  In this work, we report on the sensitivity potential of complementary
muon-on-target experiments to new physics using a scalar boson benchmark model
associated with charged lepton flavor violation. The NA64$\mu$ experiment at
CERN uses a 160-GeV energy muon beam with an active target to search for excess
events with missing energy and momentum as a probe of new physics. At the same
time, the proton beam at Fermilab, which is used to produce the neutrino beam
for the Deep Underground Neutrino Experiment (DUNE) will also produce a
high-intensity muon beam dumped in an absorber. Combined with the liquid Argon
Near Detector, the system could be used to search for similar scalar boson
particles with a lower energy but higher intensity beam. We find that both
NA64$\mu$ and DUNE could cover new, unexplored parts of the parameter space of
the same benchmark model, providing a complementary way to search for new
physics.
