Modifying and Probing Nuclear Structureusing Λ particle
M. Kimura, M. Isaka , H. Honma (Hokkaido Univ.)
A. Dote (KEK) and A. Ohnishi (YITP)Structure study of Λ hypernuclei
Study of light(s, p-shell) Λ hypernuclei
Knowledge of YN (YY) interactions– Interaction models, Lattice QCD– Exact solution of few-body problems– Increases of experimental information

Development in nuclear models– Study of unstable nuclei
Ex.: Antisymmetrized Molecular Dynamics (AMD)[4]– AMD describes dynamical changes of various structure– No assumption on clustering and deformation

Combination of modern YN(YY) interactions and nuclearstructure models will open new era of hypernuclear study[1] E. Hiyama, NPA 805 (2008), 190c, [2] Y. Yamamoto, et al., PTP Suppl. 117 (1994), 361.,[3] O. Hashimoto and H. Tamura, PPNP 57 (2006), 564., [4] Y. Kanada-En’yo et al., PTP 93 (1995), 115.

Unique Features of Λ-Hypernuclei•••

Y has no Pauli Blocking to N: a unique probe to study nuclear structure
Y supplies additional attraction to nuclear many body system
Trace the single Λ particle nature in hypernuclei allows to study the nuclear mean field

For example:1.

Λ will change nuclear structure by its attractive interaction with N
Modify nuclear properties by Λ
6Li(α+d)6Li(α+Λ+d)2.

By looking at the motion of the Λ, we’ll obtain information of nuclear mean field3.

We can bound an unbound system using Λ5H(unbound)

6H(bound)ΛStructure change by Λ: “Shrinkage effect”
Shrinkage effect: Λ hyperon makes nucleus compact
7[1,2]
Example:Li6Λ–– Li: α + d cluster structureΛ hyperon penetrates into the nuclear interiorΛ hyperon reduces α + d distanceB(E2) reduction(Observable)

B(E2) = 10.9±0.9 e2fm4[1] K. Tanida, et al., Phys. Rev. Lett. 86 (2001), 1982.

B(E2) = 3.6±0.5+0.5+0.4
24e fm[2]E. Hiyama, et al., Phys. Rev. C59 (1999), 2351.

Structure change by Λ: “Glue-like role”
Glue-like role: Λ hyperon stabilizes unbound state
Example:9ΛBe–

8Be is an unstable nucleus

8Be• Its g.s. lies at about 100 keV above α + α threshold(unbound)– 9ΛBe is bound with an α + α + Λ structure

9ΛBeCalc. [2]Expt. [1]2α + Λα + 5ΛHe(Lowest)[1] O Hashimoto et. al., Nucl. Phys. A 639 (1998) 93c. [2] Bando et. al., Prog. Theor. Phys. 69 (1982) 913.

Toward heavier and exotic Λ hypernuclei
Experiments at J-PARC, JLab and Mainz– Various Λ hypernuclei will/can be produced• p-sd shell Λ hypernuclei• neutron-rich Λ hypernuclei etc.

p-sd shell region

neutron-richregionFull of addictive issues !

O. Hashimoto and H. Tamura, PPNP 57 (2006),564.

Theoretical framework: HyperAMD

Hamiltonian

We extended the AMD to hypernuclei
HyperAMD (Antisymmetrized Molecular Dynamics forhypernuclei)

NN：Gogny D1Sˆˆˆˆˆ
H = TN + VNN + TΛ + VΛN ΛN：YNG interaction (Central force) [1]Wave function

 Nucleon part：Slater
determinant
Spatial part of single particle w.f. isdescribed as Gaussian packet

 Single particle w.f. of Λ hyperon:Superposition of Gaussian packets[ϕ N (r ) =]1
det ϕi (rj )A!

ϕi (r ) ∝ exp − ∑ν σ (r − Z i )σ2  χ iηiσ = x, y, zχi = α i χ↑ + βi χ↓ϕ Λ (r ) = ∑ cmϕ m (r )m
ϕ m (r ) ∝ exp − ∑ µν σ (r − z m )σ2  χ m
 σ = x, y, zχ m = am χ ↑ + bm χ ↓ Total w.f.：1
ψ (r ) = ∑ cmϕ m (rΛ ) ⊗det[ϕi (rj )]A!
m[1] Y. Yamamoto, T. Motoba, H. Himeno, K. Ikeda and S. Nagata, Prog. Theor. Phys. Suppl. 117 (1994), 361.

