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

TL;DR

This work presents a high-precision determination of the strong coupling $α_s$ from lattice QCD using the QCD static energy. It combines MSR-mass renormalon subtraction with R-evolution to resum large infrared logs and applies N$^3$LL ultrasoft resummation, aided by profile functions that keep perturbation theory reliable out to $r\sim0.5$ fm. Fitting to HotQCD lattice data up to $r\approx0.45$ fm yields $α_s^{(n_f=5)}(m_Z)=0.1170 \pm 0.0009$, with cross-checks against $α_s^{(n_f=3)}(m_τ)=0.3093 \pm 0.0063$, demonstrating consistency with the world average. The analysis showcases a controlled uncertainty budget and a rigorous treatment of renormalon effects, offering a competitive lattice-based determination of $α_s$.

Abstract

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.

Figures (2)

• Figure 1: (a) Dependence of the strong coupling $\alpha_s$ on the maximal distance $r_{\rm max}$ included in the dataset for N$^3$LO fits, with $r_{\rm min}=0.02\,\mathrm{fm}$. Both the central value and the total uncertainty, which includes theoretical and lattice errors, are shown. Results with a single energy offset are shown in red, while multiple-offset fits are shown in blue. (b) N$^3$LO fit results for $\alpha_s$. Uncertainties include perturbative and lattice errors added in quadrature. The dependence of single-offset results on $r_{\rm max}$ is displayed for different $r_{\rm min}$ values: $0.02$, $0.025$, $0.03$, $0.035$, and $0.04\,\mathrm{fm}$, shown in blue, red, green, yellow, and cyan, respectively.
• Figure 2: (a) Comparison of lattice data (colored dots with error bars) with our best theoretical prediction, using the central values for the strong coupling and energy offset from our final result in Eq. \ref{['eq:final3']}. Different colors indicate different lattice ensembles. The error bars show only lattice statistical uncertainties, while the gray bands represent the theoretical perturbative uncertainties. (b) Comparison of lattice data (colored dots with error bars) with our best theoretical prediction, using the central values of the strong coupling and energy offset from our final result in Eq. \ref{['eq:final3']}. Different colors indicate different lattice ensembles. The error bars show only lattice statistical uncertainties, while the gray bands represent the theoretical perturbative uncertainties