ESI The Erwin Schrodinger International Boltzmanngasse 9Institute for Mathematical Physics A-1090 Wien, Austria

Quantum Master Equation for Yang–Mills Theoryin the Exact Renormalization Group

Yuji Igarashi

Vienna, Preprint ESI 1998 (2008) January 23, 2008

Supported by the Austrian Federal Ministry of Education, Science and CultureAvailable via http://www.esi.ac.at

Quantum Master Equation for Yang-Mills Theory

in the Exact Renormalization Group∗

Yuji Igarashi

Faculty of Education, Niigata UniversityNiigata, 95-2181, JAPAN

Abstract

We discuss a general functional method for construction of the Quan-

tum Master Equation in the Batalin-Vilkovisky antifield formalism for

gauge theories in the exact renormalization group. This approach makes

it possible to provide a non-trivial realization of BRS symmetry for QED

and Yang-Mills theory even in the presence of momentum cutoffs.@ The

resulting BRS transformations depend on quantum master actions charac-

terized by a mixture of the fields and antifields. Reduction of the Quantum

Master Equation to the Ward-Takahashi identity is given.

1 Introduction

One of the most important subjects in exact renormalization group (ERG)[1-6]is to realize gauge symmetries, which are naively incompatible with the regu-larization scheme with momentum cutoffs used in the ERG. See, e.g. [7-13] forrecent reviews on the ERG or Functional Renormalization Group. There aretypically two different kinds of approaches to this problem: One is to construct amanifestly gauge invariant regularization scheme1 . Another is to use some iden-tities for Green functions. These are mostly written for the Legendre effectiveaction, Γ, the generating functional of 1PI part of the connected cutoff Greenfunctions. The identity for Γ, called as the modified Slavnov-Taylor identity[15][16], takes the form of the Zinn-Justin equation plus some contribution fromsymmetry breaking terms. There have been a lot of contributions which discussand use this identity. See e.g. [11][12] and references therein. The presence ofthese additional terms means that the standard gauge symmetries are obviouslylost in cutoff theories. However, the modified form of the Slavnov-Taylor iden-tity may not exclude the possibility that exact gauge symmetries are realized in

∗Based on joint work with Katsumi Itoh (Faculty of Education, Niigata University) andHidenori Sonoda (Physics Department, Kobe University).

1See, e.g. [14] for this approach.

1

a nontrivial way. Actually, in lattice gauge theories, exact chiral symmetry isformulated relying on some kind of the Ward-Takahashi (WT) identity called asthe Ginsparg-Wilson relation [17]. This may be the prototype of regularizationdependent realization of symmetries. The Ginsparg-Wilson relation describeshow the standard chiral symmetry installed in the continuum limit of vanishinglattice constant a → 0 is preserved while undergoing deformation due to latticeregularization with finite a. One expects that gauge symmetries in ERG can beformulated in parallel with lattice chiral symmetry.

A first step towards finding exact symmetries in cutoff theories within theframework of perturbation theory was given by Becchi in his pioneering [18].He gave a WT identity for the Wilson action S in Yang-Mills theory with aUV and an IR cutoff. The use of the Wilson action, the generating functionalof the cutoff connected Green functions, is suitable for symmetry argument.Here we wish to develop more general method for formulating regularizationdependent symmetries including gauge symmetries in the ERG. This is basedon extensive use of the Batalin-Vilkovisky (BV) antifield formalism [19][20],which is recognized to be the most general and powerful method for describingsymmetries. In the BV formalism, the Quantum Master Equation (QME) signalsthe presence of any local as well as global symmetry.

In ERG, we have shown[21] that if the QME Σ[φ, φ] = 0 holds for a cutoffremoved theory, this is also the case Σ[Φ, Φ] = 0 for the Wilson action withan IR cutoff. Our general argument has indicated the presence of exact BRSsymmetries. They are realized in a nontrivial way even for gauge theories witha momentum cutoff.

It is interesting to note, in this connection, that the Ginsparg-Wilson rela-tion, the key relation in formulating the chiral symmetry, is nothing but theQME, as shown in [22]. We have discussed how chiral symmetry is realized ina self-interacting fermionic system by solving the QME which generalizes thestandard Ginsparg-Wilson relation.

As for gauge theories, the WT identity for QED suitable for study of BRSsymmetry has was recently given by Sonoda in [23]. We have shown that thisidentity can be lifted to the QME [24]. It is found that the BRS transformationsin cutoff QED generally depend on the Wilson action and so become nonlineareven for QED. It is important to take account of the Jacobian factor associ-ated with the transformations. These aspects are automatically built in theQME. A more direct derivation of the QME for QED has been given in [25].An extension of the formulation for QED given in [23] to Yang-Mills theory hasbeen discussed in [26]. Here, we wish to generalize this method to Yang-Millstheory with two improvements: First, we use the equations of motion for ghostfields. This implies that the antighosts and anti-gauge fields are not indepen-dent variables. They are mixed and appear only with a specific combination inour master action. Second, we introduce a UV cutoff to make composite oper-ators appeared in the BRS transform well-defined. In the BV formalism, theBRS transformation for a field is given by the derivative of the action w.r.t itsantifield. Therefore, our central task is to determine the antifield dependencein the Wilson master action. We will give a general formula for construction of

2

the Wilson master action from the one without the antifields. The latter actionsatisfies the WT identity obtained from the QME by eliminating the antifields.

