Hypernucleus In A Two Frequency Model

1. Hypernucleus In A Two Frequency Model Yiharn Tzeng, S.Y.Tsay Tzeng, T.T.S.Kuo

2. B.E of - A challenging problem

3. Prescriptions • the use of an earlier Λ N central force, • the introduction of a repulsive Λ NN force • the consideration of quark effects, • the use of a modified Nijmegen NSC89 potential in the first order Bruckner-Hartree calculations, • the destruction of He-4 core structure, • the recent coherent Λ -Σ coupling . • Despite of all these efforts, it seems the end of the case still distance away

4. Shell Model calculations • Validity of hypernucleus' Shell structure has been firmly established in experiments. Hence shell model has been the framework for exploring the structure of hypernuclei. • Most shell model calculations assume same frequencies for nucleon's and hyperon's harmonic oscillator basis wave functions, i.e., = e.g., 21 MeV for , 11 MeV for .

5. Our Treatment • In this work, we set > • Our justification: • Since the rms radius • and • If the rms radii of N and Y about same • Or, if the hyperon is less bounded than the nucleon in a hypernucleus as the usual case, i.e., the rms radius for Y is greater than that for N

6. Effective Hamiltonian The full-space YN many-body problem very difficult to solve. Choose a small model space, P, then the full-space problem is formally reduced to a model-space, or P-space problem, namely • where the eigenvalues form a subset of the eigenvalues of the original full-space equation.

7. In the above, is the effective Hamiltonian which contains two parts, the unperturbed Hamiltonian and the effective interaction • is defined with the auxiliary single particle (sp) potentials, • single particle w.f. and energy defined as • ;

8. A central problem in this approach is of course how to derive , or how to derive the model-space effective interaction • The accuracy of the YN G-matrix from the realistic YN potentials is clearly crucial to our investigation. • A non-trivial problem is to treat the Pauli • operators in the G-matrix equation. • For instance, the G-matrix may be defined by the coupled integral equation

9. G-Matrix Element • Define

10. G-matrix • Define G-matrix as

11. In calculating the G-matrix elements, frequencies for a hyperon's and a nucleon's basis wave functions are set differently. • There we expanded wavefunctions with in terms of those with so that the difficulty of transformations between the CM-relative coordinates and the two particle states with different oscillator length can be removed. • The frequency for a nucleon is taken from the empirical formula = , and that for a hyperon is kept as a parameter.

12. Folded-diagram series • Using these G-matrix elements, we will be able to calculate various diagrams to be included in the -box. • An energy independent can be obtained from the -box • folded-diagram series being summed up to all orders • using the Lee-Suzuki iteration method to find

13. Lambda’s binding energy • Λ’s single-particel energy is then calculated as a function of from the box up to the diagrams of second-order in G. • The lambda's single particle energies are plotted as functions of calculated via various YN potentials. Note that each curve presents a saturation minimum at a certain . The positions of these saturation points vary with potentials. For examples, the minimum appears around = 8 MeV for the curve from NSC89, at 9.2 MeV for that from NSC97f, and at 10 MeV for NSC97d.

14. Although this is not a variation calculation, these saturation minima assure the results being stable with respect to a small variation of . Hence we take these minima as the lambda's binding energies.

15. Results • the binding energies calculated from all NSC97 potentials are between 3.8 MeV and 4.6 MeV, all too large compared to the experimental value of 3.12$\pm$ 0.02 MeV, • while the one from NSC89 is only about 1.7 MeV, much smaller than the experimental value.

16. Result consistent with \cite{halderson} where the binding energy calculated from several potentials modified from the NSC89 one and obtained the results ranging from 1.61 to 2.23 MeV. Haldeson \cite{halderson} pointed out the underbinding mainly from that the tensor force still remains too strong in both the original NSC89 and his modified potentials.

17. Akaishi and collaborators \cite{akaishi} proposed to split coupling into "coherent" and "incoherent" parts and suggested the problem could be solved by suppressing the "incoherent" part but keeping the "coherent" part. They obtained $B_{\Lambda}$ =2.38 MeV and 3.57 MeV respectively from NSC97f(S) and NSC97e(S), which are simplified potentials from NSC97f and NSC97e by including both central and tensor parts in and channels.

18. Comparison of our calculations by using the original NSC97 full potentials with theirs may provide some clues for further studies on this problem.

19. Conclusion • The binding energy $B_{\Lambda}$ is obtained from the saturation minimum of the single particle energy versus $ curve when keeping constant. • We obtain underbinding from NSC89 potential but have overbinging from all NSC97 ones. • The underbinding may come from too strong tensor force in coupling, as pointed in \cite{halserson}.

20. On the overbinding side, results from different NSC97 modes appear to have different binding strengths, with the one from NSC97d the strongest, and the one from the NSC97f the least strong but the closest to the desired experimental value. We also found that decreasing the frequency would be able to make $B_{\Lambda}$ even closer to the experimental one. This may suggest that we need a little bit large He-4 core.

21. However, this statement needs to be justified by both accurate YN realistic interaction and measured size of though it may not be easy experimentally). • Nevertheless, our work provides an alternative line of thinking for solving this problem.

22. Improvements • The fact of our energy levels' not fitting the experimental data exactly may be explained as that the $\Lambda$ particle-neutron hole formalism may not be perfectly applied to such a light hyprnucleus. The few -body effects very likely have to be taken into accounts here. • Since our results are obtained directly from the realistic Nijmegen potentials, one very possible reason for the deviation from experimental data is because of the potentials themselves.