Static force from generalized Wilson loops on the lattice using the gradient flow

The static QCD force from the lattice can be used to extract $\Lambda_{\overline{\textrm{MS}}}$, which determines the running of the strong coupling. Usually, this is done with a numerical derivative of the static potential. However, this introduces additional systematic uncertainties; thus, we use another observable to measure the static force directly. This observable consists of a Wilson loop with a chromoelectric field insertion. We work in the pure SU(3) gauge theory. We use gradient flow to improve the signal-to-noise ratio and to address the field insertion. We extract $\Lambda_{\overline{\textrm{MS}}}^{n_f=0}$ from the data by exploring different methods to perform the zero-flow-time limit. We obtain the value $\sqrt{8t_0} \Lambda_{\overline{\textrm{MS}}}^{n_f=0} =0.629^{+22}_{-26}$, where $t_0$ is a flow-time reference scale. We also obtain precise determinations of several scales: $r_0/r_1$, $\sqrt{8 t_0}/r_0$, $\sqrt{8 t_0}/r_1$ and we compare these to the literature. The gradient flow appears to be a promising method for calculations of Wilson loops with chromoelectric and chromomagnetic insertions in quenched and unquenched configurations