This paper is organized as follows. We first give a brief introduction of theBV formalism, and discuss a functional method for derivation of the QME incutoff theory. Section 3 describes general structure of the Wilson master action.We apply our results to QED and Yang-Mills theory in section 4, and discussreduction of the QME to WT in section 5. The final section is devoted tosummary and discussion.

2 A path-integral derivation of the QME in cut-

off theory

2.1 The Batalin-Vilkovisky antifield formalism

Let us consider a gauge fixed action S0 [φ] of a generic renormalizable gaugetheory in 4-dimensional Euclidean space. It is a functional of gauge and matterfields as well as ghosts, antighosts and B-fields, that are collectively denotedby φA. The index A represents the Lorentz indices of vector fields, the spinorindices of the fermions, and/or indices distinguishing different types of genericfields. The Grassmann parity for φA is expressed as ǫ(φA) = ǫA, so that ǫA = 0if the field φA is Grassmann even (bosonic) and ǫA = 1 if it is Grassmann odd(fermionic).

We assume that S0[φ] is invariant under BRS transformation δφA:

δS0 =∂rS0

∂φAδφA = 0 , (1)

and define an extended action

S[φ, φ∗] = S0[φ] + φ∗Aδφ

A , (2)

introducing antifileds (AF) φ∗A for fields φA. The AF φ∗

A have1) opposite Grassmann parity to that of φA, so that ǫ(φ∗

A) = ǫ(φA) + 1.2) carrying ghost number −gh(φA) − 1.

In the space of φA and φ∗A, one defines a canonical structure by introducing

antibracket: For any variables X and Y

(X, Y ) =∂rX

∂φA∂lY

∂φ∗A

−∂rX

∂φ∗A

∂lY

∂φA, (3)

where ∂r(l)/∂ denotes a derivative from right (left). Fields φA and their AF φ∗A

are canonical conjugates: (φA, φ∗B) = δAB, (φA, φB) = (φ∗

A, φ∗B) = 0.

Then, BRS transformation of X defined by δX = (X, S) , so that classi-cal BRS invariance of the extended action is expressed as the classical masterequation:

(S, S) = 0 . (4)

3

More generally, even for (S, S)/2 6= 0, BRS invariance is realized at quantumlevel, if the Jacobian factor associated with δφA = ∂lS/∂φ∗

A cancels it:

Σ[φ, φ∗] ≡∂rS

∂φA∂lS

∂φ∗A

−∂r

∂φAδφA =

1

2(S, S) − ∆S = 0 , (5)

where

∆ ≡ (−)ǫ(A)+1 ∂r

∂φA∂r

∂φ∗A

. (6)

The operator Σ[φ, φ∗] is called as the Quantum Master Operator, and the con-dition Σ = 0 as the Quantum Master Equation (QME).

2.2 The QME for cutoff theory

In order to regularize the theory, we introduce an IR momentum cutoff Λ anda UV cutoff Λ0 > Λ through a positive function that behaves as

K( p2

Λ2

)

∼

{

1 (p2 < Λ2)0 (p2 > Λ2)

. (7)

We use two functions K(p) ≡ K(p2/Λ2) and K0(p) ≡ K(p2/Λ20).

Let us consider the generating functional in the presence of sources JA:

Zφ[J ] =

∫

DφDφ∗ΠAδ(φ∗A) exp (−S[φ, φ∗ : Λ0] + J · φ)

=

∫

Dφ∗ΠAδ(φ∗A)Zφ[J, φ

∗] , (8)

where the bare action S is defined at the UV scale Λ0, and assumed to bedecomposed into the kinetic and interaction terms

S[φ, φ∗ : Λ0] =1

2φ ·K−1

0 D · φ+ Sint[φ, φ∗ : Λ0] , (9)

where

J · φ =

∫

p

JA(−p)φA(p),

φ ·K−10 D · φ =

∫

p

φA(−p)K−10 (p)DAB (p)φB(p). (10)

In this paper we use the gauge-fixed basis for the AF. In this basis, the AFremain intact until the final stage of functional integral. We decompose theoriginal fields φA with the propagator K0(p) (DAB(p))

−1into two classes of

fields: the IR fields ΦA with the propagator K(p) (DAB(p))−1

, and the UV

4

fields χA with (K0(p)−K(p)) (DAB(p))−1

. To this end, we substitute a gaussianintegral over new fields θA

∫

Dθ exp−1

2

(

θ− J K0(K0 −K)D−1)

·D

K0K(K0 −K)

·(

θ − (−)ǫ(J)D−1K0(K0 −K)J)

= const (11)

into the partition function Z[J, φ∗] in (8), and perform a canonical change ofvariables from {φ, φ∗, θ, θ∗} to new variables {Φ, Φ∗, χ, χ∗} by

φA = ΦA + χA, θA = (K0 −K)ΦA −KχA

φ∗A = K−1

0 [KΦ∗A + (K0 −K)χ∗

A] , θ∗A = Φ∗A − χ∗

A . (12)

Then, we obtain

Zφ[J, φ∗] = NJ

∫

DΦDχ exp−

(

1

2Φ ·K−1D ·Φ −K0K

−1 J · Φ

+1

2χ · (K0 −K)−1D · χ + Sint[Φ + χ, φ∗ : Λ0]

)

, (13)

where

NJ ≡ exp1

2(−)ǫA+1JAK0K

−1(K0 −K)(

D−1)AB

JB. (14)

For the AF, we may take a gauge choice χ∗A = 0, which gives

φ∗A = K−1

0 K Φ∗A. (15)

We will see shortly why this relation is needed. With this gauge choice, wedefine the Wilson action by

