Fixing effective range parameters in elastic $\alpha$-$^{12}$C scattering: an impact on resonant $2_4^+$ state of $^{16}$O and $S_{E2}$ factor of $^{12}$C($\alpha$,$\gamma$)$^{16}$O

Abstract

Elastic $\alpha$-$^{12}$C scattering for $l=2$ and $E2$ transition of radiative $\alpha$ capture on $^{12}$C, $^{12}$C($\alpha$,$\gamma$)$^{16}$O, are studied in cluster effective field theory. Due to the problem in fixing the asymptotic normalization coefficient (ANC) of the subthreshold $2_1^+$ state of $^{16}$O, equivalently, the effective range parameters of the $2_1^+$ state, from the elastic scattering data, we introduce the conditions that lead to a large value of the ANC. In addition, $d$-wave phase shift data of the elastic scattering up to the $\alpha$ energy, $E_\alpha=10$~MeV, which contain resonant $2_4^+$ state of $^{16}$O, are also introduced in the study. Applying the conditions, the parameters of the $S$ matrix of the elastic scattering for $l=2$ are fitted to the phase shift data, and the fitted parameters are employed in the calculation of astrophysical $S_{E2}$ factor of $^{12}$C($\alpha$,$\gamma$)$^{16}$O; we extrapolate the $S_{E2}$ factor to the Gamow-peak energy, $E_G=0.3$~MeV. We find that the conditions lead to significant effects in the observables of the $2_4^+$ state of $^{16}$O and the estimate of the $S_{E2}$ factor at $E_G$ and confirm that the ANC of the $2_1^+$ of $^{16}$O cannot be determined by the phase shift data for $l=2$.