Perturbative determination of $\mathcal{O}(am)$ improvement on the QCD running coupling

TL;DR

This work targets the mass-dependent discretization errors of order $\mathcal{O}(a m_q)$ in the lattice-QCD running coupling by computing the two-loop renormalization factor $Z_g$ using the background-field method with Clover fermions and Symanzik-improved gauge actions. By relating $Z_g$ to the BF 2-point function and employing a mass-independent renormalization scheme with a mass-dependent reparametrization $\tilde{g}_0^2= g_0^2\big(1 + b_g(g_0^2) a m_q\big)$, the authors extract the coefficients $b_g^{(1)}$ and $b_g^{(2)}$ that quantify $\mathcal{O}(a m_q)$ effects. They present explicit one- and two-loop results, including mass-dependent heavy-quark corrections, for various lattice actions and general $N_c, N_f$, highlighting how these corrections influence the matching between lattice and $\overline{MS}$ couplings. The findings are crucial for precision determinations of $\alpha_s$ and for controlled continuum extrapolations in lattice QCD, especially when heavy quarks are present. They also outline future extensions, such as stout-smearing and pure-gluon contributions, to further refine the perturbative understanding of lattice artifacts in the running coupling.

Abstract

We present the perturbative results of the discretization errors proportional to the quark mass ($\mathcal{O}(a m)$) on the QCD running coupling within lattice perturbation theory. Our analysis involves calculating the 2-loop renormalization factor $Z_g$ using improved lattice actions for the $SU(N_c)$ gauge group and $N_f$ multiplets of fermions with a finite quark mass. We employ the background field method to compute $Z_g$, by calculating quantum corrections on both the background and quantum gluon propagator, respecting the $\mathcal{O}(a)$ improvement. This allows us to evaluate the perturbative $\mathcal{O}(a m)$ lattice errors which affect the determination of the running coupling. Eliminating these $\mathcal{O}(a m)$ effects is crucial for the nonperturbative studies of precision determinations of the strong coupling constant using lattice field theory.

Figures (2)

• Figure 1: One-loop Feynman diagrams for fermion contributions to $\Gamma^{\rm BB, 1loop}_{\rm L, F}$. A solid line represents quarks. Wavy lines ending on a cross represent background gluons. Each diagram is meant to be symmetrized over the color indices, Lorentz indices, and momenta of the two external background fields.
• Figure 2: Two-loop Feynman diagrams for the fermion contributions to $\Gamma^{\rm BB, 2loop}_{\rm L, F}$. A wavy (solid) line represents gluons (quarks). Wavy lines ending on a cross represent background gluons. A solid circle is the one-loop fermion mass counterterm. Each diagram is meant to be symmetrized over the color indices, Lorentz indices, and momenta of the two external background fields.