Few-body aspects of strangeness nuclear physics

E. Hiyama (Nara Women’s Univ.)

α

Λn

p Li7Λ

Submitted to Phys. Rev. C

One of the major purpose of hypernuclear physics is

to understand the baryon-baryon interaction in unified

way.

Since the hyperon-hyperon scattering data is extremely

limited, then hyperon(Y)-nucleon(N) interaction so far

proposed have a large degree of ambiguity.

Therefore, it is very important to obtain information about

YN interaction from the spectroscopy of many single

Λ hypernuclei.

For this purpose, so far several high-resolution γ-ray spectroscopy experiment such as 9Be, 13C and 7Li have beenperformed at KEK and BNL.

Λ Λ Λ

These experiments have been done for the study of YN spin-orbit force and spin-spin force.

α

Λn

p

Li7Λ

α α

Λ

α α

Λ α

Be9Λ

C13Λ

Spin-orbit force

α α

Λ

α α

Λ α

Be9Λ

C13Λ

BNL-E930 BNL-E929

0+

2+

8Be

Λ

1/2+

5/2+

3/2+

ΔESpin-orbitsplitting

γ

γ

9BeΛ

BNL-E930

12C13CΛ

γ

γ

1/2+

3/2-

1/2-

ΔEΛ

0+

BNL-E929

E. Hiyama et al.Phys. Rev. Lett. 85, 270 (2000)

E. Hiyama et al.

Phys. Rev. Lett. 85, 270 (2000)

8Be = α α

12C = α

α α

9Be=Λ

13C=Λ

α α

Λ

α

α α

Λ

YN spin-orbit force

・ Nijmegen model

・ quark-based model (made by Fujiwara et al.)

Since explicit form of the quark-based LS and ALS

interaction was not available, we tried to use the form

of Nijmegen model for the LS force and ALS parts.

So, we tried to enlarge the strength of ALS part to be

85% of the SLS with the opposite sigh.

ΛN spin orbit force and 9Be and 13CΛ Λ

5/2+

3/2+ 80

200keV

～5/2+

3/2+35

40keV

MesonQuark

9BeΛ

～

3/2-

1/2- 360

960keV

～

150

200keV

MesonQuark

13CΛ

～1/2-

3/2-

5/2+

3/2+

31.4Exp.+2.5-3.6 keV

BNL-E930

H. Akikawa et al.Phys. Rev. Lett. 88,(2002)82501.

152Exp.

1/2-

3/2-

BNL-E929

± 54 ± 36 keVS.Ajimura et al.Phys. Rev. Lett. 86,(2001) 4255

Therefore, by comparing our theoretical calculation and

the experiments of γ-ray spectroscopy, we can

understand that the desirable strength of YN spin-orbit

force should be very small.

For the study of spin-spin force

α

Λn

p

Li7Λ

1+

3+

1/2+

3/2+

5/2+

7/2+

α+n+p

α

6Li

Λ

n p

σΛ ・

σN

7LiΛ

ΛN spin-spin splittingenergy

done by KEK-E419

0.69 MeV

done by BNL-E930

0.47 MeV

Tamura et al.

By comparing with these high-resolution γ-ray

experimental data and shell model calculation with

the restricted configuration of (0s)4(0p)n0sΛ by

Millner, we succeeded in obtaining useful information

about ΛN spin-dependent force partially.

M. Ukai et al., Phys. Rev. C 73, 012501(R) (2006)

D.J. Millener, Nucl. Phys. A754, 48c (2005)

We have 2 important issues:

(1) Can we explain consistently two spin-doublets of

3/2+-1/2+ and 7/2+-5/2+ states using ΛN spin-orbit force

and spin-spin force based on the experimental data for 9Be, 7Li and 4H?

(2) How is the level structure of the other A=7

hypernuclei, namely, 7He, 7Li (T=1) and 7Be using the

above using ΛN interaction?

Λ Λ Λ

Λ Λ Λ

My contribution

To understand the hypernuclear structure by performing our these four-body calculations and

To use this structure information to understand the

ΛN spin-spin force and spin-orbit force.

Gaussian Expansion Method Developed by Kyushu Univ. group Kamimura(1) 3-cluster structure of light nuclei(2) Coulomb 3-body muonic molecular ions appearing in the muon-catalyzed fusion cycles (1987 ～ )(3) 3-nucleon bound states with realistic NN and 3N forc

es (1988)(4)Metastable antiprotonic helium atom (He++p+e)(1995 ～ )

E. Hiyama, M. Kamimura and Y. Kino,Prog. Part. Nucl. Phys. 51 (2003), 223.

Applied to

Now, I have been applying our method to hypernuclear structure.

