Study of kaon decay to two pions

Abstract

The weak decay of the kaon to two pions is studied within the model of Nambu and Jona-Lasinio. Using the standard effective weak Hamiltonian, both the decay amplitude arising from an intermediate state $\ensuremath{\sigma}$ meson and the direct decay amplitude are calculated. The effect of final state interactions is also included. When the matching scale is chosen such that the decay amplitude with isospin $I=2$ is close to its experimental value, our model including the $\ensuremath{\sigma}$ meson contributes up to 80% of the total $I=0$ amplitude. This supports recent suggestions that the $\ensuremath{\sigma}$ meson should play a vital role in explaining the $\mathrm{\ensuremath{\Delta}}I=1/2$ rule in this system.