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WHY INTERESTING TO STUDY STRUCTURE OF LIGHT
HYPERNUCLEI?
E.HIYAMA
High Energy Accelerator Research Organization(KEK), Tsukuba 305-0801,
Japan
M. KAMIMURA
Kyushu University, Fukuoka 812-8581
T. MOTOBA
Osaka Electro-Communication University, Neyagawa 572-8530
T. YAMADA
Kanto-Gakuin University, Yokohama 236-8501
Y. YAMAMOTO
Tsuru University, Tsuru 402-8555
The structure studied of 9Λ Be and 13
Λ C related to Y N spin-orbit force are discussed.
In the microscopic α + x(= 0, n, p, d, t,3 He, α) model for 6ΛΛ He, 7ΛΛ He, 7ΛΛ Li, 8ΛΛ Li,
9 Li, 9 Be and 10 Be, we predict the binding energies and the excited-state enΛΛ
ΛΛ
ΛΛ
ergies in bound region. Also, the importance of ΛΛ − ΞN coupling in 4ΛΛ H is
discussed.
1. Introduction
Recently in hypernuclear physics, we have obtained some of epoch-making
experimental data by KEK-E373 1 , BNL-E906 16 , -E929 3 and -E930 4 .
In near future, at TJLAB, BNL, DAΦNE, J-PARC and GSI facilities, it
is planned to produce many single Λ hypernuclei and double Λ hypernuclei.
Experimental data from these facilities will continuously provide us with
many interesting and important few-body problems.
Why is it interesting and important to study structure of light hypernuclei? Why do the facilities intend to produce many hypernuclei?
1
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One of the major purpose of the strangeness nuclear physics is to understand the baryon-baryon interaction. The baryon-baryon interaction is
fundamental and important for the study of the nuclear physics. In general,
it is the most useful for determination of any interaction to perform scattering experiments. However, in hypernuclear physics, since hyperon(Y )nucleon(N) scattering experiments are diﬃcult to perform, the existing
data are very limited. Furthermore, there is no hyperon(Y )-hyperon(Y )
scattering data. The Y N and Y Y interactions so far proposed have a large
degree of ambiguity. In order to obtain useful information about Y N and
Y Y interactions, we need many Y Y and Y N scattering data. In near future, some of Y N scattering experiments will be performed at J-PARC
facility. However, it is very diﬃcult to perform Y Y scattering experiments.
Therefore, it is absolutely important to constrain ambiguity of Y N and Y Y
interactions from spectroscopic study of many kinds of single Λ and double
Λ hypernuclei.
We obtain information about the Y N and Y Y interactions by combing
theory and experiment in the following: (1) Firstly, we have candidate Y N
and Y Y interactions, which are based on the one-boson-exchange model
and the consistent quark model. (2) Secondly, we have hypernuclear γray spectroscopy experiments performed in order to provide information
about the Y N and Y Y interactions. (3)However, state-of-art structure
calculations using the models of the Y N and Y Y interactions are indispensable because the γ-ray data do not give information on the Y N and
Y Y interactions directly. This theoretical component is our calculational
contribution from few-body systems. As an example, we shall discuss γ-ray
spectroscopy experimental and theoretical structure calculations related to
the Y N spin-orbit interaction.
Secondly, the ΛΛ − ΞN coupling and structure study of 4ΛΛ H will be
discussed. It is important to obtain information on particle-conversion interactions such as ΛN −ΣN and ΛΛ−ΞN couplings in hypernuclear physics,
since the mass diﬀerence between Λ and Σ, and ΛΛ and ΞN is very small
compared with between N and ∆. Especially, ΛΛ − ΞN coupling is very
important in binding mechanism of 4ΛΛ H. Recently at BNL-E906 which was
designed to observe the production of double Λ hypernuclei, they obtained
the results which indicated the production of the double Λ hypernucleus,
4
4
ΛΛ H. However, they could not determine the binding energy of ΛΛ H. Here,
we emphasized that ΛΛ − ΞN coupling is one of the most important components to make 4ΛΛ H bound.
