Quantum Master Equation for QED in Exact Renormalization Group
aa r X i v : . [ h e p - t h ] A p r Quantum Master Equation for QEDin Exact Renormalization Group
Yuji Igarashi a , Katsumi Itoh a and Hidenori Sonoda b October 26, 2018 a Faculty of Education, Niigata University b Physics Department, Kobe University
Abstract
Recently, one of us (H.S.) gave an explicit form of the Ward-Takahashiidentity for the Wilson action of QED. We first rederive the identity us-ing a functional method. The identity makes it possible to realize thegauge symmetry even in the presence of a momentum cutoff. In thecutoff dependent realization, the abelian nature of the gauge symmetryis lost, breaking the nilpotency of the BRS transformation. Using theBatalin-Vilkovisky formalism, we extend the Wilson action by includingthe antifield contributions. Then, the Ward-Takahashi identity for theWilson action is lifted to a quantum master equation, and the modifiedBRS transformation regains nilpotency. We also obtain a flow equationfor the extended Wilson action. Introduction
One of the most important subjects in the exact renormalization group (ERG)[1, 2, 3] is to find a method to treat gauge symmetry, which is naively incom-patible with a momentum cutoff Λ introduced as a regularization (see ref. [4]and references therein for an alternative attempt to construct a gauge invari-ant regularization scheme.). Constraints on the induced symmetry breakingterms are described by some identities for Green functions, so called “the bro-ken Ward-Takahashi (WT) identities” or “the modified Slavnov-Taylor (ST)identities”[5, 6]. These identities are written either for the Wilson action S orits Legendre transformed effective action Γ. They have additional regulator de-pendent terms which are absent in the standard WT or ST identities for thecut-off removed theories. This is why they are called “broken” or “modified”identities.Even if the standard realization of the symmetry no longer works, the sym-metry can be realized in a regularization dependent way. This is most conve-niently done using the Batalin-Vilkovisky (BV) antifield formalism[7, 8](see alsoref. [9]), which has been recognized as the most general and powerful method fordealing with symmetries. Any local as well as global symmetries are describedby the quantum master equation (QME) in the BV formalism. It was shown[10] that if the QME holds for a cutoff-removed action S [ φ ], this should alsobe the case for the Wilson action S [Φ] with a finite value of momentum cutoff.Hence, at least conceptually, the presence of the symmetry along ERG flow isestablished. This result may suggest that, for consideration of symmetries, theuse of the Wilson action S is preferable to that of the Legendre action Γ.The above argument remains, however, at a formal level, and only little hasbeen known concerning how actions satisfying the QME looks like for concretecases, especially for gauge theories. So far, only for the lattice chiral symmetry,which is a prototype of the regularization dependent symmetry, it has beenshown that the QME contains the Ginsparg-Wilson relation and was solved togive an action for self-interacting fermions [14].Recently, an important result was obtained in formulating gauge symmetryin terms of ERG: one of the present authors (H.S.) has derived the WT identity Σ Φ = 0 for the Wilsonian QED action S [Φ] and its flow equation [15, 16].From the WT identity, we