Unified scheme for describing time delay and time advance in the interpolation of rotational bands of resonances

Abstract

In this paper we show how rotational bands of resonances can be described by using trajectories of poles of the scattering amplitude in the complex angular momentum plane: each band of resonances is represented by the evolution of a single pole lying in the first quadrant of the plane. The main result of the paper consists in showing that also the antiresonances (or echoes) can be described by trajectories of the scattering amplitude poles, instead of using the hard-sphere potential scattering as prescribed by the classical Breit-Wigner theory. The antiresonance poles lie in the fourth quadrant of the complex angular momentum plane, and are associated with non-local potentials which take into account the exchange forces; it derives a clear-cut separation between resonance and antiresonance poles. The evolution of these latter poles describes the passage from quantum to semi-classical physics. The theory is tested on the rotational band produced by α-α elastic scattering and on the hadronic rotational bands in π + -p elastic scattering.

Nonlocal Potentials and Complex Angular Momentum Theory

“…From the conceptual viewpoint, such a scenario based on a unified description of analytic singularities of the partial scattering amplitudes in the variables (λ, k) would be more satisfactory than the standard one based on the ad hoc adjunction of an impenetrable sphere to a given local potential. It would indeed be attractive to show that certain nonlocal potentials can produce at the same time a "Regge trajectory" λ = λ(k) with β > 0 describing a sequence of resonances, whose angular momentum increases with k, and a similar "image-trajectory" λ = λ(k) with β < 0 describing a corresponding sequence of echoes, whose angular momentum would also increase with k, in agreement with the phenomenological study of various nuclear scattering processes (see [11,12]). …”

“…Of course, the value of the previous example is limited to the description of local phenomena related to a given partial wave amplitude a ℓ (k). As a matter of fact, one should expect that, in the same way as a "Regge trajectory" λ = λ(k) with β > 0 is able to describe a sequence of resonances, whose angular momentum ℓ increases with k, a similar "image-trajectory" λ = λ(k) with β < 0 might correspondingly describe a sequence of antiresonances, whose angular momentum ℓ would also increase with k. The expectation of an alternating sequence of resonances and antiresonances associated with increasing values of ℓ and k seems indeed suggested by the phenomenological study of various nuclear scattering processes (see [10] and [11]). A strong hope exists that the theory of nonlocal potentials (in particular within the classes that have been studied in the present paper) may be able to produce such type of coupled trajectories belonging respectively to the first and to the fourth quadrant in the λ-plane, a possibility which was forbidden by the usual theory of local potentials.…”

“…Indeed, in refs. [10,11] two of us have performed an extensive phenomenological analysis, fitting the scattering data of α-α and π + -p elastic scattering by using two pole trajectories in the CAM-plane. The resonances have been described by poles in the first quadrant, the antiresonances by poles in the fourth quadrant.…”

Geometrical theory of diffracted rays, orbiting and complex rays

“…), and the angular symmetry is broken. For a detailed phenomenological analysis of the creeping waves in the π + -p elastic scattering, see [13].…”

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

“…If the walls are not completely reflecting, the lifetime of the resonance is finite. In fact, there is a wide phenomenological evidence of rotational bands of resonances in non-relativistic ion collisions [20], where the CAM theory can be applied effectively, and a clear evidence of Regge trajectories can be obtained [21]. In these rotational bands one can plot L(L + 1) versus E (see Fig.…”