O(a²) corrections to 1-loop matrix elements of 4-fermion operators with improved fermion/gluon actions

We calculate the corrections to the amputated Green's functions of 4-fermion operators, in 1-loop Lattice Perturbation theory. The novel aspect of our calculations is that they are carried out to second order in the lattice spacing, O(a^2). We employ the Wilson/clover action for massless fermions (also applicable for the twisted mass action in the chiral limit) and the Symanzik improved action for gluons. Our calculations have been carried out in a general covariant gauge. Results have been obtained for several popular choices of values for the Symanzik coefficients (Plaquette, Tree-level Symanzik, Iwasaki, TILW and DBW2 action). We pay particular attention to $\Delta F=2$ operators, both Parity Conserving and Parity Violating ($F$ stands for flavour: S, C, B). We study the mixing pattern of these operators, to O(a^2), using the appropriate projectors. Our results for the corresponding renormalization matrices are given as a function of a large number of parameters: coupling constant, clover parameter, number of colors, lattice spacing, external momentum and gauge parameter. The O(a^2) correction terms (along with our previous O(a^2) calculation of $Z_\Psi$) are essential ingredients for minimizing the lattice artifacts which are present in non-perturbative evaluations of renormalization constants with the RI'-MOM method. A longer write-up of this work, including non-perturbative results, is in preparation together with members of the ETM Collaboration.