On the finiteness of the BRS modulo-<i>d</i> cocycles

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Reference

On the finiteness of the BRS modulo- d cocycles

PIGUET, Olivier, SORELLA, Silvio Paolo

Abstract

Ladders of field polynomial differential forms obeying systems of descent equations and corresponding to observables and anomalies of gauge theories are renormalized. They obey renormalized descent equations. Moreover they are shown to have vanishing anomalous dimensions. As an application a simple proof of the non-renormalization theorem for the non-abelian gauge anomaly is given.

PIGUET, Olivier, SORELLA, Silvio Paolo. On the finiteness of the BRS modulo- d cocycles.

Nuclear Physics. B , 1992, vol. 381, no. 1-2, p. 373-393

DOI : 10.1016/0550-3213(92)90652-R

Available at:

http://archive-ouverte.unige.ch/unige:119378

Disclaimer: layout of this document may differ from the published version.

1 / 1

~,

l

Nuclear Physics B 381 (1992) 373-393 North-Holland

NUCLEAR PHYSICS B

On the finiteness of the BRS modulo-d cocycles *

livie-r :Riguet and Silvio P. Sore Ila

Departement de Physique Theorique, niuersite de Geneue, CH-1211 Geneva 4, Switzerland

Received 31 March 1992 Accepted for publication 13 May 1992

Ladders of field polynomial differential forms obeying systems of descent equations and corresponding to observables and anomalies of gauge theories are renormalized. They obey renormalized descent equations. Moreover they are shown to have vanishing anomalous dimen- sions. As an application a simple proof of the non-renormalization theorem for the non-abelian gauge anomaly is given.

1. Introduction

A whole class of observables and anomalies in gauge theories are built by solving systems of descent equations [1] for sequences - "ladders" - of classical field polynomials. This amounts to study the BRS modulo-d cohomology in the space of local field polynomials [2,3]. The nontrivial solutions of the descent equations are uniquely specified by invariant ghost monomials of the general form Tr en, n odd, where e is the usual Faddeev-Popov ghost belonging to the Lie algebra of the gauge group.

The aim of the present paper is twofold. First, we show the perturbative existence and uniqueness of quantum insertions, for any given classical ladder, which fulfil the quantum version of the descent equations. Second, we prove that these quantum insertions are "ultraviolet finite", i.e. that they are characterized by vanishing anomalous dimensions [ 4,5].

The proof of ultraviolet finiteness will be given in a particular gauge, namely the Landau gauge [6]. However, thanks to the gauge invariance of the ghost cocycle Tr en and by means of the extended BRS technique [7], its validity persists in a generic covariant gauge, as shown for a particular case in ref. [5]. The choice of the Landau gauge is the natural one for studying the ultraviolet properties of such ghost monomials. Indeed, as shown in ref. [6], the Landau gauge is characterized by the "ghost equation" which controls the dependence of the theory on the ghost

* Research supported in part by the Swiss National Science Foundation.

0550-3213/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved

374 0. Piguet, S.P. Sorel/a / BRS modulo-d cocycles

field c. It is this equation which implies the vanishing of the anomalous dimension of c and also of all cocycles Tr c^11•

These results were already proved, by using Feynman graph considerations, in the case of the ladder related to the cocycle Tr c³. This was a basic ingredient in the proof of the non-renormalization theorem of the U(l) anomaly [5] in renormal- izable non-abelian gauge theories.

We will give here the proof referring to the ladder n = 5 for simplicity, but in a way suitable for generalization to any n. This case is of particular importance for gauge theories in four-dimensional space-time due to the relation of this ladder with the non-abelian gauge anomaly. As an application of our results we present here a completely algebraic proof of the non-renormalization theorem [8-10] for the latter anomaly.

The paper is organized as follows. In sect. 2 we introduce the cocycle Tr cs in the framework of a general renormalizable gauge theory, and we write down all the functional identities characterizing the properties of the model in the classical approximation. Sect. 3 is devoted to the renormalization of that cocycle. The ultraviolet finiteness of the renormalized Tr cs is proved in sect. 4 by showing that it obeys a Callan-Symanzik equation without anomalous dimension. Sect. 5 ex- tends the proof to the whole renormalized n = 5 ladder, and in particular to the anomaly insertion. In sect. 6 we present the algebraic proof of the non-renormal- ization theorem. A useful proposition on local cohomology is given in appendix A

2. The c<⁵l cocycle

2.1. THE FUNCTIONAL IDENTITIES

We consider here a massless gauge theory in four-dimensional space-time. The gauge group is a compact Lie group G, assumed to be simple for simplicity. Its generators obey the commutation rules

(2.1) The gauge field A~ as well as the Lagrange multiplier field ba and the ghost fields ea,

c;a

belong to the adjoint representation, whereas the matter fields, collectively denoted by </>, belong to some finite unitary representation of G where the generators will also be denoted by Ta *.

