Modified Isobar Approximation forπ−ρScattering

The equations for the scattering amplitude for three pions in a $J=1^{-}$, $T=0\mathrm{}$ state, subject to the approximation of elastic unitarity, are formulated in accordance with the $\frac{N}{D}$ method. It is assumed that the incoming and outgoing pions resonate in pairs. A new method of approximation is introduced for the integration over the invariant mass of the intermediate resonance in the unitarity condition: Phase-volume factors as well as the resonance denominators are used to define the mass of the intermediate resonance, which then becomes a function of the total channel energy. Upon writing the coupled integral equations for the $\frac{N}{D}$ representation of the amplitude, using as input functions for $N$ the proximate singularities of the $\pi$, $\omega$, and $\rho$ exchange graphs, we find that the branch points for these singularities are functions of the total channel energy. That is, the unphysical singularities become movable. When we attempt to evaluate $D$ numerically and locate the $\omega$ resonance as a zero of its real part, determined by the requirement of the consistency of $M_{\omega }$ and ${\Gamma }_{\omega }$ with the values used in the input function, we find that the repulsive pion-exchange term dominates the integral, and hence it is not possible to obtain a self-consistent $\omega$ resonance. The recoupling terms are considered only qualitatively, and methods of approximation are suggested for them. The physical consequences of the movable cuts are discussed in detail.