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2. Model and Method
In order to solve three- and four-body problems precisely, we have employed
the Gaussian Expansion Method 5 . This has been successfully applied to
the bound states of various three- and four-body systems 6,7,8 . The basis
function work excellently in describing both short-range correlations and
long-range tail behavior. In Ref. [5], we proposed inﬁnitesimally-Gaussian
basis functions, which are mathematically equivalent to the previous ones
6,7
, but the diﬀerent in expression, so that the calculation of the interaction
matrix elements can be performed much more easily. Owing to the use of
these new basis functions, applicability of our calculational method is much
extended even to four-body systems with complicated interactions.
3. Y N spin-orbit force and
9
Λ Be
and
13
Λ C
One of the characteristic phenomena in non-strange nuclear physics is that
there is a strong N N spin-orbit force which leads to magic number nuclei.
How large is the spin-orbit force in comparison with the N N spin-orbit
force? However, because the there is no Y N spin-polarized scattering data,
we have no information on the strength of Y N spin-orbit force experimentally.
Two dedicated experiments with excellent energy resolution have been
performed to obtain information about the ΛN spin-orbit interaction: One
+
9
(E930 4 ) measures γ-rays from decays of the 5/2+
1 and 3/21 states in Λ Be,
−
−
and the other (E929 3 ) measures those from the 3/21 and 1/21 states in
13
Λ C.
With the prospect of new data from high-resolution γ-ray measurements, we calculated the splittings of the spin-orbit double states in 9Λ Be
8
and 13
Λ C . For this purpose we employed all available versions of the Nijmegen OBE models 9,10 , and also took quark-model interaction properties
11
into account in the separation calculations.
To perform careful structure calculations, we employ the microscopic
three-body model (2α + Λ) for 9Λ Be and the four-body model (3α + Λ) for
13
6,7
.
Λ C. The details of the calculational framework are described elsewhere
+
+
9
The calculated energy splitting of the 5/21 -3/21 doublet in Λ Be was
−
13
predicted to be 80-200 keV and that the of 3/2−
1 -1/2 1 doublet in Λ C was
predicted to be 390 - 960 keV for the NSC97a-f, ND, and NF potentials.
On the other hand, in the case of the quark-model based spin-orbit force,
we predicted signiﬁcantly small energy splitting, namely 35-40 keV in 9Λ Be
and 150-200 MeV in 13
Λ C.
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Recently data of E930 and E929 have been reported to be 43.0 ± 5.0
3
keV in 9Λ Be 4 and 152 ± 54 ± 36 keV in 13
Λ C . We see that the predicted
energy splittings using the quark-model-based spin-orbit force can explain
both data consistently. On the other hand, predictions using any of the
Nijmegen models are larger than both data.
From the comparison between the experimental data and our theoretical
calculations, we ﬁnd that the ΛN spin-orbit force is 20-30 times smaller
than the N N spin-orbit force.
4. Double Λ hypernuclei
4.1. ΛN − ΣN coupling and
4
Λ He
In non-strangeness nuclei, there is particle conversion such as N N → N ∆.
But, since the mass diﬀerence is large, it is considered that the contribution
of ∆ particle is very small.
On the other hand, in single Λ hypernuclei, the mass diﬀerence between
Λ and Σ, which is 80 MeV, is much smaller than that between N and ∆.
For this study, so far, many authors 12,8,13,14 pointed out that the Σ mixing
is signiﬁcantly important in making 4Λ H and 4Λ He bound.
Jacobian coordinates of 4Λ He are illustrated in Fig.1 of Ref. [8].
