Meson-nuclear clusters in the few-body approaches

Bound states in the 3-body φφN system are considered. Two dimensional Faddeev equations in differential form were used. Some approximations were made during the calculation. For the φ − φ interaction a new short-range D-wave potential was constructed, which fits the parameters of the f2(2010) resonance in the elastic channel (f2 −→ φ + φ). For the φ − N interaction attractive forces supporting the binding energy in the φN system equal to 9 MeV were used.


Introduction
In [1] we considered, on the basis of Faddeev equations, the bound states in the systems containing one and two φ-mesons like φN N and φφN where N is a neutron or a proton, and the potential V φφ acts in the s-state.
Below we present the result of calculations for the systems with two φ-mesons like φφN obtained in the framework of the Faddeev differential equations with V φφ acting in the d-wave state.
Attention to the φ-meson-nuclear systems is called by two facts: 1) quark structure of φ-meson wave function mainly defined by the ss -configuration of strange quarks, 2) important role of the strange ss-sea quarks in the nucleon.So one can expect exchange effects between these two hadrons.
Indications of the role of ss-sea quarks follow from different experiments such as πN scattering, pp-annihilation, strange part of form factors of nucleons and so on.
Indeed, in the eighties (see, e.g.[2]) it already became clear that the ss content of the nucleon is closely related to the so-called σ-term in πN -scattering, and is not very small.Later estimation of the strange scalar density y in the nucleon [3] gave the value y ≈ 0.5 which corresponds to the size of the πN Σ-term Σ = 64 ± 8 MeV (σ 0 = 36 ± 7 MeV is the octet baryon mass difference).However, manifestation of strange sea quarks appears not only in the internal structure of the nucleon [4], but also by the intensity of some reactions.
a e-mail: belyaev@theor.jinr.ru It turns out, for example, that there are a number of reactions which proceed with large violations of the OZI-rule (Okuba-Zweig-Izuka).
The brightest example of this phenomenon can be seen in the pp -annihilation processes, if one compares the ratio of processes with φ and ω meson production with the corresponding predictions by the OZI-rule.In some cases, a few ten times difference is reached [5].The same type of phenomena (OZI-rule violation) can be seen in the processes of φ-meson-nucleon interaction which are suppressed by the OZI-rule but are strong in reality.
Analysis of the whole picture for the hiddenstrangeness amplitudes shows that their values are mainly defined by J P C quantum numbers which have ss-pairs in the corresponding processes or systems [5].This observation makes understandable the fact that the OZI-rule is fulfilled in some processes and why it is violated in others.Now, if one comes to the few-particle systems which consist of nucleon(s) and a number of φ-mesons (1 or 2), having mainly ss quark structure, one can expect a wider range of quantum numbers carried by ss-pairs and, correspondingly, more possibilities for the manifestation of strange sea quarks in nucleons.
As it was mentioned above, in our previous work [1], we started the above program considering the 3-body systems φ + 2N and 2φ + N , with s-wave potentials.
In this report, we consider the 3-particle system 2φ + N , where V φφ acts in the d-wave.

Faddeev equations
The method of treatment is the Faddeev equations in differential form.
The Jacobi coordinates are defined as usual: where r i , m i denote the radius-vector and the mass of particle i, and indices α take on the following values: Partial wave decomposition of components of Ψ has the form: where are the bispherical harmonics.Below we will consider the s-wave φ−N interaction and the d-wave φ − φ interaction which correspond to the f 2 (2010) resonance in the d-wave [PDG].
So we consider the system with L = 2.In that case from (5), we have and Ψ 3 = Ψ 2 due to 2 identical particles (2φ-mesons).
As a result, we arrive at 7 two-dimensional equations for the partial amplitudes of the Faddeev components

Approximations
To solve the system of Eq. ( 7), let us make some approximations.Approximation 1.There are 4 equations containing operators D2λ 1 with λ > 0. The corresponding equations include terms with more centrifugal repulsion and due to this one can neglect the corresponding components of the wave function.Thus, 3 equations are left, 2 of which are identical.As a result, we arrive at the following two two-dimensional integrodifferen- 19 th International IUPAP Conference on Few-Body Problems in Physics tial equations: where indices correspond to 1 for φ-meson, 2 and 3 for nucleons.From the expression which is given in [6] one gets where 23), (12) Approximation 2. One may notice that m n ≈ m φ .Therefore, it seems reasonable to make another simplification and put m i = m = m φ .Approximation 3. Now let us reduce the system of two 2-dimensional equations (9) to the system of onedimensional equations in the variable ρ-hyperradius of the system considered.This is possible due to the following observation.The potential V φn is shortrange, strongly attractive and acts in the s-state.The potential V φφ contains a centrifugal barrier, so one can expect for the φφN -system the equilibrium configuration when nucleon is in the centrum and φ-mesons are on the opposite sides.This configuration corresponds to the values of variable ϕ different for different Jacobi sets.The equilibrium values are ϕ eq = 0 for U 1 (ρ, ϕ) and ϕ eq = π/3 for U 2 (ρ, ϕ).Expanding U 1 (ρ, ϕ) and U 2 (ρ, ϕ) around equilibrium values and putting the expansion into the equation ( 9) one immediately arrives at the system of 2 one-dimensional equations on the variable ρ (10) where ν 1 = 0.110 , ν 2 = 0.0555, which was solved for the eigenvalue problem.The program used in the computation was taken from [7].
where V 0 = −93.75MeV, r 0 = 1.2 fm.Parameters of the V φφ -potential are chosen to fit (together with the centrifugal barrier) the position and width of the f 2 (2010)-resonance which has one mode of decay into two φ-mesons.The V φN -potential supported the bound state in the φN system with an energy around 9.5 MeV.It is worth saying that 5-body calculation [9] for the phi-N system gives the same result around 9 MeV in one of the quark models.

Results
The dependence of the energy of the three-body system φφN on the depth of the φN potential |a| is given on Figure 1.One can see that the binding in the system appears only at a = −3.035hc.It is quite large with respect to the input value.

Conclusion
The study of φ-meson-nuclear systems can shed light on the structure of distributions of s and s sea quarks in the nuclei (as in the nucleons in nuclear medium) and on the possible appearance of many-body effects related to the exchange of sea s and s quarks belonging to different baryons.The study of φ-meson-nuclear systems can shed light on the structure of distributions of s and s sea quarks in the nuclei (as in the nucleons in nuclear medium) and on the possible appearance of many-body effects related to the exchange of sea s and s quarks belonging to different baryons.

Fig. 1 .
Fig. 1.The dependence of the binding energy of the φφn system on the parameter |a| of the φ − N interaction (h = c = 1).