  We investigate a model where the dark matter particle is a chiral fermion
field charged under a global $U(1)$ symmetry which is assumed to be
spontaneously broken, leading to a pseudo-Goldstone boson (PGB). We argue that
the dark matter annihilation into PGBs determine the dark matter relic
abundance. Besides, we also note that experimental searches for PGBs allow
either for a very long lived PGB, with a lifetime much longer than the age of
the Universe, or a relatively short lived PGB, with a lifetime shorter than one
minute. Hence, two different scenarios arise, producing very different
signatures. In the long lived PGB scenario, the PGB might contribute
significantly to the radiation energy density of the Universe. On the other
hand, in the short lived PGB scenario, and since the decay length is shorter
than one parsec, the s-wave annihilation into a PGB and a $CP$ even dark scalar
in the Galactic center might lead to an intense box feature in the gamma-ray
energy spectrum, provided the PGB decay branching ratio into two photons is
sizable. We also analyze the constraints on these two scenarios from thermal
production, the Higgs invisible decay width and direct dark matter searches.
