Perturbative O$(a^2)$ effects in gradient flow couplings with SF and SF-open boundary conditions

Abstract

The gradient flow provides a new class of renormalized observables which can be measured with high precision in lattice simulations. In principle this allows for many interesting applications to renormalization and improvement problems. In practice, however, such applications are made difficult by the rather large cutoff effects found in many gradient flow observables. At lowest order of perturbation theory we here study the leading cutoff effects in a finite volume gradient flow coupling with SF and SF-open boundary conditions. We confirm that O(a 2 ) Symanzik improvement is achieved at tree-level, provided the action, observable and the flow are O(a 2 ) improved. O(a 2 ) effects from the time boundaries are found to be absent at this order, both with SF and SF-open boundary conditions. For the calculation we have used a convenient representation of the free gauge field propagator at finite flow times which follows from a recently

The gradient flow coupling at high-energy and the scale of SU(3) Yang–Mills theory

“…We note that, in fact, we computed the norms Nm,e in tree-level lattice perturbation theory using the set-up described in ref [47]…”

“…. This set-up differs from ours by the way the the temporal links touching the SF boundaries are treated in the Zeuthen flow equation (see ref [47]. for the details).…”