Two-loop renormalization of fermion bilinear operators on the lattice

TL;DR

This paper presents the first two-loop lattice calculations of renormalization functions for fermion bilinear operators, providing crucial data for matching lattice QCD results with continuum schemes like MS-bar.

Contribution

It introduces the first two-loop computation of fermion bilinear operator renormalization on the lattice, including flavor singlet and non-singlet cases, using the RI' scheme and generalized to arbitrary fermion representations.

Findings

Two-loop renormalization functions for fermion bilinear operators computed.

Results expressed as functions of coupling, $c_{SW}$, $N_c$, and $N_f$ in Feynman gauge.

Results provided in both RI' and MS-bar schemes.

Abstract

We compute the renormalization functions on the lattice, in the RI' scheme, of local bilinear quark operators $\bar{\psi}\Gamma\psi$, where $\Gamma= 1, \gamma_5, \gamma_\mu, \gamma_5\gamma_\mu, \gamma_5\sigma_{\mu\nu}$. This calculation is carried out to two loops for the first time. We consider both the flavor non-singlet and singlet operators. As a prerequisite for the above, we compute the quark field renormalization, $Z_\psi$, up to two loops. We also compute the 1-loop renormalization functions for the gluon field, $Z_A$, ghost field, $Z_c$, gauge parameter, $Z_\alpha$, and coupling constant $Z_g$. We use the clover action for fermions and the Wilson action for gluons. Our results are given as an explicit function of the coupling constant, the clover coefficient $c_{SW}$, and the number of fermion colors ($N_c$) and flavors ($N_f$), in the renormalized Feynman gauge. All 1-loop quantities are evaluated in an arbitrary gauge. Finally, we present our results in the MS-bar scheme, for easier comparison with calculations in the continuum. We have generalized to fermionic fields in an arbitrary representation. Some special features of superficially divergent integrals, obtained from the evaluation of two-loop Feynman diagrams, are presented in detail in Ref. 1.