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

TL;DR

This work determines the strong coupling constant by fitting lattice QCD results for the static energy to a theoretically refined perturbative prediction. Key innovations include removing the $oldsymbol{u=1/2}$ renormalon via MSR-type subtractions with R-evolution, summing ultrasoft logs to N$^3$LL, and employing radius-dependent profile functions to stabilize the perturbative expansion over distances up to $oldsymbol{0.5\,\mathrm{fm}}$. A robust data-combination and fit methodology—marginalizing over lattice offsets and exploring multiple subtraction schemes—yields a final result of $oldsymbol{\alpha_s^{(n_f=5)}(m_Z)=0.1166(9)}$, compatible with the world average. The analysis demonstrates orderly perturbative convergence, quantifies theoretical uncertainties via a comprehensive random scan, and provides a framework adaptable to future lattice covariance implementations and refinements of renormalon subtractions.

Abstract

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.

Figures (21)

• Figure 1: Left panel: Dependence of the four-loop estimate for the renormalon normalization $N_{1/2}(n_\ell=3)$ with the dimensionless parameter $\lambda$. The green dashed lines show the position of the maximum while the red, solid, horizontal line indicates the central value. The green, dashed, double-pointed arrow shows the uncertainty on $N_{1/2}$ while the magenta vertical line corresponds to the value of $\lambda>1$ for which the central value is reproduced. Finally, the black dashed lines show the value of $N_{1/2}$ for the default value $\lambda = 1$. Right panel: Dependence of the four-loop MSR-mass R-evolution kernel with $\lambda$ for $n_\ell=3$ active flavors between $R$ and $R_0$ for various values of $R$ (20, 10, 7 and 4 GeV are shown in blue, red, yellow and green, respectively) and a fixed boundary condition $R_0=2\,$GeV. The vertical dashed black lines correspond, from left to right, to $\lambda=1$ and the position of the maxima (which does not depend on $R$) and $\lambda_{\rm mid}$. The lines are normalized to their respective value at $\lambda=0.5$ and generated with $\alpha_s^{(3)}(m_\tau)=0.305$.
• Figure 2: Left panel: Dependence of the soft and ultrasoft renormalization scales with the distance $r$ for default parameters of the profile functions. The dashed purple line represents a pure canonical choice for the soft scale $\mu_s=1.2/r$ which becomes smaller than $1\,$GeV (marked with a thin solid black line) for $r>0.2\,$fm. The solid blue line shows Eq. \ref{['eq:assimp']} for $\xi=1.2$, $\mu_0=1\,$GeV and $b=0$, and in solid red we show the result of Eq. \ref{['eq:usoftProf']} for the same values plus the choice $\kappa=1$. The dashed green line (which is plotted up to $0.4\,$fm only since for larger $r$ the strong coupling is ill-defined) corresponds to the purely canonical ultrasoft scale. Right panel: ratio of the soft over the ultrasoft scale within the random scan. Both panels use the boundary condition $\alpha_s^{(3)}(m_\tau)=0.305$.
• Figure 3: Soft (left panel) and ultrasoft (right panel) renormalization scale variation obtained by randomly scanning over the parameters that define their analytic forms. The blue bands are obtained by varying all parameters within the ranges specified in Table \ref{['tab:profiles']}. The green band for $\mu_s$ ($\mu_{\rm us}$) is obtained by fixing $\xi$ ($\kappa$) to its default value and varying the rest of parameters, while in the red band only $\xi$ ($\kappa$) is varied, with the remaining profile parameters set to their default values. The red band in panel (b) is not centered for small $r$ because $\xi$ is fixed to the default rather than to its middle value. For reference, the canonical scales $\mu_s=1.45/r$ and $\mu_{\rm us}=1.45C_A\alpha_s(1.45/r)/r$ (dashed purple lines) as well as the freeze-out scale $1\,$GeV (dashed black line) are shown. Both panels use $\alpha_s^{(3)}(m_\tau)=0.305$.
• Figure 4: Left panel: size of subtraction, soft and ultrasoft logarithms for the $R$, $\mu_s$ and $\mu_{\rm us}$ profile parameters varied within the ranges specified in Table \ref{['tab:profiles']}. Dashed horizontal black lines mark the canonical size of logarithms under standard scale variation by factors of two, and the dashed green horizontal line marks $2\log(2)$. Right panel: same as panel (a), but multiplied by $\alpha_s(\mu_s)$, $\alpha_s(R)$ and $\alpha_s(\mu_{\rm us})$ the soft, subtraction and ultrasoft logarithms, respectively. In both panels, the dashed black vertical line shows the maximal distance entering our fits. Both plots are generated with $\alpha_s^{(3)}(m_\tau)=0.305$.
• Figure 5: Static energy at LO (gray), NLO (green), N$^2$LO (blue) and N$^3$LO (red), with uncertainty bands generated by varying the profile parameters within the ranges specified in Table \ref{['tab:profiles']}. The N$^2$LO (N$^3$LO) prediction includes N$^2$LL (N$^3$LL) ultrasoft resummation. Left and right panels show the static energy in the pole and MSR schemes, respectively. Both panels use $\alpha_s^{(3)}(m_\tau)=0.305$ and $R_0=2\,$GeV.