Three-body dispersion-relation N/D equations for the coupled decay channels $\bar pp(J^{PC} = 0^{ - + } ) \to \pi ^0 \pi ^0 \pi ^0 ,\eta \pi ^0 \pi ^0 ,\eta \eta \pi ^0 ,\bar KK\pi ^0$

Abstract

For several years the data on different $\bar pp(J^{PC} = 0^{ - + } ) \to 3$ mesons channels presented by the Crystal Barrel collaboration were successfully analyzed by extracting the leading amplitude singularities—pole singularities—with the aim of obtaining information about two-meson resonances. But these analyses do not take three-body final-state interactions into account in an explicitly correct way. This paper is devoted to the consideration of this problem. It is shown how the coupled three-body equations could be written for the π ⁰ π ⁰ π ⁰, ηπ ⁰ π ⁰, ηηπ ⁰ and $\bar KK\pi ^0$ channels in the $\bar pp$ annihilation at rest using the three-body dispersion-relation N/D method.