Abstract
Abstract 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 à 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 $D=2$.
Highlights
The Wilson exact renormalization group (ERG) [1] is important because, among many other things, it provides a unique framework to consider possible quantum field theories beyond perturbation theory
Gauge symmetry is a fundamental principle and we are interested in ERG trajectories, i.e. solutions of the ERG equation, which preserve this symmetry
It is possible to define a modified gauge transformation which is consistent with the ERG evolution [9], 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
Summary
The Wilson exact renormalization group (ERG) [1] (see 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. 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/Λ2 with respect to the bare action
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