* Fields 'P in the adjoint representation will often be written as matrices: 'P = Ta'P". The Killing form is taken to be oab• i.e. the structure constants obey fabcf""" = 8~.

0. Piguet, S.P. Sorel/a / BRS modulo-d cocycles 375

The BRS transformations are

sAµ= -aµc+i[Aµ,

c]

^{= -Dµc,}

1 "{ } • 7

sc= - 2z c, c = -zc-,

sc

= b,

s<f> = -ic<f>. (2.2)

They are nilpotent.

The BRS-invariant gauge fixed classical action in the Landau gauge reads

!Landau=

f

^{d4x ( -}

4~2

^{Ft'v p:1v}

⁺

^2'matter( </>, Dµ</>) +baaµ A:+ caaµDµca). (2.3) We do not specify the part of the gauge-invariant action which depends on the matter field and on its covariant derivative Dµ</> =(aµ - iA)</>, only restricted by the usual power-counting renormalizibility condition.

Let us define

c<Sl = 2-d bcf f cacdcecfcg

40 a bde cfg (2.4)

with dabc the completely symmetric invariant tensor of rank 3. This is a BRS cocycle Cs-invariant but not of the form s( ... )) [l]. The purpose of this and the following section is to show the finiteness of c<⁵l with the help of the ghost equation governing the dependence of the theory on the ghost c. In order to control the renormalization properties of c<⁵l let us couple it to an external field p.

It will turn out to be useful to couple also certain c-monomials of degree 4, 3 and 2 to external fields TJ, w, T and <J. Coupling moreover the BRS variations of Aµ and

</> to the external fields [}IL and Y, we write the external field dependent part of the classical action as

Y _

J

^{d4 ( nµ Aa}

⁺

^{vc A.}

⁺

⁽⁵⁾

⁺

^{1 abf f c d e f}

....,,ext - X HaS µ 1S'+' pc sT/ acd befC C C C

+

₂ ^{1 (} W ^ab

+

T ab) Ca bcc!C C f c d

+

zCT ¹ ab CaCb ) (2.5) with

T/ab = T/ba, wab = wba,

7 ab= _ 7 ba, (Tab= -(Tba. (2.6)

Dimensions and ghost numbers of the different fields are given in table 1.

376

d g

Al-' 1 0

c 0

0. Piguet, S.P. Sorel/a / BRS modulo-d cocycles

c

2 -1

TABLE 1

Dimensions d and ghost numbers g

b 2 0

{21-'

3 -1

p 4 -5

'Y) 4

-4

(J)

4 -3

T

4

-3

(J"

4 -2

BRS invariance of the external field part of the action together with nilpotence is achieved by demanding the following transformation rules (!21-l and Y being kept invariant as usual):

sp = 0,

SYJab = ^2wab, swab= 0,

STab= O"ab, S<Yab = 0. (2.7)

Invariance of the total classical action

!(A, c,

c,

^b,

n,

^{Y, p, TJ, w,} T, <r) =!Landau +!ext (2.8) under the BRS transformations (2.2) and (2.7) is expressed by the Slavnov identity *

5' _ 4X _ _ _ _ ..!. a c - - - - + - - + b a -

(

o! o!

^b

o! o! o! o! o!

3'(....,) -

f

^d

o!2~ BA~ ⁺

^{zf o<rbc Bea BY} o<f> Bea

+wab _ _

o! +

^{~<Yab _ _}

o! )

=0.

OYJab - OTab (2.9)

The action is invariant under the rigid transformations o~ig of the group G:

. " f . ^o!

gf;

^1g! = L,, d⁴x 0~¹g'P_ = 0.

all fields <p O 'P

(2.10)

The Landau gauge condition is expressed by the equation o4

oba = 1-Jl-'A~. (2.11)

* The functional derivative of a functional .9T with respect to a symmetric or antisymmetric tensor field tab is defined through the variation formula 8.9" = f d⁴xi8tab8.9" /Stab·

0. Piguet, S.P. Sorel/a / BRS modulo-d cocycles 377

The antighost equation

(2.12) follows from the Slavnov identity and the gauge condition.

