# Structures of -shell double- hypernuclei studied with microscopic cluster models

###### Abstract

-orbit states in -shell double- hypernuclei (), , , , , and are investigated. Microscopic cluster models are applied to core nuclear part and a potential model is adopted for particles. The -core potential is a folding potential obtained with effective -matrix - interactions, which reasonably reproduce energy spectra of . System dependence of the - binding energies is understood by the core polarization energy from nuclear size reduction. Reductions of nuclear sizes and transition strengths by particles are also discussed.

## I Introduction

In the recent progress in strangeness physics, experimental and theoretical studies on hypernuclei have been extensively performed. Owing to experiments with high-resolution -ray measurements, detailed spectra of -shell hypernuclei have been experimentally revealed Hashimoto:2006aw (); Tamura:2010zz (); Tamura:2013lwa (). The data of energy spectra and electromagnectic transitions have been providing useful information for properties of -nucleon() interactions. They are also helpful to investigate impurity effects of a particle on nuclear systems. In a theoretical side, structure studies of -shell hypernuclei have been performed with various models such as cluster models Motoba:1984ri (); motoba85 (); Yamada:1985qr (); Yu:1986ip (); Hiyama:1996gv (); Hiyama:1997ub (); Hiyama:1999me (); Hiyama:2000jd (); Hiyama:2002yj (); Hiyama:2006xv (); Hiyama:2010zzc (); Cravo:2002jv (); Suslov:2004ed (); Mohammad:2009zza (); Zhang:2012zzg (); Funaki:2014fba (); Funaki:2017asz (), shell models Gal:1971gb (); Gal:1972gd (); Gal:1978jt (); Millener:2008zz (); Millener:2010zz (); Millener:2012zz (), mean-field and beyond mean-field models Guleria:2011kk (); Vidana:2001rm (); Zhou:2007zze (); Win:2008vw (); Win:2010tq (); Lu:2011wy (); Mei:2014hya (); Mei:2015pca (); Mei:2016lce (); Schulze:2014oia (), hyper antisymmetrized molecular dynamics (HAMD) model Isaka:2011kz (); Isaka:2015xda (); Homma:2015kia (); Isaka:2016apm (); Isaka:2017nuc (), and no-core shell model Wirth:2014apa (), and so on.

For double- hypernuclei, experimental observations with nuclear emulsion have been used to extract information of the - interaction in nuclear systems from binding energies Nakazawa:2010zzb (). Several double- hypernuclei have been observed, but precise data of binding energies are still limited because of experimental uncertainties in energies and reaction assignments. The most reliable datum is the binding energy of , which suggests an weak attractive - interaction. Another observation is a candidate event for . In order to extract information of the - interaction from the limited experimental data, systematic investigations of binding energies of -shell and have been performed with semi-microscopic cluster model Hiyama:2002yj (); Hiyama:2010zzd (); Hiyama:2010zz () and shell model calculations Gal:2011zr (). In the former calculation, dynamical effects in three-body and four-body cluster systems as well as spin-dependent contributions are taken into account in a semi-microscopic treatment of antisymmetrization effect between clusters called the orthogonal condition model (OCM). In the latter calculation, the spin-dependent and - coupling contributions in are taken into account perturbatively. A major interest concerning the - interaction is so-called - binding energy defined with masses of , , and as,

(1) |

which stands for the difference of the two- binding energy in from twice of the single- binding energy in . In Refs. Hiyama:2002yj (); Hiyama:2010zzd (); Gal:2011zr (), they have discussed systematics of comparing with available data, and pointed out that has rather strong system (mass-number) dependence because of various effects and is not a direct measure of the - interaction. For example, of is significantly deviated from global systematics in -shell double- hypernuclei because of remarkable clustering in the core nucleus, .

In our previous work Kanada-Enyo:2017ynk (), we have investigated energy spectra of low-lying -orbit states in -shell hypernuclei by applying microscopic cluster models for core nuclei and a single-channel potential model for a particle. As the core polarization effect in , the nuclear size reduction contribution by a has been taken into account. The -core potentials have been calculated with local density approximations of folding potentials using the -matrix - interactions. Since the spin-dependence of the - interactions are ignored, the spin-averaged energies of low-energy spectra have been discussed. The previous calculation describes systematic trend of experimental data for excitation energy shift by a from to , and shows that nuclear size difference between the ground and excited states dominantly contributes to the excitation energy shift. The size reduction by a particle in has been also studied, and it was found that significant size reduction occurs in and because of developed clustering consistently with predictions by other calculations Motoba:1984ri (); Hiyama:1999me (); Funaki:2014fba (). The framework developed in the previous work of hypernuclei can be applied also to double- hypernuclei straightforwardly.

