Symanzik improvement of the gradient flow in lattice gauge theories

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.