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 the Gamow-peak energy, $T_G=0.3$ MeV, fundamental quantity in nuclear-astrophysics, and we construct an EFT reaction. To fix parameters appearing Lagrangian, applied to study three reactions: elastic $\alpha$-$^{12}$C scattering energies, $E1$...