An O(a) modified lattice set-up of the Schr\"odinger functional in SU(3) gauge theory

TL;DR

This paper proposes an O(a) improved lattice setup for the Schrödinger functional in SU(3) gauge theory, reducing lattice artefacts and enabling more precise non-perturbative studies of the coupling.

Contribution

It introduces a modified lattice setup with an adjusted boundary counterterm coefficient, improving the accuracy of the Schrödinger functional in SU(3) gauge theory.

Findings

Residual cutoff effects are small in simulations.

The modified setup effectively cancels O(a) lattice artefacts.

The approach is comparable to the standard setup in accuracy.

Abstract

The set-up of the QCD Schr\"odinger functional (SF) on the lattice with staggered quarks requires an even number of points $L/a$ in the spatial directions, while the Euclidean time extent of the lattice, $T/a$, must be odd. Identifying a unique renormalisation scale, $L=T$, is then only possible up to O($a$) lattice artefacts. In this article we study such lattices in the pure SU(3) gauge theory, where we can also compare to the standard set-up. We consider the SF coupling as obtained from the variation of an SU(3) Abelian and spatially constant background field. The O($a$) lattice artefacts can be cancelled by the existing O($a$) boundary counterterm. However, its coefficient, $\ct$, differs at the tree-level from its standard value, so that one first needs to re-determine the induced background gauge field. The perturbative one-loop correction to the coupling allows to determine $\ct$ to one-loop order. A few numerical simulations serve to demonstrate that residual cutoff effects in the step scaling function are small in both cases, $T=L\pm a$ and comparable to the standard case with $T=L$.