Unitary analytic isobar model for the reaction nucleon-meson to nucleon-meson-meson

Abstract

The isobar model for a production amplitude is developed in a construction which incorporates unitarity and analyticity in each of the two-body isobar subenergy channels. The process $\mathrm{KN}\ensuremath{\rightarrow}K\ensuremath{\pi}N$ is considered in order to illustrate and deal with the complications due to spin and unequal masses. The basic aspects of the description are presented in a truncation of the problem to the production of $s$- wave isobars. The two-body discontinuities in each isobar channel have been derived previously, and are employed here in dispersion relations for the isobar factor amplitudes. A prominent complicating feature is that due to half-angle kinematics; this is investigated without approximation, and factors associated with kinematical singularities are identified and extracted. A coupled system of single-variable integral equations for the isobar factors is obtained by means of a procedure due to Pasquier and Pasquier. The kernels of the integral equations are evaluated explicity. These contain the two-body elastic amplitudes in each isobar channel, the parametrization of which propagates into the solution for the isobar production amplitudes. The formal solution of the linear system of equations for the isobar factors is indicated. The final result is unitary and analytic in each subenergy and satisfies two-body unitarity in the total energy as well.