A non-perturbative exploration of the high energy regime in $N_\text{f}=3$ QCD

TL;DR

This paper uses lattice QCD simulations with Schr"odinger functional boundary conditions to non-perturbatively determine the running coupling in three-flavor QCD over a wide energy range, achieving high precision and testing perturbation theory.

Contribution

It provides a non-perturbative determination of the QCD scale parameter in three-flavor QCD with improved accuracy and explores the validity of perturbation theory at high energies.

Findings

Precise determination of $L_0 \\Lambda^{N_f=3}_{\\overline{MS}}$

Validation of perturbation theory at high energies

Controlled systematic and statistical errors in lattice calculations

Abstract

Using continuum extrapolated lattice data we trace a family of running couplings in three-flavour QCD over a large range of scales from about 4 to 128 GeV. The scale is set by the finite space time volume so that recursive finite size techniques can be applied, and Schr\"odinger functional (SF) boundary conditions enable direct simulations in the chiral limit. Compared to earlier studies we have improved on both statistical and systematic errors. Using the SF coupling to implicitly define a reference scale $1/L_0\approx 4$ GeV through $\bar{g}^2(L_0) =2.012$, we quote $L_0 \Lambda^{N_{\rm f}=3}_{\overline{\rm MS}} =0.0791(21)$. This error is dominated by statistics; in particular, the remnant perturbative uncertainty is negligible and very well controlled, by connecting to infinite renormalization scale from different scales $2^n/L_0$ for $n=0,1,\ldots,5$. An intermediate step in this connection may involve any member of a one-parameter family of SF couplings. This provides an excellent opportunity for tests of perturbation theory some of which have been published in a letter [1]. The results indicate that for our target precision of 3 per cent in $L_0 \Lambda^{N_{\rm f}=3}_{\overline{\rm MS}}$, a reliable estimate of the truncation error requires non-perturbative data for a sufficiently large range of values of $\alpha_s=\bar{g}^2/(4\pi)$. In the present work we reach this precision by studying scales that vary by a factor $2^5= 32$, reaching down to $\alpha_s\approx 0.1$. We here provide the details of our analysis and an extended discussion.