Analysis of elastic $α$-$^{12}$C scattering with machine learning in the cluster effective field theory

TL;DR

The paper develops a data-driven framework that combines cluster EFT with a high-dimensional effective-range expansion up to angular momentum $l=6$ to describe elastic $α$-$^{12}$C scattering. A global optimization using Differential Evolution (DE) followed by Markov chain Monte Carlo (MCMC) uncertainty quantification determines 37 EFT parameters from $N=11{,}392$ differential cross-section data points, achieving $χ^2/N$ around 6.2. The resulting model reproduces differential cross sections and phase shifts consistent with $R$-matrix analyses for low $l$ and provides quantified uncertainties, improving on prior error estimates and enabling robust extrapolation to astrophysical energies. The framework demonstrates a reproducible, extensible approach to extract EFT parameters directly from data, with potential impact on low-energy nuclear astrophysics and stellar nucleosynthesis modeling.

Abstract

We analyze the elastic $α$-$^{12}$C scattering including the contribution of resonance states below the $p$-$^{15}$N breakup threshold energy. We use the cluster effective field theory in which scattering amplitude is expanded in terms of the effective range expansion parameters for the angular momentum states from $l=0$ to $l=6$. The amplitude contains 37 parameters, which are determined by fitting to 11,392 differential cross section data points of the elastic $α$-$^{12}$C scattering. To optimize the fitting process, we implement the Differential Evolution (DE) algorithm, which performs a global search over the high-dimensional parameter space and consistently converges to the same minimum $χ^{2}$ value across independent runs, suggesting proximity to the global minimum within the explored domain. In parallel, the Markov chain Monte Carlo method is used to cross-check the DE results and to estimate the parameter uncertainties. The best fit yields $χ^{2}/N\!\simeq\!6.2$ for the elastic scattering data. Using the determined 37 parameters, we calculate the differential cross sections and the phase shifts of the elastic $α$-$^{12}$C scattering and compare the results with experimental data and those of an $R$-matrix analysis. Our result of the cross section agrees with the experimental data as accurately as an $R$-matrix analysis. The results demonstrate that the cluster effective field theory, combined with machine learning based optimization and uncertainty quantification, provides a reliable and systematic framework for application to low-energy phenomena relevant to stellar evolution and nucleosynthesis.

Figures (8)

• Figure 1: Feynman diagrams for dressed $^{16}$O propagators. A thin dashed line and a thick dashed line denote the propagation of $\alpha$ and $^{12}$C, respectively. A thin-thick double-dashed line with or without a filled circle denotes a dressed or bare $^{16}$O propagators. A shaded oval denotes the Coulomb Green's function.
• Figure 2: Feynman diagram for the scattering amplitudes of elastic $\alpha$-$^{12}$C scattering in cluster EFT. A shaded oval denotes the initial or final state Coulomb wavefunction. See the caption of Fig. \ref{['fig:propagators']} as well.
• Figure 3: Differential Evolution (DE) convergence for the 37-parameter fit. The ordinate shows the chi-square, $\chi^2$, constructed from the full elastic $\alpha$-$^{12}$C dataset ($N=11{,}392$ points), while the abscissa counts objective function evaluations (“step”). The solid curve displays the running best value, $\min_{k\le\text{step}}\chi^2_k$, which decreases monotonically from $\sim 3.19\times10^{5}$ at the start to $7.10\times10^{4}$ at the end of the run. The inset (log-log scales; steps $<10^{4}$) resolves the rapid early drop and the plateau-like improvements characteristic of DE’s mutation-crossover-selection updates. Beyond this initial phase, the main panel exhibits a slow, smooth improvement and stabilization, consistent with robust global convergence.
• Figure 4: Population snapshots of the DE optimization. Populations from representative generations ($10^{4}$--$3\times10^{5}$) are projected onto the first two principal components (PC1, PC2) of a single PCA basis fitted to the combined samples. Points denote individual candidates, filled circles and stars mark the population centroids and best members, and ellipses show the 95% covariance regions. The contraction and drift of the clusters illustrate the transition from broad exploration to convergence around the global minimum. Details of the projection and interpretation are given in the text.
• Figure 5: Distribution of the total $\chi^2$ values obtained from the MCMC analysis for the elastic $\alpha$-$^{12}$C scattering. The histogram peaks around $\chi^2 \approx 7.05\times10^4$, with mean = 70,489.4, median = 70,488.8, and minimum = 70,461.7, indicate excellent convergence and statistical stability of the parameter estimation.