Clusterization and deformation of multi-$\Lambda$ hypernuclei within relativistic mean-field model

TL;DR

This study investigates the structure and deformation of multi-$ ext{Lambda}$ hypernuclei using a relativistic mean-field model, revealing cluster formations and deformation changes due to hyperon addition.

Contribution

It introduces a relativistic mean-field approach to analyze deformation and clustering in multi-$ ext{Lambda}$ hypernuclei across specific isotopic chains.

Findings

Evidence of two-$^6_{2 ext{Lambda}}$He cluster structure in $^{12}_{4 ext{Lambda}}$Be.

Addition of $ ext{Lambda}$ hyperons slightly reduces deformation in Ne isotopes.

Deformation significantly decreases or vanishes in Si hyper isotopes with hyperons.

Abstract

Deformed multi-$\Lambda$ hypernuclei are studied within a relativistic mean-field model. In this paper, we take some $N=Z$ "hyper isotope" chains, i.e., $^{8+n}_{\ \ n\Lambda}{\rm Be}$, $^{20+n}_{\ \ \ n\Lambda}{\rm Ne}$, and $^{28+n}_{\ \ \ n\Lambda}{\rm Si}$ systems where $n = 2$, $4$ for Be, and $n = 2$, $8$ for Ne and Si. A sign of two-$^6_{2\Lambda}$He cluster structure is observed in the two-body correlation in $^{12}_{4\Lambda}$Be. In the Ne hyper isotopes, the deformation is slightly reduced by addition of $\Lambda$ hyperons whereas it is significantly reduced or even disappears in the Si hyper isotopes.

Figures

Tables

Table 1. Table 1: The double- Λ Λ \Lambda binding energies B Λ ​ Λ subscript 𝐵 Λ Λ B_{\Lambda\Lambda} and Λ Λ \Lambda - Λ Λ \Lambda interaction energies Δ ​ B Λ ​ Λ Δ subscript 𝐵 Λ Λ \Delta B_{\Lambda\Lambda} obtained with the present RMF model. For comparison, the experimental data in Refs. Naka10 ; Fr95 ; Mino are also listed. Note that the values of B Λ ​ Λ subscript 𝐵 Λ Λ B_{\Lambda\Lambda} and Δ ​ B Λ ​ Λ Δ subscript 𝐵 Λ Λ \Delta B_{\Lambda\Lambda} for 2 ​ Λ 6 subscript superscript absent 6 2 Λ {}^{6}_{2\Lambda} He from Ref. Naka10 are given by the weighted average of NAGARA event ( B Λ ​ Λ = 6.91 ± 0.16 subscript 𝐵 Λ Λ plus-or-minus 6.91 0.16 B_{\Lambda\Lambda}=6.91\pm 0.16 MeV and Δ ​ B Λ ​ Λ = 0.67 ± 0.17 Δ subscript 𝐵 Λ Λ plus-or-minus 0.67 0.17 \Delta B_{\Lambda\Lambda}=0.67\pm 0.17 MeV) Nagara and MIKAGE event ( B Λ ​ Λ = 10.06 ± 1.72 subscript 𝐵 Λ Λ plus-or-minus 10.06 1.72 B_{\Lambda\Lambda}=10.06\pm 1.72 MeV and Δ ​ B Λ ​ Λ = 3.82 ± 1.72 Δ subscript 𝐵 Λ Λ plus-or-minus 3.82 1.72 \Delta B_{\Lambda\Lambda}=3.82\pm 1.72 MeV) Mikage . The former have been uniquely identified as 2 ​ Λ 6 subscript superscript absent 6 2 Λ {}^{6}_{2\Lambda} He Naka10 ; Nagara whereas the latter was only found most probable to be 2 ​ Λ 6 subscript superscript absent 6 2 Λ {}^{6}_{2\Lambda} He Naka10 ; Mikage . The data of Ref. Mino are the very recent results of MINO event in J-PARC E-07 experiment, the candidates of which are 2 ​ Λ 10 subscript superscript absent 10 2 Λ {}^{10}_{2\Lambda} Be, 2 ​ Λ 11 subscript superscript absent 11 2 Λ {}^{11}_{2\Lambda} Be (most likely), and 2 ​ Λ 12 subscript superscript absent 12 2 Λ {}^{12}_{2\Lambda} Be.

