Remarks on the renormalization of gauge invariant operators in Yang-Mills theory

Physms Letters B 313 ( 1993 ) 35-40 North-Holland

PHYSICS LETTERS B

Remarks on the renormalization of gauge invariant operators in Yang-Mills theory Marc Henneaux Facultk des Sctences, Umversltb Ltbre de Bruxelles, Campus Plame C P 231, B-1050 Bruxelles, Belgtum and Centro de Estudlos Clent[ficos de Santiago, Casdla 16443, Santiago 9, Chile

Recoved 23 June 1993 Editor: P.V. Landshoff A simplified proof of a theorem by Joglekar and Lee on the renormalizatmn of local gauge invarmnt operators m Yang-Mflls theory is given. It is based on 0) general propernes of the antlfield-annbracket formahsm and (n) well-estabhshed results on the cohomology of semi-simple Lie algebras.

1. Introduction

where J is defined by

The BRST transformatton in Y a n g - M i l l s theory was originally defined without reference to the equations o f morion and incorporated only the gauge symmetry. However, it was subsequently found necessary to take the equations of motion mto account. This can be achieved by mtroducing one "antifield" for each field appearing in the path integral, and by defining the BRST variatton of the antlfields in such a way that the BRST differential implements the equations o f motion D u F f " = 0 in cohomology. These developments were first pursued in the context o f the renormalization o f the Y a n g - M l l l s field [ 1-3 ]. They then turned out to be crucial for the BRST formulanon o f arbitrary gauge theories with "open" algebras [4], for which one cannot separate the gauge symmetry from the dynamics. The key role played by the equations o f motion and the concept of c o v a n a n t phase space [ 5 - 8 ] in the antlfield formahsm was emphasized in [9-11 ] (see also [ 12] and references therein for a related but different point o f vmw). In the case o f the pure Yang-Mills field, the complete BRST transformation reads, m the minimal sector containing the vector potential A~, the ghosts C a, and their respective antlfields A~ u, Ca,

5A~ = 0 ,

s = J + a,

sC~ = b~,

(1)

3 C a = O,

3A*au = DvF,~ u,

(2)

JC~ = DuAa u ,

and a ts the original BRST differential o f [ 1,2 ], aAau = Du Ca

,

aca=

1 f,a i,~brc "~'--bc'~ ~ ,

(3)

extended to the antifields so as to a n t i c o m m u t e with J, = -C

cA

UC c ,

=

--~ac~b '~ •

(4)

Here, Cffc are the structure constants of the Lie algebra G o f the gauge group, which we assume to be semisimple. One has j2=0,

a z=0,

3a+a3=0,

(5)

so that s z = 0.

(6)

In order to fix the gauge, one introduces also the antighosts Ca and the auxlhary fields ba, together with their respective antifields C *a and b *a. In this "nonm l m m a l " sector, the BRST differential is defined by

s b *a ~ -

-C

sb~ = O, *a .

0370-2693/93/$ 06 00 © 1993-Elsevier Science Pubhshers B.V. All rights reserved

s C *~ = 0 ,

(7) 35

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The fields and anti fields have ghost number given by ghA~ = 0 ,

g h C a = 1,

ghA*~ u = - 1 ,

(8)

ghC,~ = - 2 , ghCa = - 1 ,

ghba = 0,

g h C *a -- 0,

ghb *a = - 1 .

(9)

We shall denote by .4 the algebra o f polynomials in A~, C a, Aa , Ca, Ca, ba, C *a, b *a and their derivatives up to a finite (but arbitrary) order, and by/3 the smaller algebra of polynomials in A~, C a, A~~, C~ and their derivatives up to a finite (but arbitrary) order. The nilpotency of s enables one to define cohomological groups H* (s) in the standard manner. The following question concerning the cohomology o r s m .4 is of central interest in the analysis o f the renormalizatlon of local gauge mvariant operators [ 13-21 ]: given a polynomial in .4 that is (1) BRST-invariant and (n) of ghost number zero, can it be written as the sum of a gauge invariant polynomial and a BRST v a n a h o n ? In other words, does one have sP=0,

ghP=0,

P E `4

~ P = M + sQ,

(10) where M is a gauge m v a n a n t polynomial m the field strengths Fff, and their derivatives up to a finite order, and where Q c `4? Differently put, can one find m each equivalence class o f H ° (s) a representatwe that does not involve the ghosts and that is smctly gauge invartant 9 The answer to this question was conjectured m [ 17 ] to be the affirmative. A proof o f (10) has been given m [ 18 ] following earlier work of [ 16 ]. The purpose of this letter is to provide an alternatwe and somewhat shorter proof, which is based on known results on the anUfield formalism and the cohomology of Lie algebras.

