Theoretical study of the direct α+d 6Li +γ astrophysical capture process in a three-body model: Astrophysical S-factor, reaction rates and primordial abundance E. M. Tursunov1, S. A. Turakulov1, D. Baye2, and A. S. Kadyrov3 1.Institute of Nuclear Physics, AS, Tashkent, Uzbekistan 2. Universite Libre de Bruxelles, Brussels, Belgium 3. Curtin University, Perth, Australia 1. E.M. Tursunov, S.A. Turakulov, A.S. Kadirov, I. Bray. Phys.Rev. C94,015801(2016) 2. D. Baye, E.M. Tursunov. J. Phys. G, 45, 085102, 2018. 3. E.M. Tursunov, S.A. Turakulov, A.S. Kadyrov, I. Bray. Accepted to Phys. Rev. C 2018

OUTLOOK OUTLOOK Motivation Some remarks on the exact mass prescription in the two-body model Three-body model E1-and E2 –transition operators Wave functions Matrix elements of the isospin E1-transition operator Numerical results: Energy convergence, astrophysical S-factor, reaction rates, abundance of the 6Li element Conclusions

Motivation Primordial abundance on the base of S-factor (CS). For 6Li/7Li BBN (Standart) model yields 3 order less than astronomical observations ≈0.05!? E2 can be estimated in a 2-body model 6Li=α + d! Isospin forbidden E1 must be calculated within isospin formalism, which involves Ti=0  Tf =1 or Ti=1  Tf =0. 2-body models can not do, since Ti= Tf =0 (but people do using some not consistent trick –Exact mass prescription?). 3-body model can be applied: norm square of Tf =1 component is of order 10-3(small, but not 0).

α+d  6Li + γ d=p+n 6Li= α +p+n p   n α d 6Li

Two-body model: Exact mass prescription Exp masses are used in the effective charge N = Z nuclei: (Z1/A1-Z2/A2) is replaced by mN(Z1/M1-Z2/M2) E1 transitions would remain exactly forbidden in the d(d, γ)4He reaction, in contradiction with ab initio calculations. (ii) Using the mass expression M = AmN + (N − Z) 1/2(mn −mp) − B(A,Z)/c2, effective charges would depend on the binding energies B(A1,2,Z1,2), Binding energies per nucleon B(A,Z)/A mostly depend on the main T = 0 components of the wave functions and not on the small T = 1 components physically responsible for the non vanishing of “forbidden” E1 transitions. (iii) E1 matrix elements would be unphysically sensitive to the long Tf = 0 α+d tail of the 6Li wave function.

MODEL (1)

Jacobi coordinates p (1) α (3) n (2)

Electric transition operators E1 and E2

Initial wave functions

Final state three-body wave functions: Hyperspherical harmonics: Hyperspherical harmonics: where is Jacobi polynomial.

Matrix elements of the isospin operator

Main E1-isospin transition

Convergence of the energy of 6Li g.s. E(exp)=-3.70 MeV Model А: α –N potential of Kukulin et al. Model В: α –N potential of Kanada et al.

E1-Astrophysical S-factor in the 3-body model

Overlap integrals for the E2- S-factor Asymptotics:

E2-S-factor in the 3-body model

Astrophysical S–factor for α+d  6Li + γ process in the 3-body model: Relative contributions of E1- and E2- transitions

Astrophysical S–factor for α+d  6Li + γ process in the 3-body model <Ψ | Ψ >T=1 =5.27E-3, Model А: α –N potential of Kukulin et al. <Ψ | Ψ >T=1 =4.19E-3, Model В: α –N potential of Kanada et al.

Reaction rates for the α+d  6Li + γ direct capture process T9 is a temperature in units of 109 K.

Reaction rates for the α+d  6Li + γ direct capture process, normalized to the NACRE1999 data

Analitical approximation of the reaction rates for the α+d  6Li + γ direct capture process 1.84% (Model A) and 2.46 % (Model B)

Abundance of the 6Li element PArthENoPE public If we adopt the Planck 2015 best fit for the baryon density parameter and the neutron life time 880.3±1.1 s, for the 6Li/H abundance ratio we have an estimation (0.66 - 0.68)E-14 (MODEL A). (0.49 - 0.51)E-14 (MODEL B). 6Li/H=(0.80± 0.18)E-14 (LUNA2017) Models based on the exact mass prescription method 6Li/H=(0.90−1.8)E-14. Using this result and the estimate of the 7Li/H abundance ratio of (5.2±0.4)E-10 from literature, we get 6Li / 7Li=(1.30±0.12)E-5 which agrees with the standard estimate from the BBN

CONCLUSIONS Astrophysical S-factor and reaction rates of the α+d  6Li + γ direct capture process have been estimated in a three-body model. For the first time a contribution of “forbidden” E1-transition to the process has been calculated in a consistent model due to small isotriplet (T=1) component of the final three-body wave function of 6Li nucleus with a norm square of order 10-3. It is found that the E1 transition is dominant in the energy region up to E=100 keV. The experimental data for the S-factor and reaction rates of the direct LUNA data from 2014 and 2017 have been reproduced in the three-body model. With the help of PArthENoPE public code an estimations for the 6Li/H abundance ratio (0.66 - 0.68)E-14 have been obtained, which is consistent with the 6Li/H=(0.80± 0.18)E-14 estimation of the LUNA2017 collaboration. For the 6Li / 7Li ratio we obtained an estimation (1.30±0.12)E-5 which agrees with the standard estimate from the BBN

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