	$B_{\Lambda \Lambda }$ (MeV)				$\Delta B_{\Lambda \Lambda }$ (MeV)
	RMF	exp. Mino	exp. Naka10	exp. Fr95	RMF	exp. Mino	exp. Naka10	exp. Fr95
${}_{2\Lambda }{}^{}{}_{}{}^6$He	$4.24$	$-$	$6.93\pm 0.16$	$10.9\pm 0.6$	$1.08$	$-$	$0.70\pm 0.17$	$4.7\pm 0.6$
${}_{2\Lambda }{}^{}{}_{}{}^{10}$Be	$13.23$	$\left(15.05\pm 0.11\right)$	$11.90\pm 0.13$	$17.7\pm 0.4$	$0.59$	$\left(1.63\pm 0.14\right)$	$-1.52\pm 0.15$	$4.3\pm 0.4$
${}_{2\Lambda }{}^{}{}_{}{}^{13}$B	$-$	$-$	$23.3\pm 0.7$	$27.5\pm 0.7$	$-$	$-$	$0.6\pm 0.8$	$4.8\pm 0.7$

Table 2. Table 2: Quadrupole deformation parameters β 2 subscript 𝛽 2 \beta_{2} , root-mean-squared radii R 𝑅 R , and the distance between the two α 𝛼 \alpha ’s D α ​ α subscript 𝐷 𝛼 𝛼 D_{\alpha\alpha} of the Be hyper isotope chain. β 2 ​ N subscript 𝛽 2 𝑁 \beta_{2N} and R N subscript 𝑅 𝑁 R_{N} are deformation parameter and radius calculated from the nucleon density, and β 2 ​ Λ subscript 𝛽 2 Λ \beta_{2\Lambda} and R Λ subscript 𝑅 Λ R_{\Lambda} are calculated from the Λ Λ \Lambda hyperon density. N Λ subscript 𝑁 Λ N_{\Lambda} is the number of Λ Λ \Lambda hyperons. Λ s subscript Λ 𝑠 \Lambda_{s} and Λ p subscript Λ 𝑝 \Lambda_{p} for N Λ = 2 subscript 𝑁 Λ 2 N_{\Lambda}=2 mean that the Λ Λ \Lambda particles are in s 𝑠 s and p 𝑝 p orbitals, respectively.

$N_{\Lambda }$	${\beta }_{2N}$	${\beta }_{2\Lambda }$	$R_N$ (fm)	$R_{\Lambda }$ (fm)	$D_{\alpha \alpha }$ (fm)
Be hyper isotopes
$0$	$0.67$	$-$	$2.43$	$-$	$2.8$
$2{\Lambda }_s$	$0.64$	$0.18$	$2.36$	$2.59$	$2.6$
$2{\Lambda }_p$	$0.72$	$0.71$	$2.49$	$4.27$	$3.0$
$4$	$0.68$	$0.57$	$2.42$	$3.49$	$2.8$

Table 3. Table 3: Quadrupole deformation parameters β 2 subscript 𝛽 2 \beta_{2} and root-mean-squared radii R 𝑅 R of the Ne and Si hyper isotope chains. β 2 ​ N subscript 𝛽 2 𝑁 \beta_{2N} and R N subscript 𝑅 𝑁 R_{N} are deformation parameter and radius calculated from the nucleon density, and β 2 ​ Λ subscript 𝛽 2 Λ \beta_{2\Lambda} and R Λ subscript 𝑅 Λ R_{\Lambda} are calculated from the Λ Λ \Lambda hyperon density. N Λ subscript 𝑁 Λ N_{\Lambda} is the number of Λ Λ \Lambda hyperons.

$N_{\Lambda }$	${\beta }_{2N}$	${\beta }_{2\Lambda }$	$R_N$ (fm)	$R_{\Lambda }$ (fm)
Ne hyper isotopes
$0$	$0.43$	$-$	$2.82$	$-$
$2$	$0.40$	$0.16$	$2.81$	$2.53$
$8$	$0.38$	$0.09$	$2.82$	$3.33$
Si hyper isotopes
$0$	$-0.31$	$-$	$3.14$	$-$
$2$	$-0.22$	$-0.12$	$2.95$	$2.49$
$8$	$0.00$	$0.00$	$2.92$	$3.14$

Equations

$$\mathcal{L}$$

$$B_{\Lambda \Lambda }{(}_{2\Lambda }^{A}Z)$$

$$\Delta B_{\Lambda \Lambda }{(}_{2\Lambda }^{A}Z)$$

$${\rho }_S(\mathbf{r})$$

$$j^0(\mathbf{r})$$

$$j_3^0(\mathbf{r})$$

$${\rho }_{S\Lambda }(\mathbf{r})$$

$$j_{\Lambda }^0(\mathbf{r})$$

$${\mathbf{V}}_{T\Lambda }(\mathbf{r})$$

$$(-{\nabla }^2+m_{\sigma }^2)\sigma$$

$$(-{\nabla }^2+m_{\omega }^2){\omega }^0$$

$$(-{\nabla }^2+m_{\rho }^2){\rho }_3^0$$

$$-{\nabla }^2{A}^0$$

$$\left[-i\mathbf{\alpha }\cdot \nabla +V_N+\beta (m_N+S_N)\right]{\psi }_k^{(N)}={ϵ}_k{\psi }_k^{(N)},$$

$$S_N$$

$$V_N$$

$$\left[-i\mathbf{\alpha }\cdot \nabla +V_{\Lambda }+\beta (m_{\Lambda }+S_{\Lambda })+T_{\Lambda }\right]{\psi }_k^{(\Lambda )}={ϵ}_k{\psi }_k^{(\Lambda )},$$