obtain the BRS transformation δ that has thefollowing properties: 1) it depends on the Wilson action, and therefore, is non-linear; 2) a non-trivial Jacobian factor associated with δ is generated to cancelthe change of the action; 3) it is not nilpotent, δ = 0. The last property maybe understood as the absence of the abelian nature of the gauge algebra in thepresence of a momentum cutoff.The main aim of the present paper is to describe how these results fit in the As for global symmetries, see ref. [11]. The RG flow equation for QED was studied inref. [12] with some approximations. Ref. [13] discusses roles of the modified WT identity inrelation to numerical studies. This WT identity is expected to carry the same information as “the broken WT identity”for QED given in ref.[17, 18] for the Legendre effective action Γ. Φ = 0 for QED can be lifted to theQME Σ Φ , Φ ∗ = 0.In ref. [15], the action S [Φ] that satisfies the WT identity is obtained per-turbatively. Here we show how to construct the action S [Φ , Φ ∗ ] that fulfills theQME, assuming that we have the action satisfying the WT identity. We callthe action S [Φ , Φ ∗ ] as the master action.It is found that our master action, a formal solution to the QME, has non-trivial antifield dependence: the infinite power series expansion w.r.t. antifieldstakes the form of a Taylor expansion of the Dirac fields, which corresponds toa shift of field variables in the Wilson action. As a byproduct of introducingantifields, we employ the “quantum BRS transformation” [19], δ Q , which isassured to be nilpotent, δ Q = 0, thanks to the QME. Using this, we show theBRS invariance of the Polchinski equation for our master action. We emphasizethat the nilpotency is recovered in the BV formalism even though the BRStransformation read off from the WT identity is not nilpotent.The rest of this paper is organized as follows. In the next section, we give ageneral method for deriving the WT identity Σ Φ = 0 for a regularized theory,and apply it to QED to obtain the WT identity derived in ref. [15]. From theWT identity, we read off the BRS transformation. In section 3, after a briefexplanation of the antifield formalism, we construct a master action for QEDthat satisfies the QME Σ Φ , Φ ∗ = 0 starting from an action satisfying the WTidentity Σ Φ = 0. In the final section, the Polchinski flow equation is given forour master action, and its BRS invarinace is shown. We will derive the WT identity for the QED with a momentum cutoff usingthe path integral formalism. Our derivation is based on the method which hasalready been discussed by several authors [5, 20, 21]. The formalism given heremay be applicable to any theory with symmetry. We will first discuss a generictheory with a momentum cutoff, and then apply the results to QED. The fields are denoted collectively by φ A . The index A represents the Lorentzindices of tensor fields, the spinor indices of the fermions, and/or indices dis-tinguishing different types of fields. The Grassmann parity for φ A is expressedas ǫ ( φ A ) = ǫ A , so that ǫ A = 0 if the field φ A is Grassmann even (bosonic) and ǫ A = 1 if it is Grassmann odd (fermionic). The generating functional for this In ref.