Non-leptonic kaon decays at large $N_c$

We study the scaling with the number of colors $N_c$ of the weak amplitudes mediating kaon mixing and decay, in the limit of light charm masses ($m_u=m_d=m_s=m_c$). The amplitudes are extracted directly on the lattice for $N_c = 3-7$ (with preliminar results for $N_c = 8$ and $17$) using twisted mass QCD. It is shown that the (sub-leading) $1/N_c$ corrections to $\hat B_K$ are small and that the naive $N_c \to \infty$ limit, $\hat B_K = 3/4$, seems to be recovered. On the other hand, the ${\cal O}\, (1/N_c)$ corrections in $K \to \pi \pi$ amplitudes (derived from $K \to \pi$ matrix elements) are large and fully anti-correlated in the $I=0$ and $I=2$ channels. This may have some implications for the understanding of the $\Delta I=1/2$ rule.