Freeze-In Dark Matter from a sub-Higgs Mass Clockwork Sector via the
  Higgs Portal
  The clockwork mechanism allows extremely weak interactions and small mass
scales to be understood in terms of the structure of a theory. A natural
application of the clockwork mechanism is to the freeze-in mechanism for dark
matter production. Here we consider a Higgs portal freeze-in dark matter model
based on a scalar clockwork sector with a mass scale which is less than the
Higgs boson mass. The dark matter scalar is the lightest scalar of the
clockwork sector. Freeze-in dark matter is produced by the decay of thermal
Higgs bosons to the clockwork dark matter scalars. We show that the mass of the
dark matter scalar is typically in the 1-10 keV range and may be warm enough to
have an observable effect on perturbation growth and Lyman-$\alpha$
observations. Clockwork Higgs portal freeze-in models have a potentially
observable collider phenomenology, with the Higgs boson decaying to missing
energy in the form of pairs of long-lived clockwork sector scalars, plus a
distribution of different numbers of quark and lepton particle-antiparticle
pairs. The branching ratio to different numbers of quark and lepton pairs is
determined by the clockwork sector parameters (the number of clockwork scalars
$N$ and the clockwork charge $q$), which could therefore be determined
experimentally if such Higgs decay modes are observed. In the case of a minimal
Standard Model observable sector, the combination of nucleosynthesis and
Lyman-$\alpha$ constraints is likely to exclude on-shell Higgs decays to
clockwork scalars, although off-shell Higgs decays would still be possible.
On-shell Higgs decays to clockwork scalars can be consistent with cosmological
constraints in simple extensions of the Standard Model with light singlet
scalars.
