Asymptotic normalization coefficients for $\hat{I}\pm +{}^{12}\mathrm{C}$ synthesis and the $S$-factor for ${}^{12}\mathrm{C}(\hat{I}\pm ,{\hat{I}}^3){}^{16}\mathrm{O}$ radiative capture

Abstract

The ${}^{12}\mathrm{C}(\hat{I}\pm ,{\hat{I}}^3{)}^{16}$O reaction, determining the survival of carbon in red giants, is of interest for nuclear reaction theory and nuclear astrophysics. A specific feature of the ${}^{16}$O nuclear structure is the presence of two subthreshold bound states, (6.92 MeV, 2${}^{+}$) and (7.12 MeV, 1${}^{-}$), that dominate the behavior of the low-energy $S$-factor. The strength of these subthreshold states is determined by their asymptotic normalization coefficients (ANCs), which need to be known with high accuracy. Recently, using a model-independent extrapolation method, Blokhintsev {\it et al.} [Eur. Phys. J. A {\bf 59} (2023) 162] determined the ANCs for the $\hat{I}\pm$-particle removal taking into account three subthreshold states in ${}^{16}$O. The goal of this paper is to address four main problems elucidating the impact of the subthreshold ANCs on the low-energy $S$-factor. Firstly, we analyse the connection between variations of the subthreshold ANCs and the low-energy $S$-factor, in particular, at the most effective energy of $300$ keV. Secondly, we calculate contributions to the $S(300\mathrm{keV})$-factor from the subthreshold $1^{-}$ and $2^{+}$ resonances, that are controlled by the subthreshold ANCs. We also evaluate the contribution of the uncertainties of the subthreshold ANCs to the budget of the low-energy $S$-factor uncertainty, especially, the $S(300\mathrm{keV})$-factor. Thirdly, we analyse interference of the subthreshold resonances (SRs) with higher resonances and with the $E1$ and $E2$ direct captures to the ground state. Finally, we investigate a correlated effect of the subthreshold and ground-state ANCs on the low-energy $S$-factor and, in particular, on the $S(300\mathrm{keV})$-factor.