S[Φ, Φ∗ : Λ] ≡1

2Φ ·K−1D ·Φ + Sint[Φ, Φ∗ : Λ] , (16)

where SI is defined by

exp−Sint [Φ, Φ∗ : Λ] ≡

∫

Dχ exp−(1

2χ · (K0 −K)−1D · χ

+Sint[Φ + χ,KΦ∗ : Λ0])

. (17)

The partition function for {Φ, Φ∗},

ZΦ[K0K−1J, Φ∗] =

∫

DΦ exp(

−S[Φ, Φ∗ : Λ] +K0K−1J ·Φ

)

, (18)

is related to that for {φ, φ∗} by

Zφ[J, φ∗] = NJZΦ[K0K

−1J, Φ∗] . (19)

5

The partition function Zφ does not depend on the IR cutoff Λ so that∂Zφ/∂t = 0, where t = log(Λ/µ). This yields the Polchinski flow equation:

∂tS = −

∫

p

[

ΦA(p)(

K−1K)

(p)∂lS

∂ΦA(p)− Φ∗

A(p)(

K−1K)

(p)∂lS

∂Φ∗A(p)

]

(20)

+1

2

∫

p

[

∂rS

∂ΦA(p)

(

KD−1(p))AB ∂lS

∂ΦB(−p)− (−)ǫA

(

KD−1(p))AB ∂l∂rS

∂ΦB(−p)∂ΦA(p)

]

.

We shall not discuss RG flows in this paper, and now turn to show how theBRS symmetry installed in the original φ theory is preserved in the Φ theory,while affected by the IR cutoff. For the UV action S[φ, φ∗ : Λ0], the QuantumMaster Operator is given by

Σ[φ, φ∗ : Λ0] ≡ exp(S)∆φ exp(−S) =1

2(S, S)φ − ∆φS. (21)

We consider the functional average of Σ[φ, φ∗ : Λ0] in the presence of the sources

〈Σ[φ, φ∗ : Λ0]〉φ, J =

∫

Dφ Σ[φ, φ∗ : Λ0] exp (−S[φ] + J · φ)

= (−)ǫAJA∂r

∂φ∗A

Zφ[J, φ∗]

= NJ(−)ǫAK0K−1 JA

∂r

∂Φ∗A

ZΦ[K0K−1J, Φ∗]

= NJ 〈Σ[Φ, Φ∗ : Λ]〉Φ, K0K−1J , (22)

where

Σ[Φ, Φ∗ : Λ] =1

2(S, S)Φ − ∆ΦS , (23)

is the Quantum Master Operator for the Wilson action S. The antibracket andthe ∆ derivatives in (23) are defined w.r.t {ΦA, Φ∗

A}. Note that we have usedthe relation for the AF φ∗ = K−1

0 KΦ∗ to obtain (22). This justifies the gaugechoice χ∗ = 0 in (12).

Our basic requirement is that the action S for the original renormalizabletheory satisfies the QME in such a way that

(S, S)|Λ0→∞ = ∆S|Λ0→∞ = 0 ⇒ Σ[φ, φ∗ : Λ0 → ∞] = 0 . (24)

The relation (22) tells us that the BRS symmetry is preserved even in thepresence of a finite IR cutoff Λ in the UV-cutoff removing limit:

Σ[φ, φ∗ : Λ0 → ∞] = 0 ⇒ Σ[Φ, Φ∗ : Λ] = 0 . (25)

For large but finite UV cutoff Λ0, the conditions described above can bereplaced by

(S, S) ≈ O(1/Λ20), ∆S = 0

⇒ Σ[φ, φ∗ : Λ0] = Σ[Φ, Φ∗ : Λ] ≈ O(1/Λ20) . (26)

6

For QED, the BRS transformations are known to be linear in the UV theory,and one can take the Λ0 → ∞ limit without having UV divergence. For Yang-Mills theory, however, one needs to keep Λ0 finite in construction of Σ[Φ, Φ∗ : Λ],for nonlinearity of the BRS transformations.

Before closing this section, we give the definition of the quantum BRS trans-formation δQ for any X:

δQX ≡ (X, S) − ∆X

= − exp(S)∆(

X exp(−S))

+XΣ . (27)

We find identities even for Σ 6= 0:

δ2QX = −(Σ, X) ,

δQΣ = − exp(S)∆2 exp(−S) = 0 . (28)

Our discussion given in this section for renormalizable gauge theory is quitegeneral. The Wilson action (17) can be used to construct the Quantum MasterOperator and to see how the IR cutoff affects the realization of symmetry.Such a deformation of the symmetry realization is described by appearance ofa nontrivial AF dependence in the Wilson action via the IR cutoff.

We next consider how to determine the AF dependence.

3 The structure of the Wilson master action

This section describes general structure of the action S[Φ, Φ∗ : Λ] call as theWilson master action for a given S[φ, φ∗ : Λ0]. In order to discuss the AFdependence, we use the method developed in ref.[25]. We consider here theUV actions which are linear in the AF. In the gauge-fixing sector, it gener-ally happens that antighosts φA0 with ghost number -1 and some AF φ∗

A1can

be mixed. They are not independent variables, and appear only in a specificcombination (φ∗

A1+ K−1

0 φA0RA0A1). Decomposing the fields and the AF as

φA = {φA0, φA = φA−A0} and φ∗A = {φ∗

A1, φ∗

A′ = φ∗A−A1

}, we consider thefollowing bare action with the form

S[φ, φ∗ : Λ0] = S[φA, φ∗A1

+K−10 φA0RA0A1

, φ∗A′ : Λ0]

