We consider the light $Z$ explanation of the muon $g-2$ anomaly. Even if such a $Z$ has no tree-level coupling to electrons, in general one will be induced at loop-level. We show that future beam dump experiments are powerful enough to place stringent constraints on$-$or discover$-$a $Z$ with loop-suppressed couplings to electrons. Such bounds are avoided only if the $Z$ has a large interaction with neutrinos, in which case the scenario will be bounded by ongoing neutrino scattering experiments. The complementarity between beam dump and neutrino scattering experiments therefore indicates that there are good prospects of probing a large part of the $Z$ parameter space in the near future.
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