  This work explores scattering amplitudes that couple two-particle systems via
a single external current insertion, $2+\mathcal{J}\to 2$. Such amplitudes can
provide structural information about the excited QCD spectrum. We derive an
exact analytic representation for these reactions. From these amplitudes, we
show how to rigorously define resonance and bound-state form-factors.
Furthermore, we explore the consequences of the narrow-width limit of the
amplitudes as well as the role of the Ward-Takahashi identity for conserved
vector currents. These results hold for any number of two-body channels with no
intrinsic spin, and a current with arbitrary Lorentz structure and quantum
numbers. This work and the existing finite-volume formalism provide a complete
framework for determining this class of amplitudes from lattice QCD.
