$\eta \pi \pi$ Model of $X^0$ and a Possible $\eta -\pi$ Resonance

An $\eta \pi \pi$ model of the $X^0$ meson at 960 MeV is proposed through factorable $\pi -\pi$ and $\eta -\pi$ $s$-wave forces. The over-all dynamics is described by a three-body Schrödinger equation under a nonrelativistic approximation which is justified by the relatively low $Q$ value (∼120 MeV) of the particle with respect to decay into an $\eta \pi \pi$ system. The $\pi -\pi$ interaction, which is used as input information, is taken to be of two types: (i) a smooth attractive force which falls short of the requirements of a $\pi -\pi$ bound state (called ABC-type), and (ii) a force characterized by a long-range barrier surrounding an inner well, which is capable of producing a $\pi -\pi$ resonance at several hundred MeV (called $\sigma$-type interaction). Using three-body techniques proposed earlier by one of us (A.N.M.), the nature and strength of the $\eta -\pi$ interaction is deduced from a knowledge of the energy of the $\eta \pi \pi$ system at the experimental position of the $X^0$-mass. It is found that a $\sigma$-type $\pi -\pi$ interaction, designed to produce a $\pi -\pi$ resonance at 400-600 MeV, requires the existence of a bound $\eta -\pi$ system. On the other hand, an ABC-type $\pi -\pi$ interaction demands an $\eta -\pi$ resonance in the BeV region (via a $\sigma$-type $\eta -\pi$ interaction). The physical implications of these results are discussed.