The ghost equation, usually valid in the Landau gauge [6], extends to the present case as

(2.13) where

+

^{1 d be}

+

( 0 0 )

zfab Y/ O(J)dc

07dc

(2.14)

(2.15) At last it is easy to see, using the Jacobi identity obeyed by the structure constants, that the following "7-equation" holds:

3'"",,.!

= f a b c - -

a.!

= 0.

OT be

(2.16) The "linearized" Slavnov operator

(2.17) is nilpotent if the functional

.!

is a solution, not only of the Slavnov identity (2.9) but also of the T-equation (2.16):

3"(

.!)

= 0, (2.18)

378 0. Piguet, S.P. Sore/la / BRS modulo-d cocycles 2.2. THE FUNCTIONAL ALGEBRA

Since BRS nilpotency holds only up to the r-equation (2.16), our first task in sect. 3 will be to prove the validity of the latter to all orders. We thus will be interested, in the following, in the space of functionals y restricted by

9'";,y = 0. (2.19)

In this space the functional operators introduced above obey a non-linear algebra whose relevant part reads

(i) (ii) (iii) (iv) (v) (vi)

(vii) (viii) (ix)

D ( Dy ) _

-S"(y) _Dba -Y'. - -aA =.?Jy

y Dba a a '

Y";,S"(y)=O,

D ( Dy )

- ( :z;

y - L\ ) - Jl - - aA = 0

Dba b b b Dba a '

Y-a(Jlby-L\b) =0, {.?Ja, .?b} =0.

3. Renormalization

(2.20)

We want to show that the functional identities of sect. 2 hold to all orders for the vertex functional

r -

which coincides with the classical action .! (2.8) in the tree graph approximation.

Let us consider generic classical identities

ST;.!= 0. (3.1)

According to the quantum action principle [11,12], the possible breakings of such identities by the radiative corrections are given at their lowest non-vanishing loop

0. Piguet, S.P. Sorel/a / BRS modulo-d cocycles 379

order by local field functionals Ll; with quantum numbers and dimensions pre- scribed by those of the corresponding functional operators:

Y;I' = 11; + 0( h11). (3.2)

The functional operator algebra implies a set of consistency conditions on the Ll's which one has to solve. If the general solution has the form

L1 =

y-J

\;Ji

l l ' (3.3)

then

J

can be absorbed as a counterterm in the action, and the identities (3.1) are renormalizable *.

The gauge fixing condition (2.11) and the antighost equation (2.12) are known to be renormalizable [13] and thus will be assumed to hold:

- - a A _{oba -}

or

_{a' JJ'aI' =} 0.

The same can be assumed for the rigid invariance [14] (see (2.10)):

gf~igI'= 0.

3.1. THE T-EQUATION

(3.4)

(3.5)

The most general quantum breaking of the T-equation (2.16) - of dimension 0 and ghost number 3 - has the form

(f!a

=

tabcdcbcccd Using the first footnote in sect. 2 we can write this as

Hence the T-equation

""' - I G.T

f

^{d4 jdef t g It I}

i;za -

:zu

a X Tef dgh/C C C .

:7;,I'= 0

(3.6)

(3.7)

(3.8) can be assumed to hold to all orders. As a consequence we shall be able to exploit the functional algebra displayed at the end of sect. 2.

* In (3.1) we have omitted a possible classical breaking, i.e. a breaking linear in the quantum fields like in the ghost equation (2.13). If the operator 5j is non-linear, it is its linearized form (cf. (2.17)) which appears in (3.3).

380 0. Piguet, S.P. Sore/la / BRS modulo-d cocycles

3.2. THE GHOST EQUATION

To discuss the renormalization of the ghost equation (2.13) let us write it as

JJaI' = Lla +Ea, (3.9)

where Ea represents the possible breaking induced by the radiative corrections.

Ea is an integrated local functional with dimensions four and ghost number - 1 which, according to the non-linear algebra (2.20) satisfies the conditions

(JR - a

obc = 0,

.!if~igEb ⁼ -fabcsc,

JJaEb

+

^JJbEa = 0.

ffecEa = 0,

.:TaEb =

o,

(3.10) (3.11) (3.12) Eq. (3.10) implies that Ea is b-independent and depends on the fields

c

and [},JJ-

only through the combination

Yµ, = aµ,c

+

nµ,. (3.13)

It then follows that ^Ea can be parametrized as

';::: - f

^d4

^(t

b c d e

+

be d e f

+

^{be d e}

- a - X abcdePC c C c mabcdefT/ c c c rabcdew C c

+

_nabcde'T ^{be d e} _{c c}

+

qabcd<Y ^{be d} c

+

PabcY Eiµ, ^{bµ, ,;c}

+

u ^{YT ,/,.)} a'+' , (3.14) where, from eq. (3.11), (t, m, r, n, q, p) are invariant tensors in the adjoint representation, and

