  An effective field theory (EFT) for a nuclear reaction at low energies is
studied. The astrophysical $S$-factor of radiative $\alpha$ capture on $^{12}$C
at the Gamow-peak energy, $T_G=0.3$ MeV, is a fundamental quantity in
nuclear-astrophysics, and we construct an EFT for the reaction. To fix
parameters appearing in the effective Lagrangian, the EFT is applied to the
study for three reactions: elastic $\alpha$-$^{12}$C scattering at low
energies, $E1$ transition of radiative $\alpha$ capture on $^{12}$C, and
$\beta$ delayed $\alpha$ emission from $^{16}$N. We report an estimate of the
$S_{E1}$-factor of the reaction through the $E1$ transition at $T_G$ by
employing the EFT. We also discuss applications of EFTs to nuclear reactions at
low energies.