The four-body wavefunction are written as
ΨJ M (4Λ H,4Λ He) =
4 Y =Λ,Σ
c=1 I,j l,L,λ s0 ,s,S,t0,t0 n,N,ν
×AN
(c)
AY,nlN LνλIjs 0sSt0 t
(c)
(c)
(c)
[φnl (rc )ψN L (Rc )]I ξνλ (ρc )
j
(σ)
(σ)
(σ)
(σ)
× [χ1/2 (N1 )χ1/2 (N2 )]s0 χ1/2 (N3 ) s χ1/2 (Y )
(τ)
(τ)
(τ)
(τ)
× [χ1/2 (N1 )χ1/2 (N2 )]t0 χ1/2 (N3 ) t χtY (Y )
S JM
T =1/2
, (1)
where AN are the three-nucleon antisymmetrization operator and the
isospin tY = 0 for Y = Λ and tY = 1 for Y = Σ. Here, n, N and ν denote
the size of the Gaussian basis functions for four-body systems. The eigenenergies of Hamiltonian and coeﬃcients C are determined by Rayleigh-Ritz
variational method.
As for the Y N interaction with ΛN − ΣN coupling terms, we consider
those of Nijmegen soft core ’97f, but we simulate them by using a soft-core
potential with central, spin-orbit and tensor terms 15 which reproduce the
scattering phase shifts obtained by the original interaction. In the following,
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we refer to this phase-shift equivalent soft-core potential as NSC97f. We
employ the AV8 potential as the N N interaction.
All the calculations shown below have been done both for 4Λ H and 4Λ He,
but we present only the case of the former since the results very similar to
each other. Calculated result of 4Λ He is shown in Fig.1. In the case of only
(i): (NNN) Λ
–BΛ (MeV)
(ii): (NNN)Σ
+
1
3
0
(unbound)
+
0
He+ Λ
(unbound)
–0.54
–1
–1.15
–2
–2.39
1
+
0
+
Exp.
–2.28
(i)
1
+
0
+
(i)+(ii)
4
Λ He
Figure 1.
Calculated energy levels of
4 He
Λ
the 3N +Λ channel adopted, both of the two states are unbound. When the
N N N Σ channel is included, 0+ and 1+ states become bound with respect
to 3 He + Λ threshold. Therefore, by solving the coupled four-body problem
of 4Λ He, we found that the Σ channel component plays an essential role in
the binding mechanism of the 4Λ He.
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4.2. ΛΛ − ΞN coupling and
4
ΛΛ H
In S = −2 nuclei, since the mass diﬀerence between ΛΛ and ΞN is only
28 MeV, then ΛΛ − ΞN particle conversion must be more important. For
this study, we obtained three epoch-making experimental data on double Λ
hypenuclei at KEK-E373 and BNL-E906. First one is observation of 6ΛΛ He,
which is called the NAGARA event 1 . This event has been uniquely identiﬁed on the basis of observing the sequential weak decay, and the precise
experimental value of the ΛΛ binding energy BΛΛ = 7.25 ± 0.19+0.18
−0.11MeV
10
has been obtained. Second one is the observation of ΛΛ Be, which is called
exp
, obtained is
the Demachi-Yanagi event 16,17 . The binding energy ,BΛΛ
+0.35
12.33 −0.21 MeV. It has not been, however, determined whether this event
was the ground state or the excited state. Thirdly, at BNL-E906 which
was designed to observe the production of the double Λ hypernuclei, they
obtained the results which indicated the production of the double Λ hypernucleus, 4ΛΛ H. However, they could not determine the binding energy of
4
ΛΛ H.
Analysis of KEK-E373 experiment is still in progress. Furthermore, it is
planned to produce many double Λ hyperuclei at J-PARC, GSI and BNL.
Here, the important issues are: (1) Does the Y Y interaction which is
designed to reproduce the binding energy of 6ΛΛ He make 4ΛΛ H bound? If so,
how large is binding energy? And how important is the eﬀect of ΛΛ − ΞN
coupling in making 6ΛΛ He and 4ΛΛ H bound? (2) Does the Y Y interaction
which is designed to reproduce the binding energy of 6ΛΛ He, also reproduce
the Demachi-Yanagi event consistently? (3) If this Y Y interaction is used
for the other double Λ hypernuclei, how is the theoretical prediction of the
level structure?
