Studying the triple - $\alpha$ reaction in hyperspherical harmonic approach

The triple-$\alpha$ reaction is studied by using hyperspherical harmonic (HH) method [1]. Starting from three body model, we determine the $2^+$ state and the $0^+$ resonance as well as the quadrupole strength function B(E2). The triple-$\alpha$ reaction rate are calculated. We also carefully consider the contributions of the nonresonant continuum states to the reaction rate at low temperature ($T \le 10^8 K$). The results are compared with [2,3].\\[4pt] [1] I. J. Thompson, F. M. Nunes, B. V. Danilin, Comput.Phys.Comm. {\bf 161, 87-107 (2004).}\\[0pt] [2] K.Ogata, M.Kan, M.Kamimura, Prog. Theor. Phys. {\bf 122, 1055 (2009).}\\[0pt] [3] R. de Diego, E. Garrido, D.V. Fedorov, A.S. Jensen, {\bf arXiv:1005.5647v1.}