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phase shift up. In potential scattering, an increasing phase is associated with attraction. 

The Dyson model therefore applies to systems with attraction where scattering phase shifts increase with in- creasing energy. 

The nucleon-nucleon phase shifts, conversely, decrease with increasing energy and provide evidence for repul- sion. 

In Refs. [1–3] ℑ D ( s ) is strictly positive. Softening this constraint to ℑ D ( s ) ≥ 0 allows extension of the Dyson model to systems with repulsion: 

Let us consider the scattering of two nucleons through compound states, dibaryons, with form factors F ( s ) that have a simple zero at s = s$_{p}$ > s$_{0}$ = 4 m $^{2}$. Consequently, ℑ D ( s ) ∼ ( s − s$_{p}$ ) $^{2}$. Such behavior is presupposed in the quark compound bag (QCB) model developed by Simonov for the description of nucleon-nucleon interac- tions [6]. In the QCB model, the separable potential generated by the compound 6-quark bags is restricted to the bag surfaces. The S -wave form factor has the form F ( s ) = sin( kb ) /kb , where k is the center-of-mass momentum and b is the effective interaction radius. In relativistic notation, the D function of the model has the form of Eq. (1), while Λ( s ) and the self-energy operator Π( s ) are equivalent to Eqs. (2) and (3), respectively. 

This analogy allows the techniques developed in Ref. [1] to be used to parametrize nucleon-nucleon scat- 

tering amplitudes with functions that have the correct analytical properties. 

If ℜ D ( s$_{p}$ ) ̸ = 0, the phase touches, at s = s$_{p}$ , one of the δ ( s ) = 0 mod ( π ) levels without crossing. However, if Eq. (5) holds at s = s$_{p}$ for both the real and the imag- inary parts, the phase crosses one of the levels δ ( s ) = 0 mod ( π ) with a negative slope . This can be verified by ex- panding D ( s ) around s = s$_{p}$ . By taking Eq. (4) and the conditions ℑ D ( s$_{p}$ ) ′′ > 0 and ℜ D ( s$_{p}$ ) ′ > 0 into account, one gets 

In potential scattering, a negative slope of the phase shift is associated with repulsion. 

The Low scattering equation [2] is modified in the pres- ence of primitives. In the systems with Π( s ) given by Eq. (3), the scattering amplitude A ( s ) = e iδ ( s ) sin δ ( s ) can be represented as follows: 

This amplitude obeys the generalized Low scattering equation 

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which is essentially the dispersion integral representation for the inverse denominator function D ( s ) that accounts for the poles, which correspond to the bound states and primitives. A ( s ) and F ( s ) have simple zeros at s = s$_{p}$ , so the integrand in Eq. (7) is a regular function at s ′ = s$_{p}$ . The bound states and the primitives generate poles at s$_{b}$ < s$_{0}$ and s$_{p}$ > s$_{0}$ on the real axis, the coefficient 

is positive. 

In the QCB model, the P matrix takes the form 

where P$_{free}$ is the free P matrix. For the S wave, P$_{free}$ = kb cot( kb ) and P = kb cot( kb + δ ( s )). The value of κ is fixed by the normalization of D ( s ). The compound states of masses M$_{α}$ show up as poles of the P matrix. The poles of the P matrix split into two groups according to their physical nature: 

The first group is related to the bound states and res- onances. 

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One bound state always exists at D ( s$_{0}$ ) > 0. Addi- tional bound states can be generated by compound states with masses M$_{α}$ < √ s$_{0}$ . 

A characteristic feature of a resonance is the condition F ( s ) ̸ = 0 in the neighborhood of s = M 2 $_{α}$. Equation (5) can then be used to find a simple pole of the S matrix. The roots of Eq. (5) that lie on the real half axis ( −∞ , s$_{0}$ ) of the second Riemann sheet are virtual states that can be related to the compound states also. 

The poles of the second group are related to roots of Eq. (5) on the unitary cut in the neighborhood of s = M 2 $_{α}$. Such poles do not show up as S -matrix poles and cannot be treated as resonances. They are called primitives according to Jaffe and Low [4]. If a resonance moves from the second Riemann sheet to the unitary cut, its singular effect on the S matrix cancels out. As dis- tinct from resonances, primitives drive the phase shift down and mimic repulsion. 

In the Dyson model, there exist at most one bound state and at most one resonance, which are not associated with the CDD poles. In the QCB model, there are CDD poles related to primitives that do not give rise to bound 

