Bayesian Framework for the E1 and E2 Astrophysical Factors at 300 keV from Subthreshold and Ground-State Asymptotic Normalization Coefficients

TL;DR

This work develops a Bayesian framework to infer the astrophysical $S$-factors $S_{E1}(300~\mathrm{keV})$ and $S_{E2}(300~\mathrm{keV})$ for the important nuclear reaction ^12C(α,γ)^16O by propagating subthreshold and ground-state ANCs through calibrated $R$-matrix mappings. The analysis explicitly incorporates existing ANC constraints on $C_{1}$, $C_{2}$, and $C_{0}$, and propagates parameter correlations, particularly the interference between subthreshold resonances and direct capture amplitudes. The resulting posteriors show that, even with ANC priors, the 68% intervals for $S_{E1}$ and $S_{E2}$ remain broad, underscoring persistent astrophysical uncertainties; the a posteriori distributions reveal strong sensitivity to the chosen priors and the correlations among ANCs. Importantly, the work connects the inferred total $S(300)=S_{E1}(300)+S_{E2}(300)$ to massive-star evolution and black-hole remnant masses, illustrating how nuclear reaction constraints influence the black-hole mass spectrum observed by gravitational-wave detectors. The framework thus provides robust, correlated posterior distributions for the $S$-factors and clarifies the path toward tighter nuclear-physics constraints needed for precise nucleosynthesis and stellar-remnant predictions.

Abstract

The $^{12}\mathrm{C}(α,γ)^{16}\mathrm{O}$ reaction governs the carbon-to-oxygen ratio set during helium burning, shaping white-dwarf structure and Type~Ia supernova yields. At the astrophysical energy $E \approx 300~\mathrm{keV}$, the cross section is controlled by the subthreshold $1^{-}$ (7.12~MeV) and $2^{+}$ (6.92~MeV) states, whose contributions depend on their asymptotic normalization coefficients (ANCs) $C_{1}$ and $C_{2}$, respectively. We perform a Bayesian analysis of the $S_{E1}(300~\mathrm{keV})$ and $S_{E2}(300~\mathrm{keV})$ factors using calibrated $R$-matrix mappings and experimental ANC constraints for the $1^{-}$, $2^{+}$, and $0^{+}$ ground state. For $S_{E1}(300~\mathrm{keV})$, flat prior on the $1^{-}$ ANC lead to broad posterior with $68\%$ credible interval spanning $ [71.4,\,93.4]$~keV\,b, while Gaussian priors concentrate weight near the reported ANC values and yield narrower posteriors. For $S_{E2}(300~\mathrm{keV})$, the analysis includes the interference of the radiative transition through the subthreshold resonance with the direct capture to the ground-state, which depends on the ground-state ANC $C_{0}$, giving broad posterior with $68\%$ credible interval spanning $[30.7,\,50.5]$~keV\,b. The Gaussian priors centered near anchor values. The resulting posteriors quantify both correlations and uncertainties: despite incorporating the published ANC constraints, the $68\%$ intervals remain broad, showing that present ANC determinations do not yet reduce the astrophysical uncertainty. Overall, the Bayesian framework provides statistically robust posteriors for $S_{E1}(300~\mathrm{keV})$ and $S_{E2}(300~\mathrm{keV})$, improving the reliability of extrapolations for stellar modeling and nucleosynthesis.

Figures (10)

• Figure 1: Posterior of the $E1$ astrophysical factor obtained by propagating a flat prior in the ANC $C_{1}$ through the exact calibration map \ref{['eq:E1map_exact']}. Because the mapping is monotone with $\beta_{1}<0$, the Jacobian factor $|dS_{E1}/dC_{1}|^{-1}$ enhances probability density at larger $C_{1}$, resulting in a right--skewed posterior. The grey band shows the 68% Bayesian sredible interval.
• Figure 2: Posterior distributions of the $E1$ astrophysical factor at $E=300$ keV for Gaussian priors in $C_{1}$. The red curve corresponds to a low--Gaussian prior centered at the $C_{1}=1.83\times10^{14}$ fm$^{-1/2}$, while the black curve corresponds to a high--Gaussian prior centered at $C_{1}=2.20\times10^{14}$ fm$^{-1/2}$. Shaded bands indicate 68% Bayesian credible intervals.
• Figure 3: Three--dimensional surface of $S_{E2}(300~\mathrm{keV})$ as a function of the subthreshold ANC $C_{2}$ and the ground--state ANC $C_{0}$, computed using the calibrated mapping of Eq. (\ref{['eq:SE2mapping']}). The surface reproduces the $R$-matrix anchor points with better than $0.004~\mathrm{keV\,b}$ accuracy. The steep increase with $C_{2}$ reflects the dominant radiative capture through the subthreshold $2^{+}$ state, while the interference between the subthreshold resonance and the external nonresonant capture to the ground state generates the bilinear $C_{2}C_{0}$ dependence responsible for the downward slope with increasing $C_{0}$. The positive quadratic term in $C_{0}$ leads to the mild upturn at large $C_{0}$.
• Figure 4: $S_{E2}(300~\mathrm{keV})$ as a function of $(C_{2},C_{0})$. The diagonal interference corridor marks the combinations of ANCs for which the subthreshold resonance $2^{+}$ contribution and the external direct-capture amplitude interfere in a manner that keeps $S_{E2}$ nearly constant. Motion across this corridor leads to rapid changes in $S_{E2}$.
• Figure 5: Posterior probability density for $S_{E2}(300~\mathrm{keV})$ obtained from Eq. (\ref{['eq:SE2mapping']}) using the prior defined in Eqs. \ref{['C2prior']} and \ref{['C0prior']}. The red band denotes the 68% credible interval and the dashed lines mark the 95% limits. The MAP value is $S_{E2}(300)=30.6~\mathrm{keV\,b}$, with a median of $29.9~\mathrm{keV\,b}$. The 68% and 95% credible intervals are $[18.5,\,43.5]$ and $[12.6,\,59.4]~\mathrm{keV\,b}$, respectively.