Theoretical framework: HyperAMDProcedure of the calculationdX i κ ∂H ±=• Imaginary time development methoddt ∂X i*• Variational parameters: X i = Z i , zi , α i , β i , ai , bi ,ν i , ci

Variational Calculationκ <0Λ hyperon

Initial w.f.
nucleons(Described byGaussian wave packets)

Energy variation

Cluster

Shell

Theoretical framework: HyperAMD
Procedure of the calculationdX i κ ∂H ±=• Imaginary time development methoddt ∂X i*• Variational parameters: X i = Z i , zi , α i , β i , ai , bi ,ν i , ci

Variational Calculation

κ <0Angular Momentum Projection

J*(Ω )R(Ω ) Φ s +Φ sK ; JM = ∫ dΩDMKGenerator Coordinate Method(GCM)•Superposition of the w.f. with different configuration
J±J±•Diagonalization of H sK ,s′K ′ andN sK ,s′K ′J±s±s′±
ˆ
H sK=
Φ;
JM
H
Φ;
JM
KK′, s ′K ′J±s±s′±=
Φ;
Φ;
N sKJ
M
J
M
KK′, s ′K ′

Ψ J ± M = ∑ g sK Φ sK ; J ± M
sKApplication of HyperAMD: 7ΛLiB(E2)243.6±0.5 +0.5+0.4 e fm

B(E2)
4.8 e2fm4

E. Hiyama, et al., PRC 74, 054312

Modifying Nuclear Structure using ΛHow Λ modifies nuclear deformation?

12C (Pos)13
Λ

C12C(Pos)⊗Λ(p)β = 0.27

β = 0.30

9

ΛBe12C(Pos)⊗Λ(s)β = 0.00

adding a Λ in p orbit
adding a Λ in s orbit

20

ΛNe

quadrupole deformation parameter β21

ΛNeBackup: Reason of deformation change
Binding Energy of Λ hyperon
Λ hyperon in s orbit is deeply bound at smaller deformation
Λ hyperon in p orbit is deeply bound at larger deformation
Binding energy of Λ

12C(Neg)⊗Λ(s)12C(Pos.)⊗Λ(s)12C(Pos)⊗Λ(p)13 CΛ

E energy (MeV)Λ binding energy
[MeV]

13 CΛ

Energy curves

12C Pos.

12C(Pos)⊗Λ(p)12C(Pos)⊗Λ(s) + 8.0MeVΛ hyperon in p orbit enhances the nuclear deformation,
while Λ hyperon in s orbit reduces itModifying Nuclear Structure using ΛParity inversion of the 11
4 Be7 ground state– The ground state of 11Be is the 1/2+,
while ordinary nuclei have a 1/2− state as the ground state

Vanishing of the magic number N=81/2+ state

1/2− state

Inversion
1/2− state
1/2+ state

Excitation spectra of 11Be
13C(Exp) 11Be(Exp)11Be(AMD)

11Be 1/2−Extra neutrons in πorbit[1](small deformation)
11Be 1/2+Extra neutrons in σorbit[1](large deformation)
 Deformation of the 1/2− state is smaller than that of the 1/2+ state
Difference in the orbits of extra neutrons
11Be has 2α clusters with 3 surrounding neutrons

Excitation spectra of 11Be
13C(Exp) 11Be(Exp)11Be(AMD)

11Be 1/2−

β=0.521
11Be 1/2+β=0.724

 Deformation of the 1/2− state is smaller than that of the 1/2+ state

Excitation spectra of 11Be
13C(Exp) 11Be(Exp)11Be(AMD)

11Be 1/2−

β=0.521
11Be 1/2+BΛBΛ

β=0.724

Reversion12 −Be
 Deformation of the 1/2Λ state is smaller than that of the 1/2+ state?
 Λ hyperon in s orbit is deeply bound at smaller deformation Λ hyperon in s orbit is weakly bound at larger deformation

Parity reversion of the 12ΛBe ground state may occur by Λ in s orbit

Parity reversion of 12ΛBeGround state of 12ΛBe

Excitation Energy (MeV)– The parity reversion of the 12ΛBe g.s. occurs by the Λ hyperon

3.0

2.0

13C7(Exp.)

