  We explore the possibility that a new interaction between muons and protons
is responsible for the discrepancy between the CODATA value of the proton
radius and the value deduced from the measurement of the Lamb shift in muonic
hydrogen. We show that a new force carrier with roughly MeV mass can account
for the observed energy shift as well as the discrepancy in the muon anomalous
magnetic moment. However, measurements in other systems constrain the couplings
to electrons and neutrons to be suppressed relative to the couplings to muons
and protons, which seems challenging from a theoretical point of view. One can
nevertheless make predictions for energy shifts in muonic deuterium, muonic
helium, and true muonium under the assumption that the new particle couples
dominantly to muons and protons.
