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Generalized Pauli-Villars Regularization and the Covariant Form of Anomalies


UT-678 April ’94

Generalized Pauli-Villars Regularization

arXiv:hep-th/9405166v2 5 Jul 1994

and the Covariant Form of Anomalies
Kazuo Fujikawa Department of Physics,University of Tokyo Bunkyo-ku,Tokyo 113,Japan

Abstract In the generalized Pauli-Villars regularization of chiral gauge theory proposed by Frolov and Slavnov , it is important to specify how to sum the contributions from an in?nite number of regulator ?elds. It is shown that an explicit sum of contributions from an in?nite number of ?elds in anomaly-free gauge theory essentially results in a speci?c choice of regulator in the past formulation of covariant anomalies. We show this correspondence by reformulating the generalized Pauli- Villars regularization as a regularization of composite current operators. We thus naturally understand why the covariant fermion number anomaly in the Weinberg-Salam theory is reproduced in the generalized Pauli-Villars regularization. A salient feature of the covariant regularization,which is not implemented in the lagrangian level in general but works for any chiral theory and gives rise to covariant anomalies , is that it spoils the Bose symmetry in anomalous theory. The covariant regularization however preserves the Bose symmetry as well as gauge invariance in anomaly-free gauge theory.

1

1

Introduction

An interesting regularization of chiral gauge theory in the Lagrangian level has been proposed by Frolov and Slavnov [1]. This scheme incorporates an in?nite number of bosonic and fermionic regulator ?elds,and as such how to sum the contributions from the in?nite number of regulator ?elds constitutes an essential part of this regularization ; a formal introduction of an in?nite number of regulator ?elds in the Lagrangian does not completely specify the theory. Detailed analysis of this regularization scheme have been also performed by several authors [2][3] : For example , the covariant form of anomaly for the fermion number current in the Weinberg-Salam theory [4] is naturally reproduced in the generalized Pauli-Villars regularization[3]. The purpose of the present paper is to analyze the basic mechanism of this regularization scheme from a general view point of regularization and anomalies [5]. We show that the generalized Pauli-Villars regularization , after one sums the contributions from an in?nite number of ?elds , essentially corresponds to a speci?c choice of regulator in the calculational scheme of covariant anomalies [6] [7]. The covariant form of fermion number anomaly is thus naturally understood. A reformulation of the genaralized Pauli-Villars regularization as a regularization of composite current operators is crucial in this analysis. The calculational scheme of covariant anomalies , which works for any chiral gauge theory,was introduced as a convenient means in the path integral formulation of anomalous identities [6] [8].This regularization is not implemented in the Lagrangian level,but rather it regularizes various currents and amplitudes directly. In terms of Feynman diagrams , this regularization imposes the gauge invariance on all the vertices except for the one corresponding to the Noether current generated by the change of path integral variables.The anomaly produced at the Noether current is thus ”gauge covariant” , but it explicitly spoils the Bose symmetry of the underlying Feynman diagrams. If one applies this regularization to non-anomalous diagrams , however , the Noether current is conserved and the Bose symmetry in Feynman diagrams is preserved. In the following , we show the above correspondence ( and also some di?erence ) between the generalized Pauli-Villars regularization and the calculational scheme of covariant anomalies by explicitly evaluating several anomalous as well as non-anomalous

2

diagrams.

2

Generalized Pauli-Villars Regularization

We ?rst recapitulate the essence of the generalized Pauli-Villars regularization and reformulate it as a regularization of composite current operators. The starting theory which we want to regularize is de?ned by 1 + γ5 ψ 2

L = ψi D where

(2.1)

a D = γ ? (?? ? igAa ? (x)T )

≡ γ ? (?? ? igA? (x)) and T a is the hermitian generator of a compact semi-simple group, 1 [T a , T b ] = if abc T c , T rT aT b = δ ab . 2

(2.2)

(2.3)

In the main part of this paper , we treat the gauge ?eld A? (x) as a background ?eld,and the dynamical aspects of A? will be brie?y commented on later. In the Euclidean metric we use , the γ - matrices satisfy

{γ ? , γ ν } = 2g ?ν , g ?ν = (?1, ?1, ?1, ?1)
? (γ ? )? = ?γ ? , γ5 = γ5

(γ 5 )2 = 1 . The Dirac operator D is formally hermitian for the natural inner product of Euclidean theory

(Φ, D Ψ) ≡

d4 xΦ? D Ψ (2.4)

= (D Φ, Ψ). The generalized Pauli-Villars regularization of (2,1) is de?ned by 3

L = ψi Dψ ? ψ L MψR ? ψ R M ? ψL +φi Dφ ? φM ′ φ where (2.5)

1 1 ψR = (1 + γ5 )ψ , ψL = (1 ? γ5 )ψ 2 2 and the in?nite dimensional mass matrices in (2.5) are de?ned by

(2.6)

?

0 0 0 ···

M =

? ? ? ? ? ? ? ? ?

2 0 0 ··· ? ? 0 4 0 ··· ? ? 0 0 6 ···

? ?Λ ? ? ? ? ? ? ? ? ? ? ? 2 ?Λ ? ? ? ? ? ?

? ? ? ? ? ? ? ? ? ? ? ? ?

M ?M =

? 0

22 42 0

0

62 .. .

? ? ? ? ? ? ? ? ? ? ? ? ?

MM ? =

2 ? 2

42 62 0

0 ..

?

.

?

M′ =

? 1 ? ? ? ? ? ? ? ? ? ? ? ?

3 5 0

0 ..

?

? ? ? ? ? ? 2 ?Λ ? ? ? ? ? ?

.

? ? ? ? ? ? ?Λ ? ? ? ? ? ?

= (M ′ )?

