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We consider the non-oscillatory explanations of the low energy excess of events detected by MiniBooNE. We present a systematic search for phenomenological scenarios based on new physics which can produce the excess. We define scenarios as series of transitions and processes which connect interactions of accelerated protons in target with single shower events in the MiniBooNE detector. The key elements of the scenarios are production and decay of new light $\mathcal{O}(\text{keV}-100\,\text{MeV})$ particles (fermions or/and bosons). We find about $20$ scenarios with minimal possible number of new particles and interaction points. In practice, they are all reduced to few generic scenarios and in this way we develop the effective theory of the MiniBooNE excess. We consider tests of the scenarios with near or close detectors in neutrino experiments T2K ND280, NO$ν$A, MINER$ν$A as well as in NOMAD and PS191. The scenarios immediately connect the MiniBooNE excess and expected numbers of new physics events in these detectors. We compute the expected numbers of events as functions of lifetimes and masses of new particles and confront them with the corresponding experimental bounds. We indicate scenarios that are excluded or strongly disfavored by one or several experiments. Given our general approach, this work can also be regarded as the effective theory of new physics at accelerator based neutrino experiments, being relevant for future projects such as DUNE.