the NATO grant PST.CLG.978785. We would also like to thank CERN, and M.B. andthe Perimeter Institute, for hospitality during part of the work.

Appendix A. Consistent truncation of type IIB supergravity

In this appendix, we present details of the consistent truncation of type IIB supergravthe effective 5d system(3.12). In particular, we shall show how the constraints(3.10)and(3.11)and the 5d equations of motion derivable from(3.12)arise from the 10d supergravity equatioWhen comparing the results of this appendix with(3.12)–(3.14), one has to bear in mind that womitted the tildes in those formulas for esthetic reasons.

To start, let us identify our conventions and present some useful formulas. We use theand curvature conventions of MTW, Polchinski and Wald, i.e., the signature is mostly plus

RMNPQ = ∂NΓ

QMP − ∂MΓ

QNP + Γ S

MP ΓQSN − Γ S

NP ΓQSM,

(A.1)RMN = RMPNP ,

where the Christoffel symbols are defined as

(A.2)Γ PMN = 1

2gPQ(∂MgNQ + ∂NgMQ − ∂QgMN).

With these conventions one has the following transformation rules of the Ricci tensor andscalar of aD-dimensional manifold under a conformal transformationgMN = Ω2gMN , cf. Ap-pendix D of[55]

RMN = RMN − (D − 2)∇M∂N lnΩ − gMNgPQ∇P ∂Q lnΩ

+ (D − 2)(∂M lnΩ)(∂N lnΩ) − (D − 2)gMNgPQ(∂P lnΩ)(∂Q lnΩ),

R = Ω−2[R − 2(D − 1)gPQ∇P ∂Q lnΩ

(A.3)− (D − 2)(D − 1)gPQ(∂P lnΩ)(∂Q lnΩ)].

Moreover, our conventions for the Hodge star of ap-form ωp are

(A.4)ωp = 1ωM1...MpεM1...Mp

N1...ND−pdxN1 ∧ · · · ∧ dxND−p .

p!(D − p)!


-ecte

al-

in

e RR

nd

ction

Finally, we adopt the convention to adorn with a tilde objects derived from the metric ds25 =

gµν dyµ dyν (again, note the difference in notation compared to(3.8); in the main text we suppressed the tildes for readability). For example,∇ denotes the covariant derivative with respto gµν , andF µij = gµνgikgjlFνkl . Note the relation ofgµν to the external components of th

metric,g(ext)µν = e2p−xgµν , as follows from(3.8). g(int) denotes the remaining internal part,

though we usually omit the superscript(int) if it is clear from the indicesi, j, . . . that we mean aninternal component. Also note that the usage of the indexi to label the internal coordinatesthe 10-dimensional metric(3.8), i.e., i ∈ ψ,θ1, θ2, φ1, φ2, differs from the usage in Section4andAppendices B, C and D.

Let us turn to the analysis of the 10d equations of motion. The equation of motion for thscalarC, (3.3), is satisfied, becauseHM1...M3F

M1...M3 = 0.The equation of motion forF5, (3.7), leads to

(A.5)∂µK = 2P∂µ(h1 + bh2),

from which the constraint(3.10)follows.The second constraint,(3.11), arises from(3.5), in particular, from the mixed components

(A.6)∂M

(e−ΦHMµi√−g

) = ∂k

(e−ΦHkµi√−g

) = 0.

Eq. (A.6) follows from (3.5), becauseεM1...M10FM1...M5FM6...M8 has no mixed components, aC ≡ 0. Furthermore, in the first equality of(A.6) we have used that the components ofH have atmost one external index. One can show that all components of(A.6) are fulfilled, once(3.11)isimposed.27

The equation of motion for the dilaton can be checked as follows. If we denote

I1 := 2e8ph22 + 2∂µh2∂

µh2

+ 4(1+2e−2ga2)∂µh1∂µh1 +8e−2ga2∂µh2∂

µh2 +8a[e−2g(a2 +1)+1]∂µh1∂µh2

e2g + 2a2 + e−2g(1− a2)2

(A.7)= 1

6e2p+xHMNP HMNP ,

I2 := P 22∂µb∂µb + e8p[e2g + e−2g

(a2 − 2ab + 1

)2 + 2(a − b)2](A.8)= 1

6e2p+xFMNP FMNP ,

then the dilaton-dependent terms in the 5d action(3.13)are given by

(A.9)Sdil5 =

∫d5y

√g

(1

8∂µΦ∂µΦ + 1

8e−Φ−2xI1 + 1

8eΦ−2xI2

).

The equation of motion that follows from(3.2)and the constraint(3.11)is

(A.10)ex−2p∇2Φ = −1

2e−Φ−2p−xI1 + 1

2eΦ−2p−xI2.

Obviously,(A.10) is precisely the equation of motion that one would derive from the 5d a(A.9).

27 For this and most of the following calculations we used symbolic computation software, in particular MAPLE and theGRTENSORpackage[86].


at

d

es

e

gles,

Let us next consider the equation of motion forF3, (3.4), which reads

(A.11)∇M

(eΦFMNP

) = − 1

3!√g(ext)Fy1...y5Hijkε

ijkNP .

The right-hand side is only non-vanishing ifN andP are internal indices. One can verify ththe same holds also for the left-hand side. To see this, note that from the ansatz ofF3 and theblock structure of the metric, one only has to check

(A.12)∂k

(Fkµi sinθ1 sinθ2

) = 0,

becauseF3 can have at most one external index. The validity of(A.12) can easily be checkewith the help of a computer. Thus, the non-trivial part of the equation of motion forF3 boilsdown to

(A.13)∇M

(eΦFMlm

) = − 1

3!Ke3p− 32xHijkε

ijklm,

where we have made use ofFy1...y5 = Ke3p− 32x

√g(ext). It turns out that the angle dependenc

of the left- and right-hand sides in(A.13) coincide for each value ofl andm. Moreover, thecomponentsonlydiffer in their angle dependence. More precisely, on the one hand we hav

∇M

(eΦFMlm

) = eΦ

sinθ1 sinθ2∂k

(sinθ1 sinθ2F

klm) + ex−2p∇µ

(eΦFµlm

)= P

eΦ+6p−x−2g

[e2g(b − a) − a

(a2 − 2ab + 1

)](A.14)− ex−2p∇µ

(eΦ−2x ∂µb

)f lm(ψ, θ1, θ2, φ1, φ2),

wheref lm is some simple rational expression involving trigonometric functions of the anwhose precise form depends onl andm. On the other hand,

(A.15)− 1

3!Ke3p− 32xHijkε

ijklm = −Ke6p−3xh2flm(ψ, θ1, θ2, φ1, φ2).

Taking(A.14) and (A.15)together leads to the equation of motion forb

(A.16)P 2∇µ

(eΦ−2x ∂µb

) = PKe8p−4xh2 + P 2eΦ+8p−2x[b − a − ae−2g

(a2 + 1− 2ba

)],

which is exactly what one would derive from(3.12).Now, we come to the equation of motion forH3, (3.5), which is equivalent to

(A.17)∇M

(eΦHMNP

) = 1

3!√g(ext)Fy1...y5Fijkε

ijkNP .

Again, the right-hand side is only non-vanishing for internal componentsN andP . As we alreadysaid above, the same is true for the left-hand side after imposing the constraint(3.11), cf. (A.6).Thus, the non-trivial part of(A.17) becomes

∇M

(e−ΦHMlm

) = e−Φ

sinθ1 sinθ2∂k

(sinθ1 sinθ2H

klm) + ex−2p∇µ

(e−ΦHµlm

)(A.18)= 1

3!Ke3p− 32xFijkε

ijklm.


.

d,otice

nts

The expressions for∇µ(e−ΦHµlm) are much more involved than the case ofF3 discussed aboveAlso the general structure of the equations is more complicated. In particular, we have

1

3!Ke3p− 32xFijkε

ijklm

(A.19)= PKe6p−3x[bf lm

1 (ψ, θ1, θ2, φ1, φ2) + f lm2 (θ1, θ2, φ1, φ2)

],

wheref1 andf2 differ in such a way thatf1 always (i.e., for all values ofl andm) contains afactor cos(ψ) or sin(ψ), whereasf2 is independent ofψ . Furthermore,

(A.20)e−Φ

sinθ1 sinθ2∂k

(sinθ1 sinθ2H

klm) = −e−Φ+6p−xh2 f lm

1 (ψ, θ1, θ2, φ1, φ2),

and

ex−2p∇µ

(e−ΦHµlm

)= ex−2p∇µ

e−Φ−2x

[2a(1+ e−2g(1+ a2))

e2g + 2a2 + e−2g(1− a2)2∂µh1

+ e2g + 2a2 + e−2g(1+ a2)2

e2g + 2a2 + e−2g(1− a2)2∂µh2

]f lm

1

+ 2e−Φ−2x

[1+ 2a2e−2g

e2g + 2a2 + e−2g(1− a2)2∂µh1

(A.21)+ a(1+ e−2g(1+ a2))

e2g + 2a2 + e−2g(1− a2)2∂µh2

]f lm

2

.

It is not difficult to verify that the coefficients off lm1 in (A.19), (A.20) and (A.21), when inserted

into (A.18), add up to give the equation of motion forh2, as derived from(3.12). That is,

∇µ

e−Φ−2x

[2a(1+ e−2g(1+ a2))

e2g + 2a2 + e−2g(1− a2)2∂µh1 + e2g + 2a2 + e−2g(1+ a2)2

e2g + 2a2 + e−2g(1− a2)2∂µh2

](A.22)= e−Φ+8p−2xh2 + PKe8p−4xb,

whereas the coefficients off lm2 give the equation of motion forh1, as derived from(3.12), i.e.,

∇µ

2e−Φ−2x

[1+ 2a2e−2g

e2g + 2a2 + e−2g(1− a2)2∂µh1 + a(1+ e−2g(1+ a2))

e2g + 2a2 + e−2g(1− a2)2∂µh2

](A.23)= PKe8p−4x .

Finally, we consider Einstein’s equation,(3.1). The mixed components are trivially satisfiebecause both sides of(3.1) are identically zero. For the relevant internal components we nthat

(A.24)Rij = Rikjk + Riµj

µ.

Using the fact that the only non-vanishing Christoffel symbols besides the pure componeΓ ijk

andΓµνρ are

(A.25)Γµij = −1

2gµν∂νgij , Γ i

jµ = 1

2gil∂µgjl = Γ i

µj ,


s

but

econdction

we derive

(A.26)Rij = R(int)ij − 1

4

(∂µgij

)(∂µ lng(int)) + 1

2(∂µgik)

(∂µgjl

)gkl − 1

2∇µ∂µgij ,

whereg(int) denotes the internal block of the metric, andR(int)ij is the Ricci tensor that follow

from it. Notice the absence of tildes in(A.26).Hence, the internal components of Einstein’s equation are

0= R(int)ij − 1

4

(∂µgij

)(∂µ lng(int)) + 1

2(∂µgik)

(∂µgjl

)gkl − 1

2∇µ∂µgij

− 1

96Fim1...m4Fj

m1...m4 − 1

4

(e−ΦHiPQHj

PQ + eΦFiPQFjPQ

)+ 1

8gij

(e−Φ−2p−xI1 + eΦ−2p−xI2

)(A.27)≡ Sij ,

whereI1 and I2 were defined in(A.7) and (A.8). The expressions are quite complicated,using MAPLE we checked that all components of(A.27) are satisfied onceSψψ , Sθ1θ1, Sθ2θ2 andSφ1φ2, for instance, are zero. Moreover, taking these four components and solving for the sderivatives∇2p, ∇2x, ∇2a and∇2g, leads to the same expressions as derived from the a(3.12), i.e.,

∇µ∂µp = −1

6

e2p−2x−g

[e2g + (

1+ a2)] + 1

2e−4p−4x

[e2g + (

a2 − 1)2e−2g + 2a2]

− 2a2e−2g+8p − 2e−Φ−2x+8ph22 − e8p−4xK2

(A.28)− P 2eΦ−2x+8p[e2g + e−2g

(a2 − 2ab + 1

)2 + 2(a − b)2],

∇µ∂µx = e2p−2x−g[e2g + (

1+ a2)] − 1

2e−4p−4x

[e2g + (

a2 − 1)2e−2g + 2a2]

− 1

4P 2eΦ−2x

e8p

[e2g + e−2g

(a2 − 2ab + 1

)2 + 2(a − b)2] + 2∂µb∂µb

− 1

2e8p−4xK2 − 1

4e−Φ−2x

2e8ph2

2 + 2∂µh2∂µh2

(A.29)

+ 4(1+ 2e−2ga2)∂µh1∂µh1 + 8e−2ga2∂µh2∂

µh2 + 8a[e−2g(a2 + 1) + 1]∂µh1∂µh2

e2g + 2a2 + e−2g(1− a2)2

,

∇µ

(e−2g∂µa

)= −e−Φ−2x

[e2g + 2a2 + e−2g

(1− a2)2]−24ae−2g

(a2 − 1

)× [

1+ (a2 + 1

)e−2g

]∂µh1∂

µh1 + 4a[e−4g

(a4 − 1

) − 1]∂µh2∂

µh2

+ 2[e−4g

(a6 + 5a4 − 5a2 − 1

) − e2g + e−2g(a4 − 1

) − a2 − 1]∂µh1∂

µh2

− 2ae2p−2x−g + ae−4p−4x[(

a2 − 1)e−2g + 1

] + ae−2g+8p

(A.30)+ P 2(a − b)eΦ−2x+8p[e−2g

(a2 − 2ab + 1

) + 1],


-

ulated

d

∇µ∂µg = −e−2g∂µa∂µa − 2e−Φ−2x[e2g + 2a2 + e−2g

(1− a2)2]−2

× [e2g +4a2 +e−2g

(3a4 +2a2 −1

)]∂µh1∂

µh1 +4a2(1+a2e−2g)∂µh2∂

µh2

+ 2a[e2g + 2

(a2 + 1

) + e−2g(a4 + 4a2 − 1

)]∂µh1∂

µh2

− e2p−2x−g[e2g − (

1+ a2)] + 1

2e−4p−4x

[e2g − (

a2 − 1)2e−2g

] − a2e−2g+8p

(A.31)+ 1

2P 2eΦ−2x+8p

[e2g − e−2g

(a2 − 2ab + 1

)2].

Thus, the equations of motion for(p, x, a, g) arising from(3.12)guarantee that all internal components of Einstein’s equation are satisfied.

For the external components of Einstein’s equation we note that

(A.32)Rµν = Rµkνk + Rµρν

ρ = −3

2∇µ∂ν(x − 2p) − 1

4gmlgik(∂µgil)(∂νgmk) + R(ext)

µν ,

whereR(ext)µν stands for the purely external part of the Ricci-tensor, i.e., the one that is calc

solely withΓµνρ . Using(A.3) and

1

4gmlgik(∂µgil)(∂νgmk)

(A.33)= e−2g∂µa∂νa + 3

2∂µp∂νx + 3

2∂µx∂νp + ∂µg∂νg + 9∂µp∂νp + 5

4∂µx∂νx,

one arrives at

Rµν = Rµν −2∂µx∂νx −12∂µp∂νp −e−2g∂µa∂νa − ∂µg∂νg − gµν∇ρ ∂ρp + 1

2gµν∇ρ∂ρx

= 1

2∂µΦ∂νΦ + 1

96FµρσαβFν

ρσαβ + 1

4e−ΦHµmnHν

mn + 1

4eΦFµmnFν

mn

(A.34)− 1

48gµν

(e−ΦHMNP HMNP + eΦFMNP FMNP

).

Finally, using

(A.35)1

96FµρσαβFν

ρσαβ = −1

4K2e8p−4xgµν,

HµmnHνmn = 8e−2x

e2g + 2a2 + e−2g(1− a2)2

(1+ 2e−2ga2)∂µh1∂νh1

+ 1

2

[e2g + 2a2 + e−2g

(1+ a2)2]

∂µh2∂νh2

(A.36)+ a[e−2g

(a2 + 1

) + 1](∂µh1∂νh2 + ∂µh2∂νh1)

,

(A.37)FµmnFνmn = 4P 2e−2x∂µb∂νb,

the relations(A.7) and (A.8), as well as(A.28) and (A.29)in order to dispose of the seconderivatives ofx andp in (A.34), one verifies that

(A.38)Rµν = 2Gab∂µφa∂νφb + 4

3gµνV ,

with Gab andV given in(3.13) and (3.14).


’sn

ehe

s,

usede of

related

ace–

e

ies

Appendix B. Geometric relations for hypersurfaces

The time-slicing (or ADM) formalism[54,55], which we employ in our analysis of Einsteinequations, makes essential use of the geometry of hypersurfaces[87]. Therefore, we shall begiwith a review of the basic relations governing their geometry.

A hypersurface in a space–time with coordinatesXµ (µ = 0, . . . , d) and metricgµν is definedby a set ofd + 1 functions,Xµ(xi) (i = 1, . . . , d), where thexi are a set of coordinates on thhypersurface (note the difference toAppendix A, wherei labeled the internal coordinates of t10-dimensional space–time). The tangent vectors,X

µi ≡ ∂iX

µ, and the normal vector,Nµ, ofthe hypersurface can be chosen such that they satisfy the following orthogonality relation

(B.1)gµνXµi Xν

j = gij , Xµi Nµ = 0, NµNµ = 1,

wheregij represents the (induced) metric on the hypersurface. Henceforth, a tilde will beto label quantities characterizing the (d +1)-dimensional space–time manifold, whereas thosthe hypersurface remain unadorned.

The equations of Gauss and Weingarten define the second fundamental form,Kij , of thehypersurface,

(B.2)∂iXµj + Γ µ

λνXλi Xν

j − Γ kijX

µk = KijN

µ,

(B.3)∂iNµ + Γ µ

λνXλi Nν = −Kj

i Xµj .

The second fundamental form describes the extrinsic curvature of the hypersurface, and isto the intrinsic curvature by another equation of Gauss,

(B.4)RµνλρXµi Xν

j XλkX

ρl = Rijkl +KilKjk −KikKj l .

Furthermore, it satisfies the equation of Codazzi,

(B.5)RµνλρXµi Xν

j NλXρk = ∇iKjk − ∇jKik.

The symbol∇ denotes covariant derivatives with respect to the induced metricgij .The above formulas simplify if (as in the familiar time-slicing formalism), we choose sp

time coordinates such that

(B.6)X0 = const, Xi = xi.

Then, the tangent vectors are given byX0i = 0 andX

ji = δ

ji . One conveniently splits up th

space–time metric as (shown here for Euclidean signature)

(B.7)gµν =(

nini + n2 nj

ni gij

),

whose inverse is given by

(B.8)gµν = 1

n2

(1 −nj

−ni n2gij + ninj

).

The matrixgij is the inverse ofgij , and is used to raise hypersurface indices. The quantitn

andni are the lapse function and shift vector, respectively.The normal vectorNµ satisfying the orthogonality relations(B.1) is given by

(B.9)Nµ = (n,0), Nµ = 1(1,−ni

).

n


Usingollows,

iablesn

rface,l

em from

Then, one can obtain the second fundamental form from the equation of Gauss(B.2) as

(B.10)Kij = nΓ 0ij = − 1

2n(∂0gij − ∇inj − ∇j ni).

We are interested in expressing all bulk quantities in terms of hypersurface quantities.the equations of Gauss and Weingarten, some Christoffel symbols can be expressed as f

(B.11)Γ kij = Γ k

ij − nk

nKij ,

(B.12)Γ 0i0 = 1

n∂in + nj

nKij ,

(B.13)Γ ki0 = ∇in

k − nk

n∂in − nKij

(gjk + njnk

n2

).

The remaining components,Γ 000 andΓ k

00, are easily found from their definitions using(B.7)and (B.8),

(B.14)Γ 000 = 1

n

(∂0n + nj∂jn + ninjKij

),

(B.15)Γ k00 = ∂0n

k + ni∇ink − n∇kn − 2nKk

i ni − nkΓ 0

00.

Appendix C. Intermediate steps

In this appendix, we provide the equations of motion in terms of the geometric varcharacterizing the time-slice hypersurfaces introduced inAppendix B. The equations of motiothat follow from the action(2.1)are28

(C.1)∇2φa + Gabcg

µν(∂µφb

)(∂νφ

c) − V a = 0

for the scalar fields, and Einstein’s equations

(C.2)Eµν = −Rµν + 2Gab

(∂µφa

)(∂νφ

b) + 4

d − 1gµνV = 0.

In terms of hypersurface quantities,(C.1) takes the form∂2r − 2ni∂i∂r + n2∇2 + ninj∇i∂j − (

nKii + ∂r lnn − ni∂i lnn

)∂r

+ [n∇ in − ∂rn

i + nj∇j ni + ni

(nKj

j + ∂r lnn − nj∂j lnn)∂i

]φa

+ Gabc

[(∂rφ

b)(

∂rφc) − 2ni

(∂iφ

b)(

∂rφc) + (

n2gij + ninj)(

∂iφb)(

∂jφc)] − n2 ∂V

∂φa

(C.3)= 0.

Eq. (C.2) splits into components that are normal, mixed, and tangential to the hypersuobtained by projecting withNµNν − gijX

µi Xν

j , NµXνi andX

µi Xν

j , respectively. The normacomponents become

28 Note that, as opposed to the main text, we use a tilde here to denote 5d quantities in order to distinguish ththe hypersurface quantities.


ricful. The

)

ms, in

(nKi

j

)(nKj

i

) − (nKi

i

)2 + n2R − 4n2V

+ 2Gab

[(∂rφ

a)(

∂rφb) − 2ni

(∂iφ

a)(

∂rφb) + (

ninj − n2gij)(

∂iφa)(

∂jφb)]

(C.4)= 0.

The mixed components are

(C.5)∂i

(nKj

j

) − ∇j

(nKj

i

) − nKjj ∂i lnn + nKj

i ∂j lnn − 2Gab

(∂rφ

a − nj∂jφa)(

∂iφb) = 0.

For the tangential components one obtains

−∂r

(nKi

j

) + nk∇k

(nKi

j

) + nKij

(nKk

k + ∂r lnn − nk∂k lnn) + n∇ i∂j n

(C.6)+ nKik∇j n

k − nKkj∇kn

i − n2Rij + 2n2Gab

(∇ iφa)(

∂jφb) + 4n2V

d − 1δij = 0.

The equations of motion given in Section4.4are obtained from(C.3)–(C.6)upon expanding thefields and using the substitution rules(4.35). For this, the following expressions for geomethypersurface quantities, up to quadratic order in the gauge-invariant fluctuations, are useextrinsic curvature tensor is

nKij → 2

d − 1Wδi

j − 1

2∂re

ij + 1

2

(∂idj + ∂jd

i + 2∂i∂j

c

)+ 1

2eik∂re

kj

(C.7)− 1

2

[eik

(∂kdj + ∂jd

k + 2∂j ∂

k

c

)+

(dk + ∂k

c

)(∂iejk + ∂j e

ik − ∂ke

ij

)],

and its trace is

(C.8)nKii → 2d

d − 1W + c + 1

2eik∂re

ki − ei

k

(∂kdi + ∂i∂

k

c

).

The intrinsic Ricci tensor is replaced by

Rij → −1

2e−2A

[ei

j + ekl

(∂i∂ke

lj + ∂j ∂

leik − ∂k∂

leij − ∂i∂j e

lk

) − eikek

j

(C.9)− 1

2

(∂iek

l

)(∂j e

lk

) + (∂le

ik

)(∂kel

j

) − (∂le

kj

)(∂lei

k

)],

and the Ricci scalar becomes

(C.10)R → e−2A

[eije

ji + 3

4∂ie

jk∂

iekj − 1

2∂ie

kj ∂

j eik

].

Appendix D. Quadratic source terms

In this appendix, we provide the explicit expressions for the source termsJa , J , J i andJ ij

to quadratic order, which appear in the equations of motion(4.38), (4.39), (4.40) and (4.49,respectively. The fielddi has been dropped everywhere, since its solution(4.43) is of secondorder. Moreover, we have used the linear equations of motion in order to eliminate terparticular the relation

(D.1)∂rc − 2d

d − 1W c − e−2Ab = 0,


which follows from(4.41), (4.42) and (4.45).

J a = 1

2

[V a |bc −Ra

bcdV d − (Ra

bcd|e −Radeb|c

)WdWe

]abac − 2Ra

bcdWd(Dra

b)ac

+ 2V a |babb + (Dra

a)(c + ∂rb) + 2V ab2 + 2

(∂i

c

)∂iDra

a

− e−2A(2baa − ei

j ∂i∂jaa

) − V ab2

(D.2)+ Wa

[−b∂rb + 1

2eij ∂re

ji − ei

j

∂i∂j

c − (∂ib)∂i

c

],

J = 2Va|baaab − 2(Dra

a)(Draa) + 2Ra

bcdWaWcabad + 2e−2A

(∂iaa

)(∂iaa)

+ 8Vaaab + 4Wa

(∂ia

a)∂i

c + 8V b2 − 4V b2 + (Π

ji c

)(∂i∂j

c

)

+(

∂i∂j

c

)(∂re

ij − 4W ei

j

) − 1

4

(∂re

ij

)(∂re

ji

) + 2W eij ∂re

ji

(D.3)− e−2A

[eije

ji + 3

4

(∂ie

jk

)(∂iek

j

) − 1

2

(∂ie

jk

)(∂kei

j

)],

Ji = 2(∂ia

a)Draa − 2Wb∂ib + (∂jb)Π

ji c + 1

2(∂jb)∂re

ji − 1

2

(∂j

c

)e

ji − 1

4∂i∂r

(ejke

kj

)(D.4)+ 1

2ejk∂r∂j e

ki + 1

4

(∂ie

jk

)(∂re

kj

).

The terms underlined in(D.2), (D.3) and (D.4)can be eliminated by the field redefinitions

(D.5)b → b + 1

2b2,

(D.6)c → c − 1

2eij ∂re

ji + (∂ib)

∂i

c + eij

∂i∂j

c.

J ij = Πik

jl

2

(∂l∂m c

)(∂m∂k c

)− 2

(∂l∂k c

)(c + ∂rb) + 2

(∂m∂re

lk

)(∂m

c

)

+ (∂re

lk

)(c + ∂rb) + (

∂relm

)(∂re

mk

)+ e−2A

[2(∂lb

)(∂kb) − 4

(∂laa

)(∂kaa) − 2bel

k − 2emn ∂l∂men

k

(D.7)+ emn ∂m∂nel

k − 1

2

(∂lem

n

)(∂ke

nm

) − (∂mel

n

)(∂nem

k

) + (∂men

k

)(∂mel

n

)].

The underlined terms in(D.7) can be eliminated by the field redefinition

(D.8)eij → ei

j + 1

2Πik

jl

(elmem

k

).


wing

e

Appendix E. Matrices for the KS background

In this appendix we give the explicit form of the matrices appearing in(6.25)in the KS back-ground. In order to keep the formulas under (typographical) control, we introduce the follonotation

H1 = e6q+4f , H2 = e10q+6f , H3 = e−4q+4f ,

Υ1 = e−Φ[2 cosh(2y)P + sinh(2y)(2P − N2 + N1)

],

Υ2 = e−Φ[2 sinh(2y)P + cosh(2y)(2P − N2 + N1)

],

Υ3 = Q − P(s − N1 − N2) + 1

2

(N2

1 − N22

),

Υ4 = −4e6q − 6e10f +6q coshy + 5√

27e6f Υ3,

Υ5 = coshy − e−10f , Υ6 = e−Φ(N1 + N2),

(E.1)Υ7 = − 2

15

[8e−10f + 12 coshy − 25

√27Υ3

H1

],

where the fields denote the background values given in Section6.1. With these abbreviations, thmatrices are given by

GabcW

c = H3 ×

0 0 0 −√

2710

PH1

0√

2710

N1+P

H1−

√27

10N2−P

H1

0 0 0 −√

2710

PH1

0√

2710

N1+P

H1−

√27

10N2−P

H1

0 0 0 −√27 P

H10 −√

27N2−P

H1

√27

N1+P

H1

9Υ1 6Υ132Υ1

√27Υ3−2e6(q−f )

H13Υ2 −3sinhy −3sinhy

0 0 0 −√

272

N1−N2+2P

H10

√27 P

H1

√27 P

H1

− 92 (Υ2 + Υ6) −3(Υ2 + Υ6) − 3

4 (Υ2 − Υ6) − 32 sinhy − 3

2Υ1

√27Υ3−2e6(q−f )

H10

− 92 (Υ2 − Υ6) −3(Υ2 − Υ6) − 3

4 (Υ2 + Υ6) − 32 sinhy − 3

2Υ1 0√

27Υ3−2e6(q−f )

H1

,

∂bWa = H3 ×

Υ785Υ5 0

√3P

H125 sinhy −√

3N1+P

H1

√3

N2−P

H1125 Υ5 − 6

5 (2coshy + 3e−10f ) 0 0 − 35 sinhy 0 0

0 0 0 0 0 0 0

12Υ1 −12Υ1 −3Υ1 0 −6Υ2 −3sinh(2y)eΦ 3sinh(2y)eΦ

12sinhy −12sinhy 0 0 −3coshy 0 0

−6(Υ2 + Υ6) 6(Υ2 + Υ6) 32 (Υ2 − Υ6) 0 3Υ1

32 [e−Φ + eΦ cosh(2y)] 3

2 [e−Φ − eΦ cosh(2y)]−6(Υ2 − Υ6) 6(Υ2 − Υ6) 3

2 (Υ2 + Υ6) 0 3Υ1 − 32 [e−Φ − eΦ cosh(2y)] − 3

2 [e−Φ + eΦ cosh(2y)]


s. 2

s. Lett.

0.276.4, hep-

Nucl.

09191.

, hep-

ep-

s,

l.

ev. D 61

2, hep-

eory,

1)

WaWb = H 23 ×

Υ 24

15H22 H2

3

2Υ4Υ55H2H3

0

√3PΥ4

10H1H2H3

Υ4 sinhy

10H2H3−

√3(N1+P )Υ410H1H2H3

√3(N2−P )Υ410H1H2H3

3Υ4Υ55H2H3

185 Υ 2

5 03√

27PΥ510H1

910 sinhyΥ5 − 3

√27(N1+P )Υ5

10H1

3√

27(N2−P )Υ510H1

0 0 0 0 0 0 0

3Υ1Υ4H2H3

18Υ1Υ5 03√

27PΥ12H1

92 sinhyΥ1 − 3

√27(N1+P )Υ1

2H1

3√

27(N2−P )Υ12H1

3Υ4 sinhy

H2H318Υ5 sinhy 0 3

√27P sinhy2H1

92 sinh2 y − 3

√27(N1+P )sinhy

2H1

3√

27(N2−P )sinhy

2H1

− 3Υ4(Υ2+Υ6)

2H2H3−9Υ5(Υ2 + Υ6) 0 − 3

√27P (Υ2+Υ6)

4H1− 9

4 (Υ2 + Υ6)sinhy3√

27(N1+P )(Υ2+Υ6)

4H1− 3

√27(N2−P )(Υ2+Υ6)

4H1

− 3Υ4(Υ2−Υ6)

2H2H3−9Υ5(Υ2 − Υ6) 0 − 3

√27P (Υ2−Υ6)

4H1− 9

4 (Υ2 − Υ6)sinhy3√

27(N1+P )(Υ2−Υ6)

4H1− 3

√27(N2−P )(Υ2−Υ6)

4H1

.

References

[1] J.M. Maldacena, The largeN limit of superconformal field theories and supergravity, Adv. Theor. Math. Phy(1998) 231–252, hep-th/9711200.

[2] S.S. Gubser, I.R. Klebanov, A.M. Polyakov, Gauge theory correlators from non-critical string theory, PhyB 428 (1998) 105–114, hep-th/9802109.

[3] E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253–291, hep-th/980215[4] M. Bianchi, D.Z. Freedman, K. Skenderis, How to go with an RG flow, JHEP 0108 (2001) 041, hep-th/0105[5] M. Bianchi, D.Z. Freedman, K. Skenderis, Holographic renormalization, Nucl. Phys. B 631 (2002) 159–19

th/0112119.[6] D. Martelli, W. Mück, Holographic renormalization and Ward identities with the Hamilton–Jacobi method,

Phys. B 654 (2003) 248–276, hep-th/0205061.[7] M. Berg, H. Samtleben, Holographic correlators in a flow to a fixed point, JHEP 0212 (2002) 070, hep-th/02[8] I. Papadimitriou, K. Skenderis, AdS/CFT correspondence and geometry, hep-th/0404176.[9] I. Papadimitriou, K. Skenderis, Correlation functions in holographic RG flows, JHEP 0410 (2004) 075

th/0407071.[10] H. Lin, O. Lunin, J. Maldacena, Bubbling AdS space and 1/2 BPS geometries, JHEP 0410 (2004) 025, h

th/0409174.[11] O. Lunin, J. Maldacena, Deforming field theories withU(1) × U(1) global symmetry and their gravity dual

JHEP 0505 (2005) 033, hep-th/0502086.[12] U. Gursoy, C. Nuñez, Dipole deformations ofN = 1 SYM and supergravity backgrounds withU(1) × U(1) global

symmetry, hep-th/0505100.[13] N. Halmagyi, K. Pilch, C. Romelsberger, N.P. Warner, Holographic duals of a family ofN = 1 fixed points, hep-

th/0506206.[14] L. Girardello, M. Petrini, M. Porrati, A. Zaffaroni, The supergravity dual ofN = 1 super-Yang–Mills theory, Nuc

Phys. B 569 (2000) 451–469, hep-th/9909047.[15] C.V. Johnson, A.W. Peet, J. Polchinski, Gauge theory and the excision of repulsion singularities, Phys. R

(2000) 086001, hep-th/9911161.[16] R.C. Myers, Dielectric-branes, JHEP 9912 (1999) 022, hep-th/9910053.[17] S. Hong, S. Yoon, M.J. Strassler, On the couplings of vector mesons in AdS/QCD, hep-th/0409118.[18] S. Hong, S. Yoon, M.J. Strassler, On the couplings of theρ meson in AdS/QCD, hep-ph/0501197.[19] T. Sakai, S. Sugimoto, More on a holographic dual of QCD, hep-th/0507073.[20] J. Polchinski, M.J. Strassler, Deep inelastic scattering and gauge/string duality, JHEP 0305 (2003) 01

th/0209211.[21] S. Kachru, R. Kallosh, A. Linde, J. Maldacena, L. McAllister, S. Trivedi, Towards inflation in string th

JCAP 0310 (2003) 013, hep-th/0308055.[22] I.R. Klebanov, M.J. Strassler, Supergravity and a confining gauge theory: Duality cascades andχSB-resolution of

naked singularities, JHEP 0008 (2000) 052, hep-th/0007191.[23] M. Strassler, The duality cascade, hep-th/0505153.[24] C.P. Herzog, I.R. Klebanov, P. Ouyang, Remarks on the warped deformed conifold, hep-th/0108101.[25] J.M. Maldacena, C. Nuñez, Towards the largeN limit of pure N = 1 super-Yang–Mills, Phys. Rev. Lett. 86 (200

588–591, hep-th/0008001.


ys.

86,

s,

02.

48,

Thesis,

s.

uantum

v. D 69

, hep-

1,

, hep-

group

, Phys.

–500,

ons in an

, Phys.

ows,

4) 062,

68.B 686

51.cl.

(2005)

9–114,

[26] R. Apreda, F. Bigazzi, A.L. Cotrone, M. Petrini, A. Zaffaroni, Some comments onN = 1 gauge theories fromwrapped branes, Phys. Lett. B 536 (2002) 161–168, hep-th/0112236.

[27] P. Di Vecchia, A. Lerda, P. Merlatti,N = 1 andN = 2 super-Yang–Mills theories from wrapped branes, Nucl. PhB 646 (2002) 43–68, hep-th/0205204.

[28] M. Bertolini, P. Merlatti, A note on the dual ofN = 1 super-Yang–Mills theory, Phys. Lett. B 556 (2003) 80–hep-th/0211142.

[29] W. Mück, Perturbative and non-perturbative aspects of pureN = 1 super-Yang–Mills theory from wrapped braneJHEP 0302 (2003) 013, hep-th/0301171.

[30] O. Aharony, A. Buchel, A. Yarom, Holographic renormalization of cascading gauge theories, hep-th/05060[31] M. Krasnitz, A two point function in a cascadingN = 1 gauge theory from supergravity, hep-th/0011179.[32] M. Krasnitz, Correlation functions in a cascadingN = 1 gauge theory from supergravity, JHEP 0212 (2002) 0

hep-th/0209163.[33] M. Krasnitz, Correlation functions in supersymmetric gauge theories from supergravity fluctuations. Ph.D.

Princeton, 2003, UMI-30-68792.[34] I.R. Klebanov, A.A. Tseytlin, Gravity duals of supersymmetricSU(N) × SU(N + M) gauge theories, Nucl. Phy

B 578 (2000) 123–138, hep-th/0002159.[35] G. Papadopoulos, A.A. Tseytlin, Complex geometry of conifolds and 5-brane wrapped on 2-sphere, Class. Q

Grav. 18 (2001) 1333–1354, hep-th/0012034.[36] D.Z. Freedman, C. Nuñez, M. Schnabl, K. Skenderis, Fake supergravity and domain wall stability, Phys. Re

(2004) 104027, hep-th/0312055.[37] S.S. Gubser, Supersymmetry and F-theory realization of the deformed conifold with three-form flux

th/0010010.[38] M. Graña, J. Polchinski, Supersymmetric three-form flux perturbations onAdS(5), Phys. Rev. D 63 (2001) 02600

hep-th/0009211.[39] M. Bianchi, M. Prisco, W. Mück, New results on holographic three-point functions, JHEP 0311 (2003) 052

th/0310129.[40] O. DeWolfe, D.Z. Freedman, Notes on fluctuations and correlation functions in holographic renormalization

flows, hep-th/0002226.[41] G. Arutyunov, S. Frolov, S. Theisen, A note on gravity-scalar fluctuations in holographic RG flow geometries

Lett. B 484 (2000) 295–305, hep-th/0003116.[42] W. Mück, Correlation functions in holographic renormalization group flows, Nucl. Phys. B 620 (2002) 477

hep-th/0105270.[43] J.M. Bardeen, Gauge invariant cosmological perturbations, Phys. Rev. D 22 (1980) 1882–1905.[44] J.M. Bardeen, P.J. Steinhardt, M.S. Turner, Spontaneous creation of almost scale—free density perturbati

inflationary universe, Phys. Rev. D 28 (1983) 679.[45] O. DeWolfe, D.Z. Freedman, S.S. Gubser, A. Karch, Modeling the fifth dimension with scalars and gravity

Rev. D 62 (2000) 046008, hep-th/9909134.[46] M. Bianchi, O. DeWolfe, D.Z. Freedman, K. Pilch, Anatomy of two holographic renormalization group fl

JHEP 0101 (2001) 021, hep-th/0009156.[47] F. Larsen, R. McNees, Holography, diffeomorphisms, and scaling violations in the CMB, JHEP 0407 (200

hep-th/0402050.[48] P.K. Kovtun, A.O. Starinets, Quasinormal modes and holography, hep-th/0506184.[49] W. Mück, M. Prisco, Glueball scattering amplitudes from holography, JHEP 0404 (2004) 037, hep-th/04020[50] M. Bianchi, A. Marchetti, Holographic three-point functions: One step beyond the tradition, Nucl. Phys.

(2003) 261–284, hep-th/0302019.[51] W. Mück, Progress on holographic three-point functions, Fortschr. Phys. 53 (2005) 948–954, hep-th/04122[52] L. Ametller, J.M. Pons, P. Talavera, On the consistency of theN = 1 SYM spectra from wrapped five-branes, Nu

Phys. B 674 (2003) 231–258, hep-th/0305075.[53] E. Caceres, C. Nuñez, Glueballs of super-Yang–Mills from wrapped branes, hep-th/0506051.[54] C.W. Misner, K.S. Thorne, J.A. Wheeler, Gravitation, Freeman, San Francisco, 1973.[55] R.M. Wald, General Relativity, University of Chicago Press, Chicago, 1984.[56] A.B. Clark, D.Z. Freedman, A. Karch, M. Schnabl, The dual of Janus: An interface CFT, Phys. Rev. D 71

066003, hep-th/0407073.[57] A.Z. Petrov, Einstein Spaces, Pergamon Press, Oxford, 1969.[58] P. Breitenlohner, D.Z. Freedman, Stability in gauged extended supergravity, Ann. Phys. 144 (1982) 249.[59] I.R. Klebanov, E. Witten, AdS/CFT correspondence and symmetry breaking, Nucl. Phys. B 556 (1999) 8

hep-th/9905104.


2) 006,

2002,

76, hep-

hys.

2001)

(2003)

s.

tum

0)

onifold,

(2004)

0103

ution:

escrip-

79–745,

1994.

r. Math.

. Phys.

. Phys.

) 64–70,

1 (2004)

[60] M. Berg, H. Samtleben, An exact holographic RG flow between 2d conformal fixed points, JHEP 0205 (200hep-th/0112154.

[61] M. Berg, Holography and infrared conformality in two dimensions, in: L. Baulieu, et al. (Eds.), CargeseProgress in String and Particle Theory, Kluwer Academic, 2003, hep-th/0212197.

[62] K. Skenderis, Lecture notes on holographic renormalization, Class. Quantum Grav. 19 (2002) 5849–58th/0209067.

[63] A.V. Belitsky, V.M. Braun, A.S. Gorsky, G.P. Korchemsky, Integrability in QCD and beyond, Int. J. Mod. PA 19 (2004) 4715–4788, hep-th/0407232.

[64] L.A. Pando Zayas, A.A. Tseytlin, 3-branes on resolved conifold, JHEP 0011 (2000) 028, hep-th/0010088.[65] M. Haack, J. Louis, M-theory compactified on Calabi–Yau fourfolds with background flux, Phys. Lett. B 507 (

296–304, hep-th/0103068.[66] M. Berg, M. Haack, H. Samtleben, Calabi–Yau fourfolds with flux and supersymmetry breaking, JHEP 0304

046, hep-th/0212255.[67] B. de Wit, I. Herger, H. Samtleben, Gauged locally supersymmetricD = 3 nonlinear sigma models, Nucl. Phy

B 671 (2003) 175–216, hep-th/0307006.[68] O. Hohm, J. Louis, SpontaneousN = 2 to N = 1 supergravity breaking in three dimensions, Class. Quan

Grav. 21 (2004) 4607–4624, hep-th/0403128.[69] A. Ceresole, G. Dall’Agata, General matter coupledN = 2, D = 5 gauged supergravity, Nucl. Phys. B 585 (200

143–170, hep-th/0004111.[70] S.S. Gubser, C.P. Herzog, I.R. Klebanov, Symmetry breaking and axionic strings in the warped deformed c

JHEP 0409 (2004) 036, hep-th/0405282.[71] S.S. Gubser, C.P. Herzog, I.R. Klebanov, Variations on the warped deformed conifold, C R Physique 5

1031–1038, hep-th/0409186.[72] O. Aharony, A note on the holographic interpretation of string theory backgrounds with varying flux, JHEP

(2001) 012, hep-th/0101013.[73] A. Butti, M. Graña, R. Minasian, M. Petrini, A. Zaffaroni, The baryonic branch of Klebanov–Strassler sol

A supersymmetric family ofsu(3) structure backgrounds, hep-th/0412187.[74] S.S. Gubser, A.A. Tseytlin, M.S. Volkov, Non-Abelian 4d black holes, wrapped 5-branes, and their dual d

tions, JHEP 0109 (2001) 017, hep-th/0108205.[75] S.D. Mathur, Where are the states of a black hole?, hep-th/0401115.[76] S.D. Mathur, The fuzzball proposal for black holes: An elementary review, hep-th/0502050.[77] S.S. Gubser, Curvature singularities: The good, the bad, and the naked, Adv. Theor. Math. Phys. 4 (2002) 6

hep-th/0002160.[78] I.S. Gradshteyn, I.M. Ryzhik, Table of Integrals, Series and Products, fifth ed., Academic Press, New York,[79] M. Abramowitz, I.A. Stegun (Eds.), Handbook of Mathematical Functions, Dover, New York, 1965.[80] E. Witten, Anti-de Sitter space, thermal phase transition, and confinement in gauge theories, Adv. Theo

Phys. 2 (1998) 505–532, hep-th/9803131.[81] R. Apreda, Non-supersymmetric regular solutions from wrapped and fractional branes, hep-th/0301118.[82] F. Bigazzi, L. Girardello, A. Zaffaroni, A note on regular type 0 solutions and confining gauge theories, Nucl

B 598 (2001) 530–542, hep-th/0011041.[83] I.R. Klebanov, E. Witten, Superconformal field theory on three branes at a Calabi–Yau singularity, Nucl

B 536 (1998) 199–218, hep-th/9807080.[84] E. Caceres, R. Hernandez, Glueball masses for the deformed conifold theory, Phys. Lett. B 504 (2001

hep-th/0011204.[85] X. Amador, E. Caceres, Spin two glueball mass and glueball Regge trajectory from supergravity, JHEP 041

022, hep-th/0402061.[86] GRTensor, MAPLE package,http://grtensor.phy.queensu.ca/.[87] L.P. Eisenhart, Riemannian Geometry, Princeton Univ. Press, Princeton, 1964.

http://grtensor.phy.queensu.ca/

y

Ocharge. These

nerated.

largebreaksn may-nd

chester


SO(10)-GUT coherent baryogenesis

Björn Garbrechta,∗,1, Tomislav Prokopecb, Michael G. Schmidta

a Institut für Theoretische Physik, Universität Heidelberg, Philosophenweg 16, 69120 Heidelberg, Germanb Institute for Theoretical Physics (ITF) & Spinoza Institute, Utrecht University,

Leuvenlaan 4, Postbus 80.195, 3508 TD Utrecht, The Netherlands

Received 11 October 2005; accepted 31 October 2005


Abstract

A model for GUT baryogenesis, coherent baryogenesis within the framework of supersymmetric S(10),is considered. In particular, we discuss the Barr–Raby model, where at the end of hybrid inflationasymmetries can be created through the time-dependent higgsino–gaugino mixing mass matrixasymmetries are processed to Standard Model matter through decaysvia nonrenormalizable(B–L)-violating operators. We find that a baryon asymmetry in accordance with observation can be geAn appendix is devoted to provide useful formulas and concrete examples for calculations within SO(10). 2005 Published by Elsevier B.V.

1. Introduction

Grand Unified Theories (GUTs) generically predict a scalar potential and thereby aamount of vacuum energy when the scalar fields are displaced from the minimum whichthe symmetry down to the Standard Model. It is hence often argued that cosmic inflatiobe implemented by the scalar field dynamics of a GUT[1]. A well-known paradigm is supersymmetric (SUSY) hybrid inflation[2,3], and it has led to various models using different graunified gauge groups, see e.g.[4–6].

* Corresponding author.E-mail addresses:[email protected], [email protected](B. Garbrecht),

[email protected](T. Prokopec),[email protected](M.G. Schmidt).1 Affiliation since October 1st, 2005: Department of Physics and Astronomy, University of Manchester, Man

M13 9PL, United Kingdom.

0550-3213/$ – see front matter 2005 Published by Elsevier B.V.doi:10.1016/j.nuclphysb.2005.10.042






134 B. Garbrecht et al. / Nuclear Physics B 736 (2006) 133–155

e vi-ndetry of

into aeutri-cation,bridalism

a

theymme-ation

ithin ale andch Kyaead hoc

nt foraugino

e applysilye

ryon

nt massdied for

ns.ir masshargeecies.rticleasis in

izes thed

rse,

Another feature of GUTs, which is of possible relevance for cosmology, is obviously tholation of baryon minus lepton number,B–L, due to the unification and mixing of baryons aleptons, since this can lead to mechanisms for generating the observed baryon asymmthe Universe (BAU). While leptogenesis is strictly speaking not necessarily implementedGUT—it can be operative within the Standard Model minimally extended by right-handed nnos with Majorana mass terms—we suggested a scenario relying on baryon–lepton unificoherent baryogenesis[7,8], and implemented it within a Pati–Salam supersymmetric hyinflationary model. We give a brief review of this mechanism and our calculational forminspired by kinetic theory in Section2.

Being a product group, the Pati–Salam gauge group[9] gives, strictly speaking, not rise toGUT, and it is therefore desirable to devise models based on the proper GUT SO(10). A way ofbreaking the SO(10)-symmetry, which is particularly suitable for SUSY hybrid inflation, ismechanism proposed by Barr and Raby. Kyae and Shafi extended this model by global stries which restrict the superpotential to contain only couplings consistent with hybrid infl[6]. This model has however a rather complicated Higgs sector.

The purpose of this paper is to show that coherent baryogenesis naturally occurs wSUSY-SO(10)-framework. Therefore, we want to keep the discussion as simple as possibuse the minimal superpotential suggested by Barr and Raby, leaving aside the issues whiand Shafi focus on. In turn, we also point out that we do not extend the minimal model byterms just in order to make our mechanism viable.

In Section3 we put together the features of the Barr–Raby model which are importaour baryogenesis scenario and present in some detail the derivation of the higgsino–gmixing mass matrix. We also devote an appendix to the conventions and techniques wfor calculations within SO(10), with the intention to make this article self-contained and eacomprehensible to the reader who is not familiar with SO(10)-model building, and furthermorwe want to provide a useful help for accessing the papers mentioned above.

Putting our considerations into work, we present a numerical study of SO(10)-coherentbaryogenesis in Section4. There, we also discuss the decay processes of the higgsinosvia non-renormalizable,(B–L)-violating couplings. The result is an estimate of the produced baasymmetry for a particular set of parameters.

2. Coherent baryogenesis

Coherent baryogenesis relies on the production of particles due to a time dependeterm, a phenomenon which we refer to as preheating. Preheating has been extensively stuscalars[10–13]and for fermions[14–16], using the technique of Bogolyubov transformatioA special feature of fermionic preheating is that fermions can be amply produced when theterm crosses zero. When one considers instead of a mass term a mixing mass matrix, cC

and parityP may be violated and an asymmetry can be stored within different fermionic spA nonsymmetric mass matrix however leads to the violation of the orthogonality of paand antiparticle modes. One therefore needs a formalism which is independent of a bterms of particle and antiparticle creation and annihilation operators and thereby generalBogolyubov transformation approach. This is developed in Ref.[17] and shall be briefly reviewein the following.

We consider several fermionic flavours, mixing through a mass matrixM(η), which is a func-tion of the conformal timeη. In a spatially flat Friedmann–Lemaître–Robertson–Walker unive

B. Garbrecht et al. / Nuclear Physics B 736 (2006) 133–155 135

or

s. We

eeal phasenditions,

ut inthe

ates,ss

c-

ons of

lsoeous

id

described by the metricgµν = a2(η) × diag(1,−1,−1,−1), we rescale the fields such that fthe mass terms, there is the replacementM(η) → a(η)M(η).

Our goal is to compute the charge density, which is a bilinear form in the fermionic fieldtherefore introduce the Wigner function

(1)iS<ij (k, x) = −

∫d4r eik·r ⟨ψj (x − r/2)ψi(x + r/2)

⟩,

wherei, j are flavour indices and(iγ 0S<)† = iγ 0S< is Hermitean. The Wigner transform is thFourier transformation of the two point functionw.r.t. its relative coordinate while keeping thcenter of mass coordinate fixed. One can hence consider it as an analogue to a classicspace density defined in quantum theory. Since we assume here spatial homogeneous coone can ignore the center of mass coordinate in the following and consider iS< as a Fouriertransform. Our formalism is applicable for a 2-point function with general density matrix, bview of our applications in inflation we prefer to write it with respect to the vacuum fromoutset.

When decomposing the mass matrixM into its Hermitean and anti-Hermitean parts,

MH = 1

2

(M + M†), MA = 1

2i

(M − M†),

we find that iS< obeys the Wigner space Dirac equation

(2)

(/k + i

2γ 0∂η − (MH + iγ 5MA

)e− i

2

←∂η∂k0

)il

iS<lj = 0.

The mass matrixM emerges generically from Yukawa couplings to scalar field condensLYu = −yφψRψL +h.c. In the model we consider here,M is the higgsino–gaugino mixing mamatrix.

A crucial point is the time-dependence ofM , which is not only the source of particle prodution. The matricesM anddM/dη both contribute toCP -violating phases, which—providedManddM/dη are linearly independent—cannot be removed by time-independent redefinitithe fermionic fields.

In order to simplify the Wigner–Dirac equation(2), which is, besides the flavour indices, aendowed with a 4× 4 spinor structure, we make use of the fact that for spatially homogeniγ0S

<h , the helicity operatorh = k · γ 0γ γ 5 commutes with the Dirac operator in(2) and decom-

pose the Wigner function as[18,19]

(3)−iγ0S<h = 1

4(1 + hk · σ ) ⊗ ρµgµh,

where we have omitted the flavour indices,k = k/|k| andσµ, ρµ (µ = 0,1,2,3) are the Paulmatrices, andh = ±1 are the eigenvalues ofh. We multiply (2) by ρµ, take the Dirac trace anintegrate the Hermitean part overk0. Introducing the 0th momenta ofgµh, fµh = ∫ (dk0/2π)gµh,

we note that the functionsf ijµh explicitly read

fij

0h(x,k) = −∫

dk0

2π

∫d4r eik·r ⟨ψhj (x − r/2)γ 0ψhi(x + r/2)

⟩,

fij

1h(x,k) = −∫

dk0

2π

∫d4r eik·r ⟨ψhj (x − r/2)ψhi(x + r/2)

⟩,

fij

2h(x,k) = −∫

dk0∫

d4r eik·r ⟨ψhj (x − r/2)(−iγ 5)ψhi(x + r/2)

⟩,

2π


ively.s:

ialthatt-lated.e-ecome

ssible

d ofzero

alnal, butn. Ifry

e

ns de-.

(4)fij

3h(x,k) = −∫

dk0

2π

∫d4r eik·r ⟨ψhj (x − r/2)γ 0γ 5ψhi(x + r/2)

⟩.

Therefore, thefµh(x,k) can be interpreted as follows:f0h is the charge density,f3h is the axialcharge density, andf1h andf2h correspond to the scalar and pseudoscalar density, respect

From the Wigner–Dirac equation(2), we can now derive the following system of equation

f ′0h + i[MH ,f1h] + i[MA,f2h] = 0,

f ′1h + 2h|k|f2h + i[MH ,f0h] − MA,f3h = 0,

f ′2h − 2h|k|f1h + MH ,f3h + i[MA,f0h] = 0,

(5)f ′3h − MH ,f2h + MA,f1h = 0,

where the prime denotes a derivative with respect toη. It is understood thatM and thefµh areflavour matrices. Note that the commutators in(5), which mix particle flavours, are essentfor the production of the chargesf0h, and thus for our scenario. Moreover, one can infer,a necessary condition forf ′

0h = 0 is a nonsymmetricM . We already anticipated this when noing that for such a mass matrix the orthogonality of particle and antiparticle modes is vioNote that the tree level dynamics given by Eqs.(5) closes forfµh. When rescatterings, as dscribed through nonlocal quantum loop corrections, are included, off-shell effects may bimportant, and one would have to solve for the full dynamics of thegµh. In the nonrelativisticregime and close to equilibrium however, in which off-shell effects are suppressed, it is poto include rescatterings into Eqs.(5) and still retain closure for the equations for thefµh.

Now we fix the initial conditions for a universe, which is void of fermions at the eninflation. For an initially diagonal slowly evolving mass matrix, the Wigner functions for aparticle state are (cf. Ref.[17]):

f ab0h = La∗

h Lbh + Ra∗

h Rbh, f ab

1h = −2(LahR

∗hab),

(6)f ab3h = La∗

h Lbh − Ra∗

h Rbh, f ab

2h = 2(L∗ah Rb

h

),

with

Labh = δab

√ωa + hk

2ωa

, Rabh = δab

M∗aa√

2ωa(ωa + hk),

whereωa = √k2 + |Maa|2. Note, that for the case of realMaa , this reduces just to the usuchoice of the components of the basis spinors in chiral representation. For a nondiagoHermitean,M , one obtains the initial conditions by an appropriate unitary transformatioadditionallyMA = 0, as is the case for the SO(10) example discussed in the following, a biunitatransformation is necessary for diagonalization.

Sincef0h is the zeroth component of the vector current, the charge of the speciesa carriedby the mode with momentumk and helicityh is simply qah(k) = f aa

0h − 1. Note also, that thLagrangean

L= ψa/∂ψa − ψb

(MH + iγ 5MA

)ba

ψa

is U(1) symmetric, and thus∑

a qah(k) is conserved, as we shall verify explicitly for the SO(10)example discussed here.

The scenario for coherent baryogenesis is as follows: initially, there are zero fermioscribed by appropriate initial conditions for thefµh, andM is approximately constant in time


c-arges-can be

tsking,d Rabyr ourgsino–ular

,

ondi-

t–right

Then a phase transition occurs during whichM changes rapidly, which leads to fermion prodution. Eventually,M stops evolving and the produced number of fermions as well as the chf0h stored within the different species are frozen in. We emphasise thatf ii

0h should not be confused with the number of produced particle pairs at preheating, which in our languageexpressed in terms of thefih (i = 1,2,3) as given in Ref.[17].

3. The Barr–Raby model

One possibility to break SO(10) down to the Standard Model is to use a Higgs multipleA inthe adjoint representation45 and another pair of HiggsesC andC in the spinor representation16 and16. The apparently most simple implementation of this pattern of symmetry breawhich is in accordance with particle physics observations, has been suggested by Barr an[20]. In the following, we review the features of this model as far as they are relevant fobaryogenesis scenario, in particular we present in some detail the derivation of the higgaugino mass matrix. InAppendix A, we give account of the conventions we use, in partichow the charges under the Standard Model group

(7)GSM = SU(3)C × SU(2)L × U(1)Y

are assigned to the various multiplets of SO(10).We consider the superpotential

W ⊃ κS(CC − µ2)+ α

4MS

trA4 + 1

2MA trA2 + T1AT2 + MT T 2

2

(8)+ C′[ζ

PA

MS

+ ζZZ1

]C + C

[ξPA

MS

+ ξZZ2

]C′ + MC′C′C′,

where the additional fieldsS, P , Z1, Z2 are singlets,T1 andT2 10-plets of SO(10). Furthermorethere are the spinorC′ and the conjugate spinorC′.

Let us first discuss the purely adjoint sector. The potential is at its minimum, when the ction

(9)−F ∗A = ∂W

∂A= 0

is met. When〈A〉 = diag(a1, a2, a3, a4, a5) ⊗ iσ2, it follows

(10)α

MS

a3i − MAai = 0.

This can be solved by eitherai = 0, orai = a, where

(11)a = ±√

MAMS

α.

In order to step towards the Standard Model, it is possible to break SO(10) down to the lefsymmetric group

(12)GLR = SU(3)C × SU(2)L × SU(2)R × U(1)B−L


ations

e

nd

to thegru-ldstone

l,

nors

e

-odel

ix

by choosing for〈A〉 the Dimopoulos–Wilczek (DW) form

(13)〈A〉 =

a

a

a

00

⊗ iσ2.

Note, that〈A〉 being of DW form is proportional to the(B − L) operator given in Eq.(A.26). InAppendix A, we give account of the explicit construction of the tensor- and spinor representof SO(10) and the conventions we use.

The two Higgs doublets of the MSSM are contained withinT1 and are identified with thfour components which remain massless by the superpotential(8) when using the DW-formfor 〈A〉. The additional six degrees of freedom ofT1, two colour triplets, become heavy ahence invisible at low energies, such that there is doublet–triplet splitting. The second10-pletT2 becomes necessary since a direct mass-term for the triplet components ofT1 would lead to adisastrous rapid higgsino-mediated proton decay.

The Higgs fieldsC and C reduce the SO(10) symmetry to SU(5). When minimizing thescalar potential, the absolute values of their VEVs are〈C〉 = 〈C〉 = µ, and they point in theSU(5)-singlet direction with the quantum numbers of a right-handed neutrino.

Both sectors, the spinorial and the adjoint, in combination reduce the SO(10)-symmetryStandard Model groupGSM. However, they need to be linked together in order to get a conency of the assignment of Standard Model quantum numbers and to remove all pseudo-Gomodes from the particle spectrum. The obvious candidate term to add to the superpotentiaCAC,however destabilizes the DW form(13) by altering the expression for theF -term(10) when thespinors get a nonzero VEV. Barr and Raby therefore suggested to add the additional spiC′andC′, which get a zero VEV. The conditions for potential minimization now become

(14)−F ∗C′ =[ζ

PA

MS

+ ζZZ1

]C + MC′C′ = 0,

(15)−F ∗C′ = C

[ξPA

MS

+ ξZZ2

]+ MC′C′ = 0.

When comparing with Eq.(A.26), we note that in the DW-form(13) we can identify〈A〉 ≡32ia(B − L). If we assume that the VEV ofP is fixed, thenZ1 andZ2 settle to

(16)Z1 = −3

2iζ/ζZ

〈P 〉aMS

,

(17)Z2 = −3

2iξ/ξZ

〈P 〉aMS

,

sinceC andC point into the right-handed neutrino direction, whereB − L = 1. We have hencachieved a link between the spinorial and adjoint sector without changing the form of−F ∗

A.In our model,CP -violation arises from the phase betweenζ andξ and therefore from cou

plings of the adjoint to the spinor multiplets. Let us label the multiplets of the Standard Mgroup(7) by K . The representations16 and45 harbour as multiplets with commonGSM quan-tum numbersK = (3,2, 1

6), K = (3,1,−23) andK = (1,1,1). These multiplets therefore m

through the higgsino mass matrix. The corresponding conjugate multiplets in16 and45 are la-beled byK . Furthermore, all these representations contain the singlet(1,1,0).


) down

les

modes

oth

com-

ro

tion

s, we

The spinor pair with 32 complex degrees of freedom breaks the 45-dimensional SO(10to the 24-dimensional SU(5). The 21 Goldstone modes come from the multipletsK = (3,2, 1

6),K = (3,1,−2

3), K = (1,1,1) plus one linear combination of the singletsK = (1,1,0) within 16and16. The 45-dimensional adjoint reduces the SO(10)-symmetry to the 15-dimensionaGLR.Because of the DW VEV being proportional to the(B − L) operator, the 30 Goldstone modcan be identified with the multiplets for whichB − L = 0, that are all colour triplets.

Hence, by the supersymmetric Higgs-mechanism, there is a mixing of the higgsinowith the gaugino sector, through the Lagrangean terms

(18)√

2gϕ∗T aψλa + h.c.,

whereT a is a generator of SO(10), normalized as tr(T a)2 = 1, λa a gaugino andϕ the scalarsuperpartner of theψ -fermion. This will induce higgsino–gaugino mixing mass terms for bmultiplets,A andC whenK = (3,2, 1

6) or K = (3,1,−23), and only forC, whenK = (1,1,1).

Let us consider possible mass terms involving only the adjoint Higgs. If we denote theponents of either(3,2, 1

6) or (3,1,−23) by bK , we have (cf.Appendix A)

(19)trA2 = −6a2 − 2bKbK ,

(20)trA4 = 6a4 + 4a2bKbK + b2Kb2

K.

Hence, the portion α4MS

trA4 + 12MA trA2 of the superpotential(8) gives for these modes a ze

mass term

(21)mK = αa2

MS

− MA = 0 for K =(

3,2,1

6

)andK =

(3,1,−2

3

),

where we have used the VEV(11) for a. This result is expected, since the multiplets in quesare Goldstone. In contrast, we find

(22)mK = αa2

MS

for K = (1,1,1), (1,3,0),

(23)mK = −2αa2

MS

for K = (8,1,0).

Let us now discuss the mixing of the adjoints and spinors.ψAKandψC′

Kget a mixing mass

term through

(24)δ2W

δAKδC′K

= ξ〈C〉〈P 〉√

2MS

.

The derivation of this term is instructive and works as follows: In the block-diagonal basihave

(25)ABL10

=(

0 010 0

),

which transforms according to(A.30) to the off-diagonal basis as

(26)A10 = U−1BLOCKABL

10UBLOCK = 1

2

(10 i10i10 −10

).


ixf the

ved

sites for

The single degrees of freedom10 are represented by the ten antisymmetric 5× 5 matrices withtwo nonvanishing entries of the value 1/

√2. Without loss of generality, we pick the matr

with −1/√

2 in the first row, fourth column, corresponding to one degree of freedom oK = (3,2,−1

6)-multiplet. In order to let the tensorA act on a spinor, we make use of Eq.(A.24)and represent it in terms of theΓ -operators as

(27)A = −√

2

16

([Γ1,Γ4] + i[Γ1,Γ9] + i[Γ6,Γ4] − [Γ6,Γ9]).

When paired with the spinor

(28)Ψ = 1

2

[χ

†2χ

†3χ

†5 − χ

†3χ

†2χ

†5

]|0〉,aGSM singlet is formed, and after anticommuting theχi -operators, we find

(29)〈0|χ1χ2χ3χ4χ5AΨ = 1√2,

from which we immediately obtain Eq.(24). The higgsino–gaugino mixing terms can be derifrom (18) in a very similar way.

Finally, for the mixing among the spinors we have

(30)δ2W

δCKδC′K

= ξαK

a〈P 〉MS

,

whereαK = 32i[(B − L)K − 1] or explicitly,

(31)αK =

−1 for K = (3,2, 16

),

−2 for K = (3,1,−23

),

0 for K = (1,1,1).

We have used here the VEVs(16), (17)and again the proportionality of〈A〉 to the (B − L)

operator.We are now in the position to write down the higgsino–gaugino mass matrix:

(ψλKψAK

ψCKψC′

K)

(32)×

0 −i√

2γKga g〈C〉 0

i√

2γKga mK 0 ξ〈C〉〈P 〉√

2MS

g〈C〉 0 κ〈S〉 iαKξa〈P 〉MS

0 ζ〈C〉〈P 〉√

2MSiαKζ

a〈P 〉MS

MC′

ψλK

ψAK

ψCK

ψC′K

+ h.c.,

where

(33)γK =

12 for K = (3,2, 1

6

),

1 for K = (3,1,−23

),

0 for K = (1,1,1).

The mass matrix is nonsymmetric, therefore being endowed with the necessary prerequicoherent baryogenesis.


etryservedted. Weic

fieldserameter, cf.

mpor-eating:n thisg rate

i-ludesto

lds

4. Simulation of coherent baryogenesis

The superpotential(8) is of the type suitable for hybrid inflation. We assume that symmbreaking by the adjoint sector has already taken place before or during inflation and is prethroughout the subsequent history of the universe, such that possible monopoles are dilutherefore do not consider the dynamics of the fieldA. For a discussion of the role of cosmstrings formed at the transitionGLR → GSM after inflation, we refer to Ref.[21].

The part of the scalar potential relevant for hybrid inflation reads

(34)V = κ2∣∣C2 − µ2

∣∣2 + 2κ2|SC|2,where we have usedC = C∗ due to the vanishing of theD-terms and have writtenC ≡ C.

During inflation, the VEVs ofC andC are sitting at a minimum located atzero, andS rollsdown a logarithmic slope until reaching the critical valueScr = µ, such that the valuezerofor C

andC becomes a maximum. The waterfall regime begins, at the end of which the scalarsettle down to the supersymmetric (V = 0) minimumS = 0, |C| = |C| = µ. This is a rapid phastransition which brings coherent baryogenesis along. We simulate this scenario for the paκ = 0.05, a damping rateΓ = 0.02µ and also take account of the expansion of the universeFig. 1.

Damping partly comes into play because of the perturbative decay of the inflaton. More itant at the beginning of the waterfall regime is however the phenomenon of tachyonic prehSince the fieldsC andC attain a negative mass square term, modes with momenta less thamass get produced exponentially fast. Here we mimic this effect by introducing the dampinΓ . For numerical studies of this process, see Refs.[22–28]. Damping will also receive a contrbution from fermionic preheating. A proper treatment of fermionic preheating, which increscatterings, would require techniques used in Ref.[29], which have so far not been appliedthe question of inflaton thermalisation through decay into fermions.

In order to keep the discussion simple, we do not take the dynamics of the singlet fieZ1andZ2 into account here. In principle, their VEVs only get fixed whenC andC acquire nonzero

Fig. 1. Epoch of phase transition in the SO(10)-model.


ors

is

heeigen-

s whichours,

Fig. 2. The produced charges for the multiplet(3,2, 16).

Fig. 3. The produced charges for the multiplet(3,1,− 23).

VEVs. A possible way to fixZ1 andZ2 already during inflation is for example to shift the spinaway from the zero VEV, as proposed in Ref.[5] and is also applicable to SO(10)-models[6].

For the remaining parameters, we chooseµ = 0.5 × 1016 GeV, MS = 550µ, MC′ = 0.02µ,g = 0.2, ζ = −0.02,ξ = 0.05i, a = 25µ and〈P 〉 = 24µ.

The charge numbers which are plotted inFigs. 2 and 3refer to the mass matrix, whichdiagonal after the phase transition is completed. Hence, there is mixing amongψλK

, ψAK, ψCK

andψC′K

. The mass matrix is diagonalizedvia a biunitary transformation. It turns out that thiggsino–gaugino mass matrix in the supersymmetric vacuum has two heavy and two lightvalues. We only display the charge production corresponding to the light mass eigenvaluewe label byq1 andq2. There is no substantial charge asymmetry stored within the heavy flav|q3|, |q4| |q1|, |q2|. The apparent symmetry

∑4i=1 qi = 0 results from the overall U(1) sym-


ssument. Thisare not

eneric

ingatrices

compo-ay

ds

ecayseavierse

utri-

metry of the fermionic fields and is a useful check for the numerical results. We shall ahere that the resonant decay into gauge bosons and scalar Higgs particles is not importacan be justified by noting that the necessary conditions for a resonant decay into bosonsmet for our choice of parameters.

