U(1)′solution to theμ-problem and the proton decay problem in supersymmetry withoutR-parity

The minimal supersymmetric standard model (MSSM) is plagued by two major fine-tuning problems: the $\mu$-problem and the proton decay problem. We present a simultaneous solution to both problems within the framework of a $U(1{)}^{\prime }$-extended MSSM (UMSSM), without requiring $R$-parity conservation. We identify several classes of phenomenologically viable models and provide specific examples of $U(1{)}^{\prime }$ charge assignments. Our models generically contain either lepton number violating or baryon number violating renormalizable interactions, whose coexistence is nevertheless automatically forbidden by the new $U(1{)}^{\prime }$ gauge symmetry. The $U(1{)}^{\prime }$ symmetry also prohibits the potentially dangerous and often ignored higher-dimensional proton decay operators such as $QQQL$ and $U^c{U}^c{D}^c{E}^c$ which are still allowed by $R$-parity. Thus, under minimal assumptions, we show that once the $\mu$-problem is solved, the proton is sufficiently stable, even in the presence of a minimum set of exotics fields, as required for anomaly cancellation. Our models provide impetus for pursuing the collider phenomenology of $R$-parity violation within the UMSSM framework.