  As the majority of LHC searches are focused on prompt signatures, specific
long-lived particles have the potential to be overlooked by the otherwise
systematic new physics programs at ATLAS and CMS. While in many cases
long-lived superparticles are now stringently constrained by existing exotic
searches, we point out that the highly motivated model of gauge mediation with
staus as the next-to-lightest superparticle (NLSP) has received less attention.
We recast LHC searches for heavy stable charged particles, disappearing tracks,
and opposite-flavor leptons with large impact parameters to assess current
constraints on a variety of spectra that contain an NLSP stau, and find that
portions of the parameter space motivated by naturalness are still
experimentally unexplored. We additionally note a gap in the current
experimental search program: same-flavor leptons with large impact parameters
evade the suite of existing searches for long-lived objects. This gap is
especially noteworthy as vetoes on displaced leptons in prompt new physics
searches could be systematically discarding such events. We discuss several
motivated models that can exhibit same-flavor displaced leptons: gauge
mediation with co-NLSP sleptons, extended gauge mediation, R-parity violation,
and lepton-flavored dark matter that freezes in during a matter-dominated era
of the early universe. To address this gap, we propose a straightforward
extension of the CMS search for leptons with large impact parameters, and
project sensitivity to these scenarios at 13 TeV. Throughout this analysis, we
highlight several methods whereby LHC searches for exotic long-lived objects
could potentially improve their sensitivity to the displaced leptons
originating from gauge mediation and beyond.
