Non-leptonic kaon decays at large $N_c$

TL;DR

This paper investigates how kaon decay amplitudes scale with the number of colours, $N_c$, using lattice evaluations for $N_c=3$ to 7, revealing small corrections for $ ext{B}_K$ but large, anti-correlated corrections for $K ightarrow \pi\pi$ amplitudes.

Contribution

It provides the first lattice-based analysis of $N_c$ scaling for kaon decay amplitudes, highlighting the different behaviors of corrections in $ ext{B}_K$ and $K ightarrow \pi\pi$ processes.

Findings

Small $1/N_c$ corrections in $ ext{B}_K$

Large, anti-correlated corrections in $K ightarrow \pi\pi$ amplitudes

Implications for the $ riangle I=1/2$ rule

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 $K \rightarrow \pi\pi$ 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.