Proton-to-Alpha branching ratio in the $^{12}$C+$^{12}$C fusion reaction at astrophysical energies
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, $\langle \sigma v \rangle_p / \langle \sigma v \rangle_\alpha$, 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 $\langle \sigma v \rangle_p / \langle \sigma v \rangle_\alpha$ ratio for stellar nucleosynthesis and white-dwarf evolution are also discussed.