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

The strong coupling $α_s$ 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 $α_s$ 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 $α^{(n_f=5)}_s(m_Z)=0.1166\pm 0.0009$, compatible with the world average with similar incertitude.

💡 Research Summary

The paper presents a high‑precision determination of the strong coupling constant αₛ from lattice QCD calculations of the static energy, employing a novel “R‑improved” perturbative framework. The authors start from the three‑loop fixed‑order expression for the static potential Vₛ(r, μ) and the associated ultrasoft contribution δ_us(r, μ). The dominant u = ½ renormalon, which spoils the convergence of the perturbative series, is removed by a renormalon subtraction performed in the MSR mass scheme. This subtraction introduces a new scale R, and the authors evolve the resulting short‑distance quantity using R‑evolution equations that resum the large logarithms associated with the renormalon. They point out that the standard four‑loop R‑evolution treats N⁴LL and higher terms asymmetrically, and they correct for this bias by random scans over the parameters governing the evolution.

A central technical innovation is the use of radius‑dependent renormalization scales, called profile functions. These functions smoothly interpolate between the canonical μ ∼ 1/r behavior at short distances and a frozen scale at larger r, thereby minimizing large logarithms that would otherwise appear in the perturbative expansion. The profile functions are parametrized by several continuous variables; the authors perform a Monte‑Carlo‑like scan over these parameters to estimate the perturbative uncertainty.

The ultrasoft logarithms, which first appear at O(αₛ⁴), are resummed to next‑to‑next‑to‑next‑to‑leading‑log (N³LL) accuracy using renormalization‑group evolution. The authors also explore the impact of truncating this resummation at lower orders (N²LL) and find that N³LL provides a noticeable reduction in the theoretical error.

On the lattice side, the analysis incorporates data from multiple HotQCD ensembles with different lattice spacings (a ≈ 0.025–0.061 fm) and from JLQCD simulations that reach larger distances (r ≈ 0.4 fm). To handle the fact that each ensemble may have an additive constant offset, the authors develop an analytical marginalization technique for the χ² function, allowing them to fit the strong coupling while simultaneously profiling over one or several offset parameters. Moreover, they devise a statistical recombination algorithm that merges all individual lattice datasets into a single, reduced‑size dataset without loss of information; this greatly speeds up the fitting procedure.

A thorough systematic study is performed. The authors vary the minimum and maximum distances (r_min, r_max) included in the fit, test different renormalon subtraction schemes (MSR vs. PS), examine the effect of different R‑evolution implementations, and assess the sensitivity to the choice of profile‑function parameters. The resulting spread of αₛ values is used to construct a robust error budget that includes both perturbative uncertainties (from the random scans) and lattice statistical/systematic errors.

The final result, after evolving the extracted αₛ from the low‑energy scale to the Z‑pole using four‑loop running with proper threshold matching, is

αₛ^{(n_f=5)}(m_Z) = 0.1166 ± 0.0009.

This value is fully compatible with the current Particle Data Group world average and has an uncertainty comparable to the most precise lattice determinations. The authors also compare their outcome with previous static‑energy based analyses (e.g., Refs.