  The idea that dark matter forms part of a larger dark sector is very
intriguing, given the high degree of complexity of the visible sector. In this
paper, we discuss lepton jets as a promising signature of an extended dark
sector. As a simple toy model, we consider an $\mathcal{O}(\text{GeV})$ DM
fermion coupled to a new $U(1)'$ gauge boson (dark photon) with a mass of order
GeV and kinetically mixed with the Standard Model photon. Dark matter
production at the LHC in this model is typically accompanied by collinear
radiation of dark photons whose decay products can form lepton jets. We analyze
the dynamics of collinear dark photon emission both analytically and
numerically. In particular, we derive the dark photon energy spectrum using
recursive analytic expressions, using Monte Carlo simulations in Pythia, and
using an inverse Mellin transform to obtain the spectrum from its moments. In
the second part of the paper, we simulate the expected lepton jet signatures
from radiating dark matter at the LHC, carefully taking into account the
various dark photon decay modes and allowing for both prompt and displaced
decays. Using these simulations, we recast two existing ATLAS lepton jet
searches to significantly restrict the parameter space of extended dark sector
models, and we compute the expected sensitivity of future LHC searches.
