  Observation of charge asymmetry by comparing electron and positron, or muon
and anti-muon, scattering on a hadronic target presently serves as an
experimental tool to study two-photon exchange effects. In addition to
two-photon exchange, real photon emission also contributes to the charge
asymmetry. We present a theoretical formalism, explicit expressions, and a
numerical analysis of hard photon emission for the charge asymmetry in lepton-
and antilepton-proton scattering. Different kinematic conditions are
considered, namely, either fixed transferred momentum squared or a fixed lepton
scattering angle. The infrared divergence from real photon emission is treated
by the Bardin-Shumeiko technique and canceled with the soft part of the
two-photon exchange contribution extracted and calculated using Tsai approach.
All final expressions are obtained beyond the ultrarelativistic approximation
with respect to the lepton mass that allows to evaluate numerically of the
considered effects not only for ultrarelativistic leptons (JLab) and but for
moderately relativistic (MUSE) kinematics, too.
