$S$ factor of ${}^{13}$C($\alpha$,$n$)${}^{16}$O at low energies in cluster effective field theory

Abstract

The ${}^{13}$C($\alpha$,$n$)${}^{16}$O reaction at low energies is studied by constructing an effective field theory. We choose a separation scale at $E=1$~MeV, where $E$ is the initial $\alpha$-${}^{13}$C energy in center-of-mass frame, just below the sharp resonant $5/2^{+}$ state of ${}^{17}$O, and include two open channels, $\alpha$-${}^{13}$C and $n$-${}^{16}$O, and resonant $1/2^{+}$, $5/2^{-}$, $3/2^{+}$ states of ${}^{17}$O in the study. Parameters of the theory are fitted to experimental data, $S$ factor of ${}^{13}$C($\alpha$,$n$)${}^{16}$O at the energies below $E=1$~MeV, including the data sets recently reported by the LUNA and JUNA collaborations, and the $S$ factor of ${}^{13}$C($\alpha$,$n$)${}^{16}$O is extrapolated to the Gamow peak energy $E_G=0.19$~MeV in the low mass AGB stars. We discuss an uncertainty in the estimate of the $S$ factor and confirm that the main part of the uncertainty emerges from the parameter fit of the near-breakup threshold $1/2^{+}$ state of ${}^{17}$O.

Comments: 28 pages, 4 figures