Gradient flow exact renormalization group -- inclusion of fermion fields

The gradient flow exact renormalization group (GFERG) is an exact renormalization group motivated by the Yang--Mills gradient flow and its salient feature is a manifest gauge invariance. We generalize this GFERG, originally formulated for the pure Yang--Mills theory, to vector-like gauge theories containing fermion fields, keeping the manifest gauge invariance. For the chiral symmetry we have two options: one possible formulation preserves the conventional form of the chiral symmetry and the other simpler formulation realizes the chiral symmetry in a modified form \`a la Ginsparg--Wilson. We work out a gauge-invariant local Wilson action in quantum electrodynamics to the lowest nontrivial order of perturbation theory. This Wilson action reproduces the correct axial anomaly in~$D=2$.

The gradient flow exact renormalization group (GFERG) is an exact renormalization group motivated by the Yang-Mills gradient flow and its salient feature is a manifest gauge invariance. We generalize this GFERG, originally formulated for the pure Yang-Mills theory, to vector-like gauge theories containing fermion fields, keeping the manifest gauge invariance. For the chiral symmetry we have two options: one possible formulation preserves the conventional form of the chiral symmetry and the other simpler formulation realizes the chiral symmetry in a modified formà la Ginsparg-Wilson. We work out a gauge-invariant local Wilson action in quantum electrodynamics to the lowest nontrivial order of perturbation theory. This Wilson action reproduces the correct axial anomaly in D = 2.

Introduction
The Wilson exact renormalization group (ERG) [1] (see also Refs. [2,3]; for reviews, see Refs. [4][5][6][7][8]) is important because, among many other things, it provides a unique framework to consider possible quantum field theories beyond perturbation theory. Given specific field contents, all possible quantum field theories are obtained by the continuum limit around each fixed point of the ERG equation with that field contents. In this sense, we may regard ERG as a theory of theories.
In particle physics, gauge symmetry is a fundamental principle and we are thus interested in ERG trajectories, i.e. solutions of the ERG equation, which preserve this symmetry. Traditional formulations, however, employ the momentum cutoff to define the ERG transformation, and this cutoff explicitly breaks the gauge symmetry in the conventional form. Although it is possible to define a modified gauge transformation which is consistent with the ERG evolution [9] (see also Refs. [6,10] and references cited therein), and in principle one can maintain the gauge invariance in the modified form, since such a transformation depends on the Wilson action itself, it appears very hard to determine a nonperturbative truncation of the Wilson action being consistent with this exact symmetry of ERG. For nonperturbative applications of ERG in particle physics, therefore, a manifestly gauge-invariant ERG formulation is highly desirable. Such formulations have been developed, for instance in Refs. [11][12][13][14][15][16][17].
One direct connection between ERG and a diffusion equation may be observed as follows [18,34]. The ERG evolution of the Wilson action S τ is described by the Wilson-Polchinski equation [1,35]. For the scalar field theory in D-dimensional spacetime, in dimensionless variables, it reads 1 where τ parametrizes the ERG evolution and the functions K(p) and k(p) specify the ERG transformation; ∆(p) ≡ −2p 2 (∂/∂p 2 )K(p). 