Proton-to-Alpha branching ratio in the ${}^{12}$C+${}^{12}$C fusion reaction at astrophysical energies

Abstract

The unique resonance features in the ${}^{12}$C+${}^{12}$C fusion reaction lead to significant fluctuations in the branching ratio $R_{p/\alpha }={\sigma }_p/{\sigma }_{\alpha }$, making it difficult to determine the $R_{p/\alpha }$ at astrophysical energies. By combining Hauser--Feshbach statistical-model calculations with constraints from direct charged-particle and gamma-ray measurements, we investigate the energy dependence of the averaged $R_{p/\alpha }$ and predict its behavior within the Gamow window. Owing to the strong energy dependence of $R_{p/\alpha }$, the corresponding reaction-rate ratios, $⟨\sigma v{⟩}_p/⟨\sigma v{⟩}_{\alpha }$, during core and shell carbon burning are determined to be 0.29, 0.45, and 0.52 at $T_9=0.5$, 1.0, and 1.2, respectively, significantly lower than the widely adopted CF88 constant value of 0.79. The implications of the revised $⟨\sigma v{⟩}_p/⟨\sigma v{⟩}_{\alpha }$ ratio for stellar nucleosynthesis and white-dwarf evolution are also discussed.