Chiral anomaly as a composite operator in the gradient flow exact renormalization group formalism

The gradient flow exact renormalization group (GFERG) is an idea that incorporates gauge invariant gradient flows into the formalism of the exact renormalization group (ERG). GFERG introduces a Wilson action with a cutoff while keeping vector gauge invariance manifestly. The details of the formalism are still to be worked out. In this paper, we apply GFERG to construct the Wilson action of massless Dirac fermions under the background chiral gauge fields. By formulating the chiral anomaly as a ``composite operator,'' we make the scale invariance of the anomaly manifest. We argue that the same result extends to QCD.


Introduction
The gradient flow has been introduced to lattice gauge theories as an alternative to the renormalization group transformation [1][2][3][4][5][6][7][8][9].It has already established itself as a practical tool to calibrate the physical scale of lattice simulations (see Ref [10] for a recent review), but its relation to the renormalization group transformation has not yet been fully understood.What we call the gradient flow exact renormalization group (GFERG) [11][12][13][14][15] is an attempt to construct a Wilson action [16][17][18][19][20][21][22] whose cutoff dependence is governed by the gradient flow;1 even the formalism is still at an infant stage, however, and has been constructed in some details only for QED so far.
In the GFERG formalism, we work with continuous fields in a continuous space.The role of a physical cutoff is played by the diffusion time.As in lattice gauge theories, the vector gauge invariance is manifestly preserved in GFERG.With the advantage of a continuum language, we remain hopeful that the formalism would give us new insights into the renormalization group flows among the gauge theories.
The finite momentum cutoff of a Wilson action often leads to a misconception that the action may miss the physics at short distances, such as the axial or chiral anomalies.In fact the interaction vertices of a Wilson action result from the physics at distances shorter than the cutoff distance (the inverse of the momentum cutoff), and the Wilson action keeps the anomalies intact under the lowering of the momentum cutoff.The axial anomaly of QED and the chiral anomaly of the free massless fermions under the external chiral gauge fields have been calculated in the conventional exact renormalization group (ERG) formalism [20,26,27].
The biggest advantage in calculating anomalies using a finite cutoff theory is the absence of any subtlety in the calculations.There is no need to take the momentum cutoff to infinity, and the calculations tend to be straightforward.In this paper, we would like to consider the massless fermions under the external chiral gauge fields, and derive the chiral anomalies using the GFERG formalism.
Two notions play important roles: Composite operators and their scaling dimensions.
Both are standard notions in quantum field theory, but they carry specific meanings in the ERG and GFERG formalisms.A composite operator [18] is a functional of fields that can be considered as an infinitesimal change of the exponentiated Wilson action.If S τ is the Wilson action at a logarithmic distance scale τ , and O τ a composite operator with scaling dimension −y, then e yτ O τ satisfies the same ERG or GFERG equation as e Sτ .
We will formulate the chiral anomaly as a composite operator that generates the chiral transformation of the fermion and external gauge fields.Its non-vanishing results in the anomalous Ward-Takahashi identities.We show that the anomaly is a composite operator with zero scaling dimension, meaning that the anomaly, inherited from the short distances, is independent of the scale.This is reminiscent of Zee's proof [28] of the non-renormalization of the axial anomaly in QED [29]. 2n the GFERG formalism, we diffuse not only the chiral fermion fields but also the external chiral gauge fields.Since the anomaly is scale invariant, it is more natural to express the anomaly in terms of the undiffused gauge fields which we call the −1 variables.The advantage of GFERG is the manifest invariance under the vectorial gauge transformations.
Hence, the chiral anomaly is given most naturally in what is known as the Bardeen form [33].
The present paper is organized as follows.In Sect.2, we introduce GFERG for chiral fermions interacting only with external chiral gauge fields.In Sect.3, we introduce the chiral BRST transformation and the chiral anomaly.In particular, we show that the anomaly is a composite operator whose scale dimension vanishes.In Sect.4, we introduce the oneparticle-irreducible (1PI) formalism and use it to calculate the anomaly.Our calculation is based upon the previous work by one of us [15].After commenting on the chiral anomaly in QCD in Sect.5, we conclude the paper in Sect.6.We have prepared two appendices.In Appendix A, we introduce the −1 variables which are the undiffused fields corresponding to the bare fields at short distances.In Appendix B, we derive useful identities for functional integrals over fermions.
Throughout the paper, we work in the D = 4 dimensional Euclidean space, but we keep writing D to make apparent the dependence on space dimensions.We use the following notation for momentum integrals:

