Gradient flow scale setting with tree-level improvement

TL;DR

This paper compares different gradient flow scales and operators in lattice QCD, finding Zeuthen flow with Symanzik operators most consistent and showing that tree-level improvement reduces operator spread but not cutoff effects.

Contribution

It provides a detailed comparison of gradient flow scales and operators, highlighting the effectiveness of Zeuthen flow with Symanzik operators and evaluating the impact of tree-level improvement.

Findings

Zeuthen flow with Symanzik operators shows minimal cutoff corrections.

Tree-level improvement reduces operator spread but not cutoff effects.

No overall reduction in cutoff effects from tree-level improvement for gradient flow scales.

Abstract

Lattice scales defined using gradient flow are typically very precise, while also easy to calculate. However, different definitions of flows and operators can differ significantly, suggesting possible systematical effects. Using a subset of RBC-UKQCD's 2+1 flavor domain wall fermion and Iwasaki gauge action ensembles, we explore differences between $\sqrt{t_0}$ and $w_0$ gradient flow scales, compare the impact of different operators to define the energy density, and study the effect of using tree-level improvement for the gradient flow. We find that for this set of gauge field ensembles Zeuthen flow with Symanzik operators has the most consistent approach to the continuum limit and exhibit very small cutoff corrections. Tree-level improvement, traditionally used in step-scaling studies, significantly reduces the spread between different operators, but does not lead to an overall improvement when it comes to reducing cutoff effects for gradient flow scales $\sqrt{t_0}$ or $w_0$.