Gradient flow step-scaling function for SU(3) with $N_f=8$ fundamental flavors

TL;DR

This paper computes the step-scaling function for SU(3) gauge theory with eight flavors using lattice simulations, exploring systematic effects and phase structure to understand the renormalization group flow.

Contribution

It provides the first detailed lattice calculation of the step-scaling function for SU(3) with eight flavors, including analysis of systematic uncertainties and phase structure.

Findings

Limited reach in coupling due to unphysical phase transition

Systematic effects checked via different flows and operators

Comparison with other flavor numbers enhances understanding of conformal window

Abstract

The step-scaling function, the lattice analog of the renormalization group $\beta$ function, is presented for the SU(3) gauge system with eight flavors in the fundamental representation. Our investigation is based on generating dynamical eight flavor gauge field configurations using stout-smeared M\"obius domain wall fermions and Symanzik gauge action. On these gauge field configurations we perform gradient flow measurements using the Zeuthen, Wilson, or Symanzik kernel and consider the Symanzik, Wilson plaquette, or clover operators to determine step-scaling functions for a scale change $s=2$ including large, up to $48^4$, volumes. Considering different flows and operators as well as the optional use of tree-level improvement allows us to check for possible systematic effects. Our result covers the range of renormalized coupling up to $g_c^2 \lesssim 10$. In the case of $N_f=8$ we observe that the reach in $g_c^2$ is limited due to an unphysical first order bulk phase transition caused by large ultra-violet fluctuations. We compare our findings to $N_f=4$, 6, 10 or 12 flavors results that are obtained using the same lattice action and analysis. In addition we investigate the phase structure for simulations with different number of flavors using stout-smeared M\"obius domain wall fermions and Symanzik gauge actions to shed some light on the limited reach in $g_c^2$.