Proof of the renormalizability of the gradient flow

Abstract

We give an alternative perturbative proof of the renormalizability of the system defined by the gradient flow and the fermion flow in vector-like gauge theories.Comment: 29 pages, the final version to appear in Nuclear Physics

Gradient flow representation of the four-dimensional $\mathcal{N}=2$ super Yang–Mills supercurrent

“…See also Ref. [16]. Moreover the flowed gauge field does not need the wave function renormalization [12].…”

“…The idea is that since composite operators of fields evolved by the flow automatically become finite renormalized operators [12,13] (see also Ref. [16]), the expression of the supercurrent in terms of flowed fields is independent of regularization (in the limit in which the UV cutoff is removed); thus the expression is universal. In this way, one can have, a priori, an expression that becomes automatically the properly normalized conserved supercurrent in the continuum limit.…”

Supersymmetric gradient flow in the Wess-Zumino model

“…Here we do not consider the flow equation of G andḠ. An on-shell SUSY transformation δ ′ ξ for the flowed fields is given by (5) with the replacement (16). The commutation relation between the flow derivative and the on-shell SUSY transformation does not vanish in general but is proportional to δS/δh for h = ψ, A,ψ, A * .…”

Energy-momentum tensor on the lattice: recent developments

“…[19] (see also Ref. [37]), is that any composite operator of the flowed gauge field for a positive flow time t > 0 automatically becomes a renormalized finite operator; moreover it does not produce any new UV divergences even if other composite operators collide with it. Thus let us suppose that we take certain composite operators composed from the flowed gauge field as the probe operatorsÔ(y)Ô(z) in Eq.…”

“…(6.2) for t > 0 automatically becomes a renormalized finite operator, if the flowed fermion fields are multiplicatively renormalized [19,40] (see also Ref. [37]):…”