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

TL;DR

This work addresses the renormalization of local fermion bilinears on the lattice at two-loop order in perturbation theory, focusing on $\Gamma=\mathbb{1},\gamma_5$ for both flavor-singlet and nonsinglet cases and computing the prerequisite $Z_{\psi}$. Using the Wilson gauge action with clover-improved Wilson fermions, the authors obtain $Z_{\psi}$, $Z_S$, and $Z_P$ (and $Z_g$, $Z_A$) in the RI$^{\prime}$ scheme and translate them to $\overline{\mathrm{MS}}$, presenting results as polynomials in the clover coefficient $c_{\rm SW}$ for both bare and renormalized couplings; they also provide $Z_m$ for the singlet scalar. The pseudoscalar sector requires the finite factor $Z_5$ from dimensional regularization to ensure correct renormalization, while the scalar flavor-singlet case includes an additional finite piece beyond the non-singlet result. The paper validates known one-loop results, reports comprehensive two-loop corrections, and extends the formalism to fermions in arbitrary representations via an Appendix, enabling improved matching between lattice and continuum QCD and informing precise quark-mass determinations in lattice simulations.

Abstract

We compute the two-loop renormalization functions, in the RI $^\prime$ scheme, of local bilinear quark operators $\barψΓψ$, where $Γ$ 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, $Z_ψ$, 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. Finally, we present our results in the $\bar{MS}$ scheme, for easier comparison with calculations in the continuum. The corresponding results, for fermions in an arbitrary representation, are included in an Appendix.