At ﬁrst, we discuss subject (1). Recently, Filikhin and Gal 18 calculated N N ΛΛ four-body system. The ΛΛ interaction was improved so as
to reproduce the binding energy of 6ΛΛ He which was observed recently as
NAGARA event. They concluded that the ground state of 4ΛΛ H was not
bound. Recently, Nemura and Akaishi 19 performed more precise four-body
calculation of 4ΛΛ H and they had the same conclusion as Filikhin and Gal
as long as the ΛN interaction is adjusted so as to reproduce the observed
splitting energy of 4Λ H. Here, it should be noted that they did not included
ΛΛ − ΞN coupling in this system. This mass diﬀerence is much smaller
than the mass diﬀerence between Λ and Σ. Therefore, it is expected that
the eﬀect is not so important in 6ΛΛ He. On the other hand, it is expected
that the eﬀect is not so important in 6ΛΛ He.
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The reasons is as follows: In the case of 6ΛΛ He, two protons, two neutrons
and two Λs occupy in the lowest s-shell and this s-shell is closed. When
two Λs in 6ΛΛ He are converted into ΞN by ΛΛ − ΞN coupling, a additional
nucleon is forbidden to occupy the s-shell due to Pauli blocking. Therefore,
the eﬀect of ΛΛ − ΞN coupling is small in 6ΛΛ He and p-shell double Λ
hypernuclei. This Pauli suppression eﬀect in p-shell double Λ hypernuclei
was already pointed out in Ref. [20,21,22].
On the other hand, in the s-shell double Λ hypernucleus, 4ΛΛ H, the Ξ and
the additional nucleon converted by the ΛΛ − ΞN coupling are allowed to
occupy the lowest s-shell. Then, since there is no Pauli blocking, the ΛΛ −
ΞN eﬀect can be large. This is also pointed out in Ref. [23]. For this study,
it is highly desirable to perform 4-body calculation taking N N ΛΛ and
N N N Ξ channel explicitly. In the present calculation, all the interactions
are taken to be of central type. As for the N N interaction, Minnesota
potential which reproduce the binding energies of 3 H, 3 He and 4 He was
used. As for the ΛN interaction, simulated Nijmegen model D modiﬁed so
as to reproduce the binding energies of hypertriton and 4Λ H. In the case of
Y Y interaction, ΛΛ, ΛΛ − ΞN coupling and ΞN − ΞN potentials are used.
Here, Y Y interactions are improved so as to reproduce the binding energy
of 6ΛΛ He within the framework of α + Λ + Λ and α + Ξ + N coupled threebody model. In this three-body calculation of 6ΛΛ He, Pauli blocking eﬀect
is included by the orthogonal condition model. Here four types of ΛΛ − ΞN
potentials are employed. The strength of case (1) is the smallest of four
potentials. This strength is corresponding to that of original Nijmegen
model D. The strength of case (2) is lager than case (1). This strength is
corresponding to that of Nijmegen soft core ’89. The strength of case (3)
is larger than case (2). This strength is corresponding to that of Nijmegen
model F. The strength of case (4) is the largest of four potentials. This
strength is corresponding to that of the extended soft core potential which
was recently proposed by Nijmegen group. The calculated binding energies
of 6ΛΛ He using four types of ΛΛ − ΞN coupling are well reproduced the
observed binding energy. The calculated Ξ mixing probabilities in 6ΛΛ He are
0.1 %, 0.6 %, 1.1 % and 1.6 % in the case of (1), (2), (3) and (4), respectively.
The calculated binding energies of 4ΛΛ H are as follows: The case (1) ∼ (3) do
not make 4ΛΛ H bound, while case (4) makes this hypernucleus weakly bound.
At present, we have no experimental information about the strength of ΛΛ−
ΞN coupling. If the binding energy of 4ΛΛ H is determined experimentally
in the future, we can obtain useful information on the ΛΛ − ΞN coupling
potential.
