Slow running of the Gradient Flow coupling from 200 MeV to 4 GeV in $N_{\rm f}=3$ QCD

TL;DR

This study non-perturbatively investigates the slow running of the gradient flow coupling in three-flavor QCD across a broad energy range, emphasizing the importance of continuum extrapolations for high accuracy.

Contribution

It provides the first high-precision non-perturbative determination of the gradient flow coupling's running from 200 MeV to 4 GeV in $N_f=3$ QCD using Schrödinger Functional boundary conditions.

Findings

Coupling runs between one-loop and two-loop predictions.

No clear evidence of two-loop running contact.

Successful non-perturbative matching to SF coupling up to 100 GeV.

Abstract

Using a finite volume Gradient Flow (GF) renormalization scheme with Schr\"odinger Functional (SF) boundary conditions, we compute the non-perturbative running coupling in the range $2.2 \lesssim {\bar g}_\mathrm{GF}^2(L) \lesssim 13$. Careful continuum extrapolations turn out to be crucial to reach our high accuracy. The running of the coupling is always between one-loop and two-loop and very close to one-loop in the region of $200\,{\rm MeV} \lesssim \mu=1/L \lesssim 4\,{\rm GeV}$. While there is no convincing contact to two-loop running, we match non-perturbatively to the SF coupling with background field. In this case we know the $\mu$ dependence up to $\sim 100\,{\rm GeV}$ and can thus connect to the $\Lambda$-parameter.