1. 多体共鳴状態の境界条件によって解析した3α共鳴状態の構造
C. Kurokawa1 and K. Kato2
Meme Media Laboratory, Hokkaido Univ., Japan1
Div. of Phys., Grad. Sch. of Sci., Hokkaido Univ., Japan2

2. Theoretical studies of 12C
　　D.M.Brink in Proceedings of the Fifteen Solvay Conference on Physics (19070)
○Microscopic 3α model (RGM・GCM・OCM)
　　Y.Fukushima and M.Kamimura in Proceedings of the International Conference on Nuclear Structure (1977)
　　M.Kamimura, Nucl. Phys. A351(1981),456
Y.Fujiwara, H.Horiuchi, K.Ikeda, M.Kamimura, K.Katō, Y.Suzuki and E.Uegaki, Prog Theor. Phys. Suppl.
　　 68 (1980)60.
E.Uegaki, S.Okabe, Y.Abe and H.Tanaka, Prog. Theor. Phys. 57(1977)1262; 59(1978)1031; 62(1979)1621.
H.Horiuchi, Prog. Theor. Phys. 51(1974)1266; 53(1975)447.
K.Fukatsu, K.Katō and H.Tanaka, Prog. Theor. Phys.81(1988)738.
○3α+p3./2Closed shell
N.Takigawa, A.Arima, Nucl. Phys. A168(1971)593.
N.Itagaki Ph.D thesis of Hokkaido University (1999)
Y.Kanada-En’yo, Phys. Rev. Lett. 24(1998)5291.
○Deformation　（Mean-Field）
G.Leander and S.E.Larsson, Nucl. Phys.A239(1975)93.
○Faddeev
Y.Fujiwara and R.Tamagaki Prog. Theor. Phys. 56(1976)1503.
H.Kamada and S.Oryu, Prog. Theor. Phys 76(1986)1260.　　　　　　　　 α α α
Γ=34keV
31－ α
02+ 3
Γ=8.7eV α α
01+
Excited states of cluster states?

3. Situation around Ex= 10 MeV
Energy level of 12C a
02+ :
l=0 a
L=0 a
Alpha-condensed state
0+, 2+
A.Tohsaki et al., PRL87(2001)192501
Can 3αModel reproduce both of the 22+ and the 03+ states ?
What kind of structure dose the 03+ state have ?
Why 03+ has such a large width ?
0+ : Er= MeV, G= MeV
2+ : Er= MeV, G= MeV
[Ref.]: M.Itoh et al., NPA 738(2004)268
[Ref.] E.Uegaki et al.,PTP57(1979)1262
Boundary condition for three-body resonances
Analysis of decay widths

4. Our strategy
In order to taking into account the boundary condition for three-body
resonances, we adopted the methods to 3 Model;
Complex Scaling Method (CSM)
[Ref.] J.Aguilar and J.M.Combes, Commun. Math. Phys., 22(1971),269
E.Balslev and J.M.Combes, Commun. Math. Phys., 22(1971),280
Analytic Continuation in the Coupling Constant
[Ref.] V.I.Kukulin, V.M.Krasnopol’sky, J.Phys. A10(1977),
combined with the CSM (ACCC+CSM)
[Ref.] S.Aoyama PRC68(2003),034313
Both enables us to obtain not only resonance energy but also total decay width

5. Model : 3  Orthogonality Condition Model (OCM)
folding for Nucleon-Nucleon interaction(Nuclear+Coulomb)
[Ref.]: E. W. Schmid and K. Wildermuth, Nucl. Phys. 26 (1961) 463
, -parity )
μ=0.15 fm-2
: OCM [Ref.]: S.Saito, PTP Supple. 62(1977),11
Phase shifts and Energies of 8Be, and Ground band states of 12C a1 a2 a3
c=3 a1 a2 a3
c=2 a1 a2 a3
c=1 ,
[Ref.]: M.Kamimura, Phys. Rev. A38(1988),621

6. Methods for treatment of three-body resonant states
CSM
It is sometimes difficult for CSM
　 to solve states with quite large
　 decay widths due to the limitation
　 of the scaling angle  and finite
basis states.
In order to search for the broad 0+ state, we employed …
ACCC+CSM 2θ
Exp. Broad state
Im(k) k
δ→0
: Atractive potential with < 0
Re(k)
Resonance

7. Energy levels obtained by CSM and ACCC+CSM
G= MeV
Γ=0.12 MeV
(2+)
0+ : Er= MeV, G= MeV
2+ : Er= MeV, G= MeV
[Ref.]: M.Itoh et al., NPA 738(2004)268
03+: Er=1.66 MeV, Γ=1.48 MeV
22+: Er=2.28 MeV, Γ=1.1 MeV
3α Model reproduce 22+ and 03+ in the same energy region by taking into account the correct boundary condition
ACCC+CSM
E.Uegaki et al.,PTP(1979)

