  The Chromomagnetic operator (CMO) mixes with a large number of operators
under renormalization. We identify which operators can mix with the CMO, at the
quantum level. Even in dimensional regularization (DR), which has the simplest
mixing pattern, the CMO mixes with a total of 9 other operators, forming a
basis of dimension-five, Lorentz scalar operators with the same flavor content
as the CMO. Among them, there are also gauge noninvariant operators; these are
BRST invariant and vanish by the equations of motion, as required by
renormalization theory. On the other hand using a lattice regularization
further operators with $d \leq 5$ will mix; choosing the lattice action in a
manner as to preserve certain discrete symmetries, a minimul set of 3
additional operators (all with $d<5$) will appear. In order to compute all
relevant mixing coefficients, we calculate the quark-antiquark (2-pt) and the
quark-antiquark-gluon (3-pt) Green's functions of the CMO at nonzero quark
masses. These calculations were performed in the continuum (dimensional
regularization) and on the lattice using the maximally twisted mass fermion
action and the Symanzik improved gluon action. In parallel, non-perturbative
measurements of the $K-\pi$ matrix element are being performed in simulations
with 4 dynamical ($N_f = 2+1+1$) twisted mass fermions and the Iwasaki improved
gluon action.
