In this work we calculate the corrections to the amputated Green’s functions of four-fermion operators, in 1-loop lattice perturbation theory. One of the novel aspects of our calculations is that they are carried out to second order in the lattice spacing, $\mathcal{O}(a^2)$. We employ the Wilson/clover action for massless fermions (also applicable for the twisted mass action in the chiral limit) and a family of Symanzik improved actions 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). While our Green’s function calculations regard any pointlike four-fermion operators which do not mix with lower dimension ones, we pay particular attention to $\Delta F=2$ operators, both parity conserving and parity violating ($F$ stands for flavor: $S$, $C$, $B$). By appropriately projecting those bare Green’s functions we compute the perturbative renormalization constants for a complete basis of four-fermion operators and we study their mixing pattern. For some of the actions considered here, even $\mathcal{O}(a^0)$ results did not exist in the literature to date. The correction terms which we calculate (along with our previous $\mathcal{O}(a^2)$ calculation of $Z_{\Psi }$ [M. Constantinou, V. Lubicz, H. Panagopoulos, and F. Stylianou, J. High Energy Phys. 10 (2009) 064.][M. Constantinou, P. Dimopoulos, R. Frezzotti, G. Herdoiza, K. Jansen, V. Lubicz, H. Panagopoulos, G. C. Rossi, S. Simula, F. Stylianou, and A. Vladikas, J. High Energy Phys. 08 (2010) 068.][C. Alexandrou, M. Constantinou, T. Korzec, H. Panagopoulos, and F. Stylianou (unpublished).]) are essential ingredients for minimizing the lattice artifacts which are present in nonperturbative evaluations of renormalization constants with the $\mathrm{R}{\mathrm{I}}^{\prime }-\mathrm{MOM}$ method. Our perturbative results, for the matrix elements of $\Delta F=2$ operators and for the corresponding renormalization matrices, 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: 4-fermi.m; the file is best perused as Mathematica input. The main results of this work have been applied to improve nonperturbative estimates of the $B_K$-parameter in $N_F=2$ twisted mass lattice QCD [M. Constantinou, P. Dimopoulos, R. Frezzotti, K. Jansen, V. Gimenez, V. Lubicz, F. Mescia, H. Panagopoulos, M. Papinutto, G. C. Rossi, S. Simula, A. Skouroupathis, F. Stylianou, and A. Vladikas, arXiv:1009.5606.].

• Received 8 December 2010

DOI:https://doi.org/10.1103/PhysRevD.83.074503