Theqi are charges stored within the mass eigenstates which are Dirac fermions of the gmixing form

(35)Ψi =(

iαLλK

λK + iαLAK

ψAK+ iα

LCK

ψCK+ iα

LC′

K

ψC′K

iβRλK

λK + iβLAK

ψAK+ iβ

L

CK

ψCK+ iβ

L

C′K

ψC′K

), i = 1,2,3,4,

where the coefficientsiαLX andiβ

RX are determined by the biunitary transformation diagonaliz

the higgsino–gaugino mass-matrix. For the example we discuss here, the transformation mand the coefficients are determined numerically.

Now, we discuss how theqi charges get transformed toB–L charge stored within fermionimatter. Decay into SUSY Standard Model particles can take place through the gaugino cnentsλK , λK and through the higgsinosψCK

, ψCK. The relevant operator for gaugino dec

from the gauge supermultiplet Lagrangean is

(36)√

2gλFF + h.c.,while the higgsinos decay through the dimensionfivecouplings

(37)iγ1CF CF

MS

,

(38)iγ2CΓ aFCΓ aF

MS

,

(39)iγ3CΓ aΓ bFCΓ aΓ bF

MS

,

which are added to the superpotential(8), and whereF are the Standard Model matter fieland the right-handed neutrino, contained in16, andΓ denotes the operators defined in(A.6) and(A.7). The indexi = 1,2,3 denotes the matter generation.

While the coupling(36) is universal for all three generations of Standard Model matter,iγ1,iγ2 and iγ3 may be different for the three generations. We assume that3γ2,3 2γ2,3 1γ2,3,such that only the3γ2,3 are of relevance for the decays. In contrast, we require that the dthrough2γ1 and3γ1 are not possible since the corresponding right-handed neutrinos are hthan the decaying particle, such that only1γ1 is relevant. We now argue of which order thecouplings should be for realistic scenarios.

For nonthermal leptogenesis[33], one usually assumes that two of the three Majorana nenos from the different generations of matter fermions are heavier than half of the mass

(40)mI = √2κµ

of the inflaton fields, which are theνc-like components ofC andC and the singletS, such thattheir decay into right-handed neutrinos is kinematically forbidden. Through the coupling(37)the right-handed neutrinos acquire Majorana masses

(41)imνc = iγ1〈C〉2

MS

,


eating

ting.ed

,This is

for all

ayana

escay into

-

such that the requirement2,3mνc > mI/2 reads

(42)2,3γ1 >1√2κ

MS

µ,

where we have used〈C〉 = µ. It appears to be reasonable that also the couplings2,3γ2,3 are ofthe same order, as we shall assume.

For the scenario we discuss, the lightest right-handed neutrino is important for the rehprocess. The coupling(37)also allows for the decay of the inflaton fields at the rate

(43)Γν = 1

8πmI

(1γ1〈C〉MS

)2

.

The Universe becomes radiation dominated and entropy production stops, whenΓν = H , whereH denotes the Hubble expansion rate. The reheat temperature at this time is

(44)TR = 0.55g− 1

4∗√

ΓνmPl,

and we take the estimateg∗ = 220, the number of relativistic degrees of freedom after reheaThe value for the Planck mass ismPl = 1.2 × 1019 GeV. The mass of the lightest right-handneutrino is therefore proportional to the reheat temperature:

(45)1mνc = 7.7× g14∗√κ

µ

mPlTR.

Taking for the highest reheat temperature allowed by the gravitino boundTR = 1011 GeV andfor the parameters we use, we find1mνc < 5×109 GeV while2,3mνc > 2×1014 GeV. Thereforea fortuitous hierarchy of five orders of magnitude for the Majorana masses is required.usually assumed for scenarios of nonthermal leptogenesis.

For the coherent baryogenesis model we consider here however, we can also allowMajorana masses to be larger thanmI/2. Under these circumstances,S and theνc-like compo-nents ofC andC cannot decay through the term(37) into two right-handed neutrinos but decto three particlesvia the operators(38) and (39). Since these processes involve no Majorparticles, leptogenesis is absent for this scenario.

We also have to deal with the fact, that forαK = 0, namely forK = (1,1,1), ψCKandψC′

K

do not mix, cf. the mass matrix(32). Therefore we assume that also the fieldsC′ and C′ maydecay through couplings of the above type, suppressed however by additional powers of〈R〉/MS ,whereR is some singlet with a VEV.

Note however, that forK = (1,1,1) the mass matrix(32) is block-diagonal, such that only thpairsλK–ψCK

andψAK–ψC′

Kare mixed. Only for the second pair, theCP -violating parameter

ξ andζ are relevant and an asymmetry is generated, which vanishes however after the dematter.

The chargesqi gets processed differently toB–L whenΨi decays through its various components. Let us denote theB–L number resulting from the decay of a componentX of a Ψi

quantum byTX. By the couplings(36), the reactions

(46)λK → F ∗K

+ νc∗,

(47)λK → F ∗K + νc∗


ermion.

get

are induced, where one of the particles on the right-hand side is a scalar, the other one a fDue to its Majorana mass term coming from the operator(37), the right-handed neutrinoνc isits own antiparticle and therefore carries effectivelyB − L = 0 at tree-level. The resultingB–L-charge is therefore the one stored withinF ∗

KandF ∗

K and we find

(48)TλK= 1

3, K =

(3,2,

1

6

),

(49)TλK= −1

3, K =

(3,1,−2

3

),

(50)TλK= 1, K = (1,1,1),

(51)TλK= −1

3, K =

(3,2,−1

6

),

(52)TλK= 1

3, K =

(3,1,

2

3

),

(53)TλK= −1, K = (1,1,−1).

Similarly, the coupling(37)allows for the decay reaction

(54)ψCK→ F ∗

K + νc∗.

Hence, the charges get transformed to

(55)TCK= −1

3, K =

(3,2,−1

6

),

(56)TCK= 1

3, K =

(3,1,

2

3

),

(57)TCK= −1, K = (1,1,−1).

We can calculate the term(38) ∝ γ2 using the techniques explained inAppendix A. It ishowever easier to note that (cf. Ref.[30])

(58)

(3,2,

1

6

)⊗(

3,1,1

3

)⊃(

1,2,1

2

)⊂ 10,

as well as

(59)(1,1,0) ⊗(

1,2,−1

2

)=(

1,2,−1

2

)⊂ 10.

The components ofψCKfor K = (3,2, 1

6) therefore decay to

(60)ψCd→ dc∗ + e∗,

(61)ψCu → dc∗ + ν∗,

whereψCddenotes thed-quark like higgsino,ψCu

theu-quark like one. The charges hencetransformed as

(62)TCK= 4

3, for K =

(3,2,

1

6

).


o-

e the. As a

of the

The uc-quark like higgsino withK = (3,1,−23) decays through theγ3-coupling (39). We

note (cf. Ref.[30])

(63)

(3,1,−2

3

)⊗(

3,1,1

3

)=(

3,1,−1

3

)⊕(

6,1,−1

3

)⊂ 120,

(64)(1,1,0) ⊗(

3,1,1

3

)=(

3,1,1

3

)⊂ 120,

and therefore have the reaction

(65)ψCuc → dc∗ + dc∗,and the charge conversion

(66)TCK= 2

3, for K =

(3,1,−2

3

).

Finally, theec like higgsinoK = (1,1,1) turns into mattervia theγ2-coupling(38), as can beseen by

(67)(1,1,1) ⊗(

1,2,−1

2

)=(

1,2,1

2

)⊂ 10,

(68)(1,1,0) ⊗(

1,2,−1

2

)=(

1,2,−1

2

)⊂ 10.

Consequently, the decay reaction is

(69)ψC′ec

→ e∗ + ν∗,

and the resulting asymmetry

(70)TCK= 2, for K = (1,1,1).

The procedure to obtain the producedB–L-density is as follows: We first integrate the prduced charge numbersqi over momentum space in order to obtain charge densitiesQi . Fromthe biunitary diagonalization ofM in the supersymmetric minimum the contributions ofλK ,λK , ψCK

andψCKto theΨi are determined, which gives the branching ratios and therefor

respective contributions for the decays of the Dirac fermions to Standard Model matterformula, this reads

(71)

n0K =

4∑i=1

Qi

3|iαLλK

|22g2TλK+ |iαL

CK|2( 3γj 〈C〉

MS

)2TCK

− 3|iβRλK

|22g2TλK− |iβR

CK

|2( 1γ1〈C〉MS

)2TCK

3|iαLλK

|22g2 + |iαLCK

|2( 3γj 〈C〉MS

)2 + 3|iβRλK

|22g2 + |iβR

CK

|2( 1γ1〈C〉MS

)2 ,

where the factors of three come from the presence of three generations of matter and

(72)j =

2 for K = (3,2, 16

),

3 for K = (3,1,−23

),

2 for K = (1,1,1).

The total(B − L) number density produced at the phase transition is taking accountmultiplicity of colour and flavour given by

(73)n0B−L = 6n0

1 + 3n0¯ 2 .

(3,2, 6 ) (3,1,− 3 )


of this

typical

heate

nset of

inr boundate of

iation-less

A study of the parametric dependence of the produced asymmetry is beyond the scopepaper, which is to show that coherent baryogenesis is viable with the gauge group SO(10), andshall be discussed elsewhwere. Therefore, we content ourselves with presenting just onenumerical example here. The parameters yet to be specified are the3γ2,3 and1γ1. We can seteffectively 1γ1 = 0 because it is either very small due to the restrictions given by the retemperature(44) and the gravitino bound or, in the case of1mνc > mI/2 decays through thcoupling(37) do not take place. In accordance with the relation(42) we choose3γ2 = 3γ1 =0.05MS/µ. Then, we find

(74)n0B−L = 2.5× 10−7µ3.

In order to estimate the baryon to entropy ratio, we express the entropy densitys through thereheat temperatureTR as

(75)s = 2π2g∗T 3R/45,

and the Hubble expansion rate is given by

(76)H = 1.66√

g∗T 2

R

mPl,

wheremPl = 1.22× 1019 GeV is the Planck mass.During the epoch of coherent oscillations, that is between the end of inflation and the o

radiation era, the universe is matter dominated and expands by a factor

(77)a

a0=(

H0

H

) 23

,

whereH0 is the expansion rate at the end of inflation, given by

(78)H0 =√

8π

3

V

mPl2.

Putting everything together, we find

(79)nB

s≈ 1

3

n0B−L

s

(a0

a

)3

≈ 1

4

n0B−L

V0TR,

where we have taken account of a division by three for sphaleron transitions promoting(B–L)

to B asymmetry.The value for the vacuum energy at the end of inflation isV0 = κ2µ4, and by Eq.(79), we find

(80)nB

s= 1.0× 10−10,

where we have chosenTR = 2 × 1010 GeV. Hence, it appears that in order to get a BAUaccordance with observation, there has to be a reheat temperature of order of the uppeallowed by the requirement that gravitinos shall not be overproduced. However, our estimentropy production is rather crude. It is conceivable that initially the decay ofC, C andS isenhanced due to tachyonic and also fermionic preheating. This could lead to an initial radlike equation of state[27] or shorten the matter-dominated era and would therefore lead todilution of the initial asymmetry.


-

neu-When

s exam-toowever,

echa-

This is-ce theryogen-esultsat-arios.yogene-oavinga

thermalf right-esismass,

. In con-gestedogen-ns of

be con-frmalclud-mass

For the case1mνC< mI/2, it is of interest to compare the result(80) with the baryon asym

metry resulting from nonthermal leptogenesis, which is given by[31]

(81)nB

s= 0.5

ε1

mI

TR,

where

(82)ε1 = 2× 10−10( 1mνC

106 GeV

)(mν3

50 meV

)

is the maximalCP -violation which may arise from the decay of the lightest right-handedtrino [32], andmν3 denotes the heaviest mass eigenvalue of the light neutrino mass matrix.we assumemν3 = 50 meV and use the same parameters as for the coherent baryogenesiple and Eq.(45) for 1mνC

, we findnB/s = 6 × 10−12. Therefore our example correspondsa point in parameter space where coherent baryogenesis dominates over leptogenesis. Hwe expect that also the opposite case may occur for a different set of parameters.

5. Conclusions

In this paper, we show that during the phase transition terminating SUSY SO(10) hybridinflation, a charge asymmetry within the higgsino sector may be produced through the mnism of coherent baryogenesis and subsequently turned into baryons.CP -violation is providedby the couplings of the spinorial to the adjoint representations and occurs at tree-level.very different from leptogenesis, a one loop effect, whereCP -violation is sourced by the matrix of Yukawa couplings between the neutrinos and the Standard Model Higgs field. Sinspinor-adjoint couplings are an indispensable part of the Barr–Raby model, coherent baesis naturally occurs at the end of SUSY-SO(10) hybrid inflation. Together with similar rwhich we found for the Pati–Salam group[7], this indicates that effects from fermionic preheing are generically of importance for the generation of the BAU in hybrid-inflationary scen

Coherent baryogenesis however has been neglected in the standard picture of barsis in SUSY-GUT hybrid inflation so far, which is as follows[33]: The inflaton decays intright-handed neutrinos which then decay out-of equilibrium into Standard Model matter, lebehind aB–L asymmetryvia the leptogenesis mechanism[34]. Since the decaying Majoranneutrinos are not produced by the thermal background, this scenario is often called nonleptogenesis. However, it imposes strong constraints on the hierarchy of the masses ohanded neutrinos, as we discuss in Section4. We also emphasize that coherent baryogenrelaxes this constraint and allows for all Majorana masses to be larger than the inflatonsince the mechanism does not rely on leptogenesis and the decay of Majorana particlesclusion, the relations of the parameters of hybrid inflationary models to the BAU as suge.g. in Refs.[5,6,33] by considering leptogenesis, should be altered, since coherent barysis turns out to be an additional source of the BAU, which may dominate in some regioparameter space.

We emphasize that nonthermal leptogenesis and coherent baryogenesis should notfused with the often discussed thermal leptogenesis mechanism[34–38], an appealing feature owhich is that the BAU is generated from a universe which is—within horizon scale—in theequilibrium. Leaving aside primordial density fluctuations, all cosmological observables ining the BAU would then be predictable from an effective theory valid up to the Majoranascale of the neutrinos.


and itallow

endersnario of

ze for

atisfy asenta-caréces. In

uetation

in con-to the

ators

p-ive workten-

locks ofis not

, fol-ting onand thester

.lepton

r

Grand unified theories open up many possible paths for the generation of the BAUis yet not known where the actual asymmetry originates from. The various mechanismsto establish relations to the paramters of the underlying models. While leptogenesis rconstraints on the neutrino sector, coherent baryogenesis is of interest since it is a sceGUT-baryogenesis and thereby related to the dynamics of symmetry breaking.

Acknowledgements

We would like to thank Qaisar Shafi for interesting discussions and Zurab Tavartkiladuseful comments on the manuscript.

Appendix A. SO(10)

Besides by tensors, orthogonal groups may also be represented by spinors, which sClifford algebra. In order to construct group-transformation invariants, both types of repretions need to be linked togethervia Dirac gamma matrices. For the familiar case of the Poingroup SO(3,1), S it is often convenient to use a specific representation for these matricontrast, one better circumvents the tedious task of explicitly constructing ten 32× 32 gammamatrices for SO(10). Mohapatra and Sakita[39] have therefore devised a very useful techniqfor performing calculations involving spinors and tensors, employing just abstract commuand anticommutation relations.

On the other hand, when it comes to symmetry breaking, one has to choose a certavention, that is a certain basis, how the particles of the Standard Model are assignedrepresentation16 of SO(10). This assignment fixes in turn the definition of the charge operand hence the quantum numbers of certain entries in vectors and tensors of SO(10).

While the paper by Mohapatra and Sakita[39] does not provide much details of tensor reresentations and symmetry breaking, such a discussion can be found in the comprehensby Fukuyama et al.[30], where in turn spinors are neglected. The coupling of spinors tosors is explained for SO(10) by Nath and Syed[40]. In the paper by Barr and Raby[20], whichcontains the model we consider here, a basis where tensors nicely decompose into bSU(5)-representations is chosen. Unfortunately, the choice of basis and normalizationsexplicitly given, but has to be inferred by the reader.

In the following, we give some detailed account of the construction of SO(10)-singletslowing the conventions of Barr and Raby. Explicit expressions for the charge operators acspinors and tensors as well as for the accommodation of the Standard Model particlesright-handed neutrino in the representation16 are given, which shall ensure an easier and facomprehensibility of the Barr and Raby analysis as well as of our calculations.

A.1. Charge assignments

We denote byQ the electric charge, byY the weak hypercharge and byI3L the weak isospin

The charges which are not gauge symmetries of the Standard Model are baryon minusnumberB − L as well as the SU(2)R-isospinI3

R and the less known chargeX. There are lineadependencies among these charges, which are given by

(A.1)Q = I3L + Y, B − L = 2

(Y − I3

R

), B − L = 1

5(4Y − X).


e

right-

is

ich

Table 1Quantum numbers of matter

Q I3L

I3R

Y B − L X

Q =(

u

d

)2/3 1/2 0 1/6 1/3 −1

−1/3 −1/2 0 1/6 1/3 −1

uc −2/3 0 −1/2 −2/3 −1/3 −1dc 1/3 0 1/2 1/3 −1/3 3

L =(

ν

e

)0 1/2 0 −1/2 −1 3

−1 −1/2 0 −1/2 −1 3

νc 0 0 −1/2 0 1 −5ec 1 0 1/2 1 1 −1

Note that, when comparing to the conventions by Fukuyama et al.[30], we have twice as largvalues for(B − L), such that for a single lepton, we have(B − L) = −1.

In Table 1, we give the charge numbers of the Standard Model particles and of thehanded neutrino.

A.2. SO(2N ) in an SU(N) basis

This section contains a review of the paper by Mohapatra and Sakita[39], but adopts the basconventions of Barr and Raby[20].

Let us introduceN operatorsχi (i = 1, . . . ,N), acting on an antisymmetric Fock space, whobey the following anticommutation relations:

(A.2)χi,χ

†j

= δij ,

(A.3)χi,χj = 0.

The operators defined as

(A.4)T ij = χ

†i χj

satisfy the SU(N) algebra:

(A.5)[T i

j , Tkl

]= δkjT

il − δi

j Tkj .

We now introduce the 2N operators

(A.6)Γj = −i(χj − χ

†j

), j = 1, . . . ,N,

(A.7)ΓN+j = χj + χ†j ,

which obey by Eqs.(A.2), (A.3) the Clifford algebra

(A.8)Γi,Γj = 2δij , i, j = 1, . . . ,2N,

and hence, the algebra of generators of SO(2N ) is given by

(A.9)Σij = 1

2i[Γi,Γj ].


nne

rm

h

Since the dimension of the spinor representation of SO(2N ) is 2N , a concrete representatiocould be constructed for SO(10) in terms of 32× 32-matrices, which however shall not be dohere.

The spinor states can be constructed by letting theN creation operatorsχ†i act on the “vacu-

um” |0〉, such that the spinor representation is 2N -dimensional, as it should.It is well known, that the spinor representation of SO(2N ) is reducible. We therefore define

(A.10)Γ0 = iN2N∏i=1

Γi =N∏

j=1

(1− 2nj ),

where we have introduced the number operators

(A.11)nj = χ†j χj .

The chiral projectors12(1 ± Γ0) give therefore rise to the two irreducible 2N−1-dimensional

representations containing only even (case “+”) or only odd (case “−”) numbers of creationoperators.

Now letΨ be an SO(2N ) spinor state. We are interested in calculating products of the fo

(A.12)Ψ T BΓi1 . . . ΓiM Ψ,

involving a certain number ofΓ matrices. The matrixB is necessary sinceΨ T does not transformas a conjugate spinor when acted upon with an infinitesimal SO(10)-transformationεij :

(A.13)δΨ = iεijΣijΨ, δΨ † = −iεijΨ†Σij , δΨ T = iΨ T εijΣ

Tij .

We require fromB the property

(A.14)B−1ΣTij B = −Σij ,

such that

(A.15)δ(Ψ T B)= iεijΨ

T BB−1ΣTij B = −iεij

(Ψ T B)Σij ,

i.e.Ψ T B transforms as a conjugate spinor. The condition(A.14) can be met if

(A.16)B−1Γ Ti B = ±Γi.

By choosing the minus-sign in the latter equation, we find

(A.17)B =N∏

i=1

Γi,

because fori = 1, . . . ,N theΓi are represented by antisymmetric matrices, while fori = N +1, . . . ,2N by symmetric ones.

ForN = 5, we can arrange the Standard Model particles in the spin-16 representation, whicis projected out of the 32-dimensional spinorΨ by 1

2(1− Γ0)Ψ . Defining

ui = 1

2εikl45χ

†k χ

†l χ

†5 |0〉, di = 1

2εikl45χ

†k χ

†l χ

†4 |0〉, uc

i = χ†i χ

†4χ

†5 |0〉,

dci = χ

†i |0〉, ν = χ

†5 |0〉, e = χ

†4 |0〉,

(A.18)νc = χ†χ

†χ

†χ

†χ

†|0〉, ec = χ†χ

†χ

†|0〉,
1 2 3 4 5 1 2 3


d.sociated

Eq.

spinor

operator

wherei, k, l = 1,2,3. Cf. also Ref.[41], where the doublet and triplet blocks are interchangeThe next task is to construct the charge operators. For example, the ladder operators as

with the left isospin takeu ↔ d andν ↔ e. They are therefore given by

I+L = χ

†5χ4,

(A.19)I−L = χ

†4χ5.

The weak isospin operator is hence

(A.20)I3L = 1

2

[I+L , I−

L

]= 1

2(n5 − n4).

By comparison with the charge numbers inTable 1, we can identify

Y = 1

3

3∑i=1

ni − 1

2

5∑j=4

nj

(A.21)= 1

12i

([Γ1,Γ6] + [Γ2,Γ7] + [Γ3,Γ8])− 1

8i

([Γ4,Γ9] + [Γ5,Γ10]),

where we have used

(A.22)[Γ5+j ,Γj ] = −4inj + 2i.

When identifying the indices of theΓ operators with matrix rows and columns as implied by(A.9), we can explicitly write down the suitably normalizedY in tensor representation:

(A.23)Y = diag(1/3,1/3,1/3,−1/2,−1/2) ⊗ σ2.

Generally, we use as rule for conversion of the operator to the tensor representation2

(A.24)

[1

4[Γi,Γj ]

]ab

= δiaδjb − δjaδib,

which reads for the special case of the charge operators

(A.25)i

4[Γ5+i , Γi] = ni − 1

2= diag(δ1i , δ2i , δ3i , δ4i , δ5i ) ⊗ σ2.

Now, we easily find the other charge operators. Putting everything together, we have inand in tensor representation

Q = 1

3

3∑i=1

ni − n4 = diag(1/3,1/3,1/3,−1,0) ⊗ σ2,

I3L = 1

2(n5 − n4) = diag(0,0,0,−1/2,1/2) ⊗ σ2,

I3R = 1

2(1− n4 − n5) = diag(0,0,0,−1/2,−1/2) ⊗ σ2,

B − L = 2

3

3∑i=1

ni − 1= diag(2/3,2/3,2/3,0,0) ⊗ σ2,

2 This is of course strictly speaking no equality but an assignment of an operator acting in Fock space to anacting in tensor space.


n-

ariant

nd

Y = 1

3

3∑i=1

ni − 1

2

5∑j=4

nj = diag(1/3,1/3,1/3,−1/2,−1/2) ⊗ σ2,

(A.26)X = −25∑

i=1

ni + 5= diag(−2,−2,−2,−2,−2) ⊗ σ2,

where we have used the normalization convention(A.1).The operator representation for the charge operatorsQ is suitable for finding the charge eige

valuesq of the spinors throughQΨ = qΨ .Tensors can be constructed from the fundamental 10-dimensional vectorΦ10 by taking an-

tisymmetric products, such that a rankn tensor is of dimension 10· 9 · . . . · (10− n + 1)/n!.Explicitly, for the vector and the rank two tensor, the charges implied by the gauge-covderivatives are given by the eigenvalue equations

(A.27)QqΦ10 = qΦ10, [Qq,Φ45] = qΦ45,

whereQ is acting here by matrix multiplication.

A.3. The tensor representations

In order to perform calculations such as16.45.16, tr454 and16.10.16, we need to identify theStandard Model multiplets within10 and45, just as we did for the16 in (A.18). We first note,that under SU(5), the fundamental representation of SO(10) decomposes as10 = 5 ⊕ 5. Let usdenote an element of5 in the representation10 of SO(10) byΦ5

10, an element of5 by Φ 510. Since

they obey

(A.28)XΦ510 = 2Φ5

10 and XΦ 510 = −2Φ 5

10,

they are of the form

(A.29)Φ510 = 1√

2

a1...

a5−ia1

...

−ia5

and Φ 510 = 1√

2

a1...

a5ia1...

ia5

,

with∑5

i=1 |ai |2 = 1 and∑5

i=1 |ai |2 = 1. To remove this inconvenient mixing of the upper alower five-blocks, we introduce the unitary transformation

(A.30)UBLOCK = 1√2

(15 i1515 −i15

), U−1

BLOCK = 1√2

(15 15

−i15 i15

),

such that

(A.31)UBLOCKΦ510 =

a1...

a50...

and UBLOCKΦ 510 =

0...

0a1...

.

0 a5


er

etric

v. D 23

, Phys.

1011.998)

3, hep-

rison, in:

By this change of basis, the charge operators become diagonal, for example

(A.32)UBLOCKXU−1BLOCK = 2

(15 00 −15

).

We can therefore immediately see how the entries of45 transform under SU(5), namely

(A.33)UBLOCKΦ45U−1BLOCK =

(24 ⊕ 1 10

10 24 ⊕ 1

),

where the single entries represent 5× 5-blocks and the blocks in the upper left and the lowright are to be related to each other by the factor of minus one. The SU(5)-singlet1 has herethe form of the matrix 1/

√515. The arrangement of theGSM-multiplets contained in24 can be

schematically written as

(A.34)

1 2 3 4 5123

(8,1,0) ⊕ (1,1,0)(3,2,−5

6

)

45(3,2, 5

6

)(1,3,0)

and finally10 of SU(5) decomposes into

(A.35)

1 2 3 4 5123

(3,1,−2

3

) (3,2, 1

6

)

45

−(3,2, 16

)(1,1,1)

where the matrix is imposed to be antisymmetric, since it is identified with the antisymmpart of5 ⊗ 5 of SU(5).

References

[1] A.H. Guth, The inflationary universe: A possible solution to the horizon and flatness problems, Phys. Re(1981) 347.

[2] G.R. Dvali, Q. Shafi, R.K. Schaefer, Large scale structure and supersymmetric inflation without fine tuningRev. Lett. 73 (1994) 1886, hep-ph/9406319.

[3] E.J. Copeland, A.R. Liddle, D.H. Lyth, E.D. Stewart, D. Wands, Phys. Rev. D 49 (1994) 6410, astro-ph/940[4] L. Covi, G. Mangano, A. Masiero, G. Miele, Hybrid inflation from supersymmetric SU(5), Phys. Lett. B 424 (1

253, hep-ph/9707405.[5] R. Jeannerot, S. Khalil, G. Lazarides, Q. Shafi, Inflation and monopoles in supersymmetric SU(4)c × SU(2)L ×

SU(2)R, JHEP 0010 (2000) 012, hep-ph/0002151.[6] B. Kyae, Q. Shafi, Inflation with realistic supersymmetric SO(10), hep-ph/0504044.[7] B. Garbrecht, T. Prokopec, M.G. Schmidt, Coherent baryogenesis, Phys. Rev. Lett. 92 (2004) 06130

ph/0304088.[8] B. Garbrecht, T. Prokopec, G. Schmidt, Coherent baryogenesis and nonthermal leptogenesis: A compa

Proceedings SEWM ’04, Helsinki, 16–19 June 2004, hep-ph/0410132.


ev. D 42

1997)

0400.) 547,

rticles0437.enesis,

5, hep-

ce for

k

k

7162.g, Phys.

aking

B 536

v. D 65

reheat-

1, hep-

ilding,

s, Phys.

mmetric

) 5047,

1, hep-

. Phys.

[9] J.C. Pati, A. Salam, Lepton number as the fourth color, Phys. Rev. D 10 (1974) 275.[10] J.H. Traschen, R.H. Brandenberger, Particle production during out-of-equilibrium phase transitions, Phys. R

(1990) 2491.[11] L. Kofman, A.D. Linde, A.A. Starobinsky, Towards the theory of reheating after inflation, Phys. Rev. D 56 (

3258, hep-ph/9704452.[12] T. Prokopec, T.G. Roos, Lattice study of classical inflaton decay, Phys. Rev. D 55 (1997) 3768, hep-ph/961[13] R. Micha, M.G. Schmidt, Bosonic preheating in left–right-symmetric SUSY GUTs, Eur. Phys. J. C 14 (2000

hep-ph/9908228.[14] P.B. Greene, L. Kofman, Preheating of fermions, Phys. Lett. B 448 (1999) 6, hep-ph/9807339.[15] D.J.H. Chung, E.W. Kolb, A. Riotto, I.I. Tkachev, Probing Planckian physics: Resonant production of pa

during inflation and features in the primordial power spectrum, Phys. Rev. D 62 (2000) 043508, hep-ph/991[16] G.F. Giudice, M. Peloso, A. Riotto, I. Tkachev, Production of massive fermions at preheating and leptog

JHEP 9908 (1999) 014, hep-ph/9905242.[17] B. Garbrecht, T. Prokopec, M.G. Schmidt, Particle number in kinetic theory, Eur. Phys. J. C 38 (2004) 13

th/0211219.[18] K. Kainulainen, T. Prokopec, M.G. Schmidt, S. Weinstock, First principle derivation of semiclassical for

electroweak baryogenesis, JHEP 0106 (2001) 031, hep-ph/0105295.[19] T. Prokopec, M.G. Schmidt, S. Weinstock, Transport equations for chiral fermions to orderh-bar and electrowea

baryogenesis. Part I, Ann. Phys. 314 (2004) 208, hep-ph/0312110;T. Prokopec, M.G. Schmidt, S. Weinstock, Transport equations for chiral fermions to orderh-bar and electroweabaryogenesis. Part II, Ann. Phys. 314 (2004) 267.

[20] S.M. Barr, S. Raby, Minimal SO(10) unification, Phys. Rev. Lett. 79 (1997) 4748, hep-ph/9705366.[21] R. Jeannerot, M. Postma, Leptogenesis from reheating after inflation and cosmic string decay, hep-ph/050[22] B.R. Greene, T. Prokopec, T.G. Roos, Inflaton decay and heavy particle production with negative couplin

Rev. D 56 (1997) 6484, hep-ph/9705357.[23] T. Prokopec, Negative coupling instability and grand unified baryogenesis, hep-ph/9708428.[24] G.N. Felder, J. Garcia-Bellido, P.B. Greene, L. Kofman, A.D. Linde, I. Tkachev, Dynamics of symmetry bre

and tachyonic preheating, Phys. Rev. Lett. 87 (2001) 011601, hep-ph/0012142.[25] J. Garcia-Bellido, E. Ruiz Morales, Particle production from symmetry breaking after inflation, Phys. Lett.

(2002) 193, hep-ph/0109230.[26] E.J. Copeland, S. Pascoli, A. Rajantie, Dynamics of tachyonic preheating after hybrid inflation, Phys. Re

(2002) 103517, hep-ph/0202031.[27] D.I. Podolsky, G.N. Felder, L. Kofman, M. Peloso, Equation of state and beginning of thermalization after p

ing, hep-ph/0507096.[28] R. Micha, I.I. Tkachev, Turbulent thermalization, Phys. Rev. D 70 (2004) 043538, hep-ph/0403101.[29] J. Berges, S. Borsanyi, J. Serreau, Thermalization of fermionic quantum fields, Nucl. Phys. B 660 (2003) 5

ph/0212404.[30] T. Fukuyama, A. Ilakovac, T. Kikuchi, S. Meljanac, N. Okada, SO(10) group theory for the unified model bu

J. Math. Phys. 46 (2005) 033505, hep-ph/0405300.[31] V.N. Senoguz, Q. Shafi, GUT scale inflation, non-thermal leptogenesis, and atmospheric neutrino oscillation

Lett. B 582 (2004) 6, hep-ph/0309134.[32] K. Hamaguchi, Cosmological baryon asymmetry and neutrinos: Baryogenesis via leptogenesis in supersy

theories, hep-ph/0212305.[33] G. Lazarides, Q. Shafi, Origin of matter in the inflationary cosmology, Phys. Lett. B 258 (1991) 305.[34] M. Fukugita, T. Yanagida, Baryogenesis without grand unification, Phys. Lett. B 174 (1986) 45.[35] W. Buchmuller, M. Plumacher, Neutrino masses and the baryon asymmetry, Int. J. Mod. Phys. A 15 (2000

hep-ph/0007176.[36] W. Buchmuller, R.D. Peccei, T. Yanagida, Leptogenesis as the origin of matter, hep-ph/0502169.[37] A. Pilaftsis, CP violation and baryogenesis due to heavy Majorana neutrinos, Phys. Rev. D 56 (1997) 543

ph/9707235.[38] A. Pilaftsis, T.E.J. Underwood, Resonant leptogenesis, Nucl. Phys. B 692 (2004) 303, hep-ph/0309342.[39] R.N. Mohapatra, B. Sakita, SO(2N) grand unification in an SU(N) basis, Phys. Rev. D 21 (1980) 1062.[40] P. Nath, R.M. Syed, Complete cubic and quartic couplings of 16 and 16-bar in SO(10) unification, Nucl

B 618 (2001) 138, hep-th/0109116.[41] F. Wilczek, A. Zee, Families from spinors, Phys. Rev. D 25 (1982) 553.

ulationT-charge

omologyuces theantum

heoryced toaranesalutions

.nfigura-etchedds to auge,order in


The BRST treatment of stretched membranes

Jonas Björnsson∗, Stephen Hwang

Department of Physics, Karlstad University, SE-651 88 Karlstad, Sweden

Received 2 December 2005; accepted 8 December 2005


Abstract

The BRST-invariant formulation of the bosonic stretched membrane is considered. In this formthe stretched membrane is given as a perturbation around zero-tension membranes, where the BRSdecomposes as a sum of a string-like BRST-charge and a perturbation. It is proven, by means of cohtechniques, that there exists to any order in perturbation theory a canonical transformation that redfull BRST-charge to the string-like one. It is also shown that one may extend the results to the qulevel yielding a nilpotent charge in 27 dimensions. 2005 Elsevier B.V. All rights reserved.

1. Introduction

Membranes are interesting from many points of view, it may have a connection to M-t[1,2] and it is a generalization of the string action. In the lightcone gauge it can be redua matrix model[3–5] which is conjectured to be M-theory[6]. It is probably also relevant asD2-brane, being part of the strongly coupling region of string theory. The relevance of D-bfor string theory at strong coupling was first realized in[7]. It is also interesting by itself and astesting ground to see if methods in string theory generalize to higher extended objects. Soof the equations of motions are rare because of the highly non-linear equations of motion

In [8] we proposed to study so-called stretched membrane configurations. These are cotions which arise, in a partial fixing of the gauge, for weak tensions of the membrane. Strmembranes may be treated perturbatively around a zero tension limit, which corresponstring-like theory. In[9] we proved that, by fixing the gauge completely to the lightcone gathere is a canonical equivalence between the two theories i.e. the membrane is, to any

* Corresponding author.E-mail addresses: [email protected](J. Björnsson),[email protected](S. Hwang).





J. Björnsson, S. Hwang / Nuclear Physics B 736 (2006) 156–167 157

s may,nic astends to27 and

to seelly, oneevel thiseneral,wn duetween

rge, weer one.e.

oinstead,of the

aysokingology

seemsng theknown

ologyceour case,ut to be

ill showheransfor-theoriesgh the

gaugeeen theiscusslevant

etweentization

perturbation theory, equivalent to a string-like theory. Properties of stretched membranetherefore, be inferred from those of the string-like theory. The equivalence holds for bosowell as supersymmetric membranes. It was also shown that the canonical equivalence exa unitary one at the quantum level, yielding, among other results, the critical dimensions11 for the bosonic and supersymmetric cases, respectively.

In this article we continue our analysis of the bosonic stretched membrane. The aim iswhether the equivalence may be proven without the use of the light-cone gauge. Classicamay argue that this must be the case, at least locally in phase-space. But at the quantum lneed not be true. Proving unitary equivalence for a fully gauged fixed theory does not, in gimply that the same is true without gauge fixing, since the gauge symmetry may break doto anomalies. It is rather the converse that is true. By proving the unitary equivalence bethe BRST-charge of the stretched membrane and the unperturbed string-like BRST-chacan conclude that, since the latter theory is non-anomalous, this is also true for the formFrom this it follows that we can impose any particular gauge and still maintain equivalenc

The problem to solve is, therefore, the following. Given a BRST-charge of the form

(1.1)Q = Q0 + Q′,

whereQ0 is the unperturbed string-like BRST-charge andQ′ the perturbation, is it possible tfind a canonical transformation which takesQ into Q0? Unfortunately, the techniques used[9] do not readily generalize to the present case due to the complexity of the problem. Inwe will use another approach. As we will see, it is possible to restate the problem as onecohomology ofQ0. If this cohomology is trivial for ghost number one then there will alwexist, to any order in perturbation theory, a canonical transformation of the kind we are lofor. In fact, as we will show, the restatement of the perturbation problem as one in cohomis not something particular for stretched membranes, but is quite general.

SinceQ0 is essentially the BRST-charge of the bosonic string, the cohomology problemalready to have been solved. This is not entirely correct. First of all, the basic fields includighosts, are fields defined on the world-volume rather than the world-sheet. Secondly, theproofs of the cohomology of string theory do not directly apply to our case. The cohomw.r.t. the quantum string state-space is well known[10–14]. Using the one-to-one correspondenbetween operators and states, one may also deduce the cohomology of the operators. Inwe need to analyze the classical cohomology of the phase-space functions, which turns oa little bit different.

Having established the canonical equivalence one can turn to the quantum case. We wthat the quantization procedure proposed in[9] can, in a straightforward way, be applied to tpresent case. By this procedure one defines a specific ordering whereby the canonical tmations turn into unitary ones so that the equivalence of the perturbed and unperturbedis maintained at the quantum level. This then will show that quantum consistency, throunilpotency of the membrane BRST-charge, requires 27 dimensions.

The paper is organized as follows. In section two we consider the BRST treatment oftheories formulated as perturbation theories. Here we also show the connection betwexistence of canonical transformations and the BRST cohomology. In the third section we dthe cohomology of the BRST-charge locally in phase-space. The cohomology problem reto the membrane is treated in section four. This will also show the canonical equivalence bthe string-like theory and the stretched membrane. In the last section we discuss the quanof our model.

158 J. Björnsson, S. Hwang / Nuclear Physics B 736 (2006) 156–167

nicald and arturbedlume

ed in

ra

raints

. To

y

2. BRST treatment of perturbatively formulated gauge theories

In this section we will, in more general terms, formulate the problem of finding canotransformations which canonically map constraint theories formulated as an unperturbeperturbed part. We start with a general theory of this form. In our particular case, the unpetheory is a string-like theory, which is the standard bosonic string theory with extra world voparameter dependence. The perturbed theory is the stretched membrane theory formulat[8].

Consider the general situation where we have a theory with first-class constraints,φa ≈ 0,formulated as a perturbation theory

(2.1)φa

[pi, q

i] = ψa

[pi, q

i] +

N∑n=1

gnλ(n)a

[pi, q

i],

whereg 1 is the perturbation parameter. Since we have a closed Poisson bracket algeb

(2.2)φa,φb = Uabcφc

to any order ing, it follows that the unperturbed part,ψa , also satisfies a closed algebra

(2.3)ψa,ψb = U ′ab

cψc,

where, in general,Uabc andU ′

abc can depend on the phase-space variables and

(2.4)U ′ab

c ≡ Uabc|g=0.

The BRST-charge is generally of the form

(2.5)Q =∑n=0

(n)

Q,

where

(2.6)(0)

Q= φaca,

(n)

Q= Ab1,...,bna1,...,an+1

ca1 · . . . · can+1bb1 · . . . · bbn,

and the functions,Ab1,...,bna1,...,an+1, are determined by the Poisson bracket algebra of the const

and the nilpotency condition on the BRST-charge.In the assumed perturbation theory we can also expand in terms ofg

(2.7)Q = Q0 +N ′∑i=1

giQi.

The nilpotency condition of the full BRST-charge now yields relations to each order ing. Thezeroth order relation gives thatQ0, the BRST-charge for the unperturbed theory, is nilpotentfirst order ing we have

(2.8)Q0,Q1 = 0.

Thus, we know thatQ1 is in the cohomology ofQ0. If Q1 is a trivial element in this cohomologthen there exists a functionG1 satisfying

(2.9)Q1 = −Q0,G1

.


al

nnonicalerto anyr

as well.alencebrane,at the

ll havemotinguence

ase-e type

atendentnicns-theory

, whichlogy of

Let us assume thatG1 exists. Then we are free to interpretG1 as a generator of an infinitesimcanonical transformation. This transformation shiftsQ to

(2.10)QG1−→ Q0 + g2

(Q2 − 1

2

Q0,G1

,G1

) + · · · .

The nilpotency condition to second order ing is

(2.11)

Q0,Q2 − 1

2

Q0,G1

,G1

= 0.

This implies thatQ′2 ≡ Q2 − 12Q0,G1,G1 is in the cohomology ofQ0 and we have a

problem of the same type as in Eq.(2.8). If Q′2 is a trivial element in the cohomology, thewe may repeat the above argument and conclude that there exists an infinitesimal catransformation that transformsQ to Q0 to second order ing. One may continue this to any ordin g. Thus, we see that the problem of proving that there exists a canonical transformationorder in perturbation theory may be solved by proving that theQ0 cohomology at ghost numbeone is trivial.

It should be remarked that the above argument goes through for the quantum caseReplacing all Poisson brackets with commutators shows that the problem of unitary equivcan be restated in terms of the cohomology of the BRST-operator. For the stretched memhowever, this is not helpful. The argument requires that one can establish the nilpotencyquantum level of the BRST-operator and this we cannot do from the outset. Instead we wito proceed through the classical analysis and, using this, define a quantum theory by prothe canonical transformations to unitary ones. The nilpotency will then follow as a conseqof the unitary equivalence.

3. Local existence of a canonical transformation

In this section we will continue to consider the general situation, but only locally in phspace. We will show that in this case there always exists a canonical transformation of thdiscussed above.

The starting point is the again a theory as in section two with constraintsφa . We will herefirst use the Abelization theorem[15] (for a short proof of it, see[16]). The theorem states thfor all constraint theories there exists, locally in phase-space, an invertible coordinate-depmatrix Ka

b such thatFa ≡ Kabφb are Abelian. For explicit constructions for the free boso

string theory, see[17,18]. A theorem by Henneaux[19] shows that there exists a canonical traformation,G, in the extended phase-space such that the BRST-charge of the unperturbedis canonically equivalent to an Abelian one,

(3.1)Q0 G−→ Q0 = Faca.

Applying this canonical transformation to the full theory yields a BRST-charge

(3.2)QG−→ Q = Q0 +

N ′∑i=1

giQi,

whereQ0 is given by Eq.(3.1).As we discussed in the previous section, the existence of a canonical transformation

maps the perturbed BRST-charge to the unperturbed one, is determined by the cohomo


belian

.mation

hase-is willstricted

sense,

an

the unperturbed BRST-charge. Let us, therefore, study the cohomology of the simple Amodel in more detail.

Assume that there existm first class constraints,Fa ≈ 0, in a theory withn degrees of freedomSince the theory is Abelian there exists locally, by Darboux’s theorem, a canonical transforfrom (qi,pi) to (χa, q∗ j ,Fa,p

∗j ), wherej = 1, . . . , n − m and χa,Fb = δa

b . The BRST-charge for the Abelian model is

(3.3)QA = Faca.

One can also add ghost momenta,ba , that satisfy

(3.4)ca, bb

= δab .

We will restrict our study of the BRST cohomology to the space of polynomials in the pspace coordinates. The proof we will give will not depend on the assumption of locality. Thbe important as we will need the result in the next section, where the treatment is not reto being local.

Let us constructm charges fromQA (no summation overa)

(3.5)Na ≡ QA,χaba

= χaFa − caba.

A non-trivial BRST-invariant function has to have zero eigenvalues, in the Poisson bracketw.r.t. any of these charges. Otherwise, if a BRST-invariant functionO satisfies

(3.6)Na,O = naO,

it is BRST-trivial

(3.7)O = 1

na

QA,

O, χaba

.

The fundamental fields in this theory with non-zero eigenvalues ofNa are(Fa, ba) with eigen-value+1 and(χa, ca) with eigenvalue−1. Thus, non-trivial BRST-invariant polynomials cdepend on(χa,Fa, c

a, ba) only through the combinations(Faca,Faχ

a, baca, baχ

a) (no sum-mation overa). Define these linear combinations (no summation overa)

sa ≡ 1

2baχ

a,

ta ≡ 1

2

(bac

a + Faχa),

ua ≡ 1

2

(bac

a − Faχa),

(3.8)va ≡ Faca.

They satisfy (no summation overa)

(3.9)saQA−→ ta

QA−→ 0,

(3.10)uaQA−→ va

QA−→ 0,

(3.11)s2a = v2

a = 0,

(3.12)t2a − u2

a = 2sava,

(3.13)(ua + ta)va = 0.


oft

Eqs.(3.11) and (3.12)imply that we can reduce any polynomial to be at most linear insa , va

andua .Let us determine the cohomology of the BRST-charge by first fixing to a generic valuea

and suppress the indices of the fields(sa, va, ua, ta). Let f (s, t, u, v, x) be a BRST-invarianfunction wherex indicates dependence on other fields. Expand first thes-dependence off

(3.14)f = sf1(t, u, v, x) + f2(t, u, v, x).

The BRST-invariance off implies

(3.15)QA,f1 = 0,

(3.16)QA,f2(t, u, v, x)

+ tf1(t, u, v, x) = 0.

The second equation implies thatf2 can be split into two parts

(3.17)f2 = tG1(t, u, v, x) + f3(t, u, v, x),

whereQA,G1(t, u, v, x)

= −f1,QA,f3(t, u, v, x)

= 0.

This is always possible, because otherwisef1 = 0. Inserting Eq.(3.17)into Eq.(3.14)yields

(3.18)f = −sQA,G1 + tG1(t, u, v, x) + f3(t, u, v, x) = QA, sG1 + f3(t, u, v, x).

Thus, non-trivial functions in the cohomology ofQA are independent ofs. We can expandf3 as

(3.19)f3 = uvf 14 (t, x) + uf 2

4 (t, x) + f5(t, v, x).

The BRST-invariance off3 implies

(3.20)

QA,f 1

4 (t, x) = 0,

QA,f 2

4 (t, x) = 0,

QA,f5(t, v, x)

+ vf 24 (t, x) = 0.

The first equation shows us thatuvf 14 is trivial

(3.21)QA,vsf 1

4

= −tvf 14 = uvf 1

4 ,

where the last equality follows from Eq.(3.13). Eq.(3.20)also implies that one can splitf5 intotwo parts

(3.22)f5 = vG2(t, x) + f6(t, v, x),

with

(3.23)QA,G2 = f 24 , QA,f6 = 0.

Extracting thev-dependence of the functionf6,

(3.24)f6 = vf7(t, x) + f8(t, x),

shows us that the BRST-invariance off implies that both functions,f7 and f8, are BRST-invariant. The first term is BRST-trivial,vf7 = QA,uf7. Expanding thet-dependence off8

(3.25)f8 =∞∑

tj hj (x),

j=0


gy are

t only

ation.such aen the

dlr, exist

locally, aill nowt restrictded asas not

of the

ly based

ghostsfor this

yields that eachhj has to be BRST-invariant. This implies that

(3.26)f8 = QA,G3 + h0(x),

where

(3.27)G3 =∞∑

j=1

stj−1hj (x).

Collecting all parts we have

(3.28)f = QA, sG1 + uG2 + vsf 1

4 + uf7 + G3 + h0(x).

Concluding, we have shown that all non-trivial phase-space polynomials in the cohomoloindependent ofta , ua , va andsa , for a fixed value ofa. This is true for all values ofa. Thus,the only non-trivial elements in the cohomology are ghost number zero polynomials thadepend on(q∗ j ,p∗

j ), the coordinates that span the physical phase-space.Let us now return to our problem of proving the existence of a canonical transform

From our results of the cohomology and from the previous section we have proven thattransformation exists to all orders in perturbation theory. This implies that we have provexistence of the canonical transformationsG andG′ such that

(3.29)QG−→ Q

G′−→ Q0 G−1−→ Q0,

whereG transforms the unperturbed constraints to Abelian ones,G′ transforms the perturbeBRST-charge to the unperturbed Abelian one and, finally,G−1 transforms us to the originaunperturbed BRST-charge. These statements are true locally. There may still, howeveobstructions preventing the results to hold globally.

4. Application to the stretched membrane

We have seen from the previous section that we are assured that there exists, at leastcanonical transformation transforming the full BRST-charge to the unperturbed one. We wconsider what happens in the specific case of the stretched membrane when we do noourselves to local considerations. Although the results of the local case will not be neesuch, we do need to use the result from the analysis of the Abelian BRST-charge, which wrestricted to be local in phase-space.

For the stretched membrane the unperturbed BRST-charge is of the form Eq.(1.1) whereQ0 is that of a free string with an extra world-parameter dependence. The cohomologystate-space of the free quantum string theory is well known, see[10–14]. The cohomology of theclassical theory has, however, not to our knowledge been solved. Using techniques largeon [14], we will analyze the cohomology of our string-like model.

If we reduce one of the three constraints for the membrane theory and introduce twoand ghost momenta for the remaining constraints, one can construct a BRST-chargetheory1

(4.1)Q =∫

d2ξ Q,

1 We have corrected a sign error in[8].


eory,

s

Q= φ1c1 + φ2c

2 + ∂1c1c1b1 + ∂1c

2c2b1 + ∂1c2c1b2 + ∂1c

1c2b2

+ g[P∂2X∂2c

1c2b1 − ∂1X∂2X∂2c2c2b1 + (∂2X)2∂1c

2c2b1

(4.2)+ 2P∂2X∂2c2c2b2 − 2∂2c

1∂2c2c2b1b2

],

where

φ1 = P∂1X,

(4.3)φ2 = 1

2

P2 + (∂1X)2 + g

[(∂1X)2(∂2X)2 + (P∂2X)2 − (∂1X∂2X)2].

We can split this BRST-charge into two parts, one free part, which is that of a string-like thand a perturbation

(4.4)Q = Q0 + gQ1.

If we make a change of variables from(Xµ,Pµ), µ = 0, . . . ,D − 2, to the Fourier coefficient(q

µn , q

µn ,α

µm,n, α

µm,n), we have the non-zero Poisson brackets

αµm,n,α

νp,q

= αµ

m,n, ανp,q

= −imηµνδm+p,0δn+q,0,

(4.5)qµm,αν

0,n

= qµm, αν

0,n

= ηµνδm+n,0.

To simplify the equations, we redefine our ghosts and ghost momenta

c = c1 + c2, c = c1 − c2,

(4.6)b = 1

2(b1 + b2), b = 1

2(b1 − b2).

Fourier expanding these fields we find the non-zero Poisson brackets

(4.7)cm,n, bp,q = cm,n, bp,q = δm+p,0δn+q,0.

Choose lightcone coordinates

(4.8)A+ = 1√2

(AD−2 + A0), A− = 1√

2

(AD−2 − A0)

and introduce a grading by

Nlc =∑

m =0,n

1

im

(α+−m,−nα

−m,n + α+−m,−nα

−m,n

)

(4.9)+∑n=0

(α−

0,−nq+n + α−

0,−nq+n − α+

0,−nq−n − α+

0,−nq−n

) + α−0,0q

+0 + α−

0,0q+0 .

Nlc acts diagonally, within Poisson brackets, on the basic fields.q−n=0, q−

n=0, α−m,n andα−

m,n haveeigenvalue+1; q+

n , q+n for all n, α+

m,n andα+m,n for |m| + |n| = 0 have eigenvalue−1. All other

fields have eigenvalue zero.The string-like BRST-charge may now be split into two parts

(4.10)Q0 = Q1 + Q0,

where the lower index indicates the eigenvalue w.r.t.Nlc. The nilpotency ofQ0 implies

(4.11)Q1,Q1 = Q0,Q1 = Q0,Q0 = 0.


In or-

hat

e

n-trivial

One

mST-lue ofe isr there

r

of

Thus, the two separate terms inQ0 are nilpotent by themselves. The explicit form ofQ1 is simple

(4.12)Q1 =∑m,n

(α+

0,0α−m,nc−m,−n + α+

0,0α−m,nc−m,−n

),

and it is the BRST-charge of an Abelian theory. One may, as we will see below, useQ1 to studythe BRST-cohomology of the full theory. This requires us to determine theQ1-cohomology,which we can do using the analysis of the Abelian case given in the previous section.der to apply this analysis we need the existence of gauge fixing functionsχm,n and χm,n suchthat χm,n,Q1 = cm,n and χm,n,Q1 = cm,n. Such functions exist if we assume thatα+

0,0

and α+0,0, which are conserved quantities, are non-zero. Thenχm,n = i

mα+0,0

α+m,n for m = 0 and

χ0,n = 1α+

0,0q+n , etc. forχm,n. We will, in analyzing the cohomology, only consider functions t

are finite degree polynomials in the basic fields, exceptα+0,0 andα+

0,0, where we permit invers

powers as well. Furthermore, we will assume no dependence onq−0 andq−

0 , which is sufficientfor our case.

We can now proceed and use the results of the previous section. This yields that the nopolynomials in the cohomology ofQ1 have zero ghostnumber and have the dependence

(4.13)hQ1 = hQ1(qIn , qI

n , q−n=0, q

−n=0, α

Im,n, α

Im,n,α

+0,n, α

+0,n

),

whereI = 1, . . . ,D − 3. Let us now study the cohomology of the string-like BRST-charge.may expand a general BRST-invariant polynomial,K , in terms of its eigenvalues ofNlc definedin Eq.(4.9)

(4.14)K = KN + KN−1 + · · · + KI ,

where

(4.15)Nlc,Kn = nKn.

By assumption,N andI are finite. The BRST-invariance implies

0= Q,K =0︷︸︸︷

Q1,KN +0︷︸︸︷

Q0,KN + Q1,KN−1+ · · ·

(4.16)+0︷︸︸︷

Q0,KN−i+1 + Q1,KN−i+ · · · +0︷︸︸︷

Q0,KI ,wherei = 1,2, . . . . Thus, the highest order term,KN , is BRST-invariant w.r.t.Q1. Using thecohomology ofQ1, there exists two possibilities. EitherK has a ghost number different frozero which, by our analysis of theQ1-cohomology, implies that the highest order term is BRtrivial. This in turn implies, by the same reasoning, that all other terms with lower eigenvaNlc are trivial as well. Another possibility is thatK has zero ghost number. Although this casnot needed for our problem, we consider it out of general interest. For zero ghost numbecan exist a non-trivial part inKN

(4.17)KN = h(N)N + Q1,CN−1,

whereh(N)N is a non-trivial function in the cohomology ofQ1 and CN−1 has ghost numbe

−1 and eigenvalue(N − 1) of Nlc. The phase-space functionh(N)N depends, by the analysis

the cohomology ofQ1, only on the fields(qIn , qI

n , q− , q− , αIm,n, α

Im,n,α

+ , α+ ). Inserting
n=0 n=0 0,n 0,n


er

end is

en-

and

ction

havedence

withuse

brane

mials.

Eq.(4.17)into the equation for the next order yields

(4.18)Q0, h

(N)N

+ Q1,KN−1 − Q0,CN−1

= 0.

One can splitKN−1 into two partsh(N)N−1 + K ′

N−1 such that

(4.19)Q1, h

(N)N−1

= −Q0, h

(N)N

.

This equation can always be solved since the right-hand side isQ1-exact and has ghost numbequal to one. Thus, from theQ1-cohomology, there will always exist a functionh(N)

N−1. Eq.(4.18)now implies

(4.20)Q1,K

′N−1 − Q0,CN−1

= 0,

which is of a similar form as the equation previously solved. Consequently, the solution toKN−1is

(4.21)KN−1 = h(N)N−1 + h

(N−1)N−1 + Q1,C(N−2) + Q0,C(N−1),

whereh(N−1)N−1 is a function of(qI

n , qIn , q−

n=0, q−n=0, α

Im,n, α

Im,n,α

+0,n, α

+0,n). One can proceed in th

same way to any order inNlc. This yields the same kind of equations and the result in the e

(4.22)K =N∑

i=I

N∑j=i

h(i)j +

Q0,C,

where we have defined

(4.23)C ≡N∑

i=I

Ci−1.

The functionsh(j)i , wherei N andj i, are determined from the term with the highest eig

value ofNlc, thus, byh(i)i . This term only depends on the fields(qI

n , qIn , q−

n=0, q−n=0, α

Im,n, α

Im,n,

α+0,n, α

+0,n). Collecting the termsh(j)

i , we can construct functions that are BRST-invariantnon-trivial w.r.t. the full string-like BRST-charge

(4.24)h(j) ≡j∑

i=−I

h(j)i .

These functions are such that the term which has the highest value w.r.t.Nlc is non-trivial in thecohomology ofQ1 and terms with lower eigenvalue, are correction terms such that the funis in the cohomology ofQ0.

Let us now conclude the analysis of the cohomology of the string-like BRST-charge. Wefound that in the space of finite degree polynomials of the basic fields, excluding depenon q−

0 andq−0 , and assumingα+

0,0 andα+0,0 to be non-zero as they enter in the expressions

inverse powers, the cohomology is non-trivial only for zero ghost number. We may nowthe result of section two to conclude that, provided our assumptions are valid, the memBRST-charge is canonically equivalent to the string-like one.

Considering our assumptions, we have first of all the restriction to finite degree polynoThis is always true within our perturbation theory. Secondly, the assumption thatα+ andα+

0,0 0,0


mologyitions.

ore

t there

order

her. We

annerand

eory, atheory.g-like

r

hichfy the

ion

mem-in the

are non-zero is basically the same assumption one has for the known proof of the cohoof string theory. As these fields are conserved quantities, this restricts possible initial cond

The final assumption, namely the exclusion ofq−0 and q−

0 dependence, requires some melaborate discussion. The zeroth and first order perturbation does not involveq−

0 and q−0 . This

implies, by the proof of the cohomology in the Abelian case in the previous section, thaexists an infinitesimal canonical transformation to first order. Using the gradation w.r.t.Nlc de-fined in Eq.(4.9), one may construct the generator of the canonical transformation order byin Nlc. It is straightforward to see that this generator will not depend onq−

0 and q−0 , which in

turn implies that no higher order terms that are generated will depend on these fields, eitcan proceed in this way order by order proving the assertion.

5. Quantization

We will in this section discuss the quantization of our model. This is done in the same mas in the lightcone formulation in[9]. We will, therefore, only repeat the essential featuresdiscuss the differences of the two formulations.

We have in the previous section proven that there exits, to any order in perturbation thcanonical transformation connecting the stretched membrane model to the free string-likeWe will now define the quantum theory for the stretched membrane from the free strintheory by lifting the canonical transformations to unitary ones.

We define the unitary transformations by an iterative procedure. At some arbitrary ordeN inperturbation theory we define a unitary operator

(5.1)UN ≡ exp(−i:N−1GN :N−1).

HereGN is theN th order contribution to the generator of infinitesimal transformations, wwe, from the previous section, know exists classically. At the quantum level we specicorresponding operator by the ordering:N−1, which is the normal ordering w.r.t. the(N − 1)thorder vacuum. This vacuum is defined by

(5.2)∣∣0, k+ = 0

⟩N−1 = UN−1 · . . . · U1

∣∣0, k+ = 0⟩0,

where the zeroth order vacuum,|0, k+ = 0〉0, is defined in the usual way. Note that the conditk+ = 0 is slightly different from the one in the lightcone formulation in[9]. The full unitarytransformation to orderN in perturbation theory is then

(5.3)U(N) = UN · . . . · U1.

From the vacuum it is straightforward to construct the physical states for the stretchedbrane theory to any finite order in perturbation theory. This is done in the same way aslightcone formulation. As an example, the new oscillators to orderN are defined as

(5.4)α(N),µm,n ≡ UN · . . . · U1α

µm,nU

†1 · . . . · U†

N.

Through our construction it follows immediately that to any orderN in perturbation theory

(5.5)(Q)2 = 1

2[Q,Q] = 1

2

[U(N)Q0U(N)†,U(N)Q0U(N)†] = 1

2

[Q0,Q0] = 0,

where the last equality is true only forD = 27.


o-

ofder

ollowse

e full

D thesis,

ridge,

The partial gauge has singled out the(D − 1)-direction and the corresponding field compnents are given by

(5.6)XD−1 = 1√g

ξ2, PD−1 = −√gB,

where we have defined

(5.7)B = Pµ∂2Xµ.

One of the relevant physical operators found in[9] involved the integrated lightcone versionB. If we integrateB, denote it byB0, then it is gauge invariant, but not BRST-invariant. In orto construct a BRST-invariant expression one has to add ghosts toB0. One will find the followingBRST-invariant expression ofB0

2

(5.8)B0 =∫

d2ξPµ∂2X

µ − i∂2cb − i∂2cb.

B0 has the property that it is invariant under the constructed unitary transformations. This fdirectly from the fact thatB0 is an eigenvalue operator which counts the mode number in thξ2-direction, and that the net the mode number of the unitary operators are zero.

A final comment is that the BRST operator is only covariant w.r.t. the(D − 1)-dimensionalsubgroup of the full Lorentz group. Consequently, it is still an open question whether thLorentz group is anomaly free.

References

[1] P.K. Townsend, Phys. Lett. B 350 (1995) 184, hep-th/9501068.[2] E. Witten, Nucl. Phys. B 443 (1995) 85, hep-th/9503124.[3] J. Goldstone, unpublished.[4] J. Hoppe, Quantum theory of a massless relativistic surface and a two-dimensional bound state problem, Ph

MIT, 1982.[5] B. de Wit, J. Hoppe, H. Nicolai, Nucl. Phys. B 305 (1988) 545.[6] T. Banks, W. Fischler, S.H. Shenker, L. Susskind, Phys. Rev. D 55 (1997) 5112, hep-th/9610043.[7] J. Polchinski, Phys. Rev. Lett. 75 (1995) 4724, hep-th/9510017.[8] J. Björnsson, S. Hwang, Nucl. Phys. B 689 (2004) 37, hep-th/0403092.[9] J. Björnsson, S. Hwang, Nucl. Phys. B 727 (2005) 77, hep-th/0505269.

[10] M. Kato, K. Ogawa, Nucl. Phys. B 212 (1983) 443.[11] M.D. Freeman, D.I. Olive, Phys. Lett. B 175 (1986) 151.[12] I.B. Frenkel, H. Garland, G.J. Zuckerman, Proc. Natl. Acad. Sci. 83 (1986) 8442.[13] C.B. Thorn, Nucl. Phys. B 286 (1987) 61.[14] J. Polchinski, An Introduction to the Bosonic String, String Theory, vol. 1, Cambridge Univ. Press, Camb

1998.[15] I.A. Batalin, E.S. Fradkin, J. Math. Phys. 25 (1984) 2426.[16] M. Henneaux, C. Teitelboim, Quantization of Gauge Systems, Princeton Univ. Press, Princeton, 1992.[17] H. Aratyn, R. Ingermanson, Class. Quantum Grav. 5 (1988) L213.[18] S. Hwang, Nucl. Phys. B 351 (1991) 425.[19] M. Henneaux, Phys. Rep. 126 (1985) 1.

2 To get this expression we have redefined the ghost momenta by a factor−i, such that one has the conventionalanti-commutation relations.

es:

ofr order,elds, the

ntumicunc-eto anybe the

Nuclear Physics B 736 [FS] (2006) 169–198

Exact form factors in integrable quantum field theorithe scalingZ(N)-Ising model

Hratchya Babujiana, Angela Foersterb, Michael Karowskic,∗

a Yerevan Physics Institute, Alikhanian Brothers 2, Yerevan 375036, Armeniab Instituto de Física da UFRGS, Av. Bento Gonçalves 9500, Porto Alegre, RS, Brazil

c Institut für Theoretische Physik, Freie Universität Berlin, Arnimallee 14, 14195 Berlin, Germany

Received 28 October 2005; received in revised form 28 November 2005; accepted 1 December 2005


Abstract

A general form factor formula for the scalingZ(N)-Ising model is constructed. Exact expressionsall matrix elements are obtained for several local operators. In addition, the commutation rules fodisorder parameters and para-Fermi fields are derived. Because of the unusual statistics of the fiquantum field theory seems not to be related to any classical Lagrangian or field equation. 2005 Elsevier B.V. All rights reserved.

PACS:11.10.-z; 11.10.Kk; 11.55.Ds

Keywords:Integrable quantum field theory; Form factors

1. Introduction

The ‘form factor program’ is part of the so-called ‘bootstrap program’ for integrable quafield theories in 1+ 1 dimensions. This programclassifiesintegrable quantum field theoretmodels and in addition it provides their explicit exact solutions in terms of all Wightman ftions. This means, in particular, that we do notquantizea classical field theory. In fact thquantum field theory considered in this paper is not related (at least to our knowledge)classical Lagrangian or field equations of massive particles. The reason for this seems to

* Corresponding author.E-mail addresses:[email protected](H. Babujian),[email protected](A. Foerster),

[email protected](M. Karowski).






170 H. Babujian et al. / Nuclear Physics B 736 [FS] (2006) 169–198

fasci-ted by

he addi-

ctions

i-to

arcity

el. In

e

unusual anyonic statistics of the fields, turning this form factor investigation even morenating. The bootstrap program consists of three main steps: First the S-matrix is calculameans of general properties as unitarity and crossing, the Yang–Baxter equations and ttional assumption of ‘maximal analyticity’. Second, matrix elements of local operators

out〈p′m, . . . ,p′

1|O(x)|p1, . . . , pn〉in

are calculated using the 2-particle S-matrix as an input. As a third step the Wightman funcan be obtained by inserting a complete set of intermediate states.

The generalized form factors[1] are defined by the vacuum—n-particle matrix elements

〈0|O(x)|p1, . . . , pn〉inα1...αn

= e−ix(p1+···+pn)FOα1...αn

(θ1, . . . , θn),

where theαi denote the type (charge) and theθi are the rapidities of the particles(pi =Mi(coshθi,sinhθi)). This definition is meant forθ1 > · · · > θn, in the other sectors of the varables the functionFO

α (θ) = FOα1...αn

(θ1, . . . , θn) is given by analytic continuation with respect

theθi . General matrix elements are obtained fromFOα (θ) by crossing which means in particul

the analytic continuationθi → θi ± iπ . Using general LSZ assumptions and maximal analytiin [2] the following properties of form factors have been derived1:

(o) The form factor functionFOα (θ) is meromorphic with respect to all variablesθ1, . . . , θn.

(i) It satisfies Watson’s equations

FO...αiαj ...(. . . , θi, θj , . . .) = FO

...αj αi ...(. . . , θj , θi , . . .)Sαiαj

(θij ).

(ii) The crossing relation means for the connected part (see e.g.[4]) of the matrix element

α1〈p1|O(0)|p2, . . . , pn〉in,conn.α2...αn

= σO(α1)FOα1α2...αn

(θ1 + iπ, θ2, . . . , θn)

= FOα2...αnα1

(θ2, . . . , θn, θ1 − iπ),

whereσO(α) is the statistics factor of the operatorO with respect to the particleα.(iii) The function FO

α (θ) has poles determined by one-particle states in each sub-channparticular, ifα1 is the anti-particle ofα2, it has the so-called annihilation pole atθ12 = iπ

which implies the recursion formula

Resθ12=iπ

FOα (θ1, . . . , θn) = 2iCα1α2F

Oα3...

(θ3, . . . , θn)

× (1 − σO(α2)Sα1αn(θ2n) . . . Sα2α3(θ23)

).

(iv) Bound state form factors yield another recursion formula

Resθ12=iη

FOαβ...(θ1, θ2, θ3, . . .) = √

2FOγ ...(θ(12), θ3, . . .)Γ

γαβ

if iη is the position of the bound state pole. The bound state intertwinerΓγαβ (see e.g.[4,5])

is defined by

i Resθ=iη

Sαβ(θ) = Γ βαγ Γ

γαβ.

1 The formulae have been proposed in[3] as a generalization of formulae in[1]. The formulae are written here for thcase of no backward scattering, for the general case see[4].

H. Babujian et al. / Nuclear Physics B 736 [FS] (2006) 169–198 171

form

elds

zfactors

-tailvectorcalout

ling

oryeltigate

odel

(v) Since we are dealing with relativistic quantum field theories Lorentz covariance in the

FOα (θ1, . . . , θn) = esµFO

α (θ1 + µ, . . . , θn + µ)

holds if the local operator transforms asO → esµO wheres is the “spin” ofO.

Note that consistency of (ii), (iii) and (v) imply a relation of spin and statisticsσO(α) = e−2πis

and alsoσO(α) = 1/σO(α) whereα is the anti-particle ofα, which has the same charge asO.All solutions of the form factor equations (i)–(v) should provide the matrix elements of all fiin an integrable quantum filed theory with a given S-matrix.

