  Lepton-nucleon elastic scattering, using the one-photon and one-scalar-boson
exchange mechanisms considering all possible polarizations, is used to study
searches for a new scalar boson and suggest new measurements of the nucleon
form factors. A new light scalar boson, which feebly couples to leptons and
nucleons, may account for the proton radius and muon $g-2$ puzzles. We show
that the scalar boson produces relatively large effects in certain kinematic
region when using sufficient control of lepton and nucleon spin polarization.
We generalize current techniques to measure the ratio $G_E/G_M$ and present a
new method to separately measure $G_M^2$ and $G_E^2$ using polarized incoming
and outgoing muons.
