Vector dark matter production from catalyzed annihilation
  We provide a simple model of vector dark matter (DM) which can realize the
recently proposed freeze-out mechanism with catalyzed annihilation. In our
setup, a vector DM field $X_\mu$ and a catalyst field $C_\mu$ is unified by an
SU(2)$_D$ gauge symmetry. These gauge fields acquire their masses via
spontaneously symmetry breaking triggered by a doublet and a real triplet
scalar fields. The catalyst particle is automatically lighter than the DM since
it only acquires mass from the vacuum expectation value of the doublet scalar.
We also introduce a dimension-5 operator to generate a kinetic mixing term
between $C_\mu$ and the U(1)$_Y$ gauge field $B_\mu$. This mixing term is
naturally small due to a suppression with a high UV completion scale, and thus
it allows the catalyst to decay after the DM freeze-out. We derive the
annihilation cross sections of processes $X^\ast+X\to 2C$ and $3C\to X^\ast+X$
and solve the Boltzmann equations for both the DM and the catalyst. We develop
the analytical approximate solutions of the equations and find them matching
the numerical solutions well. Constraints from relic abundance and indirect
detection of DM are considered. We find that the DM with a mass
$m_X\gtrsim4.5$~TeV survives in the case of a long-living catalyst. On the
other hand, if the catalyst decays during the catalyzed annihilation era, then
the bound can be released. We also discuss two paradigms which can maintain the
kinetic equilibrium of DM until the DM freeze-out. In both cases, the
freeze-out temperature of DM is an order of magnitude higher than the original
model.