Since

and

fmbcnabcde = 0.

f

^{d4x p} abc ^ybµ,Ac µ, ^{= J1} a

f

^{d4x cmp} mbc ^ybµ,Ac µ,

f

^d4x YTa<f> =:Ya

f

^d4x cmYTm<f>,

it follows that the nontrivial part of Ea reduces to

';::: - f

^d4

^(t

b c d e

+

be d e f

+

^{be d e}

- a - X abcdePC c c C mabcdefT/ c c c rabcdew c c

+

_nabcde'T ^{be d e} _{c c}

+

qabcd(J" ^{be d)} c '

(3.15)

(3.16)

(3.17)

(3.18)

0. Piguet, S.P. Sorel/a / BRS modulo-d cocycles 381

i.e. Sa depends only on the variables (c, p, 71, w, r, CT) and does not contain any space-time derivative. To study the condition (3.12) on the local functional space (3.18) we introduce a dimensionless space-time independent parameter

ga

with zero ghost number and we consider the operator

(3.19) which, due to the relation (ix) of eq. (2.20), turns out to be nilpotent:

g2=0. (3.20)

The introduction of the operator !:» allows us to transform the equation (3.12) into a cohomology problem. Indeed, it is easy to check that if the general solution of the equation

!:»X=O, (3.21)

where X is an integrated local functional in the variable (g, c, p, 71, w, r, CT) with dimensions 4 and ghost number -1, is of the form

X=!:»X, (3.22)

then the general solution of eq. (3.12) reads

(3.23) which implies the absence of anomalies for the ghost equation (2.13). The most general form for the X-space is given by

(3.24) where the invariant tensors L, R, S, U, V are power series in

g.

To characterize the cohomology of !:» we introduce the filtering operator

1 ab ab 1 ab ab

( 8 8 ) ( 8 8 ))

+ 2 77

071^ab + ^{w owab} + 2 ⁷ 07^ab + o- ^{oCTab '} (3.25) according to which the operator !:» decomposes as

(3.26)

382 0. Piguet, S.P. Sore/la / BRS modulo-d cocycles

where

(3.27) It is now apparent that, due to the absence of space-time derivatives in the general expression (3.24), the cohomology of -9'⁰ vanishes and, since the cohomology of -9' is isomorphic to a subspace of the cohomology of -9'⁰ [15], it follows that also -9' has trivial cohomology on X. This proves that the ghost equation (2.13) can be implemented to all orders of perturbation theory.

3.3. THE SLA VNOV IDENTITY

Having renormalized the r and the ghost equations, let us discuss the quantum extension of the Slavnov identity (2.9). From the non-linear algebra (2.20) one has that the breaking of the Slavnov identity

(3.28) has to satisfy the conditions

(3.29)

Y:L1=0, (3.30)

3"~L1 = 0, (3.31)

where 3"~ is the linearized operator defined in (2.17) and L1 is an integrated local polynomial in the fields with ghost number 1 and dimension 4. The index n appearing in (3.28) denotes the lowest non-vanishing order for L1.

As in subsect. 3.2, the conditions (3.29) imply that L1 is b-independent and that the fields

c

and Dµ enter only in the combination 'Yµ' eq. (3.13), so that L1 can be parametrized as

A-""(

A

r1.)+jd4(//

a b c d e f + , - ; ; , a b c d e f g

LI - J 4 f C, f.L' 'I' X ./t'tabcdefPC C C C C C =abcdefgT/ C C CC C

(3.32) where L, gf, 3", 2", 'Y and .Y are invariant tensors in the adjoint representation

0. Piguet, S.P. Sore/la / BRS modulo-d cocycles 383

and .W(c, A,,,, </J) depends only on the ghost c, on the gauge field A,,, and on the matter fields </J. It is easy to see that the ghost condition in (3.30) implies that

//!=9? =.97= V= '//=5!'= 0,

a= f3 = ,.\ = 0,

f

d x - = 0 ⁴

^o.w

oc⁰ ,

(3.33) (3.34) (3.35) from which it follows that .W depends only on the space-time derivatives of c.

Finally, the Slavnov condition (3.31) reads

(

0 0 0 )

f

^d4xTr - ( D c ) - - i c²--ic<(J- .W=O.

