
DOI: 10.22661/AAPPSBL.2020.30.3.34
Gaussian Expansion Method and its Application to Nuclear Physic with Strangeness
EMIKO HIYAMA^{1,2}
^{1}DEPARTMENT OF PHYSICS, KYUSHU UNIVERSITY, FUKUOKA, 8190395, JAPAN
^{2}STRANGENESS NUCLEAR PHYSICS LABORATORY, RIKEN NISHINA CENTER, WAKO, 3510198, JAPAN
Communicated by Tohru Motobayashi
ABSTRACT
We have proposed one of the most powerful methods to calculate fewbody problems in physics, called Gaussian Expansion Method. The method has been successfully applied to various hypernuclei such as Λ, double Λ and Ξ hypernuclei. In this paper, we describe the structure of _{}He, which is a neutronrich hypernucleus, as a successful example.
INTRODUCTION
Many important problems in physics can be treated by solving accurately the SchrÃ¶dinger equation for three and fourbody interacting particles. By solving this equation we can predict various observable, before performing its measurement, and obtain new understandings by comparing the observed data and our theoretical prediction.
For this purpose, it is necessary to develop a method to calculate precisely the three and fourbody problems, and to apply it to various fields such as nuclear physics as well as atomic physics. Along this line, we have been developing our fewbody calculation method, Gaussian Expansion method (GEM), which is a kind of variational approach. This method fist was proposed by Kamimura and collaborators [1] and has been successfully applied to various threebody systems such as muonic molecules, [1, 2] threenucleon bound states (^{3}H, ^{3}He), [3] neutronrich nuclei [4] and antiprotonic helium atoms. [5] Afterwards, the present author and collaborators proposed to modify the GEM with a newtype of basis functions, called infinitesimallyshiftGaussian Lobe basis function, [6] and it became in this way much easier to apply the GEM to the solution of the four and fivebody problems. To show the accuracy of our method, we performed a fourbody benchmark test calculation of the ^{4}He ground state with realistic NN interaction among seven fewbody groups using different approaches. The calculated binding energy, radii and density distribution were found in good agreement among each other. [7]
GEM has been most extensively applied to the study of hypernuclear physics by the present author and collaborators. One of the important goals in the hypernuclear physics is to produce neutronrich Λ hypernuclei and to study the dynamical change of the structure by comparing the core nuclei. Since the Λ hyperon is free from the nuclear Pauli principle, it is considered to be responsible for a sizeable dynamical contraction of hypernuclear systems due to the addition of a Λ. This is currently known as the "gluelike" role of the Λ particle.
Using this phenomena, the present author has studied with some detail several neutronrich Λ hypernuclei. In the light nuclei sector, near the neutron drip line, some new interesting phenomena concerning the neutron halo have been observed. If a Λ particle is added to such a halo nucleus, which is a very weakly bound system, the resulting hypernucleus will become stable against the neutron decay. To investigate the structure of neutronrich Λ hypernuclei, we have been studied several systems, like nnΛ, [12] _{}He, [11] A = 7 hypernuclei, [810] etc.
Fig. 1: Three sets of Jacobi coordinates of threebody system.
METHOD
We shortly explain in this section the formalism of GEM for the threebody system. We consider the case of central forces only for simplicity of the expressions (see Ref. [6] for more complicated cases including the solutions of fourbody systems). The SchrÃ¶dinger equation is written as
 (2Â·1)

with obvious notation. The threebody total wave function Î¨_{JM} is described as a sum of amplitudes of three rearrangement channels c = 1  3 (Fig. 1).
 (2Â·2)

Each amplitude is expanded in terms of the Gaussian basis functions written in Jacobi coordinates r_{c} and R_{c}:
 (2Â·3)

 (2Â·4)

 (2Â·5)

N_{nl} (N_{NL}) denotes the normalization constant. The Gaussian ranges Î½_{n} and λ_{N} are given by a geometric progression.
The eigenenergies E and the coefficients A_{} are determined by means of the RayleighRitz variational principle.
Since the GEM obtained results are reliable, our predictions have been in good agreement with the data. In the next section, a successful example is shown.
STRUCTURE OF NEUTRONRICH HYPERNUCLEUS, _{}HE
The observed data of the core nucleus ^{6}He reported a 0_{} ground state and the second 2_{} state. In 2012, see Ref. [13], a second 2^{+} resonant state was observed in the chargeexchangereaction ^{6}Li(t, ^{3}He)^{6}He with parameters E_{x} = 2.6 Â± 0.3 and Γ = 1.6 Â± 0.4 MeV.
From the theoretical point of view, many authors have studied this system. [1416] Among these studies, Myo et al. [16] calculated the ^{6}He spectrum using the complex scaling method (CSM) to obtain the energy together with the decay width of the lowlying states. They reproduced the observed ground state and the first 2^{+} state. They also obtained a second 2^{+} state. Then, it is interesting to add a Λ particle in ^{6}He. Due to the "gluelike" role of Λ, it is likely to have narrower 3/2_{} and 5/2_{} resonant states in _{}He. It was also observed that other states, the 1/2^{+} ground and the 3/2_{} and 5/2_{} excited states, become more stable than the corresponding states of the core nucleus ^{6}He.
To investigate the structure of _{}He, we have performed a fourbody cluster model calculation of this system described as Î±+Λ+n+n and using GEM. In particular, we expected to found some resonant states in this hypernucleus. In order to obtain these resonant states, we employed the complex scaling method, in a similar way of how was applied by Myo et al. to determine the resonant states of the core nucleus ^{6}He.
In ^{6}He, we found a bound ground state with 0^{+} which is in good agreement with the observed energy. We found also two 2^{+} states at E_{r} = 0.96 MeV with Γ = 0.14MeV and E_{r} = 2.81 MeV with Γ = 4.63 MeV, respectively. The calculated energy of the first 2^{+} is also in good agreement with the data. The SPIRAL data reported an energy for the second 2^{+} state E_{r} = 1.63 MeV with Γ = 1.6 MeV, which is much lower than the calculated E_{r} energy and had narrower decay width than the calculated one.
In the _{}He system, and due to the "gluelike" role of Λ particle, all the states become more stable than the corresponding states of ^{6}He. The calculated Λseparation energy of B_{Λ} = E(^{6}He)E(_{}He) = 5.36 MeV, which is consistent with the recent observed data of JLab E01011 experiment: B_{} = 5.68Â±0.03(stat.)Â±0.25(sysm.) MeV. The recent analysis of the JLab E05E115 experiment [17] also reports an excitedstate B_{} = 3.65Â±0.20(stat.)Â±0.11(sysm.) MeV, which is also in good agreement with our calculations (see Fig. 2). In addition, we predict additional 5/2^{+} and 3/2^{+} states having respectively E_{r} = 0.03 MeV with Γ = 1.13 MeV and E_{r} = 0.07 MeV with Γ = 1.01 MeV.
To encourage future experimental activity at JLab, we propose to perform the experiments ^{7}Li(γ, K^{+})_{}He producing these states. The calculated cross sections are 3.4 and 4.3 nb+sr for 3/2_{} and 5/2_{}, respectively, which seem to be an accessible value for a future experiment. This experiment is of crucial importance in order to confirm the existence of the second 2^{+} state in ^{6}He.
Fig. 2: The calculated energy spectra of ^{6}He and _{}He. The states of 2_{} , 2_{} and
1^{+} of ^{6}He and 3/2_{} and 5/2_{} are obtained using CSM. The values in parentheses are decay widths Γ in MeV. The figure is taken from Ref. [10]
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