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Buchbinder, I. L.; Гитман, Д. М.; Shelepin, A. L.

doi:10.1023/a:1015244830241

Cited by 20 publications

(50 citation statements)

References 41 publications

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“…This is true for the Poincaré group [14] as well as for continuous [15] and discrete [5] internal symmetries. Thus, understanding outer automorphisms is essential to understand CP.…”

Section: Introductionmentioning

confidence: 86%

Symmetries of symmetries and geometrical CP violation

Fallbacher

Trautner

2015

Nuclear Physics B

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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

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“…This is true for the Poincaré group [14] as well as for continuous [15] and discrete [5] internal symmetries. Thus, understanding outer automorphisms is essential to understand CP.…”

Section: Introductionmentioning

confidence: 86%

Symmetries of symmetries and geometrical CP violation

Fallbacher

Trautner

2015

Nuclear Physics B

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“…This will be investigated in more detail by the present authors in a forthcoming publication [13]. This is of particular physical interest, because parity (P), charge conjugation (C), and time reversal (T) transformations are all known to correspond to outer automorphisms [14][15][16], which is specifically true for finite groups [17]. For these, physically very interesting situations can arise where CP is explicitly violated by the Clebsch-Gordan coefficients of a finite group [18,19] or CP transformations may be forced to be of order larger than two for specific groups [19] (see also [20] for a brief introduction, and [21] for a specific phenomenological model).…”

Section: Introductionmentioning

confidence: 91%

Simultaneous Block Diagonalization of Matrices of Finite Order

Bischer,

Döring,

Trautner

2020

Preprint

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It is well known that a set of non-defect matrices can be simultaneously diagonalized if and only if the matrices commute. In the case of non-commuting matrices, the best that can be achieved is simultaneous block diagonalization. Here we give an efficient algorithm to explicitly compute a transfer matrix which realizes the simultaneous block diagonalization of unitary matrices whose decomposition in irreducible blocks (common invariant subspaces) is known from elsewhere. Our main motivation lies in particle physics, where the resulting transfer matrix must be known explicitly in order to unequivocally determine the action of outer automorphisms such as parity, charge conjugation, or time reversal on the particle spectrum.

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“…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.…”

mentioning

confidence: 88%

“…The physically interesting case of the Poincaré or Lorentz group has been investigated in [5] and it has been found that there exists a complex conjugation Out which serves as a model independent physical CP transformation.…”

Section: Cp Transformations For Lie Groupsmentioning

confidence: 99%

CP as a Symmetry of Symmetries

Trautner

2017

J. Phys.: Conf. Ser.

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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

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Untitled

Cited by 20 publications

References 41 publications

Symmetries of symmetries and geometrical CP violation

Symmetries of symmetries and geometrical CP violation

Simultaneous Block Diagonalization of Matrices of Finite Order

CP as a Symmetry of Symmetries

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