Direct triple-$\alpha$ process in non-adiabatic approach

Triple-$\alpha$ reaction rates have been determined well with the sequential process via the narrow resonances. However, the direct triple-$\alpha$ process at off-resonant energies still remains in unsolved problems. In the present report, the direct triple-$\alpha$ contribution is estimated with a non-adiabatic method, and it is found to be 10$^{-15}$--10$^{-3}$ pb order in photodisintegration cross sections of $^{12}$C(2$^+_1 \rightarrow$ 0$^+$) for $0.15 < E < 0.35$ MeV. This is far below the values predicted by the recent adiabatic models. In spite of the large difference, the derived rates are found to be concordant with NACRE at the helium burning temperatures.