Volume 201, number I PHYSICS LETTERS B 28 January 1988

T H E ALGEBRA OF T H E GAUSS LAW IN d r 4 MASSLESS AXIAL QED W I T H A WZ TERM

Dan LEVY The Blackett Laboratory, Imperial College, Prince Consort Road, London SW7 2BZ, UK

Received 9 November 1987

The consequences of adding an anomaly-reproducing Wess-Zumino action to d= 4 massless axial QED are examined in the Weyl gauge. It is found that to the relevant leading order in h, the Jacobi identity for the hamiltonian and the Gauss law remains broken.

A common belief concerning gauge theories with chiral fermions is that the only consistent way of quantizing them is to choose in advance an anomaly- free representation for the fermions. During the last few years another approach has attracted some atten- tion, namely, that one can arrive at a consistent the- ory by adding new bosonic terms to the classical action. These terms are designed to compensate for the fermion loop anomaly by being explicitly (i.e. on the level of tree diagrams) gauge non-invariant. The way to calculate them was given, long ago, by Wess and Zumino [ 1 ]. The idea of using them in order to arrive at a consistent quantization of anomalous gauge theories was proposed by Faddeev and Shatashvilli [ 2 ]. Contributions ofbosonic terms to anomalies are also at the core of the Green-Schwarz anomaly can- cellation mechanism [ 3 ].

The main effort so far, was concentrated on two- dimensional models, notably, the chiral Schwinger model (see ref. [ 4] and references given therein). The conclusion seems to be that while adding a WZ term can indeed lead to a consistent theory, the reason which lies behind this succes may be special to d = 2. For the chiral Schwinger model, for instance, it was shown through the use of fermion-boson equiva- lence, that anomaly cancellation by adding a WZ term is equivalent to the standard approach of cancelling the anomaly by extra fermions [ 5 ]. Attempts were also made to look at the d = 4 case [6-10]. While the question of renormalizability is left aside, most of the

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answers to the consistency question seem to be opti- mistic [ 6-9].

The simplest d= 4 anomalous gauge theory to study is massless axial QED (AQED). In the hamiltonian formalism, consistency means (for an abelian the- ory) that the Gauss law, G ( x ) , should commute with itself and with H, otherwise, we cannot demand it to annihilate physical states. In particular, if a WZ term leads to a consistent theory we expect

JAQED+wz(H, G(X), G ( y ) ) = 0 , (1)

where J stands for the LHS of the Jacobi identity. Eq. (1) can be examined directly in perturbation

theory where we can calculate the relevant commu- tators. We employ for this purpose the well-known BJL prescription [ 11,12], which defines a matrix element of a commutator of two operators between two physical states from their T-product by taking the limit Po--' im where p~, is the appropriate external mo- mentum. In a similar manner, taking two external momenta successively to infinity in a T-product in- volving three operators, defines their double com- mutator. I f we want to evaluate the LHS of (1) for a given order, we have to calculate the BJL limit for all anomalous Feynman diagrams which appear in our theory to this order and may contribute to the com- mutators in (1).

We start by finding H and G(x ) for the WZ modi- fied AQED. The lagrangian is given by

L m LAQED -I" La~ , (2)

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LAOED = - l F u , ( x ) F ~ ' ( x ) + i ~ (x )~ ~ ( x )

- e J ~ ( x ) A ~ ' ( x ) , (3a)

L ~ = ½ [ OuCI)(x ) - m A u ( x ) ] 2

- ( h e 3 / 4 8 r ~ 2 r n ) ~ ( x ) e ~ " P F ~ ( x ) F , ~ p ( x ) , (3b)

In the above q~(x) is the WZ scalar field. With the present choice o fLe , q~ and m have mass dimension 1 in the natural system of units. The first term in L~, is a gauge invariant kinetic term. The second term is gauge non-invariant, and reproduces the anomaly in AQED (which fixes its coefficient). The h in this term is explicitly written in order to remind us that the anomaly comes from a one-loop diagram, using the fact that the loop expansion is equivalent to a power expansion in h ~t

The conjugate momentum to Ai is modified by L~. It reads (after fixing the Weyl gauge Ao=0)

I1A, = E i - ( he3 /6zrZ m ) ~ B i . (4)

The conjugate momentum to • is

I I ~ = 0 o ~ • (5)

The Gauss law, derived from the Ao equation of mo- tion and then reexpressed in terms of the conjugate momenta instead of time derivatives, reads

G = V "HA - e J S o - m H ~ • (6)

The hamiltonian is given by

H = ½ [112 + B z + mZA 2 +H2¢, + (V~)2]

- i ~ Y i O i ~ + m V ~ . A + e J S i A i

+ (he3/67r 2 m) t2~11A "B+ 21 [( he3]67t 2 m ) ~.[)B i ] 2 .

