Conserved vector current in QCD-like theories and the gradient flow

TL;DR

This paper analytically studies the behavior of the flavor-singlet vector current in QCD-like theories under the gradient flow, revealing how it requires multiplicative renormalization and its implications for current conservation and operator-product expansion.

Contribution

It provides the first explicit derivation of the asymptotic solution for evolved vector current correlators at next-to-leading order in perturbation theory.

Findings

Evolved correlators require multiplicative renormalization.

Conservation of the vector current is softly violated at small flow times.

The leading OPE coefficient for evolved currents is explicitly derived.

Abstract

We present analytical results for the Euclidean 2-point correlator of the flavor-singlet vector current evolved by the gradient flow at next-to-leading order ($O(g^2)$) in perturbatively massless QCD-like theories. We show that the evolved 2-point correlator requires multiplicative renormalization, in contrast to the nonevolved case, and confirm, in agreement with other results in the literature, that such renormalization ought to be identified with a universal renormalization of the evolved elementary fermion field in all evolved fermion-bilinear currents, whereas the gauge coupling renormalizes as usual. We explicitly derive the asymptotic solution of the Callan-Symanzik equation for the connected 2-point correlators of these evolved currents in the limit of small gradient-flow time $\sqrt{t}$, at fixed separation $|x-y|$. Incidentally, this computation determines the leading coefficient of the operator-product expansion (OPE) in the small $t$ limit for the evolved currents in terms of their local nonevolved counterpart. Our computation also implies that, in the evolved case, conservation of the vector current, hence transversality of the corresponding 2-point correlator, is no longer related to the nonrenormalization, in contrast to the nonevolved case. Indeed, for small flow time the evolved vector current is conserved up to $O(t)$ softly violating effects, despite its $t$-dependent nonvanishing anomalous dimension.