We analyze patterns of remnant discrete symmetries that arise from U(1) N theories by spontaneous breaking. We describe a simple, geometrical way to understand these patterns and provide methods for identifying the discrete symmetries and bringing them to the simplest possible form. Applications in GUT and string model building are briefly discussed. © SISSA 2009.

We consider the flavor structure of the minimal supersymmetric standard model (MSSM) in the framework of 'minimal flavor violation' (MFV). We show that, if one imposes the MFV structure at some scale, to a good accuracy the MFV decomposition works at all other scales. That is, quantum effects can be described by running coefficients of the MFV decomposition. We find that the coefficients get driven to non-trivial fixed points. © 2008 Elsevier B.V. All rights reserved.

Recently it was shown that there is a unique Z4R symmetry for the MSSM which allows for the Yukawa couplings and dimension five neutrino mass operator, forbids the μ term and commutes with SO(10). This Z4R symmetry contains matter parity as a subgroup and forbids dimension four and five proton decay operators. We show how to construct string vacua with discrete R symmetries in general and this symmetry in particular, and present an explicit example which exhibits the exact MSSM spectrum, the Z4R symmetry as well as other desired features such as gauge-top unification. We introduce the Hilbert basis method for determining all D-flat configurations and efficient algorithms for identifying field configurations with a desired residual symmetry. These methods are used in an explicit example, in which we describe in detail how to construct a supersymmetric vacuum configuration with the phenomenologically attractive Z4R symmetry. At the perturbative level, this is a supersymmetric Minkowski vacuum in which almost all singlet fields (moduli) are fixed. © 2011 Elsevier B.V.

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.