Isotopic Spin inK→3π

Because recent data on ${K_2}^0\to {\pi }^{+}{\pi }^{-}{\pi }^0$ are at variance with the $\Delta T=\frac{1}{2}$ rule while the data on $K^{+}\to 3\pi$ are not, the charge space kinematics of $K\to 3\pi$ are re-examined. Matrix elements are assumed to be at most linearly dependent on the usual variables $s_i$, and it follows that only four of the seven possible $3\pi$ states can contribute to the decay. Of these states, two have $T=1\mathrm{}$, the third has $T=2\mathrm{}$ and the fourth $T=3\mathrm{}$. The possible values of $\Delta T$ are ½, $\frac{3}{2}$, $\frac{5}{2}$, $\frac{7}{2}$, and accordingly, the most general interaction Hamiltonian is written as the sum of four parts $H_{\frac{n}{2}}$, each corresponding to $\Delta T=\frac{n}{2}$ ($n=1, 3, 5, 7$). It is then possible to express the matrix elements, rates and spectra of all the modes of $K\to 3\pi$ in terms of the reduced matrix elements of $H_{\frac{n}{2}}$ between the four $3\pi$ states and the $K$ meson. The analysis reveals that, provided the branching ratio of ${K_2}^0\to 3{\pi }^0$ to ${K_2}^0\to {\pi }^{+}{\pi }^{-}{\pi }^0$ is $\frac{3}{2}$, the present data are consistent with an interaction Hamiltonian containing only $\Delta T=\frac{1}{2} \mathrm{and} \frac{3}{2}$, and a $3\pi$ final state of isotopic spin one.