"' oA,,, oc o</J (3.36)

The most general solution of eq. (3.36) has been given in ref. [14] and coincides, modulo a 5'1: co-boundary, with the well-known expression for the gauge anomaly [16]:

,-.4 - µ,vpuj d4

a

a(d

a

AbAc - .l"" AbAc Ad)

..)4( - C: X µ, C abc v p u 12= a bed v p <J" ' (3.37)

with

f:gabcd = dn abfncd

+

dn acfndb

+

dn adfnbc · (3.38) To summarize this section one has that the vertex functional I'

I'=2:

+

O(h), (3.39)

obeys

(i) the gauge-fixing condition and the anti-ghost equation ob

or

=aAa,

a

~I'=O·

a ' (3.40)

(ii) the rigid gauge invariance and the T-equation

m;igr= o,

Y-J=O; (3.41)

(iii) the ghost equation

:§'aI' = Lla; (3.42)

384 0. Piguet, S.P. Sorel/a / BRS modulo-d cocycles

(iv) the anomalous Slavnov identity

Y'(I') = hnrssr

+

O(hn+^1), (3 .43) where r is a coefficient directly computable in terms of Feynman diagrams.

Moreover, as we shall see in the next sections, the coefficient r turns out to satisfy a non-renormalization theorem [8-10].

4. The Callan-Symanzik equation

This section is devoted to the derivation of the Callan-Symanzik equation obeyed by the vertex functional

r.

Eqs. (3.40)-(3.43) show that, besides the gauge-fixing condition, the anti-ghost equation and the T-equation, the functional I' obeys the ghost equation (3.42).

This equation governs the dependence of I' on the ghost field c and will impose quite strong constraints on the Callan-Symanzik equation; it will imply the absence of anomalous dimensions for the ghost and for the cocycle c<⁵l of eq. (2.4).

Moreover, the presence of the gauge anomaly at the order hn in the Slavnov identity (3.43) does not allow for a Callan-Symanzik equation which is invariant to all orders of perturbation theory: one expects a Slavnov-invariant Callan-Symanzik equation up to the order hn -^1.

However, it will be shown that the Callan-Symanzik equation will extend to the order hn, which is the order of the gauge anomaly; this property will be crucial for the non-renormalization theorem of the anomaly coefficient r in eq. (3.43) [8-10].

To characterize the scaling properties of the model we look for a basis of insertions which are invariant under the set of equations satisfied by the functional I': i.e. we look at the most general integrated local polynomial in the fields ..f 1₀c with dimension four and zero ghost number which satisfies the stability conditions [6,13]:

o..f loc = 0, obc

f7l1rigf - 0

Jla ,._,lac - '

- f - .Wc-"'loc - 0,

CT Y - 0

u a,...,loc - '

Y.s..floc = 0.

( 4.1)

.'!fa..f loc = 0, ( 4.2) ( 4.3) Proceeding as in the previous sections, it is not so difficult to show that the most general expression for ..f lac reads

z ..f loc = -

4; ₂

J

^d4

^x

^{F:v F:v}

⁺

^Y_s

J

^d4

^x(

ZAf'aµ.Aaµ. - Zq,Y</>)

+

"[,zi..fi( </>), ( 4.4)

l

0. Piguet, S.P. Sorel/a / BRS modulo-d cocycles 385

where zv zA, zq, and Z; are arbitrary parameters and the

i;

are all possible invariant matter self-interactions. Expression (4.4) represents the most general invariant counterterm. Note that there is no term in the external fields p, YJ, w, T

and cr: this is due to the last of conditions (4.2). Thus one sees that the conditions (3.40)-(3.43) determine the model up to a renormalization of the gauge coupling constant g (given by zg), of the matter self-coupling (given by z) and to a redefinition of the gauge field amplitude (zA) and of the matter fields (zq,).

Expression (4.4) can be rewritten as

( 4.5) where .! is the classical action (2.8) and Jflh Jf/q, are the Slavnov-invariant counting operators defined by

( 4.6)

( 4.7) and A; are the matter self-coupling constants.

It is apparent from eq. (4.5) that the set

( ar

(Jg,

ar

( 4.8)

can be taken as a basis for the insertions of dimension four and ghost number zero which are Slavnov invariant and which are compatible with the gauge-fixing condition and the anti-ghost equation (3.40) as well as with the T and the ghost equations (3.41) and (3.42).

The insertion defined by BT /Bµ,, where µ, is the normalization point, obeys to the same conditions. Expanding it in the basis (4.8) we get then the Callan-Sym- anzik equation,

( 4.9) where Ll~ is an integrated local polynomial corresponding to the lack of the Slavnov invariance at the order li¹¹ according to eq. (3.43).

