Non-leptonic kaon decays at large $N_c$

Abstract

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 $1/N_c$ corrections in ${\hat{B}}_K$ 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 $\Delta I=1/2$ rule.

Expanded discussion. Conclusions unchanged. Published version

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• FLAG Review 2019
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• The Large $N_c$ limit of QCD on the lattice
• SU(3) analysis of four-quark operators: $K\to \pi \pi$ and vacuum matrix elements
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