[5], it is called “the quantum action principle”. J A is given by Z φ [ J ] = Z D φ exp ( −S [ φ ] + J · φ ) , (1)where the action S is decomposed into the kinetic and interaction terms S [ φ ] = 12 φ · D · φ + S I [ φ ] . (2)In this paper we use the matrix notation in momentum space: J · φ = Z d d p (2 π ) d J A ( − p ) φ A ( p ) ,φ · D · φ = Z d d p (2 π ) d φ A ( − p ) D AB ( p ) φ B ( p ) . (3)We now introduce an IR momentum cutoff Λ through a positive function thatbehaves as K (cid:16) p Λ (cid:17) → (cid:26) p < Λ )0 ( p → ∞ ) (4)where the function goes to 0 sufficiently rapidly as p → ∞ . For simplicity,we write the function as K ( p ) in the rest of the paper. Using this function,we decompose the original fields φ A with the propagator ( D AB ( p )) − into twoclasses of fields: the IR fields Φ A with the propagator K ( p ) ( D AB ( p )) − , andthe UV fields χ A with (1 − K ( p )) ( D AB ( p )) − . To this end, we substitute agaussian integral over new fields θ A Z D θ exp − (cid:16) θ − J (1 − K ) D − (cid:17) · DK (1 − K ) · (cid:16) θ − ( − ) ǫ ( J ) D − (1 − K ) J (cid:17) = const (5)into the path-integral (1), and introduce new variables Φ and χ by φ A = Φ A + χ A , θ A = (1 − K )Φ A − Kχ A . (6)Then, we obtain Z φ [ J ] = N J Z D Φ D χ exp − (cid:18)
12 Φ · K − D · Φ − J · K − Φ+ 12 χ · (1 − K ) − D · χ + S I [Φ + χ ] (cid:19) , (7)where N J ≡ exp 12 ( − ) ǫ A J A (1 − K − ) (cid:0) D − (cid:1) AB J B . (8)4he Wilson action is given by S [Φ] ≡ Φ · K − D · Φ / S I [Φ] (9)where S I [Φ] is defined byexp − S I [Φ] ≡ Z D χ exp − (cid:16) χ · (1 − K ) − D · χ + S I [Φ + χ ] (cid:17) . (10)Note that the gaussian integral (5) is chosen in such a way that the UV fields χ A do not couple to source terms, and hence the Wilson action S I depends onlyon the IR fields Φ A . The partition function for Φ A Z Φ [ J ] = Z D Φ exp (cid:0) − S [Φ] + K − J · Φ (cid:1) (11)is related to that for φ by Z φ [ J ] = N J Z Φ [ J ] . (12)This implies that the full generating functional Z φ can be constructed from theWilson action S [Φ]. In (11), note that the source to the IR field Φ A is multipliedby K − . Therefore, the correlation functions in two theories are related as (cid:10) φ A · · · φ A N (cid:11) φ | J =0 = (cid:10) ( K − Φ A ) · · · ( K − Φ A N ) (cid:11) Φ | J =0 ( N ≥ . (13)As for the two-point functions, there are extra contributions from the factor N J .Now we consider how the IR cutoff affects the realization of symmetry. Sup-pose the original gauge-fixed action S [ φ ] is invariant under the BRS transfor-mation φ A → φ A ′ = φ A + δφ A , δφ A = R A [ φ ] λ , (14)where λ is an anticommuting constant. Hence, δ S = ∂ r S ∂φ A δφ A ≡ Σ φ λ = 0 . (15)Assuming the invariance of the functional measure D φ , we obtain the standardWT identity for Z φ : h Σ φ i φ, J = Z − φ [ J ] Z D φ J · R [ φ ] exp ( −S [ φ ] + J · φ )= Z − φ [ J ] (cid:16) J · R [ ∂ lJ ] Z φ [ J ] (cid:17) = 0 . (16) The construction of the Wilson action via similar techniques can be found in refs. [20, 22,23]. The idea of the decomposition of fields was also discussed in a non-local regularizationscheme [24]. We assume the presence of BRS invariant regularization scheme such as the dimensionalregularization in order for the Z φ theory to be well-defined. However, the knowledge of the Z φ theory is only used as the boundary condition for the Z Φ theory at Λ → ∞ . h Σ φ i φ, J = Z − φ J · R [ ∂ lJ ] Z φ [ J ] = Z − (cid:0) N − J J · R [ ∂ lJ ] N J Z Φ [ J ] (cid:1) . (17)We expect that the last expression in the above can be written as the expectationvalue of an operator in the cutoff theory: its vanishing is a consequence ofthe symmetry of the