In the present paper, we extend the previous calculation to -shell double- hyper nuclei. We solve motion of two -wave particles around a core nucleus in the -core potential calculated by folding the -matrix - interactions in the same way as the previous calculation for . For the effective - interaction, we adopt the parametrization used in Refs. Hiyama:2002yj (); Hiyama:2010zzd () with a slightly modification to reproduce the - binding energy in . We investigate systematics of the - binding energies in -shell double- hypernuclei. System dependence of the - binding energies is discussed in relation with the core polarization. We also discuss the size reduction and excitation energy shift by particles in and .

In the present calculation, -wave s are assumed and the spin dependence of the - and - interactions are disregarded. In , the total spin of is given just by the core nuclear spin as . In with a non-zero nuclear spin (), spin-doublet states completely degenerate in the present calculation. For simplicity, we denote the spin-doublet states in hypernuclei with the label of the core nuclear spin as .

## Ii Framework

### ii.1 Microscopic cluster models for core nuclei

Core nuclei in and are calculated with the microscopic cluster models in the same way as the previous calculation for Kanada-Enyo:2017ynk (). The generator coordinate method (GCM) Hill:1952jb (); Griffin:1957zza () is applied using the Brink-Bloch cluster wave functions Brink66 () of , , , , , , , and clusters for , , , , , , , and , respectively. , , , and clusters are written by harmonic oscillator configurations with a common width parameter fm. For and , the configurations are taken into account by adding the corresponding shell model wave functions to the , and cluster wave functions as done in Refs. Kanada-Enyo:2017ynk (); Suhara:2014wua (). For Be, the wave functions adopted in Ref. Kanada-Enyo:2016jnq () are superposed in addition to the wave functions.

Let us consider a -nucleon system for a mass number nucleus consisting of clusters. is the number of clusters. The Brink-Bloch cluster wave function is written by a -body microscopic wave function parametrized by the cluster center parameters (). To take into account inter-cluster motion, the GCM is applied to the spin-parity projected Brink-Bloch cluster wave functions with respect to the generator coordinates . Namely, the wave function for the state is given by a linear combination of the Brink-Bloch wave functions with various configurations of as

(2) |

where is the spin-parity projection operator. The coefficients are determined by solving Griffin-Hill-Wheeler equations Hill:1952jb (); Griffin:1957zza (), which is equivalent to the diagonalization of the Hamiltonian and norm matrices. For the and wave functions, and are chosen to be with fm}. For the wave functions, fm} are adopted to obtain a bound state solution for the resonance state corresponding to a bound state approximation. For the configurations of , , , and , are chosen to be

(3) | |||

(4) |

with fm}, fm}, and . Here is the mass number of the cluster. For the cluster, a larger model space of fm}, fm}, and are used to describe remarkable clustering in Be.

The Hamiltonian of the nuclear part consists of the kinetic term, effective nuclear interactions, and Coulomb interactions as follows,

(5) | |||||

(6) | |||||

(7) | |||||

(8) | |||||

(9) |

where the kinetic term of the center of mass (cm) motion, , is subtracted exactly. and are the effective - central and spin-orbit interactions, respectively. The nuclear energy and nuclear density of the core nuclei are calculated for the obtained GCM wave function . Here the radial coordinate in is the distance measured from the cm of core nuclei.

### ii.2 Folding potential model for particles

Assuming two -orbit particles coupling to the spin-singlet state, states of are calculated with a folding potential model in a similar way to the previous calculation for Kanada-Enyo:2017ynk ().

The Hamiltonian of states in is given by the nuclear Hamiltonian for the core nucleus and the single-particle Hamiltonian for a particle around the core as

(10) | |||||

(11) | |||||

(12) | |||||

(13) |

The Hamiltonian of states in is written straightforwardly as

(14) | |||||

(15) |

where is the even-part of the - interactions. Here, the recoil kinetic term is dropped off for states.

The -core potential is calculated by folding the - interactions with the nuclear density as

(16) | |||||

(17) | |||||

(18) | |||||

(19) | |||||

(20) |

where and are even and odd parts of the effective - central interactions, respectively, including the parameter for density dependence. The nuclear density matrix in the exchange potential is approximated with the density matrix expansion using the local density approximation Negele:1975zz () as done in the previous paper.

