Zeros in the Scattering Amplitudes of Two Pseudoscalar Mesons in the Nonet States

The low-energy pseudoscalar-meson-pseudoscalar-meson scatterings are investigated phenomenologically in terms of the forward dispersion relations. Using the scattering lengths given by the current algebra, it is shown that there exists one general feature common to the scattering amplitudes in the nonet states, namely, that the effective-range functions $\left(\frac{2p}{W}\right)cot\delta$ decrease rather rapidly above threshold. It is suggested that this behavior can be explained by assuming the existence of a zero in the scattering amplitude. Another argument which requires these nonet zeros is given in the framework of the theory of complex angular momenta.