(2.7)

4

where Λ is a parameter with dimensions of mass. The ?elds ψ and φ in (2.5) then contain an in?nite number of components , each of which is a conventional 4-component Dirac ?eld;ψ (x) consists of conventional anticommuting (Grassmann) ?elds , and φ(x) consists of commuting bosonic Dirac ?elds. The regularization (2.5) corresponds to the so-called ”vector - like” formulation [2]. The Lagrangian (2.5) is invariant under the gauge transformation ψ (x) → ψ ′ (x) = U (x)ψ (x)≡exp[iw a (x)T a ]ψ (x) ψ (x) → ψ (x) = ψ (x)U (x)? φ(x) → φ′ (x) = U (x)φ(x) φ(x) → φ (x) = φ(x)U (x)? D → D ′ = U (x) DU (x)? . (2.8)
′ ′

The Noether current associated with the gauge coupling in (2.5) is de?ned by the in?nitesimal change of matter variables in (2.8) with D kept ?xed :
′ ′ L′ = ψ i Dψ ′ ? ψ L MψR ? ψ R M ? ψL ′ ′ ′

+φ i Dφ′ ? φ M ′ φ′ = ?(D? w )aJ ?a (x) + L with J ?a (x) = ψ (x)T a γ ? ψ (x) + φ(x)T a γ ? φ(x). Similarly , the U(1) transformation ψ (x) → eiα(x) ψ (x) , ψ (x) → ψ (x)e?iα(x) φ(x) → eiα(x) φ(x) , φ(x) → φ(x)e?iα(x) (2.11) gives rise to the U(1) fermion number current J ? (x) = ψ (x)γ ? ψ (x) + φ(x)γ ? φ(x). The chiral transformation 5 (2.9)





(2.10)

(2.12)

ψ (x) → eiα(x)γ5 ψ (x) , ψ → ψ (x)eiα(x)γ5 φ(x) → eiα(x)γ5 φ(x) , φ → φ(x)eiα(x)γ5 gives the U(1) chiral current
? J5 (x) = ψ (x)γ ? γ5 ψ (x) + φ(x)γ ? γ5 φ(x).

(2.13)

(2.14)

Considering the variation of action under the transformation (2.9) and (2.11) , one can show that the vector currents (2.10) and (2.12) are naively conserved
?

?c (D? J ? )a (x) ≡ ?? J ?a (x) + gf abc Ab ? (x)J (x) = 0,

?? J ? (x) = 0 whereas the chiral current (2.14) satis?es the naive identity
? ?? J5 (x) = 2iψ L MψR ? 2iψ R M ? ψL + 2iφM ′ γ5 φ.

(2.15)

(2.16)

The quantum theory of (2.5) may be de?ned by the path integral as Z = ≡ and , for example , < ψ (x)T a γ ? ψ (x) >= d?ψ(x)T a γ ? ψ (x)exp[ Ld4x]. DψDψDφDφexp[ Ld4 x] d?exp[ Ld4 x] (2.17)

(2.18)

The path integral over the bosonic variables φ and φ for the Dirac operator in Euclidean theory needs to be de?ned via a suitable rotation in the functional space.

?

The fact that the regularized currents satisfy anomaly-free relations (2.15) shows that the regular-

ization (2.5) is ine?ective for the evaluation of possible anomalies in these vector currents.

6

De?nition of Currents in Terms of Propagators We now de?ne the currents in terms of propagators. The basic idea of this approach is explained for the un ? regularized theory in (1) as follows : We start with the current associated with the gauge coupling

< ψ (x)T a γ ? (

1 + γ5 )ψ (x) > 2 1 + γ5 = lim < T ? ψ (y )T a γ ? ( )ψ (x) > y →x 2 ? 1 + γ5 )δβ ψβc (x)ψ αb (y ) > = ? lim < T ? (T a )bc γαδ ( y →x 2 1 + γ5 1 ) δ (x ? y )] = lim T r [T aγ ? ( y →x 2 iD

(2.19)

where we used the anti-commuting property of ψ and the expression of the propagator < T ? ψ (x)ψ (y ) >= ( 1 + γ5 (?1) ) δ (x ? y ) 2 i Dx

(2.20)

The trace in (2.19) runs over the Dirac and Yang-Mills indices. We now notice the expansion

1 1 = iD i ?+gA 1 1 1 + (?g A) = i? i? i? 1 1 1 + (?g A) (?g A) +··· i? i? i?

(2.21)

When one inserts (2.21) into (2.19) and retains only the terms linear in Ab ν (x) , one obtains

1 + γ5 (?1) ν b b 1 ) γ T gAν (x) δ (x ? y )] y →x 2 i? i? 1 + γ5 (?1) ) = lim d4 zT r [T a γ ? ( y →x 2 i? 1 δ (x ? y )]gAb × δ (x ? z )T b γ ν ν (z ) i? lim T r [T a γ ? (

(2.22)

where the derivative ?? acts on all the x- variables standing on the right of it in (2.22). If one takes the variational derivative of (2.22) with respect to gAb ν (z ) , one obtains 7

y →x

lim T r [T a γ ? (

1 + γ5 (?1) 1 ) δ (x ? z )γ ν T b δ (x ? y )] 2 i? i? 1 + γ5 (?1) b ν 1 ?iq(x?z ) ?ik(x?y) d4 q d4 k T r [T a γ ? ( ) T γ ]e e = lim 4 4 y →x (2π ) (2π ) 2 k+ q k d4 q ?iq(x?z ) 1 + γ5 1 1 + γ5 1 d4 k = e ( ? 1) T r [T a γ ? ( ) T bγ ν ( ) ] 4 4 (2π ) (2π ) 2 k+ q 2 k 4 d q ?iq(x?z ) ab e Π?ν (q ) (2.23) ≡ (2π )4

where we used the representations of δ -function d4 q ?iq(x?z ) e (2π )4 d4 k ?ik(x?y) e . (2π )4

δ (x ? z ) = δ (x ? y ) =

(2.24)

