elastic scattering amplitudes in the conventional renormalization method are summarized.

[**2. EFT for the radiative capture and elastic scattering at low energies**]{}

In the study of the radiative capture process, $^{12}$C($\alpha$,$\gamma$)$^{16}$O, at $T_G=0.3$ MeV employing an EFT, at such a low energy, we regard the ground states of $\alpha$ and $^{12}$C as point-like particles whereas the first excited state energies of $\alpha$ and $^{12}$C are chosen as irrelevant degrees of freedom, by which a large scale of the theory is determined. The effective Lagrangian for the process is constructed in terms of two spinless scalar fields for $\alpha$ and $^{12}$C, and the terms of the Lagrangian are expanded in terms of the number of derivatives. An expression of the effective Lagrangian has been obtained in Eq. (1) in Ref. [@a-epja16]. The expansion parameter of the theory is $Q/\Lambda_H \sim 1/3$ where $Q$ denotes a typical momentum scale $Q\sim k_G$: $k_G$ is the Gamow peak momentum, $k_G = \sqrt{2\mu T_G}\simeq 41$ MeV, where $\mu$ is the reduced mass of $\alpha$ and $^{12}$C. $\Lambda_H$ denotes a large momentum scale $\Lambda_H\simeq \sqrt{2\mu_4 T_{(4)}}$ or $\sqrt{2\mu_{12} T_{(12)}}\sim 150$ MeV where $\mu_4$ is the reduced mass of one and three-nucleon system and $\mu_{12}$ is that 