Symanzik improvement of the gradient flow in lattice gauge theories

TL;DR

This paper applies Symanzik improvement to the gradient flow in lattice gauge theories, eliminating leading cutoff effects and classifying necessary counterterms for precise continuum extrapolation.

Contribution

It demonstrates that all $ ext{O}(a^2)$ cutoff effects can be removed at the classical level and provides a complete classification of counterterms needed for on-shell improvement.

Findings

All $ ext{O}(a^2)$ effects eliminated at classical level

Only one additional counterterm needed compared to 4D case

Perturbative test confirms counterterm coefficients at lowest order

Abstract

We apply the Symanzik improvement programme to the 4+1-dimensional local re-formulation of the gradient flow in pure $SU(N)$ lattice gauge theories. We show that the classical nature of the flow equation allows to eliminate all cutoff effects at $\mathcal O(a^2)$ which originate either from the discretized gradient flow equation or from the gradient flow observable. All the remaining $\mathcal O(a^2)$ effects can be understood in terms of local counterterms at the zero flow time boundary. We classify these counterterms and provide a complete set as required for on-shell improvement. Compared to the 4-dimensional pure gauge theory only a single additional counterterm is required, which corresponds to a modified initial condition for the flow equation. A consistency test in perturbation theory is passed and allows to determine all counterterm coefficients to lowest non-trivial order in the coupling.