=1

2φAK−1

0 DABφB +

(

φ∗A1

+K−10 φA0RA0A1

) (

R−1)A1A

DABφB

+SI [φ, φ∗ : Λ0] , (29)

where

SI [φ, φ∗ : Λ0] =

(

φ∗A1

+K−10 φA0RA0A1

)

×(

K0R(1)A1

BφB +

1

2K0φ

BR(2)A1

BCφC)

+φ∗A′

(

R(1)A′

BφB +

1

2φBR

(2)A′

BCφC)

+ SI [φA : Λ0] . (30)

7

The coefficients R and R may depend on the UV scale Λ0, but do not dependon {φ, φ∗}. For the interaction terms, we find

SI [φ, φ∗ : Λ0]|φ∗=0 = SI [φ : Λ] = Sint[φ, φ

∗ : Λ0]|φ∗=0

= φA0φA0RA0A1

(

R(1)A1

BφB +

1

2φBR

(2)A1

BCφC)

+SI [φA : Λ0] . (31)

The equations of motion for the fields φA0 can be expressed as

∂lS

∂φA0

−K−10 RA0A1

∂lS

∂φ∗A1

= 0 . (32)

This leads to[

(−)ǫ(A0)JA0+K−1

0 RA0A1

∂l

∂φ∗A1

]

Zφ[J, φ∗]

= NJ

[

(−)ǫ(A0)JA0+ RA0A1

K−1 ∂l

∂Φ∗A1

]

ZΦ[K0K−1J,Φ∗] = 0 , (33)

and therefore to the corresponding equations of motion for the Wilson masteraction:

K−10 K

∂lS

∂ΦA0

−RA0A1K−1 ∂lS

∂Φ∗A1

= 0 . (34)

For the AF dependence, we find

S[Φ, Φ∗ : Λ] = S[ΦA, K−10 K2Φ∗

A1+ ΦA0RA0A1

, Φ∗A′ : Λ]

=1

2ΦAK−1DABΦB

+(

K−10 KΦ∗

A1+K−1ΦA0RA0A1

)(

R−1)A1A

DABφB

+SI [ΦA, K−1

0 K2Φ∗A1

+ ΦA0RA0A1, Φ∗

A′ : Λ] . (35)

Thus, our task is to determine Φ∗A′ dependence in the Wilson master action. To

this end, we first consider the Wilson action without AF

S[Φ : Λ] =1

2Φ ·K−1D · Φ + SI [Φ : Λ]

= S[ΦA, ΦA0RA0A1: Λ] , (36)

where SI [Φ : Λ] = Sint[Φ, Φ∗ : Λ]|Φ∗=0. Note that the antighosts ΦA0 arealways multiplied by the factor RA1A0

in the Wilson action S[ΦA : Λ]. Theinteraction terms are given by

exp−SI [Φ : Λ] =

∫

Dχ exp−(1

2χ · (K0 −K)−1D · χ+ SI [(Φ + χ) : Λ0]

)

, (37)

8

Then, the AF dependent action SI [ΦA, Φ∗

A′ : Λ] = SI [Φ, Φ∗A′ : Λ] with out

Φ∗A1

can be expressed as

exp−SI [Φ, Φ∗A′ : Λ] =

∫

Dχ exp−(1

2χ · (K0 −K)−1D · χ

+SI [(Φ + χ), K−10 KΦ∗

A : Λ0]|Φ∗

A1=0

)

. (38)

We rewrite the integrand as

1

2χ · (K0 −K)−1D · χ+ SI [(Φ + χ), K−1

0 KΦ∗A′ : Λ0]

=1

2χ · (K0 −K)−1D · χ+K−1

0 KΦ∗A′

(

R(1)A′

B(Φ + χ)B

+1

2(Φ + χ)BR

(2)A′

BC(Φ + χ)C)

+ SI [(Φ + χ) : Λ0]

= K−10 KΦ∗

A′

(

R(1)A′

BΦB +1

2ΦBR

(2)A′

BCΦC)

+1

2χ · (K0 −K)−1D · χ

+JAχA +

1

2K−1

0 KΦ∗A′χBR

(2)A′

BCχC + SI [(Φ + χ) : Λ0] , (39)

where the effective sources coupled to χA are given by

JA = K−10 KΦ∗

B

(

R(1)B

A+ R

(2)BCA

ΦC)

, JA0= 0 . (40)

We define an action W as

SI [Φ, Φ∗A′ : Λ] = W [Φ, Φ∗

A′ : Λ]

+K−10 KΦ∗

A′

(

R(1)A′

BΦB +1

2ΦBR

(2)A′

BCΦC)

, (41)

with

exp−W [Φ, Φ∗A′ : Λ]

=

∫

Dχ exp−(1

2χ · (1 −K)−1D · χ+ J · χ+ SI [Φ + χ]

+K−1

0 K

2Φ∗

A′χBR(2)A′

BCχC)

= exp

(

−K−1

0 K

2Φ∗

A′

∂l

∂JB

R(2)A′

BC

∂l

∂JC

)

×

∫

Dχ exp−(1

2χ · (1 −K)−1D · χ + SI [Φ + χ : Λ0] + J · χ

)

. (42)

We may rewrite

1

2χ · (K0 −K)−1D · χ + J · χ

=1

2χ′ · (K0 −K)−1D · χ′ −

1

2(−)ǫ(J )J · (K0 −K)D−1J , (43)

9

where

χ′ = χ+ J · (K0 −K)(D−1) = χ+ (−)ǫ(J )D−1(K0 −K) · J . (44)

We then introduce new variables {Φ′A, Φ′A0RA0A1} as

Φ′A = ΦA − JB(K0 −K)(D−1)BA (45)

Φ′A0RA0A1= K−1

0 K2Φ∗A1

+ ΦA0RA0A1− JA(K0 −K)(D−1)AA0RA0A1

.