$$S_{\Lambda }$$

$$V_{\Lambda }$$

$$T_{\Lambda }$$

$${\rho }_n$$

$${\rho }_{\Lambda }$$

$$U_n$$

$$U_{\Lambda }$$

$${\beta }_2=\frac{4\pi }{5}\frac{\int d^{3}r{r}^2{Y}_{20}\rho }{\int d^{3}r{r}^2\rho },$$

$$R=\sqrt{\frac{\int d^{3}r{r}^2\rho }{\int d^{3}r\rho }},$$

$${\mathcal{C}}_{q\sigma }(\mathbf{r})={\left[1+{\left(\frac{{\rho }_{q\sigma }{\tau }_{q\sigma }-{\mathbf{j}}_{q\sigma }^2-\frac{1}{4}(\nabla {\rho }_{q\sigma }{)}^2}{{\rho }_{q\sigma }{\tau }_{q\sigma }^{\mathrm{TF}}}\right)}^2\right]}^{-1}.$$

$${\tau }_{q\sigma }(\mathbf{r})$$

$${\mathbf{j}}_{q\sigma }(\mathbf{r})$$

$${\overline{\mathcal{C}}}_{q\sigma }(\mathbf{r})={\mathcal{C}}_{q\sigma }(\mathbf{r})\frac{{\rho }_{q\sigma }(\mathbf{r})}{\max {\rho }_{q\sigma }(\mathbf{r})},$$

Full text

I Introduction

Hypernuclear physics studies have been made both theoretically and experimentally HT06 ; GHM16 . A primary motivation is the unified understanding of the baryon-baryon interaction based on the flavor SU(3) symmetry. The knowledge of general baryon-baryon interactions is essential for the property of neutron-star matter, in which the hyperons would emerge at high densities. Another importance of hypernuclear physics is that a hyperon could be a non-destructive probe of nuclear structure with the strong interaction since it is free from the Pauli principle of nucleons.

In any physics case, precise information on hyperon-nucleon and hyperon-hyperon interactions is essential. However, it is difficult to perform nucleon-hyperon or hyperon-hyperon scattering experiments because of the short life times of hyperons. The main strategy is to produce hyperon(s) in an ordinary nucleus via $(K^{-},\pi^{-})$, $(\pi^{+},K^{+})$, $(e,e^{\prime}K^{+})$, or $(K^{-},K^{+})$ reactions to extract information on the interactions HT06 ; GHM16 ; Fr95 ; Naka10 . Most of the experimental data are limited to single-$\Lambda$ hypernuclei, and there are also a few data for double-$\Lambda$ and $\Xi$ nuclei Fr95 ; Naka10 ; Da63 ; Pr66 ; Nagara ; Mikage ; Mino ; Yo17 . Double-$\Lambda$ hypernuclei are of special importance since it provides information of hyperon-hyperon interaction.

Besides the original motivations of hypernuclear physics, the property of nuclear system with one or more hyperons itself has been found to be interesting. In particular, so-called “impurity effects” of $\Lambda$ hyperon when a few of them are added to an ordinary nucleus have been of great interest. The impurity effects have been extensively studied with various theoretical models. The attractive $N$-$\Lambda$ and $\Lambda$-$\Lambda$ interactions and the absence of the Pauli principle lead to a variety of impurity effects in single- and double-$\Lambda$ nuclei, such as shrinkage of inter-cluster distance MBI83 ; Mo85 ; IBM85 ; HKMM99 ; HKYMR10 ; En17 , modification of deformation MH08 ; SMHS10 ; MHK11 ; Lu11 ; Is11 ; Is12 ; Is13 ; Cui17 ; Mei18 , extension of the drip line VPLR98 , emergence of new collective modes MH12 , change of fission barrier MCH09 ; MC11 , and existence of “supersymmetric” or “genuine hypernuclear” states DG76 ; MBI83 .

The multi-$\Lambda$ system with more than two $\Lambda$ particles also is intriguing. We emphasize that hypernuclei with multiple strangeness are expected to be produced in future experiments Sa10 although it is not feasible at the moment to produce it experimentally. In such systems, the $\Lambda$ hyperon is no longer a small amount of “impurity”, but it would be regarded as a third constituent of the nuclear many-body system, giving another dimension to the nuclear chart. The properties of the hyperon different from the nucleon would bring about nontrivial structures of nuclear system. As a matter of fact, multi-strangeness nuclei have been discussed in some theoretical calculations BIM ; MIB83 ; IBM85 ; Rufa90 ; Sch92 ; MZ93 ; Sch94 ; LM02 ; FSV08 ; SS09 ; KMGR15 ; MKG17 ; GBKM18 . Those include the followings: ${}^{6}_{2\Lambda}$He clusters in hypernuclei BIM ; MIB83 ; IBM85 , hyperon halo Rufa90 ; MZ93 ; Sch92 ; LM02 , competition between $\Lambda$ and $\Xi$ hyperons in finite nuclei MKG17 ; Sch94 , shell structure Sch94 , dense hypernuclei Sch94 , binding energy systematics KMGR15 , $\Lambda$-$\Lambda$ pairing correlation GBKM18 , and stability of light multi-strange nuclei FSV08 ; SS09 . So far, to the best of our knowledge, the calculations have been limited to spherical systems except for the ones in Refs. BIM , MIB83 , and SS09 , where a microscopic cluster model, a molecular orbital model, and a cluster model were applied to the Be hyper isotopes, respectively. In Ref. MIB83 , it was demonstrated that the $\Lambda$ hyperons in ${}^{12}_{4\Lambda}$Be favor a localization around the two $\alpha$ clusters.

