Determination of the QCD $\Lambda$-parameter and the accuracy of perturbation theory at high energies

TL;DR

This paper accurately determines the QCD $\Lambda$-parameter at high energies using lattice computations, confirming perturbation theory's reliability at small coupling constants, and highlighting its limitations at larger couplings.

Contribution

It introduces a scheme enabling non-perturbative lattice calculations at very high energies, demonstrating perturbation theory's precision in this regime for the first time.

Findings

Perturbation theory yields a 3% error in $\Lambda$-parameter at $\alpha_s=0.1$.

Data at $\alpha_s \\approx 0.2$ is insufficient for high-precision results.

The scheme used has advantageous properties for applying perturbation theory.

Abstract

We discuss the determination of the strong coupling $\alpha_\mathrm{\overline{MS}}^{}(m_\mathrm{Z})$ or equivalently the QCD $\Lambda$-parameter. Its determination requires the use of perturbation theory in $\alpha_s(\mu)$ in some scheme, $s$, and at some energy scale $\mu$. The higher the scale $\mu$ the more accurate perturbation theory becomes, owing to asymptotic freedom. As one step in our computation of the $\Lambda$-parameter in three-flavor QCD, we perform lattice computations in a scheme which allows us to non-perturbatively reach very high energies, corresponding to $\alpha_s = 0.1$ and below. We find that (continuum) perturbation theory is very accurate there, yielding a three percent error in the $\Lambda$-parameter, while data around $\alpha_s \approx 0.2$ is clearly insufficient to quote such a precision. It is important to realize that these findings are expected to be generic, as our scheme has advantageous properties regarding the applicability of perturbation theory.