Here the AF Φ∗A1

are included by additional shifting of the variables ΦA0RA0A1.

It follows from (37) that

exp−W [Φ, Φ∗ : Λ]

= exp

(

−K−1

0 K

2Φ∗

A′

∂l

∂JB

R(2)A′

BC

∂l

∂JC

)

exp(1

2(−)ǫ(J )J · (K0 −K)D−1 · J

)

×

∫

Dχ′ exp−(1

2χ′ · (K0 −K)−1D · χ′ + SI [Φ

′ + χ′ : Λ0])

= exp

(

−K−1

0 K

2Φ∗

A′

∂l

∂JB

R(2)A′

BC

∂l

∂JC

)

× exp(1

2(−)ǫ(J )J · (K0 −K)D−1 · J − SI [Φ

′ : Λ])

. (46)

Collecting the above results, we obtain our final expression for the Wilsonmaster action:

S[Φ, Φ∗ : Λ] =1

2ΦAK−1DABΦB

+(

K−10 KΦ∗

A1+K−1ΦA0RA0A1

)(

R−1)A1A

DABΦB

+SI [Φ, Φ∗ : Λ] , (47)

where

SI [Φ, Φ∗ : Λ] = SI [Φ′ : Λ] +K−1

0 KΦ∗A′

(

R(1)A′

BΦB +1

2ΦBR

(2)A′

BCΦC)

−1

2(−)ǫ(J )J · (K0 −K)D−1 · J

− log

[

exp(

−1

2(−)ǫ(J )J · (K0 −K)D−1 · J + SI [Φ

′ : Λ])

× exp

(

−K−1

0 K

2Φ∗

A′

∂l

∂JB

R(2)A′

BC

∂l

∂JC

)

× exp(1

2(−)ǫ(J )J · (K0 −K)D−1 · J − SI [Φ

′ : Λ])

]

. (48)

This formula enables us to construct the action SI [Φ, Φ∗] from SI [Φ]. The AFdependence is partly generated by replacing variables from Φ to Φ′. The explicitΦ∗

A′ dependence in SI [Φ, Φ∗] which is not absorbed into the fields Φ′ and J is

10

solely generated from the derivative terms, K−10 KΦ∗R(2)∂/∂J ∂/∂J /2 in the

functional integral (48). The BRS transformations for the fields ΦA′

becomecomplicated due to the presence of these terms. However, we have a closedexpression for those in AF removing limit, Φ∗ → 0 specified by

[

∂lW [Φ′, Φ∗ : Λ]

∂Φ∗A′

]

Φ∗→0

=

[

∂lSI [Φ′ : Λ]

∂Φ∗A′

]

Φ∗→0

+1

2K−1

0 KR(2)A′

CD(K0 −K)(

D−1)CE

×(K0 −K)(

D−1)DF

(

∂lSI [Φ : Λ]

∂ΦE

∂lSI [Φ : Λ]

∂ΦF−∂l∂lSI [Φ : Λ]

∂ΦE∂ΦF

)

. (49)

This equation is needed for reduction of the QME to the WT identity. We wishto apply these results first to QED and then Yang-Mills theory.

4 The Wilson master action for QED

We consider QED described by

Zφ[J, φ∗] =

∫

DφD exp (−S[φ, φ∗] + J · φ) , (50)

where fields, antifields, and sources are collectively denoted by φA = {aµ, b, c, c ψ, ψ},φ∗A = {a∗µ, b

∗, c∗, c∗, ψ∗, ψ∗}, and JA = {Jµ, JB , Jc, Jc, Jψ, Jψ}. The bareaction is given by

S[φ, φ∗ : Λ0] =1

2φK−1

0 ·D · φ+ φ∗AR

A[φ] + SI [φ : Λ0]

=1

2φAK−1

0 DABφB +

(

φ∗A1

+K−10 φA0RA0A1

)

R(1)A1

BφB

+φ∗A′

(

R(1)A′

BφB +

1

2φBR

(2)A′

BCφC)

+ SI [φ : Λ0] . (51)

Note that the antigost φA0 = c is mixed with the longitudinal component ofand the antifields φ∗

A1= a∗µ. They appear only in the combination φA1 +

K−10 RA1A0

φA0 = a∗µ(−p) − K−10 (p)pµc(−p). Therefore, the bare action takes

the form

S[φ, φ∗ : Λ0] =

∫

p

K−10 (p)

[

1

2aµ(−p)(p

2δµν − pµpν)aν(p)

−b(−p){

ipµaµ(p) +ξ

2b(p)

}

+ ψ(−p)(/p + im)ψ(p)

+(

a∗µ(−p) −K−10 (p)pµ c(−p)

)

(−i)c(p) + c∗(−p) · ib(p)

+ie

∫

q

ψ∗(−p)c(q)ψ(p − q) + ie

∫

q

ψ(−p − q)c(q)ψ∗(p)

]

+SI [Aµ, ψ, ψ : Λ0] . (52)

11

Note that the ghost and antighost do not interact and remain free-fields in thecutoff theory. This simplifies our general results (47) and (48) in the previoussection. For the cutoff theory with the IR fields ΦA = {Aµ, B, C, C, ψ, ψ} andtheir AF Φ∗

A = {A∗µ, B

∗, C∗, C∗, ψ∗, ψ∗}, the Wilson master action is given by

S[Φ, Φ∗ : Λ] =1

2ΦAK−1DABΦB

+(

K−10 KΦ∗

A1+K−1ΦA0RA0A1

R(1)A1

B

)

ΦB

+K−10 KΦ∗

A′

(

R(1)A′

BΦB +1

2ΦBR

(2)A′

BCΦC)

−1

2(−)ǫ(J )J · (K0 −K)D−1J + SI [Φ

′ : Λ] . (53)

Note that the derivative terms, K−10 KΦ∗R(2)∂/∂J ∂/∂J /2 in (48), give rise to

no contributions. Therefore, the AF dependence of the Wilson master action ispurely generated by the shift of variables, Φ → Φ′.

