Scale Setting and Strong Coupling Determination in the Gradient Flow Scheme for 2+1 Flavor Lattice QCD

We report on the determination of the gradient flow scales in $N_f=2+1$ QCD using highly improved staggered quark (HISQ) ensembles generated by the HotQCD Collaboration for bare gauge couplings ranging from $\beta = 6.423$ to $8.400$. Using bottomonium splittings, kaon decay constant, the decay constant of unmixed $\eta_s$ meson and the $\phi$ meson mass we obtained the values of the gradient flow scales in physical units, $\sqrt{t_0} = 0.14428(48)$~fm and $w_0 = 0.17391(52)$~fm. Using the same physical inputs we revisit the determination of the potential $r_1$ scale and find $r_1 = 0.3112(24)$~fm. As a byproduct of our study we obtain the running of the gauge coupling in the gradient flow scheme. We find that within the uncertainties the running of the gradient flow coupling obtained on the lattice is compatible with the perturbative results up to flow radius $\sqrt{8 \tau_F}=0.15$ fm.