  Fixed target missing-momentum experiments such as LDMX and M$^3$ are powerful
probes of light dark matter and other light, weakly coupled particles beyond
the Standard Model (SM). Such experiments involve $\sim$ 10 GeV beam particles
whose energy and momentum are individually measured before and after passing
through a suitably thin target. If new states are radiatively produced in the
target, the recoiling beam particle loses a large fraction of its initial
momentum, and no SM particles are observed in a downstream veto detector. We
explore how such experiments can use kinematic variables and experimental
parameters, such as beam energy and polarization, to measure properties of the
radiated particles and discriminate between models if a signal is discovered.
In particular, the transverse momentum of recoiling particles is shown to be a
powerful tool to measure the masses of new radiated states, offering
significantly better discriminating ability compared to the recoil energy
alone. We further illustrate how variations in beam energy, polarization, and
lepton flavor (i.e., electron or muon) can be used to disentangle the possible
the Lorentz structure of the new interactions.
