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 ∆(27), T , and Σ(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.

Flavor symmetry plays a crucial role in the standard model of particle physics but its origin is still unknown. We develop a new method (based on outer automorphisms of the Narain space group) to determine flavor symmetries within compactified string theory. A picture emerges where traditional (discrete) flavor symmetries, CP-like symmetries and modular symmetries (like T -duality) of string theory combine to unified flavor symmetries. The groups depend on the geometry of compact space and the geographical location of fields in the extra dimensions. We observe a phenomenon of "local flavor groups" with potentially different flavor symmetries for the various sectors of quarks and leptons. This should allow interesting connections to existing bottom-up attempts in flavor model building.

Modular transformations of string theory (including the well-known stringy dualities) play a crucial role in the discussion of discrete flavor symmetries in the Standard Model. They are at the origin of CP-transformations and provide a unification of CP with traditional flavor symmetries. Here, we present a novel, fully systematic method to reliably compute the unified flavor symmetry of the low-energy effective theory, including enhancements from the modular transformations of string theory. The unified flavor group is non-universal in moduli space and exhibits the phenomenon of "Local Flavor Unification" where different sectors of the theory can be subject to different flavor structures.

The Standard Model (SM) is amended by one generation of quarks and leptons which are vectorlike (VL) under the SM gauge group but chiral with respect to a new Uð1Þ 3−4 gauge symmetry. We show that this model can simultaneously explain the deviation of the muon g − 2 as well as the observed anomalies in b → sμ þ μ − transitions without conflicting with the data on Higgs decays, lepton flavor violation, or B s −B s mixing. The model is string theory motivated and Grand Unified Theory compatible, i.e. UV complete, and fits the data predicting VL quarks, leptons, and a massive Z 0 at the TeV scale, as well as τ → 3μ and τ → μγ within reach of future experiments. The Higgs couplings to SM generations are automatically aligned in flavor space.

We point out that, for Dirac neutrinos, in addition to the standard thermal cosmic neutrino background (CνB) there could also exist a non-thermal neutrino background with comparable number density. As the right-handed components are essentially decoupled from the thermal bath of standard model particles, relic neutrinos with a non-thermal distribution may exist until today. The relic density of the non-thermal (nt) background can be constrained by the usual observational bounds on the effective number of massless degrees of freedom N eff , and can be as large as nν nt 0.5 nγ. In particular, N eff can be larger than 3.046 in the absence of any exotic states. Non-thermal relic neutrinos constitute an irreducible contribution to the detection of the CνB, and, hence, may be discovered by future experiments such as PTOLEMY. We also present a scenario of chaotic inflation in which a non-thermal background can naturally be generated by inflationary preheating. The non-thermal relic neutrinos, thus, may constitute a novel window into the very early universe.

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

We describe a new, generally applicable strategy for the systematic construction of basis invariants (BIs). Our method allows one to count the number of mutually independent BIs and gives controlled access to the interrelations (syzygies) between mutually dependent BIs. Due to the novel use of orthogonal hermitian projection operators, we obtain the shortest possible invariants and their interrelations. The substructure of non-linear BIs is fully resolved in terms of linear, basis-covariant objects. The substructure distinguishes real (CP-even) and purely imaginary (CP-odd) BIs in a simple manner. As an illustrative example, we construct the full ring of BIs of the scalar potential of the general Two-Higgs-Doublet model.Everybody is used to the conventional way of setting up quantum field theory models: One picks fields in certain representations of symmetries and the Lagrangian is parametrized as linear combination of all symmetry invariant operators up to a certain dimension. However, if there are multiple fields with exclusively the same quantum numbers, these fields are physically indistinguishable, implying that they may be mixed at will, without observable consequences. On the Lagrangian level, such a mixing of fields does, in fact, correspond to a mixing of symmetry invariant operators, thereby parametrizing the Lagrangian in different ways. This arbitrariness in parametrization (or basis choice, in different words) must not affect physical statements derived from a model. Notwithstanding this, the presence of large basis change freedoms often obscures the physical properties of a model.In order to make the physical discussion as general and transparent as possible, it seems worthwhile to use basis invariant (BI) objects. An original arena for BI techniques was the detection of CP violation in the Standard Model (SM) [1] and extensions [2][3][4][5]. Here, a formulation in terms of basis invariants (BIs) immediately gets rid of spurious rephasings, thereby allowing direct access to physical properties of the model. Many more applications of BIs are conceivable and -ultimately -it should be possible to describe and relate all physical observables, say S-matrix elements, correlation functions etc., in terms of BI objects. Having such a formulation would be wonderful, but has to date not been put forward in general.Here we solve a major technical problem which arises as the first step along the way to any BI formulation: Given a theory formulated in an arbitrary basis, how does one obtain basis independent quantities in a controlled manner? Several different ways have been used to construct invariants in the literature (for a certainly incomplete list see e.g. [2][3][4][5][6][7][8][9][10][11][12][13]), none of which is entirely satisfactory for varying reasons. Several occurring shortcomings of previous approaches are at the same time advantages of our newly proposed method:• It is completely clear for us when we can stop looking for new invariants, i.e. when we have found a complete set of independent invaria...

Physical observables cannot depend on the basis one chooses to describe fields. Therefore, all physically relevant properties of a model are, in principle, expressible in terms of basisinvariant combinations of the parameters. However, in many cases it becomes prohibitively difficult to establish key physical features exclusively in terms of basis invariants. Here, we advocate an alternative route in such cases: the formulation of basis-invariant statements in terms of basis-covariant objects. We give several examples where the basis-covariant path is superior to the traditional approach in terms of basis invariants. In particular, this includes the formulation of necessary and sufficient basis-invariant conditions for various physically distinct forms of CP conservation in two-and three-Higgs-doublet models. I. INTRODUCTIONWhen describing the Standard Model (SM) or building models beyond the SM, one always faces the notorious freedom of basis-choices. Complex fields can be rephased, yet the physics emerging from the model must be invariant under these rephasings. In models with several fields with identical quantum numbers, the freedom of basis choices is even larger and includes arbitrary rotations in the space of these fields. One may fix a basis for the initial fields, then arrive at the physical (mass eigenstate) fields and explore their phenomenology. Or one can switch to a different basis and explore the phenomenology there. Although the Lagrangian and the intermediate calculations may look vastly different in different bases, the observables must be the same.This utterly obvious statement may look less obvious when one actually gets down to practical calculations. Parameters of the Lagrangian depend on the basis choice, and attributing physical *