The last expression in (2.23) stands for the vacuum polarization tensor. Namely
γ5 )ψ in the , one can generate the multiple correlation functions of currents ψT a γ ? ( 1+ 2

perturbative sense by taking the variational derivative of (2.19) with respect to gauge ?elds Aa ? . This idea also works for the non-gauge currents (2.12) and (2.14). We emphasize that we always take the limit y = x ?rst before the explicit calculation , and thus (2.19) di?ers from the point-splitting de?nition of currents. We now generalize the above de?nition of currents for the theory de?ned by (2.5). For this purpose , we rewrite (2.5) as L = ψiD ψ + φiD ′ φ with D ≡ D + iM ( D′ 1 + γ5 1 ? γ5 ) + iM ? ( ) 2 2 ≡ D + iM ′ . (2.25)

(2.26)

The gauge current (2.10) is then de?ned by

J ?a (x) = lim {< T ? ψ (y )T a γ ? ψ (x) > + < T ? φ(y )T aγ ? φ(x) >}
y →x

= y lim {? < T ? T a γ ? ψ (x)ψ (y ) > + < T ? T a γ ? φ(x)φ(y ) >} →x = y lim T r [T a γ ? ( →x 1 1 ? ′ )δ (x ? y )] iD iD 8 (2.27)

where trace includes the sum over the in?nite number of ?eld components in addition to Dirac and Yang-Mills indices. The anti-commuting property of ψ (x) and the commuting property of φ(x) are used in (2.27). We next notice the relations

1 1 = D? D D?D =
2

1 D′

1 D? 1 + γ5 ) + 2 MM ? (1 ? γ5 ) D + 1 + γ5 1 ? γ5 1 1 = [( ] +( ) 2 ) 2 ? 2 2 D +M M D + MM ? 1 + γ5 1 ? γ5 ×[D ? iM ? ( ) ? iM ( )] 2 2 1 = (D ′ )? ′ (D )? D ′ 1 = (D ? iM ′ ). 2 ′ 2 D + (M )
1 M ? M (1 2

(2.28)

We thus rewrite (2.27) as

T r ?iT a γ ? (

1 1 ? ′ )δ (x ? y ) D D 1 1 + γ5 ∞ a ? ) = T r ?iT γ ( 2 2 2 n=0 D + (2nΛ) 1 ? γ5 ∞ 1 +( ) 2 2 2 n=1 D + (2nΛ) ? = 1 Dδ (x ? y ) 2 n=0 D + [(2n + 1)Λ]
2 ∞

∞ (?1)n D 2 1 1 T r ?iT a γ ? δ (x ? y ) 2 2 2 n=?∞ D + (nΛ) D

1 1 + T r ?iT a γ ? γ5 δ (x ? y ) 2 D 1 1 T r T a γ ? f (D 2 /Λ2) δ (x ? y ) = 2 iD 1 1 + T r T a γ ? γ5 δ (x ? y ) 2 iD

(2.29)

where we explicitly evaluated the trace over the in?nite number of components and used the fact that the trace over an odd number of γ -matrices vanishes. We also de?ned f (x2 ) by 9

f (x2 ) ≡ =

(?1)n x2 2 2 n=?∞ x + (nΛ) (πx/Λ) . sinh(πx/Λ) (2.30)



This last expression of (2.30) as a sum of in?nite number of terms is given in ref.[1]. The regulator f (x2 ), which rapidly approaches 0 at x2 = ∞, satis?es

f (0) = 1 x2 f ′ (x2 ) = 0 f or x → 0 f (+∞) = f ′ (+∞) = f ′′ (+∞) = · · · = 0 x2 f ′ (x2 ) → 0 f or x → ∞. (2.31)

The essence of the generalized Pauli-Villars regularization (2.5) is thus summarized in terms of regularized currents as follows:

< ψ (x)T a γ ? ( = lim
y →x

1 + γ5 )ψ (x) >P V 2

1 1 T r T a γ ? f (D 2 /Λ2 ) δ (x ? y ) 2 iD 1 1 + T r T a γ ? γ5 δ (x ? y ) 2 iD 1 + γ5 )ψ (x) >P V < ψ (x)γ ? ( 2 1 1 = lim T r γ ? f (D 2 /Λ2 ) δ (x ? y ) y →x 2 iD 1 1 δ (x ? y ) + T r γ ? γ5 2 iD 1 + γ5 < ψ (x)γ ? γ5 ( )ψ (x) >P V 2 1 1 T r γ ? γ5 f (D 2 /Λ2 ) δ (x ? y ) = lim y →x 2 iD 1 1 + T r γ? δ (x ? y ) . 2 iD

(2.32)

In the left-hand sides of (2.32), the currents are de?ned in terms of the original ?elds appearing in (2.1). The axial-vector and vector U (1) currents written in terms of the 10

original ?elds in (2.1) are identical , but the regularized versions (i.e. the last two equations in (2.32)) are di?erent. In particular , the vector U (1) current(i.e. , the second equation in (2.32)) is not completely regularized. See also refs.[2] and [3]. This re?ects the di?erent form of naive identities in (2.15) and (2.16) ; if all the currents are well regularized , the naive form of identities would also coincide. We emphasize that all the one-loop diagrams are generated from the (partially) regularized currents in (2.32) ; in other words , (2.32) retains all the information of the generalized Pauli-Villars regularization (2.5). The trace of energy-momentum tensor generated by the matter ?eld in (2.1) is also interesting. This is related to the variation of ?eld variables[8] ψ (x) → e? 2 α(x) ψ (x) , ψ (x) → ψ (x)e? 2 α(x) which is the ?at space-time limit of the variation of the weighted variables
1 ? ?(x) ≡ g 1 4 ψ (x) , ψ (x) ≡ g 4 ψ (x) ψ 1 1

(2.33)

(2.34)

under the Weyl transformation

ψ (x) → e 2 α(x) ψ (x) , ψ (x)→ψ (x)e 2 α(x) g?ν (x) → e?2α(x) g?ν (x) , g = detg?ν . (2.35) The Noether density generated by the (in?nitesimal) transformation (2.33) in (2.1) is given by

3

3

1 + γ5 ′ )ψ (x) 2 i ? 1 + γ5 = ? d4 xα(x)ψ (x) D ( )ψ (x) 2 2 1 + γ5 )ψ (x) + d4 xψ (x)i D ( 2 d4 xψ (x)i D (


(2.36)

. Following the same procedure as in (2.32), we ?nd

11

i ? 1 + γ5 )ψ (x) >P V < ψ (x) D ( 2 2 i 1 + γ5 )ψ (x) >P V = lim ( ){< T ? ψ (y ) D x ( y →x 2 2 ← 1 + γ5 ? < T ? ψ (y ) D y ( )ψ (x) >P V }. 2 1 = lim ( )T r f (D2 /Λ2 )δ (x ? y ) . y →x 2

(2.37)

In summary, eqs (2.30),(2.32) and (2.37) are the basic results of the generalized PauliVillars regularization (2.5).