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4.3. p-shell double Λ hypernuclei
8
9
Be
Λ
Be
(MeV)
2
+
0
+
10
ΛΛBe
3.00
2.0
0.09
α+ α
5
Λ He +
α
+
–4.0
α + α+ Λ+Λ
α+ α + Λ
BΛΛ(2 ) = 12.28 (exp. 12.33)
–2.0
BΛ = 6.73 (exp. 6.71)
0.0
+
3/2
+
5/2
2.83 (3.05)
–6.0
+
1/2
–8.0
0.00 (0.00)
–6.64
cal.(exp.)
Ex
B ΛΛ (0 ) = 15.14
–10.0
+
–12.0
–14.0
–16.0
Figure 2.
Calculated energy levels of 8 Be,
9 Be
Λ
6
ΛΛ He+ α
2
+
0
+
and
2.86
–15.05
0.00
Ex
10 Be.
ΛΛ
In order to answer second issue(2), we made structure calculation of
based on an α + α + Λ + Λ four-body model.
The calculated BΛΛ for the 2+ excited state in 10
ΛΛ Be, 12.28 MeV, agree
exp
+0.35
with the experimental value BΛΛ = 12.33 −0.21MeV in the Demachi-Yanagi
event as shown in Fig.1. We therefore interpret this event as the observation
of the 2+ excited state of 10
ΛΛ Be, which is consistent with NAGARA event
10
ΛΛ Be
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(MeV)
2.0
7
ΛΛHe
6
ΛΛHe
0.0
α +Λ+Λ
α +n+ Λ + Λ
7
ΛΛLi
8
ΛΛLi
9
ΛΛLi
α +p+Λ + Λ
α+d+Λ+Λ
α+ t+ Λ+Λ
9
ΛΛBe
10
ΛΛBe
α+ 3 He+Λ+Λ
α+α+Λ+Λ
–2.0
–4.0
–6.0
6
+
0
–8.0
6
ΛΛHe+n
–7.25
ΛΛHe+p
–
3/2
–7.48
6
+
2
ΛΛHe+d
ΛBe+ Λ
8
ΛLi +Λ
–8.47
–
–10.0
5/2
–
+
–12.0
3
1.36
5.96
5/2
–
+
–14.0
1
–12.91
0.00
7/2
5.92
–
4.54
7/2
+
2
4.55
–
0.73
1/2
–
3/2
1/2
–
3/2
0.71
–16.00
0.00
–17.05
0.00
+
0
–15.05
0.00
Ex
Ex
Ex
Figure 3. Calculated energy levels of the double Λ hypernuclei,
8 Li, 9 Li, 9 Be and 10 Be.
ΛΛ
ΛΛ
ΛΛ
ΛΛ
of
2.86
Ex
–
–16.0
6
ΛΛ He +α
3
ΛΛHe+ He
8
–
3/2
6
6
ΛΛHe + t
5.63
6 He, 7 He, 7 Li,
ΛΛ
ΛΛ
ΛΛ
6
ΛΛ He.
Then since our four-body calculations are predictive, hoping to have
the data of many double Λ hypernuclei in the future experiments at JPARC, GSI, we have predicted level structure for double Λ hypernuclei in
Fig. 2 with A = 7 ∼ 10 within the framework of an α + x + Λ + Λ fourbody model, where x = n, p, d, t,3He and α. This is corresponding to the
third issue. It is our intention that these extensive four-body cluster model
calculations should contribute to stimulating comprehensive spectroscopic
studies of double Λ hypernuclei.
5. Conclusion
We have shown that there are a lot of interesting and important few-body
problems in hypernuclear physics. As examples, we reported three- and
4
7
four-body structure studies of single Λ hypernuclei of 9Λ Be, 13
Λ C, ΛΛ H, ΛΛ He,
7
8
9
9
10
ΛΛ Li, ΛΛ Li, ΛΛ Li, ΛΛ Be and ΛΛ Be. In the future, at TJLAB, BNL, DAΦNE,
J-PARC and GSI facilities, they are planning to produce many single Λ
and double Λ hypernuclei. This will provide our few-body community with
encouraging subjects continuously.
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