"Old" Resonance Model for $3\pi$ Decays of $K$ and $\eta$

In view of the current experimental interest in the three-pion modes of $K$ and $\eta$ decays, a phenomenological model of a $\pi -\pi$ resonance in $I=0\mathrm{}$, proposed several years ago by one of us, is re-examined in the context of the present experimental data. It is found that the model satisfactorily explains all the available features of $\tau$ and ${\tau }^{\prime }$ decays if the resonance occurs at an energy around 400 MeV with a width 85-90 MeV. The corresponding parameters with a somewhat smaller width (∼75-80 MeV) also give satisfactory agreement with the experimental results for $\eta$ decay. This model corresponds to a repulsive $I=0\mathrm{}$ $\pi -\pi$ interaction at low energies and is therefore not in disagreement with the earlier dispersion theoretical predictions for $\tau$ decay, using final-state interactions. It also has a strong resemblance to the $\sigma$ model of Brown and Singer.