Addressing μ-b_μand proton lifetime problems and active neutrino masses in a U(1)^\prime-extended supergravity model

We present a locally supersymmetric extension of the minimal supersymmetric Standard Model (MSSM) based on the gauge group $SU(3)_C\times SU(2)_L\times U(1)_Y\times U(1)^\prime$ where, except for the supersymmetry breaking scale which is fixed to be $\sim 10^{11}$ GeV, we require that all non-Standard-Model parameters allowed by the {\it local} spacetime and gauge symmetries assume their natural values. The $U(1)^\prime$ symmetry, which is spontaneously broken at the intermediate scale, serves to ({\it i}) explain the weak scale magnitudes of $μ$ and $b_μ$ terms, ({\it ii}) ensure that dimension-3 and dimension-4 baryon-number-violating superpotential operators are forbidden, solving the proton-lifetime problem, ({\it iii}) predict {\it bilinear lepton number violation} in the superpotential at just the right level to accommodate the observed mass and mixing pattern of active neutrinos (leading to a novel connection between the SUSY breaking scale and neutrino masses), while corresponding trilinear operators are strongly supppressed. The phenomenology is like that of the MSSM with bilinear R-parity violation, were the would-be lightest supersymmetric particle decays leptonically with a lifetime of $\sim 10^{-12}-10^{-8}$ s. Theoretical consistency of our model requires the existence of multi-TeV, stable, colour-triplet, weak-isosinglet scalars or fermions, with either conventional or exotic electric charge which should be readily detectable if they are within the kinematic reach of a hadron collider. Null results of searches for heavy exotic isotopes implies that the re-heating temperature of our Universe must have been below their mass scale which, in turn, suggests that sphalerons play a key role for baryogensis. Finally, the dark matter cannot be the weakly interacting neutralino.