Remarks on the origin of Castillejo-Dalitz-Dyson poles M. I. Krivoruchenko Institute for Theoretical and Experimental Physics , B. Cheremushkinskaya 25 117218 Moscow , Russia 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. Low’s 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. PACS numbers: 03.65.Nk, 11.55.Fv, 13.75.Cs are the nucleon and the pion masses. The D function of the process can be written as where, in the relativistic notations, Here, Φ$_{2}$( s ) = $_{πk/}$√$_{s}$ is the relativistic two-body phase space, k is the center-of-mass momentum, g$_{α}$ is the cou- pling constant, and F ( s ) is the form factor of the N$_{α}$Nπ vertex. The S matrix has the form The poles of Λ( s ) are the CDD poles. They are located between the zeros of Λ( s ) (i.e., between M 2 α and M 2 α $_{+1}$). The D function constructed in such a way is the gener- alized R function [1]. It has no complex zeros on the first Riemann sheet of the complex s plane. It also has no zeros on the real half axis ( −∞ , s$_{0}$ ), which corresponds to bound states, provided D ( s$_{0}$ ) < 0 and s$_{0}$ < M 2 $_{α}$. The simple roots of the equation In this Brief Report, we clarify the link between the CDD poles and the primitives, which can be useful for modeling the nucleon-nucleon interactions in the P - matrix formalism. CDD poles arise when the interaction of particles in- cludes intermediate states that are internally different from combined-particle states. These are discrete eigen- states of the system and basically form other channels in the scattering problem. They have also been referred to as elementary particle or compound states. In the Dyson model, one starts from the scattering of two particles, e.g., a nucleon and a pion. A nucleon can absorb a pion and can turn into an excited compound state N$_{α}$ of mass M$_{α}$ > √ s$_{0}$ = m + µ, where m and µ Such behavior is in agreement with the Breit-Wigner for- mula according to which isolated resonances drive the The poles introduced by Castillejo et al. in Ref. [1] are known as the Castillejo-Dalitz-Dyson (CDD) poles. They describe ambiguities in solutions to the Low scattering equation [2] for amplitudes that satisfy correct analytical properties and unitarity. To clarify the physical meaning of the CDD poles, Dyson constructed a model [3] that demonstrates the relation of the CDD poles to bound states and resonances. Some time ago, Jaffe and Low [4] proposed a method for identifying exotic multiquark states with primitives that appear as poles of the P matrix rather than the S matrix. The analysis, performed for scalar mesons [4] and nucleon-nucleon scattering [5], revealed primitives, in agreement with expectations from the MIT bag model. A dynamical model of the P matrix was developed by Si- monov [6] and was applied to the description of nucleon- nucleon scattering [6–11]. Other microscopic models of nucleon-nucleon forces have also been discussed [12–14]. The recent interest in the problem of nucleon-nucleon interactions is connected to new constraints on the equa- tion of state (EOS) of nuclear matter, obtained from col- lective flow data and subthreshold kaon production in heavy-ion collisions [15, 16] as well as astrophysical ob- servations of massive neutron stars [17, 18]. One-boson exchange models are in reasonable agreement with the laboratory data but predict surprisingly low masses for neutron stars in the β equilibrium [19, 20]. Microscopic models can provide better insight into the short-range dynamics of nucleon-nucleon interactions and the high- density EOS. located on the second Riemann sheet below the unitary cut, are identified as resonances. In the limit of small g$_{α}$ , roots of Eq. (5) are localized in the neighborhood of s = M 2 $_{α}$. ℜ s gives the renormalized resonance mass, while ℑ s determines the decay width Γ$_{α}$ = g 2 α ℑ D ( M 2 $_{α}$) /M$_{α}$ . At the CDD poles δ ( s ) = 0 mod ( π ), the slope of the phase is positive. If s$_{γ}$ is a CDD pole, then Eq. (2) gives Λ − $^{1}$( s$_{γ}$ ) = 0 and Λ − $^{1}$( s$_{γ}$ ) ′ < 0. By expanding the D function around s = s$_{γ}$ and by using Eq. (4), one finds δ ( s$_{γ}$ ) ′ = −ℑ D ( s$_{γ}$ )Λ − $^{1}$( s$_{γ}$ ) ′ > 0 . D ( s ) = 0 , (5) S = e 2 iδ ( s ) = D ( s − i 0) D ( s + i 0) . (4) Π( s ) = − 1 π ∫ + ∞ s 0 Φ$_{2}$( s $^{′}$) F $^{2}$( s $^{′}$) s ′ − s ds $^{′}$. (3) Λ − $^{1}$( s ) = ∑ α g 2 α s − M 2 α , (2) D ( s ) = Λ( s ) − Π( s ) , (1) arXiv:1001.1659v2 [nucl-th] 16 Jul 2010