Radiative proton capture on ${}^{15}\mathrm{N}$ within effective field theory

The astrophysical $S$ factor for the radiative proton capture process on the ${}^{15}\mathrm{N}$ nucleus, i.e., ${}^{15}\mathrm{N}(p,\gamma ){}^{16}\mathrm{O}$, at stellar energies are studied within the framework of the cluster effective field theory. The thermonuclear ${}^{15}\mathrm{N}(p,\gamma ){}^{16}\mathrm{O}$ reaction links the type-I to type-II cycles of the carbon-nitrogen-oxygen cycle and affects the abundances of elements in the universe. For investigating this reaction in the effective field theory formalism, we first construct an effective Lagrangian that is appropriate for this reaction at low energies. Since the intermediate excited states of the ${}^{16}\mathrm{O}$ nucleus have a crucial role in this reaction, we include these resonances in the formalism. The corresponding radiative capture amplitudes and cross section are calculated, which lead to the astrophysical $S$ factor. The low-energy constants introduced in the effective Lagrangian are determined by fitting the theoretical results to the observed $S$ factors in the range of $130\mathrm{keV}<E_p<2500\mathrm{keV}$ using three different experimental data sets. Considering the recent data sets, we obtain $S\left(0\right)=29.8–34.1\mathrm{keV}\mathrm{b}$, which is in a good agreement with the estimates from $R$-matrix approaches in the literature. The values of $S$ at the Gamow energy are found to be larger than $S\left(0\right)$ values by about 10%.

Figure 1

Collected experimental data for (a) the cross sections and (b) the astrophysical $S$ factors $S\left(E\right)$ of the ${}^{15}\mathrm{N}(p,\gamma ){}^{16}\mathrm{O}$ reaction. The data from Refs. [10, 11, 17, 18, 19, 25] are compiled in Ref. [23].

Figure 2

Fully dressed propagator of the difield. In the right-hand side, the double line stands for the bare propagator and the shaded region represents the Coulomb interaction between two nuclei.

Figure 3

Feynman diagrams for the radiative capture process of ${}^{15}\mathrm{N}(p,\gamma ){}^{16}\mathrm{O}$. The thin and thick solid lines denote the proton and the ${}^{15}\mathrm{N}$ nucleus, respectively, and the wavy line stands for the out-going photon. The dotted line represents the ground state of the ${}^{16}\mathrm{O}$ nucleus in the final state. The shaded blobs represent the Coulomb repulsion between the proton and ${}^{15}\mathrm{N}$. The circle with a cross in (g) is introduced as a counterterm of the loop diagrams.

Figure 4

The astrophysical $S$ factor. The parameters are fitted by the three experimental data sets from Ref. [11] (RR), [18] (LeB), and [25] (Imb). The bands indicate the error range obtained in the MCMC calculations.

Figure 5

Same as Fig. 4 but with logarithmic scales in both axes.

Figure 6

The obtained $S\left(0\right)$ values of the present work compared with the values in the literature. The abbreviated references are as follows: Heb [10], Ba [15], RR [11], Muk [14], HBG [16], LeB [18], MLK [24], deB [26], Xu [22], DD [27], Caci [19], SAO [47], and Imb [25]. The references in the upper part indicate the sources of the experimental data and the references in the lower part indicate the sources of the calculated results.