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We compute the renormalization functions on the lattice, in the RI' scheme,of local bilinear quark operators $\bar{\psi}\Gamma\psi$, where $\Gamma= 1,\gamma_5, \gamma_\mu, \gamma_5\gamma_\mu, \gamma_5\sigma_{\mu\nu}$. Thiscalculation is carried out to two loops for the first time. We consider boththe flavor non-singlet and singlet operators.  As a prerequisite for the above, we compute the quark field renormalization,$Z_\psi$, up to two loops. We also compute the 1-loop renormalization functionsfor the gluon field, $Z_A$, ghost field, $Z_c$, gauge parameter, $Z_\alpha$,and coupling constant $Z_g$.  We use the clover action for fermions and the Wilson action for gluons. Ourresults are given as an explicit function of the coupling constant, the clovercoefficient $c_{SW}$, and the number of fermion colors ($N_c$) and flavors($N_f$), in the renormalized Feynman gauge. All 1-loop quantities are evaluatedin an arbitrary gauge.  Finally, we present our results in the MS-bar scheme, for easier comparisonwith calculations in the continuum. We have generalized to fermionic fields inan arbitrary representation. Some special features of superficially divergentintegrals, obtained from the evaluation of two-loop Feynman diagrams, arepresented in detail in Ref. 1.

Two-loop renormalization of fermion bilinear operators on the lattice