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Manifestly gauge invariant exact renormalization group for quantum electrodynamics

We formulate quantum electrodynamics on the basis of gauge (or BRST) covariant diffusion equations of fields. This is a particular example of the gradient flow exact renormalization group (GFERG). The resulting Wilson action fulfills a simple gauge Ward--Takahashi identity. We solve the GFERG equation around the Gaussian fixed point to the second order in gauge coupling and obtain the 1-loop beta function and anomalous dimensions. The anomalous dimension of the electron field coincides with that of the fermion field diffused by a gauge covariant flow equation of L\"uscher.


Introduction
A Wilson action is a functional of field variables with a finite momentum cutoff, say Λ [1].
If the underlying theory is a continuum limit, the theory is defined to all momentum scales. We obtain the interaction vertices of the Wilson action S Λ by integrating out the fields with momenta larger than Λ. It is then natural to expect that only the correlations of the fields with momenta smaller than Λ are kept, but those with momenta larger than Λ are lost from S Λ . In the exact renormalization group (ERG) formalism [1], in which a sharp momentum cutoff is replaced by a smooth function of momentum, this is not the case: we can still reconstruct the full correlation functions using the Wilson action. This makes gauge invariance compatible with a momentum cutoff. This viewpoint was first adopted for QED in Ref. [2]. A general framework for constructing non-abelian gauge theories along this line was given in Ref. [3].
The realization of gauge invariance with a Wilson action has a long history starting in the 1980's. The early works in the 1990's such as Refs. [4][5][6][7][8][9] established the possibility of constructing gauge theories in the ERG formalism. (Ref. [5] gives references to the earlier works from the 1980's.) What is common in the realization of gauge invariance in the ERG formalism is that the gauge invariance is not what one expects naturally. For Λ > 0, the gauge transformation is modified so that the Jacobian is non-vanishing, and the resulting expression of gauge invariance is by no means manifest. This has been an obstacle for any calculation of the Wilson action beyond perturbation theory, since it is difficult to truncate the action keeping the non-manifest gauge invariance.
The original formulation of ERG is based on the diffusion of the fields [1]. Recently a proposal was made that we may be able to construct a manifestly gauge invariant Wilson action by replacing the diffusion equation by a gauge invariant diffusion equation [10]. This was inspired by the gauge invariant diffusion that generates a gradient flow of gauge fields, first discussed for lattice gauge theory in Refs. [11][12][13] and then by Lüscher and Weisz [14] for perturbative non-abelian gauge theory. We call this new type of ERG by the gradient flow exact renormalization group (GFERG). The present paper is a sequel to Ref. [15] where GFERG for fermions is discussed.
The paper is organized as follows. In Sect. 2 we first review the relation between a diffusion equation and the exact renormalization group transformation using a generic real scalar theory. We follow the discussions given in Ref. [16]; see also Ref. [17]. We then introduce a particular set of diffusion equations for QED that is consistent with the BRST invariance of the theory. We base our construction of GFERG on these diffusion equations. In Sect. 3 we construct a Wilson action S Λ of QED with momentum cutoff Λ that keeps its BRST invariance as we lower Λ. We derive the cutoff dependence of S Λ as a differential equation, and also derive an expression for the BRST invariance. The BRST transformation acts linearly on the action, and it is far simpler than the BRST invariance of the Wilson action in the ERG formulation, which is briefly reviewed in Appendix C. In Sect. 4 we introduce a dimensionless framework by measuring dimensionful fields and parameters in units of appropriate powers of the cutoff. We then construct the BRST invariant Wilson action perturbatively in Sect. 5. We only consider the Wilson action for the continuum limit parametrized by the gauge coupling, gauge fixing parameter, and the electron mass parameter. Since the ghost fields are free, we can reduce the BRST invariance to the Ward-Takahashi (WT) identity. This WT identity can be interpreted as manifest gauge invariance even though the transformation of the gauge field is somewhat modified. 1 We construct the Wilson action satisfying the WT identity to second order in the gauge coupling. We conclude the paper in Sect. 6.
We work in the D-dimensional Euclidean space, where D = 4 − ǫ. We use the shorthand notation for the momentum integrals: We also use the convention that the momentum cutoff decreases along the flow of the renormalization group, and the beta functions and anomalous dimensions may have the opposite signs to what the reader is familiar with.

