The gallium anomaly reassessed using a Bayesian approach

The solar-neutrino detectors GALLEX and SAGE were calibrated by electron-neutrino flux from the $^{37}$Ar and $^{51}$Cr calibration sources. A deficit in the measured neutrino flux was recorded by counting the number of neutrino-induced conversions of the $^{71}$Ga nuclei to $^{71}$Ge nuclei. This deficit was coined ``gallium anomaly'' and it has lead to speculations about beyond-the-standard-model physics in the form of eV-mass sterile neutrinos. Notably, this anomaly has already defied final solution for more than 20 years. Here we reassess the statistical significance of this anomaly and improve the related statistical approaches by treating the neutrino experiments as repeated Bernoulli trials taking into account the fact that the number of the detected $^{71}$Ge nuclei is quite small, thus calling for a Bayesian statistical approach. In addition, we take into account the systematic errors of the experiments, their correlations, theoretical uncertainties and the number of background solar-neutrino events as a Poisson-distributed random variable. To compare with the previously reported statistical significances of the anomaly we convert the posterior intervals of our Bayesian approach to standard deviations $\sigma$ of the frequentist approach. We find that our approach reduces the statistical significance of the anomaly by $0.8\,\sigma$ for all the adopted theoretical approaches. This renders the gallium anomaly a statistically weakly supported concept. Furthermore, the implications of our approach go far beyond the gallium anomaly since the results of many rare-events experiments should be reassessed for their limited number of recorded events.