A Precise $\alpha_s$ Determination from the R-improved QCD Static Energy

The strong coupling $\alpha_s$ is determined with high precision from fits to lattice QCD simulations on the static energy. Our theoretical setup relies on R-improving the three-loop fixed-order prediction for the static energy by removing its $u=1/2$ renormalon and summing up the associated large (infrared) logarithms which, in combination with radius-dependent renormalization scales (called profile functions) extends the validity of perturbation theory to distances up to $\sim 0.5\,$fm. Furthermore, we resum large ultrasoft logarithms to N$^3$LL accuracy using renormalization group evolution. We have checked that the standard four-loop R-evolution treats N$^4$LL and higher remnants in a non-symmetric way, hence we also account for this potential bias. Our estimate of the perturbative uncertainty is based on a random scan over the parameters specifying the profile functions and the treatment of R-evolution. We also devise a method to statistically combine into a single dataset results from independent simulations which use different lattice spacing and cover various ranges, which can be used to carry out fits in a much faster way. We explore the dependence of the extracted $\alpha_s$ value on the smallest and largest distances included in the dataset, on how R-evolution is treated, on how the fit is performed, and on the accuracy of ultrasoft resummation. From our final analysis, after evolving to the $Z$-pole we obtain $\alpha^{(n_f=5)}_s(m_Z)=0.1166\pm 0.0009$, compatible with the world average with similar incertitude.