Breaking classical Lie groups to finite subgroups - an automated approach

The decomposition of representations of compact classical Lie groups into representations of finite subgroups is discussed. A Mathematica package is presented that can be used to compute these branching rules using the Weyl character formula. For some low order finite groups including $A_4$ and $\Delta(27)$ general analytical formulas are presented for the branching rules of arbitrary representations of their smallest Lie super-groups.Comment: 21 pages, minor changes, matches published versio

Symmetries of symmetries and geometrical CP violation

We investigate transformations which are not symmetries of a theory but nevertheless leave invariant the set of all symmetry elements and representations. Generalizing from the example of a three Higgs doublet model with $\Delta(27)$ symmetry, we show that the possibility of such transformations signals physical degeneracies in the parameter space of a theory. We show that stationary points only appear in multiplets which are representations of the group of these so-called equivalence transformations. As a consequence, the stationary points are amongst the solutions of a set of homogeneous linear equations. This is relevant to the minimization of potentials in general and sheds new light on the origin of calculable phases and geometrical CP violation.Comment: 20+9 pages, 1 figure; v1: minor changes, added clarification, matches the published versio

On predictions from spontaneously broken flavor symmetries

We discuss the predictive power of supersymmetric models with flavor symmetries, focusing on the lepton sector of the standard model. In particular, we comment on schemes in which, after certain `flavons' acquire their vacuum expectation values (VEVs), the charged lepton Yukawa couplings and the neutrino mass matrix appear to have certain residual symmetries. In most analyses, only corrections to the holomorphic superpotential from higher-dimensional operators are considered (for instance, in order to generate a realistic $\theta_{13}$ mixing angle). In general, however, the flavon VEVs also modify the K\"ahler potential and, therefore, the model predictions. We show that these corrections to the naive results can be sizable. Furthermore, we present simple analytic formulae that allow us to understand the impact of these corrections on the predictions for the masses and mixing parameters.Comment: 12 pages, 4 figures; improved version matching PLB articl

Predictivity of models with spontaneously broken non-Abelian discrete flavor symmetries

In a class of supersymmetric flavor models predictions are based on residual symmetries of some subsectors of the theory such as those of the charged leptons and neutrinos. However, the vacuum expectation values of the so-called flavon fields generally modify the K\"ahler potential of the setting, thus changing the predictions. We derive simple analytic formulae that allow us to understand the impact of these corrections on the predictions for the masses and mixing parameters. Furthermore, we discuss the effects on the vacuum alignment and on flavor changing neutral currents. Our results can also be applied to non--supersymmetric flavor models.Comment: 34 pages, 4 figures, related Mathematica package can be found at http://einrichtungen.ph.tum.de/T30e/codes/KaehlerCorrections/, updated version with added reference, matching NPB articl

CP Violation from Finite Groups

We discuss the origin of CP violation in settings with a discrete (flavor) symmetry $G$. We show that physical CP transformations always have to be class-inverting automorphisms of $G$. This allows us to categorize finite groups into three types: (i) Groups that do not exhibit such an automorphism and, therefore, in generic settings, explicitly violate CP. In settings based on such groups, CP violation can have pure group-theoretic origin and can be related to the complexity of some Clebsch-Gordan coefficients. (ii) Groups for which one can find a CP basis in which all the Clebsch-Gordan coefficients are real. For such groups, imposing CP invariance restricts the phases of coupling coefficients. (iii) Groups that do not admit real Clebsch-Gordan coefficients but possess a class-inverting automorphism that can be used to define a proper (generalized) CP transformation. For such groups, imposing CP invariance can lead to an additional symmetry that forbids certain couplings. We make use of the so-called twisted Frobenius-Schur indicator to distinguish between the three types of discrete groups. With $\Delta(27)$, $T^{\prime}$, and $\Sigma(72)$ we present one explicit example for each type of group, thereby illustrating the CP properties of models based on them. We also show that certain operations that have been dubbed generalized CP transformations in the recent literature do not lead to physical CP conservation.Comment: 45 pages, 3 figure

Anomaly-safe discrete groups

We show that there is a class of finite groups, the so-called perfect groups, which cannot exhibit anomalies. This implies that all non-Abelian finite simple groups are anomaly-free. On the other hand, non-perfect groups generically suffer from anomalies. We present two different ways that allow one to understand these statements.Comment: 11 page

No-go theorems for R symmetries in four-dimensional GUTs

We prove that it is impossible to construct a grand unified model, based on a simple gauge group, in four dimensions that leads to the exact MSSM, nor to a singlet extension, and possesses an unbroken R symmetry. This implies that no MSSM model with either a Z_{M>=3}^R or U(1)_R symmetry can be completed by a four-dimensional GUT in the ultraviolet. However, our no-go theorem does not apply to GUT models with extra dimensions. We also show that it is impossible to construct a 4D GUT that leads to the MSSM plus an additional anomaly-free symmetry that forbids the mu term.Comment: 11+1 page

SUPERSYMMETRIC UNIFICATION AND R SYMMETRIES

We review the role of R symmetries in models of supersymmetric unification in four and more dimensions, and in string theory. We show that, if one demands anomaly freedom and fermion masses, only R symmetries can forbid the supersymmetric Higgs mass term mu. We then review the proof that R symmetries are not available in conventional grand unified theories (GUTs) and argue that this prevents natural solutions to the doublet-triplet splitting problem in four dimensions. On the other hand, higher-dimensional GUTs do not suffer from this problem. We briefly comment on an explicit string-derived model in which the mu and dimension five proton decay problems are solved simultaneously by an order four discrete R symmetry. We also comment on the higher-dimensional origin of this symmetry.Comment: 17 pages, based on invited talks given at String Phenomenology 2012, CETUP 2012, PACIFIC 2012, the 4th Bethe Center Workshop and the Universe Cluster Science Week 201