Pseudo-Nambu-Goldstone dark matter in SU(7) grand unification

We propose a grand unified theory (GUT) pseudo-Nambu-Goldstone boson (pNGB) dark matter (DM) model based on SU(7) gauge symmetry. In the GUT model, the Standard Model (SM) gauge symmetry GSM colon equals SU(3)C x SU(2)L x U(1)Y and the "dark" gauge symmetry SU(2)D are unified, where the SU(2)D symmetry plays an important role in the stability of DM. The unification of SM fermions and dark sector fermions is partially realized. The gauge symmetry SU(7) is spontaneously broken to SU(5) x SU(2) x U(1) gauge symmetry at the GUT scale by the nonvanishing vacuum expectation values of an SU(7) adjoint scalar field, where the SU(5) gauge symmetry is not usual SU(5) GUT gauge symmetry. The symmetry is further broken to GSM x SU(2)D at an intermediate scale. Furthermore, the SU(2)D symmetry is broken by the SU(2)D doublet and triplet scalar fields at the TeV scale. In the pNGB DM model based on GSM x SU(2)D, the residual global U(1)V dark custodial symmetry guarantees DM stability. On the other hand, in the SU(7) pNGB DM model, this global symmetry is explicitly broken by the Yukawa interaction and the effective Majorana mass terms. In the scalar sector, the cubic coupling constants of the SU(2)D doublet and triplet scalar fields are the order parameters of the U(1)V symmetry breaking. To maintain U(1)V symmetry and thus the DM stability, we need to tune Yukawa coupling constants and cubic scalar couplings at high accuracy. We find that the allowed DM mass region is quite restricted as the gauge coupling constant of SU(2)D is determined by the condition of the gauge coupling unification. To satisfy gauge coupling unification and the current experimental constraint on proton lifetime, we find that three generations of SU(3)C adjoint fermions and another three generations of SU(2)L adjoint fermions with the intermediate mass scale are required. We also find that there is no other solution to satisfy simultaneously the gauge coupling unification and the proton decay constraint if one assumes the other symmetry breaking schemes.