In this paper, we focus on the clusterization and deformation properties of multi-$\Lambda$ systems. We employ a relativistic mean field (RMF) model, without any assumption on spatial symmetry and cluster structure. The RMF is one of the widely used models for hypernuclei (see Ref. HY16 for a review). We take the following $N=Z$ hyper isotope chains: ${}^{8+n}_{\ \ n\Lambda}{\rm Be}$, ${}^{20+n}_{\ \ \ n\Lambda}{\rm Ne}$, and ${}^{28+n}_{\ \ \ n\Lambda}{\rm Si}$ systems where $n=2$, $4$ for Be, and $n=2$, $8$ for Ne and Si. In particular, we make an analysis on the cluster structure using the two-body correlation as well as the one-body density distribution.

The paper is organized as follows. In Sec. II, we introduce the RMF model employed in this work. In Sec. III, we present the results of the Be hyper isotopes and discuss their structures with a special attention to the cluster structure. In Sec. IV, we discuss the deformations of the Ne and Si hyper isotopes. In Sec. V, we give the summary and perspectives.

II Model and numerical details

To describe the multi-$\Lambda$ nucleus, we use a meson-exchange model with non-linear couplings for RMF theory. The Lagrangian density is given by

[TABLE]

where $\psi_{N}$ and $\psi_{\Lambda}$ are nucleon and $\Lambda$ hyperon fields, and $G^{\mu\nu}=\partial^{\mu}\omega^{\nu}-\partial^{\nu}\omega^{\mu}$, $\vec{R}^{\mu\nu}=\partial^{\mu}\vec{\rho}^{\nu}-\partial^{\nu}\vec{\rho}^{\mu}$, and $F^{\mu\nu}=\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu}$ are the field tensors of the vector mesons $\omega$ and $\rho$, and the photon, respectively. $\vec{\tau}$ is the Pauli matrix in the isospin space. We take PK1 parameter set pk1 for the nucleon-meson couplings and a parameter set in Ref. pk1y for $\Lambda$-meson couplings. In Ref. pk1y , $g_{\sigma\Lambda}$ is fitted to the binding energy of ${}^{40}_{\Lambda}$Ca ($g_{\sigma\Lambda}=0.618g_{\sigma N}$). The naive quark model value $g_{\omega\Lambda}=(2/3)g_{\omega N}$ is taken for the Yukawa coupling between $\Lambda$ and $\omega$. As for the tensor $\omega-\Lambda$ coupling, $f_{\omega\Lambda}=-g_{\omega\Lambda}$ from a quark model CW91 is used as in Ref. MJ94 . The tensor coupling modifies the spin-orbit potential and reduces the spin-orbit splittings of the single-particle energies of $\Lambda$ hyperon MJ94 ; pk1y ; TH12 . The $\Lambda$ hyperon mass $m_{\Lambda}$ is taken to be $1115.6$ MeV. This model with the parameters thus determined reproduces the observed binding energies of light to heavy single-$\Lambda$ hypernuclei reasonably well MJ94 ; pk1y . In the present study, we do not take into account the pairing correlations among nucleons and $\Lambda$ hyperons.

Here we make a remark on the $\Lambda$-$\Lambda$ interaction. There have been a discussion that the $\sigma$-$\omega$ RMF model underestimates the observed double-$\Lambda$ binding energy $B_{\Lambda\Lambda}$ and $\Lambda$-$\Lambda$ interaction energy $\Delta B_{\Lambda\Lambda}$ Sch94 ; MLM98 . $B_{\Lambda\Lambda}$ and $\Delta B_{\Lambda\Lambda}$ are defined as follows,

[TABLE]

