Symanzik Improvement of the gradient flow in lattice gauge theories

Abstract

We apply the Symanzik improvement progra- mme 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 one to eliminate all cutoff effects at O ( a 2 ) , which originate either from the discretised gradient flow equation or from the gradi- ent flow observable. All the remaining 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. Com- pared 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 consis- tency test in perturbation theory is passed and allows one to determine all counterterm coefficients to lowest non-trivial order in the coupling.