3

Covariant Regularization and Covariant Anomalies

The calculational scheme of covariant anomalies starts with regularized current operators [9] for the theory in (2.1) as follows:

1 + γ5 )ψ (x) >cov 2 1 1 + γ5 )f (D 2 /Λ2 ) δ (x ? y ) = lim T r T a γ ? ( y →x 2 iD 1 + γ5 2 1 = φn (x)? T a γ ? ( φn (x) )f (λ 2 n /Λ ) 2 iλn n 1 + γ5 )ψ (x) >cov < ψ (x)γ ? ( 2 1 + γ5 1 = lim T r γ ? ( )f (D 2 /Λ2 ) δ (x ? y ) y →x 2 iD 1 + γ5 2 1 φn (x) )f (λ 2 = φn (x)? γ ? ( n /Λ ) 2 iλn n i ? 1 + γ5 < ψ (x) D ( )ψ (x) >cov 2 2 1 = lim T r ( )f (D2 /Λ2 )δ (x ? y ) y →x 2 1 2 = φn (x)? f (λ2 n /Λ )φn (x) 2 n < ψ (x)T a γ ? ( where the complete set {φn (x)} is de?ned by

(3.1)

Dφn (x) ≡ λn φn (x) 12

φm (x)? φn (x)d4 x = δm,n δαβ δ (x ? y ) → φn (x)α φn (y )? β
n

(3.2)

with α and β including Dirac and Yang-Mills indices. The function f (x2 ) in (3.1) is any smooth function which satis?es the condition (2.31). For the moment, we assume that there is no zero eigenvalue in (3.2). One recognizes a close relation between the generalized Pauli-Villars regularization ( (2.32) and (2.37) ) and the present covariant calculational scheme. The characteristic feature of (3.1) is that it treats the vector and axial-vector components on an equal footing and regularizes them simultaneously. In principle , one could apply di?erent regulator functions , for example , f (D2 /Λ2 ) and g (D2 /Λ2 ) respectively to vector and axial-vector components in (3.1) instead of using f (D2 /Λ2 ) for both of them. In this case , (2.32) is obtained as a special case of (3.1) by taking the limit of either g (D2 /Λ2 ) = 1 or f (D2 /Λ2 ) = 1. In this limit , however , not all the currents are completely regularized. The anomaly for the current in (3.1) is evaluated as

D? < ψ (x)T a γ ? (

1 + γ5 )ψ (x) >cov 2 1 + γ5 c ? 1 + γ5 ≡ ?? < ψ (x)T a γ ? ( )ψ (x) >cov +gf abc Ab )ψ (x) >cov ? < ψ (x)T γ ( 2 2 1 ? γ5 2 1 (Dφn (x)) )f (λ 2 φn (x)? T a ( = n /Λ ) 2 iλn n 1 + γ5 2 1 )f (λ 2 φn (x) ?(Dφn (x))? T a ( n /Λ ) 2 iλn 1 ? γ5 1 + γ5 2 = (?i)φn (x)? T a ( )?( ) f (λ 2 n /Λ )φn (x) 2 2 n = i
n 2 φn (x)? T a γ5 f (λ2 n /Λ )φn (x)

= i
n

φn (x)? T a γ5 f (D2 /Λ2 )φn (x) d4 k ?ikx a e T γ5 f (D 2 /Λ2 )eikx (2π )4 (3.3)

= iT r = (

ig 2 )T rT a ??ναβ F?ν Fαβ f or Λ → ∞ 32π 2

where we used the relation (3.2). We also normalized the anti-symmetric symbol as ?1230 = ?1234 = 1. 13

(3.4)

In the last step of the calculation in (3.3), we replaced the complete set {φn (x)} by the

plane wave basis for the well-de?ned operator f ( D 2 /Λ2 ). The ?nal result in (3.3) with
a abc b c F?ν = (?? Aa A? Aν )T a holds for any function f (x2 ) which satis?es (2.31). ν ? ?ν A? + gf

This fact is explained in Appendix for the sake of completeness. Eq.(3.3) shows that only the axial-vector component contributes to the anomaly. Similarly, the U(1) current in (3.1) satis?es the identity

?? < ψ (x)γ ? ( =
n

= i
n

1 + γ5 )ψ (x) >cov 2 1 + γ5 2 1 ?(Dφn (x))? ( )f (λ 2 φn (x) n /Λ ) 2 iλn 1 ? γ5 2 1 (Dφn (x)) )f (λ 2 +φn (x)? ( n /Λ ) 2 iλn 2 φn (x)? γ5 f (λ2 n /Λ )φn (x) d4 k ?ikx e γ5 f (D 2 /Λ2 )eikx (2π )4 (3.5)

= iT r

ig 2 )T r??ναβ F?ν Fαβ f or Λ → ∞ = ( 2 32π

which is the result used in the analysis of baryon number violation [4] and naturally agrees with the result on the basis of the last current in (2.32) in the generalized Pauli-Villars regularization [3]. Again, only the axial component contributes to the anomaly. The Weyl anomaly in the last relation in (3.1) is evaluated as

1 d4 k ?ikx i ? 1 + γ5 < ψ (x) D ( )ψ (x) >cov = Tr e f (D 2 /Λ2 )eikx 2 2 2 (2π )4 1 g2 ( )T rF ?ν F?ν f or Λ → ∞ = 2 24π 2

