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

Abstract: We analyze the elastic $\alpha$-$^{12}$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 $\alpha$-$^{12}$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 $\chi^{2}$ 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 $\chi^{2}/N!\simeq!6.2$ for the elastic scattering data. Using the determined 37 parameters, we calculate the differential cross sections and the phase shifts of the elastic $\alpha$-$^{12}$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.