Study of kaon decay to two pions

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 $\sigma$ 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 $\sigma$ meson contributes up to 80% of the total $I=0$ amplitude. This supports recent suggestions that the $\sigma$ meson should play a vital role in explaining the $\mathrm{\Delta }I=1/2$ rule in this system.

Figure 1

Contribution of the $\sigma$ meson to $K\to \pi \pi$.

Figure 2

Diagrammatic representation of the inhomogeneous Bethe-Salpeter equation for the different quark-antiquark bound states (mesons) of total 4-momentum $q$.

Figure 3

Illustration of the $\sigma \to \pi \pi$ process at the quark level, where the solid lines represent $u$ or $d$ quarks.

Figure 4

Wilson coefficients, $z_i(\mu )$. The abscissa represents the energy scale, $\mu$, in units of MeV.

Figure 5

Illustration of the $K\sigma$ transition, where the solid line with an $s$ represents the $s$ quark and the other solid lines represent $u$ or $d$ quarks.

Figure 6

The possible diagrams for the direct weak decay. The solid line with $s$ represents the $s$ quark, the other solid line represents the $u$ or $d$ quark. For the operator $Q_1$ and $Q_2$, only Fig. 6(a) contributes while Fig. 6(b) and Fig. 6(c) vanish in the NJL model.