A model of pseudo-Nambu-Goldstone dark matter from a softly broken $SU(2)$ global symmetry with a $U(1)$ gauge symmetry

A model of the pseudo-Nambu-Goldstone (pNG) dark matter (DM) is proposed. We assume that there is an $SU(2)_g$ global symmetry and a $U(1)_X$ gauge symmetry in the dark sector, and they are spontaneously broken into a $U(1)_D$ global symmetry after a scalar field develops a vacuum expectation value. We add a soft symmetry breaking term that breaks the $SU(2)_g$ global symmetry into the $U(1)_g$ global symmetry explicitly. Our model predicts a stable complex pNG particle under the $U(1)_D$ global symmetry. One of the virtues of the pNG DM models is that the models can explain the current null results in the direct detection experiments. The small momentum transfer suppresses the scattering amplitudes thanks to the low energy behavior of the Nambu-Goldstone boson. In our model, the soft symmetry breaking term is uniquely determined. This is the advantage of our model to some earlier works in which some soft symmetry breaking terms cannot be forbidden but are simply assumed to be absent to avoid the constraints from the direct detection experiments. We calculate the thermal relic abundance of the pNG DM and find that model can explain the measured value of the DM energy density under some constraints from the perturbative unitarity.