Variational inference of effective range parameters for ${}^3$He-${}^4$He scattering

We use two different methods, Monte Carlo sampling and variational inference (VI), to perform a Bayesian calibration of the effective-range parameters in ${}^3$He-${}^4$He elastic scattering. The parameters are calibrated to data from a recent set of $^{3}$He-${}^4$He elastic scattering differential cross section measurements. Analysis of these data for $E_{\rm lab} \leq 4.3$ MeV yields a unimodal posterior for which both methods obtain the same structure. However, the effective-range expansion amplitude does not account for the $7/2^-$ state of ${}^7$Be so, even after calibration, the description of data at the upper end of this energy range is poor. The data up to $E_{\rm lab}=2.6$ MeV can be well described, but calibration to this lower-energy subset of the data yields a bimodal posterior. After adapting VI to treat such a multi-modal posterior we find good agreement between the VI results and those obtained with parallel-tempered Monte Carlo sampling.