Relativistic three-pion dynamics generated by two-body unitarity and analyticity

Abstract

Linear single-variable integral equations are derived for 'generalized isobar amplitudes' describing the interactions of pairs of pions in a three-pion state. Discontinuity relations deduced from two-body unitarity in the pair channels are combined with dispersion relations to give generalized isobar amplitudes satisfying two-body unitarity and having no spurious singularities. The integral equations are identical to those derived using the Khuri-Treiman approach. Explicit expressions are given for all the kernels in the omega to pi rho and Al to pi rho , pi epsilon channels. The equations give information both about the pair sub-energy variation, and about the three-body mass dependence.