O(a^2) corrections to the one-loop propagator and bilinears of clover fermions with Symanzik improved gluons

TL;DR

This work extends lattice perturbation theory to ${O}(a^2)$ for Wilson/clover fermions with Symanzik-improved gluons, computing the one-loop corrections to the fermion propagator and to all local fermion bilinears. A novel method is developed to extract the ${O}(a^2)$ terms in the presence of IR divergences that persist up to $D>6$ and non-Lorentz-invariant structures, enabling a consistent treatment beyond ${O}(a^1)$. The results display a rich dependence on the coupling $g$, number of colors $N$, lattice spacing $a$, external momentum $p$, clover coefficient $c_{SW}$, gauge parameter $\lambda$, and gluon-Symanzik coefficients, with detailed coefficients tabulated for several actions and delivered in an accompanying Oa2results.m file. These expressions support improved RI-MOM renormalization constants for quark bilinears and provide a framework extendable to higher orders in $a$ and to other operator bases and actions, enhancing the reliability of continuum extrapolations in lattice QCD.

Abstract

We calculate corrections to the fermion propagator and to the Green's functions of all fermion bilinear operators of the form $\barΨΓΨ$, to one-loop in perturbation theory. We employ the Wilson/clover action for fermions and a family of Symanzik improved actions for gluons. The novel aspect of our calculations is that they are carried out to second order in the lattice spacing, $O(a^2)$. Consequently, they have addressed a number of new issues, most notably the appearance of loop integrands with strong IR divergences (convergent only beyond 6 dimensions). Such integrands are not present in $O(a^1)$ improvement calculations; there, IR divergent terms are seen to have the same structure as in the $O(a^0)$ case, by virtue of parity under integration, and they can thus be handled by well-known techniques. We explain how to correctly extract the full $O(a^2)$ dependence; in fact, our method is generalizable to any order in $a$. The $O(a^2)$ corrections to the quark propagator and Green's functions computed in this paper are useful to improve the nonperturbative RI-MOM determination of renormalization constants for quark bilinear operators. Our results depend on a large number of parameters: coupling constant, number of colors, lattice spacing, external momentum, clover parameter, Symanzik coefficients, gauge parameter. To make these results most easily accessible to the reader, we have included them in the distribution package of this paper, as an ASCII file named: Oa2results.m ; the file is best perused as Mathematica input.