The effective sources in (40) read

Jc(−p) = −ie

∫

q

K−10 (q)K(q)

(

ψ∗(−q)ψ(q − p) − ψ(−q − p)ψ∗(q))

JB(−p) = iK−10 (p)K(p)C∗(−p)

Jψ(−p) = ie

∫

q

K−10 (q)K(q)ψ∗(−q)C(q − p)

Jψ(−p) = −ie

∫

q

K−10 (q)K(q)C(q − p)ψ∗(−q) , (54)

The shifted variables are given by

A′µ(p) = Aµ(p) −

ipµp2

(K0(p) −K(p))JB(p)

ψ′(p) = ψ(p) +(K0(p) −K(p))

/p + imJψ(p)

ψ′(−p) = ψ(−p) −Jψ(−p)(K0(p) −K(p))

/p + im

pµC′(−p) = pµC(−p) −K−1

0 (p)K2(p)A∗µ(−p) . (55)

The shifted variable C ′ does not contain contribution from the Jc term, becausethat SI [Φ

′] does not depend on the B, C, C:

SI [Φ′ : Λ] = SI [A

′µ, ψ

′, ψ′ : Λ] . (56)

The quadratic terms of the effective sources takes the form

−1

2(−)ǫ(J )J · (K0 −K)D−1J = Jψ(−p)

(K0(p) −K(p))

/p+ imJψ(p) . (57)

12

The total Wilson master action reads

S[Φ, Φ∗] =

∫

p

(

1

K(p)

[

1

2Aµ(−p)(p

2δµν − pµpν)Aν(p) + C(−p)ip2C(p)

−B(−p)(

ipµAµ(p) +ξ

2B(p)

)

+ ψ(−p)(/p + im)ψ(p)

]

+K(p)

K0(p)

[

C∗(−p) · iB(p) + A∗µ(−p)(−i)pµC(p)

+ie

∫

q

{

ψ∗(−p)C(q)ψ(p − q) + ψ(−p − q)C(q)ψ∗(p)}

]

+Jψ(−p)(K0(p) −K(p))

/p + imJψ(p)

)

+ SI [A′µ, ψ

′, ψ′] . (58)

In order to get rid of the quadratic terms in AF, We may perform a canonicaltransformation from {Aµ, C, ψ, ψ} to {Aµ, C Ψ, Ψ} generated by

G =

∫

p

[

A∗µ(−p)

(

Aµ(p) +pµp2K−1

0 (p)K(p)(K0(p) −K(p))C∗(p)

)

+C∗(−p)C(p) + Ψ∗(−p)ψ(p) (59)

+Ψ∗(−p)

(

ψ(p) − ieK0(p) −K(p)

/p+ im

∫

q

K−10 (q)K(q)C(p− q)Ψ∗(q)

)]

,

which gives

Aµ(p) = Aµ(p) +pµp2K−1

0 (p)K(p)(K0(p) −K(p))C∗(p) = A′µ(p)

C(−p) = C(−p) +pµp2K−1

0 (p)K(p)(K0(p) −K(p))A∗µ(−p)

Ψ(p) = ψ(p) − ieK0(p) −K(p)

/p+ im

∫

q

K−10 (q)K(q)C(p− q)Ψ∗(q) = ψ′(p)

Ψ(−p) = ψ(−p) − ieK−10 (p)K(p)

∫

q

Ψ∗(−q)K0(q) −K(q)

/q + imC(q − p)

= ψ′(−p) + ie

∫

q

Ψ∗(−q)U(−q, p)C(q − p) , (60)

where

U(−q, p) =

[

K(q)(K0(p) −K(p))

K0(q)(/p+ im)−K(p)(K0(q) −K(q))

K0(p)(/q + im)

]

. (61)

For the AF,

A∗µ(p) = A∗

µ, C∗(p) = C∗(p)

ψ∗(p) = Ψ∗(p), ψ∗(p) = Ψ∗(p) . (62)

13

In terms of new variables Φ ≡ {Aµ, B, C, C, Ψ, Ψ} and Φ∗ = {A∗µ, B

∗, C∗, C∗, Ψ∗, Ψ∗},our master action reads

S[Φ, Φ∗] =

∫

p

(

K−1(p)

[

1

2Aµ(−p)(p

2δµν − pµpν)Aν(p) + C(−p)ip2C(p)

−B(−p)(

ipµAµ(p) +ξ

2B(p)

)

+ Ψ′(−p)(/p + im)Ψ(p)

]

+ C∗(−p)iB(p) + A∗µ(−p)(−i)pµC(p)

+ie

∫

q

{

Ψ∗(−p)K0(q)K(p)

K(q)K0(p)C(p− q)Ψ(q)

+Ψ(−p)C(p− q)K0(p)K(q)

K(p)K0(q)Ψ∗(q)

