Performing Bayesian analyses with AZURE2 using BRICK: an application to the ${}^7$Be system

Phenomenological $R$-matrix has been a standard framework for the evaluation of resolved resonance cross section data in nuclear physics for many years. It is a powerful method for comparing different types of experimental nuclear data and combining the results of many different experimental measurements in order to gain a better estimation of the true underlying cross sections. Yet a practical challenge has always been the estimation of the uncertainty on both the cross sections at the energies of interest and the fit parameters, which can take the form of standard level parameters. Frequentist ($\chi^2$-based) estimation has been the norm. In this work, a Markov Chain Monte Carlo sampler, \texttt{emcee}, has been implemented for the $R$-matrix code \texttt{AZURE2}, creating the Bayesian $R$-matrix Inference Code Kit (\texttt{BRICK}). Bayesian uncertainty estimation has then been carried out for a simultaneous $R$-matrix fit of the $^3$He$(\alpha,\gamma)^7$Be and $^3$He$(\alpha,\alpha)^3$He reactions in order to gain further insight into the fitting of capture and scattering data. Both data sets constrain the values of the bound state $\alpha$-particle asymptotic normalization coefficients in $^7$Be. The analysis highlights the need for low-energy scattering data with well-documented uncertainty information and shows how misleading results can be obtained in its absence.