Gradient flow step-scaling function for SU(3) with $N_f$ = 6 or 4 fundamental flavors

TL;DR

This paper nonperturbatively investigates the step-scaling $eta$ function for SU(3) gauge theories with 4 and 6 fundamental flavors, revealing slower running than perturbative predictions, and carefully analyzing lattice discretization errors.

Contribution

It provides the first detailed nonperturbative determination of the RG $eta$ function for SU(3) with 4 and 6 flavors, including systematic error analysis and comparison of multiple gradient flows.

Findings

Nonperturbative $eta$ functions run slower than perturbative predictions.

Careful analysis of lattice discretization errors using multiple gradient flows.

Study extends to the boundary of chiral symmetry breaking regimes.

Abstract

Nonperturbative determinations of the renormalization group (RG) $\beta$ function are crucial to understand properties of gauge-fermion systems at strong coupling and connect lattice simulations and the perturbative ultraviolet regime. Choosing well-understood, QCD-like systems with SU(3) gauge group and either six or four fundamental flavors, we investigate their step-scaling $\beta$ function. In both cases we push the simulations to the boundary of chiral symmetry breaking and study the regime $g^2_{GF} \lesssim 8.2$ with six, and $g^2_{GF} \lesssim 6.6$ with four flavors. We carefully consider the lattice discretization errors by comparing three different gradient flows (GF), and for each flow three operators to estimate the renormalized finite volume coupling. We also consider the tree level improvement of the coupling. Noteworthy outcome is that nonperturbatively determined $\beta$ functions run much slower than perturbatively predicted.