Measurement of the Nucleon Nucleon Scattering Length with the ESC 04 Interaction

We have determined a value for the S0 neutron-neutron scattering length (ann). The scattering length result is presented for the extended-soft-core (ESC04) interaction. The value obtained in the present work is ann = −18.6249 fm. The method of solution of the radial Schrödinger equation with nonlocal potential for nucleonnucleon pairs is described and the result is consistent with previous determinations of ann = −18.63 ± 0.10 (statistical) ± 0.44 (systematic) ± 0.30 (theoretical) fm. The nonlocal potentials are of the central, spin-spin, spin-orbital, and tensor type. The analysis from the ESC04 interaction is done at energies 0 ≤ Tlab ≤ 350 MeV. We compare the present result with experimental S-wave phase shifts analysis and agreement is found.


Introduction
In nuclear physics, important information can be obtained from the scattering length associated with lowenergy nucleon-nucleon scattering.At these energies, the nucleon-nucleon interaction can be treated non-relativistically and the scattering was studied by means of a single particle Schrödinger equation which involves a nonlocal effective potential, derived from [1][2][3][4] using an extended soft-core model (ESC interaction).In the present manuscript, we consider a potential that involves a central part, a spin-spin interaction, a spin-orbital interaction and a tensor part and perform a numerical study of the associated Schrödinger equation.Also, we determine a numerical value for proton-proton and neutron-proton scattering lengths.
The present work is realized by considering energies in the range of 0 ≤ T lab ≤ 350 MeV.For nucleon-nucleon scattering, it has been demonstrated that the interaction from the ESC model gives a description that is in good agreement with the nucleon-nucleon data.The extended soft-core model, also known as ESC, is used for nucleonnucleon (NN), hyperon-nucleon (YN), and hyperonhyperon (YY) scatterings.The particular version of the model ESC, called ESC04 [T. A. Rijken, Phys.Rev. C 73, 04007 (2006)], describes NN and YN interaction in an unified way using broken SU (3) symmetry.
A good fit with the experimental data is obtained by using the ESC04 model.The manuscript is organized as follows: in Section II, we give a theoretical review of the model; in Section III, we present our numerical results and in Section IV, we draw our conclusions.

The Schroedinger Equation with Non-Local Potential
The model we are going to study numerically involves a radial Schrödinger equation with ESC04 potential; namely where ( )( ) ( )   is a second rank tensor operator.
For an S-state we introduce ( ) u r , where For a given value of the quantum number J, ( ) where we introduce ( ) where the symbol ( ) denotes a Clebsch-Gordan coefficient, and Y LML are the spherical harmonics, and ( ) The subscript on χ refers to the magnetic projection quantum number M S of the spin-1 state, while α and β represent spin up and spin down for the particular spin-½ nucleon indicated by the subscript.
The Equation ( 2) forms an orthonormal set spanning the space of spin-1 functions and functions of the direction r.The normalization of ( ) requires that the radial functions satisfy, ( ) The Schrödinger equation [Equation (1)] is processed by the method of separation of variables, we obtain as its radial component, We use the parametrized potential ) , , and S 12 may be written as an operator of the form ( ) with λ = 2 and j 1 = j 2 = 1.Here ( ) Using Racha algebra (see appendix A of [6]) we can show that ( )

Numerical Solution of the Schrödinger Equation
Considering the single state for the 1 S 0 wave, Equation ( 6) for the neutron-neutron system has the form (S = J = L = 0, where S 00-1 = S 001 = 0, S 000 = 2 are calculated from Equation (7).
For the proton-proton system we add the Coulomb effect to Equation ( 8), The numerical techniques necessary to solve equation (8) with this ESC04 potential are explained in chapter 3, Equation (3.28) of [7].The solutions of u 0 from Equation ( 8) are introduced in the S matrix (Equation (10.58) of [7], which is,

r h kr U r r h kr S U r r h kr U r r h kr
where the S matrix is evaluated in the last two points on a mesh of size ε ( 0, , 2 , r N ). U l are the solutions to Equation ( 8) with the ESC04 potential previously calculated and h l are the spherical Hankel functions defined in Equation ( 10.52) of [7].
We insert the numerical solution of the S matrix in the solution of the S matrix for a real potential where δ l is real and is known as the phase shift.
Once the δ 0 phase shift is found the a nn scattering length and the effective range r nn are calculated.For l = 0 the expression for ( ) can be parameterized in the following form, ( ) The quantity a is called the scattering length and r 0 is known as the effective range.
In the limit of low energies the scattering length is given in terms of the s-wave phase shift (see appendix B of [8]), ( ) where 2 is the center-of-mass momentum (the wave number) and ℜ indicates the real part.

Extended Soft-Core Potential (ESC04)
An Extended Soft-core potential is calculated consisting of a central, spin-spin, spin-orbital, and a tensor part.The potential of the ESC04 model is generated by one-bosonexchange (OBE), two-meson-exchange (TME) and meson-pair-exchange (MPE); this potential is calculated and explained in [1][2][3][4].In Figure 1 the total ESC04 potential is plotted as a function of the r distance.In Figure 2 we show the central, spin-spin, spin-orbital, and tensor part of this total potential.
The algoritms for the YN potential are found in [9].

Results
The a nn Scattering Length The a nn scattering length is calculated obtaining a numerical value a nn = −18.62497fm and an effective range of r nn = 2.746615 fm.We use an ESC04 potential below 350 MeV.In Figures 3 and 4 the phase shift ( ) S δ is plotted for the proton-proton and neutron-proton case.
Table 1 shows the results for the low-energy parameters from the scattering lengths and the effective ranges for neutron-proton, proton-proton and neutron-neutron system using the ESC04 interaction.

Conclusions
In the present work, we have numerically solved the Schrödinger equation with an ESC04 potential and obtained the nucleon-nucleon scattering lengths.Summarizing our main conclusions: 1) Recent calculations using the ESC04 interaction for nucleon-nucleon dispersion have been realized [4], and reproduced with the Schrödinger equation.
2) The numerical solution of the radial Schrödinger equation has been realized and has been demonstrated to give a good fit to the nucleon-nucleon data.
3) The scattering lengths a pp , a np and a nn have been calculated and are consistent with the experimental re-     The presented ESC model is thus successful in describing the NN data.

is the reduced mass of the nucleons whose individual masses are m 1 and m 2 ,
the distance between the nucleons.The potential is parameterized as

Figure 1 .
Figure 1.Total potential in the partial wave 1 S 0 , for I = ½.

Figure 3 .
Figure 3. Solid curve, proton-proton I = 1 phase shifts (degrees), as a function of T lab (MeV), numerical solution for the ESC04 model.Dots, phases of the Rijken analysis [4].Circles, s.e.phases of the Nijmegen93 PW analysis.Triangles, the m.e.phases of the Nijmegen93 PW analysis [10].