We study the scaling with the number of colors 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 $N_c = 3 - 8$ colors using twisted mass QCD. It is shown that the (subleading) $1/N_c$ 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.

Weak Decays and Matrix Elements