1 In momentum space, we adopt the convention As pointed out in Ref. [36], the ERG evolution of the Wilson action S τ under Eq. (1.2) can be neatly formulated as an equality, between the modified correlation functions defined by where the correlation function on the right-hand side is the conventional one with respect to the action S. The anomalous dimension η τ in Eq. (1.2) and the wave function renormalization factor Z τ in Eq. (1.3) are related by shows that the field variable is multiplicatively renormalized by Z τ under the ERG evolution, when it is viewed in terms of the modified correlation function. In this sense, what is suitable to characterize the scaling or critical behavior under the ERG transformation is the modified correlation function rather than the conventional correlation function. This fact explains why in Eq. (1.2) the anomalous dimension η τ , which is related to a "rescaling" of the field variable, should appear not only in the coefficient of the first-order functional derivative, but also in the coefficient of the second-order functional derivative. On this issue, see Refs. [37,38]. Now, it can be readily seen that Eq. (1.3) in coordinate space 2 is represented in terms of a functional integral as Here, we assume a particular form of K and k [1], 3 The point is that in Eq. (1.7), the field ϕ ′ (t, x) inside the delta function is given by the solution of the diffusion equation where the initial configuration for the diffusion is given by the integration variable φ ′ in the functional integral in Eq. (1.7); the dimensionless diffusion or flow time t and the ERG 2 We define . We note that the structure of Eq. (1.7) is very simple: it consists of exponential functions of the second-order functional derivative and the delta function which imposes the equality of the argument of the Wilson action and the diffused field. The diffused field ϕ ′ is rescaled in the normalization by e τ (D−2)/2 Z 1/2 τ , where (D − 2)/2 is the canonical mass dimension of the field, and in the spacetime coordinate as x → xe τ .
Considering the continuum limit around a fixed point of the ERG equation, the above connection relates the correlation function given by the functional integral with respect to the Wilson action with a finite momentum cutoff Λ and the correlation function of the diffused field at the (dimensionful) diffused or flow time t = 1/Λ 2 with respect to the bare action (with the parameter renormalization, such as the one considered in Ref. [39]) [34]. This relation provides an intuitive understanding [34] of the fact that the renormalization of parameters and the wave function of the diffused elementary scalar field automatically make the equal-point product of diffused fields finite; the reason is that the functional integral with respect to the Wilson action possesses an ultraviolet (UV) cutoff Λ. This finiteness is analogous to a remarkable property [22] of the gauge field diffused by the Yang-Mills gradient flow. These observations motivated a proposal of GFERG in the pure Yang-Mills theory in Ref. [18].
In the present paper, we generalize the GFERG in Ref. [18] to vector-like gauge theories containing fermion fields. As a natural generalization, we can maintain the manifest gauge invariance. For the chiral symmetry, we have two options: one possible formulation (see Appendix A) preserves the conventional form of the chiral symmetry, while the other simpler formulation presented in Sect. 2 realizes the chiral symmetry in a modified form known as the Ginsparg-Wilson (GW) relation [40]. Our derivation of the GW relation in the present manifestly gauge-invariant ERG formulation is very simple. In Sect. 3, to have some idea how the GFERG equation works, we compute a gauge-invariant local Wilson action in quantum electrodynamics (QED) to the lowest nontrivial order of perturbation theory. Section 4 is devoted to our conclusion. In Appendix B we compute the axial anomaly in D = 2 by using our gauge-invariant local Wilson action obtained in Sect. 3.

