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

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• Effective field theory
• Energy levels
• H & He induced nuclear reactions
• Hydrostatic stellar nucleosynthesis
• Nuclear astrophysics
• Nuclear reactors
• Resonance reactions
• S-factor
• 6 ≤ A ≤ 19
• Cluster models
• Models based on symmetries