Triple-$\alpha$ reaction rates below $T_9=3$ by a non-adiabatic three-body model

TL;DR

This study calculates the triple-alpha reaction rates at low temperatures using a non-adiabatic three-body model, revealing significantly lower rates than previous models but consistent with current evaluations at certain temperature ranges.

Contribution

The paper introduces a non-adiabatic three-body model for the triple-alpha process, providing more accurate reaction rates at low temperatures and analyzing their astrophysical implications.

Findings

Reaction rates are much smaller than adiabatic models at certain energies.

Derived rates are consistent with evaluated rates for 0.08 ≤ T_9 ≤ 3.

Rates are reduced by a factor of 10^{-4} at T_9=0.05.

Abstract

The triple-$\alpha$ reaction from the ternary continuum states at off-resonant energies, $\alpha+\alpha+\alpha\rightarrow^{12}$C, remains an open question. This direct process is scrutinized using a non-adiabatic Faddeev hyperspherical harmonics $R$-matrix expansion method, and the derived reaction rates are discussed. After reviewing the model, the resultant photo-disintegration of $^{12}$C($2^+_1\rightarrow 0^+$) is shown to be much smaller than the values predicted by the adiabatic models for $0.15 \le E \le 0.35$ MeV. Despite the large difference, the derived reaction rates are illustrated to be concordant with the current evaluated rates for $0.08 \le T_9 \le 3$. The difference below $E=0.20$ MeV can be seen in the rates for $T_9 \le 0.07$. In comparison with the calculations, the rates are found to be reduced by a factor of 10$^{-4}$ at $T_9=0.05$, because of an accurate description for $^8$Be break-up. Uncertainties of the rates are also estimated by examining the sensitivity to the 3$\alpha$ potentials. By introducing three-body $S$-factors and a resonant term, the present rates are expressed in an analytic form, and they are provided in a tabular form for astrophysical applications. To update the evaluated rates, non-resonant sequential process between $\alpha+^8$Be could be removed. The astrophysical impact is not expected to be large, although the rates are reduced around $T_9=0.05$.