  In this paper, we point out a novel signature of physics beyond the Standard
Model which could potentially be observed both at the Large Hadron Collider
(LHC) and at future colliders. This signature, which emerges naturally within
many proposed extensions of the Standard Model, results from the multiple
displaced vertices associated with the successive decays of unstable,
long-lived particles along the same decay chain. We call such a sequence of
displaced vertices a "tumbler." We examine the prospects for observing tumblers
at the LHC and assess the extent to which tumbler signatures can be
distinguished from other signatures of new physics which also involve multiple
displaced vertices within the same collider event. As part of this analysis, we
also develop a procedure for reconstructing the masses and lifetimes of the
particles involved in the corresponding decay chains. We find that the
prospects for discovering and distinguishing tumblers can be greatly enhanced
by exploiting precision timing information such as would be provided by the CMS
timing layer at the high-luminosity LHC. Our analysis therefore provides strong
additional motivation for continued efforts to improve the timing capabilities
of collider detectors at the LHC and beyond.
