A Precise $α_s$ Determination from the R-improved QCD Static Energy
The strong coupling $α_s$ is extracted with high precision through fits to lattice-QCD data for the static energy. Our theoretical framework is based on R-improving the three-loop fixed-order prediction for the static energy: we remove the $u=1/2$ renormalon and resum the associated large infrared logarithms. Combined with radius-dependent renormalization scales (the so-called profile functions), this procedure extends the range of validity of perturbation theory to distances as large as $\sim 0.5\,$fm. In addition, we resum large ultrasoft logarithms to N$^3$LL accuracy using renormalization-group evolution. Since the standard four-loop R-evolution treats N$^4$LL and higher-order contributions asymmetrically, we also incorporate this potential source of bias in our analysis. Our estimate of the perturbative uncertainty is obtained through a random scan over the parameters controlling the profile functions and the implementation of R-evolution. We analyze how the extracted value of $α_s$ depends on the shortest and longest distances included in the fit, on the details of the R-evolution procedure, on the fitting strategy itself, and on the accuracy of ultrasoft resummation. From our final analysis, and after evolution to the $Z$ pole, we obtain $α^{(n_f=5)}_s(m_Z)=0.1170\pm 0.0009$, a result fully compatible with the world average and with a comparable uncertainty.