  A persistence of several anomalies in muon physics, such as the muon
anomalous magnetic moment and the muonic hydrogen Lamb shift, hints at new
light particles beyond the Standard Model. We address a subset of these models
that have a new light scalar state with sizable couplings to muons and
suppressed couplings to electrons. A novel way to search for such particles
would be through muon beam-dump experiments by (1) missing momentum searches;
(2) searches for decays with displaced vertices. The muon beams available at
CERN and Fermilab present attractive opportunities for exploring the new scalar
with a mass below the di-muon threshold, and potentially covering a range of
relevant candidate models. For the models considered in this paper, both types
of signals, muon missing momentum and anomalous energy deposition at a
distance, can probe a substantial fraction of the unexplored parameter space of
the new light scalar, including a region that can explain the muon anomalous
magnetic moment discrepancy.
