Spectral features in LHC dileptonic events may signal radiative corrections coming from new degrees of freedom, notably dark matter and mediators. Using simplified models, we show how these features can reveal the fundamental properties of the dark sector, such as self-conjugation, spin and mass of dark matter, and the quantum numbers of the mediator. Distributions of both the invariant mass $m_{ell ell}$ and the Collins-Soper scattering angle $costheta_{CS}$ are studied to pinpoint these properties. We derive constraints on the models from LHC measurements of $m_{ell ell}$ and $costheta_{CS}$, which are competitive with direct detection and jets + Missing Energy searches. We find that in certain scenarios the $costheta_{CS}$ spectrum provides the strongest bounds, underlying the importance of scattering angle measurements for non-resonant new physics.
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