We compute the two-loop renormalization functions, in the $R{I}^{\prime }$ scheme, of local bilinear quark operators $\overline{\psi }\Gamma \psi$, where $\Gamma$ 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, $Z_{\psi }$, 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.

• Received 18 August 2007

DOI:https://doi.org/10.1103/PhysRevD.76.094514