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

TL;DR

This paper analyzes the leading cutoff effects in gradient flow couplings with SF boundary conditions at the perturbative level, confirming O(a^2) improvement and simplifying calculations with a new propagator representation.

Contribution

It provides a perturbative study of cutoff effects in gradient flow observables with SF boundary conditions, confirming O(a^2) improvement and introducing a practical propagator representation.

Findings

O(a^2) improvement achieved at tree-level with proper O(a^2) improvement

O(a^2) effects from time boundaries are absent at this order

A new propagator representation simplifies lattice perturbation theory at finite flow times

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 proposed set-up by L\"uscher and renders lattice perturbation theory more practical at finite flow time and with SF, open, SF-open or open-SF boundary conditions.