The algebra of polynomials m the fields ~bA and their derivatives up to a finite order is denoted by Y. The gauge fixed action is the integral of an element of 5r obtained by ehminatlng the anufields q~] through ck*a = d ~ / 5 ~ A , where ~u is the so-called gauge fixing fermlon [4]. The gauge-fixed BRST symmetry s~, is defined m the algebra jr through s~,q~A = sdpA(cks,(p*8 = 5~u/d~ B) and is a symmetry of the gauge-fixed action. In the case of the Yang-Mills theory considered here, s~, does not depend on the gauge fixing since sO A does not depend on the antifields, s~,~bA = s4~a. Thus, s~, is mlpotent off-sheU (although, in general, se is only ndpotent on-shell). The gauge-fixed BRST cohomology H~ (s) ~s defined to be the set of equivalence classes of weakly BRST m v a n a n t elements of .T, sR ~O,

RE.T,

(11)

modulo weakly BRST exact ones, R ~ R' iff R ' ~ R' + s T ,

R, R', T E F.

(12)

One says that two polynomials m U are "weakly equal" (for the gauge condmons under consideration) if they coincide when the equations of motion following from the gauge-fixed action hold. They then differ by a combination of these equauons of motion and their spacetime derivatives. As shown in [ 10,11 ], the gauge-fixed BRST cohomology H~ (s) defined m the algebra 5r of the fields is isomorphic wnh the BRST cohomology H* (s) defined m the algebra `4 of the fields and the antlfields. Furthermore, if P E .4 defines an element of H* (s), then R = P(~,~* = d~/dc~) C Y defines the corresponding element of the gauge-fixed BRST cohomology H~,(s). The question (10) is thus equivalent to: does sR~O,

2. Digression

imply

The question (10) was actually formulated originally in terms of the gauge-fixed action. In order to make contact with the original formulation, let us denote collectively all the fields Aau, C a, Ca and ba by 0 a and all the antifields Aa*u , Ca, • -C- *a and b *a by q~A. •

R~M+sT,

36

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(13)

RC.T

TEl,

(14)

where M is a gauge-invarlant polynomial xn the field strengths and their derivatives up to a finite order? In some gauges ("linear gauges"), eqs. (13) and (14)

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can be further simplified because the equation o f motion o f the antighost can be easily handled. However, even though H~ (s) refers explicitly to a definite gauge fixation through the gauge-fixed equations o f motion used in eq. (12), it does not depend on the choice o f gauge since it is isomorphic to H* (s). Hence, its calculation can be carried out independently of the gauge fixation. This leads to the formulation ( I 0 ) in terms o f the antlfields. The gauge independence o f the cohomologlcal questions b e h m d renormahzation theory has been particularly stressed in [20,22]. After this brief digression, we can return to the p r o o f o f eq. (10).

3. Removing the non-minimal sector The first step in the p r o o f o f eq. (10) consists in removing the non-minimal sector.

Theorem 1. In each equivalence class o f H* (s), one can find a representative that does not depend on the variables o f the n o n - m i m m a l sector. That is, if sP = O, P E .4, then P = R + sQ' wtth R E /3, sR=O,Q'cA. Proof See [9-11] (the BRST transformation m the non-minimal sector takes the characteristic contractible form sx, = y,, sy, = 0) and [23] for the implementation of locahty. We shall thus assume from now on that P in eq. (10) belongs to the polynomial algebra/3 generated by the variables o f the m i n i m a l sector and their derivatives. In that algebra, we introduce a second grading, called the "antighost number" through antxghA~ = 0,

a n t i g h C a = 0,

antighCa =2,

a n t l g h 0 u = O.

antighAa ~ = 1,

(15)

The splitting ( I ) o f the differential s simply corresponds to the sphttlng according to definite ant~ghost number, ant~ghJ=-l,

antightr = 0 .

(16)

As shown an [ 9 - I I ], the differential J is the K o s z u l Tate differential associated with the gauge-mvariant equations o f m o t i o n D u F aJu" = 0. One has

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H k ( J ) = 0,

fork ~ 0

(17)

both in the algebra o f all functionals [9-11 ] and in the algebra/3 o f local polynomials [23] *~1, while H0 ( J ) is given by the equivalence classes o f polynomials in A~, C a and their derivatives that coincide when DuF aJu" = O.