Notice that the validity of the ghost equation (3.42) has led to the absence of anomalous dimensions for the ghost field c.

386 0. Piguet, S.P. Sore/la / BRS modulo-d cocycles

From the algebraic relation

( 4.10) it follows that, applying the Callan-Symanzik operator to the anomalous Slavnov identity (3.43) and retaining the lowest order in Ii, one gets the equation

(4.11) which, due to the fact that r is a dimensionless coefficient, implies that

( 4.12) This condition tells us that the local breaking Ll~, in spite of the presence of the gauge anomaly, is Slavnov invariant and that it can be expanded in the basis (4.8).

This amounts to extend the Callan-Symanzik equation in a Slavnov-invariant way up to the order fin, i.e.

( 4.13) with Ll~ + ¹ local.

Deriving this equation with respect to the source p(x) we get

(4.14) which expresses the finiteness, i.e. the vanishing of its anomalous dimension, of the ghost cocycle (2.4) whose quantum extension is defined by

[ cCS) -I'] = -

or

.

op

5. The descent equations

( 4.15)

The purpose of this section is to show that the gauge anomaly operator (3.37), here written in terms of differential forms *,

.W'(A, c) =

J

^w~,

(5.1) w~ = ~Tr(dc(A dA

+

dA A - iA³) ),

* The symbole Tr means the normalized trace Tr(TaTb) = oa_1,. The symmetric tensor dabc in (2.4) is defined by dabc = 1Tr(T0{Th, Tc}). UJ% indicates a p-form of ghost number q, the ghost number of c being equal to I. The gauge 1-form is A= A,,. dx", its BRS transformation reads sA =de - i[c, A].

By convention dUJ = dx" a,,.(JJ for any form UJ, and s anticommutes with the exterior derivative d.

Brackets [ ] denote graded commutators.

0. Piguet, S.P. Sore/la / BRS modulo-d cocycles 387

can be renormalized in such a way that it obeys a Callan-Symanzik equation with vanishing anomalous dimension. This will follow from the vanishing of the anoma- lous dimension of the cocycle insertion (4.15) proved in sect. 4.

Throughout this section we shall assume the validity of the gauge condition and antighost equation (3.40), and of the rigid invariance. This means that we shall deal with functionals of A, </J, c, y (see eq. (3.13)), Y and u. u is the external field coupled to the BRS variation of c, see eq. (5.8) below.

5.1. RENORMALIZATION OF THE ANOMALY OPERATOR

The starting point is the relation of the ghost cocycle (2.4) to the gauge anomaly through the descent equations [1]

swg_q+dw4:'.:~=0, q=l, ... ,4,

sw6 = 0, (5.2)

where

w~ = -Tr( ( dc)²

A),

w~ = Tr( (de )²

c),

(5.3)

w⁴ = _ li Tr(dc c³)

1 2 ' _ws _ 1 T ( s)

0 - - 10 r c . We have rewritten cC⁵l in the matrix notation as

cC5)

=

w6. _{( 5 .4)}

What we need is a ladder of operator insertions {.Og _q} which is a quantum extension of the classical ladder {wg_q} and obeys the quantum descent equations - valid to the same order as the Slavnov identity (3.43) since the nilpotency of Yr is broken by the anomaly:

SCrDg_q

+

dD4:'.:~ = 0( hn), q = 1, ... , 4,

(5.5) SCrD6 = 0( hn).

The renormalized anomaly operator insertion will be defined as

SifR=fD!

^(5.6)

and by

D6

^{= [ cC5> ·I']} (5.7)

which is the finite insertion constructed in sect. 4.

388 0. Piguet, S.P. Sorel/a / BRS modulo-d cocycles

In order to construct the renormalized insertions D~ -q we introduce external fields u;!._1, with u;:!.₁ a (q - 1)-form of ghost number -q, singlet of the gauge group [5]. We therefore start now with the classical action

.! =.!Landau+

J

^d4x(Tr(

^y1LsAµ,

⁺

^Ys<P

^{+ crsc)}

"<"5 -q q )

+4q~1uq-1Ws-q ' (5.8)

with .!Landau given by (2.3). We have discarded the external fields used in sect. 2 for controlling the finiteness of c<⁵l, but we have introduced the external field <r

coupled to the BRS variation of c. This action is BRS invariant, the external fields u transforming as

su0^{1 = 0,}

SU-qr - -du-q+I

q~ q~2 ' q = 2, ... ,5.

The classical Slavnov identity now reads

( (

o.! o.! o.! o.! o.! ) o.! o.!

