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Previous studies of the reaction ${}^{3}\mathrm{He} +{}^{4}\mathrm{He}\rightarrow {}^{7}\mathrm{Be} + \gamma$ have focused on providing the bestcentral value and error bar for the $S$ factor at solar energies. Measurementsof this capture reaction, the ${}^{3}\mathrm{He}$-${}^{4}\mathrm{He}$scattering phase shifts, as well as properties of ${}^{7}\mathrm{Be}$, havebeen used to constrain employed theoretical models. Here we show that much moreinformation than was previously appreciated can be extracted fromangle-integrated capture data alone. We use the next-to-leading-order (NLO)amplitude in an effective field theory (EFT) for the reaction to perform theextrapolation. At this order the EFT describes the reaction using an s-wavescattering length and effective range, the asymptotic properties of the finalbound states, and short-distance contributions to the $E1$ capture amplitude.We extract the multi-dimensional posterior of all these parameters via aBayesian analysis. We find that properties of the ${}^{7}\mathrm{Be}$ groundand excited states are well constrained. The total $S$ factor $S(0)=0.578^{+0.015}_{-0.016}$ keV~b, while the branching ratio for excited- toground-state capture at zero energy, $Br(0)=0.406^{+0.013}_{-0.011}$, both at68\% degree of belief. This $S(0)$ is broadly consistent with other recentevaluations, including the previously recommended value $S(0)=0.56 \pm 0.03$ eVb, but has a smaller error bar. We also find significant constraints on thescattering parameters, and we obtain constraints on the $S(E)$'s angulardependence. The path forward seems to lie with better measurements of thescattering phase shift and $S(E)$'s angular dependence, together with betterunderstanding of the asymptotic normalization coefficients of the ${}^7$Bebound states' wave functions. Data on these could further reduce $S(0)$'suncertainty.

$S$-factor and scattering-parameter extractions from ${}^{3}\mathrm{He} +{}^{4}\mathrm{He} \rightarrow {}^{7}\mathrm{Be} + \gamma$