Geometrical CP Violation
aa r X i v : . [ h e p - ph ] J un CFTP/10-007, DO-TH 11/19
Geometrical CP Violation
Ivo de Medeiros Varzielas , ∗ and David Emmanuel-Costa † Departamento de F´ısica and Centro de F´ısica Te´orica de Part´ıculas (CFTP)Instituto Superior T´ecnico, Av. Rovisco Pais, 1049-001 Lisboa, Portugal and Fakult¨at f¨ur Physik, Technische Universit¨at Dortmund D-44221 Dortmund, Germany (Dated: November 7, 2018)Spontaneous CP-violating phases that do not depend on the parameters of the Higgs sector - theso-called calculable phases - are investigated. The simplest realization is in models with 3 Higgsdoublets, in which the scalar potential is invariant under non-Abelian symmetries. The non-Abeliandiscrete group ∆ ( ) is shown to lead to the known structure of calculable phases obtained with ∆ ( ). We investigate the possibily of accommodating the observed fermion masses and mixings. PACS numbers: 11.30.Hv, 12.15.Ff, 12.60.FrKeywords: CP violation; Fermion masses and mixings; Flavor symmetries; Extensions of Higgs sector
Since the discovery of CP violation in 1964, its originremains a fundamental open question in particle physics.In the context of the Standard Model (SM), CP symme-try is explicitly broken at the Lagrangian level throughcomplex Yukawa couplings which lead to CP violationin charged weak interactions via the Cabibbo-Kobayashi-Maskawa (CKM) matrix. Among many mechanisms thatgenerate CP asymmetry beyond the SM, the possibil-ity that CP is spontaneously broken together with thegauge symmetry group is a very attractive scenario [1, 2].One remarkable phenomenological implication of Sponta-neous CP violation (SCPV) is that it provides an appeal-ing solution to the strong CP problem [3], since the onlysource of CP violation are the vacuum phases. SCPVcan also soften the well known SUSY CP problem [4]. Fi-nally, it is relevant to point out that in perturbative stringtheory CP asymmetry can in principle only arise spon-taneously through complex VEVs of moduli and matterfields [5].In models of spontaneous CP violation (SCPV) onestarts from a Lagrangian that conserves CP, which im-plies that all parameters of the scalar potential are real.Then, the CP asymmetry is achieved spontaneously whenthe gauge interactions are broken through complex vac-uum expectation values (VEVs) of Higgs multiplets. Infact, just having complex VEVs is not sufficient to guar-antee CP violation in the model. One has further toverify that it is not possible to find a unitary transfor-mation, U , acting on the Higgs fields as φ i −→ φ ′ i = U ij φ i , (1)such that the following condition holds U ij h φ j i ∗ = h φ i i , (2)while leaving the full Lagrangian invariant. If such atransformation is found, CP is a conserved symmetryeven in the presence of complex Higgs VEVs.The main purpose of this Letter is the search for a dis-crete symmetry that leads to a framework of SCPV where the VEVs of the Higgs multiplets have geometrical val-ues, independently of any arbitrary coupling constants inthe scalar potential - i.e. calculable phases [6]. If such asymmetry exists calculable phases are stable against ra-diative corrections [7]. It has been shown in Ref. [6] thatcalculable phases leading to geometrical SCPV requiremore than two Higgs doublets and non-Abelian symme-tries, otherwise it is always possible to find an unitarytransformation, U , which is a symmetry of the potentialand fulfills Eq. (2). The authors found an interesting ex-ample of calculable phases with SCPV in the case of threeHiggs doublets transforming under the discrete symme-try ∆ ( ). In order to find symmetries that generate cal-culable SCPV, we start by considering the most general SU ( ) × U ( ) potential V ( φ ) with three Higgs doublets φ i , having identical hypercharge, V ( φ ) = X i h − λ i φ † i φ i + A i ( φ † i φ i ) i + X i A18 , 1697 (2003), arXiv:hep-ph/0212050.[5] E. Witten, Phys. Lett., B149 , 351 (1984); A. Stromingerand E. Witten, Commun. Math. Phys., , 341 (1985);M. Dine, R. G. Leigh, and D. A. MacIntire, Phys. Rev.Lett., , 2030 (1992), arXiv:hep-th/9205011.[6] G. C. Branco, J. M. Gerard, and W. Grimus, Phys. Lett., B136 , 383 (1984).[7] S. Weinberg, Phys. Rev., D7 , 2887 (1973); H. Georgiand A. Pais, Phys. Rev. D, , 1246 (1974).[8] E. Derman, Phys. Rev., D19 , 317 (1979); E. Dermanand H.-S. Tsao, ibid ., D20 , 1207 (1979).[9] C. Luhn, S. Nasri, and P. Ramond, J. Math. Phys., ,073501 (2007), arXiv:hep-th/0701188.[10] H. Ishimori et al.et al.