Generalized form factors are of the form[1]

(1)FOα (θ) = KO

α (θ)∏

1i<jn

F (θij ) (θij = θi − θj ),

whereF(θ) is the ‘minimal’ form factor function. It is the solution of Watson’s equation[6] andthe crossing relation forn = 2

(2)F(θ) = F(−θ)S(θ), F (iπ − θ) = F(iπ + θ)

with no poles and zeros in the physical strip 0< Im θ π (and a simple zero atθ = 0). In [4] ageneral integral representation for theK-functionKO

α (θ) in terms of theoff-shell Bethe Ansat[7,8] has been presented, which transforms the complicated equations (i)–(v) for the formto simple ones for thep-functions(see[4] and(41)below)

(3)KOα (θ) =

∑m

cnm

∫Cθ

dz1 · · ·∫Cθ

dzm h(θ, z)pO(θ, z)Ψα(θ, z).

The symbolsCθ denote specific contours in the complexzi -planes. The functionh(θ, z) is scalarand encodes only data from the scattering matrix. The functionpO(θ, z) on the other hand depends on the explicit nature of the local operatorO. We discuss these objects in more debelow. For the case of a diagonal S-matrix, as in this paper, the off-shell Bethe AnsatzΨα(θ, z) is trivial. The K-function KO

α (θ) is an meromorphic function and has the ‘physipoles’ in 0< Im θij π corresponding to the form factor properties (iii) and (iv). It turnsthat for the examples we consider in this paper there is only one term in the sum of(3).

In this paper we will focus on the determination of the form factors of the scalingZ(N)-Isingquantum field theory in 1+ 1 dimensions. An Euclidean field theory is obtained as the scalimit of a classical statistical lattice model in 2-dimensions given by the partition function

Z =∑σ

exp

(− 1

kT

∑〈ij〉

E(σi, σj )

), σi ∈

1,ω, . . . ,ωN−1, ω = e2πi/N

as a generalization of the Ising model. It was conjectured by Köberle and Swieca[9] that thereexists aZ(N)-invariant interactionE(σi, σj ) such that the resulting massive quantum field theis integrable. In particular forN = 2 the scalingZ(2)-Ising model is the well investigated mod[10–13] which is equivalent to a massive free Dirac field theory. In this paper we investhe generalZ(N)-model. It has also been discussed as a deformation[14,15] of a conformalZ(N) para-Fermi field theory[16]. TheZ(N)-Ising model in the scaling limit possessesN − 1types of particles:α = 1, . . . ,N − 1 of chargeα, massMα = M sinπ α

Nand α = N − α is the

antiparticle ofα. Then-particle S-matrix factorizes in terms of two-particle ones since the m


rle

. Weytes

mulans of

i)

ovork

lings.tion to

ttof ultra-ave been

la

ectionee resultsail.ons ofeviously

is integrable. The two-particle S-matrix for theZ(N)-Ising model has been proposed by Köbeand Swieca[9]. The scattering of two particles of type 1 is

(4)S(θ) = sinh12(θ + 2πi

N)

sinh12(θ − 2πi

N).

This S-matrix is consistent with the picture that the bound state ofN − 1 particles of type 1 isthe anti-particle of 1. This will be essential also for the construction of form factors belowconstruct generalized form factors of an operatorO(x) andn particles of type 1 and for simplicitwe writeFO

n (θ) = FO1...1(θ). Note that all further matrix elements with different particle sta

of the field operatorO(x) are obtained by the crossing formula (ii) and the bound state for(iv). As an application of this form factor approach we compute the commutation relatiofields. In particular, we consider the fieldsψ

QQ(x), (Q,Q = 0, . . . ,N − 1) with chargeQ and

‘dual charge’Q. There are in particular the order parametersσQ(x) = ψQ0(x), the disorderparametersµ

Q(x) = ψ0Q

(x) and the para-Fermi fieldsψQ(x) = ψQQ(x). We show that theysatisfy the space like commutation rules:

σQ(x)σQ′(y) = σQ′(y)σQ(x),

µQ

(x)µQ′(y) = µ

Q′(y)µQ

(x),

µQ

(x)σQ(y) = σQ(y)µQ

(x)eθ(x1−y1)2πiQQ/N ,

(5)ψQ(x)ψQ(y) = ψQ(y)ψQ(x)eε(x1−y1)2πiQ2/N .

It turns out that the nontrivial statistics factorsσO(α) in the form factor equations (ii) and (iilead to the nontrivial order–disorder and the anyonic statistics of the fields.

The form factor bootstrap program has been applied in[13] to theZ(2)-model. Form factorsfor the Z(3)-model were investigated by one of the present authors in[17]. There the formfactors of the order parameterσ1 were proposed for up to four particles. Kirilov and Smirn[18] proposed all form factors of theZ(3)-model in terms of determinants. Some related wcan be found in[19]. For generalN form factors for charge-less states (n particles of type 1 andn particles of typeN − 1) were calculated in[20]. In the present paper we present for the scaZ(N)-Ising model integral representations for all matrix elements of several field operator

Recently, there has been a renewed interest in the form factors program in conneccondensed matter physics[21–23] and atomic physics[24]. In particular, applications to Moinsulators and carbon nanotubes as well as in the field of Bose–Einstein condensatescold atomic gases have been discussed and in some instances correlation functions hcomputed.

The paper is organized as follows: In Section2 we construct the general form factor formufor the simplestN = 2 case, which corresponds to the well-known scalingZ(2)-Ising model,and show that the known results can be reproduced by our new general approach. In S3we construct the general form factor formula for theZ(3) case, which is more complex duto the presence of bound states, and discuss several explicit examples. We extend thesin Section4, where the general form factors forZ(N) are constructed and discussed in detSection5 contains the derivation of the commutation rules of the fields and some applicatithis formalism are presented. Some results of the present article have been published pr[25] without proofs.


to cal-

nves-essesh

) withonseasilyleeneral

se

ereas

2. Z(2)-form factors

To make our method more transparent and with the hope that our construction will helpculate form factors for all primary and descendant fields, we start with the simplest caseN = 2,which corresponds to the well-known Ising model in the scaling limit. This model, already itigated in[10,11,13,20], is equivalent to a massive free Dirac field theory. The model possone particle with massM and the 2-particle S-matrix isS = −1. In [13] the form factor approachas been applied to this case with the result for the order parameter fieldσ(x)

(6)Fσn (θ) = (2i)(n−1)/2

∏1i<jn

tanh1

2θij

for n odd. It is easy to see that this expression satisfies the form factor equations (i)–(iiistatistics factorσσ = 1. For theZ(2) case in this section we skip the proof that the functiobtained by our general formula satisfy the form factor equations (i)–(v), the reader mayreduce the proofs for theZ(3) and the generalZ(N) case of the following sections to this simpone. Rather, we present the results for several fields, in particular, we will show that our gformula reproduces the known results.

2.1. The general formula for n-particle form factors

We propose then-particle form factors of an operatorO(x) as given by(1) with the minimalform factor function

(7)F(θ) = sinh1

2θ

which is the minimal solution of Watson’s equations and crossing(2) for S = −1. The K-function is given by our general formula(3) where the Bethe Ansatz vector is trivial (becauthere is no backward scattering) and the sum consists only of one term

(8)KOn (θ) = NnInm

(θ,pO)

.

Thefundamental building blocksof form factors are

(9)Inm(θ,pO) = 1

m!∫Cθ

dz1

R· · ·

∫Cθ

dzm

Rh(θ, z)pO(θ, z),

(10)h(θ, z) =n∏

i=1

m∏j=1

φ(zj − θi)∏

1i<jm

τ(zi − zj ).

Theh-function does not depend on the operator but only on the S-matrix of the model, whthep-function depends on the operator. Both are analytic functions ofθi (i = 1, . . . , n) andzj

(j = 1, . . . ,m) and are symmetric underθi ↔ θj andzi ↔ zj . For all integration variableszj

the integration contoursCθ = ∑Cθi

enclose clock wise oriented the pointszj = θi (i = 1, . . . , n).This means we assume thatφ(z) has a pole atz = 0 such thatR = ∫

Cθdzφ(z−θ). The functions

φ(z) andτ(z) are given in terms of the minimal form factor function as

(11)φ(z) = 1 = −2i,

F(θ)F (θ + iπ) sinhz


therve to

sidue

f

s

en

er to

. We

(12)τ(z) = 1

φ(z)φ(−z)= 1

4sinh2 z.

The following properties of thep-functions guarantee that the form factors satisfy (i)–(iii)

(i′2) pOnm(θ, z) is symmetric underθi ↔ θj

(ii ′2) σOpOnm(θ1 + 2πi, θ2, . . . , z) = pO

nm(θ1, θ2, . . . , z)

(iii ′2) if θ12 = iπ

pOnm(θ, z)

∣∣z1=θ1

= σOpOnm(θ, z)

∣∣z1=θ2

= σOpOn−2m−1(θ3, . . . , θn, z2, . . . , zm) + p,

whereσO is the statistics factor of the operatorO with respect to the particle. The functionpmust not contribute after integration, which means in particular that is does not depend onzi

(in most casesp = 0). For convenience we have introduced the indicesnm to denote the numbeof variables. For the recursion relation (iii) in addition the normalization coefficients hasatisfy

(13)Nn = iNn−2.

One may convince oneself that the form factor satisfies (i) and (ii). Not so trivial is the rerelation (iii), however, it follows from a simplified version of the proofs for theZ(3) and thegeneralZ(N) case below.

2.2. Examples of fields and theirp-functions

We present the following correspondence of operators andp-functions which are solutions o(i′2)–(iii ′2). For the order parameterσ(x), the disorder parameterµ(x), the Fermi fieldψ(x), andthe higher conserved currentsJ

µL (x) (L ∈ Z) we propose the followingp-functions and statistic

parameters

σ(x) ↔ pσnm(θ, z) = 1 for n = 2m + 1 with σσ = 1,

µ(x) ↔ pµnm(θ, z) = ime(

∑zi− 1

2

∑θi ) for n = 2m with σµ = −1,

ψ±(x) ↔ pψ±nm (θ, z) = e±(

∑zi− 1

2

∑θi ) for n = 2m + 1 with σψ = −1,

(14)J±L (x) ↔ p

J±L

nm(θ, z) =∑

e±θi∑

eLzi for n = 2m with σµ = 1.

Note thatp = 0 in (iii ′2) only for J±L . The motivation of these choices will be more obvious wh

we investigate the commutation rules of fields in Section5 and will follow from the propertiesof the form factors which we now discuss in more detail.

Explicit expressions of the form factorsNow we have to check that the proposedp-functionsreally provide the well-known form factors for the order, disorder and Fermi fields. In ordget simple expressions for these form factors, we have to calculate the integral(9) with (10) foreachp-function separately.

For the order parameter Only for odd numbers of particles the form factors are non-zerocalculate

(15)Inm(θ,1) = 2m∏ 1

cosh12θij

for n = 2m + 1.

1i<jn


ero.

ctorscheck

er

atrixne

The proof of this formula can be found inAppendix B. The general formulae(1), (3), (7), and(13)with Nn = i(n−1)/2 then imply forn odd

(16)Fσn (θ) = 2m

∏1i<jn

F (θij )

cosh12θij

= (2i)(n−1)/2∏

1i<jn

tanh1

2θij ,

which agrees with the known result(6). The proof that the integralInm(θ,1) vanishes for evennandm > 0 is simple: Ifm(m−1) < n we may decompose for realθi the contoursCθ = C0−C0−iπ

where ReC0 goes from−∞ to ∞ and Im(θi + iπ) < ImC0 < Im(θi). The shiftzi → zi − iπ

impliesh(θ, z) → (−1)nh(θ, z) such that the integrals alongC0 andC0−iπ cancel for evenn.

For the disorder parameter Only for even numbers of particles the form factors are non-zWe calculate withpµ as given in(14)

Inm

(θ,pµ

) = 2m∏

1i<jn

1

cosh12θij

for n = 2m.

The proof of this formula is analog to that inAppendix B, therefore withNn = in/2 the formfactors are forn even

(17)Fµn (θ) = (2i)n/2

∏1i<jn

tanh1

2θij .

Similar as above for the order parameter one can show that the integralInm(θ,pµ) vanishes foroddn andm > 0. It is also interesting to investigate the asymptotic behavior of the form fawhen one of the rapidities goes to infinity. From the integral representation it is easy tothat

Fσn (θ)

θ1→∞→ Fµn−1(θ

′) θ2→∞→ 2iF σn−2(θ

′′).

Of course, this result follows easily from the explicit expressions(16) and (17). This asymptoticbehavior is another motivation for our choice(14)of thep-function forσ(x) andµ(x).

For the Fermi field Only for n = 1 the form factors are non-zero. We calculate withpψ±as

given in(14)

Inm

(θ,pψ±) = δn1e

∓ 12θ for n = 2m + 1.

The proof thatInm(θ,pψ±) = 0 for n = 2m + 1 odd andm > 0 is the same as for the disord

parameter. Therefore with the normalizationN1 = √M we obtain

(18)Fψ

1 (θ) = 〈0|ψ(0)|θ〉 = u(θ) = √M

(e− 1

2θ

e12θ

).

The property that all form factors of the Fermi field vanish except the vacuum one-particle melement reflects the fact thatψ(x) is a free field, in particular for the Wightman functions oeasily obtains

〈0|ψ(x1) · · ·ψ(xn)|0〉 = 〈0|ψ(x1) · · ·ψ(xn)|0〉free.


o.

ion

n

eticle 2,f typehouldicles of

1 aren the

ndn

For the infinite set of conserved higher currentsOnly for n = 2 the form factors are non-zerWe calculate withpJ±

L as given in(14)

Inm

(θ,pJ±

L) = δn2

(e±θ1 + e±θ2

)2i

(eLθ1

sinhθ12+ eLθ2

sinhθ21

)for n = 2m.

The proof thatInm(θ,pJ±L ) = 0 for n = 2m > 2 is again similar as above. With the normalizat

c21 = ±iM the form factors are

〈0|J±L (0)|θ1, . . . , θn〉in = ∓δn22M

(e±θ1 + e±θ2

)(eLθ1 − eLθ2

) 1

sinhθ12

such that as in[26] the chargesQL = ∫dx J 0

L(x) satisfy the eigenvalue equation(QL −

n∑i=1

eLθi

)|θ1, . . . , θn〉in = 0 for L = ±1,±3, . . . .

Obviously we get the energy–momentum tensor fromJ±±1(x).

The propertyFψn = 0 for oddn > 1 andFJ

n = 0 for evenn > 2 is related to the fact that ithe recursion relation (iii) the factor(1− σO ∏

S) is zero in both cases.

3. Z(3)-form factors

Let us now considerN = 3, which corresponds to the scalingZ(3)-Ising model. In this caswe have two particles, 1 and 2, and the bound state of two particles of type 1 is the parwhich in turn is the anti-particle of particle 1. Conversely, the bound state of two particles o2 is the particle of type 1, which in turn is again the anti-particle of 2. So, our construction stake into account this structure of the bound states. We construct the form factors for parttype 1, the others can then be obtained by the form factor bound state formula (iv).

3.1. The general formula forn-particle form factors

In order to obtain a recursion relation where only form factors for particles of typeinvolved, we have to apply the bound state relation (iv) to get the anti-particle and thecreation annihilation equation (iii) to obtain

Resθ23=iη

Resθ12=iη

FO1111...1(θ1, . . .) = Res

θ(12)3=iπFO

211...1(θ(12), θ3, . . .)√

2Γ

(19)= 2iFO1...1(θ4, . . .)

(1− σO

1

n∏i=4

S(θ3i )

)√2Γ,

whereθ(12) = 12(θ1 + θ2) is the bound state rapidity,η = 2

3π is the bound state fusion angle aΓ = i|Resθ=iη S11(θ)|1/2 is the bound state intertwiner (see[4,5]). In the following we use agaithe short notationFO

1...1(θ1, . . . , θn) = FOn (θ) and also write the statistics factor asσO(1) = σO

1 .We write the form factors again in the form(1) where minimal form factor function

(20)F(θ) = c3 exp

∞∫dt

t

2 cosh13t sinh2

3t

sinh2 t

(1− cosht

(1− θ

iπ

))
0


or

t-

is the solution of Watson’s equations(2) with the S-matrix(4) for N = 3. The constantc3 isgiven by(A.1) in Appendix A. Similar as above we make the Ansatz for theK-functions

(21)KOn (θ) = NnInmk

(θ,pO)

with the fundamental building blocks of form factors

(22)Inmk(θ,p) = 1

m!k!∫Cθ

dz1

R· · ·

∫Cθ

dzm

R

∫Cθ

du1

R· · ·

∫Cθ

duk

Rh(θ, z,u)p(θ, z,u),

h(θ, z,u) =n∏

i=1

(m∏

j=1

φ(zj − θi)

k∏j=1

φ(uj − θi)

)

(23)×∏

1i<jm

τ(zij )∏

1i<jk

τ (uij )∏

1im

∏1jk

(zi − uj ).

Again the integration contoursCθ = ∑Cθi

enclose the pointsθi such thatR = ∫Cθ

dzφ(z − θ).Equations (iii) and (iv), in particular(19) lead to the relations

(24)1∏

k=0

φ(θ + kiη)

2∏k=0

F(θ + kiη) = 1,

(25)τ(z)φ(z)φ(−z) = 1, (z)φ(z) = 1,

as an extension of(11) and (12)for theZ(2) case. The solution forφ is

(26)φ(z) = 1

sinh12zsinh1

2(z + iη)

if the constantc3 is fixed as in(A.1). The phi-function satisfies the ‘Jost function’ property

(27)φ(−θ)

φ(θ)= S(θ).

We will now show again that by the Ansatz(21)–(23)we have transformed the form factequations (i)–(v), in particular equation(19) to simple equations for thep-functions.

Assumptions for thep-functions The functionpOnmk(θ, z, u) is analytic in all variables and sa

isfies:

(i′3) pOnmk(θ, z, u) is symmetric underθi ↔ θj ,

(ii ′3) σO1 pO

nmk(θ1 + 2πi, θ2, . . . , z, u) = pOnmk(θ1, θ2, . . . , z, u),

(iii ′3) if θ12 = θ23 = iη

(28)pOnmk(θ, z, u)

∣∣z1=θ1u1=θ2

= σO1 pO

nmk(θ, z, u)∣∣

z1=θ2u1=θ3

= σO1 pO

n−3m−1k−1(θ′, z′, u′) + p,

(iv′3) pO

nmk(θ + µ,z + µ,u + µ) = esµpOnmk(θ, z, u),

whereθ ′ = (θ4, . . . , θn), z′ = (z2, . . . , zm) and u′ = (u2, . . . , uk). In (ii ′3) and (iii′3) σO

1 is thestatistics factor of the operatorO with respect to the particle of type 1 and in (v′ ) s is the spin
3


ri)

theand

heion to be

of the operatorO. Again p must not contribute after integration (in most casesp = 0). Againone may convince oneself that the form factor satisfies (i) and (ii) ifh(θ, z) is symmetric undeθi ↔ θj and periodic with respect toθi → θi + 2πi. Not so trivial is the residue relation (iiwhich is proved in the following lemma.

Lemma 1. The form factorsFOn (θ) defined by(1) and (20)–(23)satisfies(i)–(v), in particular

(19), if thep-functions satisfy(i′3)–(v′5) of (28), the functionsφ, τ and the relations(24), (25)

and the normalization constants in(21) the recursion relation

(29)Nn

(Resθ=iη

φ(−θ))2

φ(iη) F 2(iη)F (2iη) = Nn−32i√

2Γ.

Proof. The form factor equations (i), (ii), and (v) follow obviously from the equations forp-functions (i′3), (ii ′3), and (v′3), respectively. As already stated, we will prove properties (iii)(iv) together in the form of(19). Taking the residues Resθ23=iη Resθ12=iη there will bemk equalterms originating from pinchings forzi andui . We pick them fromz1 andu1 and rewrite theproducts that appear in the expression forInmk in a convenient form, such that the location of tpoles turns out to be separated from the general expression. Then, the essential calculatperformed is

Resθ23=iη

Resθ12=iη

Inmk

(θ,pO) = mk

m!k!∫Cθ ′

dz2

R· · ·

∫Cθ ′

dzm

R

∫Cθ ′

du2

R· · ·

∫Cθ ′

duk

R

×n∏

i=4

(m∏

j=2

φ(zj − θi)

k∏j=2

φ(uj − θi)

)

×∏

2i<jm

τ(zij )∏

2i<jk

τ (uij )

m∏i=2

k∏j=2

(zi − uj )

×3∏

i=1

(m∏

j=2

φ(zj − θi)

k∏j=2

φ(uj − θi)

)r

with

r = Resθ23=iη

Resθ12=iη

∫Cθ

dz1

R

∫Cθ

du1

R

n∏i=1

(φ(z1 − θi)φ(u1 − θi)

)

×m∏

j=2

τ(z1j )

k∏j=2

τ(u1j )(z1 − u1)

k∏j=2

(z1 − uj )

m∏j=2

(zj − u1)pOn (θ, z, u).

ReplacingCθ by Cθ ′ whereθ ′ = (θ4, . . . , θn) we have usedτ(0) = τ(±iη) = (0) = (−iη) = 0and the fact that thez1-, u1-integrations give non-vanishing results only for(z1, u1) at (θ1, θ2)

and(θ2, θ3). This is because forθ12, θ23 → iη pinching appears atz1 = θ2, u1 = θ3 andz1 = θ1,u1 = θ2. Defining the function

f (z1, u1) =m∏

τ(z1j )

k∏τ(u1j )(z1 − u1)

k∏(z1 − uj )

m∏(zj − u1)p

On (θ, z, u)

j=2 j=2 j=2 j=2


trix el-vates

and using the property thatf (z, z) = f (z, z − iη) = 0, we calculate

r = Resθ23=iη

Resθ12=iη

∫Cθ

dz1

R

∫Cθ

du1

Rf (z1, u1)

n∏i=1

(φ(z1 − θi)φ(u1 − θi)

)

= (Resθ=iη

φ(−θ))2

φ2(iη)

n∏i=4

(φ(θ2i )φ(θ3i )

)(f (θ2, θ3) − f (θ1, θ2)

n∏i=4

S(θ3i )

).

We have used the symmetries of the phi-functionφ(θ) = φ(θ + 2πi) = φ(−θ − iη) the Jostproperty(27)andθ12, θ23 = iη = 2

3iπ which imply that

Resθ12=iη

φ(θ21) Resθ23=iη

φ(θ32) = Resθ12=iη

φ(θ13) Resθ23=iη

φ(θ21) = (Resθ=iη

φ(−θ))2

,

φ(θ23)φ(θ31) = φ(θ12)φ(θ23) = φ2(iη),

φ(θ1i )

φ(θ3i )= φ(θ3i − iη)

φ(θ3i )= φ(−θ3i )

φ(θ3i )= S(θ3i ).

With the help of the defining equations(25) for τ and which imply(3∏

i=1

φ(z − θi)

)−1

= τ(θ1 − z)(z − θ2) = τ(θ2 − z)(θ1 − z)

= τ(θ2 − z)(z − θ3) = τ(θ3 − z)(θ1 − z),

we obtain the relations forf (θ2, θ3) andf (θ1, θ2)

3∏i=1

(m∏

j=2

φ(zj − θi)

k∏j=2

φ(uj − θi)

)f (θ1, θ2) = (θ12)p

On (θ, θ1, z

′, θ2, u′),

3∏i=1

(m∏

j=2

φ(zj − θi)

k∏j=2

φ(uj − θi)

)f (θ2, θ3) = (θ23)p

On (θ, θ2, z

′, θ3, u′).

Finally we obtain using the defining relation(24) for the phi-function

Resθ23=iη

Resθ12=iη

Inmk

(θ,pO) = (

Resθ=iη

φ(−θ))2

φ2(iη)(iη)

3∏i=1

n∏j=4

(F(θij )

)−1

× In−3m−1k−1(θ ′,pO)(

1− σO1

n∏i=4

S(θ3i )

)

if the p-function satisfies (iii′3). Therefore the form factor given by(1) and (20)–(23)satisfies(14)and (iii) if the normalization constants satisfy(29). 3.2. Examples of fields andp-functions

We present solutions of the equations for thep-functions (i′3)–(v′3) of (28) and some explici

examples of the resulting form factors. We identify the fields by the properties of their matements. In Section5 we show that the field satisfy the desired commutation rules. This motito propose a correspondence of fieldsφ(x) andp-functionspφ

(θ, z,u).
nmk


s

en

d

The order parameterσQ(x) We look for a solution of (i′3)–(v′3) with

chargeQ = 1,2,

spins = 0,

statisticsσσQ

1 = 1.

Since the fields carry the chargeQ the only non-vanishing form factors withn particles of type 1are the ones withn = Q mod3. We propose the correspondence of the field and thep-function:

σQ ↔ pσQ

nmk = 1 with n = 3l + Q,

m = l + 1, k = l for Q = 1,

m = l + 1, k = l + 1 for Q = 2.

The normalization constantsNn follow from (29).

Examples forQ = 1 The form factors of the order parameterσ1(x) for one and four particleof type 1 are

Fσ11 = 〈0|σ1(0)|p〉1 = N1I110= 1,

Fσ11111(θ) = 〈0|σ1(0)|p1,p2,p3,p4〉in

1111= N4I421(θ,1)∏

1i<j4

F(θij ),

where we calculate from our integral representation(22)

I421(θ,1) = const×(∑

e−θi∑

eθi − 1)∏

i<j

1

sinh12(θij − iη)sinh1

2(θij + iη).

This result has already been obtained in[17] where also the form factor equation (iv) has bediscussed, in particular (up to normalizations)

Resθ34=2πi/3

〈0|σ1(0)|p1,p2,p3,p4〉in1111= const× 〈0|σ1(0)|p1,p2,p3 + p4〉in

112

with

〈0|σ1(0)|p1,p2,p3〉in112= const× F(θ12)

sinh12(θ12 − iη)sinh1

2(θ12 + iη)

2∏i=1

F min12 (θi3)

cosh12θi3

,

whereF min12 is the minimal form factor function for the S-matrixS12. Further it has been foun

in [17] that

Resθ12=2πi/3

〈0|σ1(0)|p1,p2,p3〉in112= const× 〈0|σ1(0)|p1 + p2,p3〉in

22

with

(30)〈0|σ1(0)|p1,p2〉in22 = const× F(θ12)


2(θ12 + iη)

and the form factor equation (iii) has been checked

Resθ23=iπ

〈0|σ1(0)|p1,p2,p3〉in112= const× (

S(θ12) − 1).


on.

d

eter

Example forQ = 2 The form factor of the order parameterσ2(x) for two particles of type 1 is

Fσ211(θ) = 〈0|σ2(0)|p1,p2〉in

11 = N2I211(θ,1)F (θ12),

where we calculate

I211(θ,1) = const× 1


2(θ12 + iη),

which agrees with the result (30) of[17]. This is to be expected because of charge conjugati

The disorder parameterµQ

(x) We look for a solution of (i′3)–(v′3) with

chargeQ = 0,

spins = 0,

statisticsσµ

Q

1 = ωQ,

whereω = eiη, η = 2/3. We call the numberQ = 1,2 the ‘dual charge’ of the fieldµQ

(x). Sincethe fields carry the chargeQ = 0 the only non-vanishing form factors withn particles of type 1are the ones withn = 0 mod3. We propose the correspondence of the field and thep-functionQ

µQ

↔

pµ1nmk = ρ exp

(∑mi=1 zi − 1

3

∑ni=1 θi

)p

µ2nmk = ρ exp

(∑mi=1 zi + ∑k

i=1 ui − 23

∑ni=1 θi

) with

n = 3m,

k = m,

whereρ = √ω

Q(Q−N+2)m. Again the normalization constantsNn follow from (29).

Examples forQ = 1,2 The form factors of the disorder parameterµQ

(x) for 0 and 3 particlesof type 1 are

Fµ

Q = 〈0|µQ

(0)|0〉 = 1,

Fµ

Q

111(θ) = 〈0|µQ

(0)|p1,p2,p3〉in111= N3I311

(θ,p

µQ) ∏

1i<j3

F11(θij ).

We calculate from our integral representation(22)

I311(θ,p

µQ) = const× e

∓ 13

n∑i=1

θi3∑

i=1

e±θi∏i<j

1


2((θij + iη),

where the upper sign is forQ = 1 and the lower one forQ = 2. Using the form factor bounstate formula (iv) we obtain (up to a constant)

Fµ

Q

12 (θ) = e∓ 16θ12

1

cosh12θ12

F min12 (θ12).

It is interesting to note that for Reθ1 → ∞ we have the relation of order and disorder paramform factors (up to constants)

limReθ1→∞〈0|σ1(0)|p1,p2,p3,p4〉in

1111= 〈0|µ2(0)|p2,p3,p4〉in111,


sthe

f

d the

which follows from the asymptotic behavior

F(θ1i ) → e23θ1i ,

I421(θ,1) → eθ1

(4∑

i=2

e−θi

)4∏

j=2

e−θ1j∏

1<i<j

1

sinh12(θij − a)sinh1

2(θij + a).

The para-Fermi fieldψQ(x) We look for a solution of (i′3)–(v′3) with

chargeQ = Q,

spins = Q(3− Q)/3,

statisticsσψQ

1 = ωQ.

These fields have chargeQ = 1,2 and dual chargeQ = Q. The only non-vanishing form factorwith n particles of type 1 are the ones withn = Q mod3. We propose the correspondence offield and thep-function:

ψQ ↔

pψ1nmk = ρ exp

(∑mi=1 zi − 1

3

∑ni=1 θi

)p

ψ2nmk = ρ exp

(∑mi=1 zi + ∑k

i=1 ui − 23

∑ni=1 θi

) with

n = 3l + Q,

m = l + 1,

k = l + Q − 1,

whereρ = √ω

Q(Q−1)l . Again the normalization constantsNn follow from (29).

Examples forQ = 1,2 The form factors of the para-Fermi fieldψ1(x) for 1 and 4 particles otype 1 are

Fψ11 (θ) = 〈0|ψ1(0)|p〉1 = N1I110

(θ,pψ1

) = e23θ ,

Fψ11111(θ) = 〈0|ψ1(0)|p1,p2,p3,p4〉in

1111= N4I421(θ,pψ1

) ∏1i<j4

F(θij )

= const× e− 2

3

4∑i=1

θi ∑i<j

eθi+θj∏

1i<j4

F(θij )


2(θij + iη)

and the one of the para-Fermi fieldψ2(x) for 2 particles of type 1 is

Fψ211 (θ) = 〈0|ψ2(0)|p1,p2〉11 = N2I211

(θ,pψ2

)F(θ12)

= const× e13 (θ1+θ2)

F (θ12)


2(θ12 + iη).

All these examples agree with the results of[18].

The higher currentsJ±L (x) We look for a solution of (i′3)–(v′

3) with

chargeQ = 0,

spins = L ± 1,

statisticsσJ±L

1 = 1.

Since the currents areZ(3)-charge-less the only non-vanishing form factors withn particles oftype 1 are the ones withn = 0 mod3. We propose the correspondence of the currents an


be-

for

heare

p-functions forL ∈ Z

J±L ↔ p

J±L

nmk = ±(

n∑i=1

e±θi

)(m∑

i=1

eLzi +m∑

i=1

eLui

)for

n = 3m,

k = m.

Note that for this case the functionp in (28) is non-vanishing, however, it does not contributecauseInmm(θ,1) = 0 for n = 3m. The proof of this fact is similar to the one given inAppendix B.The higher chargesQL = ∫

dx J 0L(x) satisfy the eigenvalue equations(

QL −n∑

i=1

eLθi

)|p1, . . . , pn〉in = 0.

Obviously, fromJ±±1(x) we obtain the energy–momentum tensor.

Examples The form factors of the energy momentum tensor that isJ±L (x) for L = ±1 for 0 and

3 particles of type 1 are

FJ±L = 〈0|J±

L (0)|0〉 = 0,

FJ±L

111(θ) = 〈0|J±L (0)|p1,p2,p3〉in

111= c311I311(θ,pJ±

L) ∏

1i<j3

F(θij )

= ±const× (e±θ1 + e±θ2 + e±θ3

)(eLθ1 + eLθ2 + eLθ3

)×

∏1i<j3

F(θij )


2(θij + iη).

By (iv) we obtain the bound state form factor (up to a normalization) forL = ±1

FJ±L

12 (θ) = ±const× e12 (L±1)(θ1+θ2)F min

12 (θ12).

Notice that this last expression agrees with the results of[20] whenN = 3.

4. Z(N)-form factors

The scalingZ(N)-Ising model possesses particles of typeα = 1, . . . ,N −1 withZ(N)-chargeQα = α such that the anti-particle ofα is α = N − α. The bound state fusion rules are(αβ) =α + β modN , in particular the bound state ofN − 1 particles of type 1 is the anti-particle1.Therefore applyingN − 2 times formula (iv) and once (iii) we obtain the recursion relationsform factors where only particles of type 1 are involved

ResθN−1N=iη

. . . Resθ12=iη

Fn(θ1, . . . , θn)

(31)= 2iFn−N(θN+1, . . . , θn)

(1− σO

1

n∏i=N+1

S(θNi)

)N−2∏α=1

√2Γ 1+α

1α ,

whereη = 2πiN

and theΓ 1+α1α = i|Resθ=iηα S1α(θ)|1/2 are the bound state intertwiners of t

fusion (1α) = 1+ α. We will construct the form factors of particles of type 1, all the othersthen obtained by the bound state formula (iv).


r large

4.1. The general Z(N)-form factor formula

Following [1] we write the form factors again in the form(1)

(32)FOα (θ) = KO

α (θ)∏

1i<jn

F (θij ),

where minimal form factor function[17]

(33)F(θ) = cN exp

∞∫0

dt

t

2 cosh1N

t sinhN−1N

t

sinh2 t

(1− cosht

(1− θ

iπ

))

is the solution of Watson’s equations(2) with the S-matrix(4). The constantcN is given by(A.1)in Appendix A. The K-function KO

n (θ1, . . . , θn) is totally symmetric in the rapiditiesθi , 2πi

periodic, containing the entire pole structure and determines the asymptotic behavior fovalues of the rapidities. Similar as above we make the Ansatz for theK-functions

(34)KOn (θ) = NnInm

(θ,pO)

with the fundamental building blocks of form factors

(35)Inm(θ,pO) =(

N−1∏k=1

1

mk!mk∏j=1

∫Cθ

dzkj

R

)h(θ, z)pO(θ, z),

(36)

h(θ, z) =N−1∏k=1

(mk∏j=1

n∏i=1

φ(zkj − θi)∏

1i<jmk

τ(zki − zkj )

)

×∏

1k<lN−1

mk∏i=1

ml∏j=1

(zki − zlj ),

wherem = (m1, . . . ,mN−1) andz = (zki), k = 1, . . . ,N − 1, i = 1, . . . ,mk . Again the integra-tion contoursCθ = ∑

Cθienclose the pointsθi such thatR = ∫

Cθdzφ(z− θ). Equations (iii) and

(iv), in particular(31) lead to the relations

(37)N−2∏k=0

φ(z + kiη)

N−1∏k=0

F(z + kiη) = 1,

(38)τ(−z)φ(z + iπ)φ(z) = 1, (z)φ(z) = 1,

(39)Nn

(Resθ=iη

φ(−θ))N−1

N−2∏k=1

φk(kiη)

N−1∏k=1

FN−k(kiη) = Nn−N2i

N−2∏k=1

N−2∏α=1

√2Γ 1+α

1α ,

wherem − 1= (m1 − 1, . . . ,mN−1 − 1). The solution of(37) for φ is again

(40)φ(z) = 1

sinh12zsinh1

2(z + iη)


rty

ed

h

r,rticles.

tx ele-vates

ce

if the constantcN is fixed as in(A.1). The phi-function satisfies again the ‘Jost function’ propeφ(−θ)/φ(θ) = S(θ). Thep-functionpO

nm(θ, z) is analytic in all variables and satisfies:

(i ′) pOnm(θ, z) is symmetric underθi ↔ θj ,

(ii ′) σO1 pO

nm(θ1 + 2πi, θ2, . . . , z) = pOnm(θ1, θ2, . . . , z),

(iii ′) if θkk+1 = iη = 2πi/N for k = 1, . . . ,N − 1

(41)pOnm(θ, z)|zk1=θk

= σO1 pO

nm(θ, z)|zk1=θk+1 = σO1 pO

n−Nm−1(θ′, z′) + p,

(v′) pOnmk(θ + µ,z + µ) = esµpO

nmk(θ, z),

whereθ ′ = (θN+1, . . . , θn), z′ = (zki), k = 1, . . . ,N − 1, i = 2, . . . ,mk . In (ii ′) and (iii′) σO1 is

the statistics factor of the operatorO with respect to the particle of type 1 and in (v′) s is the spinof the operatorO. Again p must not contribute after integration (in most casesp = 0).

By means of the off-shell Bethe Ansatz(3) and (35)we have transformed the complicatform factor equations (i)–(v) to simple equations for thep-functions (i′)–(v′). Again one mayconvince oneself that the form factor satisfies (i) and (ii) ifh(θ, z) is symmetric underθi ↔ θj

and periodic with respect toθi → θi + 2πi. Not so trivial is again the residue relation (iii) whicis proved in the following lemma.

Lemma 2. The form factors given by equations(32)–(36)satisfy the form factor equations(i)–(v) if the functionsφ, τ , satisfy(37)and (38), the normalization constants satisfy(39)and thep-functions satisfy(i ′)–(v′) of (41).

The proof of this lemma follows the same strategy of the previousZ(3) case. Here, howevethe essential calculation is much more involved, due to the existence more types of paDetails of this proof can be found inAppendix C.

4.2. Examples of fields andp-functions

We present solutions of the equations for thep-functions (i′)–(v′) of (41) and some expliciexamples of the resulting form factors. We identify the fields by the properties of their matriments. In Section5 we show that the fields satisfy the desired commutation rules. This motito propose a correspondence of fieldsφ(x) andp-functionspφ(θ, z).

The fieldsψQ,Q

(x) These fields have the chargeQ = 0, . . . ,N − 1 and the dual chargeQ =0, . . . ,N − 1. We look for a solution of (i′)–(v′) with

(42)

chargeQ modN,

spinsψ = min(Q, Q) − QQ/N,

statisticsσψ

1 = ωQ

with ω = eiη = e2πi/N . The phase factorσψ

1 is the statistics factor of the fieldψQ,Q

(x) withrespect to the particle of type 1. Since the fields carry the chargeQ the only non-vanishing formfactors withn particles of type 1 are the ones withn = Q modN . We propose the corresponden


ldstheoryof

f

of the field and thep-function:

ψQ,Q

↔ pQQnm = ρ exp

(Q∑

k=1

mk∑j=1

zkj − Q

N

n∑i=1

θi

)

(43)with n = 3l + Q, l = 0,1,2, . . . and

mk = l + 1 for k Q,

mk = l for Q < k,

whereρ = √ω

Q(Q−N+2)n/N . One easily checks that thisp-function satisfies the equations (i′)–(v′) and the requirements(42). The normalization constantsNn follow from (39). In particularwe have for

Q = 0 the order parametersσQ(x) = ψQ0(x),

Q = 0 the disorder parameterµQ

(x) = ψ0Q(x),

Q = Q the para-Fermi fieldsψQ(x) = ψQQ(x).

They satisfy space like commutation rules(5), derived in the next section. The para-Fermi fieψQ(x) are the massive analogs of the para-Fermi fields in the conformal quantum fieldof [14,15]. One obtains a second set of fieldsψ

Q,Q(x) by changing the sign in the exponent

(43).

The higher currentsJ±L (x) These fields are charge-less, have bosonic statistics and spinL± 1.

The only non-vanishing form factors withn particles of type 1 are the ones withn = 0 modN .We propose the correspondence of the currents and thep-functions forL ∈ Z

J±L ↔ p

J±L

nm = ±n∑

i=1

e±θi

N−1∑k=1

m∑j=1

eLzkj for n = 3m.

The higher chargesQL = ∫dx J 0

L(x) satisfy again the eigenvalue equations(QL −

n∑i=1

eLθi

)|p1, . . . , pn〉in = 0.

Obviously, fromJ±±1(x) we obtain the energy–momentum tensor.

Examples Up to normalization constants we calculate for the order parametersσ1(x) andσ2(x)

〈0|σ1(0)|θ〉1 = 1,

〈0|σ2(0)|θ1, θ2〉in11 = F(θ12)

sinh12(θ12 − 2πi/N)sinh1

2(θ12 + 2πi/N)

and for the para-Fermi fieldsψQ(x) andψ2(x)

〈0|ψQ(0)|θ〉Q = eQ(N−Q)

Nθ ,

(44)〈0|ψ2(0)|θ1, θ2〉in11 = e(1− 2

N)(θ1+θ2)F (θ12)

sinh12(θ12 − 2πi/N)sinh1

2(θ12 + 2πi/N),

where|θ〉Q denotes a one-particle state of chargeQ and |θ1, θ2〉in11 a state of two particles o

charge 1.


case oftesnerald state

ctor ofion

x

y the

-

5. Commutation rules

5.1. The general formula

Techniques similar to the ones used in this section have been applied for the simplerno bound states and bosonic statistics in[3,27]. A generalization for the case of bound stahas been discussed in[28]. Here we generalize these techniques for the case of more gestatistics and also discuss the contribution of poles related to the double poles of bounS-matrices.2 In order to discuss commutation rules of two fieldsφ(x) and ψ(y) we have touse a general crossing formula for form factors which was derived in[4] (see also[3]). Forquantum field theories with general statistics we introduce assumptions on the statistics faa fieldψ(x) and a particleα. It is easy to see that for consistency of (ii) and (iii) the conditσψ(α)σψ(α) = 1 has to hold ifα is the anti-particle ofα. We assume that

(45)σψ(α) = σψ(Qα)

depends on the charge of the particle such thatσψ(Q + Q′) = σψ(Q)σψ(Q′). A stronger as-sumption (which holds for theZ(N)-model) is the existence of a ‘dual charge’Qψ of the fieldssuch that

(46)σψ(α) = ωQψQα ,

where|ω| = 1.In order to write the following long formulae we introduce a short notation: For a fieldO(x)

and for ordered sets of rapiditiesθ1 > · · · > θn andθ ′1 < · · · < θ ′

m we write the general matrielement ofO(0) as

(47)Oβα (θ ′

β, θα) := out⟨βm(θ ′m), . . . , β1(θp

′1)

∣∣O∣∣α1(θ1), . . . , αn(θn)⟩in

,

whereθα = (θ1, . . . , θn) andθ ′β = (θ ′

1, . . . , θ′m). The array of indicesα = (α1, . . . , αn) denote a

set of particles (αi ∈ types of particles) and correspondingly forβ (we also write|α| = n, etc.).Similar as for form factors this matrix element is given for general order of the rapidities bsymmetry property (i) for both the in- and out-states which takes the general form:

Oβα (θ ′

β, θα) = Sβ

δ (θ ′δ)O

δγ (θ ′

δ, θγ )Sγ

α (θα)

if θγ is a permutation ofθα andθ ′δ a permutation ofθ ′

β . The matrixSγ

α (θα) is defined as the rep

resentation of the permutationπ(θα) = θγ generated by the two-particle S-matricesSγ2γ1α1α2(θ12),

for exampleSγ3γ1γ2α1α2α3(θ1, θ2, θ3) = S

γ3γ1α1λ3

(θ13)Sλ3γ2α2α3(θ23) (cf. [4]).

We consider an arbitrary matrix element of products of fieldsO = φ(x)ψ(y) and O =ψ(y)φ(x). Inserting a complete set of intermediate states|θ γ 〉in

γ we obtain

(48)(φ(x)ψ(y)

)β

α(θ ′

β, θα) = eiP ′

βx−iPαy 1

γ !∫θ γ

φβγ (θ ′

β, θγ )ψγ

α (θγ , θα)e−iPγ (x−y),

2 For bound state form factors there are also higher order ‘physical poles’ (see e.g.[29–32]).


r-andula to

e

t

ofch

s

where φ = φ(0), ψ = ψ(0), Pα = the total momentum of the state|θα〉inα etc. and

∫θ γ

=∏|γ |k=1

∫dθk

4π. Einstein summation convention over all setsγ is assumed. We also defineγ ! =∏

α nα! wherenα is the number of particles of typeα in γ . We apply the general crossing fomula (31) of[4] which is obtained by taking into account the disconnected terms in (ii)iterating that formula. Strictly speaking, we apply the second version of the crossing formthe matrix element ofφ

(49)φβγ (θ ′

β, θγ ) =∑

θ ′ρ∪θ ′

τ =θ ′β

θς∪θ =θ γ

Sβρτ (θ

′ρ, θ ′

τ )φςρ(θς , θ ′ρ − iπ−)Cρρ1τ

σ (θ ′τ , θ)S

ςσγ (θγ ),

whereρ = (ρ|ρ|, . . . , ρ1) with ρ = antiparticle ofρ andθ ′ρ − iπ− means that all rapidities ar

taken asθ ′ − i(π −ε). The matrix1τσ (θ ′

τ , θ) is defined by(47)with O = 1 the unit operator. Thesummation is over all decompositions of the sets of rapiditiesθ ′

β andθ γ . To the matrix elemenof ψ we apply the first version of the crossing formula

(50)

ψγ

α (θγ , θα) = σψ

(γ )

∑θ ν∪θπ=θ γ

θµ∪θλ=θα

Sγ

νπ (θ ν, θπ )1νµ(θ ν, θµ)Cππψπλ(θ π + iπ−, θλ)S

µλγ (θγ ),

where we assume that the statistics factorσψ

(γ ) of the fieldψ with respect to all particles inγ

is the same for allγ which contribute to(48) (see below). Inserting(49) and (50)in (48) we

use the product formulaSςσγ (θγ )S

γ

νπ (θ ν, θπ ) = Sςσνπ (θν, θπ ). Let us first assume that the sets

rapidities in the initial stateθα and the ones of the final stateθ ′β have no common elements whi

implies that alsoθν ∩ θ σ = ∅. Then we may use (ii) to getSςσνπ (θ ν, θπ ) = 1 and we can perform

the θ ν - and θ -integrations. The remainingθ -integration variables areθω = θ ς ∩ θπ , then wemay write for the sets of particlesς = µω,π = ωτ andγ = µωτ and similar for the rapiditieand momenta. Eq.(48)simplifies as(

φ(x)ψ(y))β

α(θ ′

β, θα)

(51)=∑

θ ′ρ∪θ ′

τ =θ ′β

θµ∪θλ=θα

µ!τ !µωτ !S

βρτ (θ

′ρ, θ ′

τ )

∫θω

Xρτ

µλSµλα (θα)ei(P ′

ρ−Pµ)x−i(Pλ−P ′τ )y,

where

Xρτ

µλ = σψ

(γ )φµωρ(θµ, θω, θ ′ρ − iπ−)CρρCτ τ Cωω

(52)× ψτωλ(θ′τ + iπ−, θ ω + iπ−, θλ)e

−iPω(x−y).

The integrandXρτ

µλ may be depicted as


to

shift

f

e

e

Similarly, if we apply for the operator productψ(y)φ(x) again the second crossing formulathe matrix element ofφ and the first one the matrix element ofψ we obtain Eq.(51)whereX

ρτ

µλ

is replaced by

Yρτ

µλ = σψ

(β)φµωρ(θµ, θω − iπ−, θ ′

ρ − iπ−)CρρCτ τ Cωω

(53)× ψτωλ(θ′τ + iπ−, θ ω, θλ)e

iPω(x−y),

which means that onlyσψ

(γ ) is replaced byσψ

(β), Pω by −Pω and the integration variablesθω by

θ ω − iπ−.

No bound states In this case there are no singularities in the physical strip and we mayin the matrix element ofψ(y)φ(x) (51) with (53) for equal times andx1 < y1 the integrationvariables byθi → θi + iπ−. Note that the factoreiPω(x−y) decreases for 0< Reθi < π if x1 < y1.BecausePω → −Pω we get the matrix element ofφ(x)ψ(y) (51) with (52) up to the statisticsfactors. Therefore we conclude

(54)φ(x)ψ(y) = ψ(y)φ(x)σψφ for x1 < y1,

whereσψφ = σψ

(γ )/σ

ψ

(β). Using the assumption(45)we have withQγ = ∑

γ∈γ Qγ

σψ

(γ ) =∏γ∈γ

σψ(γ ) = σψ(Qγ ) = σψ(Qβ − Qφ),

which is the same for allγ , as assumed above. The last equation follows fromQγ = −Qγ

and charge conservation which means that the matrix elementsφβγ in (48) are non-vanishing i

Qβ + Qφ = Qγ . Therefore the statistics factor of the fieldsψ with respect toφ is

(55)σψφ = σψ(Qγ )

σψ(Qβ)= σψ(−Qφ) = 1/σψ(Qφ),

which is in general not symmetric under the exchange ofψ andφ. Finally, we obtain the spaclike commutation rules

(56)φ(x)ψ(y) = ψ(y)φ(x)

1/σψ(Qφ) for x1 < y1,

σ φ(Qψ) for x1 > y1,

where the second relation is obtained from(54) by exchangingφ ↔ ψ andx ↔ y. The sameresult appears when there are bound states. This is proved inAppendix Dwhere also the existencof double poles in bound state S-matrices is taken into account.

5.2. Application to theZ(N)-model

The statistics factors in this model are of the form(46) σψ(α) = ωQψQα whereQψ is the

dual charge of the fieldψ andQα is the charge of the particleα, thereforeσψφ = ω−QψQφ . Thegeneral equal time commutation rule(56) for fieldsψ

QQ(x) defined by(43) in Section4 reads

as

(57)ψQQ

(x)ψRR

(y) = ψRR

(y)ψQQ

(x)

ω−RQ = e−2πiRQ/N for x1 < y1,

QR 2πiQR/N 1 1
ω = e for x > y .


other.

licitFermie

e

ined

Notice that in this model we have a more general anyonic statistics.

Examples

(1) The order parameters have bosonic commutation rules with respect to each other

σQ(x)σQ′(y) = σQ′(y)σQ(x).

(2) The disorder parameters have again bosonic commutation rules with respect to each(3) For the order-disorder parameters we obtain the typical commutation rule

µQ

(x)σQ(y) = σQ(y)µQ

(x)

1 for x1 < y1,

ωQQ = e2πiQQ/N for x1 > y1.

(4) The para-Fermi fields have anyonic commutation rules

(58)ψQ(x)ψR(y) = ψR(y)ψQ(x)eε(x1−y1)2πiQR/N .

These results prove the commutation rules(5) in the introduction.

The 2-point Wightman functionIn order to compare these commutation rules with the expresults of the previous section we calculate the 2-point Wightman function for the para-fieldsψQ andψN−Q (with spin s = Q(N − Q)/N ) in 1-particle (chargeQ) intermediate statapproximation. Using the result(44)we obtain

〈0|ψQ(x)ψN−Q(0)|0〉 =∫

dθ

4π〈0|ψQ(x)|θ〉QQ〈θ |ψN−Q(0)|0〉 + · · ·

= 1

2π

(x− − iε

x+ − iε

)ν/2

Kν

(M

√i(x+ − iε)

√i(x− − iε)

) + · · · ,

whereν = 2Q(N − Q)/N andx± = t ∓ x. This agrees with the commutation rule(58), becausefor t = 0 andx > 0 using the symmetryQ ↔ N − Q, x → −x and translation invariance wobtain

〈0|ψQ(x)ψN−Q(0)|0〉 = 〈0|ψN−Q(x)ψQ(0)|0〉= eiπν〈0|ψN−Q(−x)ψQ(0)|0〉= eiπν〈0|ψN−Q(0)ψQ(x)|0〉,

where((x − iε)/(−x − iε))ν/2 = eiπνε(x)/2 has bee used. The asymptotic behavior is obtafrom

2Kν(z) →

(ν)(

z2

)−ν + (−ν)(

z2

)ν for z → 0,√2πz

e−z for z → ∞,

for ν = 0. Therefore the leading short distance behavior is up to constants

〈0|ψQ(x)ψN−Q(0)|0〉 ∼(x+ − iε

)−ν,

〈0|ψQ(x)ψN−Q(0)|0〉 ∼(x− − iε

)−ν,

where the fieldsψQ(x) are obtained by changing the sign in the exponent of(43).


chner,we

d-metrieparteoryntracte Tec-and

inimaltwo

te

m the

Acknowledgements

We thank V.A. Fateev, R. Flume, A. Fring, A. Nersesyan, R. Schrader, B. Schroer, J. TesA. Tsvelik, Al.B. Zamolodchikov and A.B. Zamolodchikov for discussions. In particularthank V.A. Fateev for bringing the preprint[18] to our attention and F.A. Smirnov for sening a copy. H.B. was supported by DFG, Sonderforschungsbereich 288 ‘Differentialgeound Quantenphysik’, partially by the grants INTAS 99-01459 and INTAS 00-561 and inby Volkswagenstiftung within in the project “Nonperturbative aspects of quantum field thin various space–time dimensions”. A.F. acknowledges support from PRONEX under coCNPq 66.2002/1998-99 and CNPq (Conselho Nacional de Desenvolvimento Científiconológico). This work is also supported by the EU network EUCLID, ‘Integrable modelsapplications: from strings to condensed matter’, HPRN-CT-2002-00325.

Appendix A. Some useful formulae

In this appendix we provide some explicit formulae for the scattering matrices and the mform factors which we frequently employ in the explicit computations. The S-matrix of‘fundamental’ particles (i.e. of type 1) is[9]

S(θ) = sinh12(θ + 2πi

N)

sinh12(θ − 2πi

N)

= −exp

∞∫0

dt

t2

sinht (1− 2/N)

sinhtsinhtx,

whereθ is the rapidity difference defined by

p1p2 = M2 coshθ.

A particle of typeα (0 < α < N ) is a bound stateα = (α1 · · ·αl) of particles of typeαi whereα = α1 + · · · + αl , in particularα = (1 · · ·1︸︷︷︸

α

) for all αi = 1. For the scattering of the bound sta

α andβ we have[17]

Sαβ(θ) = exp2

∞∫0

dx

x

coshx(1− |β−α|N

) − coshx(1− β+αN

)

sinhx tanh(x/N)sinhx

θ

iπ.

The minimal form factor functions, which satisfies Watson’s equations, are obtained froS-matrix formulae[1] and are given as (β > α)

F minαβ (θ) = exp

∞∫0

dt

t2

sinht (1− βN

)sinht ( αN

)

sinh2 t tanht/N

(1− cosht

(1− θ

iπ

))

in particular[17]

F min11 (θ) = exp

∞∫0

dt

t2

sinht (1− 1N

)cosht 1N

sinh2 t

(1− cosht

(1− θ

iπ

))

= −i sinh1

2θ exp

∞∫dt

t

sinht (1− 2N

)

sinh2 t

(1− cosht

(1− θ

iπ

))
0


in our

.

= −i sinh1

2θ

∞∏k=0

(k + 1− 1N

+ x2)

(k + 1N

+ x2)

(k + 2− 1N

− x2)

(k + 1+ 1N

− x2)

((k + 1

2 + 1N

)

(k + 32 − 1

N)

)2

and with1= (N − 1)

F min11

(θ) = exp

∞∫0

dt

t

sinht 2N

sinh2 t

(1− cosht

(1− θ

iπ

))

=∞∏

k=0

(k + 12 + 1

N+ θ

2πi)

(k + 12 − 1

N+ θ

2πi)

(k + 32 + 1

N− θ

2πi)

(k + 32 − 1

N− θ

2πi)

((k + 1− 1

N)

(k + 1+ 1N

)

)2

.

There are simple relations between the minimal form factors which we essentially useconstruction which are up to constants

F min11

(θ + iπ

N

)F min

11

(θ − iπ

N

)∝ sinh

1

2

(θ + iπ

N

)sinh

1

2

(θ − iπ

N

)F min

12 (θ)

N−1∏k=0

F min11

(θ + k

N2πi

)∝

N−2∏k=0

sinh1

2

(θ + k

N2πi

)sinh

1

2

(θ + k + 1

N2πi

)

F min11 (θ)F min

11(θ + iπ) ∝ sinh

1

2θ sinh

1

2(θ + 2iπ/N).

In Eqs.(20) and (33)we used the functionF(θ) = cNF min11 (θ) with

(A.1)cN = eiπ N−1N exp

( ∞∫0

dt

t sinht

((1− 2

N

)− sinht (1− 2

N)

sinht

)),

such that the normalizations in(37) and (40)hold.

Appendix B. Integrals for the Z(2)-model

The claim(15) follows from the following lemma

Lemma 3. For n = 2m + 1 odd andxi = eθi

fn(x) := Inm(θ,1) − (2i)(n−1)/2∏

1i<jn

tanh12θij

F (θij )= 0.

Proof. Again as in the proof of Lemma 2 in[33] we apply induction and Liouville’s theoremOne easily verifiesf1(x) = f3(x) = 0. As induction assumptions we takefn−2 = 0. The func-tionsfn(x) are a meromorphic functions in terms of thexi with at most simple poles atxi = −xj

since pinchings appear forzk = θi = θj ± iπ . The residues of the poles are proportional tofn−2

as follows from the recursion relations (iii) for both terms. Furthermorefn(x) → 0 for xi → ∞.Thereforefn(x) vanishes identically by Liouville’s theorem.


t

Note that the integrations in the definition(9) of Inm can easily be performed and withN =1, . . . , n and|K| = m

Inm(θ,1) =∑K⊂N

∏k∈K

∏i∈N \K

2i

sinhθki

.

Appendix C. Proof of the main lemma

In this appendix we prove the mainLemma 2which provides the generalZ(N)-form factorformula.

Proof. Similar as in the proof ofLemma 1we calculate

ResθN−1N=iη

. . . Resθ12=iη

Inm(θ,pOnm)

= m1 · · ·mN−1

m1! · · ·mN−1!

(N−1∏k=1

mk∏j=2

∫Cθ ′

dzkj

R

)

×N−1∏k=1

(n∏

i=N+1

mk∏j=2

φ(zkj − θi)∏

2i<jmk

τ(zki − zkj )

)

×∏

1k<lN−1

mk∏i=2

ml∏j=2

(zki − zlj )

(N−1∏k=1

N∏i=1

m∏j=2

φ(zkj − θi)

)r

with

r = ResθN−1N=iη

. . . Resθ12=iη

(N−1∏k=1

∫Cθ

dzk1

)N−1∏k=1

(n∏

i=1

φ(zk1 − θi)∏

2jm

τ(zk1 − zkj )

)

×∏

1k<lN−1

((zk1 − zl1)

mk∏i=2

(zki − zl1)

ml∏j=2

(zk1 − zlj )

)pO

nm(θ, z).

ReplacingCθ by Cθ ′ where θ ′ = (θN+1, . . . , θn) we have usedτ(0) = τ(±iη) = (0) =(−iη) = 0 and the fact that thezk1-integrations give non-vanishing results only forzk1 = θk

and θk+1, k = 1, . . . ,N − 1. This is because forθ12, . . . , θN−1N → iη pinching appears a(z11, . . . , zN−11) = (θ2, . . . , θN) and(θ1, . . . , θN−1). Defining the function

f (z11, . . . , zN−11) =N−1∏k=1

∏2jmk

τ(zk1 − zkj )∏

1k<lN−1

(zk1 − zl1)

×∏

1k<lN−1

(ml∏

j=2

(zk1 − zli)

mk∏i=2

(zki − zl1)

)pO

nm(θ, z)

one obtains by means of(38)after some lengthy but straightforward calculation


es

i-

tl

discussed

r = ResθN−1N=iη

. . . Resθ12=iη

(N−1∏k=1

∫Cθ

dzk1

R

)N−1∏k=1

(n∏

i=1

φ(zk1 − θi)

)f (z11, . . . , zN−11)

=(

Resθ=iη

φ(−θ)

N−2∏k=1

φ(kiη)

)N−1(N−1∏k=1

n∏i=N+1

φ(θk+1i )

)

×(

f (θ2, . . . , θN) −(

n∏i=N+1

φ(θ1i )

φ(θNi)

)f (θ1, . . . , θN−1)

).

It has been used thatf (. . . , z, . . . , z . . .) = f (. . . , z, . . . , z − iη . . .) = 0 because of(0) =(−iη) = 0. Using further the defining relation ofφ in terms ofF (37), the Jost property(27)oftheφ-function and the properties (iii′) of (41) for thep-function we get

ResθN−1N=iη

. . . Resθ12=iη

Inm

(θ,pO

nm

)

= (Resθ=iη

φ(−θ))N−1

N−2∏k=1

φk(kiη)

× In−Nm−1(θ ′,pO

n−Nm−1

)( N∏k=1

n∏i=N+1

F(θki)

)−1(1− σO

1

n∏i=N+1

S(θNi)

),

which together with the relation for the normalization constants(39)proves the claim. Appendix D. Proof of the commutation rules

In this appendix we prove that we find the same commutation rules for two fieldsφ(x) andψ(y) when there are bound states poles or even when the S-matrix has double poles.3

Bound states We now show that the same result(54) appears when there are bound stat4

which means that there are poles in the physical strip. Letγ = (αβ) be a bound state ofα andβ with fusion angleηγ

αβ which means that atθαβ = iηγαβ the S-matrixSαβ(θ) has a pole. The

momentum and the rapidity of the bound state are

pγ = pα + pβ,

θγ = θα − i(π − η

βγ α

) = θβ + i(π − ηα

βγ

),

whereηβγ α andηα

βγ are the fusion angles of the bound statesβ = (γ α) and α = (βγ ), respec-tively.

We start matrix element ofψ(y)φ(x) (given by(51) with (53)). First we consider the contrbution in the sum over the intermediate states whereα ∈ ω andβ ∈ λ. All the particles which arenot essential for this discussion will be suppressed. Then the functionψαβ(θα, θβ) has a pole aθα − θβ = iη

γαβ such that by shifting the integrationθα → θα + iπ− there will be the additiona

3 These poles appear typically for bound state–bound state scattering. The case of higher order poles may besimilarly and will be published elsewhere.

4 Here we follow the arguments of Quella[28].


n

rgument

, theyer theh

t

t

contribution

i

2Res

θα=θα

φα(θα − iπ−)Cααψαβ(θα, θβ)eiPα(x−y)e−iyPβ

= i

2φα(θα − iπ−)Cααψγ (θγ )

√2Γ

γαβeixPα−iyPγ

with θα = θβ + iηγαβ , θγ = θβ + i(π − ηα

βγ ) and the fusion intertwinerΓ γαβ (see e.g.[33]). Next

we consider the contribution to the sum over the intermediate states whereγ ∈ ω andβ = µ.

Then the functionφβγ (θβ, θγ − iπ) has a pole atθβ − θγ + iπ = iηαβγ such that by shifting the

integrationθγ → θγ + iπ− there will be the additional contribution

i

2Res

θγ =θγ

φβγ (θβ, θγ − iπ−)Cγ γ ψγ (θγ )eiPγ (x−y)e−ixPβ

= − i

2φα(θα − iπ−)

√2Γ α

βγ Cγ γ ψγ (θγ )eixPα−iyPγ

with θα, θγ as above and the fusion intertwinerΓ αβγ . From the crossing relation of the fusio

intertwiners

CααΓγαβ = Γ α

βγ Cγ γ

we conclude that these residue terms form bound state poles cancel. The steps of the amay be depicted as

Double poles5 Form factors have additional poles which are not related to bound statesbelong to higher poles of the S-matrix. First we consider the contribution to the sum ovintermediate states where the particles1,1 ∈ ω, 2 ∈ µ. Again we suppress all particles whic

are not essential for our discussion. Then the functionφ211(θ, θ − iπ, θ ′ − iπ) has a pole aθ = θ1 = θ + iπ/N which correspond to the bound state(21) = 1 with the fusion angleη1

21=

π(1− 1/N). We shift the integrationsθ → θ + iπ− andθ ′ → θ ′ + iπ− such that during the shif0< Im(θ − θ ′) < ε. From theθ -integration there will be the additional contribution

i

2Resθ=θ1

φ211(θ, θ − iπ, θ ′ − iπ)C1111ψ11(θ′, θ )eiP (x−y)e−ixP

= − i

2φ11(θ − iπ/N, θ ′ − iπ)

√2Γ 11

2 C1111ψ11(θ′, θ1)e

iP (x−y)e−ixP .

Further the functionφ11(θ − iπ/N, θ ′ − iπ) has a pole atθ ′ = θ2 = θ + iπ(1−3/N) which cor-respond to the bound state(11) = 2 with the fusion angleη2

11 = π2/N . From theθ ′-integration

5 This discussion is new.


ediate

factoris

there will be the additional contribution(− i

2

)2√2 Res

θ ′=θ2

φ11(θ − iπ/N, θ ′ − iπ)√

2Γ 112 C1111ψ11(θ

′, θ1)eiP (x−y)e−ixP

= −1

2φ2(θ − iπ2/N)Γ 2

11Γ112 C1111ψ11(θ2, θ1)e

i(P1+P2)(x−y)e−ixP .

This procedure may be depicted as

We show this the additional term is cancelled by a contribution to the sum over the intermstates where2 ∈ ω, 2∈ λ. Then the functionψ22(θ , θ) has a pole atθ = θ3 = θ + iπ(1− 2/N)

which correspond to the double pole of the S-matrix

S22(θ) =(

sin π2 ( θ

iπ+ N−2

N)

sin π2 ( θ

iπ− N−2

N)

)2 sin π2 ( θ

iπ+ N−4

N)

sin π2 ( θ

iπ− N−4

N)

at θ = iπ(1− 2/N). From theθ -integration there will be the additional contribution

i

2Resθ=θ3

φ2(θ − iπ)C22ψ22(θ , θ)eiP (x−y)e−iyP

= i

2iφ2(θ − iπ2/N)C22(−i)

(ψ11(θ2, θ1)Γ

112

C11Γ112

)eiP3(x−y)e−iyP .

This procedure may be depicted as

The crossing relation of the fusion intertwiners

Γ 211Γ

112 C11 = C22Γ 11

2C11Γ

112

implies that this contribution again cancelled the one above. It has been used that the formof bound states22 has a simple pole where the S-matrixS22 has a double pole and the residue

Resθ=θ3

ψ22(θ , θ) = −i(ψ11(θ2, θ3)Γ

112

C11Γ112

).

This may be calculated as follows. By the form factor equation (iv) we have

Resθ12=iη2

11

Resθ34=iη2

11

ψ1111(θ1, θ2, θ3, θ4) = 2ψ22(θ(12), θ(34))Γ211

Γ 211.


ners

iagram

nitialting

eation(ii) and

ics index

Therefore using the form factor equation (iii) and the definition of the fusion intertwiResiS = Γ Γ we obtain

Resθ(12)(34)=iπ(1−2/N)

ψ22(θ(12), θ(34))Γ211

Γ 211

= 1

2Res

θ14=iπRes

θ12=iη211

Resθ34=iη2

11

ψ1111(θ2, θ1, θ4, θ3)S11(θ12)S11(θ34)

= 1

2Res

θ14=iπRes

θ12=iη211

2iC11

(ψ11(θ2, θ3)S11(θ12)S11(θ34) − ψ11(θ2, θ3)

)= iC11

(ψ11(θ2, θ3)(−i)Γ 11

2Γ 2

11(−i)Γ 11

2 Γ 211

),

which implies the residue formula used above. This procedure may be depicted as

Note that the last graph, as an on-shell graph, resembles (half of) the ‘box’ Feynman dwhich was used to investigate the double poles of bound state S-matrices (see e.g.[29,30]).

The general caseFinally we consider the general case that the sets of rapidities in the istateθα and the ones of the final stateθ ′

β have also common elements. Then after inser

(49) and (50)in (48) there will be S-matricesSςσνπ (θ ν, θπ ) which produce additional poles in th

physical strip which would produce additional residue contributions while shifting the integrcontours. However, we can remove these S-matrices by using again the crossing relationmove all the lines of common rapidities to the left or right as depicted as follows

Then we can apply the procedure as above.

References

[1] M. Karowski, P. Weisz, Nucl. Phys. B 139 (1978) 455.[2] H.M. Babujian, A. Fring, M. Karowski, A. Zapletal, Nucl. Phys. B 538 (1999) 535.[3] F. Smirnov, Advanced Series in Mathematical Physics, vol. 14, World Scientific, 1992.[4] H. Babujian, M. Karowski, Nucl. Phys. B 620 (2002) 407.[5] M. Karowski, Nucl. Phys. B 153 (1979) 244.[6] K.M. Watson, Phys. Rev. 95 (1954) 228.[7] H.M. Babujian, in: Gosen 1990, Proceedings, Theory of elementary particles, 12–23 (see high energy phys

29 (1991) No. 12257).[8] H. Babujian, J. Phys. A 26 (1993) 6981.[9] R. Köberle, J.A. Swieca, Phys. Lett. B 86 (1979) 209.


995, p.

Press,

[10] R.Z. Bariev, Phys. Lett. A 55 (1976) 456.[11] B.M. McCoy, C.A. Tracy, T.T. Wu, Phys. Rev. Lett. 38 (1977) 793.[12] M. Sato, T. Miwa, M. Jimbo, Proc. Jpn. Acad. 53 (1977) 6.[13] B. Berg, M. Karowski, P. Weisz, Phys. Rev. D 19 (1979) 2477.[14] A.B. Zamolodchikov, Int. J. Mod. Phys. A 3 (1988) 743.[15] V.A. Fateev, Int. J. Mod. Phys. A 6 (1991) 2109.[16] V.A. Fateev, A.B. Zamolodchikov, Sov. Phys. JETP 62 (1985) 215.[17] M. Karowski, in: Lecture Notes in Physics, vol. 126, Springer, 1979, p. 344.[18] A.N. Kirillov, F.A. Smirnov, ITF preprint 88-73P, Kiev, 1988.[19] G. Delfino, J.L. Cardy, Nucl. Phys. B 519 (1998) 551.[20] M. Jimbo, H. Konno, S. Odake, Y. Pugai, J. Shiraishi, J. Stat. Phys. 102 (2001) 883.[21] F.H.L. Essler, R.M. Konik, in: M. Shifman, et al. (Eds.), From Fields to Strings, vol. 1, 2004, pp. 684–830.[22] A.M. Tsvelik, Quantum Field Theory in Condensed Matter Physics, Cambridge Univ. Press, Cambridge, 1

332.[23] A.O. Gogolin, A.A. Nersesyan, A.M. Tsvelik, Bosonization in Strongly Correlated Systems, Cambridge Univ.