GFERG for the free massless Dirac fermion
We consider N free massless Dirac fermions whose classical global symmetry is U(N) L × U(N) R .To consider the anomaly associated with the subgroup SU(N) L × SU(N) R , we first introduce an external (i.e., non-dynamical) gauge field L a µ (x) that couples to the SU(N) L fermion current.We postpone the introduction of the right-handed counterpart till Sect. 4.
In the dimensionless convention the Wilson action S τ is given by3 Here, τ is a logarithmic renormalization scale, and this "integral representation" relates two Wilson actions at RG scales τ and τ 0 , respectively.What we call an "unscrambler" is defined by (ŝ ′ ) −1 is obtained by replacing ψ, ψ by ψ ′ , ψ′ .In Eq. (2.1), the Wilson action is obtained by the "blocking procedure" implemented by the Gaussian integration; it is thus natural not to include the Gaussian integration for the external gauge field.The primed field variables in the exponent are given by the solutions to the diffusion equations, where the diffusion time t ′ ranges from 0 to t − t 0 , given by the difference of the logarithmic RG scales τ − τ 0 as α 0 is an arbitrary positive parameter that controls the diffusion of the unphysical longitudinal part, T a denote anti-Hermitian generators of SU(N) (satisfying [T a , T b ] = f abc T c ), and where γ 5 ≡ γ 1 γ 2 γ 3 γ 4 .The initial conditions for the above diffusion equations are given by the integration variables in Eq. (2.1), i.e., In Eqs.(2.3), the external gauge field also satisfies the diffusion equation where is the field strength, and the covariant derivative is defined by (2.9) We select a particular solution satisfying the boundary condition This determines as a functional of L a µ (x).The Wilson action S τ is constructed so that the correlation functions, modified by the insertion of ŝ−1 , are given by [34] ŝ−1 ψ(x 1 ) (2.12) (We derive this in Appendix B.) Hence, differentiating this with respect to τ , we obtain This amounts to the GFERG equation given by where the "scrambler" ŝ, the inverse of ŝ−1 , is given by reversing the sign in the exponent in Eq. (2.2) as and ∆'s are defined by the expressions in Eqs.(2.3) and (2.7) by removing all the primes.In this paper, we assume that the Wilson action be at most bi-linear in the fermion fields.
Under Eq. (2.14), this property is preserved.As discussed in Refs.[11,12] and as we explain in what follows, the above diffusion equations (referred to as gradient flow equations) commute with the external gauge transformations.On the other hand, the Gaussian integration and the unscrambler preserve only the vectorial part SU(N) V of the external gauge invariance.
We can thus take it for granted that the Wilson action is invariant under SU(N) V .

BRST transformation and the chiral anomaly
To analyze possible non-invariance of the Wilson action under the gauge transformations of L a µ (x), we introduce a Grassmann-odd ghost field χ a L (x), which simply plays the role of a transformation function of the gauge transformations.The chiral BRST transformation, corresponding to an arbitrary SU(N) L gauge transformation, is generated by the differential which is nilpotent δ2 L = 0 by construction.Using the unscrambler (2.2), we further define δL which is still nilpotent: δ2 We define the chiral anomaly-possible non-invariance of the Wilson action S τ under the BRST transformation-by Its correlation functions satisfy

Chiral anomaly as a scaling composite operator
The chiral anomaly Q Lτ is linear in χ a L .In order to promote the anomaly to a scaling composite operator, we need to diffuse the ghost field as We identify the original ghost field χ a L (x) with the boundary condition for the above diffusion equation This is analogous to Eq. (2.7).Under this setup, the anomaly (3.4) behaves in a very simple way under the GFERG transformation.
To see this, let us apply the BRST transformation, on the diffusion equations given by Eqs.(2.3), (2.7), and (3.7).After some calculations, we find that the changes in the diffusion equations are proportional to ) which vanishes thanks to Eq. (3.7).Therefore, with the adoption of Eq. (3.7) for χ ′a L , all the diffusion equations are invariant under the BRST transformation (3.9).In other words, the diffusion and the BRST transformation commute.
The BRST transformation (3.9) acts not only at the intermediate t ′ but also at the boundaries t ′ = 0 and t ′ = t − t 0 .We wish to show that the consistency of the diffusion with Eq. (3.9) implies where L a µ on the left-hand side is related to L ′a µ on the right-hand side by the diffusion equation as Eqs.(2.10) and (2.11).For brevity we merely sketch a (rather formal) derivation.
Following the definition (3.4) we obtain Applying ŝ−1 on Eq. (2.1) and using the identity (B6) obtained in Appendix B, we obtain Since the BRST transformation commutes with diffusion, we obtain δL + δ′ We then obtain where δ′ L was moved from left to right by the integration by parts.Finally, applying ŝ from the left, and using Eq.(3.11) and δ′ we obtain the desired result, Eq. (3.10 where the last term arises from the the dependence of Q Lτ on the ghost field.It is important that Eq. (3.18) is linear in Q Lτ .Hence, if Q Lτ = 0 for a certain τ , then Q Lτ = 0 for any τ .
4 Chiral anomaly in the 1PI formalism

