$S$ 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.