Radiative $\alpha$ capture on $^{12}$C in cluster effective field theory: short review

Study of radiative $\alpha$ capture on $^{12}$C, $^{12}$C($\alpha$,$\gamma$)$^{16}$O, in cluster effective field theory (EFT) is reviewed. A low energy EFT for $^{12}$C($\alpha$,$\gamma$)$^{16}$O at the Gamow-peak energy, $E_G=0.3$~MeV, is constructed, and the theory is first applied to the study of elastic $\alpha$-$^{12}$C scattering at low energies. The effective range parameters are fitted to the precise phase shift data of the elastic scattering and the astrophysical $S_{E1}$ factor of the $E1$ transition of $^{12}$C($\alpha$,$\gamma$)$^{16}$O at $E_G$ is estimated. For the study of the $E2$ transition of $^{12}$C($\alpha$,$\gamma$)$^{16}$O, we discuss a difficulty to determine the asymptotic normalization coefficient (ANC) of the subthreshold $2_1^+$ state of $^{16}$O from the elastic scattering data, and demonstrate the difficulty with the estimate of the astrophysical $S_{E2}$ factor of $^{12}$C($\alpha$,$\gamma$)$^{16}$O at $E_G$. We discuss the uncertainty in the estimate of the $S$ factors at $E_G$ in the present approach.