1PI formalism and the −1 variables
To study the structure of the anomaly systematically, it is convenient to employ the 1PI formalism of GFERG [14].The Legendre transformation from the Wilson action S τ to the 1PI action Γ τ is given by Extremizing the right-hand side with respect to ψ and ψ, we obtain Note that we do not Legendre transform with respect to the external gauge field.The inverse Legendre transform is given by Hence, we obtain At this stage, we introduce the following extremely useful "−1 variables".These are local functionals of the original field variables with the following functional dependences: These variables possess the following simple scaling properties: and As we explain in Appendix A, these variables can be obtained by solving diffusion equations backward in the diffusion time to t = −1, corresponding to the logarithmic scale τ = −∞.
The above relations allow us to rewrite the GFERG equation for the 1PI action in a simple form as where and L µ (t, x) ≡ L a µ (t, x)T a .In this expression, we have used Eq.(4.3) to simplify the last term, for which Ψ(x) ← − δ /δψ(y) is given as the inverse of ψ(x) For the Wilson action S τ at most bi-linear in the fermion fields, the corresponding 1PI action Γ τ is also at most bi-linear in the fermion fields, and thus Ψ(x) ← − δ /δψ(y) depends only on the external gauge field.
In Appendix A, we also show that the BRST transformation is inherited by the corresponding −1 variables as Using the above relations, the anomaly (3.4) is written as Now, let us split the 1PI action into two parts: The part bi-linear in the fermion fields, Γ f,τ , can be taken manifestly invariant under the BRST transformation (4.12).For instance, is a choice that solves the bi-linear part of the GFERG equation (4.8). 6With such a choice, the expression of the chiral anomaly (4.13) reduces to which is completely independent of the fermion fields.Then, the WZ condition (3.6) simplifies to and the GFERG equation (3.18) reduces to Now, we can repeat the above argument by introducing another external gauge field R a µ (x), which couples to the right-handed fermion current.All the considerations go through with trivial changes: the 1PI action now depends also on R a µ , and the associated anomaly Q R is proportional to the ghost χ a R for the SU(N) R transformation, etc.To summarize, we obtain the gauge field part of the action satisfies the GFERG equation and the anomaly, defined by satisfies both the GFERG equation and the WZ consistency condition In the following, we would like to show that Q τ is a scale invariant functional of L a −1,µ and R a −1,µ with a τ -independent overall constant.To begin with, we note that Γ f,τ (4.19) has no explicit τ dependence.Hence, neither does ψ(x) ← − δ /δΨ(y) given by Eq. (4.10).Inverting this we obtain τ -independent Ψ(x) ← − δ /δψ(y).
We can then expand where Then, Eq. (4.20) implies where the last sum is a homogeneous solution to Eq. (4.20) corresponding to local counterterms.We can choose to remove We keep A ′ 0 [L, R] to simplify the form of the anomaly later.
Thus, the τ dependent part, A 0 [L, R], of Γ g,τ has scaling dimension 0. Differentiating Eq. (4.21) with respect to τ and using Eq.(4.27), we obtain which is a τ -independent functional with scaling dimension 0. This implies we can expand where C d is a functional with scaling dimension d.Substituting this result into Eq.(4.22), we obtain implying and Thus, the anomaly is a τ -independent functional with scaling dimension 0. This is what we call the scale invariance of the anomaly.By definition, the anomaly is linear in χ a −1L and χ a −1R .Since the GFERG keeps the vector gauge invariance manifestly, we also know that the anomaly vanishes if L a µ = R a µ and χ L = χ R : We now recall that Eq. (4.20) leaves Γ τ ambiguous by the addition of a scale invariant functional A ′ 0 [L, R] in Eq. (4.27).Since the anomaly satisfies the WZ condition (4.23), by choosing A ′ 0 [L, R] appropriately, we can make Γ g,τ invariant under the vector transformations in the presence of both left and right gauge fields: Now, the anomaly arises only in the axial transformations parametrized by We thus obtain the anomaly Q τ as a scale invariant functional, denoted by Q A , of L a −1,µ , R a −1,µ , and χ a −1,A .In D = 4, such a combination is well-known, and is given by the Bardeen form of the gauge anomaly [33], written in terms of the −1 variables as7 where A is a τ -independent real constant, and

