Anomalous global currents and compensating fields in the BV formalism

17 September 1998

Ž .Physics Letters B 436 1998 125–131

Anomalous global currents and compensating fieldsin the BV formalism

Ricardo Amorim a,1, Nelson R.F. Braga a,2, Marc Henneaux b,c,3

a Instituto de Fısica, UniÕersidade Federal do Rio de Janeiro, Caixa Postal 68528, 21945-970 Rio de Janeiro, Brazil´b Faculte des Sciences, UniÕersite Libre de Bruxelles, Campus Plaine C.P. 231, B–1050 Bruxelles, Belgium´ ´

c Centro de Estudios Cientıficos de Santiago, Casilla 16443, Santiago 9, Chile´

Received 3 May 1998; revised 20 June 1998Editor: L. Alvarez-Gaume

Abstract

We compute the anomalous divergence of currents associated with global transformations in the antifield formalism, byintroducing compensating fields that gauge these transformations. We consider the explicit case of the global axial current inQCD but the method applies to any global transformation of the fields. q 1998 Elsevier Science B.V. All rights reserved.

PACS: 03.70.qk; 11.10.Ef; 11.15.-q

1. Introduction

It is well known that quantum corrections canmodify the expectation values of the divergence of

w xglobal currents 1,2 . In particular, a classically van-ishing divergence of a global current can acquire anon vanishing expectation value at the quantum level.

w xIt was shown by Fujikawa 3 that these quantumcontributions can be calculated by path integralmethods if one appropriately regularizes the func-tional measure.

Ž .The Batalin Vilkovisky BV , or field-antifieldformalism, is an extremely powerful procedure for

w xthe quantization of gauge theories 4–6 . The occur-

1 Email: [email protected] Email: [email protected] Email: [email protected].

rence of local anomalies in this formalism has beenw xdiscussed in 7 , where they have been related to the

Ž .non-invariance of the measure under rigid BRSTtransformations. The purpose of this letter is todevelop a method for computing the anomalous di-

Žvergence of currents associated with global as op-.posed to local transformations. To that end, we

w xintroduce pure gauge ‘‘compensating fields’’ 8 thatcouple to the divergence in question. We then turn tothe master equation and show that quantum correc-tions are needed in order to fulfill the quantummaster equation. These quantum corrections to the

Žsolution of the master equation do exist no gauge.anomaly and turn out to be crucial for our purposes.

Indeed, they precisely generate the quantum correc-tions to the divergence of the global current. This iseasily seen by choosing appropriately the gauge forthe new gauge freedom and using the standard Frad-kin-Vilkovisky theorem of the antifield formalism.

0370-2693r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved.Ž .PII: S0370-2693 98 00843-0

( )R. Amorim et al.rPhysics Letters B 436 1998 125–131126

wRigid symmetries have been discussed from a dif-w xferent point of view in the antifield formalism in 9 .

w xxSee also 10 .w xIn Ref. 11 a procedure for calculating anomalous

divergences in the particular case of Abelian globalsymmetries was presented. In that paper the originalsymmetry content of the action was enlarged by theintroduction of extra gauge fields that could be triv-ially removed by a gauge fixing. At the quantumlevel the symmetries introduced in that way areapparently broken but can be trivially restored withthe introduction of appropriate counterterms. Thiscould be done because of the cohomological trivial-ity of the field extension, which was easily provedonce the new fields and the corresponding ghosts arecombined in BRST doublets. The anomalous diver-gences of the currents associated with the enlargedsymmetries are then calculated by using the indepen-dence of the path integral with respect to the gaugefixing.

w xIn the present paper the treatment of 11 isgeneralized in several non trivial ways. First, we areconsidering non Abelian global transformations inthe context of an original theory that presents itself anon Abelian local symmetry. The gauging procedure,necessary for the calculation of the anomalous cur-rent divergences, mix non-trivially both kinds of

Ž .symmetries. We show in section 2 that the resultinggauge structure becomes actually a semi-direct prod-

Ž .uct of SU N with itself instead of a direct product.This non trivial algebraic structure reflects itself inthe process of quantization. For instance, the coho-mological triviality of the extension can only be

Žproved in a much more elaborated way see sectionŽ ..4 since the new fields only form BRST doublets inthe Abelian limit. Also, the form of the counterterms

w xis not a trivial generalization of the one of Ref. 11but relies heavily on peculiar aspects of the Lie

Ž .algebra cohomology of non-abelian semi-simpleLie groups.

