  The strong statistical significance of an observed electron-like event excess
in the MiniBooNE (MB) experiment, along with an earlier similar excess seen in
the Liquid Scintillator Neutrino Detector (LSND), when interpreted in
conjunction with recent MicroBooNE results may have brought us to the cusp of
new physics discoveries. This has led to many attempts to understand these
observations, both for each experiment individually and in conjunction, via
physics beyond the Standard Model (SM). We provide an overview of the current
situation, and discuss three major categories under which the many proposals
for new physics fall. The possibility that the same new, non-oscillation
physics explains both anomalies leads to new restrictions and requirements. An
important class of such common solutions, which we focus on in this work,
consists of a heavy ${\cal O}$(MeV$-$sub-GeV) sterile neutral fermion produced
in the detectors, (via up-scattering of the incoming muon neutrinos), and
subsequently decaying to photons or $e^+e^-$ pairs which mimic the observed
signals. Such solutions are subject to strong demands from a) cross section
requirements which would yield a sufficient number of total events in both LSND
and MB, b) requirements imposed by the measured energy and angular
distributions in both experiments and finally, c) consistency and compatibility
of the new physics model and its particle content with other bounds from a
diverse swathe of particle physics experiments. We find that these criteria
often pull proposed solutions in different directions, and stringently limit
the viable set of proposals which could resolve both anomalies. Our conclusions
are relevant for both the general search for new physics and for the ongoing
observations and analyses of the MicroBooNE experiment.
