$S_{E1}$ factor of radiative $\alpha$ capture on ${}^{12}\mathrm{C}$ in cluster effective field theory

The $S_{E1}$ factor of radiative $\alpha$ capture on ${}^{12}\mathrm{C}$ is studied in effective field theory up to next-to-leading order (NLO), and a modification of the counting rules for the radiative capture amplitudes is discussed. I find that only two unfixed parameters remain in the amplitudes up to NLO, and those two parameters are fitted to the experimental $S_{E1}$ data. A value of the $S_{E1}$ factor is calculated at the Gamow-peak energy as $S_{E1}=59\pm 3\mathrm{keV}\mathrm{b}$, and the result is found to be about 30% smaller than the estimates reported recently. An uncertainty of the estimate in the present work is also discussed.

Figure 1

Diagrams for dressed ${}^{16}\mathrm{O}$ propagators. A thick (thin) dashed line represents a propagator of ${}^{12}\mathrm{C}$ ($\alpha$), and a thick and thin double dashed line with and without a filled circle represent a dressed and bare ${}^{16}\mathrm{O}$ propagator, respectively. A shaded oval represents a set of diagrams consisting of all possible one-potential-photon-exchange diagrams up to infinite order and no potential-photon-exchange one.

Figure 2

Diagrams for the radiative capture process from the initial $p$-wave $\alpha -{}^{12}\mathrm{C}$ state. A wavy line denotes the outgoing photon, a thick and thin double dashed line with a filled circle in the intermediate state, whose diagrams are displayed in Fig. 1, the dressed composite ${}^{16}\mathrm{O}$ propagator for $l=1$, and a thick dashed line in the final state the ground ($0_1^{+}$) state of ${}^{16}\mathrm{O}$. See the caption of Fig. 1 as well.

Figure 3

Phase shift, ${\delta }_1$, plotted by using the fitted effective range parameters, $r_1, P_1, Q_1$ as a function of $E_{\alpha }$. The experimental phase shift data are also displayed in the figure.

Figure 4

$S_{E1}$ factor plotted by using the fitted parameters with $r_C=0.1$ fm as a function of $E$. The experimental data and our estimate of $S_{E1}$ at $E_G$ are also displayed in the figure.