Our method applies to any transformation of thefields, even those that are not symmetries of theclassical action. We shall develop the formalism by

Ž .considering the explicit case of an SU N Yang-Millstheory with fermions in the fundamental representa-tion. The non abelian chiral transformation is not asymmetry of the action, and the corresponding cur-

Žrents are covariantly conserved rather than con-

.served in the strict sense . We shall compute theircovariant divergence in the quantum theory by fol-lowing the method outlined above, and show how

w xthe standard anomalous term 12,13 arises in thatapproach.

2. Compensating fields and conservation laws

Our starting point is the Yang-Mills action

1k mn mS s d x y Tr F F q icg E y iA c ,Ž .Ž .H ž /0 mn m m4

2.1Ž .

Ž .where A is a SU N -connection and the fermionsm

are taken in the fundamental representation. Weassume the spacetime dimension k to be even inorder to have non-trivial chirality transformations.The action is invariant under the local transforma-tions

X X y1c sLc , c scL ,

AX sL A Ly1 y i E L Ly1 , 2.2Ž .Ž .m m m

Ž .where L g SU N .Ž .The chiral infinitesimal SU N transformations

are

X Xc s 1y ie P c , c sc 1y ieP ,Ž . Ž .q y

AX sA , 2.3Ž .m m

1 a aŽ .where P s 1" g . Here, ese T is a con-" 52

Ž .stant element of the SU N algebra. If the connec-Ž .tion does not vanish, the transformations 2.3 are

not symmetries of the action. However, it is straight-foward to verify that the associated chiral current

m a aJ scg T P c is coÕariantly conserved,q m q

am m a abc b m cD J ' E J qf A J s0 , 2.4Ž .Ž .m q m q m q

w a b x abcwhere T , T s if defines the structure con-stants of the algebra. Of course, if A s0, thism

relation reduces to E J m a s0, in agreement with them qfact that the chiral transformations are then symme-tries of the action.

It is possible to enlarge the gauge symmetrycontent of a field theory by introducing compensat-

w xing fields which, as discussed in 8 , may lead to a

( )R. Amorim et al.rPhysics Letters B 436 1998 125–131 127

different representation for the same theory, wheresome calculations become simpler. In the presentcase, we will make the chiral transformations ofŽ . Ž .2.3 , with ese x , become gauge symmetries ofthe action by introducing pure gauge compensating

Ž . Ž .group elements of SU N denoted as g x . Beingpure gauge, the group element will have no indepen-dent equation of motion. Rather, its equation ofmotion will be precisely the covariant conservation

Ž . Ž .law 2.4 at gs1 .The constructive way to derive the action with the

compensating field included is to replace theŽ .fermionic field c by P qP g c . If one does so,y q

one gets the extended action

1k mnS c , c , A , g s d x y Tr F FŽ .H ž1 m mn4

m ˜qicg E y iA c , 2.5Ž .ž / /m m

˜ ˜ ˜ w xwhere A stands for A sA A , g sP A qm m m m y m

P B withq m

B sgy1A gq igy1E g . 2.6Ž .m m m

Ž .By construction, the action 2.5 is invariant underthe local transformations

dcs i h x ye x P c ,Ž . Ž .Ž .q

dcsyic h x ye x P ,Ž . Ž .Ž .y

d A sE h x q i h x , A ,Ž . Ž .m m m

d gs i ge x q h x , g 2.7Ž . Ž . Ž .Ž .which include both the original symmetry and the‘‘gauged’’ chiral transformations. The complete

Ž .gauge group of 2.5 is the semi-direct product ofŽ .SU N with itself.The theory with compensating fields is clearly

classically equivalent to the original theory. Indeed,one can gauge away the compensating field g byusing the new gauge freedom. For instance, if we

Ž .choose the gauge gs1, the action 2.5 reduces toŽ .its original form 2.1 . The existence of a new gauge

freedom implies further Noether identities. Theseidentities relate the variational derivatives of theaction with respect to the compensating fields to theother variational derivatives, and imply that the g-

equations of motion are not independent. In fact, astraightforward calculation yields the interesting rela-tion

dS am< ' yi D J , 2.8Ž .Ž .bs0 m qadb

Žwhere on has set gs1qb in the vicinity of the.identity . Thus, one can say that the compensating