GFERG for vector-like gauge theories
Our idea for the construction of a GFERG equation in vector-like gauge theories would be almost obvious from the elucidation in the previous section. Imitating the structure of Eq. (1.7), we define the Wilson action by In this expression, the diffused gauge field B ′ (t, x) is the solution to the Yang-Mills gradient flow equation [19][20][21] where α 0 is a parameter and the initial configuration A ′ is given by the integration variable in Eq. (2.1). We have defined from the structure constants of the gauge group f abc defined from anti-Hermitian generators T a by [T a , T b ] = f abc T c . In Eq. (2.1) we have taken the canonical mass dimension of the gauge potential 1 and written the wave function renormalization factor of the gauge field as g −2 τ ; the reason for this convention will become clear later. 4 Similarly, for the fermion field, we use the diffusion equations in Ref. [41], As in Ref. [18], it is easy to see that the construction in Eq. (2.1) preserves the partition function: (2.6) Let us examine other properties that follow from Eq. (2.1).

Gauge symmetry
In an almost identical way to Ref. [18], we can see that the Wilson action in Eq.
if the initial action S τ =0 [A, ψ,ψ] is invariant under the above transformation with τ = 0: First, the exponential functions of the second-order functional derivatives are manifestly invariant under the gauge transformation (see Ref. [18]). Then, the gauge transformation on the argument of the Wilson action in Eq. (2.1) is transmitted, through the delta functions, to the gauge transformation on the diffused fields B ′ , χ ′ , andχ ′ . This gauge transformation is then, through the gauge covariance of the diffusion equations, transmitted to that on the initial configurations A ′ , ψ ′ , andψ ′ . Thus, the gauge invariance of the Wilson action finally depends on the gauge invariance of the initial action S τ =0 [A ′ , ψ ′ ,ψ ′ ] and of the integration measure [dA ′ dψ ′ dψ ′ ] (for which we assume its invariance).
In a similar manner, we can see the independence of S τ [A, ψ,ψ] from the parameter α 0 in the diffusion equations, Eqs. (2.2) and (2.4). To see this, let us suppose that we make an infinitesimal change of the parameter, α 0 → α 0 + δα 0 . For a fixed initial configuration A ′ , ψ ′ , andψ ′ , the solution of Eqs. (2.2) and (2.4) will change under this. On the other hand, we see that if we make the following infinitesimal transformation in Eqs. (2.2) and (2.4), where the function ω a (t, x) is defined as the solution of then the change of the parameter δα 0 in the diffusion equations can be compensated. By integrating Eq. (2.10) "backward against time" from ω(t, x) = 0 to ω(t = 0, x), we then have a gauge transformation ω(t = 0, x) on the initial configuration A ′ , ψ ′ , andψ ′ such that the solution, B ′ , χ ′ , andχ ′ , is identical to that before the change of α 0 . This shows that if the initial action S τ =0 [A ′ , ψ ′ ,ψ ′ ] and the integration measure in Eq. (2.1) are gauge invariant, then the Wilson action S τ [A, ψ,ψ] is independent of the parameter α 0 .

Modified chiral symmetry: GW relation
An important symmetry in a system containing the fermion field is the chiral symmetry. The Wilson action in Eq. (2.1) cannot be invariant under the conventional form of the chiral transformation, i.e.
This follows from the fact that, under Eq. (2.11), and thus the exponential function in Eq. (2.1), does not possess a simple transformation property under Eq. (2.11). One can avoid this drawback by putting an odd number of Dirac matrices in the expression such as , (2.14) where we have to also put the covariant derivative to preserve the gauge (and Lorentz) invariance. This "manifestly chiral-invariant formulation" is actually a possible option, and we write down the corresponding ERG equation in Appendix A.
Here, we pursue the simpler construction, Eq. (2.1). Quite interestingly, the Wilson action in Eq. (2.1) can be invariant under a modified chiral transformation; this is nothing but the chiral symmetry realized by the GW relation [40] 5 (for developments in the context of lattice gauge theory, see Refs. [42][43][44][45][46][47]; for studies in the context of ERG, we may refer, for instance, to Refs. [48,49]).
To find the exact chiral symmetry in Eq. (2.1), we introduce differential operators, In this expression,γ 5 acting on ψ andψ amounts, through the delta functions, to the chiral transformation on χ ′ andχ ′ because, e.g., δ(γ 5 ψ − e τ (D−1)/2 Z 1/2 Since the flow equations in Eq. (2.4) preserve the conventional chiral symmetry, the chiral transformation on χ ′ andχ ′ induces the transformation on the initial configuration, ψ ′ andψ ′ . Then, again using the definition ofΓ 5 , we see that 6 In this sense, our ERG evolution preserves the invariance under the modified chiral transformation generated byΓ 5 . We note that, from the definition, . (2.20) From this, we have . (2.21) This shows that, if we assume that the action is bilinear in the fermion field, S τ = − d D xψ(x)Dψ(x) + · · · , the modified chiral symmetry of the Wilson actionΓ 5 e Sτ = 0 implies the GW relation [40], Note that we have arrived at this relation by a very simple manipulation while maintaining a manifest gauge invariance; it would be interesting to see how this relation reproduces the axial anomaly in our GFERG formulation. 7