where $B(^{A}Z)$ is the binding energy of a nucleus ${}^{A}Z$, and $B_{\Lambda}(^{A-1}_{\ \ \ \ \Lambda}Z)=B(^{A-1}_{\ \ \ \ \Lambda}Z)-B(^{A-2}Z)$. In Refs. Sch94 and MLM98 , the discrepancies were attributed to contributions from strange $\sigma^{*}$ and $\phi$ mesons which are coupled only to $\Lambda$ hyperon. However, the discussion was based on the old data Pr66 ; Da63 ; Fr95 for $B_{\Lambda\Lambda}$ and $\Delta B_{\Lambda\Lambda}$, whereas the $B_{\Lambda\Lambda}$ and $\Delta B_{\Lambda\Lambda}$ values in newer data are considerably smaller Naka10 . In Table 1 are listed the values of $B_{\Lambda\Lambda}$ and $\Delta B_{\Lambda\Lambda}$ obtained with the present RMF model in comparison with the experimental values (see also Table I in Ref. MLM98 ). We see that the RMF results agree better with the newer data. Nevertheless, it has to be noted that the effect of beyond-mean-field correlation that could be significant in the light systems is missing in RMF although there is a partial cancellation of the correlation energies upon the subtraction in Eq. (2). Moreover, the experimental $B_{\Lambda\Lambda}$ and $\Delta B_{\Lambda\Lambda}$ values were deduced under some assumption on the formation and decay processes of the double-$\Lambda$ nuclei Naka10 . Therefore, we leave discussions on extra $\Lambda$-$\Lambda$ interaction and correlation for future works and adopt the phenomenological model in Eq. (1) fitted only to a single-$\Lambda$ nucleus. More of new data from the ongoing analyses of J-PARC E-07 experiment are awaited Mino ; Yo17 .

The model Lagrangian given above is solved within the mean-field and the no-sea approximations. We assume time-reversal invariance and charge conservation of the mean-field state, i.e., the time-odd or charged vector fields vanish. The only non-zero components are their time-like and neutral components, $\omega^{0}$, $\rho^{0}_{3}$, and $A^{0}$, where the subscript $3$ on the $\rho$ meson field means the third component in the isospace.

The scalar-isoscalar density $\rho_{S}$, vector-isoscalar and -isovector densities $j^{0}$ and $j_{3}^{0}$ of nucleon are defined in terms of the single-particle wave functions of nucleon $\psi_{k}^{(N)}$ as

[TABLE]

where $k$ runs over the occupied nucleon states. The scaler, vector, and tensor densities of $\Lambda$ hyperon are defined in terms of the single-particle wave function of $\Lambda$ hyperon $\psi_{k}^{(\Lambda)}$ as

[TABLE]

where $k$ here runs over the occupied $\Lambda$ hyperon states.

The equations of motion for the meson and electromagnetic fields read

[TABLE]

Notice that there are the contributions from the $\Lambda$ hyperon to the sources of the $\sigma$ and $\omega$ fields.

The Dirac equation for the nucleon single-particle wave function is given by

[TABLE]

where

[TABLE]

Note that $\tau_{3}=+1$ for neutron and $\tau_{3}=-1$ for proton. The Dirac equation for the single-particle wave function of $\Lambda$ hyperon is given by

[TABLE]

where

[TABLE]

The set of non-linear equations, Eqs. (4-20), are solved self-consistently. In our numerical calculations, the single-particle wave functions are represented on 3-dimensional lattice in the real space THL15 . To avoid the fermion doubling, the derivative in the Dirac equation is computed in the momentum space with the fast Fourier transform Ren17 , for which we use FFTW library fftw . The damped gradient iteration technique RC82 ; BLMR92 is used to solve the self-consistent mean-field equations (see also Ref. BSUR89 ). The Klein-Gordon equations for the mesons and the Poisson equation for the Coulomb field are solved in the momentum space with the fast Fourier transform. For solving the Poisson equation in the momentum space, we employ the same method as in Refs. Sky3D ; Poisson . Calculations are performed with $24^{3}$ mesh points and lattice spacing of $0.8$ fm.

III Results and discussion: BE hyper isotopes

First, we show the results of 8Be, ${}^{10}_{2\Lambda}$Be, ${}^{12}_{4\Lambda}$Be.

In the density plots in this paper, we will only show the neutron and $\Lambda$ densities because the neutron and proton densities are almost the same in the $N=Z$ hyper isotopes. From here on, the neutron and $\Lambda$ densities will be referred to as $\rho_{n}$ and $\rho_{\Lambda}$, respectively,

[TABLE]

In Fig. 1, the density distributions of neutron and $\Lambda$ hyperon are presented. The neutron densities $\rho_{n}$ are drawn by contour map, while the $\Lambda$ densities $\rho_{\Lambda}$ are drawn by color map. Fig. 1 (a) shows the ground state of 8Be, and Fig. 1 (b) and (c) show the ground state of ${}^{10}_{2\Lambda}$Be with the two $\Lambda$ hyperons in the $s$ orbital (denoted as ${}^{10}_{2\Lambda s}$Be) and ${}^{10}_{2\Lambda}$Be with the two $\Lambda$ hyperons in the $p$ orbital (denoted as ${}^{10}_{2\Lambda p}$Be), respectively , and Fig. 1 (d) shows the ground state of ${}^{12}_{4\Lambda}$Be. ${}^{10}_{2\Lambda p}$Be is obtained by having the $\Lambda$’s occupy the second lowest single-particle level during the self-consistent iterations. In Fig. 2 we show the central mean-field potentials of neutron and $\Lambda$,

[TABLE]

In Fig. 2, The dotted line represents 8Be, dot-dashed and double-dot-dashed lines represent ${}^{10}_{2\Lambda s}$Be and ${}^{10}_{2\Lambda p}$Be, respectively, and the solid line represents ${}^{12}_{4\Lambda}$Be.