(3.6)

which is also known to be independent of the choice of f (x2 ) in (2.31)[8]. The coe?cient of T rF ?ν F?ν in (3.6) gives the lowest order fermion contribution to the renormalization group β -function ; β (gr ) = ( 1 )g 3/(24π 2 ). 2 r The anomaly in (3.3) is covariant under gauge transformation , which is the reason why (3.3) is called ”covariant anomaly” [10]. From the diagramatic view point , all the vertices
2 of one-loop diagrams are regularized by the gauge invariant regulator f (λ2 n /Λ ) except

14

for the vertex corresponding to the Noether current itself.? Because of this asymmetric treatment of vertices, the anomaly (3.3) does not satisfy the so-called integrability (or Wess-Zumino consistency) condition [11]. The relation (3.3) however speci?es precisely the essence of the anomaly, namely, one cannot impose gauge invariance on all the vertices of anomalous diagrams. It is also known that one can readily convert the covariant anomaly in (3. 3) to the anomaly which satis?es the Wess-Zumino condition [10]. The anomaly in (3.3) vanishes for T rT a{T b , T c } = 0.

(3.7)

For the anomaly-free gauge theory, which satis?es (3.7), one can impose gauge invariance on all the gauge vertices, and consequently the Bose symmetry is recovered. The regularization (3.1) thus provides a natural regularization of all the one-loop diagrams in anomaly-free gauge theory. From the analysis presented above, one can understand the consistency of the ”partial regularization” of the generalized Pauli-Villars regularization (2.32) and (2.37) for anomaly-free gauge theory, except for the second expression in (2.32) which cannot produce U(1) anomaly by the naive treatment of the right-hand side. Our analysis, which is based on well-regularized operators in (3.1), provides a transparent way to understand the conclusions in refs.[1] ,[2] and [3]. We now add several comments on the covariant regularization (3.1). First of all, the regularization (3.1) should not be confused with the higher derivative regularization, for example , L = ψ (x)i D (
?

D 2 + Λ2 2 1 + γ 5 ) ( )ψ (x). Λ2 2

(3.8)

In the generalized Pauli-Villars regularization , all the vertices are treated on an equal footing and

Bose symmetrically since the regularization is implemented in the Lagrangian level. The evaluation of anomaly for gauge couplings , if it should be performed in the generalized Pauli-Villars regularization , would therefore be quite di?erent from the anomaly calculation in the covariant regularization (3.1). The axial-vector component of gauge current , which could produce gauge anomaly , is however not regularized in (2.5) , as is seen in (2.32). The regularized gauge current satis?es the naive identity (2.15) and , consequently , the regularization (2.5) as it stands is not applicable to the evaluation of possible gauge anomaly.

15

If one chooses f (x2 ) ≡ [Λ2 /(x2 + Λ2 )] in (3.1), the regularization (3.1) resembles the theory de?ned by (3.8). However, this resemblance is spurious, since the Noether current given by (3.8) contains higher derivative terms in addition to the minimal one and thus the one-loop diagrams are not regularized by (3.8). It is an interesting question whether (3.1) for anomaly-free gauge theory can be implemented in the Lagrangian level if one incorporates an in?nite number of regulator ?elds. Secondly, the regularization (3.1) can be implemented for more general theory such as

2

a L = ψiγ ? ?? ? iR? (x)T a (

1 + γ5 a 1 ? γ5 ) ? iLa ) ψ ? (x)T ( 2 2 1 + γ5 1 ? γ5 )ψ + ψiγ ? (?? ? iL? )( )ψ. = ψiγ ? (?? ? iR? )( 2 2

(3.9)

In this case, left-and right-handed components separately satisfy the identities[5, 9] D? < ψT a γ ? ( i 1 ± γ5 )ψ >cov = ±( )T rT a ??ναβ F?ν Fαβ . 2 32π 2

(3.10)

The covariant regularization is quite ?exible and works for arbitrary gauge theory ; for example, one can readily show that Yukawa couplings do not modify anomaly [9]. Finally, we comment on the treatment of zero modes of D in (3.1) in some detail ; this problem is also shared by the generalized Pauli-Villars regularization (2.32) in non-perturbative analysis. When the gauge ?eld A? (x) is topologically non-trivial, the eigenvalue equation (3.2) contains well-de?ned zero eigenvalues [12]. The de?nition of current operators thus becomes subtle, but the divergence of currents such as (3.3) and (3.5) does not contain the singular factor 1/λn and thus anomalies themselves are wellde?ned. In the path integral framework, this situation is treated in the following way: One ?rst notices that γ5 φn (x) belongs to the eigenvalue ?λn in (3.2) since {γ5 , D } = 0. We thus de?ne new complete basis sets [6]

1 + γ5 φR )φn (x) if λn > 0 n (x) ≡ ( √ 2 1 + γ5 )φn (x) if λn = 0 ≡ ( 2 1 ? γ5 )φn (x) if λn > 0 φL n (x) ≡ ( √ 2 16

≡ (

1 ? γ5 )φn (x) if λn = 0. 2

(3.11)

Note that φn (x) with λn = 0 can be chosen to be the eigenvector of γ5 . We thus expand ψR (x) = ( = ψ R (x) =
λn ≥0

1 + γ5 )ψ (x) 2 an φR n (x)
? bn φL n (x)

λn ≥0

(3.12)

where an and bn are Grassmann numbers, and the action (2.1) and the path integral measure are formally de?ned by

S =

Ld4 x =

λn bn an
λn >0

d? = D ψ R D ψR =

dbn dan
λn ≥0

(3.13)

The Jacobian factor under the (in?nitesimal) chiral U(1) transformation

′ ψR (x) = eiα(x)γ5 ψR (x) = eiα(x) ψR (x)

ψ R (x) = ψ R (x)e?iα(x) is given by



(3.14)

J = exp ??i = exp ??i
N N →∞

?

d4 xα(x)
λn ≥0

?

