  The effect of the lepton's mass is significantly enhanced when the beam's
energy is on the order of the lepton's mass. In the case of electrons, this
corresponds to beam momenta on the order of a few MeV and is negligible in
higher energy experiments. In this study, we calculate the differential cross
section $d\sigma/ d\Omega$ for the helicity-flip meson exchange interference in
elastic muon-proton $(\mu p)$ scattering $\mu^- p \rightarrow \mu^- p$. In
particular, we examine the $\pi^0$ meson exchange in the $t$-channel for a
longitudinally polarized beam and a transversely polarized target. We
demonstrate the contribution to be larger for muons due to the lepton mass
difference. Then we construct the corresponding beam-target double-spin
asymmetries for the target polarized normal and parallel to the momentum
transfer in the Breit frame, and then consider the model dependence of the
calculation from the estimation of the $\pi^0 \mu \mu$ vertex. The contribution
was found to be on the order of $\sim .15\%$ for muons in the kinematic region
of the MUSE experiment.
