S-matrix calculation of the triple-alpha reaction

Abstract

An S-matrix formalism is developed which can be applied to reactions in which electron screening is important, including three-body reactions and reactions involving weak interactions. The various regimes of the triple-alpha reactions are systematically discussed, and a new nonresonant regime at high densities is identified. Using the S-matrix formalism, an analytic expression is obtained for the screened triple-alpha reactions which is accurate for all temperatures and densities. The results are compared with those of Cameron (1959) and Nomoto et al. (1985), and the latter's expression for the unscreened reaction rate is verified. However, it is shown that the reaction rate in the pycnonuclear regime cannot be obtained from the unscreened rate using a screening factor, and that the results of Nomoto et al. therefore cannot be used in this regime.