? R L ? L ? {φR n (x) φn (x) ? φn (x) φn (x)}

?

d4 xα(x)
allλn

? φ? n (x)γ5 φn (x)

?

(3.15)

The sum of terms in (3.15) may be de?ned by


lim

φn (x)? γ5 φn (x) = lim
n=1

Λ→∞

2 φn (x)? γ5 f (λ2 n /Λ )φn (x) n=1

(3.16)

by replacing the mode cut-o?, which is natural for the de?nition (3.13), by the cut-o? in λn by using a suitable regulator satisfying (2.31). By this way, one directly obtains the anomaly factor (3.5) as a Jacobian without referring to current operators. In the 17

generalized Pauli-Villars regularization (2.17) the measure is shown to be invariant under the chiral U(1) transformation? but the variation of action, which corresponds to the right-hand side of (2.16), gives rise to the same anomaly factor as in (3.5) for Λ → ∞. By recalling that γ5 φn (x) belongs to the eigenvalue ?λn in (3.2) , the integration of (3.5) gives rise to

n+ ? n? = =

2 4 φn (x)? γ5 f (λ2 n /Λ )φn (x)d x n

g2 32π 2 = ν

T r??ναβ F?ν Fαβ d4 x (3.17)

where n± stand for the number of zero modes with γ5 = ±1 ,and ν is the Pontryagin index (or instanton number) [12]. Note that the replacement (3.16) is consistent with (3.17). From (3.15) and (3.17), we obtain

d? → d?exp [?iαν ] same phase factor as (3.18) for Λ → ∞. Eqs.(3.14) and (3.18) show that < ψR (x) · · · ψ R (y ) · · · >= d? ψR (x) · · · ψ R (y ) · · · exp Ld4 x

(3.18)

for the x-independent α in (3.14). In (2.17), the variation of the action gives rise to the

(3.19)

is non-vanishing only for the Green’s functions which contain ν more ψ variables than ψ variables. This gives the chirality selection rule and the fermion number non- conservation. Since the zero modes do not appear in the action (3.13), the path integral (i.e., leftderivative) over Grassmann variables corresponding to zero modes is completely consumed by n+ ψ -variables and n? ψ -variables appearing in Green’s functions. As a result, the Green’s function and current operators do not contain the (singular) inverse of zero eigenvalues any more, which could arise from the action. This procedure is thus consistent for anomaly-free gauge theory.
?

As for the vector-like transformation corresponding to the naive identities (2.15) , the Jacobian is

shown to be non-vanishing and give rise to (3.3) and (3.5) ; this evaluation of Jacobian corresponds to super-imposing the covariant regularization on the generalized Pauli-Villars regularization (2.5)

18

When the gauge group contains anomaly, the Jacobian for the gauge transformation (2.8) with gauge ?eld kept ?xed gives the covariant anomaly (3.3) for the Noether current. In this case, the variation of the partition function under the change of path integral variables has a de?nite meaning, but the partition function itself is ill-de?ned since the anomalous gauge theory cannot be completely regularized by gauge invariant cut-o? in terms of λn : To de?ne the partition function, one needs to use a regulator which explicitly breaks gauge invariance such as the conventional Pauli-Villars regularization [13].

4

Covariant Regularization of Anomaly-free Theory

In view of the partial regularization (2.32) of the generalized Pauli- Villars regularization, it is interesting to apply the fully regularized expressions (3.1) for the practical calculations in anomaly- free gauge theory. We then enjoy much more freedom in choosing the regulator f (x2 ) , simply because we do not require a Lagrangian-level implementation of f (x2 ). We illustrate this application by evaluating the vacuum polarization tensor. To be speci?c, we evaluate the vacuum polarization tensor for QED (in Euclidean metric) L = ψiγ ? (?? ? ieA? )ψ ? mψψ by choosing a simple regulator f (x2 ) = ( Λ2 )2 x2 + Λ2 (4.2) (4.1)

which is convenient for practical calculations and satis?es the condition (2.31). The ?nal result can be readily extended to the chiral theory de?ned by the ?rst current in (3.1). We thus start with a regularized current < ψ (x)γ ? ψ (x) >cov = lim T r {γ ?
y →x

with

Λ2 1 ( 2 )2 δ (x ? y )} 2 iD?m D +Λ

(4.3)

D = γ ? (?? ? ieA? ) obtains < ψ (x)γ ? ψ (x) >cov 19

(4.4)

By expanding (4.3) in powers of eA? and retaining only the terms linear in eA? , one

1 Λ2 1 )2 δ (x ? y ) (?e A) ( 2 = lim T r γ 2 y →x i ??m i ??m ? +Λ 2 Λ 1 1 ( 2 +γ ? )2 (ie){(?ν Aν ) + 2Aν ?ν + [γ α , γ ν ] Fαν } 2 i ??m ? +Λ 4 1 ×( 2 )δ (x ? y ) ? + Λ2 1 1 1 α ν ν ν +γ ? ( 2 )( ie ) { ( ? A ) + 2 A ? + [γ , γ ] Fαν } ν ν i ? ? m ? + Λ2 4 Λ2 )2 δ (x ? y ) ×( 2 2 ? +Λ
?

(4.5) where we used

D2 =

1 1 ? ν {γ , γ }D? Dν + [γ ? , γ ν ] D? Dν 2 2 ie = D? D ? ? [γ ? , γ ν ] F?ν 4

ie ? ν [γ , γ ] F?ν 4 ie = ?? ? ? ? ie [(?? A? ) + 2A? ?? ] ? e2 A? A? ? [γ ? , γ ν ] F?ν 4 = ?? ? ? ? ie [?? A? + A? ?? ] ? e2 A? A? ? (4.6)

with F?ν = ?? Aν ? ?ν A? . The derivative operator ?? in (4.5) and (4.6), except the one in (?? A? ) and F?ν , acts on all the x-variables standing on the right of it. By taking the variational derivative of (4.5) with respect to eAν (z ), one ?nds a regularized expression of the vacuum polarization tensor