original theory. Therefore, the operator is appropriatelyregarded as the “WT operator”. We denote it by Σ Φ so that h Σ Φ i Φ , K − J = 0 (18)In the next subsection, we obtain the WT operator explicitly for the QED witha momentum cutoff. In addition to the gauge and Dirac fields { A µ , ψ, ¯ ψ } , we consider, for the BRSsymmetry, the (non-interacting) ghost and anti-ghost { c, ¯ c } as well as the aux-iliary field B . Thus we have φ A = { A µ , B, c, ¯ c, ψ, ¯ ψ } and the correspondingsources J A = { J µ , J B , J c , J ¯ c , J ψ , J ¯ ψ } .The action is given as S [ φ ] = 12 φ · D · φ + S I [ φ ] . (19)The free part is12 φ · D · φ = Z k h A µ ( − k )( k δ µν − k µ k ν ) A ν ( k ) + ¯ c ( − k ) ik c ( k ) − B ( − k ) (cid:0) ik µ A µ ( k ) + ξ B ( k ) (cid:1)i + Z p ¯ ψ ( − p )(/ p + im ) ψ ( p ) , (20)where ξ is the gauge parameter, and S I [ φ ] gives the interaction part. We assumethat the above action is invariant under the standard BRS transformation δA µ ( k ) = − ik µ c ( k ) , δ ¯ c ( k ) = iB ( k ) , δc ( k ) = δB ( k ) = 0 ,δψ ( p ) = − ie Z k ψ ( p − k ) c ( k ) , δ ¯ ψ ( − p ) = ie Z k ¯ ψ ( − p − k ) c ( k ) . (21)The source dependent normalization factor N J in (12) can be calculated explic-itly as ln N J = ( − ) ǫ A J A (cid:16) − KK (cid:17) (cid:0) D − (cid:1) AB J B = Z k (cid:16) − KK (cid:17) ( k ) n J c ( − k ) − ik J ¯ c ( k ) − J B ( − k ) − ik µ k J µ ( k ) − J µ ( − k ) 1 k (cid:16) δ µν − (1 − ξ ) k µ k ν k (cid:17) J ν ( k ) o + Z p (cid:16) − KK (cid:17) ( p ) J ψ ( − p ) 1/ p + im J ¯ ψ ( p ) (22)6he operator that appears in eq. (17) takes the following form for QED: J · R [ ∂ lJ ] = (cid:0) J · R [ ∂ lJ ] (cid:1) gauge + (cid:0) J · R [ ∂ lJ ] (cid:1) matter , (23)where (cid:0) J · R [ ∂ lJ ] (cid:1) gauge = i Z k n − k · J ( − k ) ∂ l ∂J c ( − k ) + J ¯ c ( − k ) ∂ l ∂J B ( − k ) o , (24) (cid:0) J · R [ ∂ lJ ] (cid:1) matter = − ie Z p, k n J ψ ( − p ) ∂ l ∂J ψ ( − p + k ) − J ¯ ψ ( p ) ∂ l ∂J ¯ ψ ( p + k ) o ∂ l ∂J c ( − k ) (25)Let us now derive the WT-identity for the Wilson action (9) for QED. In thefollowing, we use the same notation for the IR fields as for the original fields:Φ A = { A µ , B, c, ¯ c, ψ, ¯ ψ } . The kinetic term is given by12 Φ · K − D · Φ = Z k K − ( k ) h A µ ( − k )( k δ µν − k µ k ν ) A ν ( k )+ ¯ c ( − k ) ik c ( k ) − B ( − k ) (cid:0) ik · A ( k ) + ξ B ( k ) (cid:1)i + Z p K − ( p ) ¯ ψ ( − p )(/ p + im ) ψ ( p ) . (26)It follows from (17) that our central task for finding Σ Φ is to compute Z − N − J J · RN J Z Φ . It is easy to realize that the non-trivial deformation fromthe standard WT identity has two origins: 1) the normalization factor N J , and2) the scale factor K − in the source terms K − J · Φ and in the kinetic termsΦ · K − D · Φ /
2. Now, from (17), we have0 = Z − h N − J ( J · R ) N J i Z Φ = Z − h ( J · R ) gauge + N − J ( J · R ) matter N J i Z Φ . (27)The second line is a result of the fact, ( J · R ) gauge N [ J ] = 0. The matter sectorwhich contains non-trivial contributions may be written as follows: Z − φ ( J · R ) matter Z φ = − ie *Z p, k n J ψ ( − p ) K ( p ) U ( − p, p − k ) J ¯ ψ ( p − k ) K ( p − k )+ J ψ ( − p ) ∂ l ∂J ψ ( − p + k ) − J ¯ ψ ( p ) ∂ l ∂J ¯ ψ ( p + k ) o c ( k ) + Φ , K − J (28)where U ( − p, p − k ) ≡ − K ( p − k )/ p − / k + im K ( p ) − − K ( p )/ p + im K ( p − k ) . (29)7sing Z − J A Z Φ = (cid:28) K ∂ r S∂ Φ A (cid:29) Φ , K − J , Z − ∂ l ∂J A Z Φ = (cid:10) K − Φ A (cid:11) Φ , K − J , (30)we obtain h Σ Φ i Φ ,K − J = 0 (31)withΣ Φ ≡ Z k n ∂S∂A µ ( k ) ( − ik µ ) c ( k ) + ∂ r S∂ ¯ c ( k ) iB ( k ) o (32) − ie Z p, k n ∂ r S∂ψ ( p ) K ( p ) K ( p − k ) ψ ( p − k ) − K ( p ) K ( p + k ) ¯ ψ ( − p − k ) ∂ l S∂ ¯ ψ ( − p ) o c ( k ) − ie Z p, k tr n(cid:16) ∂ l S∂ ¯ ψ ( − p + k ) ∂ r S∂ψ ( p ) − ∂ l ∂ r S∂ ¯ ψ ( − p + k ) ∂ψ ( p ) (cid:17) U ( − p, p − k ) o c ( k ) . From the identity (31), we note that any correlation function with a Σ Φ insertionvanishes, (cid:10) Σ Φ Φ A Φ A · · · Φ A N (cid:11) Φ | J =0 = 0 . (33)Therefore, we obtain the operator identity Σ Φ = 0, which is the WT identityderived in refs. [15, 16] for the Wilson action of QED.From eq. (32), it is easy to realize that the WT identity is nothing but theBRS invariance of the action under the standard BRS transformation (21) asfar as the gauge sector is concerned. Since the gauge sector is free, this is quitenatural.Though the matter contribution to Σ Φ is slightly complicated, we will seepresently that it also allows an interpretation as a change of the action undersome symmetry transformation. We may rewrite the matter contributions inΣ Φ as ie Z p, k ∂ r S∂ψ ( p ) c ( k ) n K ( p ) K ( p − k ) ψ ( p − k ) − U ( − p, p − k ) ∂ l S∂ ¯ ψ ( − p + k ) o − ie Z p, k n K ( p ) K ( p + k ) ¯ ψ ( − p − k ) o c ( k ) ∂ l S∂ ¯ ψ ( − p )+ ie Z p, k tr ∂ l ∂ r S∂ ¯ ψ ( − p + k ) ∂ψ ( p ) U ( − p, p − k ) c ( k ) . (34)From the first two lines of (34), we read off the BRS transformation of thefermion. Including the transformation for the gauge sector, we find δA µ ( k ) = − ik µ c ( k ) , δ ¯ c ( k ) = iB ( k ) , δc ( k ) = δB ( k ) = 0 δψ ( p ) = ie Z k c ( k ) n K ( p ) K ( p − k ) ψ ( p − k ) − U ( − p, p − k ) ∂ l S∂ ¯ ψ ( − p + k ) o ,δ ¯ ψ ( − p ) = ie Z k n K ( p ) K ( p + k ) ¯ ψ ( − p − k ) o c ( k ) . (35)8ith this transformation (35), Σ Φ is now written as [16]Σ Φ = ∂ r S∂ Φ A δ Φ A + ie tr (cid:16) ∂ l ∂ r S∂ ¯ ψ∂ψ U (cid:17) c. (36)The second term can be interpreted as the Jacobian factor associated with theBRS transformation (35).We have three remarks on the BRS transformation (35): (i) It depends onthe Wilson action S [Φ], and therefore it is non-linear. (ii) It is not unique: thenon-linear contribution could appear both in δψ and δ ¯ ψ . (iii) The nilpotency islost on ψ , though it holds for other fields.Obviously, the nilpotency is the most important property of the BRS sym-metry. It is desirable to elevate (35) to the one with nilpotency. In order toachieve this, we need to find out a way to take care of the Jacobian factorappearing in (36). This can be realized with the BV anti-field formalism. Let us first explain the antifield formalism briefly. For each IR field Φ A , weintroduce its antifield Φ ∗ A with the opposite Grassmann parity, ǫ (Φ ∗ A ) = ǫ (Φ A )+1, Φ ∗ A = { A ∗ µ , B ∗ , c ∗ , ¯ c ∗ , ψ ∗ , ¯ ψ ∗ } . (37)The canonical structure of fields and their anti-fields is specified by the anti-bracket. For any pair of operators, X and Y , it is defined as( X, Y ) ≡ ∂ r X∂ Φ A ∂ l Y∂ Φ ∗ A − ∂ r X∂ Φ ∗ A ∂ l Y∂ Φ A . (38)Consider a gauge theory with an action S [Φ , Φ ∗ ] and calculate the operatordefined as Σ[Φ , Φ ∗ ] ≡
12 (
