Follow-up on non-leptonic kaon decays at large $N_c$

Abstract

We report on the status of our dynamical simulations of a $SU(N_c)$ gauge theory with $N_c=3-6$ and $N_f=4$ fundamental fermions. These ensembles can be used to study the Large $N_c$ scaling of weak matrix elements in the GIM limit $m_c=m_u$, that might shed some light on the origin of the $\Delta I=1/2$ rule. We present preliminary results for the $K\to \pi$ matrix elements in the $N_c=3$ dynamical simulations, where we observe a significant effect of the quark loops that goes in the direction of enhancing the ratio of $A_0/A_2$ amplitudes. Finally, we present the relevant NLO Chiral Perturbation Theory predictions for the relation between $K\to \pi$ and $K\to \pi \pi$ amplitudes in the light charm limit.

Comments: 7 pages, 6 figures and 3 tables. This is the proceeding of the talk presented at the 36th Annual International Symposium on Lattice Field Theory, 22-28 July 2018, Michigan State University, East Lansing, Michigan, USA