}

)

+SI [Aµ, Ψ, Ψ′] (63)

where

Ψ′(−p) = Ψ(−p) − ie

∫

q

Ψ∗(−q)U(−q, p)C(q − p) . (64)

5 The Wilson master action for Yang-Mills the-

ory

We consider a pure Yang-Mills theory described by

Zφ[J, φ∗] =

∫

Dφ exp (−S[φ, φ∗] + J · φ) , (65)

where fields, AF, and sources are collectively denoted by φA = {aµ, b, c, c},φ∗A = {a∗µ, b

∗, c∗, c∗}, and JA = {Jµ, JB , Jc, Jc}. The bare action2 is givenby

S[φ, φ∗ : Λ0] =

∫

p

(

K−10 (p)

[

1

2aµ(−p) · (p

2δµν − pµpν)aν(p)

−b(−p) ·{

ipµaµ(p) +ξ

2b(p)

}

+(

K0(p)a∗µ(−p) − pµc(−p)

)

· (−i)pµc(p)

]

+c∗(−p) · ib(p) +z(Λ0)

2

∫

q

c∗(−p) ·{

c(p− q) × c(q)}

)

+ SI [φ : Λ0] .(66)

where we have introduced a constantz = z(Λ0).

2We suppress the group index and use A · B ≡ AaBa, A · (B × C) = fabcAaBbCc

14

The partition function (65)is related to that for the cutoff theory with theIR fields ΦA = {Aaµ, B

a, Ca, Ca}, and their AF Φ∗A = {A∗

µ, B∗, C∗, C∗}:

Zφ[J, φ∗] = NJZΦ[K0K

−1J, Φ∗]

ZΦ[K0K−1J, Φ∗] =

∫

DΦ exp(

−S[Φ, Φ∗] +K0K−1J · Φ

)

. (67)

The Wilson master action is given by

S[Φ, Φ∗ : Λ] =1

2ΦAK−1DABΦB

+(

K−10 KΦ∗

A1+K−1ΦA0RA0A1

)(

R−1)A1A

DABφB

+K−10 KΦ∗

A′

(

R(1)A′

BΦB +

1

2ΦBR

(2)A′

BCΦC)

−1

2(−)ǫ(J )J · (K0 −K)D−1J + SI [Φ

′ : Λ]

− log

[

exp(

−1

2(−)ǫ(J )J · (K0 −K)D−1 · J + SI [Φ

′ : Λ])

× exp

(

−K−1

0 K

2Φ∗

A′

∂l

∂JB

R(2)A′

BC

∂l

∂JC

)

× exp(1

2(−)ǫ(J )J · (K0 −K)D−1 · J − SI [Φ

′ : Λ])

.

]

(68)

We give concrete expressions. The effective sources read

JB(−p) = iK−10 (p)K(p)C∗(−p)

Jc(−p) = z

∫

q

K−10 (q)K(q)C∗(q) ×C(−p − q) , (69)

The shifted variables defined in (45) are given by

A′µ(p) = Aµ(p) −

ipµp2

(K0(p) −K(p))JB(p)

pµC′(−p) = pµC(−p) −K−1

0 (p)K2(p)A∗µ(−p)

+ipµp2

(K0(p) −K(p))Jc(−p) . (70)

15

Using these variables, we obtain

S[Φ, Φ∗ : Λ] =

∫

p

(

1

K(p)

[

1

2Aµ(−p) · (p2δµν − pµpν)Aν(p)

+C(−p) · ip2C(p) −B(−p) ·(

ipµAµ(p) +ξ

2B(p)

)

]

+K(p)

K0(p)

[

C∗(−p) · iB(p) +A∗µ(−p) · (−i)pµC(p)

+z

2C∗(−p) ·

∫

q

C(p− q) ×C(q)

]

)

+ SI [A′µ, C

′, C] (71)

− log

(

expSI [Φ′ : Λ] exp

[

z

2

∫

p,q

K(p+ q)

K0(p+ q)C∗(p+ q)

·

(

(K0(p) −K(p))(K0(q) −K(q))

p2q2∂l

∂C(p)×

∂l

∂C(q)

)]

exp−SI [Φ′ : Λ]

)

.

We now make a canonical transformation from {Aµ, C} to {Aµ, C} gener-ated by

G =

∫

p

[

A∗µ(−p) ·

(

Aµ(p) +pµp2K−1

0 (p)K(p)(K0(p) −K(p))C∗(p))

+C∗(−p) · C(p)

]

, (72)

which gives

Aµ(p) = Aµ(p) +pµp2K−1

0 (p)K(p)(K0(p) −K(p))C∗(p) = A′µ(p)

C(−p) = C(−p) +pµp2K−1

0 (p)K(p)(K0(p) −K(p))A∗µ(−p) . (73)

For AF,

A∗µ(p) = A∗

µ(p), C∗(p) = C∗(p) . (74)

We finally obtain the Wilson master action described by the new variables

16

Φ = {Aµ, B, C, C} and Φ∗ = {A∗µ, B

∗, C∗, C∗}:

S[Φ, Φ∗] =

∫

p

(

K−(p)

[

Aµ(−p) · (pδµν − pµpν)Aν(p) + C(−p) · ipC(p)

−B(−p) ·{

ipµAµ(p) +ξ

2B(p)

}

]

+ C∗(−p)iB(p) + A∗µ(−p)(−i)pµC(p)

+ zK(p)

K0(p)C∗(−p) ·

∫

q

C(p− q) ×C(q)

)

+SI [Aµ, C′, C : Λ] (75)

