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

Abstract

The static QCD force from the lattice can be used to extract ${\Lambda }_{\overline{\text{MS}}}$, 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 ${\Lambda }_{\overline{\text{MS}}}^{n_f=0}$ from the data by exploring different methods to perform the zero-flow-time limit. We obtain the value $\sqrt{8t_0}{\Lambda }_{\overline{\text{MS}}}^{n_f=0}=0.629_{-26}^{+22}$, where $t_0$ is a flow-time reference scale. We also obtain precise determinations of several scales: $r_0/r_1$, $\sqrt{8t_0}/r_0$, $\sqrt{8t_0}/r_1$ 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

arXiv comment27 pages, 19 figures The latest version matches the reviewed and accepted version

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