Non-leptonic kaon decays at large $N_c$

Abstract

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 $\mathcal{O}(1/N_c)$ 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.

Comments: 8 pages, 4 figures, Proceedings of the 35th International Symposium on Lattice Field Theory, June 18-24, 2017, Granada, Spain