appearing in the Lagrangian. Thus, a transition amplitude is systematically calculated by writing down Feynman diagrams, while the coefficients appearing in the Lagrangian are fixed by experiment. For review, one may refer to Refs. [@bv-arnps02; @bh-pr06; @m-15; @hjp-17]. For last two decades, various processes essential in nuclear-astrophysics have been investigated by constructing EFTs, which are $p(n,\gamma)d$ at BBN energies [@r-npa00; @aetal-prc06] and $pp$ fusion [@kr-npa99; @bc-plb01; @aetal-plb08; @cetal-plb13], $^3$He($\alpha,\gamma)^7$Be [@hrv-16] and ${}^7$Be($p$,$\gamma$)${}^7$B [@znp-prc14; @rfhp-epja14] in the Sun.

In our previous work [@a-epja16], we have constructed an EFT of the radiative capture reaction, $^{12}$C($\alpha$,$\gamma$)$^{16}$O, obtained the counting rules for the reaction at $T_G$, and fitted some parameters of the theory to the phase shift data of the elastic scattering. (We briefly review the counting rules for the radiative capture and elastic scattering reactions in the following sections.) In the parameter fitting to the phase shift data, we have introduced resonance energies of $^{16}$O as a large scale of the theory. As suggested by Teichmann [@t-pr51], below the resonance energies, the Breit-Wigner-type parameterization for resonances can be expanded in powers of the energy, and one can obtain an expression of the amplitude in terms of the effective range expansion. We have determined 