− log

(

expSI [Aµ, C′, C : Λ] exp

[

z

2

∫

p,q

K(p + q)

K0(p+ q)C∗(p+ q)

·

(

(K0(p) −K(p))(K0(q) −K(q))

p2q2∂l

∂C(p)×

∂l

∂C(q)

)]

exp−SI [Aµ, C′, C : Λ]

)

,

where

pµC′(−p) = pµC(−p) −K(p)A∗

µ(p) + ipµp2

(K0(p) −K(p))Jc(−p)

= pµC(−p) −K(p)A∗µ(p)

+izpµp2

(K0(p) −K(p))

∫

q

K(q)

K0(q)

[

C∗(q) ×C(−p− q)]

.(76)

6 Reduction of QME to WT identity for YM

theory

We now show the WT identity Σ[Φ : Λ] = 0 is obtained from the QME Σ[Φ, Φ∗ :Λ] = 0 by eliminating the antifields. What we should to compute are δΦA =[∂S[Φ,Φ∗Λ]/∂Φ∗

A](Φ∗ → 0):

[

∂lS

∂A∗µ(−p)

]

Φ∗=0

=K(p)

K0(p)

[

(−i)pµC(p) −K(p)pµp2

∂lSI

∂C(−p)

]

[

∂S

∂C∗(−p)

]

Φ∗=0

=K(p)

K0(p)

[

iB(p) −(

K0(p) −K(p))pµp2

∂SI∂Aµ(−p)

]

[

∂S

∂C∗(−p)

]

Φ∗=0

= zK(p)

K0(p)

∫

q

(

C(p− q) ×

[

1

2C(q) + i

(K0(q) −K(q))

q2∂lSI

∂C(−q)

]

−1

2

(K0(p− q) −K(p − q))(K0(q) −K(q))

(p− q)2q2

[

∂lSI∂C(−p+ q)

×∂lSI

∂C(−q)−

∂l

∂C(−p+ q)×

∂lSI∂C(−q)

]

)

. (77)

17

Note that the integral over q for ∂S/∂C∗ is potentially divergent if K0 = 1. TheUV cutoff Λ0 is needed for the expression to be finite. We have

[

∂rS

∂ΦA∂lS

∂Φ∗A

]

Φ∗=0

=

∫

p

(

[

∂rS

∂Aµ(p)

]

·

[

∂lS

∂A∗µ(−p)

]

+

[

∂rS

∂C(p)

]

·

[

∂lS

∂C∗(−p)

]

+

[

∂rS

∂C(p)

]

·

[

∂lS

∂C∗(−p)

]

)

Φ∗=0

. (78)

The contribution from functional measure is given by

−∆S = −(−)ǫA+1 ∂r

∂ΦA∂rS

∂Φ∗A

=

∫

p

(

[

∂r

∂Aµ(p)

]

·

[

∂rS

∂A∗µ(−p)

]

−

[

∂r

∂C(p)

]

·

[

∂rS

∂C∗(−p)

]

−

[

∂r

∂C(p)

]

·

[

∂rS

∂C∗(−p)

]

)

Φ∗=0

. (79)

The WT identity is given by

Σ[Φ : Λ] = Σ[Φ, Φ∗ : Λ]|Φ∗=0 =

[

∂rS

∂ΦA∂lS

∂Φ∗A

− ∆S

]

Φ∗=0

= 0 . (80)

7 Summary and Discussion

We have discussed a general method for construction of the Wilson master actionS[Φ,Φ∗,Λ] and quantum master operator Σ[Φ,Φ∗ : Λ] using the BV formalismfor gauge theories in ERG. A formal expression to determine the AF dependenceof the action is given. For QED, the AF dependence is generated by a shift ofthe variables. For Yang-Mills theory, one cannot absorb all the AF dependenceby this shift of variables. Although we have no closed expression for the AFdependence for ghosts, the WT identity Σ[Φ : Λ] = 0 obtained from the QMEtakes a closed form: it is expressed in terms of the Wilson action S[Φ : Λ]. Thiswill give an important nonperturbative constraint on ERG approach to gaugetheories.

We have found in (25) that if the QME holds at some value of the IR cutoffΛ1, Σ[Φ,Φ∗ : Λ1] = 0, it is the case for arbitrary value of Λ, Σ[Φ,Φ∗ : Λ] = 0.Using this property as well as the identity δQ Σ[Φ,Φ∗ : Λ] = 0, we may showthe existence of the QME. The identity is so-called “fine tuning condition”, firstdiscussed by Becchi [18]. See also [26] for the BV case.

Our formlaism given here also applies to global symmetries. There, ghostsare introduced not as dynamical variables but simply as constant parameters.We can construct the quantum master operator Σ, as in the gauge theory case.

Irrespective of local or global symmetry, there exits a symmetry if the quan-tum master operator vanishes, Σ = 0. Otherwise, Σ 6= 0 exactly corresponds to

18

an anomaly for the symmetry under consideration. This means that our methodcan be used to study anomalies for global as well as local symmetries.

Acknowledgments

The author would like to thank the organizers of the workshop “Renormaliza-tion in Quantum Field Theory, Statistical Mechanics” at ESI for all their effortswhich makes this workshop possible. He also would like to thank the Insti-tute of Theoretical Physics in Heidelberg and Department of Physics in ParmaUniversity for hospitality. He is very grateful to J.M. Pawlowski for enlightingdiscussion. This work is supported in part by the Grants-in-Aid for ScientificResearch No.17540242 from the Japan Society for the Promotion of Science.

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