Two-loop renormalization of vector, axial-vector, and tensor 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 $\ensuremath{\Gamma}$ corresponds to the vector, axial-vector, and tensor Dirac operators, in the lattice formulation of QCD. We consider both the flavor nonsinglet and singlet operators. 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 constant, in the renormalized Feynman 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, together with some special features of superficially divergent integrals, are included in the appendices.