(?1) ν Λ2 )2 δ (x ? y ) γ δ (x ? z )( 2 y →x i ??m ? + Λ2 1 Λ2 )2 (i) (? ν δ (x ? z )) + 2δ (x ? z )? ν +γ ? ( 2 2 i ??m ? +Λ 1 α ν 1 + [γ , γ ] (?α δ (x ? z )) × ( 2 )δ (x ? y ) 2 ? + Λ2 1 1 ( 2 )(i) (? ν δ (x ? z )) + 2δ (x ? z )? ν +γ ? i ? ? m ? + Λ2 1 Λ2 + [γ α , γ ν ] (?α δ (x ? z )) × ( 2 )2 δ (x ? y ) 2 ? + Λ2 lim T r γ ? (4.7) 20

We now use (2.24) in (4.7), and we obtain the momentum representation of the vacuum polarization tensor (see also (2.23))

Π?ν (q ) =

(?1) 1 Λ2 d4 k ? ν T r γ γ ( )2 (2π )4 k + q ? m k ? m ?k 2 + Λ2 1 Λ2 1 ( )2 q ν + 2k ν + [γ α , γ ν ] qα +γ ? 2 2 k + q ? m ?(k + q ) + Λ 2 1 ) ×( 2 ?k + Λ2 1 1 1 +γ ? ( ) q ν + 2k ν + [γ α , γ ν ] qα 2 2 k + q ? m ?(k + q ) + Λ 2 2 Λ ×( 2 )2 . ?k + Λ2

(4.8)

The ?rst term in (4.8) stands for the naive momentum cut-o? by a form factor, which generally spoils gauge invariance. The remaining two terms in (4.8) recover gauge invariance spoiled by the ?rst term. After the standard trace calculation and using the Feynman parameters, the ?rst term in (4.8) gives

(

Λ4 ) 4π 2

1 0



1?α 0

dβ ?g ?ν

+ ?m2 g ?ν ? α(1 ? α)g ?ν q 2 + 2α(1 ? α)q ? q ν × → ( 1 ) 4π 2 [?α(1 ?
1 0

?α(1 ?

α )q 2

β + (1 ? β )m2 + β Λ2

α )q 2

β + (1 ? β )m2 + β Λ2 ]2

dα2α(1 ? α)(q ? q ν ? g ?ν q 2 )ln

+(

1 1 ?1 2 ) (Λ ? m2 )g ?ν + g ?ν q 2 2 4π 2 6 1 ? ν 5 ? ν ? q q ? (q q ? g ?ν q 2 ) 3 18

Λ2 ?α(1 ? α)q 2 + m2

(4.9)

for Λ → large. Similarly, the second and third terms in (4.8) together give Λ4 ) 4π 2
1 0 1?α 0

(



dβ g ?ν

? α(2α ? 1)q q + α(g ?ν q 2 ? q ? q ν )
? ν

?α(1 ?

α )q 2

1?β + βm2 + (1 ? β )Λ2

21

1?β [?α(1 ? α)q 2 + βm2 + (1 ? β )Λ2 ]2 1 5 1 → ( 2 ) (Λ2 ? m2 )g ?ν + g ?ν q 2 4π 2 36 1 ? ν 1 ? ν + q q + (q q ? g ?ν q 2 ) 36 4 ×

(4.10)

for Λ → large. These two expressions in (4.9) and (4.10) put together ?nally give rise to the familiar gauge invariant result 1 Λ2 . ? 2 2 ?α(1 ? α)q + m 3

(

1 )(q ? q ν ? g ?ν q 2 ) 4π 2

1 0

dα2α(1 ? α)ln

(4.11)

The result for the chiral gauge theory (2.1) is obtained from (4.11) by setting m = 0 and
1 multiplying it by 2 T rT a T b .

The covariant regularization scheme thus gives rise to a gauge invariant result on the basis of well-regularized ?nite calculations. It is important that we always stay in d=4 dimensional space-time in this calculation. This property is crucial for a reliable treatment of the anomaly. The coe?cient of lnΛ2 in (4.11), which is related to the renormalization group β -function, is independent of the choice of f (x2 ) in (4.2). For example, one can con?rm that f (x2 ) = ( Λ2 )n , n ≥ 2 x2 + Λ2

(4.12)

gives the same numerical coe?cient of lnΛ2 by dividing (4.9) and (4.10) by Λ4 and taking suitable derivatives with respect to Λ2 . The ?nite term, -1/3, in (4.11) depends on the speci?c regulator ; this is not a drawback since the ?nite term is uniquely ?xed by the renormalization condition in renormalizable theory. The regulator independence of the coe?cient of lnΛ2 is also expected from the fact that the Weyl anomaly in (3.6) (and related β -function)is independent of f (x2 ). The present covariant regularization can be readily applied to the calculations of higher point functions and to practical calculations in chiral gauge theory such as the WeinbergSalam theory ; the covariant regularization can handle gauge anomalies in a reliable way , and thus one can treat lepton and quark sectors separately without grouping the fermions into a multiplet of SO(10). The Higgs coupling, which mixes left-and righthanded components, is readily handled by the present method as is explained in [9]. 22

5

Discussion and Conclusion

Motivated by the interesting suggestion of generalized Pauli-Villars regularization, we reexamined the regularization and anomalies in gauge theory. The generalized Pauli-Villars regularization as reformulated as a regularization of component operators in this paper will perhaps make the covariant regularization, which has been known for some time, more acceptable ; the Lagrangian level realization of the covariant regularization for anomalyfree gauge theory however remains as an open question. The covariant regularization spoils the Bose symmetry for anomalous gauge theory, but it preserves the Bose symmetry as well as gauge invariance for anomaly-free gauge theory. Our analysis here is con?ned to one-loop level calculations, though certain nonperturbative aspects such as instantons are also involved. As for multi-loop diagrams, the higher derivative regularization in the sector of gauge ?elds [14], for example, can render all the multi-loop diagrams ?nite. The one-loop diagrams which include only the gauge ?elds cannot be regularized by the higher derivative regularization but they can be covariantly regularized if one uses the covariant background gauge technique[15]. In fact, we recently illustrated a simple non-diagramatic calculation of one-loop β -function of QCD by using the method of covariant anomaly[16]. The generalized Pauli-Villars regularization is also known to have interesting implications on lattice gauge theory [17], but its analysis is beyond the scope of the present paper. In conclusion, we have shown that the basic mechanism of generalized Pauli-Villars regularization of continuum theory is made transparent if one looks at it from the view point of a regularization of composite current operators ; by this way , one can readily compare the generalized Pauli-Villars regularization with the covariant regularization of chiral gauge theory. The covariant regularization scheme, which is quite ?exible, has been also shown to be useful in practical calculations.