Ž .field couples to the covariant divergence of thechiral current. This property will turn out to becrucial in the computation of the quantum correc-tions to D J m

a.Ž .m q

The same procedure can be followed for anygroup of rigid transformations of any local action.One may introduce the group parameters as dynami-cal variables by parametrizing the fields f i as f i s

iŽ y1 X. iŽ X.f g ,f where f g,f is the transformed off iX

under the transformation g. One takes as newiX Ž X.variables g and f and drop the . The action is

invariant under the gauge transformations that shiftin a spacetime-dependent way the group variable g

Ž .by arbitrary left multiplication on the group andtransform f iX

accordingly. One can use this symme-try to eliminate g and recover the original action. Inthe extended formulation, the compensating fieldcouples to the divergence of the current associatedwith the original rigid transformations. Indeed, theEuler-Lagrange equations for g are equivalent toE j m s0 if the transformations are symmetries of them

original action, where j m is the Noether current,which is conserved by Noether theorem. This isbecause the g ’s are then ‘‘ignorable coordinates’’ of

Žthe extended action the original action is invariantunder constant transformations and thus only deriva-

w x.tives of g can occur; see e.g. 14 . If the originaltransformations are not global symmetries of theoriginal action, g will couple to a generalized ‘‘co-variant’’ divergence of j m rather than to the ordinarydivergence, as in the specific example given above.

3. Quantization

Ž .The action 2.1 has no gauge anomaly since thereis no chiral fermion. The current J m a associatedq

Ž .with the rigid transformation 2.3 has, however, ananomalous covariant divergence. This would seem to


ruin the extended theory, since one might fear thatthe new gauge symmetry will become anomalous. Iftrue, equivalence with the original model would bebroken at the quantum level and potential inconsis-tencies could even arise. That the new gauge symme-

w xtry is not afflicted by anomalies was discussed in 8 ,where it was shown that ‘‘compensating fields alsocompensate for the anomaly’’. This also follows

w xfrom the general cohomological investigation of 15 .Apart from global cocycles related to the De Rhamcohomology of the group manifold and presumablyirrelevant in perturbation theory, the BRST cohomol-

Žogy group at ghost number one related to the.anomaly has no cocycles involving the new ghosts.

We shall verify this property explicitly in the contextof the antifield formalism.

To that end, we first observe that the symmetriesŽ . w x i i2.7 close in an algebra, d , d f sd f for1 2 3

any field f i . The parameters of the transformationw xon the right hand side are given by h s i h , h3 1 2

w x w x w xand e s i h , e q e , h y e , e . TheŽ .3 1 2 1 2 1 2Ž .BV action at zero order in " then follows by the

standard procedure,

)k )SsS q d x ic cybP cy ic cybP cŽ . Ž .H1 q yž) w xqTr ig g bq c , gŽ .½

i) m ) w xqA D cq c c , cm 2

i) w x w xy b b , b y2 c , b , 3.1Ž .Ž . 5 /2

where we have introduced the ghosts c and b corre-sponding to the parameters h and e respectively andalso the antifields associated with each field.

We will represent the total sets of fields and� I4 � )4antifields respectively as w , w . Defining theI

BRST transformation of any quantity X as: sXsŽ . Ž . w xX , S , where X,Y is the standard antibracket 5 ,it is not difficult to see that s2 Xs0 and that the

Ž .classical master equation S , S s0 is valid. TheBRST transformations of the fields are given byŽ .2.7 with the gauge parameters replaced by the

w xghosts, as well as scs i c c, sbsyi b bq i c , b .

It is useful to introduce the invariant forms ssyigy 1sg s b y c q gy 1cg. These fulfill theMaurer-Cartan equations sss is 2. The related form

y1 w xs syig E g transforms as ss sE sy i s ,s .m m m m m

Ž y1 . w y1 xFurthermore, s g A g s i c y b, g A g qm m

gy1E c g and as B sys qgy1A g, one gets sBm m m m m

Ž . w xsE cyb q i cyb, B . This equation shows ex-m m

plicitly that the ghost associated with the connectionB is cyb.m

4. One-loop order master equation

The BV vacuum functional is defined as

EC iI )Z s dw d w y exp W , 4.1Ž .HC I I ž /"Ew

with WsSq"M . Properly speaking, one should1

include the non-minimal sector. This will be donebelow, but since these variables do not affect the

