Non-leptonic kaon decays at large $N_c$

We study the scaling with the number of colours, $N_c$, of the weak amplitudes mediating kaon mixing and decay. We evaluate the amplitudes of the two relevant current-current operators on the lattice for $N_c=3-7$. We conclude that the subleading $1\mathrm{/}{N}_c$ corrections in ${\widehat{B}}_K$ are small, but those in the $K\to \pi \pi$ amplitudes are large and fully anti-correlated in the $I=0,2$ isospin channels. We briefly comment on the implications for the $\mathrm{\Delta }I=1\mathrm{/}2$ rule.