23

Appendix
For the sake of completeness, we here quote the proof of f(x)- independence of (3.3) and (3.5)[6]. The calculation of (3.5), for example, proceeds as

iT r

d4 k ?ikx e γ5 f (D 2 /Λ2 )eikx 4 (2π ) (ik? + D? )(ik ? + D ? ) ? ig [γ ? , γ ν ] F?ν d4 k 4 γ f ( ) = iT r 5 (2π )4 Λ2 d4 k 2ik ? D? D ? D? ? = iΛ4 T r γ f ( ? k k + + 5 ? (2π )4 Λ Λ2 ig ? 2 [γ ? , γ ν ] F?ν ) 4Λ

(A1)

where we used

D2 =

1 ? ν 1 {γ , γ }D? Dν + [γ ? , γ ν ] D? Dν 2 2 ig = D? D ? ? [γ ? , γ ν ] F?ν 4

(A2)

around x = ?k? k ? = |k 2 | as

and re-scaled the variable k? → Λk? . We next expand the quantity involving f(x) in (A.1)

f (?k? k ? +

2ik ? D? D ? D? ig ? ν + ? [γ , γ ] F?ν ) 2 Λ Λ 4Λ2 ig ? ν 2ik ? D? D ? D? + ? [γ , γ ] F?ν } = f (?k? k ? ) + f ′ (?k? k ? ){ 2 Λ Λ 4Λ2 2ik ? D? D ? D? ig 1 + ? 2 [γ ? , γ ν ] F?ν }2 + ... + f ′′ (?k? k ? ){ 2 2! Λ Λ 4Λ

(A3)

When Λ → ∞, only the terms of order 1/Λ4 or larger in (A.3) survive in (A.1). Moreover, the trace T r (γ5...) is non-vanishing only for the terms with more than four γ -matrices. The only term that satis?es these two conditions is the third term in (A. 3) with ([γ ? , γ ν ] F?ν )2 . The calculation of (A.1) thus becomes

iT r

d4 k 1 ?ig ? ν γ5 f ′′ (?k? k ? ){ [γ , γ ] F?ν }2 4 (2π ) 2! 4 24

1 ?ig ? ν 1 = iT rγ5 { [γ , γ ] F?ν }2 2 4 16π 2 2 ig )T r??ναβ F?ν Fαβ = ( 2 32π

∞ 0

dxxf ′′ (x) (A4)

after taking the trace over γ -matrices: We here used d4 k = π 2 |k 2 |d|k 2|, and
∞ 0

dx xf ′′ (x) = xf ′ (x)|∞ 0 ?

∞ 0

dxf ′ (x) (A5)

= ?f (x)|∞ 0 = f (0) = 1

by noting the conditions (2.31) including xf ′ (x) → 0 for x → ∞. The result (A.4) is thus independent of the regulator f(x) ; the convenient choice of f(x) for practical calculations is (4.12) or f (x) = exp [?x]. The above analysis is also applicable to (3.3). It is known that a similar analysis holds for (3.6)[8]. Note Added After submitting the present paper , the works by Narayanan and Neuberger [18] on the Kaplan’s formulation came to my attention. These authors analyze two-dimensional gauge anomaly , in particular consistent from of anomaly , from a view point of 2+1 dimensional theory. This calculational scheme is apparently di?erent from the generalized Pauli-Villars regularization in (2.5) , which is not applicable to the evaluation of gauge anomaly.

References
[1] S.A.Frolov and A.A.Slavnov, Phys. Lett. B309 (1993)344 [2] R.Narayanan and H.Neuberger, Phys. Lett.B301(1993)62 [3] S.Aoki and Y.Kikukawa, Mod. Phys. Lett. A8(1993)3517 [4] G.’t Hooft, Phys. Rev. Lett. 37(1976)8 ; Phys. Rev. D14(1976)172 ; D18(1978)2199(E) [5] S.Adler, Phys.Rev. 177(1969)2426 J.S.Bell and R.Jackiw, Nuovo Cim, A60(1969)47 25

[6] K.Fujikawa, Phys. Rev. D21(1980)2848; 22(1980)1499(E); Phys. Rev. Lett. 42(1979)1195 [7] L.Alvarez-Gaum? e and E.Witten, Nucl. Phys. B234(1983)269 [8] K.Fujikawa, Phys. Rev. Lett. 44(1980)1733; Phys. Rev. D23(1981)2262 [9] K.Fujikawa, Phys. Rev. D29(1984)285 [10] W.A.Bardeen and B.Zumino, Nucl. Phys. B244(1984)421 [11] J.Wess and B.Zumino, Phys. Lett. 37B(1971)95 [12] M.Atiyah and I.Singer, Ann. Math. 87(1968)484 [13] W.A.Bardeen, Phys. Rev. 184(1969)1848 H.Shinke and H.Suzuki, Mod. Phys. Lett. A40(1993)3835 [14] L.D.Faddeev and A.A.Slavnov, GaugeF ields (Benjamin / Cummings, 1980) [15] G.’t Hooft, Nucl. Phys. B62(1973)444 [16] K.Fujikawa, Phys. Rev. D48(1993)3922 [17] D.B.Kaplan, Phys. Lett. B288(1992)342 [18] R.Narayanan and H.Neuberger , Phys. Rev. Lett. 71(1993)3251 ; Nucl. Phys. B412(1994)574

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