ERG
We would like to review the essence of the exact renormalization group (ERG for short). In one formulation of ERG we construct the Wilson action of a theory in terms of a field satisfying a simple diffusion equation. The flow of the Wilson action is generated by the diffusion of the field. For gauge theories, we can replace the simple diffusion equation by a covariant diffusion equation that is consistent with BRST invariance. The replacement results in the gradient flow exact renormalization group (GFERG for short). We will introduce the BRST covariant diffusion for QED in the next subsection. 1 We ask the reader to bear with the overblown title of the paper.
Let φ(x) be a real scalar field renormalized at momentum scale µ, and let S[φ] be its action. We introduce a diffused field φ(t; x) as the solution of a simple diffusion equation satisfying the initial condition We would like to construct a Wilson action equivalent to S[φ] in terms of the diffused field φ(t; x) instead of φ(x). Let Λ be a momentum scale smaller than µ given by We introduce the Wilson action S Λ [φ] for momentum cutoff Λ by satisfying the initial condition In momentum space it is trivial to solve the diffusion equation to obtain By construction, φ(p) equals z Λ φ ′ Λ (p) with a squared fluctuation of order 1/Λ 2 . It is not exactly the same as z Λ φ ′ Λ (p), but it corresponds to it. The choice of z Λ is not unique. For example, we can determine z Λ to normalize the kinetic term of the Wilson action S Λ [φ].
The Λ-dependence of the Wilson action S Λ [φ] is given by the ERG differential equation: For the correlation functions, we can give a precise relation between S Λ and S: In constructing S Λ , we have "scrambled" the field φ(p) around z Λ φ Λ (p). We need to unscramble the field to get back the same correlation functions. This is the role played by the exponentiated differential operator.
An alternative definition of the Wilson action is given by [10] exp is what we call the scrambler. The scrambler is necessary to maintain the locality of the Wilson action S Λ [φ]. The equality of the correlation functions can be written as using the unscrambler, i.e., the inverse of the scrambler.

Diffusion in QED
In constructing the Wilson action of a gauge theory, we can use the simple diffusion equation for gauge fields and matter as explained above for the scalar theory. The Wilson action, thus constructed, retains gauge invariance (BRST invariance to be more precise), but its realization is not as straightforward as we wish [3]. This is partially due to the use of a simple diffusion of fields which does not respect the gauge invariance. In Ref. [10] we have introduced an alternative Wilson action based upon a covariant diffusion of fields, consistent with the gauge invariance of the theory. 2 In this subsection we would like to introduce such diffusion explicitly for the simple case of QED.
Let us consider QED renormalized at momentum scale µ in D = 4 − ǫ dimensional Euclidean space. We denote the gauge field by A µ (x), and the electron field by ψ(x) andψ(x), and the free Faddeev-Popov ghost fields by c(x) andc(x). The dimensionless gauge coupling renormalized at µ is e. The action S is invariant under the following BRST transformation of the renormalized fields: where η is an arbitrary anticommuting number that keeps the statistics of the fields under the transformation. 3 We introduce the following diffusion equations: where the fields match the renormalized fields at t = 0: It is straightforward to check that the above diffusion equations are consistent with the BRST transformation. Namely, the BRST transformation of the fields at t = 0 implies the same BRST transformation of the diffused fields: To show this, we need to check that δ commutes with ∂ t . Let us check only two here.
is consistent with We also find is consistent with (2.22) We have thus introduced diffusion of fields preserving the form of the BRST transformation. Our aim is to construct a Wilson action of QED using the diffused fields as the elementary fields.

GFERG for QED
We introduce the Wilson action of QED as We do not introduce any factor of wave function renormalization for the ghosts since they remain free fields (see below). The diffused fields satisfy

GFERG differential equation
We wish to obtain in terms of the functional derivatives with respect to the field variables. As a preparation, we note the following correspondence: We need to be clear about what we mean by the above correspondence. Take Eq. (3.5a). The correspondence means Differentiating this once more we obtain where we have chosen x 2 = x 1 so that the second differentiation with respect to A µ (x 2 ) acts only on the exponential, but not on A µ (x 1 ) in the integrand. Taking the limit where we take the limit x ′ → x after the differentiation. We now calculate (3.10) Using the correspondence given above, we can rewrite the rhs as where the limit x ′ → x is implied, and we have defined the gauge coupling of mass dimension ǫ/2 by Note we have given Eq. (3.11) in two parts. The first part reproduces the ERG differential equation. (See Appendix C for a quick review of ERG for QED.) The second part is unique to GFERG; it comes from the BRST covariance of the electron diffusion equations (3.3).

BRST invariance
We next derive the expression of BRST invariance of S Λ , inherited from the invariance As explained in Sect. 2.2, this induces the diffused fields to transform as Hence, we obtain Dividing this by z Λ , and defining we can rewrite this as where we recall Eq. (3.12) defining e Λ . Using the differentials, we can rewrite this further as The terms with second order differentials with respect to ψ andψ cancel, and we obtain finally This has come pretty close to manifest BRST invariance. The simple expression is due to the BRST covariant diffusion of the fields, Eq. (3.3). In Appendix C we give the standard ERG formulation of QED for comparison.

