On the Symanzik improvement of gradient flow observables

Abstract

The gradient flow provides a new class of renormalised observables which can be measured with high precision in lattice simulations. This is relevant for many interesting applications. However, such applications are made difficult by the large discretisation effects observed in many gradient flow observables. We refer to the pure gauge theory at O(g02) in perturbation theory, where the structure of the Symanzik effective theory and all counterterms are known. At this order in perturbation theory, the theoretical expectation is that O(a2) Symanzik improvement is achieved when the action, the observable and the flow are O(a2) improved. We compute numerically the simplest observable, i.e. the action density, both with SF and SF-open boundary conditions. The first outcome of our computation confirms the theoretical expectation about the O(a2) improvement. Then we analyse a set-up with unimproved action. In finite volume, we study if it is possible to improve the observables by tuning the coefficient in the initial condition for the flow. Our analysis shows that very different values of cb are needed depending on the specific observable we want to improve. We show that the hypothesis that there is a hierarchy of cutoff effects between flow and non-flow observables is not valid, therefore it is not possible to find a universal value for the coefficient related to the only counterterm needed with respect to the 4 dimensional theory.