We consider the possible anomaly free Abelian discrete symmetries of the MSSM that forbid the μ-term at perturbative order. Allowing for anomaly cancellation via the Green-Schwarz mechanism we identify discrete R-symmetries as the only possibility and prove that there is a unique Z4R symmetry that commutes with SO(10). We argue that non-perturbative effects will generate a μ-term of electroweak order thus solving the μ-problem. The non-perturbative effects break the Z4R symmetry leaving an exact Z2 matter parity. As a result dimension four baryon- and lepton-number violating operators are absent while, at the non-perturbative level, dimension five baryon- and lepton-number violating operators get induced but are highly suppressed so that the nucleon decay rate is well within present bounds. © 2010 Elsevier B.V.

We determine the anomaly free discrete R symmetries, consistent with the MSSM, that commute with SU(5) and suppress the μ parameter and nucleon decay. We show that the order M of such Z{double-struck}MR symmetries has to divide 24 and identify 5 viable symmetries. The simplest possibility is a Z{double-struck}4R symmetry which commutes with SO(10). We present a string-derived model with this Z{double-struck}4R symmetry and the exact MSSM spectrum below the GUT scale; in this model Z{double-struck}4R originates from the Lorentz symmetry of compactified dimensions. We extend the discussion to include the singlet extensions of the MSSM and find Z{double-struck}4R and Z{double-struck}8Rare the only possible symmetries capable of solving the μ problem in the NMSSM. We also show that a singlet extension of the MSSM based on a Z{double-struck}24Rsymmetry can provide a simultaneous solution to the μ and strong CP problem with the axion coupling in the favoured window. © 2011 Elsevier B.V.