GFERG equation
Let us derive an ERG equation that the Wilson action in Eq. (2.1) fulfills. This is readily obtained by taking the τ derivative of Eq. (2.1) in a way analogous to the derivation of the ERG equation in the Yang-Mills theory [18]. The result is Here, we have defined the anomalous dimensions by (recall Eq. (1.5)) The ERG equation in Eq. (2.23) is the main result of the present paper. Once this GFERG equation has been obtained, we may forget about the underlying construction in Eq. (2.1). Possible requirements on the initial action S τ =0 in Eq. (2.1) discussed so far, such as the gauge invariance and the chiral invariance, become implicit. If these properties of the Wilson action are considered to be desirable, we should simply pick up a solution or the initial condition of the ERG equation which fulfills these and other physical requirements (especially the locality and the Lorentz invariance). In this way, the issue of the existence of the UV regularization which makes the initial action finite becomes irrelevant. The renormalizability, i.e. whether we can tune parameters in the solution such that the correlation functions become finite in the continuum limit, is another issue, and we think that the results in Refs. [22,41] become helpful in considering this question.
Since in Eq. (2.23) the power of the gauge potential always accompanies the power of g τ , we see that g τ plays the role of the gauge coupling as the convention indicates. This parameter can thus be used as an expansion parameter which defines the perturbative expansion at the Gaussian fixed point [18].

Perturbative solution in QED to
To have some idea how the GFERG equation in Eq. (2.23) works, in this section we consider the U (1) gauge theory with a Dirac fermion with the charge e, i.e.
T a → −ie, (3.1) and solve the GFERG equation to the lowest nontrivial order of perturbation theory, O(g 1 τ ). 9 We first note that QED possesses charge conjugation symmetry, i.e. invariance under 8 where the charge conjugation matrix satisfies C −1 γ µ C = −γ T µ . Since all elements in Eq. (2.1), especially the flow equations for QED (i.e. f abc = 0), preserve the invariance under Eq. (3.2), if the initial action S τ =0 [A ′ , ψ ′ ,ψ ′ ] is invariant under the charge conjugation, then S τ [A, ψ,ψ] is too. In particular, we can forbid terms purely consisting of an odd number of gauge potentials; this is Furry's theorem in the present ERG formulation. Taking this fact into account, we set the Wilson action as For the first term, we already imposed the gauge invariance in O(g −1 τ ), i.e. the invariance under A a µ (p) → A a µ (p) + g −1 τ ip µ ω a (p). Note that the function G(τ ; p) is not necessarily invariant under p → −p, because it may contain the Dirac matrix such as / p. In momentum space, the ERG equation in Eq. (2.23) times e −Sτ reads, when α 0 = 1, where we have retained only terms relevant to the ERG evolution of terms in Eq. (3.3).

O(g 0 τ ) terms
In the lowest order, O(g 0 τ ), the ERG equation in Eq. (3.4) requires, for the coefficient functions in Eq. (3.3), It can be seen that the general solutions to these are given by 9 where C(p) and C(p) are arbitrary functions of p 2 ; for locality of the Wilson action, however, C(p) and C(p) must be analytic at p = 0. In obtaining the above expression for G, we have assumed parity symmetry and that C does not contain γ 5 , and thus / p and C commute with each other.

GW relation in
Even in the the above lowest O(g 0 τ ) solution, it is interesting to see how the GW relation in Eq. (2.22) is realized. To this order, Eq. (2.22) implies For Eq. (3.6), on the other hand, we have Therefore, if and only if γ 5 and the function C in Eq. (3.6) commute, i.e. if and only if C does not contain / p, the Wilson action satisfies the GW relation. Note that, since the τ dependence of G arises only from the combination e −τ p in C, the GW relation is preserved under the evolution of τ , as our general discussion shows.
An interesting case in which the GW relation is not fulfilled is where m is a constant. In this case, the breaking of the GW relation in Eq. (3.8) becomes (3.10) The choice of C in Eq. (3.9) actually realizes a massive fermion. The propagator of the fermion field with respect to the Wilson action to this order is given by This is not, however, the propagator that obeys the scaling law under the ERG evolution; recall the discussion at Eq. (1.3). Such a propagator is given by the modified one [36] defined by (see Eq. (1.4) for the scalar field case) The correlation length in units of the UV cutoff is thus given by e −τ m −1 , and m is the mass parameter; the critical surface is approached by m → 0. As expected, the GW relation is broken by the amount of this mass parameter as Eq. (3.10). The GW relation in Eq. (3.7) is satisfied if C(p) is a scalar function of p 2 so that C commutes with γ 5 . In this case, the function G realizes a massless fermion.