Renormalization of the chromomagnetic operator on the lattice

We present our study of the renormalization of the chromomagnetic operator, O CM , which appears in the effective Hamiltonian describing ΔS 1⁄4 1 transitions in and beyond the Standard Model. We have computed, perturbatively to one loop, the relevant Green’s functions with two (quark-quark) and three (quark-quark-gluon) external fields, at nonzero quark masses, using both the lattice and dimensional regularizations. The perturbative computation on the lattice is carried out using the maximally twisted-mass action for the fermions, while for the gluons we employed the Symanzik improved gauge action for different sets of values of the Symanzik coefficients. We have identified all the operators which can possibly mix with O CM , including lower-dimensional and nongauge invariant operators, and we have calculated those elements of the mixing matrix which are relevant for the renormalization of O CM . We have also performed numerical lattice calculations to determine nonperturbatively the mixings of the chromomagnetic operator with lower-dimensional operators, through proper renormalization conditions. For the first time, the 1=a 2 -divergent mixing of the chromomagnetic operator with the scalar density has been determined nonperturbatively with high precision. Moreover, the 1=a-divergent mixing with the pseudoscalar density, due to the breaking of parity within the twisted-mass regularization of QCD, has been calculated nonperturbatively and found to be smaller than its one-loop perturbative estimate. The QCD simulations have been carried out using the gauge configurations produced by the European Twisted Mass Collaboration with N f 1⁄4 2 þ 1 þ 1 dynamical quarks, which include in the sea, besides two light mass degenerate quarks, also the strange and charm quarks with masses close to their physical values.