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

Abstract

Using a finite volume Gradient Flow (GF) renormalization scheme with Schrödinger 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 $200\mathrm{MeV}\lesssim \mu =1/L\lesssim 4\mathrm{GeV}$. 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 $\sim 100\mathrm{GeV}$ and can thus connect to the $\Lambda$-parameter.

Comments: 34 pages, LaTeX