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This work explores scattering amplitudes that couple two-particle systems viaa single external current insertion, $2+\mathcal{J}\to 2$. Such amplitudes canprovide structural information about the excited QCD spectrum. We derive anexact analytic representation for these reactions. From these amplitudes, weshow how to rigorously define resonance and bound-state form-factors.Furthermore, we explore the consequences of the narrow-width limit of theamplitudes as well as the role of the Ward-Takahashi identity for conservedvector currents. These results hold for any number of two-body channels with nointrinsic spin, and a current with arbitrary Lorentz structure and quantumnumbers. This work and the existing finite-volume formalism provide a completeframework for determining this class of amplitudes from lattice QCD.

On-shell representations of two-body transition amplitudes: single external current