We study the triple-$\alpha$ reaction process at low temperatures, which is known to play an important role in stellar physics. The Schrödinger equation for three $\alpha$ particles is solved by using hyperspherical coordinates, while a complex absorbing potential is introduced in order to describe correctly the three-body continuum states. We use an angular-momentum-independent $\alpha -\alpha$ potential and introduce three-body potentials to reproduce the energies of both the Hoyle state and the first $2^{+}$ state. The triple-$\alpha$ reaction rate is computed accurately at temperatures from $T=0.01$ to 10 GK and compared with those available in the literature. Our reaction rate is found to be up to three orders of magnitude larger than the NACRE rate at low temperatures $T\approx 0.01$ GK, while we find a reasonable agreement between them at higher temperatures $T\gtrsim 0.1$ GK.

• Received 17 August 2016

DOI:https://doi.org/10.1103/PhysRevC.94.054607