III.1 Structures of ${}^{10}_{2\Lambda}$Be with $\Lambda$’s in $s$ or $p$ orbitals

Here we compare the structures of 8Be, ${}^{10}_{2\Lambda s}$Be, and ${}^{10}_{2\Lambda p}$Be. Note that the two $\Lambda$ hyperons in ${}^{10}_{2\Lambda p}$Be occupy the $p$ state along the symmetry axis of the system [Fig. 1 (c)]. Such a configuration does not have an analog state in the non-strange isobars, e.g., 10Be where two neutrons are added to 8Be instead of $\Lambda$ hyperons HT06 ; MBI83 ; Is11 ; Mei14 ; DG76 . Thus this state is unique in the hypernucleus and is called “supersymmetric” or “genuine hypernuclear” state MBI83 ; DG76 .

We see in Fig. 1 (a)-(d) that the nucleon distributions in the Be hyper isotopes exhibit a well-developed $\alpha$ cluster structure. In ${}^{10}_{2\Lambda s}$Be, the $\Lambda$ particles stay around the middle between the two $\alpha$’s [Fig. 1 (b)]. Accordingly, the central potential for neutron, as seen from the dot-dashed curve in Fig. 2 (a), becomes deeper at the middle due to the additional attraction by the $\Lambda$ hyperons. The nucleons are attracted toward the middle of the system, which leads to a slight reduction of the distance between the two $\alpha$ clusters. On the other hand, in ${}^{10}_{2\Lambda p}$Be in which the two $\Lambda$ hyperons are in the $p$ orbital, the potential becomes shallower and wider [double-dot-dashed curve Fig. 2 (a)], and the nucleon distribution is stretched along the $z$ axis, which is the axis of symmetry. These effects are seen in the quadrupole deformation parameter $\beta_{2}$, the root-mean-squared (rms) radius $R$ of the system, and the $\alpha$-$\alpha$ distance $D_{\alpha\alpha}$, which are summarized in Table 2. The quadrupole deformation parameter and the rms radius are defined as

[TABLE]

and

[TABLE]

respectively, where $\rho$ is the density of the particle that is considered. The $\alpha$-$\alpha$ distance $D_{\alpha\alpha}$ is defined as the distance between the two maxima of the nucleon density distribution along $z$ axis. One observes that $\beta_{2N}(^{10}_{\Lambda s}{\rm Be})<\beta_{2N}(^{8}{\rm Be})<\beta_{2N}(^{10}_{\Lambda p}{\rm Be})$, and the same relations for $R_{N}$ and $D_{\alpha\alpha}$ as well. We note that a similar result was obtained also in Ref. Is11 with an antisymmetrized molecular dynamics model for ${}^{9}_{\Lambda}$Be when a $\Lambda$ particle occupies either $s$ or $p$ orbital.

III.2 Clusterization in ${}^{12}_{4\Lambda}$Be

In ${}^{12}_{4\Lambda}$Be with two more $\Lambda$’s, the density of $\Lambda$ hyperon becomes strongly deformed [Fig. 1 (d)], and the nucleon recover nearly the same deformation, radius, and $\alpha$-$\alpha$ distance as in the 8Be normal isotope (see Table 2). However, as we see in Fig. 3, the $\Lambda$ hyperon density distribution does not have a neck at the middle as the nucleon density distribution does. An interpretation to this behavior of the $\Lambda$ hyperon density is that the $\Lambda$’s are not as tightly bound to $\alpha$ as the nucleons are, so they can move more freely between the two clusters. As a consequence, the density distribution at the mean-field level becomes flat at the middle.

We make a further analysis on the cluster structure in ${}^{12}_{4\Lambda}$Be, using the fermion localization function BE90 ; RMUO11 ; ZSN16 ; SN17 ; EKN17 ; JSSN18 . The localization function is a measure of localization, which is related to the spatial two-body correlation between two like-spin fermions of the same kind. The localization function for a particle of kind $q=n$, $p$, or $\Lambda$ with spin $\sigma=+1/2\ (\uparrow)$ or $-1/2\ (\downarrow)$ is defined as

[TABLE]

Here, $\tau^{\rm TF}_{q\sigma}=\frac{3}{5}(6\pi^{2})^{2/3}\rho_{q\sigma}^{5/3}$ is the Thomas-Fermi kinetic energy density, and

[TABLE]

and

[TABLE]

where \Sigma^{3}=\left(\begin{array}[]{cc}\sigma_{z}&0\\ 0&\sigma_{z}\end{array}\right) is the spin operator, and $k$ runs over the occupied states of the fermion kind $q$. Note that $\bm{j}_{q\sigma}=\bm{0}$, $\rho_{q\uparrow}=\rho_{q\downarrow}$, and $\tau_{q\uparrow}=\tau_{q\downarrow}$ in the present case with the time-reversal symmetry.

