The strong running coupling at $τ$ and $Z_0$ mass scales from lattice QCD

TL;DR

The paper computes the running strong coupling from lattice QCD with 2+1+1 dynamical flavors using the ghost–gluon coupling in the MOM Taylor scheme, incorporating perturbative running and nonperturbative OPE corrections. By including a dominant $1/p^6$ power correction alongside the gluon condensate, the authors fit lattice data over a wide momentum range to extract $\Lambda_T$ and convert to $\Lambda_{\overline{MS}}$. They obtain $\alpha_s^{\overline{MS}}(m_\tau^2)\approx 0.339(13)$ and $\alpha_s^{\overline{MS}}(m_Z^2)\approx 0.1200(14)$, which agree with $\tau$-decay determinations and World averages, and demonstrate consistency with other lattice determinations. The results corroborate the ETM collaboration’s 2+1+1 lattice program and provide an independent, high-precision lattice determination of $\alpha_s$ with careful treatment of higher-twist corrections and discretization effects.

Abstract

This letter reports on the first computation, from data obtained in lattice QCD with $u,d,s$ and $c$ quarks in the sea, of the running strong coupling via the ghost-gluon coupling renormalized in the MOM Taylor scheme. We provide with estimates of $\ams(m_τ^2)$ and $\ams(m_Z^2)$ in very good agreement with experimental results. Including a dynamical c quark makes safer the needed running of $\ams$.

Figures (4)

• Figure 1: The departure of lattice data from the leading non-perturbative OPE prediction for the running coupling plotted in logarithmic scales, in terms of the momentum manifestly shows a next-to-leading $1/p^6$ behaviour; the dashed red line stands for the momentum scale, $p\simeq 1.7$ GeV, below which the lattice data do not follow the $1/p^6$ behaviour any longer.
• Figure 2: $\Lambda_{\overline{\hbox{\sc ms}}}$ obtained by applying the plateau method to the lattice data labelled in the plot. Red solid/dashed line corresponds to the plateau for $\Lambda_{\overline{\hbox{\sc ms}}}$ obtained with Eq. (\ref{['eq:alphahdp6']})/(\ref{['alphahNP']}). The black solid is for Eq. (\ref{['alphahNP']}) with $g^2 \langle A^2 \rangle = 0$, while black dashed corresponds to evaluate first Eq. (\ref{['eq:alphahdp6']}) with the best-fitted parameters in Tab. \ref{['tab:bestfit']} and take then the resulting $\alpha_T$ to obtain $\Lambda_{\overline{\hbox{\sc ms}}}$ by inverting Eq. (\ref{['alphahNP']}) with $g^2 \langle A^2 \rangle = 0$.
• Figure 3: Eq. (\ref{['alphahNP']}) (red dashed) and Eq. (\ref{['eq:alphahdp6']}) (red solid) for the parameters in Tab. \ref{['tab:bestfit']} fitted to the lattice data for $\alpha_T$ defined by \ref{['alpha']}. The black line is for Eq. (\ref{['alphahNP']}) with $g^2 \langle A^2 \rangle=0$.
• Figure 4: The strong $\overline{\hbox{\sc ms}}$ coupling running for 4 quark flavours and for $\Lambda_{\overline{\rm MS}}=316$ MeV (black) and $\Lambda_{\overline{\rm MS}}=324$ MeV (blue) below the bottom mass threshold. The dashed lines represent the one-$\sigma$ statistical deviations. The red point stand for the value of $\alpha_{\overline{\rm MS}}(m_\tau^2)$ obtained from $\tau$ decays Bethke:2011tr.