Generalized levinson’s theorem and theN/D factorization with inelasticity

Abstract

We discuss the general properties of the denominator function (in theN/D factorization) of an elastic-scattering amplitude for spinless equal-mass particles. The Levinson’s theorem for the real part of the phase shift is shown to follow in a simple manner from the conditions on the denominator function. Special attention is paid to the case where the inelasticity functionη(s) has a finite number of zeros above the inelastic threshold. Then, the construction of the appropriate denominator function leads to a discontinuous Hilbert problem. It is shown that the denominator function must in general have square-root singularities at the positions of the zeros of the functionη(s). The generalized Levinson’s theorem now relates the real part of the phase shift at infinity (or threshold), the number of stable-particle poles, the number of CDD poles, and a certain number which depends on the number of zeros of the functionη(s). It appears possible that the formalism discussed can be applicable to the problem of obtaining equivalence between single- and multichannelN/D calculations without CDD ambiguities.