A value of ${\cal C}_{q\sigma}$ close to one is the sign of localization RMUO11 , which means that the probability of finding two particles with the same spin close to each other is very low. ${\cal C}_{q\sigma}\approx 1$ simultaneously for all the spin-isospin combinations is a minimal necessary condition of $\alpha$ clusterization RMUO11 . The $\alpha$ cluster correlation in terms of the localization function can be naturally extended to ${}^{6}_{2\Lambda}$He: ${\cal C}_{q\sigma}\approx 1$ at the same spacial region for $n\uparrow$, $n\downarrow$, $p\uparrow$, $p\downarrow$, $\Lambda\uparrow$, and $\Lambda\downarrow$ implies ${}^{6}_{2\Lambda}$He clusterization. In the present case with $N=Z$ and time-reversal symmetry, the wave functions of neutron and proton are approximately the same, and the spin-up and -down components are exactly the same, so it suffices to consider only the neutron spin-up and $\Lambda$ spin-up components.

Since the localization is not a meaningful quantity in the regions where the one-body density is close to zero, we look at the localization function multiplied by the normalized one-body density

[TABLE]

as was done in Ref. ZSN16 .

In Fig. 4, the neutron and $\Lambda$ hyperon localization measures, $\overline{\cal C}_{n\uparrow}$ and $\overline{\cal C}_{\Lambda\uparrow}$ defined by Eq. (30) are plotted. The neutron localization has two strong peaks at the positions of the two $\alpha$ clusters, indicating the $\alpha$-cluster correlation. More interestingly, the localization of $\Lambda$ hyperon in Fig. 4 (b) and (d) takes the values slightly more around the $\alpha$ clusters than at the region in between, which was not seen in the one-body density distribution in Fig. 3. This implies the existence of two ${}^{6}_{2\Lambda}$He clusters, although the localization of the $\Lambda$ hyperons is much more obscure than the nucleon.

We remark also that the rms radius of $\Lambda$ hyperon in ${}^{12}_{4\Lambda}$Be is significantly larger than that of nucleon ($R_{\Lambda}/R_{N}=3.49/2.42\approx 1.4$) although the number of $\Lambda$ particles in ${}^{12}_{4\Lambda}$Be is the same as those of neutrons and protons. This is because the $\Lambda$ particles are more weekly bound in a shallow potential ($\approx-30$ MeV) as one sees in Fig. 2 (b). Especially, the last occupied orbital in${}^{12}_{4\Lambda}$Be is bound only by 1.3 MeV, which makes a halo- or skin-like structure of $\Lambda$. Similar results have been obtained also in Refs. Rufa90 ; Sch92 ; LM02 for spherical multi-$\Lambda$ systems.

IV Results and discussion: Ne and Si hyper isotopes

Next, we investigate Ne and Si hyper isotopes with two and eight $\Lambda$ particles: 20Ne, ${}^{22}_{2\Lambda}$Ne, ${}^{28}_{8\Lambda}$Ne, 28Si, ${}^{30}_{2\Lambda}$Si, and ${}^{36}_{8\Lambda}$Si.

In Fig. 5 we show the density distributions of neutron and $\Lambda$ hyperon in 20Ne, ${}^{22}_{2\Lambda}$Ne, and ${}^{28}_{8\Lambda}$Ne. In panels (a)-(e) of Fig. 5, the density distributions of individual hyper isotopes on $zx$ plane are plotted, and in panels (f) and (g), the densities of different hyper isotopes along the symmetry axis are compared. Notice that the axis of symmetry is horizontal in Fig. 5. The quadrupole deformation parameters $\beta_{2}$ and radii $R$ are tabulated in Table 3.

The nucleon in the Ne hyper isotopes has prolate deformation. The density distributions of $\Lambda$ particles are also prolately deformed so that they gain binding energy from the attractive interaction with nucleons. From Table 3, we see that the nucleon quadrupole deformation $\beta_{2N}$ decreases from 20Ne to ${}^{22}_{2\Lambda}$Ne and from ${}^{22}_{2\Lambda}$Ne to ${}^{28}_{8\Lambda}$Ne. The deformation of $\Lambda$ hyperon $\beta_{2\Lambda}$ also decreases from the $2\Lambda$ to the $8\Lambda$ isotopes.

We also observe in the density distributions that the hole at the center of neutron density becomes significantly depressed when there are eight $\Lambda$ hyperons. In Fig. 6 are shown the central mean fields of neutron and $\Lambda$ hyperon in the Ne hyper isotopes. As seen from the difference between the dotted and the dot-dashed curves in Fig. 6 (a), in ${}^{22}_{2\Lambda}$Ne, the $\Lambda$ particles in the lowest $s$-shell orbital give an extra contribution to the central potential that shifts down the bottom of it. When six more $\Lambda$ particles are added (${}^{28}_{8\Lambda}$Ne), they occupy the $p$-shell orbitals whose density is vanishing at the origin [compare the solid and dot-dashed curves in Fig. 5 (g)]. Consequently, the mean-field potential shown with the solid curve in Fig. 6 (a) is deepened at regions where $z\approx\pm 2$ fm. The similar thing happens in the other directions as well. This change in the potential induces the deeper hole in the nucleon density of ${}^{28}_{8\Lambda}$Ne.

