Discrete R symmetries for the MSSM and its singlet extensions

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 ZMR symmetries has to divide 24 and identify 5 viable symmetries. The simplest possibility is a Z4R symmetry which commutes with SO(10). We present a string-derived model with this Z4R symmetry and the exact MSSM spectrum below the GUT scale; in this model Z4R originates from the Lorentz symmetry of compactified dimensions. We extend the discussion to include the singlet extensions of the MSSM and find Z4R and Z8R are 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 Z24R symmetry can provide a simultaneous solution to the μ and strong CP problem with the axion coupling in the favoured window.