Measurement of the Nucleon Nucleon Scattering Length with the ESC04 Interaction ()
Keywords:Nucleon-Induced Reactions; S-Matrix Theory; Scattering Theory
1. 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-]">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 £ Tlab £ 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.
2. Theory
2.1. The Schroedinger Equation with Non-Local Potential
The model we are going to study numerically involves a radial Schrödinger equation with ESC04 potential; namely
, (1)
where 
is the reduced mass of the nucleons whose individual masses are m1 and m2, and have spins 
and
; r is the distance between the nucleons. The potential is parameterized as
where 
is a second rank tensor operator.
For an S-state we introduce
, where
.
For a given value of the quantum number J,
, (2)
where we introduce
, (3)
where the symbol 
denotes a ClebschGordan coefficient, and YLML are the spherical harmonics, and
;
;
.
The subscript on c refers to the magnetic projection quantum number MS of the spin-1 state, while a and b 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,
. (4)
The Schrödinger equation [Equation (1)] is processed by the method of separation of variables, we obtain as its radial component,
. (5)
We use the parametrized potential
and
for an S-state to obtain,
, (6)
where 
[5], and S12 may be written as an operator of the form
with l = 2 and j1 = j2 = 1. Here 
is the Clebsch-Gordan coefficient.
Using Racha algebra (see appendix A of [6]) we can show that
. (7)
2.2. Numerical Solution of the Schrödinger Equation
Considering the single state for the 1S0 wave, Equation (6) for the neutron-neutron system has the form (S = J = L = 0, L’ = −1, 0, 1),
(8)
where S00-1 = S001 = 0, S000 = 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 u0 from Equation (8) are introduced in the S matrix (Equation (10.58) of [7], which is,
, (9)
where the S matrix is evaluated in the last two points on a mesh of size e (
). Ul are the solutions to Equation (8) with the ESC04 potential previously calculated and hl 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
, (10)
where dl is real and is known as the phase shift.
Once the d0 phase shift is found the ann scattering length and the effective range rnn are calculated. For l = 0 the expression for 
can be parameterized in the following form,
. (11)
The quantity a is called the scattering length and r0 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]),
, (12)
where 
is the center-of-mass momentum (the wave number) and  indicates the real part.
2.3. 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 [1-4]. 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].
3. Results
The ann Scattering Length The ann scattering length is calculated obtaining a numerical value ann = −18.62497 fm and an effective range of rnn = 2.746615 fm. We use an ESC04 potential below 350 MeV. In Figures 3 and 4 the phase shift 
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.
4. 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 app, anp and ann have been calculated and are consistent with the experimental re-
Figure 1. Total potential in the partial wave 1S0, for I = ½.
Table 1. ESC04 low-energy parameters: S-wave scattering lengths and effective ranges.
Figure 2. Central (a), spin-spin (b), spin-orbital (c), and tensor (d) part of the YN potential.
Figure 3. Solid curve, proton-proton I = 1 phase shifts (degrees), as a function of Tlab (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].
Figure 4. Solid curve, neutron-proton I = 0 phase shifts (degrees), as a function of Tlab (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]. Diamonds, Bugg s.e. [11].
sults. The final value for ann from this study is ann = −18.625 fm. Results from previous studies are
[12],
[13]and
[14]The presented ESC model is thus successful in describing the NN data.
Acknowledgements
This work was partially supported by PIEIC-UNACH 2012 and SIINV UNACH 2012.
NOTES