Uncoupled-Phase Method in the MultichannelNDFormalism

The uncoupled-phase method is a nonperturbative formalism, developed by Ross and Shaw, relating the scattering amplitudes describing $n$ strongly coupled two-body channels to the "uncoupled" amplitudes describing $n-1\mathrm{}$ channels alone. The "uncoupled" scattering amplitudes are defined to be those that would exist if the couplings to the $n\mathrm{th}$ channel were switched off while the interactions among the $n-1\mathrm{}$ channels remain unchanged. The uncoupled-phase method, previously based on the potential model, is extended to the relativistic problem by considering a set of $n$ coupled $\frac{N}{D}$ partial-wave dispersion relations. For the situation in which the left-hand cut is approximated by the form $\frac{g}{\mathrm{}(s+m)\mathrm{}}$ where $g$ is an $n\times n$ matrix of constants and $s$ is square of the total energy in the center-of-mass system, the uncoupled-phase method is exact. The quantitative validity of the uncoupled-phase method for more complicated left-hand singularities is tested by performing a two-channel computer experiment. A full numerical solution of the coupled integral equations for the $N$ functions is obtained by the matrix-inversion technique. We consider the situations in which (a) the left-hand cut is replaced by a set of dipoles and (b) the left-hand cut is assumed to be given by exchange of a scalar particle in the corresponding "crossed" $t$ channel of any given reaction. The coupled-phase method is found to be quantiatively accurate under a wide range of conditions. The range parameter of the coupled-phase method is directly given by a principal-value integral, and an estimate of it can be made a priori.