# Feshbach Resonance due to Coherent - Coupling in He

###### Abstract

Coherent - coupling effect in He is analyzed within three-body framework of two coupled channels, -- and --, where represents trinulceon which is either H or He. The hyperon-trinucleon () and trinucleon-trinucleon () interactions are derived by folding -matrices of and interactions with trinucleon density distributions. It is found that the binding energy of He is 4.04 MeV below the ++ threshold without - coupling and the binding energy is increased to 4.46 MeV when the coupling effect is included. This state is 7.85 MeV above the He+ threshold and it may have a chance to be observed as a Feshbach resonance in Li He experiment done at Jefferson Lab.

###### keywords:

Feshbach resonance: coherent - coupling: hyperon-trinucleon interaction###### Pacs:

^{†}

^{†}journal: Nuclear Physics A

## 1 Introduction

Significance of - coupling effect in binding mechanism of light -hypernuclei has long been recognized and discussed in the references YN (); BF (). Admixture of states in -hypernuclei is probably an important aspect of hypernuclear dynamics. There are two coupling schemes namely incoherent and coherent - couplings YA (). Incoherent - coupling means a nucleon changes to an excited level after the interaction, while the other process where a nucleon remains in its ground state after converting to , is called coherent - coupling. In the latter case, all the nucleons have an equal chance to interact with the converted and coupling effect contributed from each nucleon is added coherently. Harada TH () has successfully fitted the experimental spectra of He (stopped ) RH () and He (in-flight TN () production reactions by taking into account the coherent - coupling effect. Furthermore, all the s-shell hypernuclear binding energies are well reproduced only after the coherent - coupling effect has been included YA (); HN (). It has been found that the coherent coupling contribution is significantly large on the order of 1 MeV in H and He ground states.

## 2 Coupled-channel three-body cluster model of He

Having considered the above mentioned findings, we analyze a structure of He in continuum by using three-body model of --, -- and -- coupled channels to investigate the coherent - coupling effect. The coupling between - gives coherent - coupling, while Lane term of - coupling plays a significant role in forming H THa (). All these couplings are included in our analysis. To solve three-body calculation, we employ Kamimura’s coupled rearrangement-channel method MK ().

Three-body Hamiltonian of the -- diagonal part, which we explicitly show for explanation here, is

(1) |

where expresses Pauli exclusion effect between two tritons. In orthogonality-condition model (OCM) SS (),

(2) |

where is the Pauli forbidden state. Total wave function of the -- channel is expanded in Gaussian bases which are spanned over three rearrangement-channels as follows,

(3) |

Wave functions of the other channels -- and -- are treated in a similar way.

State | H(0 | H(1 | ||||
---|---|---|---|---|---|---|

- | - | - | - | - | - | |

1 |
1.7284 | 9.4720 | -3.5575 | 0.36869 | 1.9558 | -0.16822 |

2 | 50.838 | 69.234 | 4.2647 | 43.237 | 65.877 | 7.1105 |

3 | -63.595 | -105.09 | 32.682 | -57.877 | -44.391 | 0.70109 |

4 | 6.2861 | 10.130 | -4.0631 | 5.1858 | 2.1056 | -1.3503 |

5 | -1.1202 | -2.3001 | 0.8537 | -0.86971 | -0.61958 | 0.12653 |

The interaction used in our computations is a phase equivalent potential of the Nijmegen model-D potential MM (). Then, hyperon-trinucleon potentials are obtained by folding the effective interaction, i.e. -matrix of the above potential with trinucleon density distributions SA (). They are expressed in five-range Gaussian form, the range and strength parameters of which are slightly modified so as to reproduce the empirical binding energy of H(0 and H(1, and the expansion coefficients for are given in Table 1. Trinucleon-trinucleon (-) interaction is obtained by doubly folding -matrix of Tamagaki’s OPEG potential with trinucleon density distributions. This - potential is spin-isospin dependent, and does not give any bound state of triton-triton two-body system in OCM treatment.

## 3 Results and discussions

From our calculation, a bound state is found to be at MeV below the threshold and about MeV above the He threshold as shown in Fig. 1. It is a Feshbach resonance state HF (), because it lies in continuum region of the open channels such as He, He, He and channels.

A possible way to populate this resonance state, He, is through electro-production reaction on Li target. Formation of He through the resonance is described with -channel interaction model as shown in Fig. 2. Formation and decay spectra are analyzed, as explained in Ref. YAk (), by using Yamaguchi-type separable (i.e. -channel) potential:

(4) |

where HHe.

Missing-mass spectrum and invariant-mass spectrum can be obtained by detecting emitted particles and and decay particle , respectively. The effect of interaction range on the missing-mass spectrum is investigated by varying the range parameter of H- interaction from 0.3 to 0.9 fm.

Figure 3 shows the missing-mass spectrum calculated with 3 MeV width of the resonance. We have compared this missing-mass spectrum with JLab experimental spectrum LY (), where a peak structure is found at about 7 MeV above the He+ threshold, which might correspond to our resonance state. A crude explanation of why a narrow peak appears in continuum region is such that; similarity in structures between - and - may give a strong population of state, while different structures between - and He ensure the formation of quasi-stable Feshbach resonance. However, a recent experimental spectrum of Li He displays only a prominent peak below the He+ threshold in bound region OH (). In order to clarify the possible existence of Feshbach resonance in He system, electro-production or equivalent experiments on Li target with high statistics are highly awaited.

Two of the authors, San San Mon and Khin Swe Myint, would like to thank the organizing committee for the support to attend the Conference.

## References

- (1) Y. Nogami and E. Satoh, Nucl. Phys. B 19 (1970) 93.
- (2) B. F. Gibson, A. Goldberg and M.S. Weiss, Phys. Rev. C 6 (1972) 741.
- (3) Y. Akaishi, T. Harada, S. Shinmura and Khin Swe Myint, Phys. Rev. Lett. 84 (2000) 3539.
- (4) T. Harada, Phys. Rev. Lett. 81 (1998) 5287.
- (5) R. Hayano et al., Phys. Lett. 231 (1989) 355.
- (6) T. Nagae et al., Phys. Rev. Lett. 80 (1998) 1605.
- (7) H. Nemura, Y. Akaishi and Y. Suzuki, Phys. Rev. Lett. 89 (2002) 142504.
- (8) T. Harada, S. Shinmura, Y. Akaishi and H. Tanaka, Nucl. Phys. A 507 (1990) 715.
- (9) M. Kamimura, Phys. Rev. A 38 (1998) 621.
- (10) S. Saito, Prog. Theor. Phys. 41 (1969) 705.
- (11) M.M. Nagels, Th. A. Rijken and J.J. de Swart, Phys. Rev. D 12 (1975) 744.
- (12) Sandar Myint Oo, PhD Thesis, University of Mandalay (2004).
- (13) H. Feshbach, Ann. Phys. 5 (1958) 357; 19 (1962) 287.
- (14) Y. Akaishi, Khin Swe Myint and T. Yamazaki, Proc. Jpn. Acad. B 84 (2008) 264.
- (15) L. Yuan et al., Phys. Rev. C 73 (2006) 044607.
- (16) O. Hashimoto, these Proceedings.