Elastic $\alpha \text{−}{}^{12}\mathrm{C}$ scattering at low energies with the bound states of ${}^{16}\mathrm{O}$ in effective field theory

The elastic $\alpha \text{−}{}^{12}\mathrm{C}$ scattering for $l=0,1,2,3$ channels at low energies is studied, including the energies of excited bound states of ${}^{16}\mathrm{O}$, in effective field theory. A new renormalization method is introduced due to the large suppression factor produced by the Coulomb interaction when the effective range parameters are fitted to the phase sift data. After fitting the parameters, the asymptotic normalization constants of the $0_2^{+},1_1^{-},2_1^{+},3_1^{-}$ states of ${}^{16}\mathrm{O}$ are calculated. The uncertainties when the amplitudes are interpolated to the stellar energy region of the ${}^{12}\mathrm{C}(\alpha ,\gamma ){}^{16}\mathrm{O}$ reaction are also discussed.

Figure 1

Diagrams for dressed ${}^{16}\mathrm{O}$ propagator. 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 blob represent a dressed and bare ${}^{16}\mathrm{O}$ propagator, respectively. A shaded blob 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

Diagram of the scattering amplitude. See the caption of Fig. 1 as well.

Figure 3

Phase shift, ${\delta }_0$, (a) and the real part of denominator, $Re{D}_0\left(k\right)$, of the amplitude (b) as functions of $T_{\alpha }$ (where $k=\sqrt{1.5\mu T_{\alpha }}$). Curves are plotted by using the effective range parameters, fitted from the $S0,S1,S2$ data sets, presented in Table 1. Experiment phase-shift data are also included in the figure. A filled box in (b) represents the excited binding energy of the $0_2^{+}$ state.

Figure 4

Phase shift, ${\delta }_1$, (a) and the real part of denominator, $Re{D}_1\left(k\right)$, of the amplitude (b) as functions of $T_{\alpha }$. Curves are plotted by using the effective range parameters, fitted from the $P0,P1,P2$ data sets, presented in Table 2. Experiment phase-shift data are also included in the figure. The filled box in (b) represents the binding energy of the $1_1^{-}$ state.

Figure 5

Phase shift, ${\delta }_2$, (a) and the real part of denominator, $Re{D}_2\left(k\right)$, of the amplitude (b) as functions of $T_{\alpha }$. Curves are plotted by using the effective range parameters, fitted from the $D0,D1,D2$ data sets, presented in Table 3. Experiment phase-shift data are also included in the figure. The filled box in (b) represents the binding energy of the $2_1^{+}$ state.

Figure 6

Phase shift, ${\delta }_3$, (a) and the real part of denominator, $Re{D}_3\left(k\right)$, of the amplitude (b) as functions of $T_{\alpha }$. Curves are plotted by using the effective range parameters, fitted from the $F0,F1F2$ data sets, presented in Table 4. Experiment phase-shift data are also included in the figure. The filled box in (b) represents the binding energy of the $3_1^{-}$ state.