Phenomenological theory of echo poles

In scattering theory the effect associated with the downward crossing of the phase-shift $\delta_\ell(k)$ ($\ell$ being the orbital angular momentum and $k$ the momentum) through $\delta_\ell={\pi}/{2}$ (mod $\pi$) is called \emph{echo}. In the standard nuclear theory (Breit-Wigner theory) the echo is described and evaluated in terms of scattering by an impenetrable sphere. However, this model holds only atsufficiently high energy, while it is inadequate at low energy. In this paper we show that the echo effect can be associated with two different regimes acting at low and high energy, respectively. At high energy the hard-sphere scattering model seems to describe appropriately the phenomenon. At low energy we propose a mechanism due to the exchange forces induced by the Pauli exclusion principle in the fermionic interaction, which leads to nonlocal potentials. These potentials admit for the scattering amplitude pole singularities in the fourth quadrant of the complex angular momentum plane. This paper analyzes the role played by this class of poles in the description of the low energy regime of echoes. A specific phenomenological analysis is performed, taking as typical example the $\alpha$-$\alpha$ elastic scattering.