A Unitarity Criterion for the Partial S-Matrix of Resonance Scattering

Abstract

We consider the influence of N long-lived states characterized by resonance energies E_i and widths Γ_i(E) on the elastic scattering process and obtain an expression for the partial S-matrix S_l(E) in the form of a sum over the resonance levels (poles) at which the residues have the form \({{\rm{\Gamma}}_i}\prod\nolimits_{\matrix{\hfill {k = 1,} \cr \hfill {k \ne i} \cr}}^N {{\gamma _{ik}}} \), where \({\gamma _{ik}} = ({z_i} - z_k^*)\imath /2({z_i} - {z_k})\) and z_i = E_i−lΓ_i/2. We show that a necessary condition for the unitarity of the partial S-matrix in the presence of N resonance levels can be written as \(\sum\nolimits_{i = 1}^N {{{\rm{\Gamma}}_i}(E)\prod\nolimits_{\matrix{\hfill {k = 1,} \cr \hfill {k \ne i} \cr}}^N {{\gamma _{ik}} = \sum\nolimits_{i = 1}^N {{{\rm{\Gamma}}_i}(E)}}} \).