11Be
7(Exp.)

11Be
7(Calc.)12 BeΛ(Improved YNG-NF[1])

1.0

0.0

[1] E. Hiyama, M. Kamimura, T. Motoba, T. Yamada and Y. Yamamoto, Prog. Theor. Phys. 97 (1997), 881.

Deformation and Λ binding energy
11Be(Calc.)

12Be
Λ(Calc.)Λ hyperon coupled to the 1/2− state is more deeply bound than that
coupled to the 1/2+ state• This is because the deformation of the 1/2− state is smaller than that of the 1/2+ stateProbing Nuclear Deformation using ΛProbing Nuclear Deformation using ΛMany nuclei manifests various quadrupole deformation(parameterized by quadrupole deformation parameters β and γ )Most of them are prolate or oblate deformed (axially symmetric)
60◦

Oblateγ = 60◦

Triaxial
γ0

Spherical

β0◦

γ ≈ 30◦

Prolateγ = 0◦

Triaxial deformed nuclei are not many and its identification is not easy.

Probing Nuclear Deformation using ΛA simple idea …
• Λ is a “distinguish particle” to N (no Pauli exclusion): a unique probe to studynuclear structure• Trace the single Λ particle nature in heavy hypernuclei allows to study thenuclear mean field• Energy of p-orbit should split into three depending on the direction of its orbital
angular momentum, if the core nucleus is triaxially deformed.

Example: p-states of 9ΛBe
In case of 9ΛBe with axially symmetric 2α clustering– Anisotropic p orbit of Λ hyperon– Axial symmetry of 2α clusteringΛ hyperon in p orbit parallel to/perpendicular to the 2α clustering generates
2 different bands [1,2][1] H. Bando, et al., PTP 66 (1981) 2118.
[2] H. Bando, et al., IJMP 21 (1990) 4021.

[3] O. Hashimoto et al., NPA 639 (1998)

Deformation of 24Mg
 24Mg is one of the candidates of triaxially deformed nuclei(axial)(axial + triaxial)

A Simple idea

Triaxial deformation

25 Mg
ΛMiddleExcitation Energy

Large overlap leads deep binding

Split into 3 states?
24Mg⊗Λ(p-orbit)

G.S.
Small overlap leads shallow binding

24Mg⊗Λ(s-orbit)

 p-orbit states split into 3 different stateObserving such 3 different states is strong evidence for triaxial deformation of 24MgResults： Single particle energy of Λ hyperon
Λ single particle energy on (β, γ) plane
25Mg (AMD, Λ in p orbit)
ΛLowest pstate

2nd lowest p state

3rd lowest p state– Single particle energy of Λ hyperon is different from each p state• This is due to the difference of overlap between Λ and nucleons

Results： Single particle energy of Λ hyperon
Λ single particle energy in axial/triaxial deformed nucleus②③①②③①②③①Highest p state

2nd lowest p state①②③Lowest p stateΛ s. p. energies are different from each other with triaxial deformation
split into 3 p-orbit states

Results： Single particle energy of Λ hyperon
Λ single particle energy in axial/triaxial deformed nucleus②③①②③①①②③①②③

Λ s. p. energies are different from each other with triaxial deformation
split into 3 p-orbit states

Results: Excitation spectra

Results: Excitation spectra–24Mg⊗Λp(lowest), 24Mg⊗Λp(2nd lowest), 24Mg⊗Λp(3rd lowest) bands will begenerated– These correspond to the direction of the p-orbit

Summary
Knowledge on YN(YY) interaction will open new era of hypernuclear study• Modifying nuclear structure using hyperons as an impurity• Investigating nuclear structure using hyperons as an probe• Bounding unbound systems using hyperons as a glue
Combination of the modern YN(YY) interactions and nuclear models• Antisymmetrized molecular dynamics + effective YN interactions
Modifiying nuclear structure using Λ as an impurity• Reduction of nuclear deformation by Λ in s-wave• Parity reversion of 12ΛBeProbing nuclear structure using Λ as an probe• Splitting of p-waves in 25ΛMg due to the triaxial deformation