Dark matter from a complex scalar singlet and the role of dark CP

A complex scalar singlet could provide a pseudo-scalar dark matter candidate, whose stability is guaranteed by a discrete symmetry: dark CP. The scalar singlet is also charged under a global U(1) symmetry, which is broken both explicitly and spontaneously. We analyze the phenomenology of different models with just one explicit symmetry breaking term and we also consider the case with more than one symmetry breaking term. Finally, we discuss the possibility that the explicit symmetry breaking term is much smaller than the symmetry breaking scale, leading to a pseudo-Nambu-Goldstone boson as a DM candidate.