Cambridge, 1999.[24] J. Links, H. Zhou, R. McKenzie, M. Gould, J. Phys. A 36 (2003) R63.[25] H. Babujian, M. Karowski, Phys. Lett. B 575 (2003) 144.[26] H. Babujian, M. Karowski, J. Phys. A 35 (2002) 9081.[27] M.Y. Lashkevich, LANDAU-94-TMP-4, 1994, unpublished.[28] T. Quella, Diploma thesis FU-Berlin, 1999, unpublished.[29] S.R. Coleman, H.J. Thun, Commun. Math. Phys. 61 (1978) 31.[30] H.W. Braden, E. Corrigan, P.E. Dorey, R. Sasaki, Nucl. Phys. B 338 (1990) 689.[31] G. Delfino, G. Mussardo, Nucl. Phys. B 455 (1995) 724.[32] C. Acerbi, G. Mussardo, A. Valleriani, J. Phys. A 30 (1997) 2895.[33] H. Babujian, M. Karowski, Phys. Lett. B 471 (1999) 53.

s, the

lege of thenction.le

--low

sed


Double scaling and finite size correctionsin sl(2) spin chain

Nikolay Gromova,b,∗, Vladimir Kazakova,1

a Laboratoire de Physique Théorique de l’Ecole Normale Supérieure et l’Université Paris VI, 75231 Paris, Franceb St. Petersburg INP, Gatchina, 188 300 St. Petersburg, Russia

Received 15 November 2005; accepted 7 December 2005


Abstract

We find explicit expressions for two first finite size corrections to the distribution of Bethe rootasymptotics of energy and high conserved charges in thesl(2) quantum Heisenberg spin chain of lengthJ

in the thermodynamical limitJ → ∞ for low lying states with energiesE ∼ 1/J . This limit was recentlystudied in the context of integrability in perturbativeN = 4 super-Yang–Mills theory. We applied the doubscaling technique to Baxter equation, similarly to the one used for large random matrices near the edeigenvalue distribution. The positions of Bethe roots are described near the edge by zeros of Airy fuOur method can be generalized to any order in 1/J . It should also work for other quantum integrabmodels. 2005 Elsevier B.V. All rights reserved.

1. Introduction

We study in this paper the integrable periodic HeisenbergXXXs chain of noncompact quantum spins transforming under the representations = −1/2 of sl(2), in the so-called thermodynamical limit of largeJ , whereJ is the length of the chain, in the ferromagnetic regime ofenergiesE ∼ 1/J .2

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

[email protected](V. Kazakov).1 Membre de l’Institut Universitaire de France.2 It is different from a more traditional regimeE ∼ J widely studied since many years, especially in the conden

matter literature.







200 N. Gromov, V. Kazakov / Nuclear Physics B 736 [FS] (2006) 199–224

ions

rys) is ofserved

ns

ase.

onerlying

of athe

alledtical

as first

The problem is known to be solvable by the Bethe ansatz approach (see for example[1]) andthe energy of a state ofS magnons in dimensionless units is given by a simple formula

(1)E =S∑

k=1

1

u2k + 1/4

,

where the Bethe rootsuj , j = 1,2, . . . , S, parametrizing the momenta of magnons, are solutof a system of polynomial Bethe ansatz equations (BAE)

(2)−(

uj − i/2

uj + i/2

)J

=S∏

k=1

uj − uk + i

uj − uk − i, j = 1, . . . , S.

It can be proven that for this model the roots are always real.In the thermodynamical limit we will also consider a large number of magnonsS ∼ J . It is

clear then from Eq.(1) that in order to focus on the low lying states with energiesE ∼ 1/J weshould take the characteristic Bethe roots of the orderuj ∼ J . It means that the chain is velong and the spins are rarely changing along it. The typical length of spin-waves (magnonthe order of the lengthJ . Our goal is to study the limitingJ → ∞ distributions of Bethe rootand the finite volume 1/J corrections to these distributions, to the energy and higher conscharges. In the main order this thermodynamical limit for the compact HeisenbergXXX1/2 chainof su(2) spins was already considered in[2], and later in[3] in relation to the integrable dilatatioHamiltonian in planar perturbative superconformalN = 4 super-Yang–Mills (SYM) theory. Itdescription and the general solution in terms of algebraic curves was proposed in[4] for thesu(2)

case3 and in[6,7] for thesl(2) chain. We will concentrate in the current paper on this last cGeneralization to thesu(2) case is straightforward.

The study of 1/J corrections in these systems was started recently in the papers[8,9] for thesimplest single support, or one cut distribution, whereas a similar quantumh correction to theclassical KdV solitons was already found earlier in the general multi-cut case in[10].

The main results of our paper are:

(1) The explicit formulas for the 1/J and 1/J 2 corrections to the general multi-cut distributiof Bethe roots and to the corresponding energy of a Bethe state in terms of the undalgebraic curve.

(2) The universal description of the distribution of Bethe roots in the vicinity of an edgesupport in terms of zeroes of the Airy function, similar to the double scaling limit inmatrix models.

(3) Asymptotics of conserved local chargesQn(S,J ) in the largen limit.(4) Asymptotics of conserved global (non-local) charges of high order.

Unlike the papers[8,9] using the method of singular integral equation corrected by so-canomaly term,4 we will use here the exact Baxter equation written directly for the analyfunction—the resolvent of the root distribution (similar approach was used in[6]). This equationis valid before any limit.

These results might be interesting for different kinds of specialists.

3 Following a similar approach of[5] to a somewhat different limit of large spin.4 This phenomenon of anomaly, or the contribution of close eigenvalues in the thermodynamical limit of BAE w

observed in[11].

N. Gromov, V. Kazakov / Nuclear Physics B 736 [FS] (2006) 199–224 201

Inhough

ouble

multi-m 2D

spon-

rrec-

-ifferentlreadyroots

duality.f anom-t alsoamical

e two

e

ion.d the

chargesell as

rof

t

First, for those who are studying largeN random matrix or random partition ensembles.particular, the distributions of eigenvalues in matrix models are described by similar (altnot the same) algebraic curves. The 1/J corrections remind the 1/N corrections, and the Airyedge distribution observed in our work is known to describe the edge behaviour in the dscaled matrix integrals as well[12]. No doubt that by choosing particular solutions of Eq.(2) ormodifying its l.h.s. (by considering inhomogeneous chains) we can also find here variouscritical phenomena similar to those found in the matrix models in the context of quantugravity applications[13].

Second, it might be interesting to those who work on various aspects of AdS/CFT corredence in supersymmetric string and gauge theories and in particular on the integrability inN = 4SYM theory (see[14] and references therein). They might shed some light on quantum cotions to the classical limit of the AdS dual of SYM, the superstring on theAdS5×S5 background,now known only for particular classical solitonic solutions[15] of rotating strings. AdS/CFT correspondence should manifest itself as the coincidence of such corrections in these so dintegrable systems. Their similarity, and even the coincidence in a certain regime, was aobserved on particular string and chain solutions, having only one support for the Bethedistribution[8,9,16–19,42,43]. 1/J corrections were first studied for BMN states in[40], wherethe integrable spin chain forN = 4 SYM was first proposed, and then in[41]. The Airy edgebehavior also seems to be universal enough to manifest itself on both sides of AdS/CFTThis behaviour observed in the present paper for spin chains describing the spectrum oalous dimensions in the perturbative SYM in thermodynamical limit, is natural to expecin the quantized string theory where the classical limit, the analogue of the thermodynlimit, is described by similar algebraic curves obtained by the finite gap method.5 The universaledge behaviour might be completely driven by the similarity of integrable structures of thesseemingly different systems.

The paper is organized as follows. In Section2 we summarize the explicit formulas for thHamiltonian and the local and non-local conserved charges of the model. In Section3 we reviewthe general solution of the model in thermodynamical limit whenJ → ∞ andE ∼ 1/J . In Sec-tion 4 we calculate the first 1/J correction to the general multi-cut solution from Baxter equatIn Section5 we apply the method of the double scaling limit to the Baxter equation and finnear branch point behavior. In Section6 we combine the results of Sections4 and 5to find theexplicit formula for the second, 1/J 2 correction. In Section7 we give the formulas for 1/J cor-rections to the energy and compute the asymptotics of high conserved local and non-localin all orders in 1/J . Conclusions are devoted to unsolved problems and perspectives, as wto the discussion of parallels with the matrix models. InAppendix Awe write explicit formulasfor 1/J expansion of BAE and inAppendix Bwe express 1/J 2 corrections to the energy foone-cut solution through certain double sums in mode numbers, generalizing the results[8].

2. Hamiltonian, transfer-matrix and higher charges of sl(2) chain

The Hamiltonian of interaction of the neighboring spinssl, sl+1 can be written in an expliciway [14]

(3)H−1/2 =J∑

l=1

Hl,l+1−1/2

5 See[11,20] for this approach on both sides of duality.


local

o-

ndharge

nn

with the Hamiltonian density

(4)Hl,l+1−1/2 |k,m − k〉 =

m∑k′=0

(δk=k′

(h(k) + h(m − k)

) − δk =k′

|k − k′|)

|k′,m − k′〉,

where|k1, . . . , kl, kl+1, . . . , kJ 〉 is a state vector labeled byJ integerskj (s = −1/2 spin compo-nents) andh(k) = ∑k

j=11j

are harmonic numbers.

The total momentumP(u)

(5)eiP (uj ) = uj − i/2

uj + i/2

satisfies the (quasi-)periodicity condition following directly from Eq.(2)

(6)Ptot =S∑

j=1

P(uj ) = 2πk/J, k ∈ Z.

In some applications, such as the spectrum of anomalous dimensions of operators6 in N = 4SYM theory, we select only purely periodic Bethe states

(7)Ptot = 2πm, m ∈ Z.

We can also study other physically interesting quantities of this model, such as theconserved chargesQr . They are defined as follows

(8)T (v) = exp

(i

∞∑r=1

Qrvr−1

),

where the quantum transfer matrixT (v) ≡ T (v;0,0, . . . ,0) is a particular case of the inhomgeneous transfer matrix

(9)T (v;v1, . . . , vJ ) = Tr0[R0,1(v − v1) · · · R0,J (v − vJ )

]andR0,j is the universalsl(2) R-matrix defined as[21]

(10)R0,1(v) =∞∑

j=0

Rj (v)P(j)

0,1, Rj (v) =j∏

k=1

v − ik

v + ik

with P(j)

01 being the operator projecting the direct product of two neighboring spinss0 = s1 =−1/2 to the representationj . Recall that

(11)[T (v;v1, . . . , vJ ), T (v′;v1, . . . , vJ )

] = 0

for any pairv, v′, due to Yang–Baxter equations on theR-matrix.The direct calculation shows thatPtot = −Q1 is the operator of the momentum, such that a

H−1/2 = Q2 is the Hamiltonian Eq.(3), etc. Those charges are local, in the sense that the cdensity ofQk contains k consecutive spins.

6 The operators of the type Tr(∇k1Z · · ·∇kJ Z) in SYM, where∇ = ∂ + A is a covariant derivative in a null directioandZ is a complex scalar, represent the state vectors|k1, . . . , kl , kl+1, . . . , kJ 〉 and the dilatation Hamiltonian is giveat one loop by theXXX−1/2 Hamiltonian.


dary

eir

s. The

itions

t

on

tion

There is a more efficient way to generate higher charges than using Eq.(8). We can use therecurrence relation (see[14])

(12)rQr+1 = [B,Qr ],whereB = i

∑Jl=1 lH

l,l+1−1/2 is the boost operator. The last formula is true up to some boun

terms destroying the periodicity, which should be dropped.Due to the integrability manifestly expressed by Eq.(11)all these charges commute and th

eigenvalues on a Bethe state characterized by a set of Bethe roots satisfying Eq.(2) (enforcingthe periodicity of the chain or the quasi-periodicity of the Bethe state) are given by[22]

(13)Qr =S∑

j=1

i

r − 1

(1

(uj + i/2)r−1− 1

(uj − i/2)r−1

).

We can also study the non-local charges. They can be defined in many different waydefinition could be similar to Eq.(8), but the expansion ofT (u) goes aroundu = ∞. However,the most natural charges are defined through the resolvent of Bethe roots

(14)G(x) =S∑

k=1

1

xJ − uk

=∞∑

n=1

dnx−n.

Thej th charge is thej th symmetric polynomial of Bethe roots

(15)dn = 1

J

S∑k=1

(uk

J

)n−1

, dn ≡∞∑

k=0

dk,nJ−k.

We will later estimate the behavior ofdk,n at n → ∞ and high orders of 1/J expansion. Thisasymptotics will be universal and a few leading terms of it will be the same for various definof non-local charges.

3. 1/J expansion of BAE

Let us start from reviewing the “old” method of solving Eq.(2) in the thermodynamical limiJ → ∞, uk ∼ J ∼ S, before sticking with the most efficient one using the Baxter equation.

As we mentioned in the introduction Eq.(2) has only real solutions, i.e., all the roots liethe real axis. We label the roots so thatuj+1 > uj . Suppose there exists a smooth functionX(x)

parametrizing the Bethe roots

(16)uk = JX(k/J ), (X(x)

) ≡ 1

X′(x) 1

uk+1 − uk

.

For largeS the function(x) has a meaning of density of Bethe roots. As follows from defini(16) its normalization is

(17)∫

dx (x) = α


d

s

aurents

lous

anom-ce

z.n-ion

e

with α = S/J . Taking log of both parts of Eq.(2) we have7

(18)2πimj + J loguj − i/2

uj + i/2= −

S∑′

k=1

2i arctan(uj − uk),

wheremj are strictly ordered integersmj+1 > mj . In terms of the logarithm we can write

(19)2πinj + J loguj − i/2

uj + i/2=

S∑′

k=1

loguj − uk + i

uj − uk − i,

wherenj = mj − j + S+12 are non-decreasing integers.8

Now we have instead of “fermionic”mj ’s the “bosonic”nj ’s which are simply orderenj+1 nj and different Bethe roots can have the same magnon numbersnj = nj+1 = · · · . In thethermodynamical limit we can rewrite Eq.(19) assumingk to be far from the edges, as follow(see alsoAppendix A)∑

j

′i log

(uj − uk + i

uj − uk − i

)

= −2∑j

′ 1

uj − uk

+ 2

3

∑j

′ 1

(uj − uk)3− 2

5

∑j

′ 1

(uj − uk)5+ 2

7

∑j

′ 1

(uj − uk)7

+ π′[coth(π)]6J

− 1

12J 3

((π′)3

[coth(π)

sinh2(π)

]2− 2π2′′′

[1

sinh(π)

]3

(20)+ π(3)[coth(π)

]4

)+O

(1

J 5

),

where we introduce the notation defined by[f ()]n ≡ f () − ∑n−1i=0 f (i)(0)

i

i! for the functionsregular at zero. For singular functions the Taylor series should be substituted by the Lseries so that[f ()]n is zero for = 0 and has firstn−1 zero derivatives at this point. The termin the second line represent the naive expansion of the l.h.s. in 1/(uj − uk). It works well for theterms in the sum withuj uk . The terms in the third and fourth lines describe the anomacontribution atuj ∼ uk , for close roots withi ∼ j . In this case we can expand

(21)uj − uk = j − k

(uj/J )+O(1/J )

and calculate the corresponding converging sum giving the terms in the second line. Thisaly was noticed in the Bethe ansatz context in[11] although this phenomenon was known sinlong in the large N matrix integrals or similar character expansions[23,24]. It was proven in[11]to happily cancel in the main order of 1/J , even for a more complicated nested Bethe ansat

In our case whenJ → ∞ it is obvious from Eqs.(20), (19)that the anomaly does not cotribute to the main order and the Bethe ansatz equation becomes a singular integral equat[2,3]

(22)2πnk − 1

x= 2

∫Ctot

dy 0(y)

x − y, x ∈ Ck, k = 1, . . . ,K.

7 Note that i2 log x+ix−i

= arctan(x)− π2 sign(x) for standard definition of the log, i.e., log(x) = − log(x − i0) for x < 0.

8 This fact is obvious whenS J , for J → ∞, when we can neglect the r.h.s. of Eq.(19). We believe that this is trualso in general, although this fact is irrelevant for the rest of the paper.


tum

wo

otstic

note

Fig. 1. Hyperelliptic Riemann surface.

Now we introduce the resolvent and the quasi-momentum

(23)G(x) ≡∑j

1

xJ − uj

, p(x) ≡ G(x) + 1

2x

as well as the standard definition of density9

(24)ρ(x) = 1

2πi

(p(x − iε) − p(x + iε)

)and rewrite it in terms of the following Riemann–Hilbert problem

(25)2/p(x) = 2πnk, x ∈ Ck, nk ∈ Z,

where/p(x) = 12[p(x+ i0)+p(x− i0)] denotes the symmetric (real) part of the quasi-momen

on a cutCk , k = 1, . . . ,K .To solve it[7] we notice thatp(x) is a function on a hyperelliptic Riemann surface with t

sheets related byK cuts along the real axis, as shown onFig. 1. It has a known behaviorp(x) P0 + O(x) at x = 0, andp(x) ∼ S/J+1/2

x+ O(1/x2) at x → ∞ on the first sheet. There are n

other singularities. However, as we see from Eq.(25)p(x) is not a single valued function but iderivative is. This information is enough to fixp′(x) as a single-valued function on hyperellipRiemann surfacef 2 = ∏2K

j=1(x − xj )

(26)p′(x) = 1

f (x)

K−1∑k=−1

akxk−1.

The single-valuedness imposes∮Al

dp = 0, l = 1, . . . ,K − 1, whereAl , l = 1, . . . ,K , are theA-cycles each surrounding a cutCl .

The BAE Eq.(25)become the integerB-period conditions

(27)∮Bj

dp = 2πnj , nj = 1, . . . ,K,

where the cycleBj starts atx = ∞+ on the upper sheet, goes through the cutCJ to the lowersheet and ends up at its infinityx = ∞−.

9 These two densitiesρ(x) and(x) are clearly different. By definition(x) is a smooth function, whereasρ(x) is asum ofδ-functions. However their 1/J expansion is the same at least for the first two orders. It is also important tothat to obtain the 1/J expansion ofρ(x) one should first expand the r.h.s. of Eq.(24) in powers of 1/J and then take thelimit ε → 0.


s

d byle ma-he

on

ning a

Fig. 2. Density of roots. The dots correspond to numerical 3-cut solution with total number of Bethe rootsS = 300 andequal fractionsαi = 1/6, andni = −1,3,1. They are fixed from the numerical values of the roots by Eq.(16). Solidline is the density atJ = ∞ computed analytically from the corresponding hyper-elliptic curve.x coordinates of the dot

areuj +uj+1

2Jso that the solitary points in the middle of empty cuts are artifacts of this definition.

From Eq.(2) we have the quasi-periodicity condition for the total momentum expresseEq. (6). In some applications, such as the above mentioned spectrum of the integrabtrix of anomalous dimensions inN = 4 SYM theory, we select only purely periodic Betstates(7). This information, together with the filling fractionsαk = Sk/J , k = 1, . . . ,K , suchthat

∑Kk=1 αk = S/J , or

(28)αk =∮Ak

dx p(x)

fixes completely a solution. For the energy we obtain in this limit

(29)E0 = −G′0(0).

A particular 3-cut solution is demonstrated onFig. 2and is compared with the numerical solutiof exact Bethe ansatz equation. We see that already for 300 roots andJ = 600 the description interms of the algebraic curve becomes excellent.

4. Large J limit and 1/J -corrections from Baxter equation

Eq. (2) can be also obtained as the condition that the transfer matrix eigenvalue defiBethe state (see for example[25])

(30)T (u) = W(u + i/2)Q(u + i)

Q(u)+ W(u − i/2)

Q(u − i)

Q(u),

whereQ(u) = ∏Sk=1(u − uk), W(u) = uJ . T (u) is a polynomial of degreeJ , which is clear

from the very construction of a Bethe state in the algebraic Bethe ansatz approach[1]. The Betheequations(2) follow immediately from Eq.(30)assuming analyticity (polynomiality) ofT (u).


-s

ws

In the

edensen

from

Introduce the notations:x = u/J , Φ(x) = 1J

∑Sk=1 log(x − xk), V (x) = logx, 2t (x) =

T (Jx)/(Jx)J and rewrite(30)as

(31)2t (x) = expJ

[Φ

(x + i

J

)− Φ(x) + V

(x + i

2J

)− V (x)

]+ c.c.

Defining the quasi-momentum (now at all orders in 1/J )

(32)p(x) ≡ Φ ′ + V ′/2

and expanding the Baxter equation in 1/J we get

t (x) = cosp(x)

[1− 1

J

(p′(x)

2− V ′′(x)

8

)+ 1

2J 2

(p′(x)

2− V ′′(x)

8

)2]

(33)+ 1

J 2sinp(x)

(p′′(x)

6− V (3)(x)

16

)+ O

(1

J 3

).

To find the 1/J corrections to the quasi-momentum, we will expandp(x) = p0(x) + 1Jp1(x) +

1J 2 p2(x) + O(1/J 3), t (x) = t0(x) + 1

Jt1(x) + 1

J 2 t2(x) + O(1/J 3) and plug it into the last equation. We assume that the coefficients of expansiont0(x), t1(x), t2(x), . . . are the entire functionon the planex with no cuts, having only an essential singularity atx = 0.

The quasi-periodicity property of the total momentum reads up to 3 first orders as follo

(34)Ptot = −∑j

1

uj

+∑j

1

12u3j

+O(1/J 4) = 2πk/J

and in the purely periodic case we select only the states withk = mJ , with integerm.

4.1. Zero order from Baxter equation

Let us restore from the Baxter equation the zero order result of the previous section.zero order approximation we get from Eq.(33)

(35)cosp0(x) = t0(x)

or

(36)p′0(x) = 2t ′0(x)√

1− t20

.

As is usual in the finite gap method[26], we expect that sincet0(x) is an entire functions all thbranch cuts of Eq.(26)come from the square root in denominator, after the Bethe roots conto a setC1, . . . ,CK of dense supports in theJ → ∞ limit. It is easy to see from the definitioEq. (32) thatp0(x) = α+1/2

x+ O(1/x2) whenx → ∞ andp0(x) ∼ 2π k

J+ O(x) whenx → 0.

Consequently, we reproduced the general solution (26)–(29) of[7].

4.2. 1/J correction from Baxter equation

To find the next, 1/J approximation to the density of roots and to the energy we deduceEq.(33)

(37)p1 = (−p′0/2+ V ′′/8)cotp0 − t1

,
sinp0


.al part

the

ut

filling

e-ic

ction.

wheret1(x) is an entire function on the plane as was mentioned before. We know aboutp0(x)

that

(38)p+0 = πnj − πiρ0, p−

0 = πnj + πiρ0, sinp+0 = −sinp−

0

and thus we have for the real and imaginary parts ofp0(x) on the cuts

(39)πiρ1 =(

V ′′

8t0 − t1

)1

sinp−0

, /p1 = −p′0 cotp0/2.

We will solve these equations below and restore the explicitp1.Sincet1(x) is a regular function on the cuts[44]

(40)/p1 = −1

2πρ′ cothπρ = −p±

0′cotp±

0 /2.

Moreover we see from Eq.(34) that

(41)p1(0) = 0

and for largex p1(x) should decreases asO(1/x2).We can write from(40) the general solution of this Riemann–Hilbert problem

(42)p1(x) = x

4πif (x)

∮C

f (y)p′0(y)cotp0(y)

y(y − x)dy +

K−2∑j=1

ajxj

f (x),

wheref 2(x) = ∏2Kj=1(x − xj ) and the contour encircles all cutsCk (but no other singularities)

The first term in the r.h.s. represents the Cauchy integral restoring the function from its reon the cuts and having a zero at the origin (the value of the quasi-momentump(x) at x = 0,∞was already fixed forp0) whereas the second one is purely imaginary on the cuts, withpolynomial in the numerator chosen in such a way that it does not spoil the behavior ofp(x) atx = 0,∞.10

Thus forK < 3 the solution is unique. In particular, forK = 1 we restore from here the 1-csolution of[8]. ForK 3 we have to fixK −2 parametersaj . To do this we can useK additionalconditions ensuring the right fractionsαj of the roots already chosen forp0:

(43)∮Cl

p1(x) dx = 0, l = 1, . . . ,K,

in fact onlyK − 2 of them are linear independent (since we have already fixed the totalfraction by the asymptotic properties of Eq.(42) at x = ∞: p1(x) = O(1/x2). Eq. (41) alsorestricts some linear combination of the conditions(43)). Hence we completely fixed all paramters of ourK-cut solution for the 1/J correctionp1 knowing the zero order solution (algebracurve) forp0.

It is also useful to rewrite Eq.(42) in the following way

(44)p1(x) = Q(x)

4πif (x)

∮C

f (y)p′0(y)cotp0(y)

Q(y)(y − x)dy,

10 We could also add terms13 , 15 , . . . but they are too singular at the branch points as we shell see in the next se

f f


cethese

eels all

phic

er,f

of the

n in

o

whereQ(x) = ∑K−2k=1 aj x

j and the contour of integration encircles all the cuts. Again,aj arefixed by Eq.(43). Equivalence of this formula to Eq.(42) can be seen from the coincidenof their analytical properties: blowing up the contour in the contour integrals of any ofrepresentations we obtain11

(45)p1(x) = −p′0(x)cotp0(x)/2+ 1

f (x)

∑n

cn

xn − x+

K−2∑j=1

ajxj

f (x),

where the first term comes from the residue aty = x. Note that only this term contributes to thr.h.s. of Eq.(40) thus showing that we satisfied this equation. The second term exactly cancthe poles of the first term, so thatp1 is regular everywhere except the cuts. It is a meromorfunction in thex plane, having no cuts. The last term reflects the freedom of addingK − 2coefficients before fixing them by conditions Eq.(43).

4.3. Equations for 1/J 2 corrections from Baxter relation

Expanding Eq.(33)up to 1/J 2 we obtain

(46)p2 = −1

2∂x

[cot(p0)I

] − 1

8x3− t2

2 sin(p0),

where

(47)I = − t1

sin(p0)= p1 + p′

0

2cotp0.

We introduced here the notations

t1 = t1 + cosp0

8x2,

(48)t2 = t2 − cosp0

128x4+ t1

8x2− cos(2p0) + 5

24 sinp0p′′

0 + cosp0

8 sin2 p0

(3(p′

0)2 + 4t 2

1

)so thatt1 andt2 are single valued functions.

Note that above the cutI+ = πiρ1. We will find the explicit solution of these equations latbut we will need for that some results of the next section where we study the behavior op(x)

near the branch points.

5. Double scaling solution near the branch point

As we stated above the branch point singularities come only from the square rootsdenominator of Eq.(36). We define an exact branch point as a pointx∗ wheret (x∗) = ±1. If weapproach one of the branch pointsx → x∗ we can expand

(49)t (x) ±[1− a(x − x∗)/2− b(x − x∗)2/2

].

Note thatx∗, a, b themselves depend onJ . We assume that they have a regular expansio1/J (justified by self-consistency of further calculations and by numerics) and definex∗ = x0 +x1/J + · · ·. We callx0 a classical branch point andx1/J a branch point displacement.

11 In fact one should take into account an infinite number of residues aty = 0, see[8]. A more regular procedure is texpress cot as a sum and then do the integration.


sationf the

tantpress

u-

Fig. 3. Quasi-momentum near branch point as a function of the scaling variablev for S = 200. The poles correspondto the positions of Bethe rootsui . Red dashed line—“exact” numerical value, light grey—zero order approximgiven by Airy function Ai(a1/3x), grey—first order and black—second order approximation. (For interpretation oreferences to colour in this figure legend, the reader is referred to the web version of this article.)

Denotingv = (x − x∗)J 2/3 which will be our double scaling variablev ∼ 1, we get fromEq.(30)up to 1/J 2 terms

(50)±2

(1− av

2J 2/3− bv2

2J 4/3

)Q(u) = Q(u + i)

W(u + i/2)

W(u)+ Q(u − i)

W(u − i/2)

W(u).

In terms of a new functionq(v) = e−nπvJ 1/3e

vJ1/32x∗ Q(x∗J + vJ 1/3), wheren is such thatt (x∗) =

eiπn, and after expansion in 1/J the last equation takes the form

(51)q ′′ − avq = 1

J 1/3

4vq ′ + q

4x2∗+ 1

J 2/3

[1

12q(4)(v) − v2q(v)

4

(1

x4∗− 4b

)]+O

(1

J

).

In fact, this equation can be easily solved in terms ofq0

(52)

q ∝[1+ v2

4x2∗J 1/3+ 1

J 2/3

(v4

32x4∗− 3b − a2

15av

)]q0

(v − 1

4ax2∗J 1/3+ a2 + 12b

60aJ 2/3v2

),

whereq0(v) = Ai(a1/3v) (the Airy function). The second solution of Eq.(51), Bi(a1/3v) has awrong asymptotic as we will see. The sign∝ means that the solution is defined up to a consmultiplier but this unknown multiplier does not affect the quasi-momentum. Now we can exthe quasi-momentum only through our scaling functionq(v)

(53)p

(x∗ + v

J 2/3

)= ∂vq(v, J )

q(v, J )J 1/3+ πn + 1

2x∗

(1

1+ v

x∗J 2/3

− 1

).

The first two terms in the r.h.s., if we substituteq(v) → q0(v), represent the principal contribtion to the double scaling limit near the edge, valid up to the corrections of the order 1/J 2/3. Wesee from the definition(23) that the zeros ofq(v) are nothing but the positionsui of Bethe roots.Thus we know these positions with a precision 1/J 2/3 (seeFig. 3).

The largev asymptotic will be very helpful in fixing some unknown constant in the 1/J 2

corrections given in the next section


ngthe

antum

p(x∗ + vJ−2/3) = πn + 1

J 1/3

(−√

av︸︷︷︸1

− 1

4v︸︷︷︸1/J

+ 5

32v2√

av︸︷︷︸1/J 2

+· · ·)

(54)+ 1

J 2/3

(1

8x2∗√

av︸︷︷︸1/J

− 1

16ax2∗v2︸︷︷︸1/J 2

+· · ·)

+ · · · ,

where the cut corresponds to negativev for a > 0. Introducing the notationy = vJ−2/3 andrearranging the terms by the powers 1/J we have

p(x∗ + y)

= πn +[−√

ay − (a2 + 12b)y3/2

24√

a+ · · ·

]+ 1

J

[− 1

4y+ 1

8x2∗√

ay+ a2 − 4b

16a+ · · ·

]

(55)+ 1

J 2

[5

32y2√ay− 1

16ay2x2∗+ 6− x4∗(a2 + 12b)

768x4∗(ay)3/2+ · · ·

]+ · · · .

Doing this re-expansion we assume thatJ−1 y 1, trying to sew together the double scaliregion with the 1/J corrections to the thermodynamical limit. This procedure is similar toone used in higher orders of the WKB approximation in the usual one-dimensional qumechanics (see for example[27]).

To compare withp0,p1 andp2 we have to re-expand aroundx0

(56)p(x0 + y) = p(x∗ + y) + x1

J

√a

2√

y+ 1

J 2

[− x1

4y2+ x1

16x20y

√ay

+√

ax21

8y√

y

]or, introducing notation

(57)x1 = 2A√a

− 1

4x20a

we get

p(x0 + y)

= πn +[−√

ay − (a2 + 12b)y2

24√

ay+ · · ·

]+ 1

J

[− 1

4y+ A√

y+ a2 − 4b

16a+ · · ·

]

(58)

+ 1

J 2

[5

32y2√ay− A

2√

ay2+

(A2

2y√

ay− b

64(ay)3/2−

√ay

768y2

)+ · · ·

]+ · · · .

Near the left branch point (i.e fora < 0 andy < 0) we have

p(x0 + y)

= πn +[√

ay + (a2 + 12b)y2

24√

ay+ · · ·

]+ 1

J

[− 1

4y− A√−y

+ a2 − 4b

16a+ · · ·

]

+ 1

J 2

[− 5

32y2√ay+ A

2√−ay2

+(

A2

2y√

ay+ b

64(ay)3/2+

√ay

768y2

)+ · · ·

](59)+ · · · .

Now we can compare it with our results of the previous sections and fixa, b andx1.


atrices

eng the

aic.

h

i-

ince

Let us note that similar Airy type oscillations were observed in the papers on random mwhere this behavior occurs near an endpoint of a distribution of eigenvalues[12]. This shows theintrinsic similarity of these two seemingly different problems. The largeJ limit for the spinchain is rather similar to the large sizeN limit in random matrices. We will see this analogy evclearer in the Section7 where we will calculate the asymptotics of conserved charges usinresults of this section.

5.1. Comparison with p0 and p1

It is instructive to establish the relations betweena, b,A and the parameters of the algebrcurve which completely defines, as we know,p = p0 + 1

Jp1 +O( 1

J 2 ) up to the first two ordersFor that we use the expansion(49)defininga, b and find from Eq.(35) for y > 0

(60)p0(x0 + y) = πn + arccost0 πn − √ay − a2 + 12b

24√

ay3/2 +O

(y5/2),

in agreement with Eqs.(58), (59). We can fixa andb up toO(1/J ) corrections from here througthe parameters of the solution forp0 given by Eq.(26).

To calculatea andb up toO(1/J ) and to fixA, we use the expansion Eq.(49)with Eq.(37).Note that we have the minus sign in front of

√ay which ensures the positivity of the density(24)

on the cut (i.e., fory < 0 anda > 0) ρ(y) √a(−y)/π . If we had Bi instead of Ai the sign

would be plus and the density would be negative.Now we compare this near-cut behaviour top1. Consider the regular part first

(61)/p1 = −1

2p′

0 cotp0 − 1

4y+ a2 − 4b

16a+O(y),

which is in full agreement with Eq.(55). From Eq.(42)we see that

(62)p1(x0 + y) − /p1(x0 + y) A√y

+O(

1

y3/2

),

whereA can be written explicitly, again using the parameters ofp0 given by Eq.(26).For the example of one-cut solution see Eq.(B.7).

6. General solution for p2 and E2

Now we have enough of information to constructp2 in the most general situation of an arbtrary number of cuts.

We start from a formula which immediately follows from Eq.(46)

(63)/p2 = −1

2∂x

[cot(p0)

(p1 + p′

0

2cotp0

)]− 1

8x3,

wherep1 is given by Eq.(42). The behaviors near zero and at infinity are the following. Sfrom (6) and (34)it follows thatG(0) − 1

24J 2 G′′(0) = 2πk/J +O( 1J 4 ) we can conclude that

(64)p2(0) = 1

24G′′

0(0).

For largex we have again

(65)p2(x) = O(1/x2).


de ofcan be

om

effi-

rac-

r

Repeating the arguments of the previous subsection we have

(66)p2(x) = x

4πif (x)

∮C

f (y)

y(y − x)

(1

4y3+ ∂y

[cot(p0)p1

]) +5K−1∑j=0

cjxj

f 5(x),

where the pathC is defined as in Eq.(42). Again the first term guarantees thatp2 satisfiesEq. (63). We drop out thep′

0 cothp0 for simplicity. We can do this since together withf (y)

it forms a single-valued function without cuts and the integral is given by the poles insithe path of integration. In fact there are only poles at each branch point so that the resultabsorbed into the second term in Eq.(66).

So far the second term in Eq.(66)was restricted only by the conditions(64) and (65). Of causethis does not explain why we should restrict ourselves by the fifth power off (x) in denominator.A natural explanation comes from the known behaviour near the branch points(58), (59)fromwhere we can see that

(67)p2(xi

0 + y) =

532y2√aiy

− Ai

2√

aiy2 + ( A2

i

2y√

aiy− bi

64(aiy)3/2 −√

aiy

768y2

) +O( 1

y

),

ai > 0, y > 0,

− 532y2√aiy

+ Ai

2√−aiy

2 + ( A2i

2y√

aiy+ bi

64(aiy)3/2 +√

aiy

768y2

) +O( 1

y

),

ai < 0, y < 0,

where all 6K constantsai, b,Ai for i = 1, . . . ,2K are known since they can be determined frthe near branch point behaviour ofp0 andp1 (58), (59). ai andbi follow from p0

(68)p0(xi

0 + y) =

−√aiy − (a2

i +12bi )y2

24√

aiy+O(y5/2), ai > 0, y > 0,

√aiy + (a2

i +12bi )y2

24√

aiy+O(y5/2), ai < 0, y < 0

andAi comes fromp1

(69)p1(xi

0 + y) =

− 14y

+ Ai√y

+O(y0), ai > 0, y > 0,

− 14y

− Ai√−y+O(y0), ai < 0, y < 0.

In fact Eq.(67) gives only two non-trivial conditions for each branch point which are the cocient before the half-integer power ofy so that we have 4K conditions. The extraK conditionscome from zeroA-period constraints signifying the absence of corrections to the filling ftionsαi .

(70)∮Cl

p2(x) dx = 0, l = 1, . . . ,K.

To reduce the number of unknown constants consider a branch pointx0. We can see that fosmally = x − x0 (for simplicity we assume that the cut is on the left, i.e.,ai > 0)

I1 ≡ x

4πif (x)

∮C

f (z)

z(z − x)

(1

4z3+ ∂z(p1 cotp0)

)

(71)= 3

16y2√ay− A

2√

ay2+ 1

y3/2

(b

32a3/2− 5

√a

128

)+O

(1

y

).


tion

ndion

s.is

n be

Fig. 4. Resolvent far from branch point as a function ofx. Red dashed line—“exact” numerical value for one cut soluwith S = 10,n = 2,m = 1, light grey—zero order approximation, grey—first order given by Eq.(42)and black—secondorder approximation given by Eq.(73). Note that near branch point (x0 = 0.02) the approximation does not work ainstead of it we should use the Airy function of Eq.(53), like in the usual WKB near a turning point. (For interpretatof the references to colour in this figure legend, the reader is referred to the web version of this article.)

Introduce the following integral

I2 ≡ x

4πif (x)

∮C

f (z)

z(z − x)

((p1 + p′

0 cotp0)p1 cotp0 − p′′0

12

)

(72)= − 1

32y2√ay+ 1

y3/2

(A2

2√

a− 3b

64a3/2+ 29

√a

768

)+O

(1

y

)so thatI1 + I2 reproduces the right series expansion near the branch points given by Eq(54)and (55). Moreover, on the cutsI2(x + i0) + I2(x − i0) = 0 since the function under integralsingle valued. We can simply take

(73)p2(x) = I1(x) + I2(x) +K−1∑j=0

cj xj

f (x),

where the remainingK constants are fixed from Eq.(70). Using thatp2(0) = G′′(0)/24 we canfix one constantc0 = G′′(0)f (0)

24 before imposing the condition(70).This is our final result for the second quantum correction to the quasi-momentum (Fig. 4). In

Appendix Bwe will specify this result for the example of the one-cut solution where it camade much more explicit.

7. Energy and higher charges

7.1. Energy

To find 1/J corrections to the energy we represent the exact formula Eq.(1) as follows

(74)E = − 1

JG′(0) + 1

24J 3G(3)(0) +O

(1

J 5

),

whereG(x) is defined in Eq.(23). We still have to expandG(x) = − 12x

+ p0(x) + 1Jp1(x) +

1J 2 p2(x) + O(1/J 3).

Finally, we obtain for the energy

(75)E = 1E0(x) + 1

2E1 + 1

3E2 +O

(14

),

J J J J


tion.

la-

thismpare

Fig. 5. Relative deviationδE(S)/E(S) of analytical computations of the energyE(S) from its “exact” valueEexact(S)

for the one cut distribution found numerically by Mathematica (solid line corresponds toδE(S) = 0), for a finite numberof rootsS and a finite lengthJ for zero order (light gray), first order (gray) and second order (black) approximaDetails are summarized inTable 1.

where

(76)E0 = −G′0(0),

(77)E1 = −p′1(0) = − Q′(0)

4πif (0)

∮C

f (y)p′(y)cotp(y)

Q(y)ydy,

andQ(x) = ∑K−2k=1 bkx

k is related to the last term in(42). For E2 we have from Eq.(73) thefollowing representation

E2 = G(3)0 (0)

24− p′

2(0)

= −1

4πif (0)

∮f (y)

y2

(1

4z3+ ∂z(p1 cotp0) − p′′

0

12+ (p1 + p′

0 cotp0)p1 cotp0

)

(78)− c1

f (0)+ G′′

0(0)f ′(0)

24f (0)+ G

(3)0 (0)

24.

Note that for 1-cut we should takec1 = 0. We can compare our results with numerical calcutions, as it is done for a few 1-cut solutions inFig. 5.

7.2. Local charges

In this and the next subsection we will calculate local and non-local, or global chargesQr inall powers of 1/J for the larger from the behavior near the relevant branch point. The idea ofcalculation is taken from the double scaling approach in matrix models. Namely, we can coit to the calculation of the resolvent of eigenvalues in a Gaussian unitary matrix ensemble

(79)

HN(x) =∫

dN2M

(2π)N2 exp

(−N

2TrM2

)Tr(x − M)−1 =

∞∑g=1

N2−2g∞∑

n=0

x−2n−1H(g,n),


r

rtnd thest the

e

s a

e

n

Table 1

# 1 2 3 4 5

m,n 1,2 2,1 1,3 2,2 1,5E0 12π2 24π2 16π2 32π2 24π2

E1 −558.4 −1563 −855.3 −2401 −1563E2 1160 5464 1592 8982 1504S 10 40 7 20 5J 20 20 21 20 25E(S) 4.66004 8.54515 5.7359 10.7876 7.0232E(2)(S) 4.670 8.619 5.752 10.912 7.070

whereM is a Hermitian matrix of large sizeN . The coefficientsH(g,n) actually give the numbeof specific planar graphs: it is given by the number of surfaces of genusg which can be done froma polygon with 2n edges, by the pairwise gluing of these edges. To extract the largen asymptoticsof H(g,n) for anyg one can use that in the largeN limit the density (which is the imaginary paof the resolvent on the support of eigenvalues) is given by the Wigner’s semi-circle law, anear-edge behavior is described by the Airy functional asymptotics[12,28] showing the traceof individual eigenvalues in the continuous semi-circle distribution. We will try to extracsimilar asymptotics for the distribution of Bethe roots. The role of 1/N expansion will be playedby the 1/J expansion, whether as the order of the 1/x expansion in the matrix model will bnow played by the labelr of the charge.

We start from expanding Eq.(13)

(80)Qr =∞∑

m=0

1

J r+2m−1

(−1)m+1G(r+2m−1)(0)

(2m + 1)!(r − 1)!22m.

As we shell see, for larger only them = 0 term contributes. We express the derivative acontour integral around cuts

(81)G(n)(0) = − n!2πi

∮C

G(x)

xn+1dx.

For largen only a small neighborhood of the closest to zero branch pointx0 contributes due to thexponential suppression by the 1/xn+1 factor. Near the branch pointx0 we have from Eq.(58)(see also Eqs.(59), (53))

(82)Gk(x) = δk0

(πni − 1

2x0

)+

ck(x − x0)12− 3k

2 |a| 12− k

2 +O((x − x0)1− 3k

2 ),

a > 0, x0 < 0,

(−1)k+1ck(x0 − x)12− 3k

2 |a| 12− k

2 +O((x0 − x)1− 3k2 ),

a < 0, x0 > 0,

where the universal constantsck can be computed from the known asymptotic of Airy functio

ck = Ai ′(z)Ai(z)

∣∣∣∣z− 3k−1

2

,

(83)Ai(z) = e− 2z3/23

2√

πz1/4

[n∑

k=0

(16)k(

56)k

k!(

− 3

4z3/2

)k

+O(

1

z3(n+1)/2

)]


r-ite

ral

in particularc0 = −1, c1 = −14, c2 = 5

32, c3 = −1564, c4 = 1105

2048, c5 = −16951024, c6 = 414125

65536 , c7 =−59025

2048 .These coefficients behave asymptotically asck ∼ (−1)kk! at k → ∞.We assume thatk n, r and expand (forx0 < 0)

0∮−y0

(y + x0)−nyβ dy = |x0|β+1−n(−1)n

0∮−y0

yβe−n log(1−y) dy

(84) |x0|β+1−n(−1)n

0∮−∞

yβeny dy.

For the last integral the path of integration starts at−∞ − i0, encircles the origin in the counteclockwise direction, and returns to the point−∞ + i0. For the first integral the path is finite:starts at some point−y0 − i0 where 0< y0 < |x0| and ends at−y0 + i0. The dependence on thy0 is exponentially suppressed. The last integral is nothing but the Hankel’s contour integ

(85)

0∮−y0

(y + x0)−nyβ dy = (−1)n|x0|β+1−nn−β−1 2πi

(−β)

(1+O

(1

n

))

similarly

(86)

y0∮0

(y + x0)−n(−y)β dy = −|x0|β+1−nn−β−1 2πi

(−β)

(1+O

(1

n

))

so that

(87)G

(n)k (0)

n! =

(−1)nck |a| 1

2− k2 n

3k2 − 3

2 |x0|12− 3k

2 −n

( 3k2 − 1

2 )(1+O( 1

n)), a > 0, x0 < 0,

(−1)k+1 ck |a| 12− k

2 n3k2 − 3

2 |x0|12− 3k

2 −n

( 3k2 − 1

2 )(1+O( 1

n)), a < 0, x0 > 0.

As we can see from here, only the term withm = 0 in Eq.(80)contributes at largen. The othersare suppressed as 1/n and the final result is

(88)Qk,r =

(−1)rck |a| 1

2− k2 r

3k2 − 3

2 |x0|32− 3k

2 −r

( 3k2 − 1

2 )(1+O(r−1/2)), a > 0, x0 < 0,

(−1)kck |a| 1

2− k2 r

3k2 − 3

2 |x0|32− 3k

2 −r

( 3k2 − 1

2 )(1+O(r−1/2)), a < 0, x0 > 0,

where we introduced the notation

(89)Qr = 1

J r−1

∞∑k=0

Qk,r

1

J k.

Note thatQk,r is similar toHg,n of the matrix model.


tough to

n-ed bye scal-for the

tin

redge

of theom

aledmialsdeed,

7.3. Non-local charges

Now we will find the coefficients of the 1/xn expansion for largen and arbitraryk

(90)G(x) =∞∑

k=0

1

J k

∞∑n=1

dk,n

xn,

in other words

(91)dk,n = 1

2πi

∮C

xn−1Gk(x)dx.

In fact only the cut with minimalni contributes for largen, or rather its branch point closestx = ∞. The contributions of other branch points are exponentially suppressed. It is enoconsider only a small neighborhood of the branch point with maximal|x0,i |. Near the branchpointpk(x) will behave as

(92)Gk(x) = δk0

(πni − 1

2x0

)+

ck(x − x0)12− 3k

2 |a| 12− k

2 +O(x − x0)1− 3k

2 ,

a > 0, x0 > 0,

(−1)k+1ck(x0 − x)12− 3k

2 |a| 12− k

2 +O(x0 − x)1− 3k2 ,

a < 0, x0 < 0

similarly to the previous section we obtain

(93)dk,n =

cka12− k

2 n3k2 − 3

2 xn+ 1

2− 3k2

0

( 3k2 − 1

2 )(1+O(n−1/2)), a > 0, x0 > 0,

(−1)k+n−1 cka12− k

2 n3k2 − 3

2 xn+ 1

2− 3k2

0

( 3k2 − 1

2 )(1+O(n−1/2)), a < 0, x0 < 0.

8. Conclusions

We showed in this paper on the example ofsl(2) Heisenberg spin chain, how to find geeral solutions to quantum integrable problems in a specific thermodynamical limit proposSutherland, and how to find various finite size corrections to it. We also propose a doubling analysis of the near edge distribution of Bethe roots giving some interesting resultsasymptotics of high conserved charges for the finite size corrections of any order.

Our methods borrow some ideas from the matrix models: the size of the chainJ is somewhasimilar to the size of a random matrixN , and the first two 1/J finite size corrections to the malimit which we calculated are of a similar mathematical nature as the 1/N corrections in matrixmodels. The asymptotics of high conserved charges found here in all orders of 1/J remind verymuch the double scaling approach in matrix models[12,29,30]. To calculate them we used fothat the Airy asymptotics on the edge of the Bethe root distribution, similar to the genericbehavior in the matrix models. However, it is obvious that by finetuning the parameterschain and of its largeJ solutions we can reach various multicritical points with a different frAiry classes of universality, probably also similar to those of the multicritical matrix models[13]or a two cut model[31,32]. Probably, some modern methods of analysis of the doubly scmatrix models, like the Riemann–Hilbert method for the asymptotics of orthogonal polyno[33] should work as well for the quantum integrable chains in this thermodynamical limit. In


torialanningplanar

l,nce ofhtlelselliptic

sof

dynam-was

somespon-

ections

spin-ods of

ibution

the restll casesnear-redders ofeeplytseytlin

ions.TN-

ed byniver-

it is known that the polynomialsQS(u) = ∏Sk=1(u − uk) are orthogonal for differentS with a

specificS-independent measure[34].Our methods might be useful for finding the asymptotics of numbers of various combina

objects on regular lattices related to the integrable models, such the tilting patterns, sptrees, dimers etc., as its analogue was useful for finding the asymptotics of various largegraphs via the matrix models.

The methods presented here can be easily carried over to thesu(2) quantum chain as welthough some peculiarities of this model, like complex distributions of roots and the prese“string” condensates with equally distributed roots[3], should be taken into account. Only sligmodifications of our results will allow to find the 1/J corrections in the non-local integrabdeformations of thesu(2) spin chain described in[35,36]. As for more complicated modesolved by nested Bethe ansatz and with the thermodynamical limit described by non-hyperalgebraic curves, the 1/J and 1/J 2 corrections are left to be established.

In the context of integrability in the 4D Yang–Mills theory withN = 4 supersymmetrie(SYM), where the Hamiltonian of thepsu(2,2|4) integrable spin chains describes the matrixanomalous dimensions, our methods could be especially useful. The appropriate thermoical limit in terms of non-hyperelliptic curve, corresponding to very long operators in SYMconstructed in particular sectors in[4,7,20,37]and for the fullpsu(2,2|4) SYM chain in[11].Finding the finite size corrections to these solutions might be extremely useful for gettingclues for the quantization of the string sigma model on the other side of AdS/CFT corredence. The finite size correction for the simplestsu(3) solution of SYM theory found in[38] arenot available yet, let alone more complicated solutions and sectors. The finite size corrto this solution, if found, could be compared to the string quantum correction computed in[39].The corrections found here for the most general multi-cut solution insl(2) sector could shedsome light on the structure of the first quantum corrections to string solitons describingning strings, where for the moment only the direct, very cumbersome quasi-classical methevaluation of the functional integral near the simplest classical solutions are known[15].

As for the double scaling limit used here for the refined analysis near the edge of a distrand giving the asymptotics of high conserved charges to any order of 1/J , it is very universaland is not based on any particular form of solutions. Near the edge we can forget aboutof the sheets and cuts in any algebraic curve, and the asymptotics will be governed in aby the Airy functions. In fact, we think that this near edge behavior will persist also in theclassical regime of the string theory onAdS5 × S5, and even for some of its recently considedeformations. We expect then that the asymptotics of high conserved charges at all or1/J will be also dominated by the same Airy type solution as found here. It is already a dquantum regime (reminding that of the non-critical string theories in 2 dimensions) and icould be extremely useful as a step to the full quantization and solution of the Metsaev–Tsupersgtring and hence of theN = 4 SYM theory.

Acknowledgements

We would like to thank N. Beisert, I. Kostov, F. Smirnov and K. Zarembo for discussThe work of V.K. was partially supported by European Union under the RTN contracts MRCT-2004-512194 and by INTAS-03-51-5460 grant. The work of N.G. was partially supportFrench Government PhD fellowship and by RSGSS-1124.2003.2. We also thank Institut Usitaire de France for a partial support.


f

rries, sole

mplestmplexula

Appendix A. BAE 1/J expansion

In this appendix we give the formulas for the expansion of the r.h.s. of BAE(2) in powers of1/J . The density(x) is defined in Eq.(16). We assumek,S − k ∼ J (i.e. far from the ends othe cut)∑

j

′i log

(uj − uk + i

uj − uk − i

) −2

∑j

′ 1

uj − uk

+ π′[coth(π)]0J

+O(

1

J 2

),

∑j

′i log

(uj − uk + i

uj − uk − i

)

−2∑j

′ 1

uj − uk

+ 2

3

∑j

′ 1

(uj − uk)3+ π′[coth(π)]2

J+O

(1

J 3

),

∑j

′i log

(uj − uk + i

uj − uk − i

)

−2∑j

′ 1

uj − uk

+ 2

3

∑j

′ 1

(uj − uk)3− 2

5

∑j

′ 1

(uj − uk)5+ π′[coth(π)]4

J

+ 1

12J 3

((π′)3

[coth(π)

sinh2(π)

]0− 2π2′′′

[1

sinh(π)

]1+ π′′′[coth(π)

]2

)

+O(

1

J 4

),

∑j

′i log

(uj − uk + i

uj − uk − i

)

−2∑j

′ 1

uj − uk

+ 2

3

∑j

′ 1

(uj − uk)3− 2

5

∑j

′ 1

(uj − uk)5+ 2

7

∑j

′ 1

(uj − uk)7

− π′[coth(π)]6J

− 1

12J 3

((π′)3

[coth(π)

sinh2(π)

]2− 2π2′′′

[1

sinh(π)

]3

(A.1)+ π′′′[coth(π)]4)

+O(

1

J 5

),

where we introduced the notation[f ()]n ≡ f () − ∑n−1i=0 f (i)(0)

i

i! for the functions regulaat zero. For singular functions the Taylor series should be substituted by the Laurent sethat [f ()]n is zero for = 0 and has firstn − 1 zero derivatives at this point. For examp[coth(π)]2 ≡ coth(π) − 1

π− π

3 .

Appendix B. Example: 1-cut

In this appendix we express corrections to the energy in terms of infinite sums for the sicase of one-cut solution. For this solution the hyperelliptic curve is a sphere. It is two coplanes connected by a single cut. The density of the Bethe roots is given by a simple form[7]

(B.1)ρ(x) =√

8πmx − (2πnx − 1)2.

2πx


that the

o not

p

anatione

We can easily find explicit expressions forai and bi of Eq. (67). With the notationM =√m(m + n), ai andbi become

a1 = − 8Mn4π3

(√

4M2 + n2 − 2M)2,

(B.2)b1 = 4π4n6

3(√

4M2 + n2 − 2M)4

(12M

√4M2 + n2 + 3n2 − 4n2π2M2 − 24M2)

and

a2 = 8Mn4π3

(√

4M2 + n2 + 2M)2,

(B.3)b2 = − 4π4n6

3(√

4M2 + n2 + 2M)4

(12M

√4M2 + n2 − 3n2 + 4n2π2M2 + 24M2).

It may be more convenient for comparison with string theory results[15] to expressA definedby Eq. (58) as an infinite sum. We have to evaluate the integral in Eq.(42) and findA fromthe behavior near a branch point. We compute the integral by poles. To that end we usesolutions to the equation sin(p0(x

±l )) = 0 are

(B.4)x±l = 1

2π

1√4M2 + n2 ∓ √

4M2 + l2, l 0.

The pointsx±l=0 are the branch points. They are inside the contour of integration and thus d

contribute.Using thatf (x±

l )/x±l = ± l

nand

1

x+l − x0,1

− 1

x−l − x0,1

= −√

l2 + 4M2

l2

1

πx20,1

,

(B.5)1

x+l − x0,2

− 1

x−l − x0,2

= −√

l2 + 4M2

l2

1

πx20,2

.

We can evaluate the integral Eq.(42) for x → x0 (we also takex inside the contour to droirrelevant symmetric part ofp1)

(B.6)1

2πi

∮C

f (y)p′(y)cotp(y)

y(y − x)dy → − 1

iπnx20

∞∑l=1

√l2 + 4M2

l

we can conclude that

A2 = − 1

2x22√

a2

∞∑l=1

√l2 + 4M2

l,

(B.7)A1 = − 1

2x21√−a1

∞∑l=1

√l2 + 4M2

l,

where the sum should be understood in the zeta-function regularization (a natural explwhy this regularization gives the right result is given in[8]. A more regular way to express th


tion and

03) 010,

0405

ientific

integral as a sum is to expand cot into the sum before integration)

(B.8)∞∑l=1

√l2 + 4M2

l≡

∞∑l=1

(√l2 + 4M2

l− 1

)− 1

2.

We can easily reproduce the result of[8] for E1 in terms of a sum from Eq.(77)

(B.9)E1 = −p′1(0) = 4π2

∞∑l=1

l√

l2 + 4M2

with ζ -function regularization assumed.We can also express our result for the next correction to the energyE2 given by Eq.(78) as a

double sum. We will need the following quantity

(B.10)

p1(x±k ) = ±1

2π(x±k )2k

[ ∞∑l=1

(l√

l2 + 4M2 − k√

k2 + 4M2

l2 − k2− 1

)+

√k2 + 4M2

2k− 1

2

].

Evaluating the integrals in Eq.(78)we expressE2 as a double sum

(B.11)E2 = −(I1 + I2 + I3 + I4),

where

I1 ≡ 1

4πif (0)

∮f (z)

z2∂z(p1 cotp0)

= −2p′1(0) +

∞∑k=1

[2π

∑±

(√4M2 + n2 ± 2

k2 + 2M2

√k2 + 4M2

)p1

(x±k

) − 4p′1(0)

],

I2 ≡ 1

4πif (0)

∮f (z)

4z5= 4π4M2(n2 + 5M2),

I3 ≡ I ′2(0) = 1

16

(1

x20,1

+ 1

x20,2

)+ 1

x0,1

(7a1

96− b1

8a1− A2

1

)+ 1

x0,2

(7a2

96− b2

8a2+ A2

2

),

(B.12)I4 ≡ −G′′0(0)f ′(0)

24f (0)− G

(3)0 (0)

24= 4

3M2(2n2 + 11M2)π4.

Note that in our new notations 1/x0,i = 4πM ± 2π√

4M2 + n2. Expressions forai, bi andAi

are given in Eqs.(B.2), (B.3) and (B.7).

References

[1] L.D. Faddeev, How algebraic Bethe ansatz works for integrable model, hep-th/9605187.[2] B. Sutherland, Low-lying eigenstates of the one-dimensional Heisenberg ferromagnet for any magnetiza

momentum, Phys. Rev. Lett. 74 (1995) 816.[3] N. Beisert, J.A. Minahan, M. Staudacher, K. Zarembo, Stringing spins and spinning strings, JHEP 0309 (20

hep-th/0306139.[4] V.A. Kazakov, A. Marshakov, J.A. Minahan, K. Zarembo, Classical/quantum integrability in AdS/CFT, JHEP

(2004) 024, hep-th/0402207.[5] N. Reshetikhin, F. Smirnov, Quantum Floquet Functions, Zapiski Nauchnikh Seminarov LOMI (Notes of Sc

Seminars of Leningrad Branch of Steklov Institute), vol. 131, 1983, p. 128 (in Russian).


(2004)

e cor-

tum cor-

) 283,

tt. A 4

) 1,

,

3, hep-

.509096.

Phys. 5

(1995)

77.ett.

a equa-

ethod,

nctional

.1326.

381.math-

conjec-

.

2004)

4.)

[6] G.P. Korchemsky, WKB quantization of reggeon compound states in high-energy QCD, hep-ph/9801377.[7] V.A. Kazakov, K. Zarembo, Classical/quantum integrability in non-compact sector of AdS/CFT, JHEP 0410

060, hep-th/0410105.[8] N. Beisert, A.A. Tseytlin, K. Zarembo, Matching quantum strings to quantum spins: One-loop vs. finite-siz

rections, Nucl. Phys. B 715 (2005) 190, hep-th/0502173.[9] R. Hernandez, E. Lopez, A. Perianez, G. Sierra, Finite size effects in ferromagnetic spin chains and quan

rections to classical strings, JHEP 0506 (2005) 011, hep-th/0502188.[10] F.A. Smirnov, Quasi-classical study of form factors in finite volume, Amer. Math. Soc. Transl. 201 (2000

hep-th/9802132.[11] N. Beisert, V.A. Kazakov, K. Sakai, K. Zarembo, Complete spectrum of long operators inN = 4 SYM at one loop,

JHEP 0507 (2005) 030, hep-th/0503200.[12] E. Brezin, V.A. Kazakov, Exactly solvable field theories of closed strings, Phys. Lett. B 236 (1990) 144.[13] V.A. Kazakov, The appearance of matter fields from quantum fluctuations of 2D gravity, Mod. Phys. Le

(1989) 2125.[14] N. Beisert, The dilatation operator ofN = 4 super-Yang–Mills theory and integrability, Phys. Rep. 405 (2005

hep-th/0407277.[15] S. Frolov, A.A. Tseytlin, Semiclassical quantization of rotating superstring inAdS5 × S5, JHEP 0206 (2002) 007

hep-th/0204226.[16] S. Schafer-Nameki, M. Zamaklar, K. Zarembo, Quantum corrections to spinning strings inAdS5 × S5 and Bethe

ansatz: A comparative study, hep-th/0507189.[17] N. Beisert, L. Freyhult, Fluctuations and energy shifts in the Bethe ansatz, Phys. Lett. B 622 (2005) 34

th/0506243.[18] N. Beisert, A.A. Tseytlin, On quantum corrections to spinning strings and Bethe equations, hep-th/0509084[19] S. Schafer-Nameki, M. Zamaklar, Stringy sums and corrections to the quantum string Bethe ansatz, hep-th/0[20] N. Beisert, V.A. Kazakov, K. Sakai, Algebraic curve for theSO(6) sector of AdS/CFT, hep-th/0410253.[21] P.P. Kulish, N.Y. Reshetikhin, E.K. Sklyanin, Yang–Baxter equation and representation theory, I, Lett. Math.

(1981) 393.[22] L.D. Faddeev, G.P. Korchemsky, High-energy QCD as a completely integrable model, Phys. Lett. B 342

311, hep-th/9404173.[23] A. Matytsin, On the largeN limit of the Itzykson–Zuber integral, Nucl. Phys. B 411 (1994) 805, hep-th/93060[24] J.M. Daul, V.A. Kazakov, Wilson loop for largeN Yang–Mills theory on a two-dimensional sphere, Phys. L

B 335 (1994) 371, hep-th/9310165.[25] I. Krichever, O. Lipan, P. Wiegmann, A. Zabrodin, Quantum integrable models and discrete classical Hirot

tions, Commun. Math. Phys. 188 (1997) 267, hep-th/9604080.[26] S. Novikov, S.V. Manakov, L.P. Pitaevsky, V.E. Zakharov, Theory of Solitons. The Inverse Scattering M

Plenum, 1984.[27] P. Bleher, Quasi-classical expansions and the problem of quantum chaos, in: Geometrical Aspects of Fu

Analysis, Israel Seminar (GAFA), in: Lecture Notes in Mathematics, vol. 1469, 1989.[28] M.L. Mehta, Random Matrices, Academic Press, New York, 1967.[29] M.R. Douglas, S.H. Shenker, Strings in less than one-dimension, Nucl. Phys. B 335 (1990) 635.[30] D.J. Gross, A.A. Migdal, Nonperturbative two-dimensional quantum gravity, Phys. Rev. Lett. 64 (1990) 127[31] V. Periwal, D. Shevitz, Unitary matrix models as exactly solvable string theories, Phys. Rev. Lett. 64 (1990)[32] M.R. Douglas, N. Seiberg, S.H. Shenker, Flow and instability in quantum gravity, Phys. Lett. B 244 (1990)[33] P. Bleher, A. Its, Double scaling limit in the random matrix model: The Riemann–Hilbert approach,

ph/0201003.[34] E. Mukhin, A. Varchenko, Multiple orthogonal polynomials and a counterexample to Gaudin Bethe ansatz

ture, math.QA/0501144.[35] N. Beisert, C. Kristjansen, M. Staudacher, The dilatation operator ofN = 4 super-Yang–Mills theory, Nucl. Phys

B 664 (2003) 131, hep-th/0303060.[36] D. Serban, M. Staudacher, PlanarN = 4 gauge theory and the Inozemtsev long range spin chain, JHEP 0406 (

001, hep-th/0401057.[37] S. Schafer-Nameki, The algebraic curve of 1-loop planarN = 4 SYM, Nucl. Phys. B 714 (2005) 3, hep-th/041225[38] J. Engquist, J.A. Minahan, K. Zarembo, Yang–Mills duals for semiclassical strings onAdS5×S5, JHEP 0311 (2003

063, hep-th/0310188.[39] S.A. Frolov, I.Y. Park, A.A. Tseytlin, On one-loop correction to energy of spinning strings inS5, Phys. Rev. D 71

(2005) 026006, hep-th/0408187.


08.406256.au–

dau–

[40] J.A. Minahan, K. Zarembo, The Bethe-ansatz forN = 4 super-Yang–Mills, JHEP 0303 (2003) 013, hep-th/02122[41] G. Arutyunov, S. Frolov, M. Staudacher, Bethe ansatz for quantum strings, JHEP 0410 (2004) 016, hep-th/0[42] J.A. Minahan, A. Tirziu, A.A. Tseytlin, 1/J2 corrections to BMN energies from the quantum long range Land

Lifshitz model, hep-th/0510080.[43] J.A. Minahan, A. Tirziu, A.A. Tseytlin, 1/J corrections to semiclassical AdS/CFT states from quantum Lan

Lifshitz model, hep-th/0509071.[44] G.P. Korchemsky, Quasiclassical QCD pomeron, Nucl. Phys. B 462 (1996) 333, hep-th/9508025.

edn inde-mall in-ts. Thepect tohat ourscriptionithmicin loga-

ov

hmicfieldan cell


Affine Jordan cells, logarithmic correlators,and Hamiltonian reduction

Jørgen Rasmussen

Department of Mathematics and Statistics, University of Concordia,1455 Maisonneuve W, Montréal, Québec, Canada H3G 1M8

Received 13 September 2005; received in revised form 8 December 2005; accepted 9 December 2005

Available online 4 January 2006

Abstract

We study a particular type of logarithmic extension ofSL(2,R) Wess–Zumino–Witten models. It is bason the introduction of affine Jordan cells constructed as multiplets of quasi-primary fields organized icomposable representations of the Lie algebrasl(2). We solve the simultaneously imposed set of conforandSL(2,R) Ward identities for two- and three-point chiral blocks. These correlators will in generavolve logarithmic terms and may be represented compactly by considering spins with nilpotent parchiral blocks are found to exhibit hierarchical structures revealed by computing derivatives with resthe spins. We modify the Knizhnik–Zamolodchikov equations to cover affine Jordan cells and show tchiral blocks satisfy these equations. It is also demonstrated that a simple and well-established prefor Hamiltonian reduction at the level of ordinary correlators extends straightforwardly to the logarcorrelators as the latter then reduce to the known results for two- and three-point conformal blocksrithmic conformal field theory. 2005 Elsevier B.V. All rights reserved.

Keywords:Logarithmic conformal field theory; Jordan cell; Wess–Zumino–Witten model; Knizhnik–Zamolodchikequations; Hamiltonian reduction

1. Introduction

In logarithmic conformal field theory (CFT), a primary field may have a so-called logaritpartner field on which the Virasoro modes do not all act diagonally. If only one logarithmicis associated to a given primary field, the two fields constitute a so-called conformal Jord

E-mail address:[email protected](J. Rasmussen).




226 J. Rasmussen / Nuclear Physics B 736 [FS] (2006) 225–258

ernedo leadfing

nsioned inion

rimaryeeme

rimary,-eneralcasesaffine

fieldsividual

rtainedtion is

may beIt also

oup’

eriva--ect to

tors inas

al onesed inasoro

and

Hamil-l Ward

r-

of rank two where the rank indicates the number of fields in the cell. We will be concwith conformal Jordan cells of rank two only. The appearance of such cells is known tto logarithmic singularities in the correlators. We refer to[1] for the first systematic study ologarithmic CFT, and to[2–4] for recent reviews on the subject. An exposition of links to strtheory may be found in[5].

The objective of the present work is to introduce and study a particular logarithmic exteof the SL(2,R) Wess–Zumino–Witten (WZW) model. Alternative extensions have appearthe literature, see[4,6–9], for example, but all seem to differ significantly from ours in foundatand approach.

Our construction is based on a generalization of the standard multiplets of Virasoro pfields organized as spin-j representation. We find that an infinite number of partner fields sto be required to complete such an indecomposable representation ofsl(2), and we refer to thesnew multiplets as affine Jordan cells.

We consider the case where the logarithmic fields in the affine Jordan cells are quasi-pand discuss the conformal andSL(2,R) Ward identities which follow. Without making any simplifying assumptions about the operator-product expansions of the fields, we find the gsolutions for two- and three-point chiral blocks. Our results thus cover all the possiblebased on primary fields not belonging to affine Jordan cells, primary fields belonging toJordan cells, and the logarithmic partner fields completing the affine Jordan cells.

Most of our computations are based on the introduction of generating functions for theappearing in the various representations. This means that the affine correlators of the indfields are obtained by expanding certain generating-function chiral blocks.

A modification of the Knizhnik–Zamolodchikov (KZ) equations[10] is required to coveaffine Jordan cells in addition to primary fields. It is demonstrated that the chiral blocks obas solutions to the Ward identities satisfy these generalized KZ equations. This verificastraightforward once it has been established that the two- and three-point chiral blocksexpressed compactly in terms of spins with nilpotent parts. We show that this is possible.follows that a two- or three-point chiral block factorizes into a ‘conformal’ part and a ‘grpart.