4. Cohomology of t7 The cohomology o f a - the usual BRST differential - has been computed by various authors [ 2 4 27] m the polynomial algebra generated by the potentials A~, the ghosts C a and their derivatives. One can easdy mclude the antifields Aa ~', Ca and their derivatives as follows. The algebra 13 is the product o f two algebras, /3 = C ® 79, one o f which, namely C, is contractible. The algebra C is the polynomial algebra in the components A~, o f the vector potential, their s y m m e t n z e d derivatives O~u~u2 ukAau) and their a-varmtions [27]. It is contractible because it takes the standard form a x , = y,, cry, = 0. The algebra 79 is the polynomial algebra generated by the components o f the field strengths F~,, their covarlant derivatives Du~Du2...Du~F~,, the components o f the antifields A~ su, their covarmnt derivatives Du~Du2...DukAa Ju, the antifields C~, their covanant derivatives Du~D~2...DukC~ and the ghosts C a (without their derivatives, which are m C). One has

t~ bc)(.A(~ , aca

=

1 t'-,a t'~bf',c ~ " ~ b c " - "~ ,

(18) (19)

where X• stands for Du~u2 ukF~a, Du~ m •kA*.u a and Du~u2 u~C *a (k = 0, 1, 2 . . . . ) and where the internal in&ces are raised with the Kilhng metric. Furthermore, *~l Smctly speaking, the relation (17 ) has been proved, for arbitrary gauge theories, m the algebra of polynomlals in the antlfields, the ghosts and their derivatives wnh coefficients that are smooth functions (not necessarily polynomials) of the original fields and their derivatives. However, In the Yang-Mflls case, the relation (17) is also easdy seen to hold in/3 because the equations of motion are polynomial in the A~ and their derivatives (even hnear in the terms w~th higher (second) derivatives). 37

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(20)

H*(a, 13) = H * ( a , D )

26 August 1993

5. Cohomology of 6 in Vo

since C is contractible. Now, each Z~ belongs to a copy of the adjoint representation of the Lm algebra 9. The algebra V of polynomials in Z~ provides therefore a representation p : 9 ~ V o f g . Since 9 is semi-simple, ~ splits as a direct sum offimte-dimensional irreducible representations,

Let P (Z~) be an invarmnt polynomial that is 5closed and of posture antighost number,

V = )20@ (@k>0Yk),

P = 6T

V0 = @~);0~,

(21)

(the space ofpolynommls o f degree ~< n w~th derivatives up to order k is invarmnt and fimte-dlmenslonal for arbitrary n and k). In (21), V0 stands for the sum of the one-dimensional trivial representations );0~, Le., contains all the mvarmnt polynomials in Z~, while Vk denotes the irreducible n o n - m w a l representations. Note that the triwal representation occurs an infinite number of times (one can form an mfimte number of linearly independent invariant polynommls in the Z~), so that V0 is infinite-dimensional. The differential a is nothing but the coboundary operator for the cohomology H* (9, V) of the Lm algebra 9 in the representatmn F. According to Whitehead theorem, only the lnvariant subspace of the trivial representation contributes to the cohomology and SO

H * ( a ) = H*(9, Vo).

(22)

As it follows from the work of [28], the cohomology H* (9, V0) is the tensor product of 1;0 by the cohomology H* (9) of the Lie algebra 9. H* (9) is lsomorphm to the algebra of the mvarmnt cocycles on 9. Both ~;0 and H* (9) are free graded-commutative algebras, and the most general element of H* (a) = H* (9, Vo) can be taken to have as representative

E

a,j P' (Z']) EJ (Ca),

(23)

8 P = 0,

antighP>0,

PEV0.

(24)

By (17), one has (25)

but there is a priori no guarantee that T belongs also to v0.

Theorem 2. One may choose T xn (25) to be in V0. Proof. The linear operator 6 is defined in the space V of the polynomials in Z] and commutes with the representation p of 9 in V, 6p = p,~.

(26)

Hence ~t maps the mvarlant subspaces );k of (21) on m v a n a n t subspaces. Sxnce Vk is irreducible for k ~ 0, the subspace 81;k lS either ~somorphic to l;k and yields an equtvalent representation, or is equal to 0. Thts ~mphes that ~ X has no component along the subspace 1;0 of the trivial representation if X C Fk, k ¢= 0, i.e. 6Vk n V0 = 0 for k i= 0. Let T = To + T1 + T2 + ... be the decomposition of T along the subspaces of (21). Since 6 T = P belongs to "1~o,it follows that 6 ( ~ k > 0 Tk) = 0, SO that one may choose T = To E V0 m (25). This proves the theorem.

6. Cohomology of s We can now prove the theorem

l,J

where (i) P' (Z~) are a set of independent generators of V0, and (iO EJ(c a) a r e a set of independent generators of H* (G), which are known to be in number equal to the rank of G ("primitive forms"). The mdependence of the generators in cohomology means that xf ~,,,j a,j P' (Z~ ) E J (C a) = a (something), then a,j = 0 for all i, J. That is, if an invanant polynomial of the type (23) is a-exact, then it is identically zero. 38

Theorem 3. Let P be a polynomial in the algebra .4 which is 0 ) BRST-invariant; and (ii) of ghost number zero. Then P = M+sQ,

(27)

where M is a gauge invariant polynomial in the field strengths F~a and their derivatives up to a fimte order, and where Q E -4.

