  We report on the status of our dynamical simulations of a $SU (N_c )$ gauge
theory with $N_c=3-6$ and $N_f =4$ fundamental fermions. These ensembles can be
used to study the Large $N_c$ scaling of weak matrix elements in the GIM limit
$m_c=m_u$, that might shed some light on the origin of the $\Delta I=1/2$ rule.
We present preliminary results for the $K \to \pi$ matrix elements in the
$N_c=3$ dynamical simulations, where we observe a significant effect of the
quark loops that goes in the direction of enhancing the ratio of $A_0/A_2$
amplitudes. Finally, we present the relevant NLO Chiral Perturbation Theory
predictions for the relation between $K \to \pi $ and $K \to \pi \pi$
amplitudes in the light charm limit.

High Energy Physics - Lattice;High Energy Physics - Phenomenology