The chiral blocks are found to exhibit hierarchical structures obtained by computing dtives with respect to the spins. This extends an observation made in[11–13]that the sets of twoand three-point conformal blocks in logarithmic CFT are linked via derivatives with respthe conformal weights.

It is noted that a simple distinction has been introduced as we refer to chiral correlathe WZW model aschiral blocks, while chiral correlators in logarithmic CFT are referred toconformal blocks. This is common practice.

A merit of our construction seems to be that the affine correlators reduce to the conformwhen a straightforward extension of the prescription for Hamiltonian reduction introduc[14,15]is employed. The idea is formulated in the realm of generating functions for the Virprimary fields in spin-j multiplets, and we find that it may be extended to affine Jordan cellsthus cover the reduction of our logarithmicSL(2,R) WZW model to logarithmic CFT.

This paper proceeds as follows. To fix our notation and to prepare for the discussion oftonian reduction, we first review the recently obtained general solutions to the conformaidentities for two- and three-point conformal blocks in logarithmic CFT[13]. This is followed bya discussion in Section3 of generating-function primary fields and their correlators inSL(2,R)

WZW models. This is the framework which we extend in Section4 and eventually use in ouanalysis of chiral blocks. In Section4, we thus describe the indecomposablesl(2) representa

J. Rasmussen / Nuclear Physics B 736 [FS] (2006) 225–258 227

e alsot in-arts, asails ared iniewed

g an

llulard asmbi-

orsnce is

tions underlying the affine Jordan cells. The correspondingly modified KZ equations arintroduced. Section5 concerns the explicit results on two- and three-point chiral blocks. Icludes a discussion of the factorization of the chiral blocks based on spins with nilpotent pwell as a discussion of the hierarchical structures of the chiral blocks. Some technical detdeferred toAppendix A. The extended prescription for Hamiltonian reduction is considereSection6, where it is demonstrated that our chiral blocks reduce to the conformal ones revin Section2. Section7 contains some concluding remarks.

2. Correlators in logarithmic CFT

2.1. Conformal Jordan cell

A conformal Jordan cell of rank two consists of two fields: a primary field,Φ, of conformalweight∆ and its non-primary, ‘logarithmic’ partner field,Ψ , on which the Virasoro algebra

(1)[Ln,Lm] = (n − m)Ln+m + c

12n(n2 − 1

)δn+m,0

generated byLn does not act diagonally. The central extension is denotedc. With a conven-tional relative normalization of the fields, we have[

Ln,Φ(z)] = (

zn+1∂z + ∆(n + 1)zn)Φ(z),

(2)[Ln,Ψ (z)

] = (zn+1∂z + ∆(n + 1)zn

)Ψ (z) + (n + 1)znΦ(z).

It has been suggested by Flohr[16] to describe these fields in a unified way by introducinnilpotent, yet even, parameterθ satisfyingθ2 = 0. We will follow this idea here, though use aapproach closer to the one employed in[13,17,18]. We thus define the field or unified cell

(3)Υ (z; θ ) = Φ(z) + θΨ (z),

which is seen to be ‘primary’ of conformal weight∆ + θ as the commutators(2) may be ex-pressed as

(4)[Ln,Υ (z; θ )

] = (zn+1∂z + (∆ + θ )(n + 1)zn

)Υ (z; θ ).

Following[13], a primary field belonging to a conformal Jordan cell is referred to as a ceprimary field. A primary fieldnot belonging to a conformal Jordan cell may be representeΥ (z;0), and we will reserve this notation for these non-cellular primary fields. To avoid aguities, we will therefore refrain from considering unified cellsΥ (z; θ ), as defined in(3), forvanishingθ .

2.2. Conformal Ward identities

We considerquasi-primary fieldsonly, ensuring the projective invariance of their correlatconstructed by sandwiching the fields between projectively invariant vacua. This invariamade manifest qua the conformal Ward identities which are given here forN -point conformalblocks:

0=N∑

∂zi

⟨Υ1(z1; θ1) · · ·ΥN (zN ; θN )

⟩,

i=1


rmal

e ind

d non-ansionsllulare on

ad

0=N∑i=1

(zi∂zi+ ∆i + θi )

⟨Υ1(z1; θ1) · · ·ΥN (zN ; θN )

⟩,

(5)0=(LN + 2

N∑i=1

θizi

)⟨Υ1(z1; θ1) · · ·ΥN (zN ; θN )

⟩.

To simplify the notation, we have introduced the differential operator

(6)LN =N∑i=1

(z2i ∂zi

+ 2∆izi

).

Information on the individual correlators may be extracted from solutions to the confoWard identities involving unified cells. In the case of

(7)⟨Υ1(z1; θ1)Υ2(z2;0)Υ3(z3; θ3)

⟩,

for example, the third conformal Ward identity(5) reads

(8)0= (L3 + 2(θ1z1 + θ3z3)

)⟨Υ1(z1; θ1)Υ2(z2;0)Υ3(z3; θ3)

⟩.

A solution to the complete set of conformal Ward identities is an expression expandablθ1andθ3. The term proportional toθ1 but independent ofθ3, for example, should then be identifiewith 〈Ψ1(z1)Υ2(z2;0)Φ3(z3)〉.

By construction, and as illustrated by this example, correlators involving unified cells ancellular primary fields may thus be regarded as generating-function correlators whose expin the nilpotent parameters give the individual correlators involving combinations of ceprimary fields, non-cellular primary fields, and logarithmic fields. Our focus will therefore bcorrelators of combinations of unified cells and non-cellular primary fields.

2.3. Two-point conformal blocks

Based on the ansatz

(9)⟨Υ1(z1; θ1)Υ2(z2; θ2)

⟩ = A(θ1, θ2) + B(θ1, θ2) ln z12

z2h12

,

where

(10)A(θ1, θ2) = A0 + A1θ1 + A2θ2 + A12θ1θ2,

and similarly forB(θ1, θ2), the general (generating-function) two-point conformal blocks re⟨Υ1(z1;0)Υ2(z2;0)

⟩ = A0V2,⟨Υ1(z1; θ1)Υ2(z2;0)

⟩ = A1θ1V2,⟨Υ1(z1;0)Υ2(z2; θ2)

⟩ = A2θ2V2,

(11)⟨Υ1(z1; θ1)Υ2(z2; θ2)

⟩ = A1θ1 + A1θ2 + (

A12 − 2A1 ln z12)θ1θ2

V2.

Here we have introduced the shorthand notation

(12)V2 = δ∆1,∆2

z∆1+∆2

.

12


ndentsednce to

lyidual

e

int

To keep the notation simple, we are using the standard abbreviationzij = zi −zj . It is understoodthat anA1, for example, appearing in one (generating-function) correlator a priori is indepeof anA1 appearing in another. Also, even thoughA2 does not appear explicitly in some of theexpressions, it may nevertheless be related toA1. For the sake of simplicity, the solutions listehere are merely indicating the general form and the degrees of freedom without referethe fate of all the various parameters appearing in the ansatz(9). Similar comments also appto the results on correlators discussed in the following. Finally, the solutions for the indivtwo-point conformal blocks are easily extracted[13].

By consideringθi as the nilpotent part of the generalized conformal weight∆i + θi [13,17],one may represent the results(11)as

⟨Υ1(z1;0)Υ2(z2;0)

⟩ = δ∆1,∆2

A0

z∆1+∆212

,

⟨Υ1(z1; θ1)Υ2(z2;0)

⟩ = δ∆1,∆2

A1θ1

z(∆1+θ1)+∆212

,

(13)⟨Υ1(z1; θ1)Υ2(z2; θ2)

⟩ = δ∆1,∆2

A1θ1 + A1θ2 + A12θ1θ2

z(∆1+θ1)+(∆2+θ2)12

.

The similar expression for the correlator〈Υ1(z1;0)Υ2(z2; θ2)〉 is obtained from the second onby interchanging the indices.

2.4. Three-point conformal blocks

Based on the ansatz⟨Υ1(z1; θ1)Υ2(z2; θ2)Υ3(z3; θ3)

⟩=

A(θ1, θ2, θ3) + B1(θ1, θ2, θ3) ln z12 + B2(θ1, θ2, θ3) ln z23 + B3(θ1, θ2, θ3) ln z13

+ D11(θ1, θ2, θ3) ln2 z12 + D12(θ1, θ2, θ3) ln z12 ln z23 + D13(θ1, θ2, θ3) ln z12 ln z13

+ D22(θ1, θ2, θ3) ln2 z23 + D23(θ1, θ2, θ3) ln z23 ln z13

(14)+ D33(θ1, θ2, θ3) ln2 z13z−h112 z

−h223 z

−h313 ,

where

A(θ1, θ2, θ3) = A0 + A1θ1 + A2θ2 + A3θ3

(15)+ A12θ1θ2 + A23θ2θ3 + A13θ1θ3 + A123θ1θ2θ3,

and similarly forBi(θ1, θ2, θ3) andDij (θ1, θ2, θ3), the general (generating-function) three-poconformal blocks read⟨

Υ1(z1;0)Υ2(z2;0)Υ3(z3;0)⟩ = A0V3,⟨

Υ1(z1; θ1)Υ2(z2;0)Υ3(z3;0)⟩ =

A0 + A1θ1 − A0θ1 lnz12z13

z23

V3,⟨

Υ1(z1; θ1)Υ2(z2; θ2)Υ3(z3;0)⟩

=A0 + A1θ1 − A0θ1 ln

z12z13 + A2θ2 − A0θ2 lnz12z23

z23 z13


repre-

.

-

+ A12θ1θ2 − A1θ1θ2 lnz12z23

z13− A2θ1θ2 ln

z12z13

z23+ A0θ1θ2 ln

z12z23

z13ln

z12z13

z23

V3,⟨

Υ1(z1; θ1)Υ2(z2; θ2)Υ3(z3; θ3)⟩

=A1θ1 + A2θ2 + A3θ3 + A12θ1θ2 − A1θ1θ2 ln

z12z23

z13− A2θ1θ2 ln

z12z13

z23

+ A23θ2θ3 − A2θ2θ3 lnz23z13

z12− A3θ2θ3 ln

z13

z12z23+ A13θ1θ3 − A1θ1θ3 ln

z23z13

z12

− A3θ1θ3 lnz12z13

z23+ A123θ1θ2θ3 − A12θ1θ2θ3 ln

z23z13

z12

− A23θ1θ2θ3 lnz12z13

z23− A13θ1θ2θ3 ln

z12z23

z13+ A1θ1θ2θ3 ln

z23z12

z13ln

z23z13

z12

(16)+ A2θ1θ2θ3 lnz12z13

z23ln

z23z13

z12+ A3θ1θ2θ3 ln

z12z23

z13ln

z12z13

z23

V3.

Here we have introduced the abbreviation

(17)V3 = 1

z∆1+∆2−∆312 z

−∆1+∆2+∆323 z

∆1−∆2+∆313

.

The remaining correlators are obtained by appropriate permutations in the indices.As in the case of two-point conformal blocks, the three-point conformal blocks may be

sented in terms of generalized conformal weights,∆i + θi :

⟨Υ1(z1;0)Υ2(z2;0)Υ3(z3;0)

⟩ = A0

z∆1+∆2−∆312 z

−∆1+∆2+∆323 z

∆1−∆2+∆313

,

⟨Υ1(z1; θ1)Υ2(z2;0)Υ3(z3;0)

⟩ = A0 + A1θ1

z(∆1+θ1)+∆2−∆312 z

−(∆1+θ1)+∆2+∆323 z

(∆1+θ1)−∆2+∆313

,

⟨Υ1(z1; θ1)Υ2(z2; θ2)Υ3(z3;0)

⟩= A0 + A1θ1 + A2θ2 + A12θ1θ2

z(∆1+θ1)+(∆2+θ2)−∆312 z

−(∆1+θ1)+(∆2+θ2)+∆323 z

(∆1+θ1)−(∆2+θ2)+∆313

,

⟨Υ1(z1; θ1)Υ2(z2; θ2)Υ3(z3; θ3)

⟩(18)= A1θ1 + A2θ2 + A3θ3 + A12θ1θ2 + A23θ2θ3 + A13θ1θ3 + A123θ1θ2θ3

z(∆1+θ1)+(∆2+θ2)−(∆3+θ3)12 z

−(∆1+θ1)+(∆2+θ2)+(∆3+θ3)23 z

(∆1+θ1)−(∆2+θ2)+(∆3+θ3)13

.

The remaining four combinations are obtained by appropriate permutations in the indices

2.5. Hierarchical structures for conformal blocks

Based on ideas discussed in[11,12], it was found in[13] that the correlators involving logarithmic fields may be represented as follows:⟨

Ψ1(z1)Υ2(z2;0)⟩ = A1V2,⟨

Ψ1(z1)Φ2(z2)⟩ = A1V2,⟨

Ψ1(z1)Ψ2(z2)⟩ = (

A12 + A2∂ ˆ + A1∂ ˆ)V2,
∆1 ∆2


refore

tained

⟨Ψ1(z1)Υ2(z2;0)Υ3(z3;0)

⟩ = (A1 + A0∂

∆1

)V3,⟨

Ψ1(z1)Φ2(z2)Υ3(z3;0)⟩ = (

A1 + A0∂∆1

)V3,⟨

Ψ1(z1)Ψ2(z2)Υ3(z3;0)⟩ = (

A12 + A1∂∆2

+ A2∂∆1

+ A0∂∆1

∂∆2

)V3,⟨

Ψ1(z1)Φ2(z2)Φ3(z3)⟩ = A1V3,⟨

Ψ1(z1)Ψ2(z2)Φ3(z3)⟩ = (

A12 + A2∂∆1

+ A1∂∆2

)V3,⟨

Ψ1(z1)Ψ2(z2)Ψ3(z3)⟩ = (

A123+ A23∂∆1

+ A13∂∆2

+ A12∂∆3

(19)+ A3∂∆1

∂∆2

+ A1∂∆2

∂∆3

+ A2∂∆1

∂∆3

)V3,

in addition to expressions obtained by appropriately permuting the indices. One may therepresent the correlators hierarchically as⟨

Ψ1(z1)Υ2(z2;0)⟩ = A1V2 + ∂

∆1

⟨Φ1(z1)Υ2(z2;0)

⟩,⟨

Ψ1(z1)Φ2(z2)⟩ = A1V2 + ∂

∆1

⟨Φ1(z1)Φ2(z2)

⟩,⟨

Ψ1(z1)Ψ2(z2)⟩ = A12V2 + ∂

∆1

⟨Φ1(z1)Ψ2(z2)

⟩ + ∂∆2

⟨Ψ1(z1)Φ2(z2)

⟩(20)− ∂

∆1∂∆2

⟨Φ1(z1)Φ2(z2)

⟩,

in the case of two-point conformal blocks, and⟨Ψ1(z1)Υ2(z2;0)Υ3(z3;0)

⟩ = A1V3 + ∂∆1

⟨Φ1(z1)Υ2(z2;0)Υ3(z3;0)

⟩,⟨

Ψ1(z1)Φ2(z2)Υ3(z3;0)⟩ = A1V3 + ∂

∆1

⟨Φ1(z1)Φ2(z2)Υ3(z3;0)

⟩,⟨

Ψ1(z1)Ψ2(z2)Υ3(z3;0)⟩ = A12V3 + ∂

∆1

⟨Φ1(z1)Ψ2(z2)Υ3(z3;0)

⟩+ ∂

∆2

⟨Ψ1(z1)Φ2(z2)Υ3(z3;0)

⟩− ∂

∆1∂∆2

⟨Φ1(z1)Φ2(z2)Υ3(z3;0)

⟩,⟨

Ψ1(z1)Φ2(z2)Φ3(z3)⟩ = A1V3 + ∂

∆1

⟨Φ1(z1)Φ2(z2)Φ3(z3)

⟩,⟨

Ψ1(z1)Ψ2(z2)Φ3(z3)⟩ = A12V3 + ∂

∆1

⟨Φ1(z1)Ψ2(z2)Φ3(z3)

⟩ + ∂∆2

⟨Ψ1(z1)Φ2(z2)Φ3(z3)

⟩− ∂

∆1∂∆2

⟨Φ1(z1)Φ2(z2)Φ3(z3)

⟩,⟨

Ψ1(z1)Ψ2(z2)Ψ3(z3)⟩ = A123V3 + ∂

∆1

⟨Φ1(z1)Ψ2(z2)Ψ3(z3)

⟩ + ∂∆2

⟨Ψ1(z1)Φ2(z2)Ψ3(z3)

⟩+ ∂

∆3

⟨Ψ1(z1)Ψ2(z2)Φ3(z3)

⟩ − ∂∆1

∂∆2

⟨Φ1(z1)Φ2(z2)Ψ3(z3)

⟩− ∂

∆2∂∆3

⟨Ψ1(z1)Φ2(z2)Φ3(z3)

⟩ − ∂∆1

∂∆3

⟨Φ1(z1)Ψ2(z2)Φ3(z3)

⟩(21)+ ∂

∆1∂∆2

∂∆3

⟨Φ1(z1)Φ2(z2)Φ3(z3)

⟩,

in the case of three-point conformal blocks. As above, the remaining correlators may be obby appropriately permuting the indices.

3. On SL(2,RRR) WZW models

The affinesl(2)k Lie algebra, including the commutators with the Virasoro modes, reads

[J+,n, J−,m] = 2J0,n+m + knδn+m,0,


e con-d

treat

p-

s of

ized in

tion

[J0,n, J±,m] = ±J±,n+m,

[J0,n, J0,m] = k

2nδn+m,0,

(22)[Ln,Ja,m] = −mJa,n+m.

Another conventional notation is obtained by replacingJ+,n,2J0,n, J−,n by En,Hn,Fn. Thelevel of the algebra is indicated byk and is related to the central charge asc = 3k/(k + 2). Thenon-vanishing entries of the Cartan–Killing form ofsl(2) are given by

(23)κ00 = 1

2, κ+− = κ−+ = 1,

and appear as coefficients to the central terms in(22). Its inverse is given by

(24)κ00 = 2, κ+− = κ−+ = 1,

and comes into play when discussing the affine Sugawara construction below. We will bcerned mainly with the ‘horizontal’ part of the affine Lie algebra, thesl(2) Lie algebra generateby the zero modesJa := Ja,0.

We will assume that the Virasoro primary fields of a given conformal weight∆ may be orga-nized in multiplets corresponding to spin-j representations of thesl(2) algebra, where

(25)∆ = j (j + 1)

k + 2.

In the following, j is taken to be real even though the general formalism is amenable toj complex as well. If 2j is a non-negative integer, we may label the 2j + 1 members of theassociated multiplet as in

(26)φ−j (z), φ−j+1(z), . . . , φj−1(z), φj (z),

where the dependence onj , often indicated byφj,m, is suppressed. A finite-dimensional reresentation like(26) is often referred to as an integrable representation. The fieldφm hasJ0eigenvaluem, while we will use the following convenient choice of relative normalizationthe fields:[

J+, φm(z)] = (j + m + 1)φm+1(z),[

J0, φm(z)] = mφm(z),

(27)[J−, φm(z)

] = (j − m + 1)φm−1(z).

If 2j is not a non-negative integer, the associated primary fields may in general be organan infinite-dimensional multiplet corresponding to ansl(2) representation.

3.1. Generating-function primary fields

A generating function for the 2j + 1 Virasoro primary fields in an integrable representamay be written[19]

(28)φ(z, x) =j∑

m=−j

φm(z)xj−m.


etherr half-ltiplet,

fields,

s

affineterms

eatedgives

of

To keep the notation simple here and in the following, we do not indicate explicitly whthe sum is over integers or half-integers as this should be obvious from the integer ointeger nature of the spin itself. For general spin and associated infinite-dimensional muthe generating function for a so-called highest-weight representation, for example, reads

(29)φ(z, x) =∑

m∈(j+Z)

φm(z)xj−m.

The adjoint action of the affine generators on the generating-function primary field reads

(30)[Ja,n,φ(z, x)

] = −znDa(x)φ(z, x),

where the differential operatorsDa(x) are defined by

(31)D+(x) = x2∂x − 2jx, D0(x) = x∂x − j, D−(x) = −∂x.

They generate the Lie algebrasl(2), and one recovers(27) from (30).A correlator like theN -point chiral block

(32)⟨φ1(z1, x1) · · ·φN (zN , xN )

⟩is seen to correspond to a generating function for the individual correlators based onφi,mi

(zi), appearing in expansions like(28) (or (29), for example). That is, theN -point chiralblock

(33)⟨φ1,m1(z1) · · ·φN ,mN (zN )

⟩appears as the coefficient to

∏Ni=1 x

ji−mi

i in an expansion of(32). The general expansion thureads⟨

φ1(z1, x1) · · ·φN (zN , xN )⟩

(34)=∑

m1,...,mN

⟨φ1,m1(z1) · · ·φN ,mN (zN )

⟩x

j1−m11 · · ·xjN −mN

N ,

where the ranges of the summation variables depend on the individual spin-ji representations.

3.2. The KZ equations

In a WZW model, the Virasoro generators are realized as bilinear expressions in thegenerators. This is referred to as the affine Sugawara construction which is here written inof modes

(35)LN = 1

2(k + 2)κab

( ∑n−1

Ja,nJb,N−n +∑n0

Ja,N−nJb,n

).

Here and in the following, we will use the convention of summing over appropriately repgroup indices,a = ±,0. Acting on a highest-weight state, the affine Sugawara constructionrise to singular vectors of the combined algebra. The decoupling of these is trivial forN > 0. ForN = 0, it reproduces the relation(25) as the eigenvalue ofL0 is equated with the eigenvaluethe normalized quadratic Casimir:

(36)∆ = κabDa(x)Db(x) = j (j + 1).

2(k + 2) k + 2


t.

iden-

ed ofordanwith

eferaffines rel-fields

in theeension

son

The condition corresponding toN = −1 leads to the celebrated KZ equations[10] which arewritten here for anN -point chiral block of generating-function primary fields

(37)0= KZi

⟨φ1(z1, x1) · · ·φN (zN , xN )

⟩, i = 1, . . . ,N ,

where

(38)KZi =(

(k + 2)∂zi−

∑j =i

κabDa(xi)Db(xj )

zi − zj

).

TheseN differential equations associated to a givenN -point chiral block are not all independenThis is easily illustrated by considering the sum

(39)N∑i=1

KZi = (k + 2)

N∑i=1

∂zi,

which merely induces translational invariance already imposed by the first conformal Wardtity (5). As we will discuss below, a simple modification of the KZ equations(37), (38)apply tocorrelators involving certain logarithmic fields to be introduced in the following.

4. Affine Jordan cells

We wish to consider the situation where every Virasoro primary field in a givensl(2) rep-resentation may have a logarithmic partner. The resulting multiplet of fields is comprisprimary fields as well as so-called logarithmic fields and will be referred to as an affine Jcell. A priori, the hosting model may consist of a family of affine Jordan cells in coexistencean independent family of multiplets of primary fields without logarithmic partners. We will rloosely to such a model as a logarithmic WZW model. Primary fields not appearing in anJordan cell will be called non-cellular primary fields. It is found that the affine Jordan cellevant to our studies contain primary fields not having logarithmic partners. These primaryare naturally included in the generating functions for the logarithmic fields rather thangenerating functions for the primary fields comprising the original spin-j representation we arextending. To reach this appreciation of the affine Jordan cells, we initially consider an extof the differential-operator realization(31)and its role in a generalization of(30).

4.1. Generating-function unified cells

The differential-operator realization

D+(x; θ) = x2∂x − 2(j + θ)x,

D0(x; θ) = x∂x − (j + θ),

(40)D−(x; θ) = −∂x,

of the Lie algebrasl(2) is designed to act on a representation of (generalized) spinj + θ , whereθ is a nilpotent, yet even, parameter satisfyingθ2 = 0. Extending the idea of organizing fieldin generating functions as in(28) satisfying(30), we introduce the formal generating-functiunified cellΥ (z, x; θ) satisfying

(41)[Ja,Υ (z, x; θ)

] = −Da(x; θ)Υ (z, x; θ).


ng

sresent

isntationso.

oneioned,l spin-c

ing

We note that this also applies to generating-function primary fields as it reduces to(30) (forn = 0) when we setθ = 0. Here and in the following, focus is on thesl(2) Lie algebra part of theaffine generators. An expansion of the generating-function unified cell with respect toθ may bewritten

(42)Υ (z, x; θ) = Φ(z, x) + θΨ (z, x),

resembling the definition of the unified cell(3) in logarithmic CFT. In terms of the new generatifunctions,Φ(z, x) andΨ (z, x), the commutators(41) read[

J+,Φ(z, x)] = −D+(x)Φ(z, x),[

J+,Ψ (z, x)] = −D+(x)Ψ (z, x) + 2xΦ(z, x),[

J0,Φ(z, x)] = −D0(x)Φ(z, x),[

J0,Ψ (z, x)] = −D0(x)Ψ (z, x) + Φ(z, x),[

J−,Φ(z, x)] = −D−(x)Φ(z, x),

(43)[J−,Ψ (z, x)

] = −D−(x)Ψ (z, x),

where the differential operators,Da(x), are given in(31).These commutators severely restrict the set ofsl(2) representations for which the two field

Φ(z, x) andΨ (z, x) can be considered generating functions. It is beyond the scope of the pwork, though, to classify these representations, even in the simple case whereΦ(z, x) is thegenerating function for a finite-dimensional representation as in(28). We hope to address thclassification elsewhere. Here we merely wish to demonstrate the existence of represecorresponding to the generating functions(42), (43)and to illustrate their complexity. We will dso by considering a particular logarithmic extension of a finite-dimensional spin-j representationMore general examples are considered in Section4.2.

We thus introduce the following expansions of the generating functionsΦ(z, x) andΨ (z, x):

(44)Φ(z, x) =j∑

m=−j

Φm(z)xj−m, Ψ (z, x) =j∑

m=−∞Ψm(z)xj−m.

The remark following(28)aboutm taking on integer or half-integer values also applies when(or even both) of the summation bounds is (either plus or minus) infinity. As already mentwe are concerned with Jordan cells whose principal parts correspond to finite-dimensionajrepresentations, here governed by the generating functionΦ(z, x). It is noted that the logarithmipart, on the other hand, consists of infinitely many fields.

With the understanding thatΦm(z) only exists form = −j, . . . , j while Ψm(z) only exists form = −∞, . . . , j , the adjoint action of thesl(2) Lie algebra on the modes of the two generatfunctions(44)may be written compactly as[

J+,Φm(z)] = (j + m + 1)Φm+1(z),[

J+,Ψm(z)] = (j + m + 1)Ψm+1(z) + 2Φm+1(z),[

J0,Φm(z)] = mΦm(z),[

J0,Ψm(z)] = mΨm(z) + Φm(z),[

J−,Φm(z)] = (j − m + 1)Φm−1(z),

(45)[J−,Ψm(z)

] = (j − m + 1)Ψm−1(z).


pears

ns

t leastt have

f

see

teda-

an-

ation ofions of

This is equivalent to simply setting a non-existing field equal to zero whenever it formally apin (45). The following diagram may help visualizing the representation:

(46)

Ψ−j−1

Φ−j

Ψ−j

Φ−j+1

Ψ−j+1

Φj

Ψj←→ ←−

←→

←→

←→

←→

←→

←→

↓ ↓ ↓

· · ·

· · ·

· · ·

· · ·J−, J+ J− J−, J+ J−, J+ J−, J+

J−, J+ J−, J+ J−, J+

J+ J+ J+ J+J0 J0 J0

Here the arrows indicate the adjoint actions ofJa (except the primary parts of the adjoint actioof J0 which are not indicated explicitly). It is observed that only 2j + 1 of the fieldsΨm(z)

are logarithmic fields, where a logarithmic field is characterized by the property that aone of the affine generators acts non-diagonally on them. Here, in particular, they do nowell-definedJ0 eigenvalues.

The naive expansion whereΨ (z, x) is a sum of 2j + 1 fields similar to the expansion oΦ(z, x) turns out to be inconsistent. The same problem occurs when trying to writeΨ (z, x) asan infinite sum from−j to ∞, as it actually occurs for all power-series expansions ofΨ (z, x)

having lowest magnetic moment,m, equal to−j . This asymmetry in extensions beyondj and−j , respectively, stems from the fact thatJ+ may act non-diagonally (in the algebraic sen(45), i.e., diagonally in the diagram(46)) while J− only acts diagonally (i.e., horizontally in thdiagram(46)). One can extend in both directions simultaneously

(47)Φ(z, x) =j∑

m=−j

Φm(z)xj−m, Ψ (z, x) =∞∑

m=−∞Ψm(z)xj−m,

in which case one obtains the following reducible extension of(46):

(48)

Ψ−j−1

Φ−j

Ψ−j

Φj

Ψj+1Ψj←→ ←−

←→

←→

←→

←→ −→ ←→

↓ ↓

· · ·

· · ·

· · ·· · · · · ·J− J−, J+ J−, J+ J+ J−, J+

J−, J+ J−, J+

J+ J+ J+J0 J0

The representation(46) is obtained from(48) by factoring out the submodule generafrom Ψj+1. The form of the structure constants in(45) allow us to indicate both representtions by the same commutator algebra(45) and in both cases write the expansion ofΨ (z, x)

as a sum over allm. Infinitely many terms will be redundant, though, when writing the expsion corresponding to(46) in this way. Other representations can be envisaged (cf. Section4.2),and as already indicated, we hope to return elsewhere with a discussion of the classificaffine Jordan cells defined as logarithmic (i.e., non-diagonal or indecomposable) extensintegrable or non-integrable (affine)sl(2) representations.


d

f.n

have

er-

e

s

se-ltonian

s

In the two examples discussed above,(44) and (47), the generating function for the unifiecell (42)may be expanded as

(49)Υ (z, x; θ) =∑m

Υm(z; θ)xj−m =∑m

Φm(z)xj−m + θ∑m

Ψm(z)xj−m,

where the ranges for the summation variables may be different in the last two sums, c(47),for example. Analogous to the discussion of unified cells following(4), we reserve the notatioΥ (z, x;0) for generating-function primary fields not belonging to a unified cell like(49). In termsof the modes of the generating-function Jordan cell given in(49), the commutators(45) read[

J+,Υm(z; θ)] = (j + m + 1+ 2θ)Υm+1(z; θ),[

J0,Υm(z; θ)] = (m + θ)Υm(z; θ),

(50)[J−,Υm(z; θ)

] = (j − m + 1)Υm−1(z; θ).

As in (4) where∆ + θ may be interpreted as a generalized conformal weight, we nowa generalized spin and associated generalized magnetic moments given byj + θ andm + θ ,respectively. This was already indicated following(40).

It is recalled that a correlator of generating functions like(32) may be regarded as a genating function for the individual conformal blocks, cf.(34). This principle extends toN -pointchiral blocks involving generating-function unified cells. If allN generating functions may bexpanded as in(49), we then have⟨

Υ1(z1, x1; θ1) · · ·ΥN (zN , xN ; θN )⟩

=∑

m1,...,mN

⟨Υ1(z1; θ1) · · ·ΥN (zN ; θN )

⟩x

j1−m11 · · ·xjN −mN

N

=∑

m1,...,mN

⟨Φ1(z1) · · ·ΦN (zN )

⟩ + θ1⟨Ψ1(z1)Φ2(z2) · · ·ΦN (zN )

⟩ + · · ·

(51)+ θ1 · · · θN⟨Ψ1(z1) · · ·ΨN (zN )

⟩x

j1−m11 · · ·xjN −mN

N

whereΥi(zi, xi; θi) denotes a generating-function primary field ifθi = 0. As above, the rangeof the summation variables depend on the individualsl(2) representations.

4.2. More on indecomposable sl(2) representations

It is stressed that the differential-operator realization(40)and the generating-function(49)donot exhaust all possible extensions of the ordinary WZW model outlined in Section3. This isillustrated by the models discussed in[4,6–9], for example, and will be addressed further elwhere. The construction developed in the present work has the virtue that it, under Hamireduction, reduces to the non-affine logarithmic CFT reviewed in Section2. This will be the topicof Section6 below. Here we wish to indicate the level of complexity of thesl(2) representationassociated to more general expansions ofΨ (z, x) than power-series expansions such as(47).

To this end, we consider the expansions

(52)Φ(z, x) =∑m

Φm(z)xj−m, Ψ (z, x) =∑m,n

Ψm,n(z)xj−m lnn x,


utators

ncee

d

on annd

where we have left the summation ranges unspecified. In terms of these modes, the comm(43) read[

J+,Φm(z)] = (j + m + 1)Φm+1(z),[

J+,Ψm,n(z)] = (j + m + 1)Ψm+1,n(z) − (n + 1)Ψm+1,n+1(z) + 2Φm+1(z),[

J0,Φm(z)] = mΦm(z),[

J0,Ψm,n(z)] = mΨm+1,n(z) − (n + 1)Ψm,n+1(z) + Φm(z),[

J−,Φm(z)] = (j − m + 1)Φm−1(z),

(53)[J−,Ψm,n(z)

] = (j − m + 1)Ψm−1,n(z) + (n + 1)Ψm−1,n+1(z),

where a field whose indices do not match the expansion(52) is set to zero. To illustrate such aindecomposablesl(2) representation, we letn run from 0 to 2 and focus on a typical sequenin the magnetic moments:m − 1, m, m + 1, where−j < m < j . The announced part of thcorresponding diagram then looks like

(54)

Φm−1

Ψm−1,0

Ψm−1,1

Ψm−1,2

Φm

Ψm,0

Ψm,1

Ψm,2

Φm+1

Ψm+1,0

Ψm+1,1

Ψm+1,2

←→ ←→

←→ ←→

←→ ←→

←→ ←→

↓ ↓ ↓

↑ ↑ ↑

↑ ↑ ↑

· · · · · ·

· · · · · ·

· · · · · ·

· · · · · ·

As in (46) and (48), the arrows indicate the adjoint actions of thesl(2) generators. The kinkearrows refer to parts of theJ0 actions.

4.3. Modified KZ equations

The logarithmic WZW model hosting the affine Jordan cells introduced above, is basedextension of the affine Sugawara construction(35). As the actions of the Virasoro modes depeon the target field being a unified cell or not (compare(4) to the first commutator in(2)), the


t have a

t

n uni-e

sat-g-

ell.

cedrrela-nd

iral

actions of the affine modes appearing in the extended affine Sugawara construction mussimilar dependence. The generalization of(36) thus reads

(55)∆ + µ = κabDa(x; θ)Db(x; θ)

2(k + 2)= (j + θ)(j + θ + 1)

k + 2,

from which it follows that the nilpotent part of the conformal weight,µ, is related to the nilpotenpart of the spin,θ , as

(56)µ = 2j + 1

k + 2θ.

Likewise, the KZ equations may be extended to cover correlators of generating-functiofied cells simply by replacingDa(xi) by Da(xi; θi) if the ith field is such a unified cell. With thunderstanding that a non-cellular generating-function primary field,Υi(zi, xi;0), corresponds tosettingθi = 0 in Υi(zi, xi; θi), the modified KZ equations read

(57)0= KZi

⟨Υ1(z1, x1; θ1) · · ·ΥN (zN , xN ; θN )

⟩, i = 1, . . . ,N ,

where

(58)KZi = (k + 2)∂zi−

∑j =i

κabDa(xi; θi)Db(xj ; θj )

zi − zj

.

As in the non-logarithmic case, theseN differential equations are not all independent as theyisfy (39). This is true for all combinations ofN generating-function fields, i.e., every generatinfunction field can be a generating-function primary field or a generating-function Jordan c

5. Correlators in logarithmic SL(2,RRR) WZW models

We now turn to the computation of correlators in the logarithmic WZW model introduabove. Focus will be on two- and three-point chiral blocks of generating-functions. The cotors are worked out asSL(2,R) group-invariant solutions to the conformal Ward identities aare subsequently demonstrated to satisfy the generalized KZ equations(57), (58). This meansthat conformal andSL(2,R) group invariance fix the form of the two- and three-point chblocks as is the case in the ordinary, non-logarithmicSL(2,R) WZW model.

5.1. SL(2,R) group invariance and conformal Ward identities

Bearing the link(56) in mind, the conformal Ward identities(5) now read

0=N∑i=1

∂zi

⟨Υ1(z1, x1; θ1) · · ·ΥN (zN , xN ; θN )

⟩,

0=N∑i=1

(zi∂zi

+ ∆i + 2ji + 1

k + 2θi

)⟨Υ1(z1, x1; θ1) · · ·ΥN (zN , xN ; θN )

⟩,

(59)0=(LN + 2

N∑i=1

2ji + 1

k + 2θizi

)⟨Υ1(z1, x1; θ1) · · ·ΥN (zN , xN ; θN )

⟩,


aidesnce.l referting-e ones

.ttors,

-

rsd ons isintpartchiralertion.aking

riori, sot chiral

would

focus

where the differential operatorLN is defined in(6). Correlators satisfying these identities are sto be projectively invariant. Likewise, invariance underSL(2,R) group transformations (the ongenerated by the horizontalsl(2) algebra) is sometimes referred to as loop-projective invariaThe corresponding Ward identities are often called affine Ward identities, though we wilto them asSL(2,R) Ward identities. For generating-function correlators involving generafunction Jordan cells (and possibly non-cellular generating-function primary fields) as thappearing in(59), they read

0=N∑i=1

∂xi

⟨Υ1(z1, x1; θ1) · · ·ΥN (zN , xN ; θN )

⟩,

0=N∑i=1

(xi∂xi− ji − θi)

⟨Υ1(z1, x1; θ1) · · ·ΥN (zN , xN ; θN )

⟩,

(60)0=(JN − 2

N∑i=1

θixi

)⟨Υ1(z1, x1; θ1) · · ·ΥN (zN , xN ; θN )

⟩.

Here we have introduced the differential operator

(61)JN =N∑i=1

(x2i ∂xi

− 2jixi

).

It is noted that the middle identities in(59) and (60)follow from the first and third identitiesThis is a simple consequence of[L1,L−1] = 2L0 and [J+, J−] = 2J0, respectively. The firsconformal andSL(2,R) Ward identities merely impose translation invariance on the correlaallowing us to express them solely in terms of differences,zi − zj andxi − xj , between coordinates of the same type.

The two sets of identities are very similar in nature, as the act of replacing(xi, ji , θi) by(zi,−∆i,−µi) in the operators appearing in(60) leads to(59). Also, one of the sets of operatodoes not depend on the group coordinatesxi , while the other set of operators does not depenthe conformal coordinateszi . We know that the form of two- and three-point conformal blockfixed by the conformal Ward identities, cf. Section2. We also know that the two- and three-pochiral blocks involving only generating-function primary fields factorize into a ‘conformal’and a ‘group’ part. It is therefore natural to expect that the general two- and three-pointblocks may factor into a conformal part and a group part. Our analysis will support this assIt is demonstrated in the process, though, that one would miss a wealth of solutions by msuch a factorization ansatz too naively. Furthermore, a factorization is not guaranteed a pwe will base our analysis on very general ansatzes for the form of the two- and three-poinblocks.

5.2. Two-point chiral blocks

Let us first comment on the factorization ansatz alluded to above. The naive approachbe to consider two-point chiral blocks constructed by multiplying expressions of the form(11)with similar expressions for the group part. To illustrate the shortage of this procedure, we


chiralloga-ntly,

r-sets

sen-

llserating-ne in

on the two-point chiral block of two generating-function unified cells:

⟨Υ1(z1, x1; θ1)Υ2(z2, x2; θ2)

⟩ = δ∆1,∆2

A1(µ1 + µ2 − 2µ1µ2 ln z12) + A12µ1µ2

z∆1+∆212

× δj1,j2

B1(θ1 + θ2 + 2θ1θ2 lnx12) − B12θ1θ2

x

j1+j212

(62)= δj1,j2

2A1B1(j1 + j2 + 1)

k + 2θ1θ2

xj1+j212

z∆1+∆212

.

This could have been the end of the story in which case the only non-vanishing two-pointblock of the individual fields would have been the one containing two generating-functionrithmic fields. Furthermore, no logarithmic singularity would occur. As we will discuss presethere are more non-trivial solutions than this one.

We will base our analysis on the following ansatz:

(63)⟨Υ1(z1, x1; θ1)Υ2(z2, x2; θ2)

⟩ = A(θ1, θ2) + B(θ1, θ2) ln z12 + C(θ1, θ2) lnx12

x2s12

z2h12

,

where

(64)A(θ1, θ2) = A0 + A1θ1 + A2θ2 + A12θ1θ2,

and similarly forB(θ1, θ2) andC(θ1, θ2). Correlators involving non-cellular primary fields corespond to setting the associatedθs equal to zero. Due to the translational invariance (in bothof coordinates) of the ansatz, it suffices to impose the two identities involvingLN andJN . Thethird conformal Ward identity(59) thus leads to the conditions

0=(

−h + ∆1 + 2j1 + 1

k + 2θ1

)A(θ1, θ2) + 1

2B(θ1, θ2),

0=(

−h + ∆2 + 2j2 + 1

k + 2θ2

)A(θ1, θ2) + 1

2B(θ1, θ2),

0=(

−h + ∆1 + 2j1 + 1

k + 2θ1

)B(θ1, θ2) =

(−h + ∆2 + 2j2 + 1

k + 2θ2

)B(θ1, θ2),

(65)0=(

−h + ∆1 + 2j1 + 1

k + 2θ1

)C(θ1, θ2) =

(−h + ∆2 + 2j2 + 1

k + 2θ2

)C(θ1, θ2),

whereas the thirdSL(2,R) Ward identity corresponds to the conditions

0= (s − j1 − θ1)A(θ1, θ2) + 1

2C(θ1, θ2) = (s − j2 − θ2)A(θ1, θ2) + 1

2C(θ1, θ2),

0= (s − j1 − θ1)B(θ1, θ2) = (s − j2 − θ2)B(θ1, θ2),

(66)0= (s − j1 − θ1)C(θ1, θ2) = (s − j2 − θ2)C(θ1, θ2).

We defer the analysis of these conditions toAppendix A. It is noted, though, that as in the caof (non-affine) conformal Jordan cells[13], one may lose solutions for correlators involving nocellular fields if one simply sets the correspondingθs equal to zero in the solution for unified ceonly. Instead, examining the conditions case by case (distinguished by the number of genfunction unified cells appearing in the generating-function two-point chiral block) as do


eds,

s. It isre

equa-

erating-ating-

Appendix Aresults in⟨Υ1(z1, x1;0)Υ2(z2, x2;0)

⟩ = A0W2,⟨Υ1(z1, x1; θ1)Υ2(z2, x2;0)

⟩ = A1θ1W2,⟨Υ1(z1, x1; θ1)Υ2(z2, x2; θ2)

⟩(67)=

A1θ1 + A1θ2 + A12θ1θ2 − 2A1θ1θ2

(j1 + j2 + 1

k + 2ln z12 − lnx12

)W2.


(68)W2 = δj1,j2

xj1+j212

z∆1+∆212

,

and used that the identityA2 = A1 is required in the last two-point chiral block. It is recallthat the weights are related to the spins according to(25). In terms of the individual correlatorit follows from (67) that⟨

Φ1(z1, x1)Υ2(z2, x2;0)⟩ = 0,

(69)⟨Ψ1(z1, x1)Υ2(z2, x2;0)

⟩ = A1W2,

and ⟨Φ1(z1, x1)Φ2(z2, x2)

⟩ = 0,⟨Ψ1(z1, x1)Φ2(z2, x2)

⟩ = A1W2,

(70)⟨Ψ1(z1, x1)Ψ2(z2, x2)

⟩ = A12 − 2A1

(j1 + j2 + 1

k + 2ln z12 − lnx12

)W2.

The remaining two-point chiral blocks are obtained by appropriately permuting the indicestressed that the structure constantA1 appearing in(69) a priori is independent of the structuconstantA1 appearing in(70).

For a translational-invariant two-point chiral block, there is only one independent KZtion. Referring to the ansatz(63) or to the (loop-)projectively invariant expressions(67), it maybe written

0=(

(k + 2)∂z12 + −x212∂

2x12

+ 2(j1 + j2 + θ1 + θ2)x12∂x12 − 2(j1 + θ1)(j2 + θ2)

z12

)(71)× ⟨

Υ1(z1, x1; θ1)Υ2(z2, x2; θ2)⟩.

Here one or both of the nilpotent parameters may vanish depending on the number of genfunction unified cells there are in the correlator. It is straightforward to verify that the generfunction two-point chiral blocks(67)satisfy this KZ equation.

5.3. Three-point chiral blocks

Here we base our analysis on the ansatz⟨Υ1(z1, x1; θ1)Υ2(z2, x2; θ2)Υ3(z3, x3; θ3)

⟩=

A(θ1, θ2, θ3) + B12(θ1, θ2, θ3) ln z12 + B23(θ1, θ2, θ3) ln z23 + B13(θ1, θ2, θ3) ln z13


ee

+ C12(θ1, θ2, θ3) lnx12 + C23(θ1, θ2, θ3) lnx23 + C13(θ1, θ2, θ3) lnx13

+ D11(θ1, θ2, θ3) ln2 z12 + D12(θ1, θ2, θ3) ln z12 ln z23 + D13(θ1, θ2, θ3) ln z12 ln z13

+ D22(θ1, θ2, θ3) ln2 z23 + D23(θ1, θ2, θ3) ln z23 ln z13 + D33(θ1, θ2, θ3) ln2 z13

+ E11(θ1, θ2, θ3) ln z12 lnx12 + E12(θ1, θ2, θ3) ln z12 lnx23




+ E33(θ1, θ2, θ3) ln z13 lnx13 + F11(θ1, θ2, θ3) ln2 x12

+ F12(θ1, θ2, θ3) lnx12 lnx23 + F13(θ1, θ2, θ3) lnx12 lnx13

+ F22(θ1, θ2, θ3) ln2 x23 + F23(θ1, θ2, θ3) lnx23 lnx13

(72)+ F33(θ1, θ2, θ3) ln2 x13x

s112x

s223x

s313

zh112z

h223z

h313

,

where

A(θ1, θ2, θ3) = A0 + A1θ1 + A2θ2 + A3θ3

(73)+ A12θ1θ2 + A23θ2θ3 + A13θ1θ3 + A123θ1θ2θ3,

and similarly for the otherθ -dependent structure constants:Bij (θ1, θ2, θ3), Cij (θ1, θ2, θ3),Dij (θ1, θ2, θ3), Eij (θ1, θ2, θ3), andFij (θ1, θ2, θ3). The conditions following from imposing thconformal andSL(2,R) Ward identities are discussed inAppendix A. This analysis leads to thfollowing generating-function three-point chiral blocks:⟨

Υ1(z1, x1;0)Υ2(z2, x2;0)Υ3(z3, x3;0)⟩ = A0W3,⟨

Υ1(z1, x1; θ1)Υ2(z2, x2;0)Υ3(z3, x3;0)⟩

=A0 + A1θ1 − A0θ1

(2j1 + 1

k + 2ln

z12z13

z23− ln

x12x13

x23

)W3,⟨

Υ1(z1, x1; θ1)Υ2(z2, x2; θ2)Υ3(z3, x3;0)⟩

=A0 + A1θ1 − A0θ1

(2j1 + 1

k + 2ln

z12z13

z23− ln

x12x13

x23

)

+ A2θ2 − A0θ2

(2j2 + 1

k + 2ln

z12z23

z13− ln

x12x23

x13

)

+ A12θ1θ2 − A1θ1θ2

(2j2 + 1

k + 2ln

z12z23

z13− ln

x12x23

x13

)

− A2θ1θ2

(2j1 + 1

k + 2ln

z12z13

z23− ln

x12x13

x23

)

+ A0θ1θ2

(2j1 + 1

k + 2ln

z12z13

z23− ln

x12x13

x23

)(2j2 + 1

k + 2ln

z12z23

z13− ln

x12x23

x13

)W3,⟨

Υ1(z1, x1; θ1)Υ2(z2, x2; θ2)Υ3(z3, x3; θ3)⟩

=A1θ1 + A2θ2 + A3θ3 + A12θ1θ2 − A1θ1θ2

(2j2 + 1

lnz12z23 − ln

x12x23)

k + 2 z13 x13


h has

− A2θ1θ2

(2j1 + 1

k + 2ln

z12z13

z23− ln

x12x13

x23

)

+ A23θ2θ3 − A2θ2θ3

(2j3 + 1

k + 2ln

z23z13

z12− ln

x23x13

x12

)

− A3θ2θ3

(2j2 + 1

k + 2ln

z12z23

z13− ln

x12x23

x13

)

+ A13θ1θ3 − A3θ1θ3

(2j1 + 1

k + 2ln

z12z13

z23− ln

x12x13

x23

)

− A1θ1θ3

(2j3 + 1

k + 2ln

z23z13

z12− ln

x23x13

x12

)

+ A123θ1θ2θ3 − A12θ1θ2θ3

(2j3 + 1

k + 2ln

z23z13

z12− ln

x23x13

x12

)

− A23θ1θ2θ3

(2j1 + 1

k + 2ln

z12z13

z23− ln

x12x13

x23

)

− A13θ1θ2θ3

(2j2 + 1

k + 2ln

z12z23

z13− ln

x12x23

x13

)

+ A1θ1θ2θ3

(2j2 + 1

k + 2ln

z12z23

z13− ln

x12x23

x13

)(2j3 + 1

k + 2ln

z23z13

z12− ln

x23x13

x12

)

+ A2θ1θ2θ3

(2j1 + 1

k + 2ln

z12z13

z23− ln

x12x13

x23

)(2j3 + 1

k + 2ln

z23z13

z12− ln

x23x13

x12

)

(74)

+ A3θ1θ2θ3

(2j1 + 1

k + 2ln

z12z13

z23− ln

x12x13

x23

)(2j2 + 1

k + 2ln

z12z23

z13− ln

x12x23

x13

)W3.


(75)W3 = xj1+j2−j312 x

−j1+j2+j323 x

j1−j2+j313

z∆1+∆2−∆312 z

−∆1+∆2+∆323 z

∆1−∆2+∆313

.

In terms of individual correlators (besides the one for non-cellular primary fields only, whicbeen already listed in(74)), we thus have⟨

Φ1(z1, x1)Υ2(z2, x2;0)Υ3(z3, x3;0)⟩ = A0W3,

(76)

⟨Ψ1(z1, x1)Υ2(z2, x2;0)Υ3(z3, x3;0)

⟩ = A1 − A0

(2j1 + 1

k + 2ln

z12z13

z23− ln

x12x13

x23

)W3,

and ⟨Φ1(z1, x1)Φ2(z2, x2)Υ3(z3, x3;0)

⟩ = A0W3,⟨Ψ1(z1, x1)Φ2(z2, x2)Υ3(z3, x3;0)

⟩ = A1 − A0

(2j1 + 1

k + 2ln

z12z13

z23− ln

x12x13

x23

)W3,⟨

Ψ1(z1, x1)Ψ2(z2, x2)Υ3(z3, x3;0)⟩

=A12 − A1

(2j2 + 1

lnz12z23 − ln

x12x23)

− A2(

2j1 + 1ln

z12z13 − lnx12x13

)
k + 2 z13 x13 k + 2 z23 x23


es.tions,

(77)+ A0(

2j1 + 1

k + 2ln

z12z13

z23− ln

x12x13

x23

)(2j2 + 1

k + 2ln

z12z23

z13− ln

x12x23

x13

)W3,

and ⟨Φ1(z1, x1)Φ2(z2, x2)Φ3(z3, x3)

⟩ = 0,⟨Ψ1(z1, x1)Φ2(z2, x2)Φ3(z3, x3)

⟩ = A1W3,⟨Ψ1(z1, x1)Ψ2(z2, x2)Φ3(z3, x3)

⟩=

A12 − A1

(2j2 + 1

k + 2ln

z12z23

z13− ln

x12x23

x13

)

− A2(

2j1 + 1

k + 2ln

z12z13

z23− ln

x12x13

x23

)W3,⟨

Ψ1(z1, x1)Ψ2(z2, x2)Ψ3(z3, x3)⟩

=A123− A12

(2j3 + 1

k + 2ln

z23z13

z12− ln

x23x13

x12

)

− A23(

2j1 + 1

k + 2ln

z12z13

z23− ln

x12x13

x23

)

− A13(

2j2 + 1

k + 2ln

z12z23

z13− ln

x12x23

x13

)

+ A1(

2j2 + 1

k + 2ln

z12z23

z13− ln

x12x23

x13

)(2j3 + 1

k + 2ln

z23z13

z12− ln

x23x13

x12

)

+ A2(

2j1 + 1

k + 2ln

z12z13

z23− ln

x12x13

x23

)(2j3 + 1

k + 2ln

z23z13

z12− ln

x23x13

x12

)

(78)+ A3(

2j1 + 1

k + 2ln

z12z13

z23− ln

x12x13

x23

)(2j2 + 1

k + 2ln

z12z23

z13− ln

x12x23

x13

)W3.

The remaining three-point chiral blocks are obtained by appropriately permuting the indicFor a translational-invariant three-point chiral block, there are two independent KZ equa

cf. (39). Referring to the ansatz(72)or to the (loop-)projectively invariant expressions(74), theymay be written

(79)0= KZi

⟨Υ1(z1, x1; θ1)Υ2(z2, x2; θ2)Υ3(z3, x3; θ3)

⟩, i = 1,2,

where

KZ1 = (k + 2)∂z1

− 2D0(x1, θ1)D0(x2, θ2) + D+(x1, θ1)D−(x2, θ2) + D−(x1, θ1)D+(x2, θ2)

z12

− 2D0(x1, θ1)D0(x3, θ3) + D+(x1, θ1)D−(x3, θ3) + D−(x1, θ1)D+(x3, θ3)

z13,

KZ2 = (k + 2)∂z2

+ 2D0(x1, θ1)D0(x2, θ2) + D+(x1, θ1)D−(x2, θ2) + D−(x1, θ1)D+(x2, θ2)

z12

(80)− 2D0(x2, θ2)D0(x3, θ3) + D+(x2, θ2)D−(x3, θ3) + D−(x2, θ2)D+(x3, θ3)

z23.


ding onward,

s withrmale

chiral

e two-

In these expressions, one, two or all three of the nilpotent parameters may vanish depenthe number of generating-function unified cells there are in the correlator. It is straightforthough rather tedious, to verify that the generating-function three-point chiral blocks(74)satisfythese KZ equations.

5.4. In terms of spins with nilpotent parts

Here we wish to extend to the logarithmic WZW model the idea put forward in[17] that corre-lators in logarithmic CFT may be represented compactly by considering conformal weightnilpotent parts∆ + θ . The most general results of this kind for two- and three-point confoblocks were found in[13] and are given above as(13) and (18). As already indicated, we will herassociate the generalized spinji + θi to the generating-function unified cellΥi(zi, xi; θi). Thecorresponding generalized conformal weight thus reads∆i +µi whereµi = (2ji +1)θi/(k +2).This allows us to express the generating-function correlators for two- and three-pointblocks in the following simple way:

⟨Υ1(z1, x1;0)Υ2(z2, x2;0)

⟩ = δj1,j2A0 x

j1+j212

z∆1+∆212

,

⟨Υ1(z1, x1; θ1)Υ2(z2, x2;0)

⟩ = δj1,j2A1θ1

x(j1+θ1)+j212

z(∆1+µ1)+∆212

,

(81)⟨Υ1(z1, x1; θ1)Υ2(z2, x2; θ2)

⟩ = δj1,j2

A1θ1 + A1θ2 + A12θ1θ2

x(j1+θ1)+(j2+θ2)

12

z(∆1+µ1)+(∆2+µ2)12

,

and

⟨Υ1(z1, x1;0)Υ2(z2, x2;0)Υ3(z3, x3;0)

⟩ = A0 xj1+j2−j312 x

−j1+j2+j323 x

j1−j2+j313

z∆1+∆2−∆312 z

−∆1+∆2+∆323 z

∆1−∆2+∆313

,

⟨Υ1(z1, x1; θ1)Υ2(z2, x2;0)Υ3(z3, x3;0)

⟩=

A0 + A1θ1 x

(j1+θ1)+j2−j312 x

−(j1+θ1)+j2+j323 x

(j1+θ1)−j2+j313

z(∆1+µ1)+∆2−∆312 z

−(∆1+µ1)+∆2+∆323 z

(∆1+µ1)−∆2+∆313

,

⟨Υ1(z1, x1; θ1)Υ2(z2, x2; θ2)Υ3(z3, x3;0)

⟩=

A0 + A1θ1 + A2θ2 + A12θ1θ2

× x(j1+θ1)+(j2+θ2)−j312 x

−(j1+θ1)+(j2+θ2)+j323 x

(j1+θ1)−(j2+θ2)+j313

z(∆1+µ1)+(∆2+µ2)−∆312 z

−(∆1+µ1)+(∆2+µ2)+∆323 z

(∆1+µ1)−(∆2+µ2)+∆313

,

⟨Υ1(z1, x1; θ1)Υ2(z2, x2; θ2)Υ3(z3, x3; θ3)

⟩=

A1θ1 + A2θ2 + A3θ3 + A12θ1θ2 + A23θ2θ3 + A13θ1θ3 + A123θ1θ2θ3

(82)

× x(j1+θ1)+(j2+θ2)−(j3+θ3)

12 x−(j1+θ1)+(j2+θ2)+(j3+θ3)

23 x(j1+θ1)−(j2+θ2)+(j3+θ3)

13

z(∆1+µ1)+(∆2+µ2)−(∆3+µ3)12 z

−(∆1+µ1)+(∆2+µ2)+(∆3+µ3)23 z

(∆1+µ1)−(∆2+µ2)+(∆3+µ3)13

.

The remaining combination in(81)and the remaining four combinations in(82)are obtained byappropriate permutations in the indices. Thus confirming our assertion, it follows that th


tations

sed in

-formalee-hical

ct

ithmic

and three-point chiral blocks factor into a conformal part and anSL(2,R) group part. The extradegrees of freedom in(81) and (82)compared to the incomplete result(62) are contained in thefact that theθ -dependent structure constants in(81) and (82)do not necessarily factor as in(62).It is noted that the present factorization into a conformal part and anSL(2,R) group part is notevident a priori, while our analysis has demonstrated its validity. These compact represenconstitute a significant simplification of the results given above (and derived inAppendix A).The verification of the KZ equations is particularly simple when the correlators are expresthis way.

5.5. Hierarchical structures for chiral blocks

Based on ideas discussed in[11,12], it was found in[13] that the conformal blocks involving logarithmic fields may be represented in terms of derivatives with respect to the conweights. This was reviewed in Section2.5. Here we wish to extend this idea to the two- and thrpoint chiral blocks of the logarithmic WZW model introduced above. It is found that hierarcstructures similar to the ones discussed in Section2.5apply in the affine case.

First, it is observed that acting on eitherW2 or W3, we may substitute derivatives with respeto the spins by multiplicative factors according to

(83)∂j1 = ∂j2 → −2j1 + j2 + 1

k + 2ln z12 + 2 lnx12

or

∂j1 → −2j1 + 1

k + 2ln

z12z13

z23+ ln

x12x13

x23,

∂j2 → −2j2 + 1

k + 2ln

z12z23

z13+ ln

x12x23

x13,

(84)∂j3 → −2j3 + 1

k + 2ln

z23z13

z12+ ln

x23x13

x12,

respectively. This simple observation allows us to represent the correlators involving logarfields as follows:⟨

Ψ1(z1, x1)Υ2(z2, x2;0)⟩ = A1W2,⟨

Ψ1(z1, x1)Φ2(z2, x2)⟩ = A1W2,⟨

Ψ1(z1, x1)Ψ2(z2, x2)⟩ = (

A12 + A2∂j1 + A1∂j2

)W2,⟨

Ψ1(z1, x1)Υ2(z2, x2;0)Υ3(z3, x3;0)⟩ = (

A1 + A0∂j1

)W3,⟨

Ψ1(z1, x1)Φ2(z2, x2)Υ3(z3, x3;0)⟩ = (

A1 + A0∂j1

)W3,⟨

Ψ1(z1, x2)Ψ2(z2, x2)Υ3(z3, x3;0)⟩ = (

A12 + A1∂j2 + A2∂j1 + A0∂j1∂j2

)W3,⟨

Ψ1(z1, x1)Φ2(z2, x2)Φ3(z3, x3)⟩ = A1W3,⟨

Ψ1(z1, x1)Ψ2(z2, x2)Φ3(z3, x3)⟩ = (

A12 + A2∂j1 + A1∂j2

)W3,⟨

Ψ1(z1, x1)Ψ2(z2, x2)Ψ3(z3, x3)⟩ = (

A123+ A23∂j1 + A13∂j2 + A12∂j3

(85)+ A3∂j1∂j2 + A1∂j2∂j3 + A2∂j1∂j3

)W3,


erefore

ined by

elsveld in

steps

in addition to expressions obtained by appropriately permuting the indices. One may threpresent the correlators hierarchically as⟨

Ψ1(z1, x1)Υ2(z2, x2;0)⟩ = A1W2 + ∂j1

⟨Φ1(z1, x1)Υ2(z2, x2;0)

⟩,⟨

Ψ1(z1, x1)Φ2(z2, x2)⟩ = A1W2 + ∂j1

⟨Φ1(z1, x1)Φ2(z2, x2)

⟩,⟨

Ψ1(z1, x1)Ψ2(z2, x2)⟩ = A12W2 + ∂j1

⟨Φ1(z1, x1)Ψ2(z2, x2)

⟩ + ∂j2

⟨Ψ1(z1, x1)Φ2(z2, x2)

⟩(86)− ∂j1∂j2

⟨Φ1(z1, x1)Φ2(z2, x2)

⟩,

in the case of two-point chiral blocks, and⟨Ψ1(z1, x1)Υ2(z2, x2;0)Υ3(z3, x3;0)

⟩ = A1W3 + ∂j1

⟨Φ1(z1, x1)Υ2(z2, x2;0)Υ3(z3, x3;0)

⟩,⟨

Ψ1(z1, x1)Φ2(z2, x2)Υ3(z3, x3;0)⟩ = A1W3 + ∂j1

⟨Φ1(z1, x1)Φ2(z2, x2)Υ3(z3, x3;0)

⟩,⟨

Ψ1(z1, x1)Ψ2(z2, x2)Υ3(z3, x3;0)⟩

= A12W3 + ∂j1

⟨Φ1(z1, x1)Ψ2(z2, x2)Υ3(z3, x3;0)

⟩+ ∂j2

⟨Ψ1(z1, x1)Φ2(z2, x2)Υ3(z3, x3;0)

⟩− ∂j1∂j2

⟨Φ1(z1, x1)Φ2(z2, x2)Υ3(z3, x3;0)

⟩,⟨

Ψ1(z1, x1)Φ2(z2, x2)Φ3(z3, x3)⟩ = A1W3 + ∂j1

⟨Φ1(z1, x1)Φ2(z2, x2)Φ3(z3, x3)

⟩,⟨

Ψ1(z1, x1)Ψ2(z2, x2)Φ3(z3, x3)⟩

= A12W3 + ∂j1

⟨Φ1(z1, x1)Ψ2(z2, x2)Φ3(z3, x3)

⟩ + ∂j2

⟨Ψ1(z1, x1)Φ2(z2, x2)Φ3(z3, x3)

⟩− ∂j1∂j2

⟨Φ1(z1, x1)Φ2(z2, x2)Φ3(z3, x3)

⟩,⟨

Ψ1(z1, x1)Ψ2(z2, x2)Ψ3(z3, x3)⟩

= A123W3 + ∂j1

⟨Φ1(z1, x1)Ψ2(z2, x2)Ψ3(z3, x3)

⟩+ ∂j2

⟨Ψ1(z1, x1)Φ2(z2, x2)Ψ3(z3, x3)

⟩ + ∂j3

⟨Ψ1(z1, x1)Ψ2(z2, x2)Φ3(z3, x3)

⟩− ∂j1∂j2

⟨Φ1(z1, x1)Φ2(z2, x2)Ψ3(z3, x3)

⟩ − ∂j2∂j3

⟨Ψ1(z1, x1)Φ2(z2, x2)Φ3(z3, x3)

⟩− ∂j1∂j3

⟨Φ1(z1, x1)Ψ2(z2, x2)Φ3(z3, x3)

⟩(87)+ ∂j1∂j2∂j3

⟨Φ1(z1, x1)Φ2(z2, x2)Φ3(z3, x3)

⟩,

in the case of three-point chiral blocks. As above, the remaining correlators may be obtaappropriately permuting the indices.

6. Hamiltonian reduction

It is well known thatSL(2,R) WZW models may be linked to conformal minimal modvia Hamiltonian reduction[20–23]. A precise description of this reduction was given at the leof correlators in[14,15], while a simple and direct proof of this description was presente[24,25]based on[25,26]. The basic idea in this context is to start with anN -point chiral block ofgenerating-function primary fields in the affine model, in which case the correspondingN -pointconformal block in the CFT is obtained by settingxi = zi for i = 1, . . . ,N [14,15]. This wasrefined a bit in[24,25]where it was discussed how the procedure may be performed in twoby first settingxi = xzi followed by fixing the common proportionality constant tox = 1, forexample.


three-d theirstudy

chiral

ld

eructure

mpactlye linke

n uni-

re-

ical

s

s areed by

Our current situation is quite simple, though, since we are only interested in two- andpoint functions and in their form rather than the relations between structure constants andependencies on the spins and conformal weights. The objective here is therefore towhether a naive extension of the Hamiltonian-reduction principle settingxi = zi applies to thelogarithmic correlators found above. That is, we wish to show that the two- and three-pointblocks(67) and (74)reduce to the two- and three-point conformal blocks(11) and (16), respec-tively, upon settingxi = zi . The conformal weights,∆i , in the resulting logarithmic CFT shouthen be given by

(88)∆i = ∆i − ji = ji(ji + 1)

k + 2− ji,

whereas the central charges are related asc = c−6k−2= 3kk+2 −6k−2. It is emphasized that w

are only concerned with the form of the correlators, not the various dependencies of the stconstants.

The reductions are straightforward to analyze when the correlators are expressed coin terms of spins and conformal weights with nilpotent parts. We thus wish to examine thbetween(81) and (82), on one hand, and(13) and (18), on the other hand. It follows that thaffine correlators reduce to the conformal ones with conformal weights given in(88), if theidentificationsxi = zi for i = 1, . . . ,N are accompanied by

(89)θi = µi − θi =(

2ji + 1

k + 2− 1

)θi, i = 1, . . . ,N ,

and (forji, ji′ = (k + 1)/2) the renormalizations

A0 = A0,

Ai = Ai

2ji+1k+2 − 1

, 1 i 3,

Aii′ = Aii′(2ji+1k+2 − 1

)(2ji′+1k+2 − 1

) , 1 i < i′ 3,

(90)A123= A123(2j1+1k+2 − 1

)(2j2+1k+2 − 1

)(2j3+1k+2 − 1

) .

The apparent subtlety in the case of two-point functions, composed of generating-functiofied cells only, is resolved by the Kronecker delta function inj1 andj2 appearing in(81).

In the exceptional case whereji = (k + 1)/2, this Hamiltonian-reduction procedure corsponds to formally replacingΥi(zi, xi; θi) by the non-cellular primary fieldΥ (zi;0), cf. (89), inwhich case the renormalizations(90) involving ji no longer apply. The representation-theoretmechanism underlying this reduction in logarithmic nature remains to be understood.

Following [27], primary fields are calledproper primaryif their operator-product expansionwith each other cannot produce a logarithmic field. It is argued in[27] (see also[28]) that thestructure constants of three-point conformal blocks not involving improper primary fieldrelated. According to[13] and in the notation used above, such conformal blocks are obtainsetting

(91)A1 = A2 = A3, A12 = A23 = A13.


tion ofin the

con-

y fieldsal

e rep-found

ins. Ad, and

ated thatlatorsresults

his islatorsormal-

yin the

izationach a

th just

A nat-n. The

This class of restricted three-point conformal blocks can be reached by Hamiltonian reduca particular subset of the three-point chiral blocks in the affine case. This is quite obviousframework with generalized spins and generalized conformal weights, cf.(90). One merely sets

A1

2j1+1k+2 − 1

= A2

2j2+1k+2 − 1

= A3

2j3+1k+2 − 1

,

(92)A12(2j1+1

k+2 − 1)(2j2+1

k+2 − 1) = A23(2j2+1

k+2 − 1)(2j3+1

k+2 − 1) = A13(2j1+1

k+2 − 1)(2j3+1

k+2 − 1) ,

in the last chiral block in(82). Hamiltonian reduction then reproduces the last three-pointformal block in(18)with (91)satisfied.

7. Conclusion

We have studied a particular type of logarithmic extension ofSL(2,R) WZW models. It isbased on the introduction of affine Jordan cells constructed as multiplets of quasi-primarorganized in indecomposable representations of the Lie algebrasl(2). We have found the genersolution to the simultaneously imposed set of conformal andSL(2,R) Ward identities for two-and three-point chiral blocks. These correlators may involve logarithmic terms and may bresented compactly by considering spins with nilpotent parts. The chiral blocks have beento exhibit hierarchical structures obtained by computing derivatives with respect to the spset of KZ equations, appropriately modified to cover affine Jordan cells, have been derivethe chiral blocks have been shown to satisfy these equations. It has been also demonstra simple and well-established prescription for Hamiltonian reduction at the level of correextends straightforwardly to the logarithmic correlators as the latter reduce to the knownfor two- and three-point conformal blocks in logarithmic CFT.

