Composite pairs and effective two-body scattering in a many-body medium

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Abstract
The generalized off-shell scattering of pairs of constituent fermions embedded in an infinite system of identical particles having thermodynamic properties characterized by arbitrary values of the temperature and chemical potential, is examined using a temperature-dependent Green's-function formalism, which treats particle-particle and hole-hole excitations symmetrically. The important features which emerge are (i) the possible existence of bound-state pairs (of quasiparticles which are linear combinations of particle and hole states) which correspond to poles in the scattering amplitude at negative total energies depending parametrically on the properties of the medium and the total momentum of the interacting pair; (ii) a generalized unitarity condition from which one obtains a parametrization of the on-shell positive-energy scattering amplitude by a set of real effective phase shifts; (iii) that these phase shifts completely determine the large-separation asymptotic behavior of the scattering (positive energy) pair wave functions in a manner which is form equivalent to two-particle scattering in free space; and (iv) that the small- and large-momentum asymptotic limits of the phase shifts are related to the number of bound-state pairs in the medium via an analogous, but modified, form of Levinson's theorem for free scattering. The effects of these pairing singularities on the single-particle excitation spectrum, and hence on the thermodynamic behavior of the system are outlined.

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