(1) Can we explain consistently two spin-doublets of3/2+-1/2+ and 7/2+-5/2+ states using ΛN spin-orbit forceand spin-spin force based on the experimental data for 9Be, 7Li and 4H?

(2) How is the level structure of the other A=7 hypernuclei, namely, 7He, 7Li (T=1) and 7Be using theabove using ΛN interaction?

Λ Λ Λ

Λ Λ Λ

ΨJM( 　 7Li)=∑ΦJM(rc,Rc,ρC)Λ C=1

9

(spatial)=φnl(c)(rc)ψνλ

(c)(ρc)χNL(c)(Rc)

φnlm(c)=rle-(r/r ) Ylm(rc), rn=r1an-1(n=1 ～ nmax)n

Ψνλμ(ρc)=ρλe-(ρ/ρ 　 ) Yλμ(ρc) 　 , ρμ=ρ1αμ-1

(μ=1 ～ μmax)

^

^μ

2

2

χNL(c)(Rc)=RLe-(R/R )YLM(Rc), RN=R1AN-1(N=1 ～ Nmax)

Geometric progression

(H-E)ΨJM=0The Schödinger equation is solved with Rayleigh-Ritz variationalmethod.For the angular-momentum component of the wavefunction, theapproximation with l,L,λ≤2 was found to be sufficient to obtainin getting satisfactory convergence of the binding energies.But, no truncation of the interaction is made in the angular-momentum space.

α

Λ

np

Li7Λ

α-N 　 interaction: potential which reproducereasonably well the low-lying states andlow-energy scattering phase shifts of theαN systems

α-Λ interaction: Nijgemen soft core ’97fYNGfolded into the density of the α cluster

α

Λ

np

Li7Λ

ΛN interaction: Nijmegen ’97f

Not original one but simulated one

The ΛN-ΣN coupling interaction can berenomalized into the ΛN-ΛN interaction effectively.

VΛN=V0+σΛ・ σNVs+(σΛ+σN)/2・ VSLS+(σΛ-σN)/2・ VALS

Made by Yamamoto so as to reproduce thephase shifts given by the original one

N

N

N

Λ

3H+Λ0MeV

-2.43 MeV

-2.05 MeV

0+

1+

0+

1+

-2.00

-1.04

Exp.

NSC97f4HΛ

V0+σΛ ・ σNVs

Adjusted so as to reproduce the observed data of 4HΛ

0+

2+

8Be

Λ

1/2+

5/2+

3/2+

γ

γ

9BeΛ

BNL-E930

31.4+2.5-3.6 keV

(σΛ+σN)/2・ VSLS+(σΛ-σN)/2・ VALS

Adjusted so as to reproduce thedata of 9Be

Λ

12C13CΛ

γ

γ

1/2+

3/2-

1/2-

Λ

0+

BNL-E929 152Exp. ± 54± 36 keV

α

α α

Λ13CΛ

Calculated energy splitting0.2 MeV → consistent with the data within the error

Here, in the study of A=7 hypernuclei based on the α+Λ+N+N4-body model, before 4-body calculation, it is absolutely necessary to examine whether the model with the interactionadopted is able to reproduce reasonably well the following observed quantities:

(i) Energies of the low-lying states and scattering phase shiftsof the α+N, NN and αNN nuclear systems

(ii) BΛ of hypernuclei composed of α+Λ, α+N+Λ

In our model, the observed low-energy properties of the α+NNuclei and the existing Λ-binding energies of the α+Λ andα+Λ+N hypernuclei have been reproduced accurately.

α

Λ

np

Li7Λ

α

Λ

np

α

Λ

np

α

Λn

p

α

Λnp

This encourages us to perform the 4-body calculation

with NO adjustable parameter at this stage, expecting

high reliability of the results.

α+Λ+n+p threshold0 MeV

α

Λn

p Li7Λ

3H+Λ0MeV

1+

-2.00

-1.04

Exp.Cal.

0+

0+

1+

-2.00

-1.04

N

N

N

Λ

4HΛ

similar

~~

1/2+

3/2+

5/2+

7/2+

α+Λ+n+p threshold0 MeV

~~

Exp

. (1

/2+)9

.28

MeV

overbound by 0.5MeV

Agree with thebinding energy of of the ground state of 1/2+ Due to the repulsive nature of NSC97f

The odd-state interactionof the other Nijmegen modelare attractive.

