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 ΛMS¯, 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 ΛMS¯nf=0 from the data by exploring different methods to perform the zero-flow-time limit. We obtain the value 8t0ΛMS¯nf=0=0.629-26+22, where t0 is a flow-time reference scale. We also obtain precise determinations of several scales: r0/r1, 8t0/r0, 8t0/r1, 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.