Background: The triple-$\alpha$ reaction is the key to our understanding about the nucleosynthesis and the observed abundance of ${}^{12}$C in stars. The theory of this process is well established at high temperatures but rather ambiguous in the low temperature regime where measurements are impossible.

Purpose: Develop a new three-body method, which tackles properly the scattering boundary condition for three charged particles and takes into account both the resonant and the nonresonant reaction mechanisms on the same footing, to compute the triple-$\alpha$ reaction rate at low temperatures.

Methods: We combine the $R$-matrix expansion, the $R$-matrix propagation method, and the screening technique in the hyperspherical harmonics basis.

Results: Both the $2_1{}^{+}$ bound state and the $0_2{}^{+}$ resonant state in ${}^{12}$C are well reproduced. We also study the cluster structure of these states. We calculate the triple-$\alpha$ reaction rate for $T=0.01–0.1$ GK.

Conclusions: We obtain the same rate as NACRE for temperatures above $0.07$ GK, but the new rate is largely enhanced at lower temperatures ($\approx$${10}^{12}$ at $0.02$ GK). The differences are caused by the direct capture contribution to the reaction when three $\alpha$ particles cannot reach the resonant energies.

• Received 22 September 2012

DOI:https://doi.org/10.1103/PhysRevC.87.054615