GFERG and BRST in the dimensionless framework
To gain more insights into the scaling properties of the Wilson action, we adopt the dimensionless framework. Instead of the momentum cutoff Λ, we use the dimensionless logarithmic scale parameter τ defined by We write the Fourier transforms of the fields in terms of the dimensionless fields (with tildes) as The dimensionless gauge coupling is defined bỹ The gauge fixing parameter remains the same: we obtain The Wilson action in the dimensionless framework is a functional of the dimensionless In order to rewrite the GFERG differential equation (3.11) in the dimensionless framework, we need to use (4.9) Since we work only in the dimensionless framework from now, we omit the tildes altogether, and we obtain the GFERG equation as . (4.10) Except for the last two integrals, the rhs coincides with the ERG equation. For the last two integrals we have kept the coordinate space notation to take the limit x ′ → x carefully. 4 Similarly, we can rewrite the BRST invariance in the dimensionless framework. In coordinate space it is given by (4.11)

Perturbative solution
In constructing the Wilson action S τ , we assume that it does not depend on τ explicitly: we assume that its τ dependence comes only through the τ dependence of three dimensionless parameters, i.e., the gauge coupling e τ , gauge fixing parameter ξ τ , and the electron mass parameter m τ . This assumption is not valid, however, for S τ that is the dimensionless version of S Λ given by Eq. (3.1). As long as the renormalization scale µ is finite, S Λ depends on Λ/µ = e −τ explicitly. To remove this, we must take the "continuum limit", i.e., we must take µ → +∞. Both the differential equation (4.10) and the BRST invariance (4.11) have been derived based on the integral formula (3.1), but neither has explicit dependence on τ . In practice we can construct the continuum limit of S τ by solving Eqs. (4.10) and (4.11) simultaneously under the above assumption. The gauge coupling e τ and the gauge fixing parameter ξ τ are introduced through the BRST invariance (4.11). We normalize the kinetic terms of the gauge and electron fields; this fixes the anomalous dimensions γ τ and γ F τ as functions of e 2 τ although they may also depend on ξ τ . (We believe neither depends on ξ τ .) We introduce a normalization condition of the electron mass term that determines the τ dependence of m τ in the form d dτ We believe β m is also independent of ξ τ . We thus obtain ∂ τ e Sτ = ∂ τ e S(eτ ,ξτ ,mτ ) ∂m τ e S(eτ ,ξτ ,mτ ) . (5.2) We will drop the suffix τ from the parameters e τ , ξ τ , and m τ . 5 Thus, our GFERG differential equation becomes This is solved by We have normalized the kinetic terms appropriately.
The ξ-dependence of the longitudinal part is determined by the BRST invariance (4.11); at tree level it gives Let us check that Eq. (5.8) satisfies this: Hence, S (0) is BRST invariant.

BRST invariance simplified to manifest gauge invariance
Before proceeding to calculate S (1) , we stop to simplify our expression for the BRST invariance given in Eq. (4.11).
GFERG equation (5.3) implies the absence of higher order corrections to the ghost dependent part of the action S. 6 Hence, the ghost part of the action is exactly as given in Eq. (5.8): k 2 e −2k 2 + k 2 c(k). (5.11) Hence, we can rewrite the BRST invariance as The integrand is proportional to c(k), and its coefficient must vanish. This results in the Ward-Takahashi (WT) identity given by is the interaction part, and we have used the BRST invariance of S (0) . Equation (5.13) differs from the classical gauge invariance merely by the k-dependent factor on the lhs.
In fact Eq. (5.13) implies that the action without the gauge fixing and ghost terms is invariant under the infinitesimal "gauge"

First order
Since the gauge coupling e in Eq. (5.3) accompanies the gauge field, the first order term S (1) must have the structure: Because of the charge conjugation symmetry of our formulation, we can exclude the term cubic in the gauge potential [15].
. (5.19) At this stage, it is very helpful to introduce new variables by 7 from Eq. (5.8). These h-functions are the high-momentum propagators satisfying  25) and . (5.26) In terms of these new variables, Eq. (5.19) becomes quite simple: We now write so that the vertex part V µ satisfies the inhomogeneous scaling equation We wish to find a local solution which can be expanded in powers of p and k at zero momenta. Equation (5.29) determines V µ (p, k) up to a constant vector. A particular solution is obtained by the formula in Appendix A. The general solution is whereṼ µ is a constant vector, and V µ is determined by imposing the WT identity (5.13), which requires This gives V µ = γ µ , and we obtain It follows from this that which will be used frequently below.
Our result for S (1) coincides with the first order term of the gauge invariant local Wilson action obtained in Ref. [15].