For a given nuclear density of a core nuclear state of , the -core wave functions in and are calculated with the Gaussian expansion method Kamimura:1988zz (); Hiyama:2003cu () so as to minimize the single- energy and two- energy , respectively. The rms radius () and the averaged nuclear density () for the distribution are calculated with the obtained -core wave function ,

(21) | |||||

(22) |

### ii.3 Core polarization

We take into account the core polarization, i.e., the nuclear structure change induced by -orbit s, in the same way as done in the previous calculation of hypernuclei. To prepare nuclear wave functions polarized by the particles, we add artificial nuclear interactions to the nuclear Hamiltonian. By performing the GCM cluster-model calculation of the core nuclear part for the modified Hamiltonian , we obtain wave function for . For the prepared nuclear wave function, we calculate the nuclear energy for the original nuclear Hamiltonian of without the artificial nuclear interactions. Using the nuclear density obtained with , the single- and two- energies ( and ) in and are calculated. Finally, the optimum value, i.e., the optimum nuclear wave function is chosen for each state in and so as to minimize the total energy

(23) |

in , and

(24) |

in .

For the artificial interaction , we use the central part of the nuclear interactions as . It corresponds to slight enhancement of the central nuclear interaction as

(25) |

where is regarded as the enhancement factor to simulate the nuclear structure change induced by the -orbit particles. The main effect of the enhancement on the structure change is size reduction of core nuclei. Therefore, it is considered, in a sense, that the present treatment of the core polarization simulates the nuclear size reduction, which is determined by energy balance between the potential energy gain and nuclear energy loss. In the optimization of , we vary only the GCM coefficients but fix the basis cluster wave functions, corresponding to the inert cluster ansatz. In this assumption, the enhancement of the central nuclear interactions acts as an enhancement of the inter-cluster potentials.

### ii.4 Definitions of energies and sizes for and systems

The binding energy () in is calculated as

(26) | |||||

(27) |

where is the unperturbative nuclear energy without the particle and stands for the nuclear energy increase by a particle in .

Similarly, the two- binding energy () in is calculated as

(28) | |||||

(29) |

where is the nuclear energy increase caused by two particles in .

The - binding energy in is given as

(30) | |||

(31) |

To discuss the binding, we also define for excited states () as

As discussed by Hiyama et al. in Refs. Hiyama:2002yj (); Hiyama:2010zzd (), is not necessarily a direct measure of the - interaction because it is contributed also by various effects such as the core polarization effect. Alternatively, the bond energy

(33) |

has been discussed in Refs. Hiyama:2002yj (); Hiyama:2010zzd (). Here is the two- binding energy obtained by switching off the - interactions (), and indicates the difference of between calculations with and without the - interactions.

For excited states and with the excited core , excitation energies measured from the ground states are denoted by and , respectively. Note that the parameter is optimized for each state meaning that the core polarization is state dependent. Because of the impurity effect of particles, the excitation energies are changed from the original excitation energy of isolate nuclei . For each , we denote the energy change caused by s as

(34) | |||||

(35) |

which we call the excitation energy shift.

The nuclear sizes in , , and are calculated with the nuclear density as

(36) |

We define nuclear size changes and by particles in and from the original size as,

(37) | |||||

(38) |

## Iii Effective interactions

### iii.1 Effective nuclear interactions

We use the same effective two-body nuclear interactions as the previous calculation; the finite-range central interactions of the Volkov No.2 force VOLKOV () with , , and and the spin-orbit interactions of the G3RS parametrization LS () with MeV except for Li and Li. For Li, we use the adjusted parameter set, , , , and MeV, which reproduces the experimental and energies measured from the threshold energy. The same set of interaction parameters is used also for Li. This interaction set gives the and threshold energies, 1.48 MeV and 2.59 MeV, for and , respectively. (The experimental threshold energies are 1.48 MeV for and 2.47 MeV for .)

### iii.2 Effective - interaction

As the effective - central interactions, we use the ESC08a parametrization of the -matrix - () interactions derived from - interactions of the one-boson-exchange model Yamamoto:2010zzn (); Rijken:2010zzb (). Since spin-dependent contributions are ignored in the present folding potential model, the -core potentials are contributed by the spin-independent central parts,

(39) | |||||

(40) | |||||

(41) | |||||

(42) |

The values of the parameters and are given in Table II of Ref. Yamamoto:2010zzn ().

As for the parameter in the interactions, we adopt three choices. The first choice is the density-dependent called “averaged density approximation (ADA)” used in Refs. Yamamoto:2010zzn (); Isaka:2016apm (); Isaka:2017nuc (). The is taken to be , where is the averaged Fermi momentum for the distribution,

(43) |

and self-consistently determined for each state. We label this choice of the interactions as ESC08a(DD) consistently to the previous paper.

The second choice is the density-independent interaction with a fixed value, . We use a system-dependent but state-independent value as the input parameter in calculation of , and use the same value in calculation of . As the value for each system, we used the averaged value of self-consistently determined by the ADA treatment in the ESC08a(DD) calculation for low-lying states. We label this choice as ESC08a(DI).