Žcohomological considerations they form trivial pairsw x.6 , they will not be written explicitly here. The BVvacuum functional is independent of the choice ofgauge fermion C if, besides the classical masterequation, the one loop order master equation is alsosatisfied

M ,S s i DS , 4.2Ž . Ž .1

d dr lwhere D' . This equation is undefined un-A )df dfA

less we regularize the action of the D operator.Ž .Using a Pauli Villars PV regularization, with usual

mass terms for the PV fermionic fields, the fourŽ .dimensional case ks4 to which we shall restrict

our attention from now on, can be written in thew xform 7

DSsa tr d4 x E cyb D m yE c D m . 4.3Ž . Ž .Ž .H m B m A

Here,

im mnrsD se A E A y A A A 4.4Ž .A n r s n r sž /2

and D m is given by a similar expression, with theBŽ .replacement A ™B given by 2.6 . In the abovem m

1expression, asy . We are assuming that the224p

Žmeasure for the g sector is BRST invariant we are.taking the Haar measure .


The term trHd4 xE cD m is the standard ABBJm AŽ .Adler-Bardeen-Bell-Jackiw anomaly for the gaugefield A . It can be rewritten in form notations asm

Ž Ž . 3. A BB JtrHdc AdAy ir2 A 'Ha and is well knownA

to be a solution of the Wess-Zumino consistency4 Ž . mcondition. Similarly, the term trHd xE cyb D ,m B

Ž .Ž Ž . 3.which can be rewritten trHd cyb BdBy ir2 B'Ha A BB J is the ABBJ anomaly with B instead ofB m

A . It also solves the Wess-Zumino consistency con-m

dition. Consequently, for the full DS, one has s DSŽ .s DS , S s0.The quantity DS represents the variation of the

path integral measure under BRST transformations.If this variation cannot be compensated by the varia-tion of some local counterterm then the theory wouldbe anomalous, and this would be a priori a disastersince it is a gauge anomaly. So, the important pointnow is to find out if there is a local counterterm M ,1

to be added to the action, whose BRST variationcancels the candidate anomaly DS. It was shown inw x11 that such a counterterm exists in the Abeliancase. We extend this result here to the non-abeliancase.

It is rather easy to see that DS is s-exact in thespace of local functionals, and thus that M exists.1

Indeed, it is well known that the ABBJ anomaly isrelated to the invariant tr F 3 in 2 dimensions higher,i.e., here, in six dimensions through a chain of

Ž w x.descent equations see e.g. 16 . Explicitly, one hastr F 3 sdQ5,0 where Q5,0 is the Chern-Simons 5-formA A A

constructed out of A, and sQ5,0 qdQ4,1 s0, whereA A

Q4,1 is the ABBJ anomaly a A BB J. In Qi, j, the firstA A

superscript is the form-degree, while the secondsuperscript is the ghost number. Similarly, one getstr F 3 sdQ5,0 where Q5,0 is the Chern-Simons 5-formB B B

constructed out of B, and sQ5,0 qdQ4,1 s0, whereB B

Q4,1 is the ABBJ anomaly a A BB J. Now, because AB BŽand B are related by a gauge transformation Eq.

Ž ..2.6 , they have field strengths related as F sBy1 3 3 Ž 5,0g F g and thus tr F s t rF . This implies d QA A B B

5,0. 5,0 5,0 4,0yQ s0, i.e., Q yQ sdM for some 4-A B A

form M 4,0. Substituting this result in the next de-Ž 4,1 4,1 4,0.scent equation yields then d Q yQ ysM sB A

0, i.e.,

a A BB J ya A BB J 'Q4,1 yQ4,1 ssM 4,0 qdM 3,1B A B A

4.5Ž .

for some 3-form M 3,1. This implies, upon integra-tion, that DS is indeed exact,

iDSs ia a A BB J ya A BB J ssM ,Ž .H B A 1

M s ia M 4,0 . 4.6Ž .H1

The explicit form of the counterterm M may be1

found either by following the above procedure, or byusing a perturbative expansion in the number offields. One gets

i5 mnrvlM sa d xe Tr s s s s sŽ .H1 m n r v lž 10MM

q d4 x e mnrsHE MM

=i

y1Tr E g g A E A y A A Am n r s n r s½ ž /21 y1 y1y E gg A E gg Am n r s4

1 y1 y1q E gg E gg A Am n r s2

iy1 y1 y1y E gg E gg E gg A , 4.7Ž .m n r s 5 /2

where the first term is the Wess-Zumino term which,as usual, is defined on an kq1 dimensional mani-fold MM, with boundary E MM given by the four-dimen-sional space on which the theory is defined.