Y(.!)=jd

⁴x Tr - - + - - + b -

+ - -

[j[}IL

oAµ, ocr oc oc oY o<P

5 1

o! ) o

= '

+ L suCf_~ ouCf_~

^I

q~2

(5.9)

(5.10)

with a corresponding nilpotent linearized form (cf. (2.17)). The rigid Ward identity (2.10), the gauge condition (2.11) and the anti-ghost equation (2.12) are left unchanged and will again be assumed to hold to all orders for the vertex functional

r.

The construction of a ladder of insertions fulfilling the quantum descent equations (5.5) will be achieved by showing the validity of the Slavnov identity (5.10) (anomalous at order n):

S"(I') =hnrsif+O(h¹¹+¹). (5.11)

We thus have to show the triviality of the cohomology of S"₂ in the sector of ghost number 1 for the u-dependent integrated local functionals

A=jxl.

^(5.12)

The condition Y₂L1 = 0 and the triviality [3,17] of the cohomology of d mean

0. Piguet, S.P. Sore/la / BRS modulo-d cocycles 389 that the 4-form

xl

belongs to a ladder of forms obeying descent equations similar to (5.2):

r.o q

+

d q + I - 0 u ~Xs-q X4-q - ,

Y~x6 = 0.

q = 1, ... ,4,

(5.13)

We have first to solve the last of these equations, which is a problem of local cohomology (see definition A.2 in appendix A). Then we have to solve the remaining equations, for increasing form degree, which again is a problem of local cohomology.

In the absence of the external fields u the local cohomology depends only on the invariants Tr en and on those made with the Yang-Mills curvature F and its derivatives [15]. This uniquely leads to the ladder (5.2) and to the chiral anomaly (5.1).

It is shown in appendix A that the dependence of the cohomology on the external fields u occurs only through the zero-form u

0

^1. The most general u-dependent expression for

x6

in the local cohomology would be the superposition of the monomials u

0

^{1 Tr c6 and u}

0

^{1 (Tr c3)2,} but they vanish due to the cyclicity of the trace and the anti-commutativity of the ghost fields. This proves the absence of u-dependent anomalies, hence of any obstruction to the construction of the renormalized ladder (5.5).

5.2. CALLAN-SYMANZIK EQUATION FOR THE ANOMALY

In order to prove the vanishing of the anomalous dimension for the renormal- ized anomaly operator defined in subsect. 5.1, (see (5.6)), we have to repeat in presence of the u-fields the construction of the Callan-Symanzik equation per- formed in sect. 4, and to use the fact that the c<⁵l insertion (5.7) has no anomalous dimension.

The basis of classical invariant insertions used in eq. (4.4) must be completed by adding to it all BRS invariants of the form

(5.14) where

x2

is a 4-form of zero ghost number, depending on the u's. Like in the case of eqs. (5.12) and (5.13), the latter belongs to a ladder obeying the descent equations

07 q +d q+1_0

u ~X4-q X3-q - , q = 0, ... ,3,

(5.15)

390 0. Piguet, S.P. Sorel/a / BRS modulo-d cocycles

The most general non trivial expression for

xci

depending only on u0 ^{1 and on c} according to the proposition of appendix A, is u0¹ Tr c⁵. It leads to the expres- sion, unique modulo Y~:

J

^x2=

J ^L

5

^u-;;:!. 1 w~-q=.Afu!,

q=l

(5.16)

with the u-counting operator defined by

(5.17) The second equality in (5.16) follows from the observation that its left-hand side consists just of the u-dependent terms of the action (5.8). The most general form for the classical invariant L1 thus is a linear superposition of (5.16) and of the

Y~-variation of an arbitrary local field polynomial. The quantum basis (4.8) can accordingly be completed by adding to it the insertion .AfuI' and the Yr-variation of an arbitrary insertion of ghost number -1. This implies that the Callan-Sym- anzik equation in presence of the fields u will take the form

(5.18) where

J

is some u-dependent insertion of ghost number -1; '?? is the Callan- Symanzik operator defined in (4.9). The fact that the anomaly starts to produce effects at order n

+

1 only follows from the same argument as the one used in sect.

4.

