Remarks on the origin of Castillejo-Dalitz-Dyson poles

Castillejo-Dalitz-Dyson (CDD) poles are known to be connected with bound states and resonances. We discuss a new type of CDD pole associated with primitives i.e., poles of the $P$ matrix that correspond to zeros of the $D$ function on the unitary cut. The Low scattering equation is generalized for amplitudes with primitives. The relationship between the CDD poles and the primitives is illustrated by a description of the $S$-wave nucleon-nucleon phase shifts.

Figure 1

$D$-function zeros in the complex $s$ plane. The unitary cut is shown by a bold solid line. Compound states 1, 2, and 3 are eigenstates of the free Hamiltonian. Upon switching on the $s$-channel interaction, zeros move to new positions shown by crosses. Compound states 1, 2, and 3 become bound state, primitive, and resonance, respectively. A pair of the CDD poles that squeezes compound state 2 of the primitive type is shown by arrows.

Figure 2

${}^3{S}_1$ and ${}^1{S}_0$ scattering phase shifts in radians (upper panel) and real and imaginary parts of the $D$ functions (lower panel) versus the proton kinetic energy. The solid curves are parametrizations within the relativistic QCB model. The experimental phase shifts [21] are shown by triangles.