Normal Scalar Nonet;K1→2πDecay andK1−K2Mass Difference

We have assumed the existence of a scalar nonet [the physical particles are designated by (${\kappa }^{+}, {\kappa }^0$); (${\overline{\kappa }}^0, {\kappa }^{-}$), (${\epsilon }^{+}, {\epsilon }^0, {\epsilon }^{-}$), ${\xi }^0$, and ${\sigma }^0$] and have considered the dynamics of $K_1\to 2\pi$ decay, $K^0\leftrightarrow {\overline{K}}^0$ transition, and $S$-wave $\pi \pi$ scattering. It is stressed that the scalar mesons, if they exist, will play a dominant role in the dynamics of each of the above three processes. We have assumed (for reasons discussed in a previous work) that all the parity-violating $\left|\Delta Y\right|=1\mathrm{}$ nonleptonic transitions are dominated by the $K_1$ tadpole, even though the latter is forbidden in the limit of $\mathrm{SU}\left(3\right)\mathrm{}$. In order to obtain a consistent theoretical picture (in the single-particle intermediate-state approximation) for the observed sign ($m_{K_2}>m_{K_1}$) and magnitude of the $K_1-K_2$ mass difference as well as the observed rate of the $K_1\to 2\pi$ decay, it appears necessary that there should exist scalar mesons lying above the mass of the $K$ meson, especially an $I=Y=0\mathrm{}$ scalar meson (called $\sigma$) lying close to the $K$ meson but above it ($510 \mathrm{MeV}<m_{\sigma }<550\mathrm{} \mathrm{MeV}$, say]. The width of the $\sigma$ meson can be small [$\Gamma (\sigma \to 2\pi )\approx 10\sim 30$ MeV, for example] and could lead to a small relative production cross section of the $\sigma$ meson in pion-nucleon processes.