(7)

The O (h), three interaction vertices are relevant:

Vl = m V ~ . A ,

V 2 - ( h e 3 / 6 7 ~ 2 m ) ~ l - l a . B , V s - - e J S . A . (8)

Vt actually belongs to the quadratic part in H. It is convenient to sum this term to all orders in m into the scalar and photon propagators using the formula for the sum of a geometric series. Accordingly, we will have photon-photon, photon-scalar and sca-

~ "Ordinary" h factors are set to 1. An anomaly in (1) for ex- ample, should have at least h 3 but a power of two comes from

the fact that J consists of double commutators.

lar-scalar propagators (denoted respectively, by D o, Di, and D) given in momentum space by

i Do(P) = p2 _ m 2 ( ~o--PiPs/p2) , (9a)

impi Oi(P) = p2(p2 _ m 2) , (9b)

i D ( p ) = p2 _ m 2 (1 - m Z / p 2) . (9c)

Two properties of J are useful in simplifying the analysis. Using the linearity of J in its arguments we have

J ( H , G(x), a ( y ) )

= J d3z[J( ½11~ (z), V "HA (x), V "HA (y)) +...] ,

(10)

i.e. the RHS of (10) is a sum of J 's whose arguments are the various operators in H and G (only the first one is explicitly written). The second useful property of J which follows from a similar property of single commutators is

J ( A B , C , D ) = A J ( B , C , D ) + J ( A , C , D ) B . (11)

Here one must be careful because short-distance sin- gularities in the product A B can invalidate (11). However, a deviation from (11) can appear only when loop corrections to the composite operators are considered. Hence, (11 ) is safe when the RHS of (11 ) is already O(h) or when the relevant loops are not anomalous.

Contributions to (1) cannot arise from canonical commutators of the new terms in G and H. Also the last term in H, being O(h2), can be neglected in all considerations.

The computation of anomalous commutators to O(h) involves the anomalous one-loop fermion dia- grams generated by V3, and tree diagrams containing V2. Vj can appear in both, but we can exclude a mix= ing of 1"2 and II3 in the same diagram since it will be of higher order in h or e or both.

Consider the contributions from one-loop fermion diagrams. We will now show that the result in this case is equal to the result one gets by taking q ~ = / / ~ = 0 in eqs. (4), (6) and (7) (in other words to the pure AQED result). Each bosonic operator of

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interest has to couple to a fermion loop via one of the propagators (9). In the BJL limit, the leading behav- iour of (9) is (c is a constant)

lim (any of the D(p) 's) =cpff 2 +O(pff 4) . (12) po ~ iov

Since the relevant fermion loops contribution in this limit is at most linear in Po, the BJL limit of the whole diagram will vanish, unless c# 0. For (9b) c= O, and hence, ¢~ or H~ coupling to a fermion loop vanishes in the BJL limit. Moreover, c for (9a) has the same value as in pure AQED. Finally, an external • line cannot couple to a fermion loop through V~ and Do(p) because

PiDij(P) I p'- = o = O. (13)

The pure AQED contributions to J are in fact non- zero. The details of the calculations are given else- where [ 12 ]. Here we quote the final result:

JAQED(H, G(x) , G(y ) )

= -- (he3/6rt2)V XE(x) "Vx~3(x-y) . (14)

Next we consider the diagrams involving only V2 and VI. V2 connects two photon lines to a scalar line. In momentum space it is given by

V~'( kl , k2) oc ( he3 /6~2 m )e°:( k l tk20-k2 lk lo) . (15)

Again, /I2 is linear in the zero components of the ex- ternal momenta, and as in the previous case, dia- grams containing the mixed propagator Di(p) vanish in the limit. This leaves for discussion only J(HA,(Z), HA:(x), H ~ ( y ) ) . The expression for the relevant T-product (keeping only the leading term in the propagators) is given in momentum space by

klo kzo qo he 3 T(HA,HA:H~)oc k 2 - - m 2 k 2 - - m 2 q2 m 2 6/r2 m

Xe~Jl(k~/k2o - kE/k~o), (16)

where q - - (kl +k2). The double commutator one gets on taking a double BJL limit in (16) is propor- tional in (x-space) to a first derivative of a 8-func- tion. However, when J is calculated, using