5. Anomalous divergence

We will now show how to get the anomalousdivergence of the chiral current from the previousresults. The quantum BV action W is the sum of thecounterterm from the last section with the BV classi-

Ž .cal action of Eq. 3.1 . Let us introduce also a trivialpair of fields p ,b, and the corresponding antifields

) )p ,b , in order to allow an appropriate gauge fix-ing of the extra symmetry. We should also introducenon minimal variables for the original gauge symme-try, but these will not be written explicitly. So we

4 )Ž .have WsSq"M qHd xTr p b . We choose a1

gauge fixing fermion of the form C s TrŽ .b gy1yb qC where b is an infinitesimalŽ .Ž .

external function and where C does not depend on


the fermionic variables or the extra fields g,b ,p andb. This choice of gauge fixing fermion enforces thegauge gs1qb and from what we have seen above,b will appear as a source for the covariant diver-gence of the chiral current.

A direct calculation gives as gauge fixed quantumaction

ECI )W sW f , w s

S I IEw

14 mn m ˜s d x y Tr F F q icg E y iA cŽ .H ž /mn m m4žw xqTr ib g bq c , gŽ .½

EC i ECm w xq D c q c , c

mE A 2 E c

yp gy1yb q"M . 5.1Ž . Ž .15 /w xNow we build up the vacuum functional Z b s

iIw x � 4H dw exp W where we are omitting the depen-S"

dence on C in the notation. Then we integrate overthe fields p and g. This amounts to substituting gby 1qb. The fermionic term of the action becomes

m micg E y iA cqcg P iE bq A , b c .Ž . Ž .m m q m m

5.2Ž .

The integration over b and b is direct and yields onetogether with the Haar measure. The vacuum func-tional becomes therefore

iAw xZ b s df exp W , 5.3Ž .H S½ 5"

� A4 � 4 Žwhere f s A ,c, . . . the dots refer to trivialm

.pairs associated to the original gauge symmetry andŽ .where W is just the sum of the extended classicalS

Ž .action and the counterterm of Eq. 4.7 in the gaugegs1qb , plus non-minimal terms that do not in-volve b. As the master equation is satisfied, thisobject is independent of the external parameter b

Ž .‘‘Fradkin-Vilkovisky theorem’’ . Thus we have

w xdZ b i d WS< ² : <s s0 . 5.4Ž .bs0 bs0a adb " db

Ž .From Eq. 4.7 we find

dM1 mnrs< sy aebs0adb

=i

aTr E A E A y A A A T .m n r s n r s½ 5ž /25.5Ž .

Ž . Ž .Using this equation as well as 5.2 and 2.8 wefinally get

w xdZ b am< ²" s D JŽ .bs0 m qadb

imnrs a :yi"ae Tr E A E A y A A A Tm n r s n r s½ 5ž /2

s0 5.6Ž .where now the expectation values are taken in theoriginal theory, with no extra variables. This repro-duces the desired anomalous divergence.

6. Conclusion

We have shown that anomalous expectation val-ues of currents associated with global transforma-tions can be calculated by introducing compensatingfields. We have performed the analysis in the anti-field formalism. The present work extends the previ-

w xous Abelian study of 11 . We have shown that thenew gauge symmetries associated with the compen-sating fields are not obstructed at the quantum levelsince they do not change the cohomology of thetheory. However, it is necessary to introduce quan-tum corrections to the BV action in order to fulfillthe quantum master equation. These quantum correc-tions precisely generate the appropriate anomalouscontribution to the divergence of the global current,although no gauge anomalies are present. Our proce-dure represents thus an interesting example wherequantum corrections have a non trivial role.

Acknowledgements

This work is partially supported by CNPq, FINEPŽ .and FUJB Brazilian Research Agencies . M.H. is

grateful to Glenn Barnich for useful comments, as


well as to CNPq and the physics departments ofUERJ and UFRJ for kind hospitality.

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Anomalous global currents and compensating fields in the BV formalism

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