Multichannel Effective-Range Theory from theNDFormalism

Abstract

An effective-range theory for systems of many coupled two-body channels is given using the $\frac{N}{D}$ formalism. The effective-range expansion is carried out in the amplitudes $M_{\mathrm{ij}}$ (where $M$ is essentially the matrix $T^{-1\mathrm{}}$ with the right-hand cut removed). Quite in analogy with the single-channel effective-range theory, the diagonal elements $M_{\mathrm{ii}}$ are given by an expression quadratic in $k_i$, the relative momentum in channel $i$. The effective ranges $R_{\mathrm{ii}}$ are given by certain principal-value integrals which depend on the position of the left-hand singularities in the corresponding channels and can be taken to be energy-independent to the same extent as in the one-channel theory. The nondiagonal elements $M_{\mathrm{ij}}$, in general, have a weak energy dependence and can approximately be treated as constants. A two-channel computer experiment is performed to test these proposals in detail. Three different situations for the left-hand cut are considered: (i) a set of monopoles, (ii) a set of dipoles, and (iii) the left-hand cut produced by the exchange of scalar particles in the "crossed" $t$ reactions. For a large number of situations considered, the simple features proposed for the multichannel effective range theory were found to exist. The above formalism is similar to the multichannel effective-range theory of Ross and Shaw in the potential model.