In the Si hyper isotope chain, the shape variation is more drastic. In Fig. 7, we show the density distributions of neutron and $\Lambda$ hyperon in 28Si, ${}^{30}_{2\Lambda}$Si, and ${}^{36}_{8\Lambda}$Si, and the quadrupole deformations and radii are tabulated in Table 3. 28Si is obtaletly deformed. With two $\Lambda$ hyperons, the oblate deformation is relaxed from $\beta_{2N}=-0.31$ to $-0.22$. With eight $\Lambda$ hyperons, the spherical magicity of $\Lambda$ overcomes the nucleon deformation, and the system becomes spherical. The nucleons are in $(1s_{1/2})^{4}(1p_{3/2})^{8}(1p_{1/2})^{4}(1d_{5/2})^{12}$ configuration in ${}^{32}_{8\Lambda}$Si. Similar results are obtained also in the previous works on single-$\Lambda$ hyper nuclei MH08 ; Lu11 . Since the potential energy surface of 28Si is rather flat as a function of the quadrupole deformation, the Si hyper isotopes with one, two, or eight $\Lambda$ hyperons, which resist against deformation, easily go towards a spherical shape MH08 ; Lu11 .

As in the case of the Be hyper isotopes, the radius of $\Lambda$ hyperon is larger than that of nucleon when $\Lambda$’s are in the $p$-shell which are more loosely bound than the nucleon [see Table 3 and Fig. 6 (b)].

V Summary and perspectives

The scope of hypernuclear physics is now being extended to multi-strangeness systems both in theoretical and experimental sides. In this work, we have investigated the clusterization and deformation properties of $N=Z$ multi-$\Lambda$ nuclei, ${}^{8+n}_{\ \ n\Lambda}{\rm Be}$, ${}^{20+n}_{\ \ \ n\Lambda}{\rm Ne}$, and ${}^{28+n}_{\ \ \ n\Lambda}{\rm Si}$, where $n=2$, $4$ for Be, and $n=2$, $8$ for Ne and Si.

We employed a relativistic mean-field model with meson-exchange interaction. One of the three coupling constants of the $\Lambda$-meson couplings is fitted to the binding energy of a single-$\Lambda$ hypernucleus ${}^{40}_{\Lambda}$Ca, and the remaining two are fixed by quark models.

In ${}^{10}_{2\Lambda}$Be nucleus, when the two $\Lambda$’s occupy the $s$ ($p$) orbital, the intercluster distance decreases (increases), as in the case of ${}^{9}_{\Lambda}$Be Is11 .

In ${}^{12}_{4\Lambda}$Be nucleus, a sign of two-${}^{6}_{2\Lambda}$He cluster structure is observed in the two-body correlation embedded in the localization function, being consistent with the result obtained in Ref. MIB83 . Notice that, to draw a conclusion on the cluster correlation, one needs to perform a beyond-mean-field calculation which properly takes into account the quantum dynamics of shape degrees of freedom, and to analyze the spatial and spin-isospin correlations in the many-body wave function.

When $\Lambda$ hyperons fill up the spherical major shells ($s$- and $p$-shells in the present study) but nucleons favor deformation, there is a competition between them due to the attractive interaction between the nucleons and the $\Lambda$ hyperons. In the Ne hyper isotopes ${}^{22}_{2\Lambda}$Ne and ${}^{28}_{8\Lambda}$Ne, the nucleon density distributions are slightly less deformed than in the normal isotope 20Ne, attracted by the $\Lambda$ hyperons that prefer a spherical shape. At the same time, the $\Lambda$ hyperon densities get deformed, attracted by the nucleons that prefer a deformed shape. The Si hyper isotopes are softer against the deformation and its relaxation. The deformation of the Si hyper isotopes changes more drastically as a function of the $\Lambda$ hyperon number.

It is also seen that, as $\Lambda$ hyperons in the $p$-shell orbitals are more weekly bound than the nucleons, their density spreads out more than that of nucleon in the deformed systems as well as in spherical systems Rufa90 ; MZ93 ; Sch92 ; LM02 .

There are several directions for future works to explore the generalized nuclear chart with the extra dimensions of hyperons. More systematic investigations on the ground-state structure of multi-$\Lambda$ nuclei would be interesting. For this purpose, it may be desirable to include $\Lambda$-$\Lambda$ and nucleon-$\Lambda$ pairing correlations as well as nucleon-nucleon pairing. Other interesting subjects are nuclei with many $\Sigma$ and/or $\Xi$ hyperons as well as $\Lambda$ hyperon. An extension of our model and numerical code to such systems is straightforward as long as the interaction is known. To further deepen our knowledge of the general multi-strangeness system, collective excitations of multi-strangeness nuclei should also be studied with, e.g., GCM (geberator coordinate method) or RPA (random phase approximation) calculations.

Acknowledgements.

The author thanks Kouichi Hagino for helpful discussions and comments on the manuscript.