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

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 (MCMC) method is used to crosscheck 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 global optimization and uncertainty quantification based on DE-MCMC methods, provides a reliable and systematic framework for applications to low energy phenomena relevant to stellar evolution and nucleosynthesis.