Effect of the Schr\"odinger functional boundary conditions on the convergence of step scaling

TL;DR

This paper investigates how boundary conditions in the Schr"odinger functional method affect the convergence of step scaling in lattice gauge theories, finding that careful boundary field choices improve convergence.

Contribution

It demonstrates that boundary field choices significantly influence the convergence rate of step scaling in lattice gauge theories with different gauge groups and fermion representations.

Findings

Improved Wilson action reduces linear terms in step scaling.

Standard boundary conditions lead to slow convergence.

Careful boundary field selection enhances convergence.

Abstract

Recently several lattice collaborations have studied the scale dependence of the coupling in theories with different gauge groups and fermion representations using the Schr\"odinger functional method. This has motivated us to look at the convergence of the perturbative step scaling to its continuum limit with gauge groups SU(2) and SU(3) with Wilson fermions in the fundamental, adjoint or sextet representations. We have found that while the improved Wilson action does remove the linear terms from the step scaling, the convergence is extremely slow with the standard choices of the boundary conditions for the background field. We show that the situation can be improved by careful choice of the boundary fields.