Lattice study of vacuum polarization function and determination of strong coupling constant

TL;DR

This study uses lattice QCD with overlap fermions to compute vector and axial-vector vacuum polarization functions, enabling a nonperturbative determination of the strong coupling constant via perturbative QCD plus the Operator Product Expansion. By nonperturbatively subtracting lattice artifacts and employing the Adler-function framework, the authors extract $\Lambda^{(2)}_{\overline{MS}}$ and estimate the gluon condensate as well as four-quark condensates from the $V-A$ channel. The results yield $\Lambda^{(2)}_{\overline{MS}} = 0.234(9)(^{+16}_{-0})$ GeV with systematic checks showing robustness against discretization and truncation effects, and provide first-principle estimates of four-quark condensates relevant to kaon decays. The approach demonstrates the utility of exact chiral symmetry in simplifying OPE analyses and outlines clear paths for extending to 2+1 flavor QCD and conserved-current implementations.

Abstract

We calculate the vacuum polarization functions on the lattice using the overlap fermion formulation.By matching the lattice data at large momentum scales with the perturbative expansion supplemented by Operator Product Expansion (OPE), we extract the strong coupling constant $α_s(μ)$ in two-flavor QCD as $Λ^{(2)}_{\overline{MS}}$ = $0.234(9)(^{+16}_{- 0})$ GeV, where the errors are statistical and systematic, respectively. In addition, from the analysis of the difference between the vector and axial-vector channels, we obtain some of the four-quark condensates.

Figures (9)

• Figure 1: Momentum dependence of $B^J_1(Q)$, $B^J_2(Q)/a^2$, and $C^J_{11}(Q)/a^2$ at $m_q=0.015$. Circles (crosses) show the vector (axial-vector) channel. The solid curves represent a polynomial fit and the dashed curves show the one-loop results.
• Figure 2: Momentum dependence of $B^J_1(Q)$, $C^J_{11}(Q)/a^2$ and $B^J_2(Q)/a^2$ calculated in perturbation theory.
• Figure 3: $\Pi_V^{(0+1)}(Q)$ from off-diagonal $\mu\ne\nu$ and diagonal $\mu=\nu$ correlators with (lower panel) and without (upper panel) the subtraction of $B^J_{1,2}(Q)$ and $C^J_{11}(Q)$.
• Figure 4: Fit range dependence of $\Lambda^{(2)}_{\overline{MS}}$, $\langle(\alpha_s/\pi)GG\rangle$ and the constant term $c$. The maximum momentum squared $(aQ)^2_{max}$ is 1.324.
• Figure 5: $\Pi^{(0+1)}_{V+A}(Q)$ as a function of $(aQ)^2$. The lattice data at different quark masses are shown by open symbols. Fit curves for each quark mass and in the chiral limit are drawn. The full result in the chiral limit (dashed-dots curves are at the finite masses, and solid curve is in the chiral limit), as well as that without $\langle\alpha_s G^2\rangle/Q^4$ term (dashed curve), are shown.