Precise calculation of the triple-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>α</mml:mi></mml:math>reaction rates using the transmission-free complex absorbing potential method
We study the triple-$\ensuremath{\alpha}$ reaction process at low temperatures, which is known to play an important role in stellar physics. The Schr\odinger equation for three $\ensuremath{\alpha}$ 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 $\ensuremath{\alpha}\ensuremath{-}\ensuremath{\alpha}$ potential and introduce three-body potentials to reproduce the energies of both the Hoyle state and the first ${2}^{+}$ state. The triple-$\ensuremath{\alpha}$ 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 $T\ensuremath{\approx}0.01$ GK, while we find a reasonable agreement between them at higher temperatures $T\ensuremath{\gtrsim}0.1$ GK.