Perturbative Renormalization Factors of Bilinear Quark Operators for Improved Gluon and Quark Actions in Lattice QCD

TL;DR

This work provides a one-loop perturbative analysis of renormalization factors for bilinear quark operators in lattice QCD with an improved gluon action that includes six-link loops and an $O(a)$-improved quark action, in the massless limit. By deriving the full action, Feynman rules, and quark self-energy alongside bilinear operator vertices, the authors compute $Z_ ext{Γ}$ factors for representative actions (Symanzik, Wilson, Iwasaki) and for general $(c_1, c_{23})$, then quantify how the finite parts of these coefficients depend on the six-link terms. They show that the one-loop finite parts are reduced by about 10–20% for Symanzik-type actions and by roughly a factor of two for renormalization-group improved actions compared to the plaquette action, with a monotonic decrease as $c_1$ or $c_{23}$ becomes more negative. These results enable more precise matching of lattice bilinear operators to the MS-bar scheme and guide action choices in precision lattice QCD calculations.

Abstract

We calculate one-loop renormalization factors of bilinear quark operators for gluon action including six-link loops and $O(a)$-improved quark action in the limit of massless quark. We find that finite parts of one-loop coefficients of renormalization factors diminish monotonically as either of the coefficients $c_1$ or $c_2+c_3$ of the six-link terms are decreased below zero. Detailed numerical results are given, for general values of the clover coefficient, for the tree-level improved gluon action in the Symanzik approach $(c_1=-1/12, c_2=c_3=0)$ and for the choices suggested by Wilson $(c_1=-0.252, c_2=0, c_3=-0.17)$ and by Iwasaki $(c_1=-0.331, c_2=c_3=0$ and $c_1=-0.27, c_2+c_3=-0.04)$ from renormalization-group analyses. Compared with the case of the standard plaquette gluon action, finite parts of one-loop coefficients are reduced by 10--20% for the Symanzik action, and approximately by a factor two for the renormalization-group improved gluon actions.

Figures (4)

• Figure 1: Quark-gluon vertices needed for our one-loop calculations.
• Figure 2: Lattice finite part $z_m$ for quark mass renormalization factor as a function of $c_1$ and $c_{23}$ for clover quark action with $c_{SW}=1$.
• Figure 3: Lattice finite part $z_V$ for vector current as a function of $c_1$ and $c_{23}$ for clover quark action with $c_{SW}=1$.
• Figure 4: Lattice finite part $z_A$ for axial vector current as a function of $c_1$ and $c_{23}$ for clover quark action with $c_{SW}=1$.