  Understanding the processes governing the dissociation of excitons to free
charge carriers in semiconductors and insulators is of central importance for
photovoltaic applications. Dyson's $\mathcal{S}$-matrix formalism provides a
framework for computing scattering rates between quasiparticle states derived
from the same underlying Hamiltonian, often reducing to familiar Fermi's golden
rule like expressions at first order. By presenting a rigorous formalism for
multi-channel scattering, we extend this approach to describe scattering
between composite quasiparticles and in particular, the process of exciton
dissociation mediated by the electron-phonon interaction. Subsequently, we
derive rigorous expressions for the exciton dissociation rate, a key quantity
of interest in optoelectronic materials, which enforce correct energy
conservation and may be readily used in ab initio calculations. We apply our
formalism to a three-dimensional model system to compare temperature-dependent
exciton rates obtained for different scattering channels.
