Step scaling and the Yang-Mills gradient flow

TL;DR

This paper investigates the use of the Yang-Mills gradient flow in step-scaling studies of lattice QCD, demonstrating that boundary conditions significantly affect systematic errors and that certain observables are insensitive to boundary effects.

Contribution

It shows that Neumann boundary conditions eliminate topology-freezing issues and that local gauge-invariant observables are robust against boundary lattice effects.

Findings

Neumann boundary conditions remove topology-freezing problems.

Local observables at positive flow time are insensitive to boundary effects.

Boundary lattice effects are manageable with appropriate boundary conditions.

Abstract

The use of the Yang-Mills gradient flow in step-scaling studies of lattice QCD is expected to lead to results of unprecedented precision. Step scaling is usually based on the Schr\"odinger functional, where time ranges over an interval [0,T] and all fields satisfy Dirichlet boundary conditions at time 0 and T. In these calculations, potentially important sources of systematic errors are boundary lattice effects and the infamous topology-freezing problem. The latter is here shown to be absent if Neumann instead of Dirichlet boundary conditions are imposed on the gauge field at time 0. Moreover, the expectation values of gauge-invariant local fields at positive flow time (and of other well localized observables) that reside in the center of the space-time volume are found to be largely insensitive to the boundary lattice effects.