We find it natural to say that our results pertain to affine Jordan cells of rank two. Tsupported in part by the fact that Hamiltonian reduction of the chiral blocks results in correof rank-two conformal Jordan cells. In order to argue more directly, we recall that a confJordan cell of rankr [11] consists of one primary field,ϕ0(z), andr − 1 logarithmic and quasiprimary partner fields,ϕ1(z), . . . , ϕr−1, satisfying

(93)[Ln,ϕi(z)

] = (zn+1∂z + ∆(n + 1)zn

)ϕi(z) + (n + 1)znϕi−1(z).

One could say that the fieldϕi(z) has degreei or is at depthi. That is, the depth is given bthe number of adjoint actions of the Virasoro modes required to reach the primary fieldcell. The rank is then given by one plus the maximum depth. If we extend this characterto the affine case, we would say that a field is at depthi if i adjoint actions of the Lie algebrgenerators (or more generally,i adjoint actions of symmetry generators) are required to reaprimary field in the affine Jordan cell. With the rank denoting one plus the maximum depdefined, the rank of our affine Jordan cells is indeed two.

It would be interesting to extend our work to higher ranks in the sense just indicated.ural construction seems to suggest itself and is based on the following simple observatioconformal Jordan cell(93)may be written compactly as

(94)[Ln,υ(z; θ )

] = (zn+1∂z + (∆ + θ )(n + 1)zn

)υ(z),


r real-

requirefound

malctionk toher.f affinele repre-d furtherhe

pearslocksip-s, oneformal

where we have introduced the generating-function unified cell as

(95)υ(z; θ ) =r−1∑i=0

θ iϕi(z).

In this section,θ is a nilpotent, yet even, parameter satisfying

(96)θ r = 0, θ r−1 = 0.

We thus suggest to generalize the affine Jordan cell by introducing the differential-operatoization

D+(x; θ) = x2∂x − 2(j + θ)x,

D0(x; θ) = x∂x − (j + θ),

(97)D−(x; θ) = −∂x,

of the Lie algebrasl(2) and the corresponding generating-function unified cellυ(z, x; θ) satisfy-ing

(98)[Ja,υ(z, x; θ)

] = −Da(x; θ)υ(z, x; θ).

In this section,θ is a nilpotent, yet even, parameter satisfying

(99)θr = 0, θr−1 = 0.

The generalization of the expansion(42)would then read

(100)υ(z, x; θ) =r−1∑i=0

θiΘi(z, x),

whereΘ0(z, x) is a generating-function primary field similar toΦ(z, x) in (42). An examinationof Hamiltonian reduction of the correlators based on these higher-rank affine Jordan cellsknowledge on higher-rank conformal Jordan cells. Partial results in this direction may bein [11,12]. Conformal Jordan cells of infinite rank have been introduced in[29].

An interesting extension of the work[13] concerns the general solution to the superconforWard identities appearing in logarithmic superconformal field theory. Results in this diremay be found in[30]. A complete solution would facilitate an extension of the present worOSp(1|2) WZW models and their Hamiltonian reduction. This deserves to be explored furt

As already mentioned, we hope to address elsewhere the classification problem oJordan cells. In particular, indecomposable representations as extensions of non-integrabsentations would be interesting to understand. These results could eventually be extendeto the higher-rank affine Jordan cells based on(97) and (98)and could be developed along tlines of Section4.

We also hope to study the four-point chiral blocks involving our affine Jordan cells. It apstraightforward to implement the Ward identities, after which the general four-point chiral bshould follow from the modified KZ equations(57), (58). To test whether the extended prescrtion for Hamiltonian reduction employed above also applies to these four-point functioncould compare the resulting correlators to the recently obtained results on four-point conblocks[31,32].


erating-

Acknowledgements

The author is grateful to M. Flohr for very helpful comments on the manuscript.

Appendix A. Analysis of Ward identities

Below are indicated some of the steps leading to the general expressions for the genfunction two- and three-point chiral blocks given in(81) and (82), respectively.

A.1. Two-point chiral blocks

We initially consider the case with two generating-function unified cells, that is,θ1, θ2 = 0.Expanding(65) leads to the conditions

0= B0 − 2(h − ∆1)A0 = B0 − 2(h − ∆2)A

0,

0= B1 − 2(h − ∆1)A1 + 2

2j1 + 1

k + 2A0 = B1 − 2(h − ∆2)A

1,

0= B2 − 2(h − ∆1)A2 = B2 − 2(h − ∆2)A

2 + 22j2 + 1

k + 2A0,

0= B12 − 2(h − ∆1)A12 + 2

2j1 + 1

k + 2A2 = B12 − 2(h − ∆2)A

12 + 22j2 + 1

k + 2A1,

0= (h − ∆1)B0 = (h − ∆2)B

0,

0= (h − ∆1)B1 − 2j1 + 1

k + 2B0 = (h − ∆2)B

1,

0= (h − ∆1)B2 = (h − ∆2)B

2 − 2j2 + 1

k + 2B0,

0= (h − ∆1)B12 − 2j1 + 1

k + 2B2 = (h − ∆2)B

12 − 2j2 + 1

k + 2B1,

0= (h − ∆1)C0 = (h − ∆2)C

0,

0= (h − ∆1)C1 − 2j1 + 1

k + 2C0 = (h − ∆2)C

1,

0= (h − ∆1)C2 = (h − ∆2)C

2 − 2j2 + 1

k + 2C0,

(A.1)0= (h − ∆1)C12 − 2j1 + 1

k + 2C2 = (h − ∆2)C

12 − 2j2 + 1

k + 2C1,

whereas an expansion of(66)yields the conditions

0= C0 + 2(s − j1)A0 = C0 + 2(s − j2)A

0,

0= C1 + 2(s − j1)A1 − 2A0 = C1 + 2(s − j2)A

1,

0= C2 + 2(s − j1)A2 = C2 + 2(s − j2)A

2 − 2A0,

0= C12 + 2(s − j1)A12 − 2A2 = C12 + 2(s − j2)A

12 − 2A1,

0= (s − j1)B0 = (s − j2)B

0,

0= (s − j1)B1 − B0 = (s − j2)B

1,


ouldto satisfy

0= (s − j1)B2 = (s − j2)B

2 − B0,

0= (s − j1)B12 − B2 = (s − j2)B

12 − B1,

0= (s − j1)C0 = (s − j2)C

0,

0= (s − j1)C1 − C0 = (s − j2)C

1,

0= (s − j1)C2 = (s − j2)C

2 − C0,

(A.2)0= (s − j1)C12 − C2 = (s − j2)C

12 − C1.

It follows immediately that a non-trivial solution requires

(A.3)s = j1 = j2, h = ∆1 = ∆2,

further implying the relations

0= A0 = B0 = B1 = B2 = C0 = C1 = C2,

(A.4)0= A1 − A2 = B12 + 22j1 + 1

k + 2A1 = C12 − 2A1.

The parameterA12 is independent of the other ones.In the case whereθ1 = 0 while θ2 = 0, the third conformal Ward identity (i.e.,(65)) yields

0= B0 − 2(h − ∆1)A0 = B0 − 2(h − ∆2)A

0,

0= B1 − 2(h − ∆1)A1 + 2

2j1 + 1

k + 2A0 = B1 − 2(h − ∆2)A

1,

0= (h − ∆1)B0 = (h − ∆2)B

0,

0= (h − ∆1)B1 − 2j1 + 1

k + 2B0 = (h − ∆2)B

1,

0= (h − ∆1)C0 = (h − ∆2)C

0,

(A.5)0= (h − ∆1)C1 − 2j1 + 1

k + 2C0 = (h − ∆2)C

1,

while the thirdSL(2,R) Ward identity (i.e.,(66)) corresponds to

0= C0 + 2(s − j1)A0 = C0 + 2(s − j2)A

0,

0= C1 + 2(s − j1)A1 − 2A0 = C1 + 2(s − j2)A

1,

0= (s − j1)B0 = (s − j2)B

0,

0= (s − j1)B1 − B0 = (s − j2)B

1,

0= (s − j1)C0 = (s − j2)C

0,

(A.6)0= (s − j1)C1 − C0 = (s − j2)C

1.

It is stressed thatB2, for example, does not exist (or is set to zero) in this case and shtherefore not be treated as a free parameter. As above, the spins and weights are seen(A.3), and it follows that

(A.7)0= A0 = B0 = B1 = C0 = C1,

while A1 is the only free parameter.


s

In the case whereθ1 = θ2 = 0, the two sets of conditions reduce to

0= B0 − 2(h − ∆1)A0 = B0 − 2(h − ∆2)A

0,

0= (h − ∆1)B0 = (h − ∆2)B

0,

(A.8)0= (h − ∆1)C0 = (h − ∆2)C

0,

and

0= C0 + 2(s − j1)A0 = C0 + 2(s − j2)A

0,

0= (s − j1)B0 = (s − j2)B

0,

(A.9)0= (s − j1)C0 = (s − j2)C

0.

Once again, the spins and weights satisfy(A.3). This time,B0 = C0 = 0 while A0 is the onlyfree parameter.

This analysis leads to the two-point chiral blocks given in(67).

A.2. Three-point chiral blocks

Based on the ansatz(72), the third conformal Ward identity(59)corresponds to the condition

0= (s1 + s3 − 2j1 − 2θ1)A + C12 + C13 = (s1 + s2 − 2j2 − 2θ2)A + C12 + C23

= (s2 + s3 − 2j3 − 2θ3)A + C23 + C13,

0= (s1 + s3 − 2j1 − 2θ1)B12 + E11 + E13 = (s1 + s2 − 2j2 − 2θ2)B12 + E11 + E12

= (s2 + s3 − 2j3 − 2θ3)B12 + E12 + E13,

0= (s1 + s3 − 2j1 − 2θ1)B23 + E21 + E23 = (s1 + s2 − 2j2 − 2θ2)B23 + E21 + E22

= (s2 + s3 − 2j3 − 2θ3)B23 + E22 + E23,

0= (s1 + s3 − 2j1 − 2θ1)B13 + E31 + E33 = (s1 + s2 − 2j2 − 2θ2)B13 + E31 + E32

= (s2 + s3 − 2j3 − 2θ3)B13 + E32 + E33,

0= (s1 + s3 − 2j1 − 2θ1)C12 + 2F11 + F13 = (s1 + s2 − 2j2 − 2θ2)C12 + 2F11 + F12

= (s2 + s3 − 2j3 − 2θ3)C12 + F12 + F13,

0= (s1 + s3 − 2j1 − 2θ1)C23 + F12 + F23 = (s1 + s2 − 2j2 − 2θ2)C23 + F12 + 2F22

= (s2 + s3 − 2j3 − 2θ3)C23 + 2F22 + F23,

0= (s1 + s3 − 2j1 − 2θ1)C13 + F13 + 2F33 = (s1 + s2 − 2j2 − 2θ2)C13 + F13 + F23

= (s2 + s3 − 2j3 − 2θ3)C13 + F23 + 2F33,

0= (s1 + s3 − 2j1 − 2θ1)Dij = (s1 + s2 − 2j2 − 2θ2)Dij = (s2 + s3 − 2j3 − 2θ3)Dij ,

0= (s1 + s3 − 2j1 − 2θ1)Eij = (s1 + s2 − 2j2 − 2θ2)Eij = (s2 + s3 − 2j3 − 2θ3)Eij ,

(A.10)

0= (s1 + s3 − 2j1 − 2θ1)Fij = (s1 + s2 − 2j2 − 2θ2)Fij = (s2 + s3 − 2j3 − 2θ3)Fij ,

whereas the thirdSL(2,R) Ward identity corresponds to the conditions

0= (−h1 − h3 + 2∆1 + 2µ1)A + B12 + B13 = (−h1 − h2 + 2∆2 + 2µ2)A + B12 + B23

= (−h2 − h3 + 2∆3 + 2µ3)A + B23 + B13,


stantsils inmployed

bers

xpands

iated to

0= (−h1 − h3 + 2∆1 + 2µ1)B12 + 2D11 + D13

= (−h1 − h2 + 2∆2 + 2µ2)B12 + 2D11 + D12

= (−h2 − h3 + 2∆3 + 2µ3)B12 + D12 + D13,

0= (−h1 − h3 + 2∆1 + 2µ1)B23 + D12 + D23

= (−h1 − h2 + 2∆2 + 2µ2)B23 + D12 + 2D22

= (−h2 − h3 + 2∆3 + 2µ3)B23 + 2D22 + D23,

0= (−h1 − h3 + 2∆1 + 2µ1)B13 + D13 + 2D33

= (−h1 − h2 + 2∆2 + 2µ2)B13 + D13 + D23

= (−h2 − h3 + 2∆3 + 2µ3)B13 + D23 + 2D33,

0= (−h1 − h3 + 2∆1 + 2µ1)C12 + E11 + E31

= (−h1 − h2 + 2∆2 + 2µ2)C12 + E11 + E21

= (−h2 − h3 + 2∆3 + 2µ3)C12 + E21 + E31,

0= (−h1 − h3 + 2∆1 + 2µ1)C23 + E12 + E32

= (−h1 − h2 + 2∆2 + 2µ2)C23 + E12 + E22

= (−h2 − h3 + 2∆3 + 2µ3)C23 + E22 + E32,

0= (−h1 − h3 + 2∆1 + 2µ1)C13 + E13 + E33

= (−h1 − h2 + 2∆2 + 2µ2)C13 + E13 + E23

= (−h2 − h3 + 2∆3 + 2µ3)C13 + E23 + E33,

0= (−h1 − h3 + 2∆1 + 2µ1)Dij = (−h1 − h2 + 2∆2 + 2µ2)Dij

= (−h2 − h3 + 2∆3 + 2µ3)Dij ,

0= (−h1 − h3 + 2∆1 + 2µ1)Eij = (−h1 − h2 + 2∆2 + 2µ2)Eij

= (−h2 − h3 + 2∆3 + 2µ3)Eij ,

0= (−h1 − h3 + 2∆1 + 2µ1)Fij = (−h1 − h2 + 2∆2 + 2µ2)Fij

(A.11)= (−h2 − h3 + 2∆3 + 2µ3)Fij .

To keep the notation simple, we have left out the explicit indications that the structure condepend on theθs. To keep the presentation simple as well, we will leave out most of the detathe analysis of these conditions. Since the approach essentially is the same as the one ein the study of two-point chiral blocks, we will merely outline the main steps.

We distinguish between the different numbers of unified cells, that is, the different numof non-vanishingθs. In every case, one finds the relations

s1 = j1 + j2 − j3, s2 = −j1 + j2 + j3, s3 = j1 − j2 + j3,

(A.12)h1 = ∆1 + ∆2 − ∆3, h2 = −∆1 + ∆2 + ∆3, h3 = ∆1 − ∆2 + ∆3.

Having split the analysis into the four cases characterized by 0, 1, 2 or 3 unified cells, one ethe conditions(A.10) and (A.11)on the set of associatedθs, where it is recalled thatµi =(2ji + 1)θi/(k + 2). Since the resulting conditions are linear in the structure constantsA0,B0

ij ,etc., it is straightforward to work out the relations between the structure constants assocthe four cases. The relations are listed below.


re

t

In the case whereθ1 = θ2 = θ3 = 0, we find

(A.13)B0ij = C0

ij = D0ij = E0

ij = F 0ij = 0,

which means thatA0 is the only free structure constant.In the case whereθ2 = θ3 = 0 while θ1 = 0, we find

0= B0ij = C0

ij = D0ij = D1

ij = E0ij = E1

ij = F 0ij = F 1

ij ,

B112 = −B1

23 = B113 = −2j1 + 1

k + 2A0,

(A.14)C112 = −C1

23 = C113 = A0,

while A1 is unconstrained. That is, we may considerA0 andA1 as the only independent structuconstants.

In the case whereθ3 = 0 while θ1, θ2 = 0, we find

0= B0ij = C0

ij = D0ij = D1

ij = D2ij = E0

ij = E1ij = E2

ij = F 0ij = F 1

ij = F 2ij ,

B112 = −B1

23 = B113 = −2j1 + 1

k + 2A0, B2

12 = B123 = −B1

13 = −2j2 + 1

k + 2A0,

B1212 = −2j2 + 1

k + 2A1 − 2j1 + 1

k + 2A2, B12

23 = −B1213 = −2j2 + 1

k + 2A1 + 2j1 + 1

k + 2A2,

C112 = −C1

23 = C113 = A0, C2

12 = C223 = −C2

13 = A0,

C1212 = A1 + A2, C12

23 = −C1213 = A1 − A2,

D1211 = −D12

22 = 1

2D12

23 = −D1233 = (2j1 + 1)(2j2 + 1)

(k + 2)2A0, D12

12 = D1213 = 0,

E1211 = −E12

22 = E1223 = E12

32 = −E1233 = −2

j1 + j2 + 1

k + 2A0,

E1212 = −E12

13 = −E1221 = E12

31 = −2j1 + 2j2

k + 2A0,

(A.15)F 1211 = −F 12

22 = 1

2F 12

23 = −F 1233 = A0, F 12

12 = F 1213 = 0,

whileA12 is unconstrained. That is, we may considerA0, A1, A2 andA12 as the only independenstructure constants.

In the case whereθ1, θ2, θ3 = 0, we find

0= A0 = B0ij = Bl

ij = C0ij = Cl

ij ,

0= D0ij = Dl

ij = Dlmij = E0

ij = Elij = Elm

ij = F 0ij = F l

ij = F lmij ,

B1212 = −2j2 + 1

k + 2A1 − 2j1 + 1

k + 2A2, B12

23 = −B1213 = −2j2 + 1

k + 2A1 + 2j1 + 1

k + 2A2,

B2323 = −2j3 + 1

k + 2A2 − 2j2 + 1

k + 2A3, B23

12 = −B2313 = 2j3 + 1

k + 2A2 − 2j2 + 1

k + 2A3,

B1313 = −2j3 + 1

k + 2A1 − 2j1 + 1

k + 2A3, B13

12 = −B1323 = 2j3 + 1

k + 2A1 − 2j1 + 1

k + 2A3,

B12312 = 2j3 + 1

A12 − 2j1 + 1A23 − 2j2 + 1

A13,
k + 2 k + 2 k + 2


B12323 = −2j3 + 1

k + 2A12 + 2j1 + 1

k + 2A23 − 2j2 + 1

k + 2A13,

B12312 = −2j3 + 1

k + 2A12 − 2j1 + 1

k + 2A23 + 2j2 + 1

k + 2A13,

C1212 = A1 + A2, C12

23 = −C1213 = A1 − A2,

C2323 = A2 + A3, C23

12 = −C2313 = −A2 + A3,

C1313 = A1 + A3, C13

12 = −C1323 = −A1 + A3,

C12312 = −A12 + A23 + A13, C123

23 = A12 − A23 + A13,

C12313 = A12 + A23 − A13,

D12311 = − (2j2 + 1)(2j3 + 1)

(k + 2)2A1 − (2j1 + 1)(2j3 + 1)

(k + 2)2A2 + (2j1 + 1)(2j2 + 1)

(k + 2)2A3,

D12322 = (2j2 + 1)(2j3 + 1)

(k + 2)2A1 − (2j1 + 1)(2j3 + 1)

(k + 2)2A2 − (2j1 + 1)(2j2 + 1)

(k + 2)2A3,

D12333 = − (2j2 + 1)(2j3 + 1)

(k + 2)2A1 + (2j1 + 1)(2j3 + 1)

(k + 2)2A2 − (2j1 + 1)(2j2 + 1)

(k + 2)2A3,

D12312 = 2

(2j1 + 1)(2j3 + 1)

(k + 2)2A2, D123

23 = 2(2j1 + 1)(2j2 + 1)

(k + 2)2A3,

D12313 = 2

(2j2 + 1)(2j3 + 1)

(k + 2)2A1,

E12311 = 2(j2 + j3 + 1)

k + 2A1 + 2(j1 + j3 + 1)

k + 2A2 − 2(j1 + j2 + 1)

k + 2A3,

E12312 = −2j2 + 2j3

k + 2A1 − 2(j1 + j3 + 1)

k + 2A2 + −2j1 + 2j2

k + 2A3,

E12313 = −2(j2 + j3 + 1)

k + 2A1 + −2j1 + 2j3

k + 2A2 + 2j1 − 2j2

k + 2A3,

E12321 = 2j2 − 2j3

k + 2A1 − 2(j1 + j3 + 1)

k + 2A2 + 2j1 − 2j2

k + 2A3,

E12322 = −2(j2 + j3 + 1)

k + 2A1 + 2(j1 + j3 + 1)

k + 2A2 + 2(j1 + j2 + 1)

k + 2A3,

E12323 = −2j2 + 2j3

k + 2A1 + 2j1 − 2j3

k + 2A2 − 2(j1 + j2 + 1)

k + 2A3,

E12331 = −2(j2 + j3 + 1)

k + 2A1 + 2j1 − 2j3

k + 2A2 + −2j1 + 2j2

k + 2A3,

E12332 = 2j2 − 2j3

k + 2A1 + −2j1 + 2j3

k + 2A2 − 2(j1 + j2 + 1)

k + 2A3,

E12333 = 2(j2 + j3 + 1)

k + 2A1 − 2(j1 + j3 + 1)

k + 2A2 + 2(j1 + j2 + 1)

k + 2A3,

F 12311 = −A1 − A2 + A3, F 123

22 = A1 − A2 − A3, F 12333 = −A1 + A2 − A3,

(A.16)F 12312 = 2A2, F 123

23 = 2A3, F 12313 = 2A1,

while A123 is unconstrained. That is, we may considerA0, Ai , Aij andA123 as the only inde-pendent structure constants.


070.26.

dings in

cien-

, 1996,

The relations corresponding to the situation whereθ1 = θ3 = 0 while θ2 = 0, for example,are obtained from the relations corresponding to the case whereθ2 = θ3 = 0 while θ1 = 0 by anappropriate permutation in the indices.

These results lead to the three-point chiral blocks given in(82).

References

[1] V. Gurarie, Nucl. Phys. B 410 (1993) 535.[2] M. Flohr, Int. J. Mod. Phys. A 18 (2003) 4497.[3] M.R. Gaberdiel, Int. J. Mod. Phys. A 18 (2003) 4593.[4] A. Nichols, SU(2)k logarithmic conformal field theory, Ph.D. thesis, University of Oxford, 2002, hep-th/0210[5] N.E. Mavromatos, Logarithmic conformal field theories and strings in changing backgrounds, hep-th/04070[6] M.R. Gaberdiel, Nucl. Phys. B 618 (2001) 407.[7] J. Fjelstad, J. Fuchs, S. Hwang, A.M. Semikhatov, I.Yu. Tipunin, Nucl. Phys. B 633 (2002) 379.[8] F. Lesage, P. Mathieu, J. Rasmussen, H. Saleur, Nucl. Phys. B 647 (2002) 363;

F. Lesage, P. Mathieu, J. Rasmussen, H. Saleur, Nucl. Phys. B 686 (2004) 313.[9] A. Nichols, J. Stat. Mech. (2004) P09006.

[10] V.G. Knizhnik, A.B. Zamolodchikov, Nucl. Phys. B 247 (1984) 83.[11] M.R. Rahimi Tabar, A. Aghamohammadi, M. Khorrami, Nucl. Phys. B 497 (1997) 555.[12] M. Flohr, Nucl. Phys. B 634 (2002) 511.[13] J. Rasmussen, Nucl. Phys. B 730 (2005) 300.[14] P. Furlan, A.Ch. Ganchev, R. Paunov, V.B. Petkova, Phys. Lett. B 267 (1991) 63;

P. Furlan, A.Ch. Ganchev, R. Paunov, V.B. Petkova, Nucl. Phys. B 394 (1993) 665.[15] A.Ch. Ganchev, V.B. Petkova, Phys. Lett. B 293 (1992) 56.[16] M. Flohr, Nucl. Phys. B 514 (1998) 523.[17] S. Moghimi-Araghi, S. Rouhani, M. Saadat, Nucl. Phys. B 599 (2001) 531.[18] J. Rasmussen, J. Stat. Mech.: Theory Exp. (2004) P09007, math-ph/0408011.[19] A.B. Zamolodchikov, V.A. Fateev, Sov. J. Nucl. Phys. 43 (1986) 657.[20] A.A. Belavin, in: Proceedings of the Second Yukawa Symposium, Nishinomiya, Japan, in: Springer Procee

Physics, vol. 31, Springer, Berlin, 1988, p. 132.[21] A.M. Polyakov, in: L. Brink, D. Friedan, A.M. Polyakov (Eds.), Physics and Mathematics of Strings, World S

tific, Singapore, 1990.[22] M. Bershadsky, H. Ooguri, Commun. Math. Phys. 126 (1989) 49.[23] B.L. Feigin, E. Frenkel, Lett. Math. Phys. 19 (1990) 307.[24] J.L. Petersen, J. Rasmussen, M. Yu, Nucl. Phys. B 457 (1995) 343;

J.L. Petersen, J. Rasmussen, M. Yu, Nucl. Phys. B (Proc. Suppl.) 49 (1996) 27.[25] J. Rasmussen, Applications of free fields in 2D current algebra, Ph.D. thesis, NBI, Univ. of Copenhagen

hep-th/9610167.[26] J.L. Petersen, J. Rasmussen, M. Yu, Nucl. Phys. B 457 (1995) 309.[27] M. Flohr, Null vectors in logarithmic conformal field theory, hep-th/0009137.[28] M.R. Gaberdiel, P. Goddard, Commun. Math. Phys. 209 (2000) 549.[29] J. Rasmussen, Lett. Math. Phys. 73 (2005) 83.[30] M. Khorrami, A. Aghamohammadi, A.M. Ghezelbash, Phys. Lett. B 439 (1998) 283.[31] M. Flohr, M. Krohn, Four-point functions in logarithmic conformal field theory, hep-th/0504211.[32] J. Nagi, Phys. Rev. D 72 (2005) 086004.

il

n alumes us

lesf thes also

ciences,


Kink scaling functions in 2D non-integrable quantumfield theories

G. Mussardoa,b, V. Rivac,d,∗, G. Sotkove,1, G. Delfinoa,b

a International School for Advanced Studies, Via Beirut 1, 34100 Trieste, Italyb Istituto Nazionale di Fisica Nucleare, Sezione di Trieste, Italy

c Rudolf Peierls Centre for Theoretical Physics, University of Oxford, 1 Keble Road, Oxford, OX1 3NP, UKd Wolfson College, Oxford, UK

e Departamento de Fisica, Universidade Federal do Espirito Santo, 29060-900 Vitoria, Espirito Santo, Braz

Received 17 October 2005; accepted 9 December 2005


Abstract

We determine the semiclassical energy levels for theφ4 field theory in the broken symmetry phase o2D cylindrical geometry with antiperiodic boundary conditions by quantizing the appropriate finite-vokink solutions. The analytic form of the kink scaling functions for arbitrary size of the system allowto describe the flow between the twisted sector ofc = 1 CFT in the UV region and the massive particin the IR limit. Kink-creating operators are shown to correspond in the UV limit to disorder fields oc = 1 CFT. The problem of the finite-volume spectrum for generic 2D Landau–Ginzburg models idiscussed. 2005 Elsevier B.V. All rights reserved.

PACS:11.10.Kk; 11.15.Kc; 11.25.Hf; 11.27.+d

Keywords:Semiclassical quantization; Kink solutions in finite volume; Scaling functions; Conformal field theory

* Corresponding author.E-mail addresses:[email protected](G. Mussardo),[email protected], [email protected](V. Riva),

[email protected](G. Sotkov),[email protected](G. Delfino).1 On leave of absence—Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of S

Tsarigradsko Chaussee 72, BG-1784 Sofia, Bulgaria.








260 G. Mussardo et al. / Nuclear Physics B 736 [FS] (2006) 259–287

avioreories:

r-r in-

tentialsels of

ctrum,(cylin-nd on2D,ermo-r,

approx-fer

pecifice kinkbativedbilitys-riodicdressssical

sin thetral

assedndinga minor

back-cription

1. Introduction

The universal thermodynamical properties of statistical systems with multicritical behare described, in mean-field approximation, by appropriate Landau–Ginzburg (LG) field th

(1.1)Vl(φ) =l∑

k=1

λkφ2k−2, l = 3,4, . . .

Structural (commensurate–incommensurate) phase transitions[1], interface phenomena in odered and disordered media[2] and phase structure of ferromagnetic systems (see fostance[3]) provide few examples for the applications of the simplestφ4 and φ6 LG modelsto statistical mechanics and condensed matter physics. In two dimensions, the LG po(1.1) appear also in the description of the relevant perturbations of Virasoro minimal modconformal field theory[4], as well as of the renormalization group flows between them.

The physical quantities associated with a field theory—partition function, energy specorrelation functions, etc.—strongly depend on the geometry of the considered problemdrical, strip, plane, etc.), on the boundary conditions chosen (periodic, Dirichlet, etc.) athe range of the values of the couplingsλk . For several integrable quantum field theories inthe above quantities have been exactly computed in finite volume with the so-called Thdynamics Bethe ansatz method[5] or Destri–deVega equations[6]. These techniques, howeverequire the integrability of the model, and cannot be applied to the LG theories(1.1), due totheir non-integrable nature. In this case, the analysis of the finite-size effects is based onimate methods as perturbative renormalization group (see[2,3] and references therein), transintegral techniques[1] and numerical methods.

The low temperature (broken symmetry) phase of these models exhibits, however, sfeatures—multiple degenerate vacua, non-trivial topological sectors and non-perturbativsolutions (domain walls)—which require certain improvements of the standard perturmethods. The non-perturbative semiclassical expansion[7] is known to be an effective methofor the quantization of the kink solutions in an infinite volume, independently of the integraof the model. Its recent extension to finite geometries[8,9] allowed us to derive analytic expresions for the scaling functions of the sine-Gordon model defined on a cylinder with quasi-peb.c. (i.e. in the one-kink sector) and on a strip with Dirichlet b.c.’s. It is then natural to adthe problem of the finite-size effects in 2D LG models within the context of the semiclaquantization of kinks in finite volume.

The present paper is devoted to the derivation of the scaling functions of the 2Dφ4 theory on acylindrical geometry withantiperiodicb.c.φ(x+R) = −φ(x), which for this model correspondto consider a single kink on the cylinder. This continues our analysis of finite-size effectsφ4 model, which begun in[10] with the derivation of the finite-volume form factors and specfunctions for the same kind of geometry.

From the mathematical point of view, the derivation of the scaling functions for theφ4 theoryon thetwistedcylinder is analogous to the one performed in[8] for the sine-Gordon model oncylinder withquasi-periodicb.c. This is due to the fact that the finite volume kinks are exprein both cases in terms of a Jacobi elliptic function, and the computation of the correspoenergy levels is therefore based on the solution of the so-called Lamé equation. Besidestechnical difference (the equation appears now in a more complicated form, the so-calledN = 2Lamé form), an important new feature emerges in the antiperiodic case: the oscillatingground cannot be defined for any value of the size of the system, so that the complete des

G. Mussardo et al. / Nuclear Physics B 736 [FS] (2006) 259–287 261

pecific

linghe

isonehav-g data

hesnthel (

s fortion ofe havepesiodiceatingi.e.ionsnticalsink-ties.e

actns are

t theseolumeieved in

in thes andlassicaltion ofvelopeddd

of the problem is achieved in this case by also including a constant background below a svalue of the size.

Our main result, presented in Section2, consists in the analytic expression of the kink scafunctions (for arbitrary value of the size of the systemR), which describes the flow between ttwisted sector ofc = 1 CFT in the UV region and the massive particles in theQ = ±1 topologicalsectors of the brokenφ4 theory in the infrared (IR) limit. This section also includes a comparbetween the large-R corrections to the kink masses, as obtained from the IR asymptotic biour of the scaling functions, and the values expected from the infinite-volume scatterinthrough Luscher’s theory[11].

A detailed study of the UV regime is left to Section3. Here we analyse the properties of tc = 1 CFT fields that play the role of creating operators for theφ4 kinks, as well as of the kinkof generic LG models. It turns out that forZ2-invariant polynomial potentials (in their brokephase) the disorder fieldµ of dimension 1/8 (and its descendants) from the twisted sector ofc = 1 CFT are the only operators local with respect to the potential and carrying topologicaZ2)charges. Therefore they must describe the UV limit of the LG-kinks.

Section3 actually begins with the more familiar discussion of soliton-creating operatorthe sine-Gordon model in the winding (i.e., quasiperiodic) sector. Due to the compactificathe field, indeed, this theory admits more types of b.c., including the antiperiodic ones. Wthen devoted Section4 to the analysis of this interestingly rich model, which displays two tyof non-trivial classical solutions in finite volume, respecting two different b.c.’s (quasiperand antiperiodic). Their UV limits are described, respectively, by the standard soliton-croperators from the winding sector ofc = 1 CFT and by the disorder field in its twisted sector,that one which creates theZ2 charged kinks. The two corresponding types of scaling functare given explicitly, and their difference is observed at any finite volume, except for their ideIR limits. It is therefore clear that passing from periodic toZ2-symmetric polynomial potentialonly the kink-type (antiperiodic) solution survives, which explain why the finite volume ktype solutions of SG andφ4 models (as well as their UV limits) share many common proper

The explicit analytic form obtained in the present paper for the scaling functions of thφ4

model (and in previous works[8,9] for the sine-Gordon model) is intrinsically related to the fthat the stability equations to be solved are of Lamé type, and the corresponding solutiowell known. As we shall show in Section5, similar construction forφ6 and higher (l 5) LGmodels leads again to Schrödinger-like equations for periodic potentials, but it turns out thaare more complicated generalizations of the Lamé equation. The derivation of the finite-venergy spectrum of these models thus depends on the further progress that will be achthe future on their analytical or numerical solutions.

2. Semiclassical quantization of the broken φ4 theory in finite volume

The standard perturbative methods of QFT’s inD-dimensions (including theD = 2 case weare interested in) are known to be inefficient for the description of the quantum effectstopologically non-trivial sectors of an important class of theories with non-linear interactionmultiple degenerate vacua. As a rule, such theories admit finite-energy non-perturbative csolutions (kinks, vortices, monopoles, etc.) carrying topological charges. The quantizathese solutions (both static and time-dependent) requires non-perturbative techniques, deby Dashen, Hasslacher and Neveu (DHN) in[7] for theories in infinite volume. The DHN methoconsists, for static backgrounds, in splitting the fieldφ(x, t) in terms of the classical solution an


c

are then

r is ob-

tiza-chlet

od for

ion

its quantum fluctuations, i.e.

φ(x, t) = φcl(x) + η(x, t), η(x, t) =∑

k

eiωktηk(x),

and in further expanding the Lagrangian of the theory in powers ofη, keeping only the quadratiterms. As a result of this procedure,ηk(x) satisfies the so-called “stability equation”

(2.1)

[− d2

dx2+ V ′′(φcl)

]ηk(x) = ω2

kηk(x),

together with certain boundary conditions. The semiclassical energy levels in each sectorbuilt in terms of the energy of the corresponding classical solution and the eigenvaluesωi of theSchrödinger-like equation(2.1), i.e.

(2.2)Eni = Ecl + h∑

k

(nk + 1

2

)ωk + O

(h2),

wherenk are non-negative integers. In particular the ground state energy in each sectotained by choosing allnk = 0 and it is therefore given by2

(2.3)E0 = Ecl + h

2

∑k

ωk + O(h2).

In our recent papers[8,9], we have extended this technique to the study of soliton quantion in the sine-Gordon model on the cylinder (with periodic b.c.) and on a strip with Dirib.c. This section is devoted to the quantization of the kinks of theφ4 theory in theZ2 brokensymmetry phase, defined by the Lagrangian

(2.4)L= 1

2(∂µφ)

(∂µφ

)− V (φ), with V (φ) = λ

4

(φ2 − m2

λ

)2

,

on a cylinder with the antiperiodic b.c.’s

(2.5)φ(x + R) = −φ(x),

imposed. In order to fix the ideas and the notations, we first shortly review the DHN meththe quantization ofφ4-kinks in infinite volume.

2.1. Infinite volume kinks

The static solutions of the equation of motion associated to the potential(2.4)can be obtainedby integrating the following first order equation

(2.6)1

2

(∂φcl

∂x

)2

= 1

4

(φ2

cl − φ20

)(φ2

cl − 2+ φ20

),

where we have rescaled the variables as

(2.7)φ =√

λ

mφ, x = mx,

2 From now on we will fixh = 1, since the semiclassical expansion inh is equivalent to the expansion in the interactcouplingλ.


of the

formetric

Fig. 1. Potential(2.4)and infinite-volume kink(2.8)with x0 = 0.

andφ0 is an arbitrary constant defined byV (φ0) = −A, i.e.

1

2

(∂φcl

∂x

)2

= V (φcl) + A.

In infinite volume we have to impose as b.c. that the classical field reaches the minimapotential atx → ±∞, i.e. φcl(±∞) = ±1. This corresponds to choosing the valueφ0 = 1 forthe arbitrary constant in(2.6), and, as a consequence, we find the well-known kink solution

(2.8)φcl(x) = tanh

(x − x0√

2

),

shown inFig. 1, which has classical energyEcl = 2√

23

m3

λ.

The stability equation(2.1) around this background can be cast in the hypergeometricin the variablez = 1

2(1+ tanh x√2), and the solution is expressed in terms of the hypergeom

functionF(α,β, γ ; z) as

η(x) = z

√1− ω2

2m2(1− z)

−√

1− ω2

2m2F

(3,−2,1+ 2

√1− ω2

2m2; z)

.

The corresponding spectrum is given by the two discrete eigenvalues

(2.9)ω20 = 0, with η0(x) = 1

cosh2 x√2

,

and

(2.10)ω21 = 3

2m2, with η1(x) =

sinh x√2

cosh2 x√2

,

plus the continuous part, labelled byq ∈ R,

(2.11)

ω2q = m2

(2+ 1

2q2)

, with ηq(x) = eiqx/√

2(

3 tanh2x√2

− 1− q2 − 3iq tanhx√2

).


d the

ce be-a mass

e

alue of, in

dimen-n in, withlue offn

HNc sector

e

The presence of the zero modeω0 is due to the arbitrary position of the center of massx0 in(2.8), while ω1 andωq represent, respectively, an internal excitation of the kink particle anscattering of the kink with mesons3 of mass

√2m and momentummq/

√2.

The semiclassical correction to the kink mass can be now computed as the differentween the ground state energy in the kink sector and the one of the vacuum sector, pluscounterterm due to normal ordering:

(2.12)

M = Ecl + 1

2m

√3

2+ 1

2

∑n

[m

√2+ 1

2q2n −

√k2n + 2m2

]− 1

2δm2

∞∫−∞

dx

[φ2

cl(x) − m2

λ

],

with

(2.13)δm2 = 3λ

4π

∞∫−∞

dk√k2 + 2m2

.

The discrete valuesqn andkn are obtained by putting the system in a big finite volume of sizR

with periodic boundary conditions:

(2.14)2nπ = knR = qn

mR√2

+ δ(qn),

where the phase shiftδ(q) is extracted fromηq(x) in (2.11)as

(2.15)ηq(x) −→x→±∞ e

i[q mx√

2± 1

2δ(q)], δ(q) = −2 arctan

(3q

2− q2

).

SendingR → ∞ and computing the integrals one finally has

(2.16)M = 2√

2

3

m3

λ+ m

(1

6

√3

2− 3

π√

2

).

Notice that, from the knowledge of this quantity, one can extract a rough estimate of the vcouplings at which the brokenφ4 theory actually describes the Ising model. It is well knownfact, that perturbing the conformal Gaussian theoryLG = 1

2(∂µφ)(∂µφ) with the potential(2.4)one can have different renormalization group trajectories depending on the values of thesionless couplingλ/m2. The universality class of the Ising model is described by the situatiowhich the infrared point is not a massive theory but rather another conformal field theorycentral chargec = 1/2. Therefore, we can estimate semiclassically the corresponding vaλ/m2 by imposing the vanishing of the mass(2.16), which givesλ/m2 2. The large value othis quantity suggests, however, that the one-loop order in the semiclassical expansion iλ/m2

can hardly be able to detect the Ising fixed point.

2.2. Classical solutions in finite volume

Before discussing the kink solution on the cylinder, it is worth briefly recalling that the Dmethod can be also applied to the constant solutions describing the vacua in the periodi

3 The mesons represent the excitations over the vacua, i.e. the constant backgroundsφ± = ± m√λ

, therefore their squar

mass is given byV ′′(φ±) = 2m2.


n the

r

y

Fig. 2. Potential(2.4)and finite-volume kink(2.20)with x0 = 0.

of the theory. In particular, for the potential(2.4)we have

φvaccl (x) ≡ (±)

m√λ

,

(2.17)ωvacn =

√2m2 +

(2nπ

R

)2

, n = 0,±1,±2 . . . .

Therefore, according to(2.2), the smallest mass gap in the system, i.e. the difference betweefirst excited state and the ground state, is given by:

(2.18)E1(R) − E0(R) = ωvac0 (R) ≡ √

2m.

This quantity, which is related to the inverse correlation lengthξ−1 on a finite size[3,13,24], isthe one that has to be used4 in the definition of the scaling variable

(2.19)r ≡ mR.

If we now want to describe a kink on a cylinder of circumferenceR, we have to look for asolution of Eq.(2.6)satisfying the antiperiodic boundary conditions(2.5). This can be found fo1< φ0 <

√2, and it is expressed as

(2.20)φcl(x) =√

2− φ20 sn

(φ0√

2(x − x0), k

),

where sn(u, k) is the Jacobi elliptic function with modulusk2 = 2φ2

0− 1 and period 4K(k2),

whereK(k2) is the complete elliptic integral of the first kind (seeAppendix Afor the definitionsand properties of elliptic integrals and Jacobi elliptic functions). As shown inFig. 2, the clas-

sical solution(2.20) oscillates between the values−√

2− φ20 and

√2− φ2

0, and the boundarconditions(2.5)are satisfied by relating the elliptic modulus to the size of the system as

(2.21)mR =√

1+ k22K(k2).

As expected,(2.20) goes to the infinite-volume kink(2.8) for k → 1 (i.e. φ0 → 1), whichcorresponds to the infrared limitmR → ∞. In the complementary limitk → 0 (i.e. φ0 → √

2 ),

4 Up to inessential numerical constants which we fix here to 1/√

2 for later convenience.


und

lated

tely

asst

Fig. 3. Classical energy(2.24).

which corresponds tomR → π , the kink(2.20)tends to the constant solution

(2.22)φcl(x) ≡ 0,

which identically satisfies the antiperiodic b.c.(2.5)and can be used, therefore, as the backgrofield configuration in the interval 0< mR < π . The choice of the background

(2.23)φcl(x) =√

2− φ20 sn

( φ0√2(x − x0), k

)for mR > π,

0 for mR < π

will be fully motivated in the following, after the discussion of the stability frequencies reto the classical solutions(2.20) and (2.22).

The classical energy of the kink(2.23)is given by

(2.24)Ecl(R) =

m3

6λ1

(1+k2)3/2

3k4K(k2) + 2k2[K(k2) + 4E(k2)] + 8E(k2) − 5K(k2)

for mR > π,

m3

4λmR for mR < π,

and it is plotted inFig. 3. From the analytic knowledge of this quantity, we can immediaextract some important scattering data of the non-integrableφ4 theory. In fact, the leading termin the kink mass is given by the classical energy, expressed for genericR by (2.24). It is easy

to see that forR → ∞ the energy indeed tends to the infinite-volume limitEcl(R) → 2√

23

m3

λ.

From its asymptotic expansion for largeR, we can also obtain the leading order of the kink mcorrection in finite volume, and compare it with Lüscher’s theory[11,12]. Taking into accounthek → 1 (k′ → 0) expansions ofE andK (seeAppendix A) and noting from(2.21)that

e−√2mR = 1

256(k′)4 + · · · ,

we derive the following asymptotic expansion ofEcl for largeR:

(2.25)Ecl(R) = Ecl(∞) − 8√

2m3

λe−√

2mR + O(e−2

√2mR

).


ss

ion

inof thel

own in

tu-

The counterpart of this leading-order behavior in Lüscher’s theory is given by

(2.26)Mk(R) − Mk(∞) = −mbRkkbe−mbR,

where the indexk refers to the kink, and the indexb refers to the elementary meson (with mamb = √

2m), which can be seen as a kink–antikink bound state withS-matrix residueRkkb. Fromthe comparison between(2.25)and(2.26)we finally extract the leading semiclassical expressfor the residue of this 3-particle process

(2.27)Rkkb = 8m2

λ,

and therefore the 3-particle coupling5

(2.28)kkb = 2√

2m√λ

.

This quantity is of particular interest, since the non-integrability of theφ4 theory prevents theknowledge of its exactS-matrix. In the different context of infinite volume form factors,[10] we proposed another way of extracting this coupling, i.e. by looking at the residuekink–antikink form factor in infinite volume, and the result obtained in[10] is consistently equato (2.28).

2.3. Semiclassical scaling functions

The stability equation(2.1)around the background(2.20)takes the form

(2.29)

d2

dx2+ ω2 + 1− 3

(2− φ2

0

)sn2(

φ0√2x, k2

)η(x) = 0,

whereω = ω/m, and it can be reduced to the Lamé equation withN = 2 (seeAppendix B). Theallowed and forbidden bands, with corresponding values of the Floquet exponent, are shFig. 4.

The boundary conditions(2.5) translate into the requirement of antiperiodicity for the flucationη

η(x + R) = −η(x),

which selects the values ofω2 for which the Floquet exponent is an odd multiple ofπ . Theseeigenvalues are the zero mode

(2.30)ω20 = 0,

the discrete value

(2.31)ω21 = 3k2

1+ k2,

and the infinite series of points (with multiplicity 2) inside the highest band

(2.32)ω2n ≡ 1− 3

1+ k2

[P(an) +P(bn)

],

5 Crossing symmetry implies the equalityRkkb = Rkkb .


-

e-

one,

of

Fig. 4. Spectrum of Eq.(2.29).

with an, bn constrained by

(2.33)

F = 2i

K[ζ(an) + ζ(bn)] − (an + bn)ζ(K)

= (2n − 1)π, n = 2,3, . . . ,

P ′(an) +P ′(bn) = 0, n = 2,3, . . . .

In the IR limit (k → 1) this spectrum goes to the one related to the standard background(2.8).

In fact, the allowed band 1− 2√

k4−k2+11+k2 < ω2 < 0 shrinks to the eigenvalueω2

0 = 0, the other

band 3k2

1+k2 < ω2 < 31+k2 shrinks toω2

1 = 32, and finallyω2 > 1+ 2

√k4−k2+11+k2 goes to the contin

uous part of the spectrumω2q = 2+ 1

2q2.In order to complete the spectrum also at valuesmR < π , we have to put together the fr

quencies(2.31) and (2.32)with the ones obtained by quantizing the constant solution(2.22). Wetherefore obtain6

(2.34)ω21 =

3k2

1+k2 for mR > π,

−1+ π2

m2R2 for mR < π,

and

(2.35)ω2n =

1− 31+k2 [P(an) +P(bn)] for mR > π,

−1+ (2n − 1)2 π2

m2R2 for mR < π.

With the explicit knowledge of the stability frequencies, and in particular of the firstplotted inFig. 5, we can now understand the physical meaning of the pointmR = π . This corre-sponds, in fact, to the limitk → 0 and this is the value below which the analytic continuation

6 To be precise, notice that the eigenvalueω21 = −1 + π2

m2R2 is double, and atmR = π it splits into the two simpleeigenvalues(2.30) and (2.31).


uarehe so-

yround

ned in

Fig. 5. The first level defined in(2.34).

Fig. 6. The first few levels defined in(2.34)and(2.35).

the classical background(2.20)becomes imaginary. Correspondingly, the first frequency sqω2

1 tends to zero, and its continuation would become negative, signaling an instability of tlution. At the same time, the constant background(2.22)is stable just up to the pointmR = π ,as it can be easily seen fromFig. 5.

Fig. 6shows the plots, for generic values ofr in (2.19), of the first few frequencies given b(2.34)and(2.35), which represent the energies of the excited states with respect to their gstateE0(R).

We have now all data to write the ground state energy in the kink sector, which is defianalogy with the infinite volume case(2.12)as

(2.36)E0(R) = Ecl(R) + 1

2

∑i

ωi(R) + C.T.− 1

2

∞∑n=−∞

ωvacn (R),


as

panding

ionof thet

in a

e

y

thegiven

where the frequenciesωi are defined in(2.34)and(2.35), and the mass counterterm is defined

C.T.= −δm2

2

R/2∫−R/2

dx

[(φkink

cl (x))2 − m2

λ

],

with

δm2 = 3

4πλ

2π

R

∞∑n=−∞

1

ωvacn

, ωvacn (R) =

√2m2 +

(2nπ

R

)2

.

A more transparent expression for the ground state energy(2.36), which explicitly shows thecancellation of the divergencies present in each term separately, can be obtained by exall quantities around some specific value ofr . In particular, in the limits of large or smallr onecan extract the asymptotic IR and UV data of the theory. We have already seen in Sect2.2how the large-r expansion of the classical energy correctly encodes the scattering datainfinite volume theory, and we will now study the UV limitr → 0, in which we can extracsome conformal data related to the theory in exam. Furthermore, inAppendix Cwe perform theexpansion around the pointr = π , where it is possible to see how the divergencies cancelmore subtle way.

The small-r expansion of(2.36)is easily obtained to be

E0(R)

m= 2π

r

[ ∞∑n=1

(n − 1

2

)−

∞∑n=1

n

]− 1

4√

2

(2.37)+ r

2π

[π

2

m2

λ−

∞∑n=1

1

2n − 1+

∞∑n=1

1

2n

]+ · · · .

The individually divergent series present in(2.37)combine to give a finite result, in virtue of threlations

∞∑n=1

(2n − 1) −∞∑

n=1

(2n) = 2[ζ(−1,1/2) − ζ(−1)

]= 2

[1

24+ 1

12

],

∞∑n=1

1

(2n − 1)−

∞∑n=1

1

(2n)=

∞∑k=1

(−1)k+1

k= log2.

The UV behaviour forr → 0 of the ground state energyE0(R) of a given off-critical theoryis related to the Conformal Field Theory (CFT) data(h, h, c) of the corresponding critical theorand to the bulk energy term as

(2.38)E0(R) 2π

R

(h + h − c

12

)+BR + · · · ,

wherec is the central charge,h + h is the lowest anomalous dimension in a given sector oftheory andB the bulk coefficient. Therefore, we estimate the semiclassical bulk term to beby

B = m2(

1

4

m2

λ− log2

2π

).


t lines

spaceoncir-

exist.nof

round,-ects

e

leoSuchof the

e

ence ofas two

hat the

tate

Furthermore, our result for the leading semiclassical term in the anomalous dimension is7

(2.39)h + h = 1

8.

As it is fully discussed in Section3, this result agrees with the CFT prediction.Finally, again in accordance with the CFT expectation, the excited levels are given by

(2.40)Ekn(R) 2π

R

[1

8+∑n

kn

(n − 1

2

)]+[B − m2

2π

∑n

kn

2n − 1

]R + · · · .

2.4. Other interpretations of the classical solution

In concluding this section, it is worth to comment how the classical solution(2.20)has beenstudied in the literature either in different contexts, or in the same as ours but along differenof interpretation.

In fact, this kind of background, regarded however as a time-dependent solution in zerodimensions, has been proposed in[15] to describe dominating contributions to the partitifunction at finite temperatureT , i.e. when the Euclidean time variable is compactified on acumferenceβ = 1

kBTwith periodicb.c. In this case, the finite value ofT which corresponds to

k = 0 is naturally interpreted as a limiting temperature, above which no periodic solutionsMoreover, the background(2.20)has also been studied in[16,17]as a static classical solutio

on a cylindrical geometry. In these works, however,periodicb.c. are considered, and the sizethe system is related to the elliptic modulus as

mR =√

1+ k24NK(k2), with N ∈ N.

This choice, which corresponds to considering the solution as a train ofN kinks andN antikinks,implies the selection ofN distinct eigenvalues withω2

n < 0 in the spectrum of Eq.(2.29). Theirimaginary contributions to the energy levels indicate the instability of the considered backgwhich is explained in[17] by noting that in thek → 1 (R → ∞) limit the solution tends to a single kink, instead of keeping its periodic nature of a train of kinks and antikinks. All this reflthe ambiguity present in the definition of the size of the systemR in terms of the elliptic mod-ulusk, simply due to the periodicity of the Jacobi function sn(u, k), and correspondingly in thinterpretation of the solution for a chosen definition ofR. However, choosing(2.21), i.e.antiperi-odicb.c., the infinite volume limit is smoothly recovered ask → 1, and the corresponding singkink solution is stable. It is then natural to expect that for any value ofR of the finite system, alstime-dependent solution exist, which describe multikink or kink–antikink configurations.solutions can be quantized in finite volume as well, although this is a subject that is outscope of the present paper.

Finally, in the recent paper[18] the orbifold geometryS1/Z2 is considered, instead of thcircle, for the worldsheet space coordinatex, and a classical background very similar to(2.23)isintroduced. The analogy with our case, however, is only apparent. In fact, due to the abstranslational invariance, on the orbifold the kink and the antikink have to be considereddistinct degenerate solutions, suggesting therefore a phase transition atmR = π . In our case,on the contrary, the lowest energy level is never degenerate, consistently with the fact t

7 Notice that the central charge contribution−c/12 is absent in(2.37), because we are subtracting the ground senergies of kink and vacuum sector, which both have the same central chargec = 1.


ldan

iann con-

ce, SG

areg be-

ticles,pec-reforens andfor the

under-ace ofpace oflimits ofocal

ritical

rd

behavior of the scaling functions atmR = π does not hint at any underlying conformal fietheory. The discontinuity of the derivative ofω1 atmR = π should be then interpreted as justeffect of the semiclassical approximation.

3. Kink-creating operators in Landau–Ginsburg models

As it is well known, starting fromc = 1 CFT in two dimensions and adding to its Lagrangdifferent relevant operators with an appropriate choice of the coupling constants, one castruct many integrable and non-integrable 2D massive QFT’s having degenerate vacua[4]. Theycan be classified according to the symmetries preserved by the perturbation. For instanand double SG models are examples ofZ ⊗ Z2-invariant theories (i.e.φ → ±φ + 2πn), whileLG models ofZ2-invariant (i.e.φ → −φ) ones. The common feature of all these modelsthe non-perturbative topologically stable classical solutions (solitons or kinks) interpolatintween two vacua. In the quantum theory they give rise to specific “strong coupling” parcarrying topological (Z or/andZ2) charges and representing an important part of their IR strum. The description of the finite volume spectrum (on the cylinder) of these models therequires both the construction of the finite volume counterparts of such topological solutiothe identification of the quantum states related to them. An important consistency checkfinite volume spectrum is provided by its UV and IR limits (in the scaling variablemR) thatshould reproduce the CFT and the massive model spectra correspondingly. In order tostand the flow between the UV theory to the IR one, i.e. the relation between the CFT spstates (and the corresponding field operators) and the infinite volume (massive) particle sstates, it is also necessary to recognize the states (and operators) that describe the UVsuch solitons and kinks in thec = 1 CFT. The soliton (and kink) creating operators are non-lfunctionals of the fieldφ that satisfy the following requirements:

(a) To carry (Z or Z2) topological charges±1 or equivalently to produce specific b.c.’s8 for φ,

(3.1)φ(ze2iπ , ze−2iπ

)= φ(z, z) + 2πnR, n = ±1

for solitons (whereR is the compactification radius ofφ, sayR= β−1 for SG), and


)= −φ(z, z)

for the (Z2) kinks.(b) To be local with respect to the perturbation (i.e.,Vl(φ) =∑l

k=1 λkφ2k−2 for the LG models)

or/and to the corresponding energy density operator in order to have well defined off-cproperties.

Before discussing the construction of the kink-creating operators for the LG models(1.1), itis worthwhile to remind how the soliton operators are derived in the case of SG model[19,20].As it well known[21], the primary fields in the untwisted (“winding”) sector9 of the (compact)c = 1 Gaussian CFT are represented by the followingdiscreteset of vertex operators

Vn,s(z, z) = :exp(ipφ + ipφ):

8 The relation between thez andz coordinates used in this section and thex andt used in all the others is the standa

plane to cylinder one, i.e.z = eiR

(x+t) andz = e− i

R(x−t).

9 Defined by the condition that the chiral U(1) currentsI (z) = ∂ϕ(z) andI (z) = ∂ϕ(z) are single valued.


e

al

op-CFT

-

.

with

p = s

R , p = 2πgnR, n, s = 0,±1,±2, . . . .

Their “chiral” dimensions10 are given byh = (p+p)2

8πgandh = (p−p)2

8πgand therefore they hav

spins = h − h and dimension∆ = h + h. We have introduced the free fieldsϕ(z) andϕ(z) suchthat φ = ϕ(z) + ϕ(z) and its dual isφ = ϕ(z) − ϕ(z). They take values on the circleS1 withradiusR = 1

βand their correlation functions have the form:

(3.3)⟨ϕ(z)ϕ(w)

⟩= − 1

4πgln(z − w),

⟨ϕ(z)ϕ(w)

⟩= − 1

4πgln(z − w).

As one can easily verify from the OPE

(3.4)φ(z, z)Vn,s(0,0) = − i

4πg

(p ln

(z

z

)+ p ln(zz)

)Vn,s(0,0) + · · · ,

the vertex operatorsVn,s for n = ±1 and for arbitrary spin s, create theZ-type b.c.’s(3.1)(in factone can take, says = 0 or s = ±1, since the onlyφ contribution is relevant). They are also loc

with respect to the SG potentialVSG= m2

β2 cos(βφ) as it follows from their OPE’s (withR= 1β

)

cos(βφ(z, z)

)Vn,s(0,0)

(3.5)= 1

2

(z

z

) pβ4πg

(zz)pβ4πg Vn,s+1(0,0) + 1

2

(z

z

)− pβ4πg

(zz)− pβ

4πg Vn,s−1(0,0) + · · · ,

i.e. we have no changes under the transformation(ze2iπ , ze−2iπ ) to (z, z), since p = 2πgnβ

and pβ4πg

= n2. Therefore forn = ±1 they represent the one soliton-creating operators. The

erators withn 2 create multi-soliton states. It should be noted that in the perturbed(i.e. in SG theory) the dual fieldφ is nonlocal in terms of the SG fieldφ, i.e. we haveφ(x, t) = ∫ x

−∞ dy ∂yφ(x, y). TheZ topological (i.e. soliton) chargeQ is defined by the eigenvalues of the well-known SG charge operator

(3.6)Q = β

2π

∞∫−∞

dx ∂xφ(x, t).

In order to describe the operators that createZ2-type (antiperiodic) b.c.’s(3.2)for the SG fieldφ we have to consider the twisted sector of thec = 1 CFT. It is defined (see Ref.[22]) by thecondition that the chiralU(1) currentsI (z) = ∂ϕ(z) and I (z) = ∂ϕ(z) are double valued, i.etheir mode expansions contain only half-integer modes

(3.7)I (z) =∞∑

m=−∞Im− 1

2z−m− 1

2 , I (z) =∞∑

m=−∞Im− 1

2z−m− 1

2 ,

10 Note that we have introduced arbitrary normalization constantg in the actionAgauss= g2

∫d2x (∂µφ)(∂µφ) and

as a consequence the chiral component of the stress-tensorT (z, z) is given byT = −2πg:(∂φ)2:. The standard CFTnormalization isg = 1 , but we shall often useg = 1.
2π


by

r

fieldseld

men-

o

d an-eory

e oneit

ng

are

ctor of(i.e.val)

E

where the modesIm− 1

2(andI

m− 12) satisfy the following Heisenberg type algebra:

(3.8)[Im− 1

2, I

l− 12] = m − 1

2

2δm+l , [I

m− 12, I

l− 12] = 0.

The primary fields in this sectorµ±k,k

, i.e.

(3.9)Im+ 1

2µ±

k,k= 0, I

m+ 12µ±

k,k= 0, m, k, k = 0,1,2, . . . ,

have “chiral” dimensionshk = (2k+1)2

16 and hk = (2k+1)2

16 and the allowed spins are givens = 0,±1

2. As one can see from the OPE

(3.10)φ(z, z)µ±0 (0,0) = √

zµ±1 (0,0) + √

zµ±1 (0,0) + · · ·

the fieldsµ±0,0(0,0) = µ±

0 (of lowest dimensionh + h = 18 and spins = 0), called disorder (o

spin) fields, create branch cut singularity forφ and thus reproduces theZ2-type b.c.’s(3.2).Their locality with respect to cos(βφ) is a consequence of the OPE(3.10)and of the followingcorrelation function⟨

µ−0 (∞,∞)eiαφ(w,w) cos

(βφ(z, z)

)µ+

0 (0,0)⟩

(3.11)= C+−2

[((√

w − √z)(

√w − √

z)

(√

w + √z)(

√w + √

z)

) αβ4πg +

((√

w − √z)(

√w − √

z)

(√

w + √z)(

√w + √

z)

)− αβ4πg].

Note that the currentI (z) does not have zero mode in the twisted sector and therefore theµ±

k,kdo not carryU(1) (andZ), but onlyZ2 charge. All these properties of the disorder fi

µ±0 (0,0) lead to the conclusion that it represents the kink-creating operator. It should be

tioned that the fieldφ in this case takes its values on the orbifoldS1Z2

and, as usually, the tw

disorder fieldsµ±0 (0,0) are related to the two fixed pointsφ = 0 andφ = πR [14]. As we shall

show in Section4, in finite volume one can have both the quasiperiodic (soliton type) antiperiodic (kink type) solutions and states, which however in the IR (infinite volume) SG thare related to the same soliton (and antisoliton) states.

The description of the kink-creating operators in the LG models is quite similar to thof the SG model. The main difference is that the fieldφ is no longer compactified, i.e.

lives now on the orbifolded lineR(1)

Z2. The untwisted (i.e.Z2-even) sector of the correspondi

(noncompact)c = 1 CFT contains twocontinuousparameters(q, q) family of vertex operators

Vq,q = :exp(iqφ + iqφ): of “chiral” dimensionsh = (q+q)2

8πgandh = (q−q)2

8πg. As in SG case the

operators withq = 0 produce certain nontrivial b.c.’s forφ, but with continuousU(1) charge. Asexpected, there is not a properly definedZ topological charge in this case. Such operatorsalso non-local with respect to the LG potential(1.1)as it can be seen from the OPE’s, say

(3.12):φ(z, z)k::eiqφ(0,0): = :(

− iq

4πgln

z

z+ φ(0,0)

)k

eiqφ(0,0): + · · · .Therefore they cannot represent kink-creating operators. The structure of the twisted sethis noncompactc = 1 CFT is quite similar to the one considered in the context of the SGcos(βφ)) perturbation above. Since in the orbifold line (as well as in orbifold finite interwe have only one fixed pointφ = 0, we have correspondingly only one disorder fieldµ0 ofdimension 1/8 and spin zero. As in the SG case, the fieldµ0 produces branch cut in the OP


it

ch

on

of

eachume?

ts

e

rd ones

fferent)such

tions

with φ and so, it implements theZ2-type (antiperiodic) b.c.’s(3.2). In order to check whetheris local with respect to the LG potential let us consider its correlation functions

(3.13)⟨µ0(∞,∞)eiαφ(w,w)eiγ φ(z,z)µ0(0,0)

⟩= C0

(√w − √

z√w + √

z

) αγ4πg(√

w − √z√

w + √z

) αγ4πg

,

⟨µ0(∞,∞)eiαφ(w,w):φ(z, z)kµ0(0,0)

⟩(3.14)= C0(−i)k

(α

4πgln

(√w − √

z√w + √

z

)(√w − √

z√w + √

z

))k

.

These can be derived from theϕ mode expansionϕ(z) = ∑∞m=−∞

Im− 1

212−m

z−m+ 12 , the algebra

(3.8)of its modes and the properties(3.9)of the disorder fieldµ0. It is now easy to see that ea(linear) combination of even powers of the fieldφ is local with respect toµ0, i.e. it does notchange under the transformation(ze2iπ , ze−2iπ ) to (z, z). It becomes clear from this discussithat the only field that can createZ2-kinks in the LG models is then the disorder fieldµ0. In the“broken phase”φ4 model(2.4)we have only one kink interpolating between the two minimathe potential. In the symmetric type LG potentials, as for example

V oddl = 1

2

l−12∏

k=1

(φ2 − a2

k

)2 for l = 3,5, . . . ,

(3.15)V evenl = 1

2φ2

l−22∏

k=1

(φ2 − a2

k

)2 for l = 4,6, . . . ,

we have instead a finite number ofl degenerate vacua and therefore different kinks relatingtwo consecutive vacua. An important question is: how to distinguish them in a finite volMoreover, in the CFT language, what are the operators which create such kinks?

To answer such questions, observe that the minima of these potentials are at the poinφk =±ak (k = 1,2, . . . , l−1

2 for l odd) and since we considera1 > a2 > · · · the kinks are interpolatingbetweenφ1 andφ2, etc. and not, as in theφ4 case, between±φ0. Therefore the analog of thantiperiodic b.c.’s(3.2) for the case of many degenerate vacua is given by


)= ak + ak+1 − φ(z, z),

i.e. we have different b.c.’s for each kink. Indeed one can reduce such b.c.’s to the standa(3.2)by introducing the “shifted” fields and the analog of the antiperiodic b.c.’s(3.2) in the caseof many degenerate vacua is given by

(3.17)Φk(z, z) = φ(z, z) − (ak + ak+1)

2.

In this scheme, however, the new fields have different vacua expectation values. Since (diorbifolds based on(3.16) have different fixed points, one can formally prescribe to eachpoint one disorder fieldµ(k)(z, z). As we shall see on the example of theφ6 model in Section5below, although all these kinks have coinciding UV data, their finite volume scaling funcare however different, with different bulk coefficients, etc.


f kinkn sine-

irringurally

e form

tic

4. Sine-Gordon model with antiperiodic b.c.

In the light of the discussion of kink-creating operators presented in Section3, it is worth toillustrate in more detail the interesting case of the sine-Gordon model, where both kinds oexist. This fact can be easily understood in the framework of the correspondence betweeGordon and Thirring models. In fact, the sine-Gordon solitons are identified with the Thfermions, for which two types of boundary conditions (periodic and antiperiodic) can be natimposed in a finite volume.

The Euler–Lagrange equation for static backgrounds in the sine-Gordon model take th

(4.1)1

2

(∂φcl

∂x

)2

= m2

β2(1− cosβφcl + A),

and it admits three kinds of solution, depending on the sign of the constantA. The simplestcorresponds toA = 0 and it describes the standard kink in infinite volume:

(4.2)φ0cl(x) = 4

βarctanem(x−x0).

The other two solutions, relative to the caseA = 0, can be expressed in terms of Jacobi ellipfunctions[23], defined inAppendix A. In particular, forA > 0 we have

(4.3)φ+cl (x) = π

β+ 2

βam

(m(x − x0)

k, k

), k2 = 2

2+ A,

which has the monotonic and unbounded behaviour in terms of the real variableu+ = m(x−x0)k

shown inFig. 7. For−2< A < 0, the solution is given instead by

(4.4)φ−cl (x) = 2

βarccos

[k sn

(m(x − x0), k

)], k2 = 1+ A

2,

and it oscillates in the real variableu− = m(x − x0) between thek-dependent valuesφ and2πβ

− φ (seeFig. 7).The solution(4.3)satisfies quasiperiodic b.c.

(4.5)φ(x + R) = φ(x) + 2π

β,

Fig. 7. Solutions of Eq.(4.1), A > 0 (left-hand side),−2 < A < 0 (right-hand side).


.

-

undnd is

provided the circumferenceR of the cylinder is identified withR = 1m

2kK(k2). The completesemiclassical quantization of this background has been performed in[8]. It is worth to recall herethe UV limit of the corresponding energy levels, given by

Ekn(R)

m= 2π

r

(π

β2+∑n

knn

)− 1

4+ 1

β2r − 1

8

(r

2π

)2

(4.6)−(

r

2π

)3[1

8ζ(3) − 1

4(2 log2− 1) − π

2β2+∑n

kn

n

4n2 − 1

]+ · · · ,

wherekn is a set of integers defining a particular excited state of the kink.We will now present a similar analysis for the solution(4.4), which satisfies antiperiodic b.c

(4.7)φ(x + R) = −φ(x) + 2π

β,

if it is defined on a cylinder of circumference

(4.8)R = 1

m2K(k2).

Similarly to the kink(2.20)studied in theφ4 case, the solution(4.4)tends to the standard infinitevolume soliton(4.2) for A → 0, whenR goes to infinity. In the other limitA → −2, whichcorresponds tomR → π , (4.4)goes to the constant solution

(4.9)φcl(x) ≡ π

β,

which identically satisfies the antiperiodic b.c.(4.7)and can be therefore used as the backgroin the interval 0< mR < π . Therefore, the classical energy associated to this kink backgrou

(4.10)Ecl(R) =

8m

β2

[E(k) − 1

2(1− k2)K(k)]

for mR > π,

2m

β2 mR for mR < π,

and it is plotted inFig. 8.

Fig. 8. Classical energy(4.10).


ting

tan-

olu-

Fig. 9. Spectrum of Eq.(4.11).

The stability equation associated to(4.4) takes the form

(4.11)

d2

dx2+ ω2 + 1− 2k2 sn2 x

η(x) = 0,

where

(4.12)x = mx, ω = ω

m.

This can be cast in the Lamé form withN = 1 (for the details, seeAppendix B), which has theband structure shown inFig. 9. Imposing then the antiperiodic boundary conditions (i.e. selecthe values ofω2 for which the Floquet exponent is an odd multiple ofπ ), we obtain the simpleeigenvaluesω2

0 = 0 and

(4.13)ω21 = k,

and the infinite series of double eigenvalues

(4.14)ω2n ≡ 2k2 − 1

3−P(iyn)

in the bandω2 > k2, with yn defined by

(4.15)F = 2Kiζ(iyn) + 2ynζ(K) = (2n − 1)π, n = 2,3 . . . .