J = - l i r a klo lim k2oT(k l , k2 ) klo ~ioo k2o ~ioo

+ lim k20 lim k loT(k~ ,k2) k20 ~ioo klO ~ioo

+ lim qo lim k 2 0 T ( k l = - ( q + k 2 ) , k 2 ) , (17) q o ~ i ~ k20~ioo

it is found to vanish. A way to understand why tree diagrams involving

/I2 do not lead to a J breaking, is to recall that in per- turbation theory the vertices for the canonical mo- menta appearing in our commutators, are taken from the free theory. For HA, for example, we really use E'= -O°A i as a vertex. Therefore, the contributions of the above-mentioned diagrams are just reproduc- ing the information in (4). A BJL calculation of [Ha~, HA ] for example is in fact equivalent to

[H.(x) , HA,(y)] = [H. (x) , E'(y)]

- (he3 /6~r2m)[Ho(x) , ~ ( y ) ] B ~ ( y )

= 0 + ( i h e 3 / 6 n 2 m ) ~ 3 ( x ' y ) B ~ ( y ) . (18)

However, equivalence to a canonical commutator immediately implies no contributions to J.

We thus find that to O(h)

JAQED+wz(H, G(X), G(y))=JAQED(H, G(x) , G(y ) )

# 0 , (19)

and therefore the consistency of the modified theory is put to question. One can expect a similar problem in the BRST formalism since in this abelian case QBRST is basically the Gauss law, but this should be checked explicitly.

Before concluding, I would like to point out a com- mon drawback to all specific discussions from the canonical point of view #2 of the Faddeev and Sha- tashvilli proposal, given so far (including the present one). They all seem to mix classical and quantum ar- guments. If as a first step towards quantization we want to regard (2) as a pure classical expression, in- cluding the O(h) term in L~, in order to define mo- menta and identify constraints, then classically this lagrangian is not gauge invariant. The fixing of the gauge Ao = 0 anticipates full gauge invariance of the quantum theory, but is done before the theory is ac- tually quantized. In the covariant approach [ 7,9 ], a similar sort of thing happens: The equation OuJ~, = anomaly is added to the classical equations of motion as extra information anticipating quantum results before quantization actually takes place. The best justification to this procedure one can come up with

~2 For a general criticism of the path integral approach see ref. [4].

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is that it seems to work in d = 2, and that nothing bet- ter has been proposed so far. In fact, the present re- sult may indicate that this procedure does not work,

at least for d = 4 AQED. While this work was being completed, I received

preprints by Della Selva et al. [9] and by Han and Pervushin [ 10]. Ref. [ 10] seems to arrive at conclu-

sions similar to mine concerning d = 4 AQED.

I wish to thank Y. Frishman, I.G. Halliday, A. Schwimmer and S. Solomon for very useful discussions.

References

[ 1 ] J. Wess and B. Zumino, Phys. Lett. B 37 (1971) 95.

[ 2 ] L.D. Faddeev and S.L. Shatashvilli, Phys. Lett. B 167 (1986) 225.

[3] M.B. Green and J.H. Schwarz, Phys. Lett. B 149 (1984) 117. [4] R. Jackiw, Update on anomalous theories, MIT preprint

CTP# 1436, in: CECS Scientific Series, ed. C. Teitelboim (Plenum, New York), to be published.

[ 5 ] I.G. Halliday, E. Rabinovici, A. Schwimmer and M. Chan- owitz, Nucl. Phys. B 268 (1986) 413.

[6] R. Rajaraman, Phys. Lett. B 184 (1987) 369. [7] S.G. Rajeev, MIT preprint CTP# 1405 (1986). [ 8 ] K. Harada and I. Tsutsui, Tokyo Institute of Technology

preprint TIT/PEP-113 (1987). [9] A. Della Selva, L. Masperi and G. Thompson, Trieste pre-

print IC/87/131 (1987). [10] N.S. Han and V.N. Pervushin, JINR preprint E2-87-279

(1987). [ 11 ] K. Johnson and F. Low, Prog. Theor. Phys. (Kyoto) Suppl.

37-38 (1966) 74. [ 12 ] D. Levy, Imperial College preprint Imperial/TP/86-87/17

(1987).

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