Differentiation of (5.18) with respect to

u4

⁵ and use of the anti-commutativity of this derivative with the linearized Slavnov operator Yr, yields the Callan-Sym- anzik equation for the insertion of the cocycle ^cC5l with anomalous dimension Yu:

The last equality follows from the fact that the most general expression for the u4⁵-dependent part of

J

reads

u4

⁵ Tr c4, which identically vanishes. But since the insertion ^cC5l was shown to have zero anomalous dimension, we conclude that

Yu= 0. (5.20)

Differentiating now (5.18) with respect to u

0

¹ and integrating over space-time we get the Callan-Symanzik equation

( 5 .21)

0. Piguet, S.P. Sore/la / BRS modulo-d cocycles 391

for the anomaly insertion, without anomalous dimension as announced, up to an irrelevant Y"rvariation following from the right-hand side of eq. (5.18) and from the anti-commutativity of

fo/ou

₀^{1 with} Y"r·

6. The nonrenormalization theorem of the gauge anomaly

As an application of the finiteness properties displayed by the quantum inser- tions {D~_q} of (5.5), let us discuss the non-renormalization theorem of the gauge anomaly.

Following ref. [10], we can extend the anomalous Slavnov identity (3.43) to the order tzn + ¹ as

Y"(I') = tznrNR

+

tzn+l!)J

+

O(tzn+2), _(6.1) where .WR is the renormalized anomaly operator insertion defined in (5.6) and 9J is an integrated local functional of ultraviolet dimension 4 and ghost number 1.

Applying the Callan-Symanzik operator to both sides of eq. (6.1) and using the Callan-Symanzik equation without anomalous dimension (5.21) for Jlf R one gets, to lowest order (i.e. order n

+

1) in tz, the equation

/3<ll_

+

'°'{3\ll_ Jlf

=

µ,- +Y"'"Ll (

ar ar ) a!JJ _

g a g '-;' ' a>..; aµ, ~ ' ^(6.2)

where Jlf is the gauge anomaly expression (5.1), f3j/l and {3)1l are respectively the one-loop beta functions for the gauge and the self matter couplings and

2f

is a local integrated functional.

Taking into account that 9J is homogeneous of degree zero in the mass parameters [10], i.e.

a!JJ ( 6.3)

µ,- = 0, aµ,

and that the gauge anomaly Jlf cannot be written as a local Y"_s-variation one has the condition

ar ar

/3(1)_

+

"/3(1)_ = 0

g ag L- ' a>... '

l l

( 6.4) which, in the generic case {3~¹l =!= 0 and f3)1l =!= 0, implies [10] that if the coefficient r vanishes at one-loop order it will vanish to all orders.

We wish to thank C. Becchi, A Blasi, R. Collina and R. Stora for useful discussions.

392 0. Piguet, S.P. Sorel/a / BRS modulo-d cocycles

Appendix A LOCAL COHOMOLOGY FOR COMPLETE LADDER FIELDS

Let

er

be the space of non-integrated local field functionals, the fields being defined on some differential manifold, and let us assume a coboundary operator

o

acting on

er,

nilpotent and anti-commuting with the exterior derivative d.

Definition A. I.

A "complete ladder field" on a D-dimensional differentiable manifold is a set

yCQl = {u~-p Ip= 0, ... , D} of D

+

1 differentiable forms *, where Q is some fixed algebraic integer. The co boundary operator

o

acts on the u 's as

Definition A.2.

OU~= 0,

" Q-p d Q-p+I 1 D

uUP = - Up-1 , p+ , ... , .

(A.1)

The "local cohomology" space is the set of equivalence classes modulo-a of solutions tff E

er

of the equation

o&=O. (A.2)

Proposition A.I.

The local cohomology space depends on the complete ladder field y<Ql only through its element u~, not derivated.

Proof

The proof is a slight generalization of the one given in ref. [3]. The space of local functionals we have to consider consists of all the polynomials in the symmetric and antisymmetric derivatives of the form components uff.~.~·J.tp]:

S^{Q-p -}

a a

UQ-p p - 1 D· r:;,, 0

J.l.[···J.l.p•v1 ... v , . - v 1 · · · (v,. J.Li) ... µp' - , . . . , ' 7 '

AQ-p =o ... a uQ-p , p=l, ... ,D;r~O,

µ.l···f..Lp•vl···vr v 1 [v,. µi] ... µ.P (A.3)

AQ v1 ... v,. =ov ... avuQ, 1 r r~ l,

*

uz

is a form of degree ^p and ghost number ^q.

O. Piguet, S.P. Sorel/a / BRS modulo-d cocycles 393

The action of 8 on these derivatives reads

p=l, ... ,D;r;;>O, (A.4)

OUQ = 0.

One sees that all fields and their derivatives - excepted the non-derivated 0-form

uQ - are displayed in BRS doublets, i.e. in couples (U, V = 8U). It is known [3,18]

that the cohomology cannot depend on such couples. This ends the proof of the proposition.

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