It is easy to see that in the IR limit (A → 0) this spectrum goes to the one related to the sdard background(4.2). In order to complete the spectrum, also at valuesmR < π , we have toglue the frequencies(4.13) and (4.14)with the ones obtained by quantizing the constant stion (4.9). We therefore obtain

(4.16)ω21 =

k for mR > π,

−1+ π2

m2R2 for mR < π,


s illus-the

erent

for thesisk

in the

me

Fig. 10. The first few levels defined in(4.16) and (4.17).

and

(4.17)ω2n =

2k2−1

3 −P(iyn) for mR > π,

−1+ (2n − 1)2 π2

m2R2 for mR < π,

which are plotted inFig. 10.The study of the corresponding scaling functions can be performed along the same line

trated for the brokenφ4 theory. One easily obtains the UV limit of the ground state energy inform

(4.18)E0(R) 2π

R

(h + h − c

12

)+BR + · · · ,

with h + h = 1/8 and

B = m2(

2

β2− log2

2π

).

Therefore, we have seen explicitly how the two types of kink(4.3) and (4.4), although theyhave the same IR limit, display different energy levels in finite volume, and in particular diffUV limits, describing both twisted and untwisted sectors ofc = 1 CFT.

5. Open problems and discussion

In this paper we have applied the semiclassical method to derive analytic expressionsenergy levels of the brokenφ4 theory on a cylinder with antiperiodic b.c. Although this analyis technically similar to the one performed in[8] for the sine-Gordon model in the one-kinsector, various conceptual differences have emerged.

The derivation of analytic expressions for the finite-volume semiclassical energy levelsφ4 model is based on two important ingredients: the explicit form of the kink solution(2.20)and the eigenvalues(2.31), (2.32)of theN = 2 Lamé equation. Therefore its extension toφ6 andhigher orderp 5 LG potentials(1.1)requires the knowledge of the corresponding finite-volukinks as well as certain properties of the solutions of their stability equations(2.1). Consider a


-

rive

family of symmetric (or “hyperelliptic”) LG potentials

V oddp = 1

2

p−12∏

k=1

(φ2 − a2

k

)2 for p = 3,5, . . . ,

(5.1)V evenp = 1

2φ2

p−22∏

k=1

(φ2 − a2

k

)2 for p = 4,6, . . . .

Their static kink solutions, i.e. the solutions of the first order equation

1

2

(dφcl

dx

)2

= Vp(φcl) + A = 1

2

p−1∏l=1

(φ2

cl − bl

),

whereA = −V (φ0), bl = bl(ak) andb1 = φ20, are given for both odd and evenp by the inverse

of the following hyperelliptic integrals:

(5.2)±2x =φ2

cl(x)∫φ2

0

dz√z∏p−1

l=1 (z − bl)

.

In the casep = 4 (i.e. for theφ6 model) the integral in(5.2) is of elliptic type and the corresponding finite-volume kink has the explicit form

(5.3)φ(p=4)

cl (x) =√

b1√1− (

1− b1b2

)sn2(√

b2(b3 − b1)x, k) ,

where

k2 =(

b3

b2

)b2 − b1

b3 − b1,

b2 = 12

(2a2

1 − b1 −√

b1(4a21 − 3b1)

),

b3 = 12

(2a2

1 − b1 + √b1(4a1 − 3b1)

).

This background satisfies the boundary conditions

φcl(R) =√b1 +√

b2 − φcl(0),

provided we identify the size of the system as

R = 1√b2(b3 − φ2

0)

K.

Although for p > 4 the kink solutions are not given in an explicit form, one can easily detheir stability equation through the change of variablez = φ2

cl(x):

(5.4)d2η(z)

dz2+ 1

2

(1

z+

p−1∑l=1

1

z − bl

)dη(z)

dz− V ′′

p (z) − ω2

2z∏p−1

l=1 (z − bl)η(z) = 0,


er

eigen-

ssary

mentsvels in

tationsmore

ccuratetiodiceriodicted

00325ne ofthe

s and

with the antiperiodic b.c. expressed as

(5.5)η(z(R)

)= −η(z(0)

),

whereR is the smallest real period of the hyperelliptic integral(5.2). The above second ordODE’s withp + 1 regular singular points (atz = 0, bl,∞) represents a generalization[29] of theLamé equation in the so-called algebraic form11

(5.6)d2η(z)

dz2+ 1

2

(1

z+ 1

z − 1+ 1

z − a

)dη(z)

dz− N(N + 1)z − λ

2z(z − 1)(z − a)η(z) = 0,

which coincides with(5.4) for N = 2, p = 3 andV ′′3 (z) = 6z − 2a2

1, i.e. for theφ4 potentialanalyzed in Section2.

Hence the derivation of the semiclassical scaling functions of the genericp 4 LG models(5.1) defined on the cylinder reduces to the problem of construction of the solutions andvalues of the generalized Lamé equation(5.4) for antiperiodic b.c.(5.5). For p 4 this is aninteresting open problem, whose analytical or numerical solutions will provide the neceingredients for calculations of the corresponding energy levels.

Finally, it is worth mentioning few more research directions that arise as natural developof the analysis carried out here. One of them consists of the determination of the energy lethe presence of different boundary conditions. Equally interesting is to extend our computo higher loop orders: although the one-loop quantization around a kink background ispowerful than standard perturbative techniques, we have seen however that it is not yet aenough to identify the Ising critical point in the phase diagram of theφ4 theory. The last poinwe would like to mention is the study of symmetry restoration in finite volume for antiperboundary conditions. This phenomenon is well understood in the vacuum sector (i.e. for pb.c.[3,24]) but it is still an open problem in the kink sector, and it may be fruitfully investigawithin the semiclassical approach.

Acknowledgements

This work is done under the European Commission TMR programme HPRN-CT-2002-(EUCLID). The work of V.R. is supported by EPSRC, under the grant GR/R83712/01. Ous (G.M.) would like to thank LPTHE and the ENS in Paris for their warm hospitality duringperiod in which this work was completed, and CNRS for partial financial support.

Appendix A. Elliptic integrals and Jacobi’s elliptic functions

In this appendix we collect the definitions and basic properties of the elliptic integralfunctions used in the text. Exhaustive details can be found in[25].

The complete elliptic integrals of the first and second kind, respectively, are defined as

(A.1)K(k2)=

π/2∫0

dα√1− k2 sin2 α

, E(k2)=

π/2∫0

dα√

1− k2 sin2 α.

11 The same equation is expressed in the alternative Weierstrass form in(B.1).


ed for

tary

e,

, is

The parameterk, called elliptic modulus, has to be bounded byk2 < 1. It turns out that the ellipticintegrals are nothing but specific hypergeometric functions, which can be easily expandsmallk:

K(k2)= π

2F

(1

2,

1

2,1; k2

)= π

2

1+ 1

4k2 + 9

64k4 + · · · +

[(2n − 1)!!

2nn!]2

k2n + · · ·,

E(k2)= π

2F

(−1

2,

1

2,1; k2

)= π

2

1− 1

4k2 − 3

64k4 + · · · −

[(2n − 1)!!

2nn!]2

k2n

2n − 1+ · · ·

.

Furthermore, fork2 → 1, they admit the following expansion in the so-called complemenmodulusk′ = √

1− k2:

K(k2)= log

4

k′ +(

log4

k′ − 1

)k′2

4+ · · · ,

E(k2)= 1+

(log

4

k′ − 1

2

)k′2

2+ · · · .

Note that the complementary elliptic integral of the first kind is defined as

K′(k2)= K(k′2).

The function am(u, k2), depending on the parameterk, and called Jacobi’s elliptic amplitudis defined through the first order differential equation

(A.2)

(d am(u)

du

)2

= 1− k2 sin2[am(u)],

and it is doubly quasi-periodic in the variableu:

am(u + 2nK + 2imK′) = nπ + am(u).

The Jacobi’s elliptic function sn(u, k2), defined through the equation

(A.3)

(d snu

du

)2

= (1− sn2 u

)(1− k2 sn2 u

),

is related to the amplitude by snu = sin(amu), and it is doubly periodic:

sn(u + 4nK + 2imK′) = sn(u).

Appendix B. Lamé equation

The second order differential equation

(B.1)

d2

du2− E − N(N + 1)P(u)

f (u) = 0,

whereE is a real quantity,N is a positive integer andP(u) denotes the Weierstrass functionknown under the name ofN th Lamé equation. The functionP(u) is a doubly periodic solutionof the first order equation (see[25])

(B.2)

(dP)2

= 4(P − e1)(P − e2)(P − e3),
du


on

o the

st

whose characteristic rootse1, e2, e3 uniquely determine the half-periodsω andω′, defined by

P(u + 2nω + 2mω′) = P(u).

The stability equation(2.29), related to the brokenφ4 theory, can be identified with Eq.(B.1)

for N = 2, u = φ0√2x + iK′ andE = (1+ k2)(1− ω2); also the stability equation(4.11), encoun-

tered in the analysis of the sine-Gordon model, can be identified with Eq.(B.1), in this case with

N = 1, u = x + iK′ andE = 2k2−13 − ω2. Both these identifications hold in virtue of the relati

betweenP(u) and the Jacobi elliptic function sn(u, k) (see formulas 8.151 and 8.169 of[25]):

(B.3)k2 sn2(x, k) = P(x + iK′) + k2 + 1

3.

Relation(B.3) is valid if the characteristic roots ofP(u) are expressed in terms ofk2 as

(B.4)e1 = 2− k2

3, e2 = 2k2 − 1

3, e3 = −1+ k2

3,

and, as a consequence, the real and imaginary half periods ofP(u) are given by the ellipticintegrals of the first kind

(B.5)ω = K(k), ω′ = iK′(k).

All the properties of Weierstrass functions that we will use in the following are specified tcase when this identification holds.

We will now present the solutions of the Lamé equation forN = 1 andN = 2, which havebeen derived in[26,27] together with more complicated cases.

In the caseN = 1 the two linearly independent solutions of(B.1) are given by

(B.6)f±a(u) = σ(u ± a)

σ (u)e∓uζ(a),

wherea is an auxiliary parameter defined throughP(a) = E, andσ(u) andζ(u) are other kindsof Weierstrass functions:

(B.7)dζ(u)

du= −P(u),

d logσ(u)

du= ζ(u),

with the properties

ζ(u + 2K) = ζ(u) + 2ζ(K),

(B.8)σ(u + 2K) = −e2(u+K)ζ(K)σ (u).

As a consequence of Eq.(B.8) one obtains the Floquet exponent off±a(u), defined as

(B.9)f (u + 2K) = f (u)eiF (a),

in the form

(B.10)F(±a) = ±2i[Kζ(a) − aζ(K)

].

The spectrum in the variableE of Eq. (B.1) with N = 1 is divided in allowed/forbidden banddepending on whetherF(a) is real or complex for the corresponding values ofa. We have thaE < e3 ande2 < E < e1 correspond to allowed bands, whilee3 < E < e2 andE > e1 are forbid-den bands. Note that if we exploit the periodicity ofP(a) and redefinea → a′ = a+2nω+2mω′,this only shiftsF to F ′ = F + 2mπ .


Fig. 11. Spectrum of Eq.(B.1) with N = 2, wheree1, e2, e3 are the roots ofP andg2 = 2(e21 + e2

2 + e23).

The solutions of the Lamé equation withN = 2 are given by

(B.11)f (u) = σ(u + a)σ (u + b)

σ 2(u)e−u[ζ(a)+ζ(b)],

wherea andb are two auxiliary parameters satisfying the constraints

(B.12)

3P(a) + 3P(b) = E,

P ′(a) +P ′(b) = 0,

andσ(u) andζ(u) are defined in(B.7). The Floquet exponent off (u) is now given by

(B.13)F = 2iK[ζ(a) + ζ(b)

]− (a + b)ζ(K).

The spectrum in the variableE of Eq.(B.1) with N = 2 is divided in allowed (A) and forbidden(F ) bands depending on whetherF is real or complex for the corresponding values ofa andb,as shown inFig. 11.

Finally, it is worth mentioning that the functionζ(u) admits a series representation[28] thatis very useful for our purposes in the text:

(B.14)ζ(u) = π

2Kcot

(πu

2K

)+(

EK

+ k2 − 2

3

)u + 2π

K

∞∑n=1

h2n

1− h2nsin

(nπu

K

),

whereh = e−πK′/K. The small-k expansion of this expression gives

ζ(u) =(

cotu + u

3

)+ k2

12

(u − 3 cotu + 3ucot2 u

)(B.15)+ k4

64

(−3u + (4u2 − 5

)cotu + ucot2 u + 4u2 cot3 u + sin2u

)+ · · ·(note thath ≈ ( k

4)2 + O(k4)). A similar expression takes place forP(u), by noting thatP(u) =− dζ(u)

du.

Appendix C. Ground state energy regularization at r ≈ π

We present in this appendix the evaluation of the ground state energy(2.36) for r π andr π , comparing the two corresponding expressions at the pointr = π .

In the caser π , we obtain

(C.1)E0

m(r) = A− + √

2

√1− r

π+ B−

(1− r

π

)+ · · · ,

where the coefficientsA− andB− are defined as

A− = m2

λ

π

4+

∞∑√(2n − 1)2 − 1+ 3

2

∞∑ 1√(2n)2 + 2

−∞∑√

(2n)2 + 2− 1

4√

2,

n=1 n=1 n=1


ned

e

ech-l

eass

B− = −m2

λ

π

4+

∞∑n=2

(2n − 1)2√(2n − 1)2 − 1

− 3

2

∞∑n=1

(2n)2

[(2n)2 + 2]3/2−

∞∑n=1

(2n)2√(2n)2 + 2

.

Expanding in 1(2n−1)

and 1(2n)

, we obtain

A− = m2

λ

π

4+

∞∑n=1

(2n − 1) −∞∑

n=1

(2n) − 1

2

∞∑n=1

1

(2n − 1)+ 1

2

∞∑n=1

1

(2n)− 1

4√

2− C−,

(C.2)

B− = −m2

λ

π

4+

∞∑n=1

(2n − 1) −∞∑

n=1

(2n) + 1

2

∞∑n=1

1

(2n − 1)− 1

2

∞∑n=1

1

(2n)− 1+ D−,

whereC− andD− are finite constants given by

C− =∞∑

k=1

(−1)k(2k − 1)!!(k + 1)!22k+1

[ζ(2k + 1,1/2)

2k+1− 3k + 1

2ζ(2k + 1)

],

D− =∞∑

k=1

(−1)k+1(2k − 1)!!k!22k+1

×[

2k + 3

k + 1

ζ(2k + 1,1/2)

2k+1+ (3k + 1)(2k + 1)

2(k + 1)ζ(2k + 1) − 2k+1

],

with numerical valuesC− 0.018,D− 0.39, and the functions in this expressions are defias

n! = 1 · 2 · · · · · n,

(2n + 1)!! = 1 · 3 · · · · · (2n + 1),

ζ(p) =∑∞

n=11kp ,

ζ(p,α) =∑∞n=0

1(k+α)p

.

The individually divergent series present in(C.2)combine to give a finite result, in virtue of threlations

∞∑n=0

(2n + 1) −∞∑

n=1

(2n) = 2[ζ(−1,1/2) − ζ(−1)

]= 2

[1

24+ 1

12

],

∞∑n=0

1

(2n + 1)−

∞∑n=1

1

(2n)=

∞∑k=1

(−1)k+1

k= log2.

Therefore, the final expressions for the coefficientsA− andB− are

A− = m2

λ

π

4+ 1

4− 1

2log 2− 1

4√

2− C−,

B− = −m2

λ

π

4+ 1

4+ 1

2log2− 1+ D−.

The other casemR π can be similarly treated, being more complicated only from the tnical point of view. In fact, it requires to compare, in the limitk → 0, the behavior of classicaenergy and stability frequencies, defined in(2.24), (2.31)and(2.32), respectively, with the onof the scaling variable, defined in(2.21). The expansions of elliptic integrals and Weierstrfunctions, necessary for this purpose, can be found inAppendices A and B. Since the scaling


t

ons and

ystems,86.

variable has the small-k behaviour

(C.3)r = π

[1+ 3

4k2 + · · ·

],

it is easy to see that

(C.4)Ecl

m= m2

λ

π

4

(1+ 3

4k2)

+ · · · = m2

λ

π

4+ m2

4λ(r − π) + · · ·

and

(C.5)ω1

m= √

3k + · · · = 2

√r

π− 1+ · · · .

The frequencies(2.32)have the most implicit expression in term ofr . Noting that in the highes

bandω2 > 1 + 2√

k4−k2+11+k2 the auxiliary parametersa and b are related asa = −b∗, we can

conveniently parameterizean andbn in (2.33)as

(C.6)

an = −xn + iyn,

bn = xn + iyn.

Expanding equations(2.33)for smallk, we obtain

(C.7)

xn = 12 arcsin

(√ 3(2n+1)2−1

)[1+ k2

4 + · · ·],yn = 1

2arcsinh(3√

(2n+1)2

[(2n+1)2−1][(2n+1)2−4])[

1+ k2

4 + · · ·],and therefore

(C.8)ω2n = [

(2n + 1)2 − 1]

1− 3

2k2 (2n + 1)2 − 2

(2n + 1)2 − 1+ · · ·

.

Comparing this with(C.3)we finally obtain

(C.9)ωn

m(r) =

√(2n + 1)2 − 1− (2n + 1)2 − 2√

(2n + 1)2 − 1

(r

π− 1

)+ · · · .

Therefore, the ground state energy has the behaviour

(C.10)E0

m(r) = A+ +

√r

π− 1+ B+

(r

π− 1

)+ · · · ,

whereA+ = A− andB+ = B−.

References

[1] J.A. Krumhansl, J.R. Schrieffer, Phys. Rev. B 11 (1975) 3535.[2] D. Jasnow, Renormalization Group Theory of Interfaces, in: C. Domb, J.L. Lebowitz (Eds.), Phase Transiti

Critical Phenomena, vol. 10, Academic Press, 1986;G. Forgacs, R. Lipowski, Th.M. Nieuwenhuizen, The Behaviour of Interfaces in Ordered and Disordered Sin: C. Domb, J.L. Lebowitz (Eds.), Phase Transitions and Critical Phenomena, vol. 14, Academic Press, 19

[3] J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, Clarendon Press, Oxford, 1989.[4] A.B. Zamolodchikov, Sov. J. Nucl. Phys. 44 (1986) 529.[5] Al.B. Zamolodchikov, Nucl. Phys. B 342 (1990) 695.


eprint

method,

[6] C. Destri, H. de Vega, Nucl. Phys. B 438 (1995) 413.[7] R.F. Dashen, B. Hasslacher, A. Neveu, Phys. Rev. D 10 (1974) 4130;

R.F. Dashen, B. Hasslacher, A. Neveu, Phys. Rev. D 11 (1975) 3424.[8] G. Mussardo, V. Riva, G. Sotkov, Nucl. Phys. B 699 (2004) 545.[9] G. Mussardo, V. Riva, G. Sotkov, Nucl. Phys. B 705 (2005) 548.

[10] G. Mussardo, V. Riva, G. Sotkov, Nucl. Phys. B 670 (2003) 464.[11] M. Lüscher, Commun. Math. Phys. 104 (1986) 177.[12] T.R. Klassen, E. Melzer, Nucl. Phys. B 362 (1991) 329.[13] J.L. Cardy (Ed.), Finite-Size Scaling, Amsterdam, North-Holland, 1988.[14] L. Dixon, D. Friedan, E. Martinec, S. Shenker, Nucl. Phys. B 282 (1987) 13.[15] B.J. Harrington, Phys. Rev. D 18 (1978) 2982.[16] N.S. Manton, T.M. Samols, Phys. Lett. B 207 (1988) 179.[17] J.Q. Liang, H.J.W. Müller-Kirsten, D.H. Tchrakian, Phys. Lett. B 282 (1992) 105.[18] H.T. Cho, Vacuum structure of two-dimensionalφ4 theory on the orbifoldS1/Z2, hep-th/0507179.[19] S. Mandelstam, Phys. Rev. D 11 (1976) 3026.[20] D. Bernard, A. LeClair, Commun. Math. Phys. 142 (1991) 99.[21] A.B. Zamolodchikov, Al.B. Zamolodchikov, Conformal field theory and 2D critical phenomena, 5A and 5B, pr

ITEP-90-91 and ITEP-90-92.[22] Al.B. Zamolodchikov, Nucl. Phys. B 285 (1987) 481.[23] K. Takayama, M. Oka, Nucl. Phys. A 551 (1993) 637.[24] E. Brezin, J. Zinn-Justin, Nucl. Phys. B 257 (1985) 867.[25] I.S. Gradshteyn, I.M. Ryzhik, Table of Integrals, Series, and Products, Academic Press, New York, 1980.[26] K. Takemura, The Heun equation and the Calogero–Moser–Sutherland system I: the Bethe ansatz

math.CA/0103077.[27] A.O. Smirnov, Elliptic solitons and Heun’s equation, math.CA/0109149.[28] H. Hancock, Lectures on the Theory of Elliptic Functions, Dover, New York, 1958.[29] E.L. Ince, Ordinary Differential Equations, Dover, New York, 1956.

ns

s

radigm.r Betheboth at

atively

-CFTnaly-ons and

its


Wrapping interactions and a new source of correctioto the spin-chain/string duality

J. Ambjørna,d, R.A. Janikb,∗, C. Kristjansenc

a The Niels Bohr Institute, Copenhagen University, Blegdamsvej 17, DK-2100 Copenhagen Ø, Denmarkb Institute of Physics, Jagellonian University, Reymonta 4, PL 30-059 Krakow, Poland

c NORDITA, Blegdamsvej 17, DK-2100 Copenhagen Ø, Denmarkd Institute for Theoretical Physics, Utrecht University, Leuvenlaan 4, NL-3584 CE Utrecht, The Netherland

Received 28 October 2005; accepted 12 December 2005


Abstract

Assuming that the world-sheet sigma-model in the AdS/CFT correspondence is an integrablequantumfield theory, we deduce that there might be new corrections to the spin-chain/string Bethe ansatz paThese come from virtual particles propagating around the circumference of the cylinder and rendeansatz quantization conditions only approximate. We determine the nature of these correctionsweak and at strong coupling in the near-BMN limit, and find that the first corrections behave qualitas wrapping interactions at weak coupling. 2005 Elsevier B.V. All rights reserved.

1. Introduction

The discovery of integrable structures underlying planarN = 4 SYM [1–4] as well as classical string theory onAdS5 × S5 [5–11] has opened up new avenues for testing the AdS/correspondence[12]. Moreover, the application of spin chain techniques has facilitated the asis on the gauge theory side. Non-trivial comparisons of gauge theory anomalous dimensistring state energies can now be carried out in three different regimes: the BMN limit[13], thenear-BMN limit [14] and the spinning string limit[15]. Whereas for the first of these three lim

* Corresponding author.E-mail addresses:[email protected](J. Ambjørn),[email protected], [email protected](R.A. Janik),

[email protected](C. Kristjansen).







J. Ambjørn et al. / Nuclear Physics B 736 [FS] (2006) 288–301 289

lagued

stringslysisesame

actionseds theto theactions

d threeher

lutionsations.gs was

y oneetheition for

d stringn

ltring

ies of

everything at the level of anomalous dimensions appears to work fine the latter two are pby vexing three-loop discrepancies[16,17].

It has been suggested that the three-loop discrepancy for near-BMN states and spinningis due to the non-commutativity of the two limits which are involved in performing the anaand which are imposed in different order on respectively the string and gauge theory sid[17].To perform an analysis on the gauge theory side in which the order of limits would be theas on the string theory side one would have to take into account so-called wrapping inter[18]. In the spin chain language wrapping interactions are interactions whose range excelength of the chain. It is a fact that wrapping interactions are important when it comesdetermination of anomalous dimensions of short operators. For instance, wrapping intercontribute to the four-loop anomalous dimension of the Konishi operator[2,19]. A systematicdescription of wrapping interactions in terms of Feynman diagrams can be found in[20]. Analternative explanation of the three-loop discrepancy was suggested in[21].

On the gauge theory side it is very hard to do any rigorous derivations beyond the two anloop ones of[2,22], see also[23]. However, assuming integrability and BMN scaling at higloop orders a conjecture for an all loop Bethe ansatz has been put forward[18,24,25]. On thestring theory side it is possible to derive integral equations encoding all classical string so[8,10,11,26]. These integral equations can be viewed as classical, continuum Bethe equInspired by these two achievements a suggestion for a Bethe ansatz for quantum strinpresented in[24,25,27]. The quantum string Bethe ansatz is identical to the gauge theorup to two loop order but differs from it beyond two loops. The all loop gauge theory Bansatz and the quantum string one are very similar in nature. They both express the condfactorized scattering for a set of elementary excitations with individual momentapk

(1)exp(iLpk) =M∏

j =k

S(pk,pj ),

whereS is the S-matrix for scattering of two of these excitations andL is the length of thespin chain. The elementary excitations have the same dispersion relation on the gauge antheory side. What differs between the two sides is the form of theS-matrices. The difference caconveniently be encoded in a so-called dressing factor

(2)Sstring= SdressingSgauge,

which can be expressed as a phase shift, i.e.

(3)Sdressing(pk,pj ) = exp(iθ(pk,pj )

),

where

(4)θ(pk,pj ) = 2∞∑

r=2

cr(λ)

(λ

16π2

)r(qr(pk)qr+1(pj ) − qr+1(pk)qr(pj )

),

with λ being the ’t Hooft coupling constant and theqr ’s certain conserved charges[18,27]. Theexpansion coefficientscr(λ) must fulfill that cr(λ) → 1 asλ → ∞ in order that the classicastring theory limit is correctly reproduced. Recently, it was found by a study of one-loop ssigma model corrections in thesl(2) sector that it is not possible to havecr(λ) = 1 for all valuesof λ [28]. The first string sigma model loop correction produces contributions to energstrings spinning with total angular momentumL which contain half-integer powers ofλ′ = λ

2
L

290 J. Ambjørn et al. / Nuclear Physics B 736 [FS] (2006) 288–301

tors

couldobtainle

e

tumnite

m fieldexcita-ersion

tions

e

tual

heninterac-

ptotictteringwrap-set in

rynein the

nsy and

[29,30]starting at order(λ′)5/2 [30] as well as non-perturbative contributions containing facof the type exp(− 1√

λ′ ) [29]. The appearance of terms of the type exp(− 1√λ′ ) as well as terms

with half-integer powers ofλ′ was earlier observed in the BMN limit in[31] in connection with astudy of the three string interaction vertex. The leading half-integer power ofλ′ in string energiescan be accounted for by the phase factor(4) if the coefficientc2(λ) has the expansion[30]

(5)c2(λ) = 1− 3

4

1√λ

+O(

1

λ

).

This opens the interesting possibility thatc2(λ) could tend to zero at weak coupling[30] andthus that the three loop discrepancy for near-BMN states as well as for spinning stringsindeed be due to an order of limits problem. Recently, it was shown that it is possible tothe classical string equations of motion of thesu(2) sector as the classical limit of an integrabquantumfield theory, namely theOsp(2m+2|2m) coset model[32]. This model is defined on thplane and not on the cylinder as needed for a quantum string theory onAdS5 × S5. The currentunderstanding is that theOsp(2m + 2|2m) coset model is capable of capturing the quaneffects introducing half-integer powers ofλ in the expansions of string energies but not the fisize corrections which appear when the theory is put on a cylinder[32].

In the present paper we will explore the consequences of putting an integrable quantutheory (IQFT) on a cylinder. In particular we shall discuss what happens if the elementarytions of the field theory in stead of the standard relativistic dispersion relation obey the disprelation implied by the conjectured quantum string Bethe equations[27]. We shall work in thenear-BMN limit and shall show that wrapping interactions generically give rise to contribuof orderλL at weak coupling and of order exp(− 1√

λ′ ) at strong coupling.One of the established properties of an integrablequantumfield theory on a cylinder is th

fact that the Bethe ansatz quantization conditions

(6)eipkL =∏j =k

S(pk,pj ),

are no longer exact (see e.g.[33] Section 5). The effect is very generic and comes from vircorrections—excitations going around the circumference of the cylinder (seeFig. 1below).

Intuitively such processes, involving a virtual particle going around the cylinder, wtranslated into Feynman graphs of the gauge theory, should correspond to wrappingtions. These types of virtual effects can be described1 throughS-matrix information, where theS-matrix is defined in the infinite volume limit. Thus, since there are proposals for the asymS-matrix for gauge theory/strings we propose to interpret them as the infinite volume scadata and use the framework of the virtual corrections to IQFT to incorporate the effect ofping interactions. We show that at weak coupling the order at which the virtual correctionsis λL just as expected for wrapping interactions.

The outline of this paper is as follows. In Section2we will review these corrections in ordinarelativistic integrable field theories. In Section3 we will motivate what changes need to be doin order to incorporate the non-standard dispersion relation characteristic of excitationsconjectured long range Bethe ansatz, then in Section4 we estimate the size of the correctioboth at weak coupling and in the near-BMN limit. We close the paper with a summaroutlook.

1 At leading order in all relativistic IQFT’s, and for some specific theories exactly.


sses/

d thee

the

les suchnimal

sheets weree wasstum

thence

2. Finite size mass shift in relativistic integrable field theories

In [34,35] Lüscher calculated the leading order corrections to the (infinite volume) maenergies of 1-particle states when put on a cylinder of circumferenceL:

(7)m(L) = m(L = ∞) + mµ(L) + mF (L).

These corrections arise from two different types of processes: the first is the so-calledµ-termwhich arises when the particle splits into a pair of virtual on-shell particles which go arouncylinder and recombine later, while the second one, theF -term comes from a virtual particlloop where the virtual particle goes around the circumference of the cylinder (seeFig. 1).

The expressions for these terms in the case of 1+ 1 dimensions and a single mass scale intheory are[38]

(8)mµ(L)

m(∞)= −

√3

2

∑b,c

Mabc(−i) resθ=2πi/3 Sabab (θ) · e−

√3

2 mL,

(9)mF (L)

m(∞)= −

∞∫−∞

dθ

2πe−mLcoshθ coshθ

∑b

(Sab

ab

(θ + i

π

2

)− 1

),

whereSabab (θ) is the (infinite volume)S-matrix, andMabc = 1 if c is a bound state ofa andb and

zero otherwise. These formulas have been checked to agree with a wide variety of exampas Ising field theory, the so-called scaling Lee–Yang model (SLYM), various perturbed mimodel CFT’s, etc.[37,38].

However, it is rather difficult to use these formulas directly in the case of the worldintegrable field theories appearing in our present context. The proofs of Lüscher’s formulaa mixture of diagrammatic analysis and analytical continuation where Lorentz invarianccrucial (although clearly many aspects remain valid on the lattice[39]). In particular, one alwayhad the dispersion relationE2 − p2 = m2 and the parametrization of energy and momen

Fig. 1. The diagram to the left (theµ-term) shows a particle splitting in two virtual, on-shell particles, traveling aroundcylinder and recombining. The diagram to the right (theF -term) shows a virtual particle going around the circumfereof the cylinder.


ithsually

for all

s ins of this

men-in are

e sizeof the

an in-l

es.r here isBethee term

ow

e

n

h

through rapidity

(10)E = mcoshθ, p = msinhθ.

For the world sheet string theories incurvedAdS space–time it was pointed out[9,14,16,40]thatthe usual light cone gauge likeτ = t (wheret is e.g. the global AdS time) is inconsistent wputting a Minkowskian metric on the world sheet. Hence the second gauge condition is utaken to be the uniform gauge where the density of a conserved charge (such as theR-chargeJ )is spread uniformly along the string.2 Then the gauge-fixed field theory isnotexplicitly Lorentz-invariant! A common ingredient of the asymptotic Bethe ansätze is the dispersion relationelementary excitations (hereE ≡ ∆ − J ):

(11)E =√

1+ 8g2 sin2 p

2, g2 = λ

8π2.

In the following we will assumethat this is indeed the true dispersion relation for excitationthe integrable quantum field theories on the world sheet and explore the consequenceassumption.

Another point which one should mention is that e.g. in the near-BMN limit the zero motum 1-particle states are protected by supersymmetry and what one is really interestedcorrections to 2-particle (or higher) states for which we lack similar expressions.

In order to attack the above problems it is convenient to look at a way in which the finitvirtual corrections can be calculated exactly, and which may shed more light on the originformulas(8) and (9).

There are basically three (interrelated) ways of computing exactly the spectrum oftegrable QFT on a cylinder: the thermodynamic Bethe ansatz (TBA)[42], non-linear integraequation (NLIE) or Destri–de-Vega (DDV) equation[43], and functional relations[33]. All thisis still an area of active research[44]. We will here concentrate on the first of these approach

Before we start let us emphasize that the thermodynamic Bethe ansatz that we considenot the thermodynamic approximation which is used for approximately solving algebraicansatz equations for a large number of roots in the spin chain/string literature. We use thin the sense that it is used in the context of relativistic integrable field theories.

Thermodynamic Bethe ansatzThe thermodynamical Bethe ansatz[42] was originally devised as a method for finding h

the ground state of an integrable field theory depends on the circumferenceL. The main idea isto consider the theory on a very elongated torus with circumferenceL and lengthR → ∞, andto compute the (Euclidean) partition function. From the point of view ofL being space andRbeing very large ‘time’ this gives the ground state energy being calculated:

(12)E0(L) = − 1

RlogZ.

On the other hand if one looks atR being space andL being the (Euclidean) time this is just ththermal free energy of the system on aninfinite line, which can be calculated by minimisingE −T S with T = 1/L. The main point is that for the theory withR → ∞, Bethe ansatz quantizatio

2 Usually one makes the conditionpφ = J keeping the size of the cylinder to 2π [9]. But this populates the action witexplicit factors ofJ . It is more convenient for our purposes to set the density to be equal to 1 and then the totalR-chargeis encoded just in the size of the cylinder like in[41].


ropriate

findingicalut first

del

-s.

-

ns

te for

remains exact. Then one introduces continuous densities for the occupied roots AND appentropy factors. The result is the set of equations3:

(13)ε(θ) = LETBA(θ) − φ ∗ L,

(14)E0 =∞∫

−∞

dθ

2πp′

TBA(θ)L(θ),

where

(15)L(θ) = log(1+ e−ε(θ)

),

(16)(φ ∗ L)(θ) =∞∫

−∞

dθ ′

2πφ(θ − θ ′)L(θ ′),

(17)φ(θ) = −id

dθlogS(θ),

andETBA(θ) = mcoshθ , andpTBA(θ) = msinhθ .As it stands the thermodynamic Bethe ansatz seems to be confined to being a tool for

just the ground state energy. However, in[45] Dorey and Tateo suggested that by analytcontinuation one could obtain the energies of excited states. This program was carried oin the scaling Lee–Yang model (SLYM)[45] and then in a series of perturbed minimal moCFT’s [46].

The underlying mechanism of the construction is thatL(θ) = log(1+e−ε(θ)) developed singularities which after a contour deformation added source terms to the right-hand sides of Eq(13)and (14). E.g. for a 1-particle state at zero momentum, themLcoshθ in (13) should be substituted by

(18)mLcoshθ + logS(θ − θ0)

S(θ + θ0),

whereθ0 is a singularity ofL(θ), which leads to an additional equationε(θ0) = iπ . This setof equations then givesexactresults for the excited state energyE1(L) for any L. Moreover,Lüscher’s formulas for(8) and (9)then immediately follow from solving that system of equatioby iteration (see[45]).

For our purposes it is interesting to obtain the form of corrections for a 2-particle stawhich we lacked an explicit expression.

From the results in[45] we obtain

E = 2 coshθB + 2 coshθB · δ

(19)−∞∫

−∞

dθ

2πcoshθe−LcoshθS

(θ + i

π

2− θB

)S

(θ + i

π

2+ θB

),

whereθB is the Bethe root which follows from solving the Bethe quantization condition

(20)eiLsinhθB = S(2θB).

3 For the simplest case of a single particle in the spectrum.


piece

me,

for at non-

states-particlee.g. thens

thate case

when

course

n themake anon-

provedc.

reIsingcan

ough an

We see that the first piece of(19) is the classical energy of the 2-particle state. The secondis aµ-term of the form

(21)δ = 3e−√

32 LcoshθB cothθB

√2 cosh 2θB + 1

2 cosh 2θB − 1,

while the last term is the analogue of theF -term. The exponential factor remains the sawhile the twoS-matrices correspond to a virtual particle of rapidityθ + iπ/2 moving around thecylinder and scattering off the two particles with Bethe roots (rapidities)±θB .

The point that we want to make here is that the structure of theF -term correction to the2-particle state is a straightforward generalization of the Lüscher mass shift formula1-particle state. In view of applications to the world sheet theory we would expect a firstrivial correction to occur at the level of 2-particle states since the mass of the one-particleare protected by supersymmetry. The absence of corrections for the ground state and 1states should be a property of the full world sheet theory and cannot be obtained just insu(2) sector. The reason is that thesu(2) excitation scatters non-trivially from other excitatio[25] which might therefore circulate in the loop.

Another motivation for discussing at length the TBA derivation of Lüscher’s formulas isthe space–time interpretation of the derivation of TBA may suggest its generalization to thof the world sheet theory which, when gauge-fixed, is no longer Lorentz-invariant.

In the next section we will use the TBA intuition to propose what changes are neededthe dispersion relation is no longer exactly relativistic but is modified to(11). We will also try tomotivate, using a solvable example, that such a procedure can be legitimate—which is ofnot completely clear a priori.

3. Non-standard dispersion relations

As we saw above, the derivation of the TBA is based on a modular transformation otorus i.e. an interchange of space and time. For a relativistic theory this does not reallydifference but it is far from clear if such a philosophy may be applied at all to the ratherstandard world-sheet theories.

In order to motivate this we will show an example where the proposed procedure can beto give the exact result despite the fact that the dispersion relation is not exactly relativisti

The Ising model off criticalityThis example has been given by Lüscher in his Cargese lectures[34]. We need to be here mo

explicit in the calculation, as the intermediate steps will be crucial for our purposes. Themodel on a finiteL × L lattice can be completely solved in terms of transfer matrices whichbe expressed through free fermion operators with the dispersion relation[36]:

(22)coshEq = coshm0 + 1− cosq ≡ coshm0 + 2 sin2 q

2.

The mass gap (i.e. the mass/energy of the lowest lying excitation) can be expressed threxact formula:

(23)m(L) = m0 + 1

2

L−1∑ν=0

EπL

(2ν+1) − 1

2

L−1∑ν=0

EπL

·2ν = m0 − 1

2

2L−1∑ν=0

eiπνEπL

·ν .


ur of

ore

re andtum in

trans-should

t in thee sameo

terms

The discrete sum can be rewritten using the (finite) Poisson resummation formula[47] as

(24)m(L) = m0 − L

∞∑ν=−∞

π∫−π

dq

2πeiqL(2ν+1)Eq.

Now after an integration by parts one changes variables to

(25)Q = −iEq,

Q will play the role of the new momentum in what follows. ForintegerL one then gets

(26)m(L) = m0 +∞∑

ν=−∞

1

2ν + 1

−iEπ∫−iE−π

dQ

2πeiq(Q)L(2ν+1).

We note that due to the form of(22)q(Q) = iEQ so the exponent simplifies to−EQL(2ν + 1).After a slightly non-trivial contour argument it turns out that one can change the contointegration to go from−π to π just as ifQ was a physical momentum. One then gets

(27)m(L) = m0 +∞∑

ν=0

2

2ν + 1

π∫−π

dQ

2πe−L(2ν+1)EQ ∼ m0 + 2

π∫−π

dQ

2πe−LEQ.

The leading correction is then the term withν = 0. The exponent is just as in the formula frelativisticF -term correction (with theS-matrix equal to−1, which is the value for the massivdeformation of the Ising model).

Space–time interchangeLet us now go back and analyze what kind of analytical continuations were made he

adopt a notation more reminiscent of the TBA expressions. What plays the role of momenthe final formula(27) is

(28)pTBA ≡ Q = −iEq,

while the energy appearing in the exponent of(27) is in fact −i times the original momentumexpressed in terms ofQ ≡ pTBA:

(29)ETBA = −iq(Q).

The above substitutions are indeed quite natural from the point of view of a modularformation exchanging space and time. Since time should be interchanged with space oneWick rotate both coordinates (hence the−i ’s in (28) and (29)), and also exchangeE andq witheach other. In the final integral(27)pTBA ≡ Q is taken to be real.

To add some more plausibility to the interpretation of the above formulas, one sees thaLorentz invariant case the dispersion relation in the double Wick rotated space remains thand the shift of the rapidity of the virtual particle byiπ/2 in theF -term exactly corresponds tthe above transformations(28)–(29).

(30)mcosh(θ + iπ/2) = imsinhθ, msinh(θ + iπ/2) = imcoshθ.

In the following section we will apply these ideas to estimate the size of the correctionfor the world sheet theory with the dispersion relation(11) but before we do that, we will showthat the above procedure works for a modified Ising model with(11).


e dis-Letnging

rmulae

ith theth atbtaineory

akingse for

csinh is

Modified Ising modelSince we lack any explicit example of a completely defined integrable theory with th

persion relation(11), we will construct a somewhat artificial but exactly solvable example.us modify the Ising model defined by the transfer matrices involving free fermions by chatheir dispersion relation from(22) to (11).

The exact mass shift can then be obtained in exactly the same fashion. It is given by foanalogous to(23)–(24), i.e.

(31)m(L) = 1− L

∞∑ν=−∞

π∫−π

dq

2πeiqL(2ν+1)

√1+ 8g2 sin2 q

2.

Let us now evaluateETBA(pTBA). Performing the substitutions(28)–(29)we obtain

(32)ETBA(pTBA) = 2 arcsinh

(1

2√

2g·√

1+ p2TBA

).

Now again the quite remarkable identity holds for integerL:

(33)−L

π∫−π

dq

2πeiqL(2ν+1)

√1+ 8g2 sin2 q

2= 1

2ν + 1

∞∫−∞

dq

2πe−L(2ν+1)·2arcsinh

(√1+p2

TBA2√

2g

),

which for ν = 0 reproduces the expected correction

(34)m(L) = 1+ 2

∞∫−∞

dq

2πe−L·2arcsinh

(√1+p2

TBA2√

2g

)

but with the somewhat odd-looking expression(32) for the energy.

4. Wrapping interactions and near-BMN limit

Let us now apply the above considerations to the case of the world sheet theory wdispersion relation(11). We will evaluate at what order does the virtual correction enter boweak and at strong gauge theory coupling (in the near-BMN limit). We will not try here to othe exact form of the correction since this would have to involve the full world sheet thwhoseS-matrix is so far not known completely. Moreover the scattering is non-diagonal mthe procedure even more complicated. The conjectured Bethe ansätze[25] may be used in thirespect but carrying out this program still remains a very non-trivial task which we leavfuture investigation.

In the following we will discuss first theF -term and then theµ-term.

TheF -termThe size of theF term at largeL is governed by the exponential factor

(35)e−LETBA = e−L·2arcsinh

(√1+p2

TBA2√

2g

).

Let us see how this expression behaves at small coupling. Then the argument of the arvery large and we may substitute it by a logarithmETBA ∼ −2 logg + · · ·. Consequently the


r sup-rticlesgauge

e

e

npair of

at an

ills in

lin

correction term is of the order

(36)g2L,

which is exactly the expected order when wrapping interactions should set in. This furtheports the intuition that virtual corrections in the world sheet theory corresponding to pamoving in loops around the cylinder should correspond to wrapping interactions on thetheory side.

Let us now proceed to take the near-BMN limit. We take4 L ∼ J → ∞ but at the same timkeepλ′ = λ/J 2. Therefore

(37)8g2 = λ′J 2

π2.

In this limit the argument of the arcsinh is small and the exponent(35) governing the size of thcorrections takes the form5

(38)e− 2π√

λ′√

1+p2TBA .

µ-termWe have much less control over the precise form of theµ-term, however its physical origi

is quite clear. It occurs when there is a possible process when a particle can decay into avirtual particles which areon-shell(but yet will have in general complex momenta).

For Lorentz-invariant theories, the exponents of theµ-terms(8) and (21)are seen to followfrom the imaginary parts of the momenta of the two virtual particles. Explicitly, suppose thon-shell particle of momentump0 disintegrates into a pair of particles with momentap1 andp2which are on-shell. From the kinematics we obtain forp1,2:

(39)p1,2 = p0

2± i

√3

2

√1+ p2

0.

We do not have a precise expression for theµ-term in the case of multi-particle states. We where just use an ad-hoc formula exp(−L · Impi), and we find that it reproduces the exponent(8) and (21).

Repeating the same kind of calculation with the dispersion relation(11) gives in generaquite complicated expressions. We may however takep0 to be the momentum of a particlea 2-particle state in the near-BMN limit:

(40)p0 = 2πn

J + 2+ 2πn√

1+ λ′n2(J + 2)2+ · · · .

Assumingpi to be small we obtain

(41)p1,2 = 1

J

(nπ ± i

√3√λ′ π

√1+ λ′n2

).

This suggests that theµ-term, if present, would also give 1/√

λ′ corrections in the near-BMNlimit.

4 For a two impurity state in thesu(2) sectorL = J + 2.5 We note that in this limit the same result would also arise from an unmodified BMN dispersion relationE =√1+ 2g2p2.


non-at thensiveith the

ve beenexamplenning-

t beens.odelantumhinski

of the

sheeterof the

d usingriantng,ulation

and just

s in theroundphs, be

inections

ll or aoset

field

various

5. Conclusions

Lacking a direct proof of the AdS/CFT correspondence, tests which are truly reflectingtrivial interactions are of utmost importance. The integrable structures discovered bothstring side and theN = 4 SYM side provide a good testing ground and there has been exteresearch comparing the predictions coming from the string world-sheet sigma model wpredictions from the spin-chain model ofN = 4 SYM.

Numerous examples of very detailed agreement between gauge and string theory hafound, nevertheless there remains some subtle but significant discrepancies. One notableis the notorious three loop discrepancy occurring both in the near-BMN limit and for spistrings. Another is the appearance half-integer powers ofλ in string sigma model loop corrections, cf. the introduction. However, it is known that at least some ingredients have noincluded in the asymptotic Bethe ansätze such as e.g. gauge theory wrapping interaction

While integrability on the string theory side is fairly developed for the classical sigma mnot much is known for the complete quantum theory. A very recent attempt to study the queffects viaS-matrix (Bethe ansatz) techniques can be found in the paper by Mann and Polc[32], who considered directly a relativistic quantum field theory onR2 (a Osp(2m + 2|2m) su-percoset model) with a knownS-matrix. They found an embedding of the ‘classical’ stringsu(2)

sector into the Bethe ansatz of their full quantum theory, and found generic correctionstype 1/

√λ, but did not consider finite size effects.

In our paper we wanted to concentrate on the specific features of an integrablequantumfieldtheory when put on a cylinder, as is always ultimately required for the closed string worldtheory. The generic feature is then that the Bethe ansatz quantization condition is no longexactand receives virtual corrections from particles moving in loops around the circumferencecylinder.

The energy shifts due to these effects in a finite volume are most conveniently calculatethe thermodynamic Bethe ansatz.6 We provided arguments that despite the non-Lorentz-invanature of the dispersion relation(11) conjectured to hold for the excitations of the AdS strione can still use TBA. It does not seem possible at this stage to perform a complete calcfor the world-sheet theory. Therefore, in this paper we have confined ourselves to understthe general nature of the corrections, using only the dispersion relation(11)as an assumption.

At weak coupling we find that the first corrections are of the orderλL. It is natural to con-jecture that the analogous term on the gauge-theory side is due to wrapping interactionspin-chain picture. This is also very intuitive as the propagation of a virtual excitation athe cylinder would presumably, when translated somehow into gauge theory Feynman gradescribed by graphs ‘wrapping’ the cylinder.

At strong coupling the contribution is of the ordere−1/√

λ′which cannot be seen directly

perturbative gauge theory. However, as mentioned earlier, in string sigma model loop corrone has found contributions to string energies involving half integer powers ofλ′ as well as

contributions of the forme−1/√

λ′[28–30], see also[31].

It would be very interesting to perform a more complete calculation, perhaps using apart of the asymptotic Bethe ansätze of[25]. The same could be also considered in the supercmodel of[32], which has the additional simplification that it is a true relativistic integrable

6 Although no general proof of this method for excited states exists, extensive tests have been performed inrelativistic integrable field theories[45,46,48].


ndardalismtrings.

try), arame-italitye and(2005–

p-

.

hep-

Mills

2,

s.

ev.

–Mills

0405

-

)

s. 2

s. Lett.

,

-

) 115,

theory—so it does not suffer from the conceptual difficulties associated with the non-stadispersion relation. Last but not least, it would be very interesting to try to extend the formof such finite size virtual corrections to many particle states dual to macroscopic spinning s

Acknowledgements

The authors were all supported by ENRAGE (European Network on Random GeomeMarie Curie Research Training Network financed by the European Community’s Sixth Fwork Programme, network contract MRTN-CT-2003-504052. R.J. thanks the NBI for hospwhen this work was carried out. R.J. was supported in part by Polish Ministry of SciencInformation Society Technologies grants 1P03B02427 (2004–2007) and 1P03B040292008).

References

[1] J.A. Minahan, K. Zarembo, The Bethe-ansatz forN = 4 super-Yang–Mills, JHEP 0303 (2003) 013, heth/0212208.

[2] N. Beisert, C. Kristjansen, M. Staudacher, The dilatation operator ofN = 4 super-Yang–Mills theory, Nucl. PhysB 664 (2003) 131, hep-th/0303060.

[3] N. Beisert, M. Staudacher, TheN = 4 SYM integrable super-spin chain, Nucl. Phys. B 670 (2003) 439,th/0307042.

[4] A.V. Belitsky, S.E. Derkachov, G.P. Korchemsky, A.N. Manashov, Quantum integrability in (super)Yang–theory on the light-cone, Phys. Lett. B 594 (2004) 385, hep-th/0403085.

[5] I. Bena, J. Polchinski, R. Roiban, Hidden symmetries of theAdS5 ×S5 superstring, Phys. Rev. D 69 (2004) 04600hep-th/0305116.

[6] G. Arutyunov, S. Frolov, J. Russo, A.A. Tseytlin, Spinning strings inAdS5×S5 and integrable systems, Nucl. PhyB 671 (2003) 3, hep-th/0307191;G. Arutyunov, J. Russo, A.A. Tseytlin, Spinning strings inAdS5 × S5: New integrable system relations, Phys. RD 69 (2004) 086009, hep-th/0311004.

[7] L. Dolan, C.R. Nappi, E. Witten, A relation between approaches to integrability in superconformal Yangtheory, JHEP 0310 (2003) 017, hep-th/0308089.

[8] V.A. Kazakov, A. Marshakov, J.A. Minahan, K. Zarembo, Classical/quantum integrability in AdS/CFT, JHEP(2004) 024, hep-th/0402207.

[9] G. Arutyunov, S. Frolov, Integrable Hamiltonian for classical strings onAdS5 × S5, JHEP 0502 (2005) 059, hepth/0411089.

[10] N. Beisert, V.A. Kazakov, K. Sakai, K. Zarembo, The algebraic curve of classical superstrings onAdS5 × S5,hep-th/0502226.

[11] L.F. Alday, G. Arutyunov, A.A. Tseytlin, On integrability of classical superstrings inAdS5 ×S5, JHEP 0507 (2005002, hep-th/0502240.

[12] J.M. Maldacena, The largeN limit of superconformal field theories and supergravity, Adv. Theor. Math. Phy(1998) 231, Int. J. Theor. Phys. 38 (1999) 1113, hep-th/9711200;S.S. Gubser, I.R. Klebanov, A.M. Polyakov, Gauge theory correlators from non-critical string theory, PhyB 428 (1998) 105, hep-th/9802109;E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253, hep-th/9802150.

[13] D. Berenstein, J.M. Maldacena, H. Nastase, Strings in flat space and pp waves fromN = 4 super-Yang–Mills,JHEP 0204 (2002) 013, hep-th/0202021.

[14] C.G. Callan, H.K. Lee, T. McLoughlin, J.H. Schwarz, I.J. Swanson, X. Wu, Quantizing string theory inAdS5 × S5:Beyond the pp-wave, Nucl. Phys. B 673 (2003) 3, hep-th/0307032.

[15] S. Frolov, A.A. Tseytlin, Semiclassical quantization of rotating superstring inAdS5 ×S(5), JHEP 0206 (2002) 007hep-th/0204226;S. Frolov, A.A. Tseytlin, Quantizing three-spin string solution inAdS5 × S5, JHEP 0307 (2003) 016, hepth/0306130.

[16] C.G. Callan, T. McLoughlin, I.J. Swanson, Holography beyond the Penrose limit, Nucl. Phys. B 694 (2004hep-th/0404007;


Nucl.

2004)

ding

hep-

0.(2004)

4.406256.

0509096..B 548

086002,

ener-

re given

le states,

, Rev.

0) 485.in QFT,

1) 151,

e–Yang

r lattice

Ising

)

9, hep-

C.G. Callan, T. McLoughlin, I.J. Swanson, Higher impurity AdS/CFT correspondence in the near-BMN limit,Phys. B 700 (2004) 271, hep-th/0405153.

[17] D. Serban, M. Staudacher, PlanarN = 4 gauge theory and the Inozemtsev long range spin chain, JHEP 0406 (001, hep-th/0401057.

[18] N. Beisert, V. Dippel, M. Staudacher, A novel long range spin chain and planarN = 4 super-Yang–Mills,JHEP 0407 (2004) 075, hep-th/0405001.

[19] B. Eden, C. Jarczak, E. Sokatchev, A three-loop test of the dilatation operator inN = 4 SYM, Nucl. Phys. B 712(2005) 157, hep-th/0409009.

[20] C. Sieg, A. Torrielli, Nucl. Phys. B 723 (2005) 3, hep-th/0505071.[21] J.A. Minahan, TheSU(2) sector in AdS/CFT, Fortschr. Phys. 53 (2005) 828, hep-th/0503143.[22] N. Beisert, Thesu(2|3) dynamic spin chain, Nucl. Phys. B 682 (2004) 487, hep-th/0310252.[23] A.V. Belitsky, G.P. Korchemsky, D. Mueller, Integrability in Yang–Mills theory on the light cone beyond lea

order, Phys. Rev. Lett. 94 (2005) 151603, hep-th/0412054;A.V. Belitsky, G.P. Korchemsky, D. Mueller, Integrability of two-loop dilatation operator in gauge theories,th/0509121.

[24] M. Staudacher, The factorized S-matrix of CFT/AdS, JHEP 0505 (2005) 054, hep-th/0412188.[25] N. Beisert, M. Staudacher, Long-rangepsu(2,2|4) Bethe ansätze for gauge theory and strings, hep-th/050419[26] V.A. Kazakov, K. Zarembo, Classical/quantum integrability in non-compact sector of AdS/CFT, JHEP 0410

060, hep-th/0410105;N. Beisert, V.A. Kazakov, K. Sakai, Algebraic curve for the SO(6) sector of AdS/CFT, hep-th/0410253;S. Schafer-Nameki, The algebraic curve of 1-loop planarN = 4 SYM, Nucl. Phys. B 714 (2005) 3, hep-th/041225

[27] G. Arutyunov, S. Frolov, M. Staudacher, Bethe ansatz for quantum strings, JHEP 0410 (2004) 016, hep-th/0[28] S. Schafer-Nameki, M. Zamaklar, K. Zarembo, Quantum corrections to spinning strings inAdS5 × S5 and Bethe

ansatz: A comparative study, JHEP 0509 (2005) 051, hep-th/0507189.[29] S. Schafer-Nameki, M. Zamaklar, Stringy sums and corrections to the quantum string Bethe ansatz, hep-th/[30] N. Beisert, A.A. Tseytlin, On quantum corrections to spinning strings and Bethe equations, hep-th/0509084[31] I.R. Klebanov, M. Spradlin, A. Volovich, New effects in gauge theory from pp-wave superstrings, Phys. Lett.

(2002) 111, hep-th/0206221.[32] N. Mann, J. Polchinski, Bethe ansatz for a quantum supercoset sigma model, Phys. Rev. D 72 (2005)

hep-th/0508232.[33] V.V. Bazhanov, S.L. Lukyanov, A.B. Zamolodchikov, Quantum field theories in finite volume: Excited state

gies, Nucl. Phys. B 489 (1997) 487, hep-th/9607099.[34] M. Lüscher, On a relation between finite size effects and elastic scattering processes, DESY 83/116, Lectu

at Cargese Summer Institute, Cargese, France, 1–15 September 1983.[35] M. Lüscher, Volume dependence of the energy spectrum in massive quantum field theories. I. Stable partic

Commun. Math. Phys. 104 (1986) 177.[36] T.D. Schultz, D.C. Mattis, E.H. Lieb, Two-dimensional Ising model as a soluble problem of many fermions

Mod. Phys. 36 (1964) 856.[37] T.R. Klassen, E. Melzer, Purely elastic scattering theories and their ultraviolet limits, Nucl. Phys. B 338 (199[38] T.R. Klassen, E. Melzer, On the relation between scattering amplitudes and finite size mass corrections

Nucl. Phys. B 362 (1991) 329.[39] G. Munster, The size of finite size effects in lattice gauge theories, Nucl. Phys. B 249 (1985) 659.[40] R.R. Metsaev, C.B. Thorn, A.A. Tseytlin, Light-cone superstring in AdS space–time, Nucl. Phys. B 596 (200

hep-th/0009171.[41] L.F. Alday, G. Arutyunov, S. Frolov, New integrable system of 2dim fermions from strings onAdS5 × S5, hep-

th/0508140.[42] A.B. Zamolodchikov, Thermodynamic Bethe ansatz in relativistic models. Scaling three state Potts and Le

models, Nucl. Phys. B 342 (1990) 695.[43] C. Destri, H.J. de Vega, Unified approach to thermodynamic Bethe ansatz and finite size corrections fo

models and field theories, Nucl. Phys. B 438 (1995) 413, hep-th/9407117.[44] See e.g., P.A. Pearce, L. Chim, C. Ahn, Excited TBA equations II: Massless flow from tricritical to critical

model, Nucl. Phys. B 660 (2003) 579, hep-th/0302093;J. Balog, A. Hegedus, TBA equations for the mass gap in the O(2r) non-linearσ -models, Nucl. Phys. B 725 (2005531, hep-th/0504186.

[45] P. Dorey, R. Tateo, Excited states by analytic continuation of TBA equations, Nucl. Phys. B 482 (1996) 63th/9607167.


98) 575,
[46] P. Dorey, R. Tateo, Excited states in some simple perturbed conformal field theories, Nucl. Phys. B 515 (19hep-th/9706140.
[47] B.J.B. Crowley, Some generalisations of the Poisson summation formula, J. Phys. A 12 (1979) 1951.[48] J. Balog, A. Hegedus, TBA equations for excited states in theO(3) andO(4) nonlinearσ -model, J. Phys. A 37

(2004) 1881, hep-th/0309009.

es

5

ories inshow

heorieserate one.hases areonly foregenerate

eYM2

nses,

n that


Large-N transitions for generalized Yang–Mills theoriin 1+ 1 dimensions

Florian Dubath

Département de Physique Théorique, Université de Genève, 24 quai Ernest-Ansermet, CH-1211 Genève 4, Switzerland

Received 18 September 2005; received in revised form 29 November 2005; accepted 13 December 200

Available online 4 January 2006

Abstract

We describe the entire phase structure of a large number of colour generalized Yang–Mills the1+1 dimensions. This is illustrated by the explicit computation for a quartic plus quadratic model. Wethat the Douglas–Kazakov and cut-off transitions are naturally present for generalized Yang–Mills tseparating the phase space into three regions: a dilute one a strongly interacting one and a degenEach region is separated into sub-phases. For the first two regions the transitions between sub-pdescribed by the Jurkiewicz–Zalewski analysis. The cut-off transition and degenerated phase arisea finite number of colours. We present second-order phase transitions between sub-phases of the dphase. 2005 Elsevier B.V. All rights reserved.

1. Introduction

Since ’t Hooft’s seminal work, the Yang–Mills theory in 1+1 dimensions (YM2) has becoma laboratory for testing ideas and concepts about Yang–Mills and also string theory. Thetheory has an exact stringy description in the limit of a large numberN of colours[1–3]. It is alsoknown that one can build generalizations of the YM2 theory[1,4] and that such generalizatiohave also a stringy behaviour at largeN [5]. It was also shown that YM2 has different phasand in particular a third-order transition was present by Douglas and Kazakov[6] (here afterDK transition). Recently, new progress has been made in YM2. In particular it was showits time evolution could be interpreted as a Brownian motion into the gauge group[7–9]. The

E-mail address: [email protected](F. Dubath).




F. Dubath / Nuclear Physics B 736 [FS] (2006) 302–318 303

ed in

whichrunningonic

systemnsitionmper-DK

lf-line

ure allpick uptudy wegeneralscribed

ainJZte

ction

-eat

il-

equivalent of the cut-off transition, well known for Brownian motion, has also been identifiYM2, and is different from the DK one.

The relevant parameter for both cut-off and DK transitions is the area of the manifold,plays the role of an inverse temperature. The phase space is then a half-line for the areafrom zero to infinity. As fermions do the YM2 state density is limited by 1 and the fermipicture can be used to help understand the phase structure: at very low temperature thebehaves as a degenerate Fermi liquid. Raising the temperature, we found the cut-off traand above it a strong interacting phase where the exclusion principle is at work. At high teature the fermions dilute and finally the density falls down below 1. Above this point (thetransition) the fermionic nature is irrelevant, and we have a weak interacting system.

Working with generalized Yang–Mills (GYM) theory the phase space opens from the haof the YM2 case to a hyper plane.

All generalized YM2 theories have the same structure. It is therefore possible to captessential features of their phase space by studying a particular model. In this paper wea quartic plus quadratic model and describe its phase space and transitions. From this sdeduce the general case. In particular, we show that the cut-off and the DK transitions arefeatures that extend into generalized YM2 and that such transitions coexist with those deby Jurkiewicz and Zalewski[13] (hereafter JZ transitions).

This paper is organized as follows. We first recall in Section2 how the generalization of YM2is obtained. We define the model we use in Section3. We present the phase space and our mresults in Section4. Detail of the computations are given in the following sections: DK andtransition in Sections5, 6 and 7, cut-off transition in Section8, transition between the degeneraphases in Section9. We draw some conclusions in Section10.

2. Generalized YM2

The action is the key for building the generalized YM2. Rather than writing the usual awith theFµνFµν term, we follow Ref.[1] and use an equivalent action with an auxiliary fieldφ.For thed = 2 case, this action is

(1)I = −1

4

∫d2x

(i∑a

φaεµνFµν a + g2

2

∑a

φaφa

).

The generalized YM2 theories (GYM2) are obtained by replacingg2

2

∑a φaφa by a sum contain

ing other terms of higher order inφ with other coupling constant. Building a generalized hkernel equation[4] and using the holonomy variable, we obtain a Hamiltonian of the form

(2)HG =∑

k

λkL

Nk−1Ck,

with a higher order Casimir operatorCk rather than only the usual quadratic one. This Hamtonian replaces the YM2 one which is1

(3)H2 = λ/2L

NC2.

1 Note that the 1/2 is re-absorbed intoλ2 for the generalized case.

304 F. Dubath / Nuclear Physics B 736 [FS] (2006) 302–318

n of the

ver the

lmir

as ae(inise in the

e GYMhe

produce

In the above expressions we have absorbed the coupling constant into the generalizatio’t Hooft couplingλk , which is held fixed at largeN [4].

The YM2 partition function[1,10] on an orientable surfaceM of genusg, with p boundariesand surfaceA, is a sum over the irreducible representationsR of the gauge group:

(4)ZM =∑R

d2−2g−pR χR(U1) · · ·χR(Up)exp

−λ2A

2NC2(R)

,

wheredR is the dimension of the representation andχR(Uj ) the character of the holonomyUj .Its generalized counterpart is simply

(5)ZM =∑R

d2−2g−pR χR(U1) · · ·χR(Up)exp

−

∑k

λkA

Nk−1Ck(R)

.

Until this point the analysis has been completely general. In order to perform the sum oirreducible representations, we now specify the gauge group. We are interested inSU(N). Thesegroups have irreducible representations labeled by maximal weighthi. A Young diagram canbe associated to each representation with rows of a length given byhi. We make the usuachange of variablesni = hi + N+1

2 − i. The computation of the symmetrized quartic Casifor SU(N) can be found in[5]. We have

(6)C2(ni

) =N∑

i=1

n2i − 1

N

(N∑

i=1

ni

)2

− N(N2 − 1)

12,

C4(ni

) =N∑

i=1

n4i − 2N2 − 3

6

N∑i=1

n2i − 4

N

N∑i=1

n3i

N∑j=1

nj + 6

N2

N∑i=1

n2i

(N∑

j=1

nj

)2

(7)+ N2 − 3

6N

(N∑

i=1

ni

)2

− 3

N

(N∑

i=1

ni

)4

+ N(N2 − 1)(11N2 − 9)

720.

3. Quartic model

We focus on the case of the sphere, i.e., we study the model for a surfaceM with g = 0 andno boundary. For YM2, the partition function reduces to

(8)ZS =∑

n1>n2>···>nN

(∏i<j

(ni − nj )

j − i

)2

exp

−λ2A

2N

∑n2

i

exp

λ2A(N2 − 1)

12

.

As the denominator∏

i<j (j − i) is the same for all the representations, one can see itnormalization constant and forget it. Note that the rescaled areaλ2A plays the role of an inverstemperature and that in the large-N limit this model is equivalent to fermions in a potentialthe sense that the state density cannot be greater than 1). The transitions we consider arlarge-N limit for very different values of the rescaled area.

Rather than dealing with the general case, one can capture the essential features of thby studying the quartic Casimir case.2 We use the case of a GYM2 model with a mix of t

2 Models where the higher Casimir is odd do not lead to an energy bounded from below and therefore do nota well defined theory.


given

2

ri-

wheresity andransitionare

-gximum

ay. Weensity

ated in

stateof the

s place

quartic and quadratic Casimir instead of only the quadratic one. The Hamiltonian is nowby

(9)Hm = λ2/2L

NC2 + λ4L

N3C4 = λ4L

(µ/2

NC2 + 1

N3C4

),

whereµ is the ratio of the 2 coupling constantsµ = λ2/λ4, and we have kept explicit the factorof λ2/2 in order to make the contact between the limitsµ → ∞ and YM2. The partition functionis

(10)ZS =∑

n1>···>nN

∏i<j

(ni − nj )2 exp

−λ4A

(µ

2NC2

(ni) + 1

N3C4

(ni))

.

From now on we will work with theSU(N) group, thereforeC2(ni) andC4(ni) are given by(6) and (7).

4. Phase space

The phase space is described by the variablesAλ4 andµ. We can also recast these two vaables into the Jurkiewicz–Zalewski description[13]:

(11)β2 = λ2A = µλ4A,

(12)β4 = λ4A.

In this paper, we will use both parameterizations.Anticipating our results, we plot the complete phase space (seeFig. 1) for theβ ’s parameter-

ization.As in the YM2 case, we have three kinds of phases.For smallβ ’s (high temperature) we have dilute phases where the state density is every

below 1. There are two different dilute phases. The first one has a continuous state denthe other one a gapped state density. Between these two phases a third-order phase ttakes place as expected from JZ work[13]. The dilute phases and the transition in betweendescribed in Section5.

Raisingβ ’s, we cross the DK transition (seeSections 5 and 6). The DK transition is a hightemperature process. It is easier to have intuition about it in the fermionic picture. Raisinβ ’scorresponds to lets the temperature go down. The dilute fermions concentrate until the maof the state density is 1. Below this critical temperature, the fermionic nature comes into plbring the DK transition to light by reversing this conceptual chain: we compute the state din a bosonic picture. The DK transition takes place for the value of theβ ’s for which the statedensity goes above 1.

After the DK transition we enter into strongly interacting phases. Again these are separdifferent subspaces, which are the continuation of the JZ ones.

Working at finiteN , we encounter another phase transition for highβ ’s. This is the cut-off transition (see Section8). This takes place at low temperature, when the fundamentalceases to dominate the partition function. This transition was discovered as an analogycut-off transition for random walk on a finite surface[7–9]. In the fermionic picture it simplycorresponds to the temperature crossing of the Fermi energy. The cut-off transition takealong the two (N -dependent) curves:

(13)β2 = 4 ln(N) − β4,

3


ts arefithr the graynces to

eratesee Sec-r

.

sition

Fig. 1. The (β2, β4) plane forN = 10000, with the typical form of the density function for each phase. The 2 red dothe DK and cut-off transitions for YM2 at the usual valuesβ2|DK = π2 andβ2|cut-off = 4 ln(N). The actual location othe thin lines sketched into the strongly interacting phase (phases with tray) areN -dependent and not computable wthe methods used in this article. The dashed lines correspond to second order phase transitions and (apart foline of the cut-off transition) the continuous lines to third order phase transitions. (For interpretation of the referecolor in this figure legend, the reader is referred to the web version of this article.)

(14)β2 = 7

24β4 − 4 ln(N) + 1

8

√β2

4 + 64β4 ln(N).

These lines draw a triangle with summit

(15)(β2, β4) =(

0,−4 ln(N)),(0,4 ln(N)

),

(4

3ln(N),8 ln(N)

).

The last summit lies on the lineµ = β2/β4 = 1/6.For largerβ ’s (and finiteN ) the system is in degenerate phases. There are two degen

phases (the fundamental state being given by the trivial or a stepped representation; stion 8), which are separated by theβ4 = 6β2 line. We show in Section9 that a second-ordephase transition between the two degenerate phases occurs along this line.

