Non-perturbative calculation of $Z_V$ and $Z_A$ in domain-wall QCD on a finite box

TL;DR

This paper presents a non-perturbative method to calculate the renormalization factors for vector and axial-vector currents in quenched domain-wall QCD using finite box techniques and Ward-Takahashi identities.

Contribution

It introduces a novel finite-volume approach with Dirichlet boundary conditions to determine $Z_V$ and $Z_A$, including an interpolation formula for $Z_V$ in the infinite volume.

Findings

Confirmed $Z_V \,\simeq\, Z_A$ relation numerically.

Provided an interpolation formula for $Z_V$ as a function of $g^2$ and $M$.

Demonstrated the effectiveness of the finite-volume method in non-perturbative renormalization.

Abstract

We report on a non-perturbative evaluation of the renormalization factors for the vector and axial-vector currents, $Z_V$ and $Z_A$, in the quenched domain-wall QCD (DWQCD) with plaquette and renormalization group improved gauge actions. We take the Dirichlet boundary condition for both gauge and domain-wall fermion fields on the finite box, and introduce the flavor-chiral Ward-Takahashi identities to calculate the renormalization factors. As a test of the method, we numerically confirm the expected relation that $Z_V \simeq Z_A$ in DWQCD. Employing two different box sizes for the numerical simulations at several values of the gauge coupling constant $g^2$ and the domain-wall height $M$, we extrapolate $Z_V$ to the infinite volume to remove $a/L$ errors. We finally give the interpolation formula of $Z_V$ in the infinite volume as a function of $g^2$ and $M$.