Radiative decay of the subthreshold $1_1^{-}$ and $2_1^{+}$ states of ${}^{16}\mathrm{O}$ in cluster effective field theory

Radiative decay of the subthreshold $1_1^{-}$ and $2_1^{+}$ states of ${}^{16}\mathrm{O}$ is studied in cluster effective field theory. The wave function normalization factors for initial and final states of the radiative decay amplitudes are deduced by using the phase shift data of elastic $\alpha \text{−}{}^{12}\mathrm{C}$ scattering for $l=0,1,2$, which are related to the asymptotic normalization coefficients of $0_1^{+},$ $1_1^{-},$ $2_1^{+}$ states of ${}^{16}\mathrm{O}$ for the two-body $\alpha \text{−}{}^{12}\mathrm{C}$ channel; then only one unfixed parameter remains in each of the radiative decay amplitudes. I fit the parameters to the experimental radiative decay rates and apply the fitted parameters to the study of radiative $\alpha$ capture on ${}^{12}\mathrm{C},$ ${}^{12}\mathrm{C}(\alpha ,\gamma ){}^{16}\mathrm{O}$. The order of magnitude of astrophysical $S$ factors of $E1$ and $E2$ transitions of ${}^{12}\mathrm{C}(\alpha ,\gamma ){}^{16}\mathrm{O}$ is reproduced compared to the experimental data, and I discuss an improvement in the calculation of $S$ factors of ${}^{12}\mathrm{C}(\alpha ,\gamma ){}^{16}\mathrm{O}$.

Figure 1

Diagrams of radiative decay amplitudes for ${}^{16}\mathrm{O}^{*}\left(1_1^{-}\right)\to {}^{16}\mathrm{O}_{g.s.}+\gamma$ and ${}^{16}\mathrm{O}^{*}\left(2_1^{+}\right)\to {}^{16}\mathrm{O}_{g.s.}+\gamma$. A thick (thin) dashed line represents a propagator of ${}^{12}\mathrm{C}$ ($\alpha$), a thick and thin double dashed line with a filled circle represents the initial and final states of ${}^{16}\mathrm{O}$, and a wavy line is the outgoing photon. A shaded blob represents a set of diagrams consisting of all possible one-potential-photon-exchange diagrams up to infinite order and the no-potential-photon-exchange one, namely, the Coulomb Greens function. A filled box represents the contact interactions of $O^{*}\gamma O$ vertex functions, which renormalize the divergent terms from the loop diagrams.

Figure 2

Diagrams for the dressed ${}^{16}\mathrm{O}$ propagator. Thick and thin double dashed lines with and without a filled circle represent dressed and bare ${}^{16}\mathrm{O}$ propagators, respectively. See the caption of Fig. 1 as well.

Figure 3

Diagram of the scattering amplitude. Shaded blobs in the initial and final states represent nonperturbative Coulomb interaction, namely, the Coulomb wave functions. See the captions of Figs. 1 and 2 as well.

Figure 4

Curves of the $S_{E1}$ factor of radiative $\alpha$ capture on ${}^{12}\mathrm{C}$ are plotted as a function of $\alpha \text{−}{}^{12}\mathrm{C}$ energy $E$ in the center-of-mass frame by using the large and small values of $h_R^{\left(1\right)}$ at $r_C=0.01$ fm in Table 1. Experimental data are displayed in the figure as well.

Figure 5

Curves of $S_{E2}$ factor of radiative $\alpha$ capture on ${}^{12}\mathrm{C}$ are plotted as a function of $\alpha \text{−}{}^{12}\mathrm{C}$ energy $E$ in the center-of-mass frame by using the small and large values of $h_R^{\left(2\right)}$ at $r_C=0.01$ fm in Table 2. Experimental data are displayed in the figure as well.

Figure 6

Phase shift ${\delta }_0$ of elastic $\alpha \text{−}{}^{12}\mathrm{C}$ scattering for $l=0$ is plotted as a function of $E_{\alpha }$ by using the fitted parameters, where $E_{\alpha }$ is the $\alpha$ energy in the laboratory frame. The experimental data are displayed in the figure as well.