Distinguishing dark matter stabilization symmetries using multiple kinematic edges and cusps

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We emphasize that the stabilizing symmetry for dark matter (DM) particles does not have to be the commonly used parity (${Z}_{2}$) symmetry. We therefore examine the potential of the colliders to distinguish models with parity stabilized DM from models in which the DM is stabilized by other symmetries. We often take the latter to be a ${Z}_{3}$ symmetry for illustration. We focus on signatures where a single particle, charged under the DM stabilization symmetry decays into the DM and standard model (SM) particles. Such a ${Z}_{3}$-charged mother particle can decay into one or two DM particles along with the same SM particles. This can be contrasted with the decay of a ${Z}_{2}$-charged mother particle, where only one DM particle appears. Thus, if the intermediate particles in these decay chains are off-shell, then the reconstructed invariant mass of the SM particles exhibits two kinematic edges for the ${Z}_{3}$ case but only one for the ${Z}_{2}$ case. For the case of on-shell intermediate particles, distinguishing the two symmetries requires more than the kinematic edges. In this case, we note that certain decay chain topologies of the mother particle which are present for the ${Z}_{3}$ case (but absent for the ${Z}_{2}$ case) generate a cusp in the invariant mass distribution of the SM particles. We demonstrate that this cusp is generally invariant of the various spin configurations. We further apply these techniques within the context of explicit models.