Precise determination of the strong coupling constant in Nf=2+1 lattice QCD with the Schrödinger functional scheme

TL;DR

The paper develops a nonperturbative determination of the running coupling for $N_f=2+1$ QCD using the Schrödinger functional scheme to bridge low-energy hadronic inputs with high-energy perturbative QCD and to convert to the $MSbar$ scheme. It employs seven SF renormalization scales and three lattice spacings to compute a nonperturbative step-scaling function and a corresponding beta function, then sets the physical scale via hadron masses and performs flavor-threshold matching to extract $\alpha_s(M_Z)$ and $\Lambda_{\overline{MS}}^{(5)}$ with a controlled error budget. The final result, $\alpha_s(M_Z)=0.12047(81)(48)^{+0}_{-173}$, is consistent with other lattice determinations and the PDG range, validating the SF approach and the continuum extrapolation. The work highlights the importance of nonperturbative renormalization, careful treatment of lattice artifacts, and perturbative matching at flavor thresholds for precision lattice QCD determinations of the strong coupling.

Abstract

We present an evaluation of the running coupling constant for Nf=2+1 QCD. The Schroedinger functional scheme is used as the intermediate scheme to carry out non-perturbative running from the low energy region, where physical scale is introduced, to deep in the high energy perturbative region, where conversion to the MS-bar scheme is safely performed. Possible systematic errors due to the use of perturbation theory occur only in the conversion from three-flavor to four-flavor running coupling constant near the charm mass threshold, where higher order terms beyond 5th order in the $β$ function may not be negligible. For numerical simulations we adopted Iwasaki gauge action and non-perturbatively improved Wilson fermion action with the clover term. Seven renormalization scales are used to cover from low to high energy region and three lattice spacings to take the continuum limit at each scale. A physical scale is introduced from the previous Nf=2+1 simulation of the CP-PACS/JL-QCD collaboration, which covered the up-down quark mass range heavier than $m_π\sim 500$ MeV.

Figures (6)

• Figure 1: Distribution of inverse of the renormalized coupling at lowest energy scale given by $\overline{g}^2(L)\sim5$, which corresponds to $L/a=16$, $\beta=2.5$ (left), $L/a=12$, $\beta=2.34652$ (middle) and $L/a=8$, $\beta=2.15743$ (right). Solid line is a fit in a Gaussian function.
• Figure 2: Polynomial fit of discrepancy $\Sigma\left(u,{a}/{L}\right)/\sigma_{\rm PT}^{(3)}(u)$ at high $\beta\gtrsim4$. The fit is given for each lattice spacings $a/L=1/4$ (left), $a/L=1/6$ (middle) and $a/L=1/8$ (right). Black dotted line is a perturbative one loop behavior and red solid line is a quadratic fit with fixed $d_1$ to its one loop value.
• Figure 3: The SSF on the lattice with its continuum extrapolation at each renormalization scale.
• Figure 4: RG flow of the SSF divided by the coupling $\overline{g}^2(L)$. Dotted line is three loops perturbative running. Solid line is a polynomial fit of the SSF.
• Figure 5: Non-perturbative $\beta$-function for $N_f=3$ and $2$ QCD. Solid lines are three loops perturbative running for comparison. Data for $N_f=2$ is reproduced from Ref. DellaMorte:2004bc.