Nonlocal Potentials and Complex Angular Momentum Theory

Abstract

The purpose of this paper is to establish meromorphy properties of the partial scattering amplitude T (λ, k) associated with physically relevant classes N γ w (ε) ,α of nonlocal potentials in corresponding domains D (δ) γ,α of the space C 2 of the complex angular momentum λ and of the complex momentum k (namely, the square root of the energy). The general expression of T as a quotient Θ(λ, k)/σ(λ, k) of two holomorphic functions in D (δ) γ,α is obtained by using the Fredholm-Smithies theory for complex k, at first for λ = integer, and in a second step for λ complex (Re λ > −1/2). Finally, we justify the "Watson resummation" of the partial wave amplitudes in an angular sector of the λ-plane in terms of the various components of the polar manifold of T with equation σ(λ, k) = 0. While integrating the basic Regge notion of interpolation of resonances in the upper half-plane of λ, this unified representation of the singularities of T also provides an attractive possible description of echoes in the lower half-plane of λ. Such a possibility, which is forbidden in the usual theory of local potentials, represents an enriching alternative to the standard Breit-Wigner hard-sphere picture of echoes.

Transition from resonances to surface waves in pi^+-p elastic scattering

“…1 A complete analysis of the conditions needed to perform a Watson-type resummation of the expansion of the partial waves generated by non-local potentials along the lines indicated here, requires very long and detailed mathematical proofs. This work has been published in a mathematical physics journal [31]. We now lay great stress on the following point: while in the case of local potentials (notably Yukawian) the poles are all located in the first quadrant of the CAM-plane [30], in the case of non-local potentials the poles can lie in both the first and the fourth quadrant of the CAM-plane.…”

“…If these conditions are satisfied 1 , then we can sum the partial wave expansion following the method of Watson, and therefore transform the series over discrete values of ℓ into an integral encircling the real positive semi-axis of the CAM-plane. Next, we can deform this integration path into a path composed by arcs of circles and two straight lines which delimit an angular sector Λ in the CAM-plane [31] (see Fig. 3).…”

Particle decay in expanding Friedmann-Robertson-Walker universes

“…New phenomena are introduced into decay rate calculations when transferring into curved space. Gravitational particle creation, lack of some conservation laws, and the possibility of a field decaying into its own quanta [9][10][11][12] all imply that one has to be very critical about the use of Minkowskian results in curved spacetime.…”