1. Faddeev three-body calculation of triple- alpha reaction Souichi Ishikawa Hosei University, Japan 1 The Fifth Asia-Pacific Conference on Few-Body Problems in Physics 2011 (APFB2011) 22~26 August 2011, Seoul, Republic of Korea

2. Triple-alpha reaction -Resonant process (T>10 8 K) 8 Be, 12 C* Resonance formula -Non-resonant process (T<10 8 K) Nuclear Astrophysics Compilation of Reaction Rates (NACRE) [1] based on Nomoto et al. [2]: Extension of the resonance formula with energy dependent width at low energies 1. INTRODUCTION 2 Refs. [1] C. Angulo et al., NPA656 (1999) 3. [2] K. Nomoto et al., A&A149 (1985) 239.

3. 3 Resonant Non-resonant =T/(10 7 K) Astrophysical input: 3  reaction rate [cm 6 /s] n 12 (n 4 ): Number density of 12 C ( 4 He)

4. (2) Ogata et al. (OKK rate) [3] Quantum 3-body calculations by the method of Continuum- Discretized Coupled-Channel (CDCC): *Normalized to the NACRE rate at T 7 =100 ~10 26 larger at T 7 =1 ~10 6 larger at T 7 =10 compared to the NACRE rate  “Severe inconsistency with the current understanding of the observations.” [4] [3] K. Ogata et al., PTP122 (2009) 1055. [4] T. Suda et al., arXiv:1107.4984, and references therein. NACRE OKK ~10 26 ~10 6

5. In the present talk: (3) Faddeev method, which was successfully applied to three-nucleon scattering systems in a sufficient accuracy with Coulomb force [5]. CONTENTS (1. Introduction) 2. Formalism 3.  -  and  -  -  potentials 4. Results 5. Summary [5] S. Ishikawa, PRC80, 054002 (2009); MPL A 24, 855 (2009) (APFB 2008); Proc. of INPC 2011 (to be published).

6. 2. Formalism Consider 12 C as an  -  -  system. The inverse process: (E2-)photodisintegration of 12 C(2 + ). 12 C(2 + ) +  (E2)   +  +  (L=0) Define a wave function for the disintegration process and apply the Faddeev 3-body formalism to calculate it. y x

7. Faddeev eq.: Multiple scattering with rearrangements 7 1 2 3 1 2 3 1 2 3 Problem in the presence of long- range Coulomb forces (1)Rearrangement at long distance  Severe singularity in kernel (2)Spectator particle should be distorted by Coulomb force Channel-1 Channel-3 Channel-2

8.    Sasakawa-Sawada method [6]: Auxiliary Coulomb potential (23)1  (12)3  Distortion of the spectator particle  (Partial) cancellation of long-range  -  Coulomb force The cancellation is not perfect for breakup channels.  treat this problem approximately by a (mandatory) cutoff procedure. [6] T. Sasakawa and T. Sawada, PRC20 (1979) 1954.

9. Shallow  -  potential (no forbidden state)[7]  -  -  Potential to reproduce the resonance energy (continuum 0 + state) and the binding energy (2 + state). [7] D.V. Fedorov and A. S. Jensen, PLB 389 (1996) 631 3.  -  and  -  -  Potentials

10. 1.E2-photodisintegration cross section of 12 C(2 + ): 2.3  reaction rate 4. Results 10

11. Photodisintegration cross section E r =0.383MeV [Exp.=0.379MeV]  =11.7eV [Exp.=8.3(1.0)eV] B(E2,0 + 2  2 + 1 ) = 9.4 e 2 fm 4 [Exp=13.3 e 2 fm 4 ]

12. is normalized with respect to the E2 transition strength, B(E2,0 + 2  2 + 1 ), (effective charge ~ 0.2). Normalization of 12 W 3 (MeV) E r (keV)  (eV) B(0 +  2 + ) (e 2 fm 4 ) -168383.211.79.4 Exp.379.88.3(1.0) 13.3(1.3)

13.  reaction rate OKK This work NACRE ~10 26 for OKK ~0.98

14. 5. SUMMARY Calculations of the 3  -reaction as a quantum mechanical three-body problem A wave function corresponding to the inverse process: 12 C(2 + ) +    +  +  applying the Faddeev three-body theory with accommodating long- range Coulomb force effect, which has been successfully applied for three-nucleon systems. Present calculations of : ~1000 times larger than the NACRE rate at T 7 =1. The result is not consistent with the CDCC calculations for T 7 < 20 (Why ?) Three-body Coulomb problem is still tough one.

15. 15

16. Photodisintegration cross section

17. OKK’s insist: Due to a reduction of Coulomb barrier of  -  subsystem between the incoming  particle for non-resonant  -  system. Enhancement at low temperature 17

18. Cancellation of the long-range character in  -  Coulomb force