The third choice is the hybrid version of the ESC08a(DD) and ESC08a(DI) interactions. In the previous study of with the ESC08a(DD) and ESC08a(DI) interactions, it was found that ESC08a(DD) fails to describe the observed excitation energy shift in , whereas ESC08a(DI) can describe a trend of the excitation energy shift but somewhat overestimates the experimental data. It suggests that a moderate density-dependence weaker than ESC08a(DD) may be favored. In the hybrid version, we take the average of the ESC08a(DD) and ESC08a(DI) interactions as

in which is self-consistently determined for each state. We label this interaction as ESC08a(Hyb).

Since all of ESC08a(DD), ESC08a(DI), and ESC08a(Hyb) are system-dependent interactions through the values determined for each system (), these interactions reasonably reproduce the binding energies of -shell hypernuclei because the interactions are originally designed so as to reproduce the systematics of experimental binding energies in a wide mass number region. It should be noted that ESC08a(DI) is state-independent, but ESC08a(DD) and ESC08a(Hyb) are state-dependent interactions. Namely, ESC08a(DI), ESC08a(Hyb), and ESC08a(DD) have no, mild, and relatively strong density dependence of the interactions, respectively.

### iii.3 Effective - interactions

For the - interaction in states in , we adopt the singlet-even part of the effective - interactions used in Refs. Hiyama:2002yj (); Hiyama:2010zzd (),

(45) |

with , , and in fm. The original values of the strength parameters in the earlier work in Ref. Hiyama:2002yj () are , , and in MeV, but a modified parameter () was used in the later work in Ref. Hiyama:2010zzd () to fit the revised experimental value of MeV Nakazawa:2010zzb (). By using with in the ESC08a(Hyb) calculation, we obtain MeV for the frozen core with the experimental size fm reduced from the charge radius data. In the ESC08a(DD) and ESC08a(DI) calculations, we readjust the parameter as and , respectively, which give almost the same values.

## Iv Results and Discussions

### iv.1 Properties of ground and excited states in

ESC08a(Hyb) | |||||||||
---|---|---|---|---|---|---|---|---|---|

0.96 | 2.83 | 0.97 | 1.46 | 3.53 | 3.12(2) | ||||

0.93 | 2.64 | 0.93 | 2.32 | 0.23 | 5.35 | 5.58(3) | 5.12(3) | ||

0.91 | 2.55 | 0.96 | 2.34 | 0.26 | 6.68 | 6.80(3) | |||

0.90 | 2.59 | 0.93 | 2.57 | 0.84 | 6.53 | 6.71(4) | |||

0.95 | 2.51 | 0.99 | 2.54 | 0.34 | 8.06 | 9.11(22) | |||

1.04 | 2.47 | 1.06 | 2.39 | 0.11 | 9.01 | ||||

1.03 | 2.44 | 1.08 | 2.34 | 0.08 | 9.31 | 10.24(5) | 10.09(5) | ||

1.07 | 2.40 | 1.13 | 2.29 | 0.05 | 10.06 | 11.37(6) | 11.27(6) | ||

1.11 | 2.41 | 1.15 | 2.31 | 0.05 | 10.44 | 11.69(12) | |||

ESC08a(DI) | |||||||||

0.96 | 2.81 | 0.97 | 1.46 | 3.60 | |||||

0.93 | 2.57 | 0.95 | 2.22 | 0.59 | 5.44 | ||||

0.91 | 2.41 | 1.00 | 2.25 | 0.79 | 7.32 | ||||

0.90 | 2.44 | 0.98 | 2.44 | 1.69 | 7.04 | ||||

0.95 | 2.39 | 1.03 | 2.43 | 0.99 | 8.69 | ||||

1.04 | 2.41 | 1.08 | 2.34 | 0.41 | 9.32 | ||||

1.03 | 2.36 | 1.10 | 2.29 | 0.37 | 9.97 | ||||

1.07 | 2.33 | 1.16 | 2.24 | 0.29 | 10.94 | ||||

1.11 | 2.35 | 1.18 | 2.26 | 0.27 | 11.07 | ||||

ESC08a(DD) | |||||||||

2.84 | 0.96 | 1.46 | 3.49 | ||||||

2.66 | 0.91 | 2.40 | 0.08 | 5.43 | |||||

2.61 | 0.93 | 2.42 | 0.06 | 6.43 | |||||

2.67 | 0.90 | 2.69 | 0.44 | 6.43 | |||||

2.57 | 0.96 | 2.62 | 0.08 | 7.84 | |||||

2.49 | 1.04 | 2.44 | 0.01 | 8.92 | |||||

2.48 | 1.06 | 2.38 | 0.00 | 8.97 | |||||

2.45 | 1.11 | 2.33 | 0.00 | 0.00 | 9.57 | ||||

2.44 | 1.13 | 2.35 | 0.00 | 0.00 | 10.13 |