8. Structures of 0+ states through Amplitudes
Wave function of 0+ states
Y (12C) Jp=0+ = al=0,L=0 j0,0 + al=2,L=2 j2,2 + al=4,L=4 j4,4
8Be
jl,L= [ 8Be (l) x L ] a l a
al,L2 : Channel Amplitudes L a
Channel Amplitudes of 01+, 02+ and 04+
E [MeV]
Rr.m.s. [fm]
a0,02
a2,22
a4,42 Er G
Re.
Im.
01+
-7.29
2.36
0.364
0.382
0.254
02+
0.76
2.4x10-3
4.29
0.29
0.775
0.033
0.149
-0.019
0.076
-0.014
04+
4.58
1.1
3.26
0.97
0.499
0.170
0.307
-0.017
0.194
-0.153

9. Feature of the broad 3rd 0+ state
Channel amplitudes as a function of 
8Be a
l=0 a
L=0 2 2
Dominated 2 a
Similar property to 02+（ Rr.m.s= 4.29 fm）
Re(Rr.m.s) (d= -140): 5.44 fm
Large component of a0,02 makes such the large width.
Wave function of 03+ shows similar properties to 02+.
03+ is considered as an excited state of Higher nodal state of 02+ ?

10. Summary of obtained 0+ states
04+
03+
L=0
but higher nodal ?
I=0
I=0
L=0
02+
r.m.s.=4.29 fm

11. Structure of the 04+ state
4th 0+ state ;
Large component of high angular momentum compared with 2nd 0+
a0,02 =0.499,　 a2,22 =0.307, a4,42 =0.194
Total decay width is sharp: Er=4.58 MeV, =1.1 MeV
3αOCM with SU(3) base : K.Kato, H.Kazama, H.Tanaka, PTP 77(1986),185.
Component of linear-chain configuration: 56%
AMD: Y.Kanada-En’yo, nutl-th/
FMD: T.Neff, H.Feldmeier, NPA 738(2004), 357.
Linear chain like structure is found α α α

12. Probability Density of 1st 0+ and 4th 0+ states (Preliminary)
r1 = r2 = r
q12
01+
r [fm]
04+
q12
q12

13. Summary and Future work
We solve states above 3αthresold energy taking into account the boundary condition for three-body resonant states.
Obtained resonance parameters of many J states reproduce experimental data well.
We obtained broad 3rd 0+ state near the 2nd 2+ state. The state has similar structure to the 2nd 0+ state. It is thus expected to be an excited state of 2nd 0+.
The 4th 0+ state has large component of high angular momentum channel, [8Be (2+) x L=2], and has a sharp decay width.
These features reflect the linear-chain like structure of 3αclusters. Members of rotational band built upon the 4th 0+ state ?
How do these states contribute to the real energy ? To investigate it we calculate the Continuum Level Density in the CSM and partial decay widths to 8Be(0+, 2+, 4+)+α in feature.
[Ref.] A.T. Kruppa and K. Arai, PLB 431(1998)237
R. Suzuki, T. Myo, and K. Kato, PTP 113 (2005) 1273

14. Probability Density of 0+ states
0２+
04+
q12

15. Contributions from resonant states to real energy
Continuum Level Density (CLD) Δ(E) [Ref.] S.Shomo, NPA 539 (1992) 17.
δl: phase shift
Discretization with a finite number N of basis functions
[Ref.] A.T. Kruppa and K. Arai, PLB 431(1998)237.
Smoothing technique is needed,
but results depend on smoothing parameter.

16. CLD in the Complex Scaling Method [Ref. ] R. Suzuki, T. Myo, and K
CLD in the Complex Scaling Method [Ref.] R. Suzuki, T. Myo, and K. Kato, PTP 113 (2005) 1273
Bound state
Resonance
Continuum
　ER, εc(θ) have complex eigenvalues in CSM
CLD in CSM:
Smoothing technique is not needed

17. Application to ３α system
CLD of 3αsystem α２ α１

18. Continuum Level Density: 0+ states
8Be(0+) +α
8Be(2+) +α
E [MeV]

19. Subtraction of contribution from 8Be+αα２
α1- α2: resonance + continuum
(α1α2)- α3: continuum α１ α３
α1- α2: continuum
(α1α2)- α3: continuum

20. Contributions from 8Be+α are subtracted‘
02+
04+
03+

21. Subtraction of contribution from 8Be+α

22. Search for broad 0+ state with
δ= －50 MeV
δ= －110 MeV
δ= －150 MeV
03+
05+
04+
04+
04+
05+
δ= －200 MeV
03+
δ= －250 MeV
04+
05+

23. Trajectories of the broad 03+ state
Complex-Energy plane
Complex-Momentum plane
Obtained resonance parameter
Present calc.
Exp. data
Er (MeV)
1.66
Γ (MeV)
1.48

24. Methods for treatment of three-body resonant states
Complex Scaling Method (CSM)
It is sometimes difficult for CSM to solve state with a quite large decay width due to the limitation of the scaling angle .
In order to search for the broad 0+ state, we employed …
Analytic Continuation in the Coupling Constant combined with the CSM (ACCC+CSM)
CSM
ACCC+CSM
Im(k) k
Im(k) k
Branch cut
Bound state
δ→0
Re(k)
Re(k) q
Anti-bound state
Resonance
Resonance