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states or resonances. The neighboring CDD poles squeeze masses of compound states that become bound states, resonances, or primitives when coupling to the continuum is switched on. This is illustrated in Fig. 1. 

The S -wave nucleon-nucleon scattering can be consid- ered as an example of dynamics influenced by the CDD poles that are connected to primitives. 

The model we discuss is the relativistic extension of the QCB model. The P -matrix formalism is recovered with 

Equation (3) for κ = 2 mb/π gives 

In addition to the interaction through compound states, we introduce a contact interaction. This amounts to a redefinition of Λ − $^{1}$( s ) as compared to Eq. (2). In the case of one CDD pole, the most general expression for Λ − $^{1}$( s ) becomes 

where c$_{p}$ is a free parameter such that κc$_{p}$r$_{p}$ = g 2 $_{α}$, r$_{p}$ = 8( π/b ) 2 is the residue of P$_{free}$ in the S wave, and s$_{d}$ = M 2 d is the deuteron pole or the threshold. The D function with the contact interaction remains the generalized R function. 

In the $^{3}$S$_{1}$ channel, the phase shift vanishes at T$_{lab}$ = 354 MeV. The S matrix according to Eq. (4) is unit in two cases: Λ( s ) = ∞ and ℑ D ( s ) ≡ −ℑ Π( s ) = 0 . The poles of Λ( s ) are the CDD poles. At the CDD poles, the slope of the phase is positive, which corresponds to attraction. The second case ℑ Π( s ) = 0 gives repulsion. T$_{lab}$ = 354 MeV is equivalent to k = 408 MeV. The 

equation F ( s ) = 0 gives kb = π . Thus, we determine b = 1 . 52 fm. Since F ( s$_{p}$ ) = 0 if and only if ℜ Π( s$_{p}$ ) = 0, Eq. (5) simplifies to Λ( s$_{p}$ ) = 0. Λ( s ) vanishes when s = M 2 $_{α}$. The compound state shows up as the primitive of mass M$_{α}$ $_{= 2}$√$_{k}$ 2 + m 2 = 2047 MeV. 

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The parametrization ensures the existence of the deuteron pole for 

Unphysical zeros of the D function are eliminated by constraining c$_{p}$ . One can easily show that ℑ D ( s ) ∼ ℑ s , and that the coefficient of proportionality is positive for positive c$_{p}$ . In this case, D ( s ) has no zeros for ℑ s ̸ = 0. The real half axis ( −∞ , s$_{0}$ ) remains to be checked. The derivative D ( s ) ′ is positive below the threshold. D ( s ) crosses the real axis at s = s$_{d}$ < s$_{0}$ . This is the unique zero of the D function, provided Λ( s ) has no poles for s < s$_{0}$ . Let us investigate the zeros of Λ − $^{1}$( s ). Since κ − $^{1}$Λ − $^{1}$( s$_{d}$ ) = − 1 /γ < 0, Λ − $^{1}$( s ) ′ < 0 and Λ − $^{1}$( s ) has no poles for s < s$_{p}$ by construction, the condition Λ − $^{1}$( −∞ ) < 0 is sufficient to exclude unphysical zeros. Finally, c$_{p}$ satisfies the constraint 

In Fig. 2 (a), we show the $^{3}$S$_{1}$ phase shift versus the proton kinetic energy for c$_{p}$ = 0 . 9 c max p . This is compared to the partial wave analysis data provided by Ref. [21]. For the pn system, s = s$_{0}$ +2 m$_{n}$T$_{lab}$ , where m$_{n}$ is neutron mass. The CDD pole is located at M = 3203 MeV. The pion production threshold is at T$_{lab}$ = 280 MeV and the inelasticity is small up to ∼ 350 MeV. In Fig. 2 (c), we show D ( s ) and D ( s ) as functions of 

ℜ ℑ T$_{lab}$ . ℜ D ( s ) has one zero below s$_{0}$ , which corresponds to the deuteron. The second zero at s > s$_{0}$ with ℑ D ( s ) ̸ = 0 and a negative slope of ℜ D ( s ) ensures the crossing of the level δ ( s ) = π . The third zero corresponds to the primitive. 

In the $^{1}$S$_{0}$ channel, the phase shift vanishes at T$_{lab}$ = 265 MeV. The same arguments as before give b = 1 . 76 fm and M$_{α}$ = 2006 MeV. − 

Λ $^{1}$( s ) has the form of Eq. (11), with s$_{d}$ replaced by s$_{0}$ = 4 m $^{2}$. Near the threshold, κD ( s ) = − γ +1+ ikb + . . . From the other side, D ( s ) ∼ 1 − iδ ( k )+ ... = 1 − ika + ..., where a = 23 . 56 fm is the scattering length. One has to require 

D ( s ) has no zeros for complex values of s . Its derivative is positive for real s < s$_{0}$ . To avoid unphysical zeros, it is sufficient to require 

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The second inequality is fulfilled, and the first one gives c$_{p}$ < min( c max p , c max p / ( γ − 1)). Since b ≪ a , this reduces to Eq. (13) with s$_{d}$ replaced by s$_{0}$ . 

In Fig. 2 (b), we show our fit of the pn $^{1}$S$_{0}$ phase shift with c$_{p}$ = 0 . 9 c max p compared to the experimental data [21]. The CDD pole occurs at M = 2916 MeV. 

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