  The effect on particle physics processes by intense electromagnetic fields in
the charge bunch collisions at future lepton colliders is considered. Since the
charge bunch fields are tied to massive sources (the $e^{+}e^{-}$ charges), a
reference frame is chosen in which the fields appear to be co-propagating.
Solutions of the Dirac equation minimally coupled to the electromagnetic fields
reasonably associated with two intense overlapping charge bunches are obtained
and found to be a Volkov solution with respect to a null 4-vector whose
3-vector part lies in the common propagation direction. These solutions are
used within the Furry interaction picture to calculate the beamstrahlung
transition rate for electron radiation due to interaction with the
electromagnetic fields of two colliding charge bunches. New analytic
expressions are obtained and compared numerically with the beamstrahlung in the
electromagnetic field of one charge bunch. The techniques developed will be
applied to other collider physics processes in due course.