Second order
We expect that the anomalous dimensions are second order in e: In what follows we denote γ ≡ γ 1 e 2 + · · · , γ F ≡ γ F 1 e 2 + · · · , β m ≡ β m1 e 2 + · · · . (5.36) Extracting the second order terms in Eq. (5.3) is already a laborious task. We obtain where we have used variables defined in Eq. (5.20). Note that we take the limit x ′ → x only after taking differentials as has been explained in Sect. 3. In Appendix B we elaborate on how this limit actually works in this case.
We have four types of terms: We compute them one by one.

5.5ψAAψ term
Let us first consider the term proportional toψAAψ: where we have used the relation (5.23). Noting further the properties (5.22), we can simplify the above to The last line can be integrated by the formula in Appendix A as where the function F (x) is defined by Eq. (5.31). Therefore, the solution to Eq. (5.41) is given by where X µν (p, k, l) satisfies This can be solved again by the formula in Appendix A to yield X µν (p, k, l) where we have used the identity has no solution analytic in momenta and m.
Since the corresponding homogeneous equation has no solution analytic in momenta and m, we get the unique local solution Now, we can study the second order correction to the AA term. We will see that the analyticity of this term determines the first nontrivial order coefficient of the anomalous dimension, γ 1 in Eq. (5.36). We first note that the WT identity (5.13) requires that this correction be transverse: . (5.58) The 4-momentum integral on the rhs is thus given by a surface integral at |p| → ∞. From the explicit form of the integrand, it is not difficult to find the surface term that contributes to the integral at |p| = ∞. For instance, the last term does not contribute because of the factor e −2p 2 . In this way, we obtain . 5.8ψψ term and γ F 1 , β m Finally, we consider the second order correction to the fermion kinetic and mass terms: The GFERG equation is given by Now, let us compute β m1 , γ F 1 for D = 4. The lhs of Eq. (5.62) gives to first order in / p and m. We can determine β m1 and γ F 1 by calculating I F (p) to the same order. We can take p · ∂/∂p + m∂/∂m = 1, and Eqs. (5.62) and (5.63) give .

(5.65)
It is again straightforward to find the surface term at |k| = ∞, which contributes to this integral, and we obtain The former is the usual mass anomalous dimension in QED. Interestingly, the latter coincides with the anomalous dimension resulting from the wave function renormalization of the flowed or diffused (i.e., not usual) fermion field; the 1-loop renormalization factor has been given in Eq. (2.16) of Ref. [26], where C F = 1 for QED with the electron. Note that this anomalous dimension is independent of the gauge-fixing parameter ξ. This is expected because the Wilson action (3.1), by construction, reproduces the correlation functions of flowed or diffused fields up to contact terms [10].

Conclusion
In this paper we have constructed the gradient flow exact renormalization group (GFERG) for QED, based on the BRST invariant diffusion equations (2.16). With the exclusion of the gauge fixing term, the Wilson action (5.16) becomes manifestly invariant under the gauge transformation (5.17). We have computed the action perturbatively in powers of the coupling e to the order e 2 , reproducing the 1-loop beta function and anomalous dimensions. It was especially pleasing to find the anomalous dimension of the electron field as gauge invariant.
Our perturbative calculations show that the Wilson action becomes complex despite the simplicity in gauge invariance. The complexity comes from that of the GFERG differential equations. But we believe that the manifest gauge invariance will turn out to be a big advantage when we attempt to solve the GFERG differential equations non-perturbatively (with some gauge invariant approximations).
Whether the GFERG differential equation (4.10) has a non-trivial fixed-point satisfying the WT identity (5.13) is of much interest to be studied in the future. See, for example, Ref. [27] and references cited therein for related studies. At present we even do not know what it means to have a fixed-point in the GFERG formalism.
The GFERG formalism was originally introduced for non-abelian gauge theories [10]. It should be interesting to extend the analysis of this paper to see how far we can simplify the realization of non-abelian gauge invariance compared with the standard ERG formalism.

Acknowledgments
This work was partially supported by Japan Society for the Promotion of Science (JSPS) Grant-in-Aid for Scientific Research Grant Number JP20H01903. Hiroshi Suzuki would like to thank Katsumi Itoh for informative discussions.

A Integration formula
A particular solution to Note that this solution is analytic in the momenta and m.  B The working of the limit x ′ → x In Sect. 3.1 we have explained how to take second and higher order functional differentials at the same point as a limit of functional differentials at different points. This careful treatment is necessary to avoid unphysical singularities. In deriving the GFERG differential equation for the second order Wilson action S (2) we need to practice the treatment. There are three integrals to consider.