CP as a Symmetry of Symmetries

Abstract

Abstract. It is explained that the Standard Model combined charge conjugation and parity transformation (CP) is a simultaneous complex conjugation outer automorphism transformation of gauge and space-time symmetries. Simple examples are given for the general concept of outer automorphisms ("symmetries of symmetries"), as well as for their possible actions on physical theories. It is highlighted that complex conjugation outer automorphisms do not, in general, exist for all symmetries. Examples are given for cases in which the physical CP transformation is violated as a consequence of requiring another symmetry. A toy model is illustrated in which CP is spontaneously violated in the broken phase of a continuous gauge symmetry, while an unbroken outer automorphism protects the topological vacuum angle at θ = 0. IntroductionOne of the main questions raised by the Standard Model of particle physics (SM) is the origin of the SM flavor structure. By observation, the violation of flavor and the violation of matter-anti matter symmetry are highly intertwined in the SM. In particular, CP violation has never been observed without flavor violation, implying that also the strong CP problem can be regarded as inherent part of the flavor puzzle. Ultimately, a complete theory of flavor should also be a theory of CP violation.Among many other approaches, the flavor puzzle has been addressed with discrete symmetries. In this context, the interplay of CP transformations and discrete flavor symmetries has been investigated [1,2]. It has been found that CP transformations are described by particular outer automorphisms of discrete symmetries [2,3]. In review, this is not surprising as it has already been understood much earlier that outer automorphisms also describe CP transformations of gauge [4] and space-time symmetries [5].A crucial difference between discrete and continuous groups is that many discrete groups are inconsistent with outer automorphisms which correspond to physical CP transformations in generic models [3]. For such groups it has been found that CP violating phases are calculable and take quantized ("geometrical") values. This has been observed in models of explicit [3,6,7] and spontaneous CP violation [8,3] alike. Moreover, very recently a model has been found in which geometrical CP violation is predicted in the dynamical breaking of a continuous gauge symmetry to a discrete subgroup, while an unbroken outer automorphism enforces θ = 0 for the topological vacuum angle of the gauge group [9].The aim of this talk is to give a brief overview of the topic of predictive CP violation and outer automorphisms. For this, the concept of an outer automorphism (a "symmetry of a symmetry") is discussed based on simple examples. Then it is explained how the well established

A string theory of flavor and CP

“…We think it is always self-explanatory and clear from the context when we refer to which object. 9 Note that the other factorized dualityM is also part of Oη(2, 2, Z) through the following relation:…”

“…For example, this can done by using GAP [33]. We note that SG(108, 17) is a group of CP-type IIA -in contrast to ∆(54) which is a CP-type I group [8,9].…”

“…[4], where a CP candidate was identified as an outer automorphism of the traditional flavor group. This provided a string theory origin of the general mechanism of group theoretical CP violation discussed earlier in [5][6][7][8][9][10]. Still, a comprehensive picture of the origin of flavor and CP remained illusive: A priori, it is not clear whether the interpretation of the geometry of compact dimensions and string theory (space group) selection rules [11][12][13] gives the complete set of symmetries.…”

Dark Matter from self-dual gauge/Higgs dynamics

“…The perturbative WW * → AA, As, ss DM annihilation cross sections are thereby equal to those given in eq. (12). Furthermore there are extra WW → W * semi-annihilations, as in DM models with an ad-hoc Z 3 symmetry [32].…”

“…Such term would violate CP at non-perturbative level. The SU(N ), SO(2N ) and E 6 groups with symmetric Dynkin diagrams admit a Z 2 outer automorphism (complex conjugation) [12] that acts on vectors by flipping the sign of some vectors, as determined by the vanishing of some f abc group structure constants. More simply, the CP-even vectors are those associated to purely imaginary generators T a in some complex representation (e.g.…”