Fully characterizing a rank-0 forbidden beta decay for the reactor $\bar{\nu}_e$ energy spectrum*

An anomalous bump-like excess in the $5-7$ MeV range of the reactor anti-neutrino ($\bar{\nu}_e$) spectra continues to persist. In this energy range first-forbidden $0^-$ to $0^+$ decays account for $\sim 30\%$ of the total $\bar{\nu}_e$ flux. The $^{92}Rb$ ground-state to ground-state (GS) branch alone accounts for $30-50\%$ of the total $0^-$ to $0^+$ $\bar{\nu}_e$ flux in this energy range. Using the TRINAT neutral $^{92}Rb$ atom trap, all final state momenta are kinematically constrained, and the beta-neutrino angular correlation coefficient $a_{\beta\nu}$ is examined experimentally. GS decays are kinematically isolated from excited state transitions in Q-value by the total energy release. The GS beta energy dependence of $a_{\beta\nu}$ is used to test the rank-0 forbidden formalism and measure the ratio of the two nuclear elements $\omega/\xi_o$. The ratio $\omega/\xi_o$ is extracted from the $^{92}Sr$ recoil ion energy spectra with sensitivity of $\pm 0.10$ at $90\%$ C.L. A deviation in $\omega/\xi_o$ from zero at our sensitivity limits would necessarily distort the allowed $\bar{\nu}_e$ spectrum in the $5-7$ MeV range by as much as $\pm 3\%$.