S, S ) − ∆ S, (39)where ∆ ≡ ( − ) ǫ A +1 ∂ r ∂ Φ A ∂ r ∂ Φ ∗ A . (40)The equation Σ[Φ , Φ ∗ ] = 0 is the quantum master equation of the BV formalism.The action satisfying the QME describes a gauge invariant system. The actionsatisfying Σ[Φ , Φ ∗ ] = 0 is called a quantum master action, or simply a masteraction. Later, we denote a master action as S M [Φ , Φ ∗ ].Our aim in this section is to construct a master action, by using the WTidentity (32) for our Wilson action S [Φ].9n a standard gauge theory with a gauge-fixed action S [ φ ] and a nilpotentBRS transformation δφ A , the master action is S [ φ ] + φ ∗ A δφ A , linear in anti-fields. To start with, let us try an extended action linear in the anti-fieldsΦ ∗ A : S lin [Φ , Φ ∗ ] = S [Φ] + Φ ∗ A δ Φ A . This action, however, does not satisfy theQME: Σ[Φ , Φ ∗ ] ∝ c c ψ ∗ U U . To cancel this contribution, one should addsuitable terms S quad [Φ , Φ ∗ ] quadratic in the anti-fields, and so on. After severaltrials, we realize that this expansion w.r.t. antifields is the Taylor expansion ofthe action, where ¯ ψ is replaced by ¯ ψ → ¯ ψ − ieψ ∗ c U .Let us assume this form for the master action and prove that it indeedsatisfies the QME. Our master action is S M [Φ , Φ ∗ ] = S [Φ ′ ] + Z k (cid:16) A ∗ µ ( − k )( − i ) k µ c ( k ) + i ¯ c ∗ ( − k ) B ( k ) (cid:17) (41)+ ie Z p,k (cid:16) ψ ∗ ( − p ) K ( p ) K ( p − k ) c ( k ) ψ ( p − k ) + ¯ ψ ( − p − k ) c ( k ) K ( p ) K ( p + k ) ¯ ψ ∗ ( p ) (cid:17) . Here we have introduced the shifted field ¯ ψ ′ and Φ ′ A ¯ ψ ′ ( − p ) ≡ ¯ ψ ( − p ) − ie Z k ψ ∗ ( − p − k ) c ( k ) U ( − p − k, p ) , Φ ′ A = { A µ , B, c, ¯ c, ψ, ¯ ψ ′ } . (42)In eq. (41), note that the second term in δψ of (35) is absorbed into S [Φ ′ ] dueto the shift.In proving the QME for S M , it is important that the action S [Φ] satisfiesthe WT identity. For convenience, we rewrite the identity for S [Φ ′ ] with theshifted fields (42). We obtain Z k n ∂S [Φ ′ ] ∂A µ ( k ) ( − ik µ ) c ( k ) + ∂ r S [Φ ′ ] ∂ ¯ c ( k ) iB ( k ) o + ie Z p,k ∂ r S [Φ ′ ] ∂ψ ( p ) c ( k ) n K ( p ) K ( p − k ) ψ ( p − k ) − U ( − p, p − k ) ∂ l S [Φ ′ ] ∂ ¯ ψ ( − p + k ) o + ie Z p,k,l ∂ r S [Φ ′ ] ∂ ¯ ψ ( − p ) K ( p ) K ( p + k ) × n ¯ ψ ( − p − k ) − ie ψ ∗ ( − p − k − l ) c ( l ) U ( − p − k − l, p + k ) o c ( k )+ ie Z p,k tr (cid:16) ∂ l ∂ r S [Φ ′ ] ∂ ¯ ψ ( − p ) ∂ψ ( p + k ) U ( − p − k, p ) (cid:17) c ( k ) = 0 . (43)Now it is straightforward to verify that the action (41) satisfies the QME.10ere we calculate the contributions to ( S M , S M ) / Z p ∂ r S M ∂ ¯ ψ ( − p ) ∂ l S M ∂ ¯ ψ ∗ ( p ) = ie Z p,k ∂ r S [Φ ′ ] ∂ ¯ ψ ( − p ) K ( p ) K ( p + k ) ¯ ψ ( − p − k ) c ( k ) , (44) Z p ∂ r S M ∂ψ ( p ) ∂ l S M ∂ψ ∗ ( − p ) (45)= ie Z p,l ∂ r S [Φ ′ ] ∂ψ ( p ) c ( l ) K ( p ) K ( p − l ) ψ ( p − l ) − U ( − p, p − l ) ∂ l S [Φ ′ ] ∂ ¯ ψ ( − p + l ) ! + e Z p,k,l K ( p + k + l ) K ( p + l ) c ( k ) c ( l ) ψ ∗ ( − p − k − l ) U ( − p − l, p ) ∂ l S [Φ ′ ] ∂ ¯ ψ ( − p ) . The quantum term may be calculated as∆ Φ S M = − ie Z p,k tr (cid:16) ∂ l ∂ r S M ∂ ¯ ψ ( − p ) ∂ψ ( p + k ) U ( − p − k, p ) (cid:17) c ( k ) . (46)Combining all the terms in (44), (45) and (46), we findΣ[Φ , Φ ∗ ] ≡
