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.

💡 Research Summary

The paper presents a high‑precision determination of the strong coupling constant α_s by fitting lattice QCD data for the static quark‑antiquark energy. The authors improve the theoretical description of the static energy in three essential ways. First, they eliminate the leading u = ½ renormalon ambiguity by expressing the pole mass in terms of the short‑distance MSR mass and performing an R‑evolution (R‑improvement) that resums the associated large infrared logarithms. This removes a non‑perturbative O(Λ_QCD) contamination and stabilises the perturbative series at distances up to about 0.5 fm. Second, they resum the large ultrasoft logarithms that appear at O(α_s⁴) using potential NRQCD renormalisation‑group equations, achieving next‑to‑next‑to‑next‑to‑leading‑log (N³LL) accuracy. This resummation is crucial for the intermediate distance region where ultrasoft gluon effects dominate. Third, they introduce radius‑dependent profile functions for the renormalisation scales μ_s(r), R(r) and μ_us(r). These functions behave canonically (μ ∼ 1/r) at short distances but freeze to safe values before hitting the Landau pole at larger r, thereby extending the perturbative validity.

To assess perturbative uncertainties, the authors perform a random scan over the parameters defining the profile functions (ξ, β, μ_∞, Δ, b, etc.) and over a dimensionless parameter λ that controls the truncation of the R‑evolution kernel. They generate 500 distinct profile sets, minimise the χ² for each, and obtain a distribution of best‑fit α_s values. The χ² includes lattice statistical errors and offsets for each of the nine HotQCD ensembles (total 2512 data points). Two offset treatments are examined: a common offset for all ensembles and independent offsets for each. The latter leads to unstable fits at short distances, so the single‑offset approach is adopted for the final result.

The lattice data cover distances from 0 to 1 fm, with the physical scale set by r₁ = 0.3093(20) fm. The fit range is varied systematically: r_min between 0.02 and 0.04 fm and r_max between 0.35 and 0.45 fm. The resulting three‑flavour coupling at the τ‑mass scale is α_s^{(3)}(m_τ) = 0.3093 ± 0.0063 (total error, including lattice, r₁, and perturbative components). This value is then evolved to the Z‑pole using the five‑loop β‑function and four‑loop matching at the charm and bottom thresholds. Matching scales are varied as μ_c = 2 ± 1 m_c and μ_b = 2 ± 1 m_b to capture scale‑dependence. The final five‑flavour result is

α_s^{(5)}(m_Z) = 0.1170 ± 0.0009,

where the uncertainty combines perturbative, lattice, r₁, and matching‑scale contributions. This determination is fully compatible with the Particle Data Group world average (0.1181 ± 0.0011) and has a comparable precision.

The authors compare their outcome with recent static‑energy based extractions, notably the TUMQCD analysis and the study of Ref.