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

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 $T_9=0.05$.