  We present our study of the renormalization of the chromomagnetic
operator,O(CM), which appears in the effective Hamiltonian describing Delta S =
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 non gauge 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 non-perturbatively 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 non-perturbatively 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 non-perturbatively 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
Nf = 2 + 1 + 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.
