  Recent measurements of the proton radius using the Lamb shift in muonic
hydrogen are troublingly discrepant with values extracted from hydrogen
spectroscopy and electron-proton scattering experiments. This discrepancy,
which differs by more than five standard deviations, may be a signal of new
physics caused by a violation of lepton universality. Another candidate for a
new physics signal is the muon anomalous magnetic moment. The measurement at
BNL differs from the standard model prediction by at least three standard
deviations. Motivated by these two puzzles, first we use polarized
lepton-nucleon elastic scattering to search for a new scalar boson, and
furthermore we suggest new measurements of the nucleon form factors. Next, we
display a method to analyze the beam dump experiments without using
approximation on phase space, and we use it to constrain all possible new
spin-0 and spin-1 particles, and a variety of other measurements to study the
possibility of the new physics. Finally, assuming a new scalar boson can solve
the two puzzles simultaneously, we present a general model-independent analysis
and constrain the existence of the new physics.
