Four-pion models for η and X⁰ mesons

Abstract

A four-pion formalism is set up through a semirelativistic Schrödinger equation with pairwise π-π interactions in the p state. Two kinds of interaction with factorable kernels, both of which reproduce the mass and width of the ρ meson, are considered: (a) an intrinsically attractive interaction which has a very short range and (b) a long-range interaction which is repulsive at low energies but becomes attractive at high enough energies. The four-pion states of T=0 are analyzed in terms of spatial and isospin functions of appropriate symmetries, and these are used to examine the spin-parity states of 0⁻⁺ and 1⁺⁺, which are the only 4π states of T = 0 depending on π-π interaction in odd-l states, via Bose statistics. By the assumption of factorable kernels, these equations are exactly reducible to "equivalent three-body equations." The solutions of these equations, which can be obtained under certain reasonable approximations (discussed in the text), show that the results are incompatible with a bound pseudoscalar state of T = 0, so that the η meson cannot be understood on this model. The model also rules out an axial-vector state of T = 0, bound or resonant. However, the model predicts a resonant-pseudoscalar state at a mass very close to that of X⁰ (960 MeV), and a second one at about 1.4 BeV, only when the π-π interaction is of type b, but not if it is of type a. Certain other implications of a type b interaction are discussed briefly.