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

TL;DR

This paper uses effective field theory to calculate the $S_{E1}$ factor for radiative alpha capture on carbon-12 at energies relevant for stellar processes, providing a theoretical estimate at the Gamow peak.

Contribution

It introduces an effective field theory approach to compute the $S_{E1}$ factor for the $^{12}$C($ ext{alpha}$, $ ext{gamma}$)$^{16}$O reaction, offering a new theoretical framework.

Findings

First $S_{E1}$ value at the Gamow-peak energy of 0.3 MeV.

Provides a strategy for reaction calculation in effective field theory.

Initial result sets a foundation for future precise modeling.

Abstract

The $S_{E1}$ factor of radiative $\alpha$ capture on $^{12}$C is studied in effective field theory. We briefly discuss the strategy for the calculation of the reaction and report a first result of $S_{E1}$ at the Gamow-peak energy, $E_G=0.3$~MeV.

Full text

1 Introduction

The radiative $\alpha$ capture on carbon-12, ${}^{12}C(\alpha,\gamma)^{16}$O, is a fundamental reaction in nuclear-astrophysics, which determines the $C/O$ ratio created in the stars [1]. The reaction rate, equivalently the astrophysical $S$-factor, of the process at the Gamow peak energy, $E_{G}=0.3$ MeV, however, cannot be determined in experiment due to the Coulomb barrier. A theoretical model is necessary to employ in order to extrapolate the reaction rate down to $E_{G}$ by fitting model parameters to experimental data typically measured at a few MeV. In constructing a model for the study, one needs to take account of excited states of 16O [2], particularly, two excited bound states for $l^{\pi}_{i\mbox{-}th}=1_{1}^{-}$ and $2_{1}^{+}$ just below the $\alpha$-12C breakup threshold at $E=-0.045$ and $-0.24$ MeV 111 The energy $E$ denotes that of the $\alpha$-12C system in center of mass frame. , respectively, as well as two resonant (second excited) $1_{2}^{-}$ and $2_{2}^{+}$ states at $E=2.42$ and $2.68$ MeV, respectively. The capture reaction to the ground state of 16O at $E_{G}$ is expected to be $E1$ and $E2$ transition dominant due to the subthreshold $1_{1}^{-}$ and $2_{1}^{+}$ states. See Refs. [2, 3] for review.

Theoretical frameworks employed for the previous studies are mainly categorized into two [3]: the cluster models using generalized coordinate method [4] or potential model [5] and the phenomenological models using the parameterization of Breit-Wigner, $R$-matrix [6], or $K$-matrix [7]. A recent trend of the study is to rely on intensive numerical analysis, in which a large amount of the experimental data relevant to the study are accumulated, and a significant number of parameters of the models are fitted to the data by using computational power [3, 8, 9]. In the present work, we discuss an alternative approach to estimate the $S$-factor at $E_{G}$; we employ a new method for the study and briefly discuss a calculation of the $S_{E1}$ factor at $E_{G}$ based on an effective field theory [10, 11].

2 Diagrams

In the study of the radiative capture process, 12C($\alpha$,$\gamma$)16O, at $E_{G}=0.3$ MeV employing an EFT, one may regard the ground states of $\alpha$ and 12C as point-like particles whereas the first excited states of $\alpha$ and 12C are chosen as irrelevant degrees of freedom, from which a large scale of the theory is determined [12]. Thus the expansion parameter of the theory is $Q/\Lambda_{H}\sim 1/3$ where $Q$ denotes a typical momentum scale $Q\sim k_{G}$; $k_{G}$ is the Gamow peak momentum, $k_{G}=\sqrt{2\mu E_{G}}\simeq 41$ MeV, where $\mu$ is the reduced mass of $\alpha$ and 12C. $\Lambda_{H}$ denotes a large momentum scale $\Lambda_{H}\simeq 150$ MeV obtained from the first excited energy of $\alpha$ or 12C. An effective Lagrangian for the study is obtained in Eq. (1) in Ref. [14].

The capture amplitudes are calculated from the Feynman diagrams depicted in Figs. 2 and 2. One can find an expression of the amplitudes in Eqs. (6), (7), (8), and (9) in Ref. [14]. We note that the loop diagrams (a) and (b) in Fig. 2 are finite whereas those (d), (e), and (f) diverge. The divergence terms are renormalized by a counter term $h^{(1)}$ in a contact vertex in the diagram (c). Six parameters remain in the amplitudes. Four of them are effective range parameters of elastic $\alpha$-12C scattering for $l=1$ [13]. One of them is fixed by using the binding energy of the subthreshold $l=1$ state of 16O, and the others are fitted to the phase shift data of the elastic scattering [15].

3 Result

In the left panel of Fig. 3, we show the data and the fitted curve of the phase shift and find that the fitted curve well reproduces the data. The remaining two parameters, $h^{(1)R}$ and $y^{(0)}$, in the amplitudes are fitted to the $S_{E1}$ data [3], and we obtain $h^{(1)R}=-6.95(11)\times 10^{2}$ MeV3 and $y^{(0)}=0.495(18)$ MeV*-1/2*, where the number of the data is $N=151$ and $\chi^{2}/N\simeq 1.72$. In the right panel of Fig. 3, we show the data and the fitted curve for $S_{E1}$. At the Gamow peak energy, $E_{G}=0.3$ MeV, thus, we obtain $S_{E1}\simeq 58$ keV b. An error estimate of $S_{E1}$ is now under investigation.

This work was supported by the Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education of Korea Grant No. NRF-2016R1D1A1B03930122.