Tension between gauge coupling unification, the Higgs boson mass, and a gauge-breaking origin of the supersymmetric μ-term

We investigate the possibility of generating the μ-term in the minimal supersymmetric standard model (MSSM) by the condensation of a field that is a singlet under the standard model (SM) gauge group but charged under an additional family-independent U(1)_X gauge symmetry. We attempt to do so while preserving the gauge coupling unification of the MSSM. For this, we find that SM nonsinglet exotics must be present in the spectrum. We also prove that the pure U(1)_X anomalies can always be solved with rationally charged fields, but that a large number of SM singlets are often required. For U(1)_X charges that are consistent with an embedding of the MSSM in SU(5) or SO(10), we show that the U(1)_X charges of the MSSM states can always be expressed as a linear combination of Abelian subgroups of E₆. However, the SM exotics do not appear to have a straightforward embedding into grand-unified theory multiplets. We conclude from this study that if this approach to the μ-term is correct, as experiment can probe, it will necessarily complicate the standard picture of supersymmetric grand unification.