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Proof From theorem l, one has P = R + sQ' where R E/3 and s R = 0. Let us decompose R according to the antighost number,

Going on in the same fashion, one then removes successively Rt_~, R t - 2 , ... up to RI. This yields

R = R'o + s T ' , R = R0+RI

+Rz +... +Rt,

antighR~ = k,

k = 0, 1..... L .

(28)

In virtue of eqs (1) and (16), the equation s R = 0 is equivalent to the chain of equations

P J ( z ~ ) E J (Ca) + a Q L ,

(30)

J where PJ C )2o and E J ( c a) are the primitive forms. By a redefinition Of R L - i if necessary, one can absorb QL in an s-variation. So, let us take QL ---- 0. The dvariation of R t is an lnvariant polynomial of the type (23), d R t = Y~j dP J (Z~J)EJ ( c a ) which, by (29), must be o.-exact. By our analysis of the cohomology of 0., it must thus vanish,

dRL = 0,

(31)

i.e., dP J = 0. By theorem 2, this implies PJ = d T J with T J E )20 and thus RL = t~TL+I, TL+I = Z

TJ (z~)EJ (Ca) '

(32)

J

with

where R6 is an invarlant polynomial in the field strengths and their derivatives. Thus one gets

P = 34 + s Q ,

(35)

(29)

l f L = 0, then R = R0 does not contain the antifields. Since its ghost n u m b e r vanishes, it does not contain the ghosts either. Our analysis of the o.-cohomology implies then that R0 is an invariant polynomial in the field strengths and their derivatives (aRo = 0), which establishes the theorem. So let us assume that L > 0. From o.Rt = 0, one gets, as explained above

RL = Z

(34)

with M = R~ and Q = Q' + T'. This proves the theorem.

o.Ro + dR~ = 0 ..... o.Rt_~ + d R t = 0, o.Rt = 0.

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O T L + 1 ~- O.

RL = sTL+I

Accordingly, (33)

and one can remove RL from R by adding an svariation. Once this is done, R contains only components of antighost numbers 0, 1. . . . up to L - 1.

7. Comments ( 1 ) The above analysis proves that in each equivalence class o f H ° (s), one can find a representative that does not involve the ghosts or the antlfields and that is strictly gauge invariant. This representative, of course, is not unique since there exist invariant polynomials in the field strengths and their derivatives which vanlsh when the equations of motion D u F au" = 0 hold, and which can thus be written as s-variations. Theorem 3 strengthens the results of [9,23] in that (i) the work of [9,23] indicates that the local BRST cohomology in degree zero is isomorphic with the set of on-shell gauge mvariant local polynomials in the field variables and their derivatives for a general gauge theory (with the identification of two such polynomials that coincide on-shell), but does not guarantee that one can find strongly gauge lnvarlant polynomials in each equivalence class; (n) the work of [9,23] shows that the equations s U -- 0, antigh U > 0 imply U = s V for some V but does not guarantee, as done in this letter, that V has a finite expansion in the antighost n u m b e r (V could a priori contain an infinite n u m b e r of terms of arbitrarily high antighost number, and so, not be a polynomial). Although it is the strong version expressed by theorem 3 that has been invoked m renormahzatlon theory, it appears that the isomorphism o f H ° (s) with the set of on-shell gauge invariant polynomials is just sufficient on physical grounds. Indeed, the physical matrix elements of BRST-exact operators (with s given by the sum (1)) vanish and so, these operators are physically irrelevant [10,11 ]. Accordingly, provided the BRST symmetry is not anomalous so that BRST 39

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mvariant operators mix only with BRST mvariant operators, and BRST exact operators mix only with BRST exact ones, then, the mixing is well defined in cohomology. By the isomorphism mentioned above, this means that only gauge-lnvarmnt operators are physically relevant. The isomorphism of H* (s) with the set of gauge m v a r i a n t functlonals admits a formal extension to non-polynomial and non-local operators [9-11]. (2) Theorem 3 can be extended to other values of the ghost number. One shows along the same lines the existence, m each equivalence class of H* (s), of a representative annihilated by a. Also, theorem 2 can be extended to any semi-simple group that may act as global symmetry group (e.g. Lorentz group). (3) Finally, our analysis does not cope with the more complicated question of computmg H ( s = + a [ d), where d is the spaceUme exterior derivative. The cohomology of H ( a [ d) has been the subject of various works ([24,29,25-27] and references therein). It would be of interest to extend those resuits to H ( s I d). We plan to return to this question elsewhere.

Acknowledgement The author is grateful to John Colhns for a question that prompted this work and to G l e n n Barnich, Michel Dubols-Vlolette, Jim Stasheff, Michel Talon, Claudio Teitelbolm and Claude Vmllet for useful conversations. This work has been supported in part by research funds from FNRS and a research contract with the Commission of the European Communities.

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