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

Abstract

We compute the two-loop renormalization functions, in the $R{I}^{\ensuremath{'}}$ scheme, of local bilinear quark operators $\overline{\ensuremath{\psi}}\ensuremath{\Gamma}\ensuremath{\psi}$, where $\ensuremath{\Gamma}$ denotes the scalar and pseudoscalar Dirac matrices, in the lattice formulation of QCD. We consider both the flavor nonsinglet 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, ${Z}_{\ensuremath{\psi}}$, 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 ${c}_{\mathrm{SW}}$, in terms of both the renormalized and bare coupling constants, in the renormalized Feynman gauge. We also confirm the one-loop renormalization functions, for generic gauge. Finally, we present our results in the $\overline{MS}$ scheme, for easier comparison with calculations in the continuum. The corresponding results, for fermions in an arbitrary representation, are included in the Appendix.