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

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?

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=2.7+0.3 MeV, G= 2.7+0.3 MeV 2+ : Er=2.6+0.3 MeV, G= 1.0+0.3 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

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

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

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

Energy levels obtained by CSM and ACCC+CSM G= 0.375+0.040 MeV Γ=0.12 MeV (2+) 0+ : Er=2.7+0.3 MeV, G= 2.7+0.3 MeV 2+ : Er=2.6+0.3 MeV, G= 1.0+0.3 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)

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

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 02+. Higher nodal state of 02+ ?

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

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/0605047. FMD: T.Neff, H.Feldmeier, NPA 738(2004), 357. Linear chain like structure is found α α α

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

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

Probability Density of 0+ states 0２+ 04+ q12

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.

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

Application to ３α system CLD of 3αsystem α２ α１

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

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

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

Subtraction of contribution from 8Be+α

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+

Trajectories of the broad 03+ state Complex-Energy plane Complex-Momentum plane Obtained resonance parameter Present calc. Exp. data Er (MeV) 1.66 2.73 + 0.3 Γ (MeV) 1.48 2.7 + 0.3

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