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

The triple-$\alpha$ reaction from ternary continuum states at off-resonant energies, $\alpha+\alpha+\alpha \rightarrow ^{12}$C, still remains in unsolved problems. This direct process is estimated with a non-adiabatic Faddeev hyper-spherical harmonics and $R$-matrix expansion method. After reviewing this model, the resultant photo-disintegration of $^{12}$C($2^+_1 \rightarrow 0^+$) is shown to be in 10$^{-15}$--10$^{-3}$ pico-barn order for $0.15 \le E \le 0.35$ MeV. This is far below the values predicted by the recent adiabatic calculations. In spite of the large difference, the derived reaction rates are illustrated to be concordant with the current evaluated rates for $0.08 \le T_9 \le 3$ including helium burning temperatures. 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 triple-$\alpha$ reaction rates are found to be reduced by about 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 sensitivity to 3$\alpha$ potentials. With 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 may be eliminated by hand.