Computation of A
The coefficient A in Eq. ( 4.36) can be determined by choosing a particular gauge field configuration, R µ = L µ = V µ .The anomaly is then simplified to We may further set the −1 variables to the lowest order in gauge fields; for the choice α 0 = 1,8 we have Hence, in momentum space, we find where the terms higher order in gauge fields are suppressed.On the other hand, Eq. (4.21) gives

.40)
The first term, being the covariant derivative of a local term [δ/δA c −1,µ (x)]Γ g,τ , gives no contribution to the lowest order in (k + l) 2 and can be neglected for the determination of A in Eq. (4.39).
For the last term of (4.40), Ψ(x) ← − δ /δψ(y) is obtained by inverting ψ(x) where ( Vµ can also be found in Ref. [14].)Since the diffusion equations (A1) do not contain any Dirac matrices, the higher order O(V 2 ) terms in Eq. (4.41) are also linear in γ µ .But the trace in the last term of Eq. (4.40) requires at least four Dirac matrices, and the O(V 2 ) term in Eq. (4.41) gives no contribution.We then obtain To the first order in k and l, the integral has been calculated in Ref. [15] as9 p e −(p−l) 2 e −(p+k) 2 tr γ 5 e −2(p+k where the integral is evaluated as

.46)
Comparing the result with Eq. (4.39), we obtain We also get the Gaussian integration over the dynamical gauge fields v A µ (x).Correspondingly, the unscrambler (2.2) must have the additional factor exp{−(1/2) d D x δ 2 /[δv A µ (x)δv A µ (x)]} for the dynamical gauge fields.The diffusion equation for the fermi fields depends also on the dynamical gauge fields, which satisfy their own diffusion equation.
The external gauge transformation remains generated by Eq. (3.1) and commutes with the diffusion equations.The chiral anomaly Q Lτ , still given by Eq. (3.4), remains a composite operator with scaling dimension 0. The GFERG equation for Q Lτ , however, is no longer given by Eq. (3.18); the scaling dimension of the fermions is changed from (D − 1)/2 to (D − 1)/2 + γ τ , and we get an additional term of functional differentiation with respect to the dynamical gauge fields.We can introduce the −1 variables for the dynamical gauge fields by repeating the argument given in Appendix A, but a simple construction such as Eq.(4.15) does not solve the GFERG equation anymore.Though we believe Eq. (4.16) is still valid, its derivation does not go through unchanged.
However, if we assume that Q τ depends only on the external gauge fields L a µ and R a µ , then Q τ again satisfies Eqs.(4.21), (4.22), and (4.23).We can now repeat the above argument and infer that Eq. (4.36) is the most general form of the anomaly in the presence of both left and right gauge fields.Then Eq. (4.22) implies that the anomaly coefficient A has no τ dependence.This implies further that A cannot depend on the τ -dependent gauge coupling constant g τ to the dynamical gauge fields.We can thus conclude that the chiral anomaly is not affected by the inclusion of the dynamical gauge fields.This is the statement of the Adler-Bardeen theorem [28].

Conclusion
In this paper we have successfully tested the GFERG formalism by deriving the chiral anomaly for the free massless fermions under arbitrary background left and right gauge fields.We have formulated the anomaly as a composite operator that generates the chiral gauge (BRST) transformations, and have shown that its scaling dimension vanishes, i.e., the chiral anomaly is scale invariant.We have also shown that the standard form of the anomaly is obtained in terms of the −1 variables which are functionals of the diffusing external gauge fields.As in the ERG formalism, the chiral anomaly is expressed by a finite loop integral that we can calculate without taking a short distance limit.

.47) 5
Comment on the inclusion of QCDSo far, we have considered the massless free fermions coupled only to external gauge fields.Before concluding the paper we would like to comment on how our argument and conclusion should (or should not) be changed when we couple the fermions to dynamical gauge fields such as the gluons in QCD[11,12].First of all, in defining the Wilson action (2.1), we should change the canonical scaling factor [(D − 1)/2](τ − τ 0 ) by including the anomalous dimension τ τ 0 dτ ′ [(D − 1)/2 + γ τ ′ ].