Higher loop renormalization of fermion bilinear operators

TL;DR

This paper calculates two-loop renormalization functions for fermion bilinear operators in lattice QCD, providing essential data for precise non-perturbative studies of quark properties.

Contribution

It presents the first two-loop renormalization functions for scalar and pseudoscalar quark operators in the RI' scheme on the lattice, including flavor singlet and non-singlet cases.

Findings

Computed $Z_ ext{psi}$ up to two loops.

Derived polynomial expressions in $c_{SW}$ for renormalization functions.

Confirmed one-loop results for generic gauges.

Abstract

We compute the two-loop renormalization functions, in the RI' 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. A longer write-up of the present work, including the conversion of our results to the MSbar scheme and a generalization to arbitrary fermion representations, can be found in arXiv:0707.2906 .