Faddeev three-body calculation of triple-alpha reaction

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

2. 1. INTRODUCTION Refs. [1] C. Angulo et al., NPA656 (1999) 3.[2] K. Nomoto et al., A&A149 (1985) 239. Triple-alpha reaction-Resonant process (T>108 K)8Be, 12C* Resonance formula-Non-resonant process (T<108 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

3. Astrophysical input: 3a reaction rate <aaa> [cm6/s] n12 (n4): Number density of 12C (4He) Resonant Non-resonant =T/(107 K)

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 T7=100~1026 larger at T7=1 ~106 larger at T7=10 compared to the NACRE rate “Severe inconsistency with the current understanding of the observations.” [4] OKK ~106 NACRE ~1026 [3] K. Ogata et al., PTP122 (2009) 1055. [4] T. Suda et al., arXiv:1107.4984, and references therein.

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

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

8. a a a Sasakawa-Sawada method [6]:Auxiliary Coulomb potential • (23)1 (12)3 • Distortion of the spectator particle • (Partial) cancellation of long-range a-a 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. 3. a-a and a-a-a Potentials • Shallow a-a potential (no forbidden state)[7] • a-a-a 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

10. 4. Results E2-photodisintegration cross section of 12C(2+): 3a reaction rate <aaa>

11. Photodisintegration cross section Er=0.383MeV [Exp.=0.379MeV] G=11.7eV [Exp.=8.3(1.0)eV] B(E2,0+22+1) = 9.4 e2fm4[Exp=13.3 e2fm4]

12. Normalization of is normalized with respect to the E2 transition strength, B(E2,0+22+1), (effective charge ~ 0.2).

13. aaa reaction rate ~1026 for OKK OKK This work ~0.98 NACRE

14. 5. SUMMARY • Calculations of the 3a-reaction as a quantum mechanical three-body problem • A wave function corresponding to the inverse process: 12C(2+) + g a + a + aapplying the Faddeev three-body theory with accommodating long-range Coulomb force effect, which has been successfully applied for three-nucleon systems. • Present calculations of <aaa> : ~1000 times larger than the NACRE rate at T7=1. • The result is not consistent with the CDCC calculations for T7 < 20 (Why ?) • Three-body Coulomb problem is still tough one.

16. Photodisintegration cross section

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

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