The $\mu\nu$SSM with an extra U(1)

The $\mu\nu$SSM solves the $\mu$ problem of the MSSM and generates correct neutrino masses by simply using right-handed neutrinos. This mechanism implies that only dimensionless trilinear terms, breaking R-parity, are present in the superpotential. We present an extension of the $\mu\nu$SSM with an extra U(1) gauge symmetry. We use the extra U(1) charges of the matter fields to forbid the presence in the superpotential of renormalizable and non-renormalizable baryon number violating operators, the trilinear operator producing a domain wall problem, and bilinear operators such as the $\mu$ term and the Majorana masses. We apply the anomaly cancellation conditions associated to the extra U(1), to constrain the values of the U(1) charges. We find that six assignments of the U(1) charges to the matter fields are viable, once extra matter is introduced. In particular, three generations of vector-like color triplets and $SU(2)_L$ doublets, as well as six Standard Model singlets are necessary. Electroweak symmetry breaking is viable in the model, with wide regions of the parameter space fulfilling the experimental constraints on the existence of a new gauge boson $Z'$. Neutrinos and the extra gaugino mix with the MSSM neutralinos, producing a generalized see-saw matrix that can reproduce the experimental results on neutrino masses. Finally, we have estimated the tree-level upper bound on the lightest Higgs mass, finding that it can be as large as about 120 GeV.