12 ( S M , S M ) Φ − ∆ Φ S M = 0 , (47)thanks to the identity (43). Therefore the action S M defined by (41) is indeeda master action. Note that the same e term appears in both (45) and (43).In summary, we have observed that the ( ∂S/∂ ¯ ψ )( ∂S/∂ψ ) term of Σ Φ is ab-sorbed into the classical part ( ∂S M /∂ψ )( ∂S M /∂ψ ∗ ) of the QME, correspondingto the shift of ¯ ψ . Likewise, the ∂∂S/∂ ¯ ψ∂ψ term of Σ Φ turns into the jacobianassociated with the BRS transformation. The shift of ¯ ψ needed for constructing S M from S now appears quite natural.In the antifield formalism, the “quantum” BRS transformation [19] is definedby δ Q X ≡ ( X, S M ) − ∆ X (48)for any operator X . For the fields in QED, it takes the following form: δ Q A µ ( k ) = − ik µ c ( k ) , δ Q ¯ c ( k ) = iB ( k ) , δ Q c ( k ) = δ Q B ( k ) = 0 ,δ Q ψ ( p ) = ie Z k c ( k ) n K ( p ) K ( p − k ) ψ ( p − k ) − U ( − p, p − k ) ∂ l S M ∂ ¯ ψ ( − p + k ) o ,δ Q ¯ ψ ( − p ) = ie Z k n K ( p ) K ( p + k ) ¯ ψ ( − p − k ) o c ( k ) . (49)This transformation has the same form as (35). However, the action on the r.h.s.of δ Q ψ is now S M [Φ , Φ ∗ ], and the BRS transformation has a non-trivial antifielddependence. The BRS transformation in the gauge sector is quite simple, whilethat of the matter sector is rather complicated.The quantum BRS transformation is nilpotent if and only if the QME holds: δ Q X = ( X, Σ[Φ , Φ ∗ ]) = 0 . (50)11n other words, the QME enables us to define the nilpotent BRS transformation.This should be compared with the classical counterpart, δX ≡ ( X, S M ) whichdoes not vanish due to the lack of the Jacobian factor, δ X = 12 ( X, ( S M , S M )) = 0 . (51) In this section, we derive the Polchinski flow equation for our master action andshow its BRS invariance.Let us begin with the well-known generic result on the Polchinski flow equa-tion for the Wilson action S [Φ] without antifields. It is given by ∂ t S [Φ] = − Z p Φ A ( p ) (cid:0) K − ˙ K (cid:1) ( p ) ∂ l S∂ Φ A ( p ) (52)+ 12 Z p h ∂ r S∂ Φ A ( p ) (cid:0) ˙ KD − ( p ) (cid:1) AB ∂ l S∂ Φ B ( − p ) − ( − ) ǫ A (cid:0) ˙ KD − ( p ) (cid:1) AB ∂ l ∂ r S∂ Φ B ( − p ) ∂ Φ A ( p ) i up to terms independent of fields. Here, we use a dimensionless parameter t = log(Λ /µ ) and ˙ K = ∂ t K .The flow equation for our master action S M [Φ , Φ ∗ ] of QED can be obtainedthrough a straightforward calculation. From the definition (41), we have ∂ t S M [Φ , Φ ∗ ] = ∂ t S [Φ] (cid:12)(cid:12) Φ=Φ ′ − ie Z p,k ψ ∗ ( − p − k ) c ( k ) ∂ t U ( − p − k, p ) ∂ l S [Φ ′ ] ∂ ¯ ψ ( − p ) (53)+ Z p,k c ( k ) h ψ ∗ ( − p ) ∂ t (cid:16) K ( p ) K ( p − k ) (cid:17) ψ ( p − k ) − ¯ ψ ( − p − k ) ∂ t (cid:16) K ( p ) K ( p + k ) (cid:17) ¯ ψ ∗ ( p ) i . In replacing S [Φ ′ ] by S M [Φ , Φ ∗ ], one takes account of the following points spe-cific to the abelian nature of QED: (1) the ghost is a free field, and the BRStransformation for the gauge and ghost sector { A µ , B, c, ¯ c } is cutoff indepen-dent; (2) the shift of the fermionic field ¯ ψ → ¯ ψ − iec ψ ∗ U generates a non-trivialantifield dependence in the flow equation. We also note the following identityfor the matrix U : ∂ t U ( − p − k, p ) = (cid:16) ˙ K ( p + k ) K ( p + k ) + ˙ K ( p ) K ( p ) (cid:17) U ( − p − k, p )+ ˙ K ( p + k ) K ( p + k ) K ( p ) 1/ p + / k + im − ˙ K ( p ) K ( p ) K ( p + k ) 1/ p + im . (54)12hen, putting altogether, we obtain ∂ t S M [Φ , Φ ∗ ]= 12 Z p ˙ K ( p ) p (cid:16) δ µν − (1 − ξ ) p µ p ν p (cid:17)(cid:16) ∂S M ∂A µ ( p ) · ∂S M ∂A ν ( − p ) − ∂ S M ∂A µ ( p ) ∂A ν ( − p ) (cid:17) − Z p ˙ K ( p ) K ( p ) h A µ ( p ) (cid:16) δ µν − p µ p ν p (cid:17) ∂S M ∂A ν ( p ) + ip ν p ξB ( p ) ∂S M ∂A ν ( p ) i + Z p ˙ K ( p ) K ( p ) h B ( − p ) (cid:0) ip · A ( p ) + ξB ( p ) (cid:1) − ip ¯ c ( − p ) c ( p ) i + Z p ˙ K ( p ) h ∂ r S M ∂ψ ( p ) 1/ p + im ∂ l S M ∂ ¯ ψ ( − p ) + tr (cid:16) p + im · ∂ l ∂ r S M ∂ ¯ ψ ( − p ) ∂ψ ( p ) (cid:17)i − Z p ˙ K ( p ) K ( p ) (cid:16) ¯ ψ ( − p ) ∂ l S M ∂ ¯ ψ ( − p ) + ∂ r S M ∂ψ ( p ) ψ ( p ) − ψ ∗ ( − p ) ∂ l S M ∂ψ ∗ ( − p ) − ∂ r S M ∂ ¯ ψ ∗ ( p ) ¯ ψ ∗ ( p ) (cid:17) − ie Z p,k ˙ K ( p ) K ( p ) K ( p − k ) c ( k ) × (cid:16) ψ ∗ ( − p ) 1/ p + im ∂ l S M ∂ ¯ ψ ( − p + k ) − ∂ r S M ∂ψ ( p ) 1/ p + im ¯ ψ ∗ ( p − k ) (cid:17) . (55)Thanks to the abelian nature of the theory, no antifields appear in the gaugeand ghost sector. The fermionic sector has explicit antifield dependence.Let us discuss the BRS invariance of the flow equation (55). For the RGflow of the WT operator, we obtain the relation, ∂ t Σ[Φ , Φ ∗ ] = ( ∂ t S M , S M ) − ∆ ∂ t S M = δ Q ∂ t S M = 0 , (56)which implies that the flow itself should be written as a quantum BRS transformof something. Actually, up to the QME, we find [10] ∂ t S M = − δ Q G, (57)where G is the generator of a canonical transformation G = G + G + G (58)that has three parts: G ≡ Z k A ∗ µ ( − k ) h
12 ˙ K ( k ) k (cid:16) δ µν − (1 − ξ ) k µ k ν k (cid:17) ∂S M ∂A ν ( − k )+ ˙ K ( k ) K ( k ) ik µ k (cid:16) ik · A ( k ) − ξB ( k ) (cid:17)i , (59) G ≡ − Z k ˙ K ( k ) K ( k ) h A ∗ µ ( − k ) A µ ( k ) + B ∗ ( − k ) B ( k )+ ¯ c ∗ ( − k )¯ c ( k ) + ψ ∗ ( − k ) ψ ( k ) + ¯ ψ ( k ) ¯ ψ ∗ ( − k ) i , (60) G ≡ Z p ψ ∗ ( − p ) ˙ K ( p )/ p + im h ∂ l S M ∂ ¯ ψ ( − p ) − ieK ( p ) Z k c ( k ) K ( p − k ) ¯ ψ ∗ ( p − k ) i . (61)13 Summary and Discussion
In this paper, we have rederived the WT identity for the Wilson action ofQED using a functional method and shown that it can be lifted to a QMEin the BV antifield formalism. The master action, our formal solution to theQME, generically has non-linear but simple anti-field dependence which appearsmerely as a shift of field variables. We have also found that the master actionis not unique, and that it can be deformed by canonical transformations in thespace of fields and antifields. No deformation can remove the non-linear anti-field dependence in the master action. We believe that the non-linear anti-fielddependence is an inherent feature of any local symmetries in cutoff field theories.We have also derived an extended flow equation for the master action. Sincethe master action is determined up to canonical transformations, the flow equa-tion is not unique, and can be modified by canonical transformations.A pair of fundamental equations, the WT identity Σ Φ = 0 and the Polchinskiequation ∂ t S [Φ] − F [Φ] = 0, can be interpreted as a gauge fixed version of theQME and extended flow equation:Σ Φ = Σ[Φ , Φ ∗ ] | Φ ∗ → = 0 ∂ t S [Φ] − F [Φ] = ( ∂ t S M [Φ , Φ ∗ ] − F [Φ , Φ ∗ ]) | Φ ∗ → = 0It should be emphasized that the QME plays a crucial role not only in construct-ing a nilpotent BRS transformation, but also in showing the BRS invariance ofthe extended flow equation. These properties imply that the exact gauge sym-metry does exist in the Wilson action of QED despite the presence of a finitemomentum cutoff.A perturbative solution to the WT identity Σ Φ = 0 and the Polchinski equa-tion ∂ t S [Φ] − F [Φ] = 0 has been obtained in refs. [15, 16]. It is straightforwardto to find the corresponding perturbative solution to the QME: Σ[Φ , Φ ∗ ] = 0and the extended flow equation: ∂ t S M [Φ , Φ ∗ ] − F [Φ , Φ ∗ ] = 0. Acknowledgments
This work is supported in part by the Grants-in-Aid for Scientific Research No.13135209, 15540262 and 17540242 from the Japan Society for the Promotion ofScience.
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