5. DK transition for GYM2

The DK transition manifests itself as a saturation of the state density. In the large-N limit,after passing to a continuous variable, one can show that the state densityρ cannot exceed 1This constraint is not built in the Gaussian matrix model that we use to computeρ [9]; we cantherefore track the state density only until its maximum reaches 1. After which the DK trantakes place. Therefore, finding a value of the parameter (the rescaled area) for whichρ reachesthe value 1 is sufficient to claim the presence of the DK transition.


y, we

gh

n that

e

teps.

r

We start by presenting the YM2 DK transition. In order to compute the state densitperform a saddle-point analysis of the partition function(8), rewriting it as

(16)ZS =∑

n1>n2>···>nN

eN2Seff,

(17)Seff = 2

N2

∑j<i

ln(|ni − nj |

) − λ2A

2N3C2.

Taking as zero the variation ofSeff with respect toni , and passing to continuous variables throuni

N= n and 1

N

∑ = ∫dnρ(n), we obtain forρ(n) a singular integral equation[11]

(18)P

∫dn′ ρ(n′)

n − n′ = λ2A

2n,

whereP∫

is the principal value integral. This equation can be solved under the assumption varies continuously on a single interval (one-cut solution) and we obtain[11,12]

(19)n ∈[− 2√

λ2A,

2√λ2A

],

(20)ρ(n) = λ2A

2π

√4

λ2A− n2.

The maximum ofρ arises forn = 0 and is equal to 1 ifλ2A = π2. The DK transition takes placfor the valueπ2 of the rescaled areaλ2A.

We tract out the DK transition for the quartic plus quadratic GYM2 following the above sThe effective action that replaces(17) is computed using(6) and (7)and is

(21)Seff = −λ4A

(1

N5C4 + µ

2N3C2

)+ 2

N2

∑j<i

ln(|ni − nj |

)

= −λ4A

[1

N

∑i

(ni

N

)4

−(

3µ − 2

6− 1

2N2

)1

N

∑i

(ni

N

)2

− 41

N

∑i

(ni

N

)3 1

N

∑j

nj

N+ 6

1

N

∑i

(ni

N

)2(

1

N

∑j

nj

N

)2

− 3

(1

N

∑i

ni

N

)4

+(

1− 3µ

6− 1

2N2

)(1

N

∑i

ni

N

)2]

(22)+ 1

N2

∑j<i

(2 ln

(|ni − nj |) + λ4A

ninj

3N2

)+ const.

After taking the variation and changing for continuous variables,3 we obtain the singulaintegral equation

P

∫dn′ ρ

(n′) 1

n − n′ − λ4A

2

(4n3 + 3µ − 2

3n

)

3 As the continuous limit is valid for largeN , we can drop the sub-leading contributions.


ndnd, we

esis

metric

t

= λ4A

2

[∫dn′ ρ

(n′)(−12n2n′ − 4n′3 + 1− 3µ

2n′

)

+ 12∫

dn′ ρ(n′)∫

dn′′ ρ(n′′)(nn′n′′ + n′2n′′)

(23)− 12∫

dn′ ρ(n′)∫

dn′′ ρ(n′′)∫

dn′′′ ρ(n′′′)n′n′′n′′′

].

In order to solve this equation forρ(n) we face two new problems. First, if we write the left-haside of this equation into a kernel form we have no longer a Cauchy type integral and, secocan learn from the study of the cut-off transition (see Section8) that the largeλ4A distributionof n may have a gap for some range ofµ. So we have to be careful where the one-cut hypothis valid and work with a two-cut case where it is not.

Looking at (22) one can see that symmetric tables (in the sensenk = −nN−k+1) are localminimum ofSeff. Therefore, in the saddle point analysis we will look at distributionsρ(n) whichare compatible with this property. That is we restrict ourself to distributions which satisfyρ(n) =ρ(−n). This symmetry hypotheses together with the fact that the integral range is symaround 0 gives for oddk

(24)∫

dn′ ρ(n′)n′k = 0,

and(23) reduces to

(25)P

∫dn′ ρ(n′)

n − n′ = λ4A

2

(4n3 + 3µ − 2

3n

).

Working with (25), we have to ensure that the symmetry condition is fulfilled.

5.1. One-cut solution

From the study of the low-temperature case, we expect that, for large enoughµ, the distri-bution ρ(n) will contain no gap. The corresponding solution for(25) is the so-called one-cusolution.4 Solving(25) can be done using the same machinery as for(18) (see[11]). We obtainthe interval[−a, a] given by

(26)a2 = 1

18

(2− 3µ +

√9µ2 − 12µ + 4+ 432

λ4A

)

and, after some computation, the state density

(27)ρ(n) = λ4A

2π

(3µ − 2

3+ 2a2 + 4n2

)√a2 − n2.

This function fulfills the symmetry condition and has a maximum atn = 0 or two maxima at

(28)n = ±1

6

√4− 6µ +

√9µ2 − 12µ + 4+ 432/λ4A.

4 One-cut into the plane describing the complexified variablen.


t

uationthis

d thehis

betion

threearated

The boundary between these two cases is a curve into the(µ,λ4A) plane given by the righbranch of

(29)B1: λ4A =(

4

µ − 2/3

)2

.

Above this curve the maximum arises atn = 0 and we can compute the value ofλ4A such thatρ(0) = 1; this gives a curveTDK,1(µ), which has the asymptotic forµ → ∞ (β4 → 0):

(30)TDK,1(µ) = π2

µ+

(2π2

3− 4

)1

µ2+O

(1

µ3

)

and crossB1 for µ ∼ 2.588. BelowB1, the two maxima ofρ are equal to 1 for a curveTDK,2(µ),which picks a maximum atµ = 2/3 and meetsTDK,1 onB1.

We have to check the one-cut condition. In the two-maxima region,ρ(0) is a minimum andwe verify that it is positive. If it is not the case we are no longer in the one-cut case. The eqρ(0) = 0 is satisfied on the left branch ofB1 and the one-cut solution breaks down abovecurve.

5.2. Two-cut solution

For smallµ we have to compute the two-cut solution. Using the symmetry condition anasymptotic conditions from the general formalism of[11], we can compute the state density. Tfunction has support on two segments[−a,−b] ∪ [b, a] and is given by[14,15]

(31)ρ(n) = 2λ4A

π|n|

√a2 − n2

√n2 − b2.

With a, b satisfying

(32)a2 = 2− 3µ

12+ 1√

λ4A,

(33)b2 = 2− 3µ

12− 1√

λ4A,

b is real above the left branch of the curveB1, exactly where the one-cut solution ceases tovalid. Finding the maximum ofρ(n) we are able to compute the area at which the DK transitakes place. We obtain the curveTDK,3 which has the asymptotic forµ → −∞:

(34)TDK,3(µ) = −π2

µ−

(2π2

3+ 4

)1

µ2+O

(1

µ3

)

and meetsTDK,2 when crossing the curveB1. Thus the whole DK transition forSU(N) takesplace along a continuous curve (seeFig. 2).

Above the DK transition (for smaller temperature, larger area), the solutionρ(n) cannot betrusted.

5.3. Jurkiewicz–Zalewski structure of the dilute phase

For small coupling values, we are below the DK transition, into dilute phases. We haveregions: the two-cut, the one-cut with two maxima and the one-cut with one maximum, sep


n

o phaseichtion is

i-

avefound

g phasesity to

sityal.

Fig. 2. The (µ,λ4A) plane with the one- and two-cut regions and typical form ofρ(n) in each region. The DK transitiolines are also plotted.

by the curveB1; that is, inβ ’s language

(35)β2 = 2

3β4 ± 1

4β

3/2a .

Along the minus branch, between the one- and two-maximum one-cut regions, we have ntransition. The densityρ and its support[−a, a] are determined by the same functions, whhave no singularity of any type along this line; the free energy in the saddle approximauniquely determined byρ.

Along the upper branch we expect from the JZ classification[13] a third-order phase transtion, since a gap opens in the support ofρ. That is easily checked by noting that

(36)ρ2-cut(a,n)∣∣B1

= ρ1-cut(a,n)∣∣B1

= 2β4

πn2

√a2 − n2,

and that the support borna1-cut and a2-cuts have the same value and first derivative but ha different second derivative. Detailed computation of the dilute phase structure can beinto [16]

6. Degree of the DK phase transition for GYM2

We study the transition between the one-cut one-maximum phase and the correspondinabove the DK transition. Crossing the DK transition causes the maximum of the state denbe replaced by a tray.

Working into the saddle approximation, the free energy depends only on the state denρ.We computeρ for the one-tray phase. The idea, see[6], is to set the density at 1 into an intervWe expect a symmetric function and we can parameterizeρ by

(37)ρ(n) = 0, |n| > a,

(38)ρ(n) = ρ(n), c |n| a,

(39)ρ(n) = 1, |n| < c.


g the

.

haseeo find a

rongly

o-tray

hich is

Making the substitution into Eq.(25), we obtain a two-cut problem forρ

(40)P

∫dn′ ρ(n′)

n − n′ = λ4A

2

(4n3 + 3µ − 2

3n

)− ln

(n + c

n − c

).

We can use the same setup as in Section5.2(see[11] for details): we compute the resolvent forρ

(41)ω0(p) =∮C

dz

2πi

λ4A2

(4z3 + 3µ−2

3 z) − ln

(z+cz−c

)p − z

√p2 − c2

√p2 − a2

√z2 − c2

√z2 − a2

.

Deforming the contour to the pole at infinity we also enclose the cut of the logarithm. Takindiscontinuity equation and coming back fromρ to ρ we eventually obtain the state density

(42)ρ(n) =√

a2 − n2√

n2 − c2

π

c∫−c

ds

(n − s)√

a2 − s2√

c2 − s2,

and from the asymptotic conditions a couple of equations

(43)0= λ4A

(a2 + c2 + 3µ − 2

6

)− 2

aK(c/a),

(44)1= λ4A

(a2c2

2+ 3

4

(a4 + c4) + 3µ − 2

12

(a2 + c2))2a

(K(c/a) − E(c/a)

),

whereK andE are the standard complete elliptic integrals. For a fixedλ4A andµ (β4, β2) wecan computea andc from the above equation. We can also fix other of these four variables

In order to study the phase transition we look at thec → 0 limit. Keepingµ fixed we expanda andλ4A = β4 in series ofc. The zeroth-order equation matches the values of the dilute pand the first correction is of orderc2. Plugging these solutions intoρ and computing the freenergy in the saddle approximation leads to a third-order phase transition. One can alsspecial case of this computation into[16]

7. Jurkiewicz–Zalewski structure of the strongly interacting phase

We present in this section the transition between the different phases with trays (stinteracting phase).

From the above section we know the form of the one-tray phase. We compute the tw(and no-gap) phase. We are looking for the state densityρ to be an even function ofn with aparameterization given by

(45)ρ(n) = 0, |n| > a,

(46)ρ(n) = ρ(n), c |n| a,

(47)ρ(n) = 1, c > |n| > d,

(48)ρ(n) = ρ(n), |n| d.

Using resolvent method we obtain the state density for the two-tray (no-gap) phase wgiven by

(49)ρ(n) = 2

π

c∫ds

√a2 − n2

√c2 − n2

√n2 − d2

(n − s)√

a2 − s2√

c2 − s2√

d2 − s2,

−c


alnothert ton-trivial

i-

er

is casectivetween

nto the

ic plusunder-

. The

tions intitione of

ion and

we have also four asymptotic conditions. Only two of them are non-trivial and are

(50)0= λ4A

(a2 + c2 + 3µ − 2

6

)− 2

c∫d

ds s√a2 − s2

√c2 − s2

√d2 − s2

,

1= λ4A

(a2c2 + a2d2 + c2d2

2+ 3

4

(a4 + c4 + d4) + 3µ − 2

12

(a2 + c2 + d2))

(51)− 2

c∫d

ds s3

√a2 − s2

√c2 − s2

√d2 − s2

.

We have two equations for the five variablesa, c, d,λ4A,µ (which reduce to Eqs.(43) and (44)in the limit d → 0). So we can expressa andc as a function of theβ ’s and ofd . For fixedβ ’swe obtain a family of solutions parameterized byd . All these solutions of the singular integrequation(23)coexist and, in order to select the one to be used in the saddle-point we need aequation. This can be done by performing the variation of the effective action with respecd .We can thus obtain the solutiondm as a function ofβ ’s. Note that without the symmetry conditioon ρ the system is under-constrained. Note also that solving the system is a highly nontask.

Finally, we have to take into account the fact that, by definition,b 0 and keepb =max(0, dm). So we are in the same situation we will encounter in Section9 and the phase transtion between the one-tray and the two-tray phases has to be of second order.

This is consistent with the JZ classification if we focus onρ. The one-tray–two-tray phastransition corresponds to the opening of a new interval into the support ofρ and is thus of ordesmaller than 3.

The same can be done for the gaped two-tray phase, looking at a four cuts solution. In thwe have three non-trivial asymptotic conditions which together with the minimum of the effeaction are enough to fix the position of the trays and gap. According to JZ, the transition bethe two-tray and gapped two-tray phases, which is given by the opening of a new gap isupport ofρ, has to be of order 3.

The fact that we can solve the position of trays and gaps is a peculiarity of the quartquadratic model under the symmetry condition. For other models multi-cuts solutions aredetermined and extra constraints have to be imposed[17].

8. Cut-off transition for GYM2

Let us start by recalling the YM2 cut-off transition. We look at the fundamental statetrivial representation corresponds to the Young diagram with no box. In terms ofni, this gives

(52)R0: ni =

N − 1

2,N − 3

2, . . . ,−N − 1

2

.

This representation minimizes the Casimir, and thus the energy. As explained in[9] the cut-offtransition takes place when this representation starts to dominate the other representathe partition function(8). This can be estimated by computing the ratio between the parfunction contribution fromR0 and the one from the “first exited” representation. In the casSU(N), above the trivial representation, we have the fundamental representationR1 (only onebox). We first compute the difference between the Casimir evaluate on this representat


nnotis not

adratic

repre-r

tal

theo setswith ameterize

he

the trivial one:

(53)∆2 = C2(R0) − C2(R1) = −N.

The ratio between theR0 andR1 contributions is

(54)ZS(R0)

ZS(R1)= 1

N2exp

−λ2A

2N∆2

= 1

N2eλ2A/2,

and the trivial representation starts to dominateR1 for the value ofλ2A given by

(55)λ2A = 4 ln(N).

So the system is in a degenerate phase when its rescaled areaλ2A is larger than 4 ln(N).If we want to track the cut-off transition for the GYM2, we have to be careful: we ca

directly extend the computation of the YM2 case, since the state with smallest energynecessarily the trivial representation. We make the computation for our quartic plus qumodel.

8.1. Fundamental-state candidate

In order to find the fundamental state, we have to minimize the Hamiltonian over thesentations. In fact it is sufficient to find the representationni that gives the smallest value fothe function

(56)E =(

µ

2NC2

(ni) + 1

N3C4

(ni))

.

As already explain in Section5 symmetric table (in the senseni = −n(N−i+1)) are local mini-mum of the effective action and also of the functionE and, we can guess that the fundamenstate is symmetric. For symmetric representations, theC4 has a term proportional to

∑n4

i minusa term in

∑n2

i and theC2 will modulate this term. Using the fermionic analogy we can seesystem as fermions in a Mexican hat potential. For low Fermi energy, we expect to find twof fermions, one around each minimum of the potential. Thus we expect a configurationgap. Such a gaped configuration corresponds to a Young diagram with a step and we parait as

(57)ni = N + 1

2− i + q

2for i N

2,

(58)ni = N + 1

2− i − q

2for i >

N

2.

Passing toQ = q/N , theQ-dependent part of the functionE takes the value

(59)E(RQ) = N2Q

8

(Q3

2+ Q2 + 3µ + 1

3Q + 6µ − 1

6

)+O(N).

This function has a minimum, solution of∂E(RQ)

∂Q= 0, which is given by

(60)Qm = −1

2+

√5

12− µ.

By definition,Q 0 and therefore forµ 16 we haveQm = 0, i.e. the fundamental state is t

trivial representation. Below1 the fundamental state is a step given by the above equation.
6


ng dia-by one

ct, theases.

e of the

tes. Theestate

Fig. 3. The four changes for which we check the stability ofRQm .

8.2. States near the fundamental-state candidate

We check that the step representation is the lowest-energy state. Looking at the Yougram, seeFig. 3, we see that there are four ways to change a generic step representationbox (cases (A)–(D)).

There is a symmetry between the cases (A) and (D) (respectively, (B) and (C)). In favariation of theE function and the dimension computation give the same result for the two cWe obtain

• µ 1/6. For this range ofµ, we haveQm = 0, only cases (A) and (D) apply. We have

(61)E(A)µ1/6 = E

(D)µ1/6 = µ

2+ 1

6.

• µ < 1/6. For this rangeQm is given by Eq.(60)and the four cases are possible:

(62)E(A)µ<1/6 = E

(D)µ<1/6 = 5

24+ 1

24

√15− 36µ − µ

2,

(63)E(B)µ<1/6 = E

(C)µ<1/6 = 5

24− 1

24

√15− 36µ − µ

2,

and we haveE(A)µ<1/6 E

(B)µ<1/6.

All these quantities are positive: the step state is the fundamental state.We now compute the ratio between the dimension of the fundamental state and the on

near-by cases. We obtain

(64)d(A) = d(D) = 1

N

2Q + 1

Q + 1,

(65)d(B) = d(C) = 1

N

2Q + 1

Q.

The cut-off transition takes place when the fundamental state dominates all the other stalast state to be dominated by the fundamental stateRQm is of type (A) or (B), depending on thvalue ofµ. The ratio between the contribution to the partition function of the fundamentaland that of the first exited state is of the form

(66)ZS(RQm)

ZS(Case( ) aroundRQm))= (d)2eλ4AE.

TheRQm representation starts to dominate when the rescaled area is

(67)λ4A = 2ln(N) + const,

E


se

. Using

e

ith

where the constant is two times the logarithm of the factor after 1/N in (64), (65)and can beneglected in the large-N limit. Using Eq.(67), the cut-off transition takes place:

• for µ 1/6 at

(68)λ4A∣∣µ1/6 = 4

µ + 1/3ln(N),

• for µ < 1/6, using the less energetic cases (B), (C), at

(69)λ4A∣∣µ<1/6 = 4

−µ + 512 + 1

12

√15− 36µ

ln(N).

Settingβ4 = λ4A andµ = β2/β4 and solving forβ2 the two above equations give the phaboundary(13), (14).

9. Phase transition between the two degenerate phases

In the degenerate phase, the partition function is dominated by the fundamental statethe saddle approximation, we will keep only this term in the partition function. We have

(70)Z ∼ exp

(2 ln

(d(RQm)

) −(

β2

2NC2(RQm) + β4

N3C4(RQm)

)),

and the free energy is the exponent divided byN2. For highµ = β2/β4 we haveQm = 0 and thefundamental state is the trivial representation, which has

(71)C2(R0) = C4(R0) = 0 and d(R0) = 1.

In this phase the free energy is identically null. For smallerµ, Qm is a function of the ratioµand goes to 0 forµ = 1/6. We can compute

(72)C2 = 1

4(Q + 1)QN3,

(73)C4 = 1

16

(Q3 + 2Q2 + 2Q + 1

)QN5 +O

(N3).

As limµ→1/6 Q = 0, we have limµ→1/6 d(RQ) = 1; from this and the above Casimir value wsee that the free energy is a continuous function.

Using Eq.(60) we have∂Qm

∂µ|µ=1/6 = −1, so that we can perform the differentiation w

respect toQ. For the Casimir we get

(74)∂QC2 = 1

4(2Q + 1)N3,

(75)∂QC4 = 1

16

(4Q3 + 6Q2 + 4Q + 1

)N5 +O

(N3)

and for the dimension using the parameterization(57) and (58)we have


nerate,l fea-ir of

low,

an

upport isal in

ses. As

rding tois

of thehe

rated by

∂Q ln(d(RQ)

)∣∣Q=0 = ∂Q

(N/2∑i=1

N∑j=N/2+1

ln(j − i + 2Q)

)∣∣∣∣∣Q=0

(76)=N/2∑i=1

N/2∑j=1

2

N/2+ j − i=

N/2−1∑=−N/2+1

2(N/2− ||)N/2+

.

Making use of the digamma functionΨ and of its asymptotic behaviour

limx→∞

(ψ(x) − ln(x)

) = 0

we obtain, for largeN ,

(77)∂Q ln(d(RQ)

)∣∣Q=0 ∼ N

(1

2− ln(2)

).

Collecting all the term we get for the derivative of the free energy

(78)∂µF∣∣µ=1/6 =

(β2

8+ β4

16

)+O(1/N),

and the derivative of the free energy is not continuous across theβ4 = 6β2 transition line. Thesystem undergoes a second-order phase transition along this line.

10. Concluding remarks

The fermionic analogy leads to the conclusion that the three kinds of phase (degestrongly interacting, and dilute) and the DK and cut-off transitions are completely generatures that we will find in any GYM2 model. Keeping in mind the fact that the higher Casimany model has to be of the formC2m in order to guarantee a Hamiltonian bounded from bethe state density will have as a support the union ofk intervals withk ∈ [1,m − 1]. The numberof intervals will be a function ofβj = λjA. Using the fermionic analogy with fermions inpotential given by a polynomial of degree 2m, we find intuitively that each local maximum carise above the Fermi level. There exists a region of phase space where the state-density sonly one interval. Moving away from this region gaps will open, splitting the unique intervdifferent pieces. In the dilute phase this structure is equivalent to the one described in[13] fromwhich we can deduce the order of the phase transitions between the different dilute phaan illustration we can consider the case of the 6th-order model developed in[17]. in this modelwe have a fixed quadratic term plus a quartic term (with its couplingg1) and a 6th-order term(with its couplingg2). We have then 4 kind of potential for our fermions as shown inFig. 4.Looking at the state-density support we deduce the order of the phases transitions acco[13]. The precise location of the phase boundary can be found in[17] and the phase structurein agreement with the one we have sketched.

This picture is also valid in the case in the strongly interacting phase, but each intervaldensity of state may show a tray. Again, argument of[13] can be used to obtain the order of tphase transitions between the five different tray-phases.

In the degenerate phase the state-density is a collection of pieces with value 1 sepagaps. The state-density support structure depends only on the ratio of the differentβ ’s and tran-sitions between this different sub-phases have to be of second order.


otations

fnt

ensimir

etive

Fig. 4. Cartoon of the dilute phase structure of the sixth order model with quadratic term fixed. We have used nof [17]. Note that the lower part of theP2 phase (withg2 < 0) is unstable.

10.1. Remark about U(N)

We have worked withSU(N) groups and want to ask now: what aboutU(N) ones? The maindifference between theU(N) and SU(N) representations is the fact thatSU(N) are invariantunder translation, i.e., the representationhi is equivalent tohi + with an integer.

U(N) can be split intoSU(N) × U(1) in the language of theni’s; the U(1) part is the“center of mass” positiony = 1

N

∑ni . TheSU(N) is given by themi = ni − y, then the sum o

themi is null. For the YM2 case, the quadratic Casimir forU(N) is given (up to some constaterm) by

C2(U(N)

) =∑

(mi + y)2 =∑

m2i + 2y

∑mi + Ny2

(79)=∑

m2i + Ny2 = C2

(SU(N)

) + Ny2.

They is decoupled from theSU(N) part. Performing the sum overy into the partition functiongives an overall normalization and all the considerations on theSU(N) case apply to theU(N)

one. For GYM2, things are different. Higher CasimirsCk (k > 2), couple theSU(N) andU(1)

parts. The principal term ofCk for U(N) is

(80)∑

nki =

∑(mi + y)k =

∑mk

i +k∑

j=1

(j

k

)yj

∑i

m(k−j)i ,

that is the term belonging to theSU(N) Casimir of orderk andy dependent terms. For a chosGYM2 model (with higher Casimir of even order) and with ratio between the higher Cacoupling and the other onesµj . We collect all they terms and obtain a polynomial of degreek

in y. Unlike k = 2 case, polynomial coefficients are given by sum ofSU(N) Casimirs of ordersmaller thank and depend onµ’s. Plugging this expression into theU(N) partition functionand performing the sum overy (which is possible untilk is even) we obtain a function of thC

SU(N), j < k which multiply theSU(N) part. This term can be re-absorbed into an effec
j


t meRNme

al fund

action which now depend on all theSU(N) Casimirs of degreek. However, theU(N) case canbe analyzed usingSU(N) one with Hamiltonian containing product and power of Casimirs.

Acknowledgements

I want to thank L. Alvarez-Gaumé for his patience and his availability, M. Blau for leknow Ref. [16]. M. Mariño, M. Weiss and M. Ruiz-Altaba for useful discussions, the CEth-unit for kindly hosting me during the redaction of this work. Alice and little Fabio for letsleep enough to be able to work. . . This work was partially supported by the Swiss nation(FNS).

References

[1] S. Cordes, G.W. Moore, S. Ramgoolam, Nucl. Phys. B (Proc. Suppl.) 41 (1995) 148, hep-th/9411210;S. Cordes, G.W. Moore, S. Ramgoolam, Commun. Math. Phys. 185 (1997) 543, hep-th/9402107.

[2] D.J. Gross, Nucl. Phys. B 400 (1993) 161, hep-th/9212149;D.J. Gross, W. Taylor, Nucl. Phys. B 400 (1993) 181, hep-th/9301068.

[3] J.A. Minahan, A.P. Polychronakos, Phys. Lett. B 312 (1993) 155, hep-th/9303153;J.A. Minahan, Phys. Rev. D 47 (1993), hep-th/9301003.

[4] O. Ganor, J. Sonnenshein, S. Yankielowicz, Nucl. Phys. B 434 (1995) 139, hep-th/9407114.[5] F. Dubath, S. Lelli, A. Rissone, Int. J. Mod. Phys. A 19 (2004) 205, hep-th/0211133.[6] M.R. Douglas, V.A. Kazakov, Phys. Lett. B 319 (1993) 219, hep-th/9305047.[7] A. D’Adda, P. Provero, Commun. Math. Phys. 245 (2004) 1, hep-th/0110243.[8] S. Haro, M. Tierz, Phys. Lett. B 601 (2004) 201, hep-th/0406093.[9] A. Apolloni, S. Arianos, A. D’Adda, hep-th/0506207.

[10] A.A. Migdal, Sov. Phys. JETP 42 (1975) 413.[11] M. Mariño, Les Houches Lectures on Matrix Models and Topological Strings, hep-th/0410165.[12] N.I. Muskhelishvili, Singular Integral Equations, Noordhoff, Groningen, 1953.[13] J. Jurkiewicz, K. Zalewski, Nucl. Phys. B 220 (1983) 167.[14] Y. Shimamune, Phys. Lett. B 108 (1982) 407.[15] G.M. Cicuta, L. Molinari, E. Montaldi, Mod. Phys. Lett. A 1 (1982) 125.[16] M. Alimohammadi, A. Tofighi, Eur. Phys. J. C 8 (1999) 711, hep-th/9807004;

M. Alimohammadi, M. Khorrami, A. Aghamohammadi, Nucl. Phys. B 510 (1998) 313, hep-th/9707081.[17] J. Jurkiewicz, Phys. Lett. B 245 (1990) 178.

r aboveansitionsd diver-of zeros

phaseiscon-ournsitionsromag-articles ap-scenarioshich


Properties of higher-order phase transitions

W. Jankea, D.A. Johnstonb, R. Kennac,∗

a Institut für Theoretische Physik, Universität Leipzig, Augustusplatz 10/11, 04109 Leipzig, Germanyb Department of Mathematics, School of Mathematical and Computer Sciences, Heriot-Watt University, Riccarton,

Edinburgh EH14 4AS, Scotland, UKc Applied Mathematics Research Centre, Coventry University, Coventry, CV1 5FB, England, UK

Received 15 October 2005; received in revised form 7 December 2005; accepted 14 December 2005


Abstract

Experimental evidence for the existence of strictly higher-order phase transitions (of order three oin the Ehrenfest sense) is tenuous at best. However, there is no known physical reason why such trshould not exist in nature. Here, higher-order transitions characterized by both discontinuities angences are analysed through the medium of partition function zeros. Properties of the distributionsare derived, certain scaling relations are recovered, and new ones are presented. 2005 Elsevier B.V. All rights reserved.

1. Introduction

In its original format, the Ehrenfest classification scheme identifies the order of atransition as that of the lowest derivative of the Helmholtz free energy which displays a dtinuity there[1]. Typical transitions which fit to this scheme are first-order solid–liquid–vaptransitions and second-order superconducting transitions. There are, however, many tracharacterised by divergent rather than discontinuous behaviour. Examples include fernetic transitions in metals and the spontaneous symmetry breaking of the Higgs field in pphysics, which display power-law or logarithmic divergent behavior as the transition iproached. The classification scheme has, in practice, been extended to encompass theseand the order of a transition is commonly given by the order of the lowest derivative in wany type of non-analytic behaviour is manifest.

* Corresponding author.E-mail address: [email protected](R. Kenna).




320 W. Janke et al. / Nuclear Physics B 736 [FS] (2006) 319–328

discon-easonpertiesus tran-

at the

priate-ordersatel

died

nction-

isthe

curved,

y-

given

licit

firstnuous,t theeld)

nmers

r

eniteoulde

It has long been suspected that transitions of Ehrenfest order greater than two (with atinuity at the transition point) do not exist in nature. However there is no obvious physical rwhy this should be the case. In fact, recent experimental observations of the magnetic proof a cubic superconductor have been ascribed to its possessing a fourth-order discontinuosition [2] (see also[3] where the existence of well-defined anomalies in the specific heattransition point was claimed). A theory for higher-order transitions was developed in[4–6] andfound to be consistent with experimental work.

Higher-order phase transitions (with either a discontinuity or a divergence in an approfree-energy derivative) certainly exist in a number of theoretical models. There are thirdtemperature-driven transitions in various ferromagnetic and antiferromagnetic spin model[7,8],as well as spin models coupled to quantum gravity[9,10]. Recent theoretical studies also indicthe presence of third-order transitions in various superconductors[11], DNA under mechanicastrain[12], spin glasses[13], lattice and continuum gauge theories[14] and matrix models linkedto supersymmetry[15]. A fourth-order transition in a model of a branched polymer was stuin [16] and the Berezinskii–Kosterlitz–Thouless transition is of infinite-order[17].

In this paper, we analyse higher-order transitions through the medium of partition fuzeros. To set the notation, lett represent a generic reduced even variable andh be the odd equivalent so thatt = T/Tc − 1 andh = H/kBT in the notation of the Ising model (i.e.,T is thetemperature, which is critical atTc, andH is the external magnetic field). The critical pointgiven by(t, h) = (0,0). This may be the end-point of a line of first-order transitions, as iscase in the Ising or Potts models. In the Potts-like case where the locus of transitions iswe may instead assume thatt andh are suitable mixed variables, so thath is orthogonal tot ,which parameterizes arc length along the transition line[18]. The free energy in the thermodnamic limit is denoted byf (t, h) and itsnth-order even and odd derivatives aref

(n)t (t, h) and

f(n)h (t, h) so that the internal energy, specific heat, magnetization and susceptibility are

(up to some inert factors) ase(t, h) = f(1)t (t, h), C(t, h) = f

(2)t (t, h), m(t,h) = f

(1)h (t, h), and

χ(t, h) = f(2)h (t, h), respectively. In the following, to simplify the notation, we drop the exp

functional dependency on a variable if it vanishes.One then commonly describes as anmth-order phase transition a situation where the

(m − 1) derivatives of the free energy with respect to the even (thermal) variable are contibut where themth thermal derivative is singular, with a discontinuity or a divergence atransition point. The lowest(m′ − 1) derivatives of the free energy with respect to the odd (fivariable may also be continuous int , with a singularity occurring in them′th derivative. Thus acontinuous specific heat is realized ifm > 2 and the susceptibility is also continuous ifm′ > 2as well. This situation, which is not normally possible in a ferromagnet,1 is the one analysed i[4,5], in whichm = m′ > 2. Such higher-order transitions may be possible in branched polyand diamagnets such as superconductors.

In the more common scenario wherem′ is not necessarily the same asm, the scaling behaviouat the transition may be described by critical exponents ath = 0:

(1.1)f(m)t (t) ∼ t−A, f

(m′)h (t) ∼ t−G, f

(1)h (t) ∼ tβ ,

1 One can readily see this by considering the Rushbrooke scaling law(1.5) together with hyperscaling which givγ /ν = d − 2β/ν (d being dimensionality andν the correlation-length critical exponent). Since for a system of filinear extentL, the magnetisation obeys〈|m|〉 ∝ L−β/ν , and since completely uncorrelated ferromagnetic spins wlead by the central limit theorem to〈|m|〉 ∝ L−d/2, we obtain the boundβ/ν < d/2, since the actual decay in thcorrelated case is slower. From this, one obtains the restriction thatγ /ν cannot be negative for a ferromagnet.

W. Janke et al. / Nuclear Physics B 736 [FS] (2006) 319–328 321

f a

Grif-

sr zeroslocus

ratureng–Leeur

f zerosectionrestric-ods forety

ions are

ter,

while, for the magnetization in field att = 0, we write

(1.2)f(1)h (h) ∼ h1/δ.

In the familiar case of a second-order transition (m = m′ = 2), the exponentsA andG become,in standard notation,α andγ , associated with specific heat and susceptibility, respectively.

In the theoretical work of[5], the following scaling relations were derived for the case odiverging higher-order transition in whichm′ = m:

(1.3)(m − 1)A + mβ + G = m(m − 1),

(1.4)G = β((m − 1)δ − 1

).

In the second-order case(1.3) and (1.4)become equivalent to the standard Rushbrooke andfiths scaling laws,

(1.5)α + 2β + γ = 2, γ = β(δ − 1),

as one would expect.Since the seminal work by Lee and Yang[19] as well as by Fisher[20], the analysis of zero

of the partition function has become fundamental to the study of phase transitions. Fishein the complex temperature plane pinch the real axis at the physical transition point. Theof Lee–Yang zeros, in the complex magnetic-field plane, is controlled by the (real) tempeparameter and in the high-temperature phase, where there is no transition, it ends at the Yaedge. Denoting the distance of the edge from the real axis byrYL , one has the generic behavio

(1.6)rYL (t) ∼ t∆/2,

at a second-order transition. The exponent∆ is related to the other exponents through∆ =2γ δ/(δ − 1).

In the remainder of this paper, a number of results concerning the locus and density oare presented. Higher-order transitions controlled by a single parameter are analysed in S2where the locii and densities of the corresponding Fisher zeros are determined. Varioustions on the properties of such transitions are established and simple quantitative methanalysing them are suggested. In Section3, the focus is on the zeros of the Lee–Yang variwhere even and odd control parameters come into play. Here, the scaling relations(1.3) and (1.4)are recovered and elucidated and a number of other ones are presented. Finally, conclusdrawn in Section4.

2. Fisher zeros

In the bulk of physical models the locus of Fisher zeros is linear in a suitable parameu,which is a function oft and can be parameterized near the transition point,uc, by [21–24]

(2.1)u(r) = uc + r exp(iφ(r)

).

This singular line in the upper half-plane has 0< φ(r) < π , while that in the lower half is itscomplex conjugate.

In the thermodynamic limit, the (reduced) free energy is

(2.2)f (t) = 2 Re

R∫ln

(u − u(r)

)g(r) dr,

0


by

ns

s

transi-

nuous

r 3in

ystemred to

os is

xcept

hilederr

whereg(r) is the density of zeros andR is a cutoff. We are interested in the moments given

(2.3)f(n)t (t) = 2(−1)n−1(n − 1)!Re

R∫0

g(r)

(u − u(r))ndr,

and consider the cases of discontinuous and divergentmth-order temperature-driven transitioseparately.

The difference in free energies on either side of the transition can be expanded asf+(t) −f−(t) = ∑∞

n=1 cn(u − uc)n, where+ and− refer to above and belowuc. For a discontinuou

transition,cn = 0 for n < m, while cm = 0 and the discontinuity in themth derivative of the freeenergy is

(2.4)f(m)t = m!cm.

Now, the real parts of the free energies must match across the singular line (otherwise thetion would be of order zero) which, from(2.1), means

∑∞n=m cnr

n cosnφ(r) = 0. Therefore theimpact angle (in the upper half-plane),φ = limr→0 φ(r), is

(2.5)φ = (2l + 1)π

2mfor l = 0, . . . ,m − 1.

It is now clear that, under these conditions, vertical impact is allowed only at any discontitransitions of odd order. A discontinuous second-order transition with impact angleπ/2 is for-bidden. Similarly an impact angle ofπ/6, for example, is only allowed at a transition of ordeor 9 or 15, etc. This recovers disparate results for first-, second- and third-order transitions[19,25] and[10] which are associated with impact anglesπ/2 (corresponding tol = 0), π/4 (l = 0)andπ/2 (l = 1), respectively. The question now arises as to the mechanism by which the sselects itsl-value. One expects that further studies of higher-order transitions will be requiprovide an answer.

Let t = u − uc, τ = te−iφ and assume that the leading behaviour of the density of zerg(r) = g0r

p , whereg0 is constant. Ifp is an integer, analytical extension of the integration(2.3)to the complex plane yields the following result for thenth derivative:

(2.6)f(n)t (t) = −2(n − 1)!g0 Re

p+1∑j=1

e−inφTj

∣∣∣∣∣R

δ

,

whereδ is a lower integral cutoff and

(2.7)Tj = p!τp+1−j (r − τ)j−n

(j − 1)!(p + 1− j)!(j − n)for j = n,

(2.8)Tn = p!τp+1−n ln (r − τ)

(n − 1)!(p + 1− n)! .One finds that allTj terms vanish as the transition is approached from above or below, e

the term for whichj = p + 1. If n p this term is constant and there is no discontinuity inf(n)t

across the transition, while forn = p + 1 it leads to a discontinuous transition withf(p+1)t =

2πg0p!sin(p + 1)φ. Therefore the firstp derivatives are continuous across the transition wthe(p + 1)th derivative is not. In other words, to generate a discontinuous transition of orm

under these assumptions, it is necessary and sufficient thatp = m− 1, i.e., the leading behaviou


ensity

as op-

the

t with

f

es on

of the density is

(2.9)g(r) = g0rm−1.

From(2.4) and (2.5), one now hascm = (−1)l2πg0/m, and the discontinuity in themth deriva-tive of the free energy is related to the density of zeros as

(2.10)f(m)t = (−1)l(m − 1)!2πg0.

This recovers the well known result that the latent heat or magnetization is related to the dof zeros at a first-order transition throughf

(1)t = 2πg0 [19].

We next consider anmth-order diverging transition where

(2.11)f(m)t (t) ∼ |t |−A,

for 0< A < 1. If A = 0, we are back to the discontinuous case or the case of a logarithmicposed to power-law divergence (see the discussion below), while ifA > 1, it is more appropriateto consider the transition as(m − 1)th order.

Considerations similar to those in[23,24] may be used to show that in order to obtainappropriate divergence it is necessary and sufficient that

(2.12)g(r) = g0rm−1−A.

Indeed, from the general expression(2.3), the form(2.11)is obtained provided (withr = tr ′)

(2.13)Re

R∫0

tAg(r)

(reiφ − t)mdr = Re

R/t∫0

tA−m+1g(tr ′)(r ′eiφ − 1)m

dr ′

is independent oft as t → 0. The further condition thatg(0) = 0 givesA < m − 1. If m = 1,this violates the condition that 0< A < 1, leading to the requirement thatm 2 for a divergingtransition. On this basis, there are no diverging first-order transitions. This is consistenexperience.

To demonstrate sufficiency, we put(2.12)into (2.3)and use the substitutionw = r exp(iφ)/|t |,to find, for thenth derivative of the free energy,

(2.14)f(n)t (t) = g0(n − 1)!|t |m−n−Ae−i(m−A)φI±,

in which

(2.15)I± = 2 Re

Reiφ/|t |∫0

wm−1−A

(1± w)ndw for t ≶ 0.

If n < m, this vanishes ast → 0, establishing the continuity of thenth derivative there, while in = m, one finds

(2.16)f(m)t (t) = −2g0|t |−AΓ (m − A)Γ (A) ×

cos(m − A)φ if t < 0,

cos((m − A)φ + Aπ) if t > 0.

In the case of a second-order transition,(2.16) recovers a result derived in[23,26]. Note that(2.16) provides a direct relationship between the impact angle and the critical amplitudeither side of the transition. These critical amplitudes coincide if the impact angle isφ = (2N −


econd-to

e

g

n-

e

us

nd

A)π/2(m−A) whereN is any integer. In particular, ifm is even an impact angle ofπ/2 results inthe symmetry of amplitudes around the transition. This result was already observed in the sorder case in[23]. The implications of(2.16)are that, while this symmetry may be extendedall even-order diverging phase transitions, it does not hold for odd ones.

If A = 0 in (2.12), the singular part of themth derivative of the free energy becomes

(2.17)f(m)t (t) = 2(m − 1)!g0 ×

cos(mφ) ln |t | if t < 0,

(cos(mφ) ln |t | + π sin(mφ)) if t > 0.

This recovers a result in[23] if m = 2. Moreover, the discontinuity in themth moment across thtransition is consistent with(2.5) and (2.10).

From(2.9) and (2.12), the integrated density of Fisher zeros isG(r) ∼ rm−A (whereA = 0in the case of a discontinuous transition). For a finite system of linear extentL, the integrateddensity is defined asGL(tj ) = (2j − 1)/2Ld [27]. EquatingG(tj ) to GL(tj ) leads to the scalinbehaviour

(2.18)|tj | ∼ L− dm−A .

In the diverging case where hyperscaling (f (t) ∼ ξ(t)d ) holds, andm − A = 2 − α = νd , thisrecovers the usual expression,|tj | ∼ L−1/ν , for finite-size scaling of Fisher zeros. In the discotinuous case, whereA = 0, (2.18)yields

(2.19)ν = m

d.

This is a generalization of the usual formal identification ofν with 1/d , which applies to afirst-order transition. Such a generalized identification was observed at the third-order (m = 3)discontinuous transition present in the spherical model in three dimensions[7] as well as in theIsing model on planar random graphs if the Hausdorff dimension is used ford [10].

3. Lee–Yang zeros

In the Lee–Yang case, where there is an edge,rYL (t), to the distribution of zeros, the freenergy is

(3.1)f (t, h) = 2 Re

R∫rYL (t)

ln(h − h(r, t)

)g(r, t) dr,

where the density of zeros is written asg(r, t) to display itst-dependency and where the locof zeros ish(r, t) = r exp(iφ(r, t)). (If the Lee–Yang circle theorem holds,φ = π/2 andR = π

[19].) Them′th field derivative of the free energy ath = 0 is

(3.2)f(m′)h (t) = 2(−1)m

′−1(m′ − 1)! cos(m′φ)

rYL (t)m′−1

R/rYL∫1

g(xrYL , t)

xm′ dx,

having used the substitutionr = xrYL (t). As in the second-order case, we assume thatrYL (t) issufficiently small near the transition point (t = 0) so that the upper integral limit diverges acompare with the limiting scaling behaviour in(1.1) to find [21,22]

(3.3)g(r, t) = t−GrYL (t)m′−1Φ

(r

),

rYL (t)


neti-

n-ese

weing the

ld

whereΦ is an unknown function of its argument. Similar considerations yield, for the magzation,

(3.4)f(1)h (t, h) = 2t−GrYL (t)m

′−1 Re

∞∫1

Φ(x)

hrYL (t)

− xeiφdx,

which we may write as

(3.5)f(1)h (t, h) = t−GrYL (t)m

′−1Ψφ

(h

rYL (t)

).

Comparison with(1.2) now givesΨ (h/rYL (t)) ∼ (h/rYL (t))1/δ . The t-dependence must ca

cel in (3.5) as t → 0, giving the small-t scaling behaviour of the Yang–Lee edge under thcircumstances to be

(3.6)rYL (t) ∼ tGδ

(m′−1)δ−1 .

Whenm′ = 2 andG = γ , this recovers the second-order transition behaviour of(1.6). Further-more,(3.3)now reads

(3.7)g(r, t) = tG

(m′−1)δ−1 Φ

(r

rYL (t)

),

and the expression for the magnetization in (3.5) gives

(3.8)f(1)h (t, h) = t

G(m′−1)δ−1 Ψφ

(h

rYL (t)

).

Strictly, this equation of state has been derived fort > 0, where there is an edge. Howevermay assume it can be analytically continued into the low temperature regime, where, takh → 0 limit and comparing with the magnetization in(1.1), it yields the scaling relation

(3.9)β = G

(m′ − 1)δ − 1.

In the situation wherem′ = m, this recovers the Griffiths-type scaling relation(1.4), derivedin [5].

Integrating(3.1)by parts gives, for the singular part of the free energy,

(3.10)f (t, h) = 2 Re

R∫rYL (t)

G(r, t)

he−iφ − rdr,

whereG(r, t) is the integrated density of zeros. From(3.3) and (3.6), the latter isG(r, t) =tG(δ+1)/((m′−1)δ−1)F (r/rYL (t)) in which F(x) = ∫ x

1 Φ(x′) dx′. Again using r = xrYL (t)

in (3.10), and taking the upper integral limit to infinity, one has, for the free energy,

(3.11)f (t, h) = tG δ+1

(m′−1)δ−1Fφ

(h

rYL (t)

),

whereFφ(w) = 2 Re∫ ∞

1 F(x)/(we−iφ −x)dx. Themth temperature derivative of the zero-fie

free energy is therefore of the formf (m)t (t) ∼ tG(δ+1)/((m′−1)δ−1)−m. Comparison with(1.1)then


and

ap-he

aw

cificto

sition

of

nts

yields the scaling relation

(3.12)A = m − Gδ + 1

(m′ − 1)δ − 1.

Together,(3.9) and (3.12)recover all four scaling relations derived in[5] in the more restrictivecase wherem′ = m. In the second-order case (m = 2), they recover the standard RushbrookeGriffiths scaling laws of(1.5).

In fact, these laws also hold in the present case, albeit with negativeα (and possiblyγ ). Tosee this, letf (n)

t (t) ∼ t−αn andf(n)h (t) ∼ t−γn (so thatα2 = α andγ2 = γ ). Sincef (m)

t (t) ∼ t−A,one has, directly, thatn − αn = m − A. Differentiating(3.11)with respect to field, now gives

(3.13)f(n)h (t) ∼ tβ−(n−1)βδ = tnβ−(n−1)(m−A),

having used(3.9) and (3.12)and seth = 0. Now, one has

(3.14)γn = (n − 1)βδ − β, (n − 1)αn + nβ + γn = n(n − 1),

which recover(1.5)whenn = 2.2

The formulae(1.1) describe the behaviour of various moments as the critical point isproached tangential to the transition line (i.e., alongh = 0). One may also be interested in torthogonal behaviour, namely, theh-dependence att = 0. In the case of theh-derivatives offree energy, this comes directly from(1.2). For thet-derivatives, we may assume the power-lbehaviour (att = 0),

(3.15)f(j)t (h) ∼ hsj .

In the second-order case,(3.15)gives theh-dependency of the internal energy and the speheat att = 0 ase(h) = f

(1)t (h) ∼ hε andC(h) = f

(2)t (h) ∼ h−σ . These exponents are related

δ andγ through (see[21] and references therein)

(3.16)ε = 2− (δ − 1)(γ + 1)

δγ, σ = (δ − 1)(γ + 2)

δγ− 2.

Following the reasoning of[21], we may argue that because there should be no phase tranaway fromh = 0 for anyt , the free energy,f (t, h) in (3.11)must be a power series int there. Soif Fφ(w) involves a term,wq , the free energy involvest−G+(m′−q)Gδ/((m′−1)δ−1)hq which mustbe an integral power,N , of t . This givesq = m′ − [(m′ − 1)δ − 1](G + N)/Gδ, or the powerseries

(3.17)f (t, h) =∞∑

N=0

antNhm′− (m′−1)δ−1

Gδ(G+N).

Differentiating appropriately, puttingt = 0 and comparing with(3.15)yields the scaling laws

(3.18)sj = m′ − (m′ − 1)δ − 1

Gδ(G + j).

In the second-order case withm′ = 2 this recovers(3.16)with s1 = ε ands2 = −σ .

2 It is interesting to note the restrictions imposed onδ at a higher-order transition coming from the first equation(3.14). For n < m′, γn should be negative, so, ifβ is positive, the best bound onδ is δ < 1/(m′ − 2). Also, the secondformula in(3.14)gives, for 2 n m−1 and henceαn < 0, δ > (m−1)/β −1 orβ > (m−1)(m′ −2)/(m′ −1). Theseare no restraints in the familiar second-order case (wherem = m′ = 2 and largeδ is common), but are severe constraiat higher order.


of theture ofe-es thethe fa-

s controlsociated

cluding

aveat-likeis nocerningit wouldrder of

4. Conclusions

Different types of higher-order phase transitions have been analysed using the zerospartition function. In the Fisher case, the impact angle is restricted by the order and nathe transition. For a transition with a discontinuity inf

(m)t (t), it is unclear how the system s

lects from them permissible angles. For a divergent transition, the impact angle determinrelevant amplitude ratios. Finite-size scaling is seen to hold at higher-order transitions andmiliar formal identification ofν with 1/d that is used at first-order transitions extends toν = m/d

for discontinuous transitions ofmth order.Lee–Yang zeros, on the other hand, are appropriate to the case where two parameter

the system. Here, they have been used as a route to derive scaling relations between aseven and odd exponents, which recover well-known formulae in the second-order case, inthe Rushbrooke and Griffiths laws.

One of the main points of[2] is that many higher-order transitions may exist which hnot yet been identified as such. Indeed determination of critical exponents or latent-hediscontinuities is notoriously difficult from numerical work on finite systems where theretrue transition and signals are smoothed out. There, amplitude ratios are often more disand here we see impact angles even more so, at least in theory. From the results herein,appear that analysis of the impact angle provides a very robust way to recognise the otransitions.

Acknowledgements

This work was partially supported by the EU RTN-Network ‘ENRAGE’:Random Geometryand Random Matrices: From Quantum Gravity to Econophysics under grant No. MRTN-CT-2004-005616. R.K. thanks Pradeep Kumar for an e-mail correspondence.

References

[1] P. Ehrenfest, Commum. Kamerlingh Onnes Lab. Leiden 20 (Suppl. 75B) (1933) 8.[2] P. Kumar, D. Hall, R.G. Goodrich, Phys. Rev. Lett. 82 (1999) 4532.[3] B.F. Woodfield, D.A. Wright, R.A. Fisher, N.E. Philips, H.Y. Tang, Phys. Rev. Lett. 83 (1999) 4622.[4] P. Kumar, J. Low Temp. Phys. 106 (1997) 705.[5] P. Kumar, A. Saxena, Philos. Mag. B 82 (2002) 1201;

P. Kumar, Phys. Rev. B 68 (2003) 064505.[6] A.K. Farid, Y. Yu, A. Saxena, P. Kumar, Phys. Rev. B 71 (2005) 104509.[7] T. Berlin, M. Kac, Phys. Rev. 86 (1952) 821.[8] K. Kanaya, S. Kaya, Phys. Rev. D 51 (1995) 2404;

H.G. Ballesteros, L.A. Fernández, V. Martín-Mayor, A. Muñoz Sudupe, Phys. Lett. B 378 (1996) 207;H.G. Ballesteros, L.A. Fernández, V. Martín-Mayor, A. Muñoz Sudupe, Nucl. Phys. B 483 (1997) 707;A.P. Gottlob, M. Hasenbusch, J. Stat. Phys. 77 (1994) 919;M. Kolesik, M. Suzuki, Physica A 216 (1995) 469;A.N. Rubtsov, T. Janssen, cond-mat/0304447.

[9] V.A. Kazakov, Phys. Lett. A 119 (1986) 140;J. Ambjørn, G. Thorleifsson, M. Wexler, Nucl. Phys. B 439 (1995) 187.

[10] J. Ambjørn, K.N. Anagnostopoulos, U. Magnea, Mod. Phys. Lett. A 12 (1997) 1605;W. Janke, D.A. Johnston, M. Stathakopoulos, Nucl. Phys. B 614 (2001) 494.

[11] C. Cronström, M. Noga, Czech J. Phys. 51 (2001) 175;R. Werner, Phys. Rev. B 67 (2003) 014505.

[12] J. Rudnick, R. Bruinsma, Phys. Rev. E 65 (2002) 030902.


in High

968,

[13] A. Crisanti, T. Rizzo, T. Temesvari, Eur. Phys. J. B 33 (2003) 203.[14] D.J. Gross, E. Witten, Phys. Rev. D 21 (1980) 446;

M.R. Douglas, V.A. Kazakov, Phys. Lett. B 319 (1993) 219;M. Caselle, A. D’Adda, L. Magnea, S. Panzeri, in: E. Gava, et al. (Eds.), Proc. 1993 Trieste Summer SchoolEnergy Physics and Cosmology, Trieste, 1993, World Scientific, Singapore, 1994;C.R. Gattringer, L.D. Paniak, G.W. Semenoff, Ann. Phys. 256 (1997) 74;J. Hallin, D. Persson, Phys. Lett. B 429 (1998) 232;L. Li, Y. Meurice, hep-lat/0507034.

[15] H. Fuji, S. Mizoguchi, Phys. Lett. B 578 (2004) 432.[16] P. Bialas, Z. Burda, Phys. Lett. B 384 (1996) 75.[17] V. Berezinskii, Zh. Eksp. Teor. Fiz. 59 (1970) 907, Sov. Phys. JETP 32 (1971) 493;

V. Berezinskii, Zh. Eksp. Teor. Fiz. 61 (1971) 1144, Sov. Phys. JETP 34 (1971) 610;J. Kosterlitz, D. Thouless, J. Phys. C 6 (1973) 1181;J. Kosterlitz, J. Phys. C 7 (1973) 1046.

[18] F. Karsch, S. Stickan, Phys. Lett. B 488 (2000) 319.[19] C.N. Yang, T.D. Lee, Phys. Rev. 87 (1952) 404;

C.N. Yang, T.D. Lee, Phys. Rev. 87 (1952) 410.[20] M.E. Fisher, in: W.E. Brittin (Ed.), Lecture in Theoretical Physics, vol. VIIC, Gordon & Breach, New York, 1

p. 1.[21] R. Abe, Prog. Theor. Phys. 38 (1967) 72.[22] M. Suzuki, Prog. Theor. Phys. 38 (1967) 289;

M. Suzuki, Prog. Theor. Phys. 38 (1967) 1225.[23] R. Abe, Prog. Theor. Phys. 37 (1967) 1070;

R. Abe, Prog. Theor. Phys. 38 (1967) 322.[24] M. Suzuki, Prog. Theor. Phys. 38 (1967) 1243.[25] R.A. Blythe, M.R. Evans, Phys. Rev. Lett. 89 (2002) 080601;

R.A. Blythe, M.R. Evans, Braz. J. Phys. 33 (2003) 464.[26] B. Widom, J. Chem. Phys. 43 (1965) 3892;

B. Widom, J. Chem. Phys. 43 (1965) 3898;L.P. Kadanoff, Physics 2 (1966) 263;C. Domb, D.L. Hunter, Proc. Phys. Soc. 86 (1965) 1147.

[27] W. Janke, R. Kenna, J. Stat. Phys. 102 (2001) 1211;W. Janke, D.A. Johnston, R. Kenna, Nucl. Phys. B 682 (2004) 618.


CUMULATIVE AUTHOR INDEX B731–B736

Ableev, V. B732 (2006) 1Albino, S. B734 (2006) 50Alcaniz, J.S. B732 (2006) 379Alexandrov, S. B731 (2005) 242Aliev, T.M. B732 (2006) 291Alimohammadi, M. B733 (2006) 123Ambjorn, J. B734 (2006) 287Ambjørn, J. B736 (2006) 288Ammosov, V. B732 (2006) 1Anguelova, L. B733 (2006) 132Antoniadis, I. B731 (2005) 164Apollonio, M. B732 (2006) 1Arakawa, G. B732 (2006) 401Arce, P. B732 (2006) 1Artamonov, A. B732 (2006) 1

Babujian, H. B736 (2006) 169Bagulya, A. B732 (2006) 1Barnes, E. B732 (2006) 89Barr, G. B732 (2006) 1Bauer, R.O. B733 (2006) 91Becirevic, D. B734 (2006) 138Behrndt, K. B732 (2006) 200Belitsky, A.V. B735 (2006) 17Beneke, M. B736 (2006) 34Benincasa, P. B733 (2006) 160Berg, M. B736 (2006) 82Björnsson, J. B736 (2006) 156Blondel, A. B732 (2006) 1Blossier, B. B734 (2006) 138Bobisut, F. B732 (2006) 1Bogdan, A.V. B732 (2006) 169Bogomilov, M. B732 (2006) 1Bolzoni, P. B731 (2005) 85Bondarev, A.L. B733 (2006) 48Bonesini, M. B732 (2006) 1Booth, C. B732 (2006) 1Borghi, S. B732 (2006) 1Boucaud, Ph. B734 (2006) 138

Buchel, A. B731 (2005) 109Buchel, A. B733 (2006) 160Bunyatov, S. B732 (2006) 1Burguet-Castell, J. B732 (2006) 1Buttar, C. B732 (2006) 1

Calvi, M. B732 (2006) 1Campanelli, M. B732 (2006) 1Canfora, F. B731 (2005) 389Cao, J. B731 (2005) 352Carlevaro, L. B736 (2006) 1Catanesi, M.G. B732 (2006) 1Cervera-Villanueva, A. B732 (2006) 1Chen, C.-M. B732 (2006) 224Chen, W. B732 (2006) 118Chimenti, P. B732 (2006) 1Chizhov, M. B732 (2006) 1Chong, Z.-W. B732 (2006) 118Chukanov, A. B732 (2006) 1Ciechanowicz, S. B734 (2006) 203Coney, L. B732 (2006) 1

Dai, J. B731 (2005) 285D’Alessandro, M. B732 (2006) 64Das, S.R. B733 (2006) 297D’Auria, R. B732 (2006) 389Dedovitch, D. B732 (2006) 1De Fazio, F. B733 (2006) 1de Haro, S. B731 (2005) 225Delfino, G. B736 (2006) 259de Medeiros Varzielas, I. B733 (2006) 31De Min, A. B732 (2006) 1Denner, A. B734 (2006) 62Derendinger, J.-P. B736 (2006) 1De Santo, A. B732 (2006) 1Dev, A. B732 (2006) 379Di Capua, E. B732 (2006) 1Dittmaier, S. B734 (2006) 62Dolgov, A. B734 (2006) 208

0550-3213/2006 Published by Elsevier B.V.doi:10.1016/S0550-3213(06)00039-3

http://dx.doi.org/10.1016/S0550-3213(06)00039-3

330 Nuclear Physics B 736 (2006) 329–332

Dore, U. B732 (2006) 1Dubath, F. B736 (2006) 302Dumarchez, J. B732 (2006) 1

Edgecock, R. B732 (2006) 1Elagin, A. B732 (2006) 1Ellis, M. B732 (2006) 1Endo, I. B732 (2006) 426

Fehér, L. B734 (2006) 304Feldmann, T. B733 (2006) 1Ferrara, S. B732 (2006) 389Ferri, F. B732 (2006) 1Foerster, A. B736 (2006) 169Forte, S. B731 (2005) 85Fré, P. B733 (2006) 334Friedrich, R.M. B733 (2006) 91Frolov, S.A. B731 (2005) 1

Gaillard, M.K. B734 (2006) 116Galleas, W. B732 (2006) 444Gapienko, V. B732 (2006) 1Garbrecht, B. B736 (2006) 133Gastaldi, U. B732 (2006) 1Genovese, L. B732 (2006) 64Giani, S. B732 (2006) 1Giannini, G. B732 (2006) 1Gibbons, G.W. B732 (2006) 118Gibin, D. B732 (2006) 1Gilardoni, S. B732 (2006) 1Giménez, V. B734 (2006) 138Giusto, S. B733 (2006) 297Gómez-Cadenas, J.J. B732 (2006) 1Gorbatov, E. B732 (2006) 89Gorbunov, P. B732 (2006) 1Gößling, C. B732 (2006) 1Gostkin, M. B732 (2006) 1Grange, P. B732 (2006) 366Grant, A. B732 (2006) 1Graulich, J.S. B732 (2006) 1Grégoire, G. B732 (2006) 1Grichine, V. B732 (2006) 1Gromov, N. B736 (2006) 199Grossheim, A. B732 (2006) 1Gruber, P. B732 (2006) 1Guglielmi, A. B732 (2006) 1Guralnik, Z. B732 (2006) 46Guskov, A. B732 (2006) 1

Haack, M. B736 (2006) 82HARP Collaboration B732 (2006) 1Hayato, Y. B732 (2006) 1Hegedus, Á. B732 (2006) 463Hodgson, P. B732 (2006) 1Hollik, W. B731 (2005) 213Howlett, L. B732 (2006) 1

Hurth, T. B733 (2006) 1Hwang, S. B736 (2006) 156

Ichikawa, A. B732 (2006) 1Ichinose, I. B732 (2006) 401Imeroni, E. B731 (2005) 242Intriligator, K. B732 (2006) 89Ivanchenko, V. B732 (2006) 1Ivanov, D.Yu. B732 (2006) 183

Jacob, P. B733 (2006) 205Jacobsen, J.L. B731 (2005) 335Jain, D. B732 (2006) 379Janik, R.A. B736 (2006) 288Janke, W. B736 (2006) 319Jantzen, B. B731 (2005) 188Jarvis, P.D. B734 (2006) 272Johnston, D.A. B736 (2006) 319Jung, E. B731 (2005) 171

Kain, B. B734 (2006) 116Kakizaki, M. B735 (2006) 84Kant, E. B731 (2005) 125Karowski, M. B736 (2006) 169Kato, I. B732 (2006) 1Kaufhold, C. B734 (2006) 1Kawano, T. B735 (2006) 1Kayis-Topaksu, A. B732 (2006) 1Kazakov, V. B736 (2006) 199Kenna, R. B736 (2006) 319Khachatryan, Sh. B734 (2006) 287Khartchenko, D. B732 (2006) 1Khorrami, M. B733 (2006) 123Kirsanov, M. B732 (2006) 1Klimov, O. B732 (2006) 1Klinkhamer, F.R. B731 (2005) 125Klinkhamer, F.R. B734 (2006) 1Kniehl, B.A. B734 (2006) 50Kobayashi, T. B732 (2006) 1Koibuchi, H. B732 (2006) 426Kokorelis, C. B732 (2006) 341Kolev, D. B732 (2006) 1Korchemsky, G.P. B735 (2006) 17Koreshev, V. B732 (2006) 1Kramer, G. B734 (2006) 50Krasnoperov, A. B732 (2006) 1Kristjansen, C. B736 (2006) 288Kühn, J.H. B731 (2005) 188Kustov, D. B732 (2006) 1

Laveder, M. B732 (2006) 1Lee, R.N. B732 (2006) 169Li, T. B732 (2006) 224Linssen, L. B732 (2006) 1Lopes Cardoso, G. B732 (2006) 200Lü, H. B732 (2006) 118Lubicz, V. B734 (2006) 138Lüst, D. B732 (2006) 243

Nuclear Physics B 736 (2006) 329–332 331

Maeda, T. B735 (2006) 96Mahapatra, S. B732 (2006) 200Manvelyan, R. B733 (2006) 104Martins, M.J. B732 (2006) 444Mass, M. B732 (2006) 1Mathieu, P. B733 (2006) 205Mathur, S.D. B733 (2006) 297Matone, M. B732 (2006) 321Matsui, T. B732 (2006) 401Matsumoto, S. B735 (2006) 84Mayr, P. B732 (2006) 243Meier, U. B731 (2005) 213Menegolli, A. B732 (2006) 1Mescia, F. B734 (2006) 138Metlitski, M.A. B731 (2005) 309Mezzetto, M. B732 (2006) 1Mills, G.B. B732 (2006) 1Minahan, J.A. B735 (2006) 127Minasian, R. B732 (2006) 366Misiaszek, M. B734 (2006) 203Morariu, B. B734 (2006) 156Morone, M.C. B732 (2006) 1Muciaccia, M.T. B732 (2006) 1Mück, W. B736 (2006) 82Müller, D. B735 (2006) 17Mussardo, G. B736 (2006) 259

Nakatsu, T. B735 (2006) 96Nakaya, T. B732 (2006) 1Nanopoulos, D.V. B732 (2006) 224Nardi, E. B731 (2005) 140Nikolaev, K. B732 (2006) 1Nishikawa, K. B732 (2006) 1Noma, Y. B735 (2006) 96Novella, P. B732 (2006) 1

Okuda, T. B733 (2006) 59Ookouchi, Y. B733 (2006) 59Ookouchi, Y. B735 (2006) 1Orestano, D. B732 (2006) 1Ozpineci, A. B732 (2006) 291

Paganoni, M. B732 (2006) 1Paleari, F. B732 (2006) 1Palladino, V. B732 (2006) 1Panman, J. B732 (2006) 1Pantev, T. B733 (2006) 233Papa, A. B732 (2006) 183Papadopoulos, I. B732 (2006) 1Papageorgakis, C. B731 (2005) 45Park, D.K. B731 (2005) 171Parkhomenko, S.E. B731 (2005) 360Pascoli, S. B734 (2006) 24Pasquali, M. B732 (2006) 1Pasternak, J. B732 (2006) 1Pastore, F. B732 (2006) 1

Pattison, C. B732 (2006) 1Pelliccia, D.N. B734 (2006) 208Penin, A.A. B731 (2005) 188Penin, A.A. B734 (2006) 185Petcov, S.T. B734 (2006) 24Piperov, S. B732 (2006) 1Polukhina, N. B732 (2006) 1Pope, C.N. B732 (2006) 118Popov, B. B732 (2006) 1Prior, G. B732 (2006) 1Prokopec, T. B736 (2006) 133Pusztai, B.G. B734 (2006) 304

Radicioni, E. B732 (2006) 1Ramgoolam, S. B731 (2005) 45Rasmussen, J. B736 (2006) 225Reffert, S. B732 (2006) 243Richard, J.-F. B731 (2005) 335Ridolfi, G. B731 (2005) 85Riva, V. B736 (2006) 259Robbins, S. B732 (2006) 1Roiban, R. B731 (2005) 1Ross, G.G. B733 (2006) 31Rühl, W. B733 (2006) 104

Sakakibara, K. B732 (2006) 401Saleur, H. B734 (2006) 221Santin, G. B732 (2006) 1Sanz-Cillero, J.J. B732 (2006) 136Sato, Y. B735 (2006) 84Schmidt, M.G. B736 (2006) 133Schmitz, D. B732 (2006) 1Schomerus, V. B734 (2006) 221Schroeter, R. B732 (2006) 1Schwetz, T. B734 (2006) 24Sedrakyan, A. B734 (2006) 287Semak, A. B732 (2006) 1Senami, M. B735 (2006) 84Serdiouk, V. B732 (2006) 1Sharpe, E. B733 (2006) 233Simone, S. B732 (2006) 1Simula, S. B734 (2006) 138Singh, H. B734 (2006) 169Smirnov, V.A. B731 (2005) 188Sobków, W. B734 (2006) 203Soler, F.J.P. B732 (2006) 1Sorel, M. B732 (2006) 1Sorin, A.S. B733 (2006) 334Sotkov, G. B736 (2006) 259SPQCDR Collaboration B734 (2006) 138Srivastava, Y. B733 (2006) 297Starinets, A.O. B733 (2006) 160Stieberger, S. B732 (2006) 243Sviridov, Yu. B732 (2006) 1

Tachikawa, Y. B733 (2006) 188Tachikawa, Y. B735 (2006) 1

332 Nuclear Physics B 736 (2006) 329–332

Takashima, S. B732 (2006) 401Tamakoshi, T. B735 (2006) 96Tarantino, C. B734 (2006) 138Taylor, T.R. B731 (2005) 164Tcherniaev, E. B732 (2006) 1Temnikov, P. B732 (2006) 1Tereshchenko, V. B732 (2006) 1Tierz, M. B731 (2005) 225Tirziu, A. B735 (2006) 127Tonazzo, A. B732 (2006) 1Tornero, A. B732 (2006) 1Tortora, L. B732 (2006) 1Tran, N.-K. B734 (2006) 246Trigiante, M. B732 (2006) 389Troquereau, S. B732 (2006) 1Tsenov, R. B732 (2006) 1Tseytlin, A.A. B731 (2005) 1Tseytlin, A.A. B735 (2006) 127Tsukerman, I. B732 (2006) 1

Uccirati, S. B731 (2005) 213Ueda, S. B732 (2006) 1

Vaman, D. B733 (2006) 132van Holten, J.W. B734 (2006) 272Vannucci, F. B732 (2006) 1

Veenhof, R. B732 (2006) 1Vergeles, S.N. B735 (2006) 172Vidal-Sitjes, G. B732 (2006) 1Volpato, R. B732 (2006) 321

Wang, X.-J. B731 (2005) 285Wang, X.R. B731 (2005) 352Wang, Y. B731 (2005) 352Wiebusch, C. B732 (2006) 1Wright, J. B732 (2006) 89Wu, X. B733 (2006) 297Wu, Y.-S. B731 (2005) 285

Xie, X.C. B731 (2005) 352Xiong, G. B731 (2005) 352

Yagi, F. B735 (2006) 1Yang, D. B736 (2006) 34

Zaets, V. B732 (2006) 1Zhemchugov, A. B732 (2006) 1Zhitnitsky, A.R. B731 (2005) 309Zhou, C. B733 (2006) 297Zuber, K. B732 (2006) 1Zucchelli, P. B732 (2006) 1Zuluaga, J.I. B731 (2005) 140

Five-brane thresholds and membrane instantons in four...

Documents

Transcript of Five-brane thresholds and membrane instantons in four...