-9.38

-8.41

0.97

-6.23

0.97

-7.13

0.86

The important role of the repulsive odd-state interactiondoes not necessarily mean that the odd-state part in NSC97fis more realistic than the other interactions.

The detailed reason is discussed later.

α+Λ+n+p threshold0 MeV

~~

-9.38

-8.41

0.97

-6.23

0.97

-7.13

0.86

Exp.0.69 MeV

Splitting energy is larger than the experimental data

The odd-state interaction is adjusted so as to reproduce the observed splittingenergy.

α+Λ+n+p threshold0 MeV

~~

0.54

1/2+

3/2+

5/2+

7/2+

Now, we come to the important stage of looking at the role ofthe SLS and ALS interactions for splitting energies.It should be noted here that SLS and ALS interactions workdifferently for two doublets states in 7Li.

Λ

The spin-orbit contribution to the ground-state doublets(1/2+-3/2+) very smallThe spin-orbit contribution to the excited-state doublets(5/2+-7/2+) large

α

Λ

n p

Li7Λ

L=0α

Λ

n p

L=2

1/2+

3/2+

5/2+

7/2+

α+Λ+n+p threshold0 MeV

~~

α

Λ

n p

Li7Λ

S=3/20.54

α

Λ

n p

S=3/2

α

Λ

n p

S=1/2

α+Λ+n+p threshold0 MeV

~~

0.54

0.54

In this way, owing to thecombined effect of SLSand ALS, our final resultreproduces nicely theobserved energies of thespin-doublet states of7Li.

α+Λ+n+p threshold0 MeV

~~

Λ

It is interesting to see the level structure of the other A=7hypernuclei such as 7He, 7Li (T=1) and 7Be. Λ Λ Λ

α

Λn n

He7Λ

α

Λn

p

Li7Λ

α

Λ p

Be7Λ

p

5HeΛ

n n

7HeΛ

5HeΛ

n p

7LiΛ

5HeΛ

p p

7 BeΛ

α Λ = 5HeΛ

E. Hiyama, M. Kamimura, T. Motoba, T. Yamada and Y. Yamamoto, Phys. Rev. C53 (1996), 2075

E. Hiyama, M. Kamimura, T. Motoba, T. Yamada and Y. Yamamoto, Phys. Rev. C53 (1996), 2075

How is the level structure based on α+Λ+N+N 4-bodymodel using ΛN interaction which was applied to thelevel structure of 7Li(T=0) 　 ?

Λ

Cal.BΛ=5.56MeVCharge symmetry breaking effect

In this way, here in this talk, I discussed about the

level structure of A=7 hypernuclei.

Comment on the role of ΛN-ΣN coupling

Our basis assumption in this work:

The ΛN-ΣN coupling interaction can be renomalized

into the ΛN-ΛN 　 interaction.

In this sprit, the even-state part of our ΛN interaction

were adjusted so as to reproduce the 0+ and 1+ state of 4H. Λ

However, the role of the ΛN-ΣN coupling may be important for 4H and 7Li.

Λ Λ

α+Λ+n+p threshold0 MeV

~~

0.54

1/2+

3/2+

5/2+

7/2+

Exp

. (1

/2+)9

.28

MeV The repulsive odd-state

interaction such as NSC97freproduce the observed binding energyof the ground state of 7Li.

Λ

But, it might be reasonable to consider that the ΛN-ΣN couplingworks more repulsively in 7Li.Λ

At the present, it is likely that the role of the odd-state repulsionin our treatment is a substitute for this effect.

・ Y. Akaishi et al. Phys. Rev. Lett. 84, 3539 (2000)・ B. F. Gibson et al. Phys. Rev. C6, 741 (1972)

N1 Λ N2 N3

Σ

N1 Λ N2 N3

The extra contribution to the 0+-1+ splitting of 4H fromthe 3-body correlated ΛN-ΣN mixing.

Λ

E. Hiyama et al., Phys. Rev. C65, 011301(R) (2001)

Obtained the value of 0.3 MeV for the three-body contribution of ΛN-ΣNcoupling in 4H

Λ

In the shell model calculation, Millner calculated the spin-doublet states in 7Liincluding ΛN-ΣN coupling and he concluded that this contribution was smallin these splitting.

Λ

However, it is an open problem to study ΛN-ΣN coupling effectconsistently for 4H and 7Li.

Λ Λ

α

Λ

n p

α

Σ

n p

+

Future my work