Two-loop renormalization of scalar and pseudoscalar fermion bilinears on the lattice

TL;DR

This paper calculates two-loop renormalization functions for scalar and pseudoscalar fermion bilinear operators in lattice QCD, providing results in both RI' and MS schemes to improve precision in theoretical predictions.

Contribution

It presents the first two-loop renormalization functions for these operators on the lattice, including flavor singlet and non-singlet cases, with results applicable to various fermion representations.

Findings

Two-loop renormalization functions computed for scalar and pseudoscalar operators.

Results expressed as polynomials in the clover coefficient $c_{SW}$.

Results provided in both RI' and MS schemes for broad applicability.

Abstract

We compute the two-loop renormalization functions, in the RI $^\prime$ scheme, of local bilinear quark operators $\bar{\psi}\Gamma\psi$, where $\Gamma$ denotes the Scalar and Pseudoscalar Dirac matrices, in the lattice formulation of QCD. We consider both the flavor non-singlet and singlet operators; the latter, in the scalar case, leads directly to the two-loop fermion mass renormalization, $Z_m$. As a prerequisite for the above, we also compute the quark field renormalization, $Z_{\psi}$, up to two loops. We use the clover action for fermions and the Wilson action for gluons. Our results are given as a polynomial in $c_{SW}$, in terms of both the renormalized and bare coupling constant, in the renormalized Feynman gauge. We also confirm the 1-loop renormalization functions, for generic gauge. Finally, we present our results in the $\bar{MS}$ scheme, for easier comparison with calculations in the continuum. The corresponding results, for fermions in an arbitrary representation, are included in an Appendix.