How to prove that the LHC did not discover dark matter

If the LHC is able to produce dark matter particles, they would appear at the end of cascade decay chains, manifesting themselves as missing transverse energy. However, such "dark matter candidates" may decay invisibly later on. We propose to test for this possibility by studying the effect of particle widths on the observable invariant mass distributions of the visible particles seen in the detector. We consider the simplest non-trivial case of a two-step two-body cascade decay and derive analytically the shapes of the invariant mass distributions, for generic values of the widths of the new particles. We demonstrate that the resulting distortion in the shape of the invariant mass distribution can be significant enough to measure the width of the dark matter "candidate", ruling it out as the source of the cosmological dark matter.