Four-Pion Models for $\eta$ and $X^0$ Mesons

A four-pion formalism is set up through a semirelativistic Schrödinger equation with pairwise $\pi -\pi$ interactions in the $p$ state. Two kinds of interaction with factorable kernels, both of which reproduce the mass and width of the $\rho$ 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 $T=0\mathrm{}$ 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 $4\pi$ states of $T=0\mathrm{}$ depending on $\pi -\pi$ 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 $T=0\mathrm{}$, so that the $\eta$ meson cannot be understood on this model. The model also rules out an axial-vector state of $T=0\mathrm{}$, bound or resonant. However, the model predicts a resonant-pseudoscalar state at a mass very close to that of $X^0$ (960 MeV), and a second one at about 1.4 BeV, only when the $\pi -\pi$ interaction is of type b, but not if it is of type a. Certain other implications of a type b interaction are discussed briefly.