  The process of electron-positron annihilation into two photons in the
presence of an intense classical plane wave of an arbitrary shape is
investigated analytically by employing light-cone quantization and by taking
into account the effects of the plane wave exactly. We introduce a general
description of second-order 2-to-2 scattering processes in a plane-wave
background field, indicating the necessity of considering the localization of
the colliding particles and achieving that by means of wave packets. We define
a local cross section in the background field, which generalizes the vacuum
cross section and which, though not being directly an observable, allows for a
comparison between the results in the plane wave and in vacuum without relying
on the shape of the incoming wave packets. Two possible cascade or two-step
channels have been identified in the annihilation process and an alternative
way of representing the two-step and one-step contributions via a "virtuality"
integral has been found. Finally, we compute the total local cross section to
leading order in the coupling between the electron-positron field and the
quantized photon field, excluding the interference between the two
leading-order diagrams arising from the exchange of the two final photons, and
express it in a relatively compact form. In contrast to processes in a
background field initiated by a single particle, the pair annihilation into two
photons, in fact, also occurs in vacuum. Our result naturally embeds the vacuum
part and reduces to the vacuum expression, known in the literature, in the case
of a vanishing laser field.
