Compatible abelian symmetries in N-Higgs-doublet models

Received: November Compatible abelian symmetries in N-Higgs-doublet C.C. Nishi 0 1 2 3 Open Access 0 1 2 3 c The Authors. 0 1 2 3 0 Maryland Center for Fundamental Physics 1 09. 210-170, Santo Andr e, SP , Brazil 2 Universidade Federal do ABC - UFABC 3 University of Maryland , College Park, MD 20742 , U.S.A We analyze the compatibility between abelian symmetries acting in two different sectors of a theory using the Smith Normal Form method. We focus on N-Higgs-doublet models (NHDMs) and on the compatibility between symmetries in the Higgs potential and in the Yukawa interactions, which were separately analyzed in previous works. It is shown that two equal (isomorphic) symmetry groups that act in two separate sectors are not necessarily compatible in the whole theory and an upper bound is found for the size of the group that can be implemented in the entire NHDM. We also develop useful techniques to analyze compatibility and extend a symmetry from one sector to another. Consequences to the supersymmetric case are briefly discussed. Discrete and Finite Symmetries; Beyond Standard Model 1 Introduction 2 Review of the method Example Notation Charges are not unique Linear independence Height of vector Z4-3HDM Z2 Z2-3HDM Matching up symmetries Maximal symmetries for three Higgs doublets 5 Discarding maximal symmetries Discarding maximal symmetries for N = 4 and N = 5 Discarding maximal symmetries for N 6 6 Building symmetric models 6.4 From reduced DY to full DY 7 Discussion and conclusion A Proof of proposition B Rephasing space From minimal to maximal number of terms From the Yukawa sector to the potential From the potential to the Yukawa sector C Backbone structure for maximal symmetry in 3HDM D Height of pi vectors for maximal DY E Height of p3 can not be reduced F Examples of full models and textures Symmetry has always played a crucial role in our understanding of fundamental physics. The construction of the current framework the Standard Model (SM) of particle physics has culminated in 2012 with the discovery of the Higgs boson [1, 2], the particle that results from the breaking of the electroweak symmetry in its simplest form. Hence, it was also a successful attempt to probe a hidden (broken) symmetry in nature and its breaking mechanism. However, as we probe higher and higher energies, new symmetries may emerge as key ingredients to understand the physics beyond the SM. As we try to guess which new symmetry governs the physics above the electroweak scale, we are also confronted with the question of what is the breaking scale and what could be the signatures after breaking. One old but fruitful example where the symmetry should linked to the smallness of neutrino masses (see, e.g., ref. [3] and references therein). In parallel to continuous symmetries, discrete symmetries are also possible ingredients with which we can understand flavor (for a review, see e.g. refs. [48]) and the stability of dark matter (with, e.g., R-parity [9] or matter parity [10]). In the effort to classify and discover useful abelian discrete symmetries, the Smith Normal Form (SNF) method has been used successfully in various contexts to find discrete symmetries arising from the breaking of continuous gauge symmetries [11, 12], find useful R-symmetries in supersymmetric extensions of the SM [13], justify two-zero textures in the neutrino mass matrix with symmetries [14] and classify abelian symmetries in multi-Higgs-doublets models [15]. The latter class of models will be the focus of this work. The N-Higgs-doublet models (NHDMs) are among the most conservative extensions of the SM and they can present additional features that are absent in the single-Higgsdoublet SM such as spontaneous CP violation [16, 17] or geometric CP violation [1821]. In many ways, these new phenomena are possible because the scalar potential has more structure to allow different symmetry breaking paths. Along with more structure comes the possibility of accommodating larger symmetries, specially discrete symmetries. One can for example impose a Z 2 symmetry to naturally suppress dangerous flavor changing currents for quarks [22] or obtain a dark matter candidate with radiative neutrino mass generation [23]. The list of all possible symmetries that can be accommodated in the 2HDM is short and the groups in it are small [2430] (more symmetries arise if we allow for accidental symmetries [3136]). More and larger discrete symmetries are possible in the 3HDM potential [37], and all possible breaking patterns were summarized recently in ref. [38]. In general, we can accommodate larger symmetries as we add more fields. If we are restricted to abelian symmetries, the maximal order of the group that can be separately implemented in the Higgs potential and in the Yukawa interactions were presented in refs. [39] and [15], respectively. In this work, we want to extend the methods of [15] to consider the compatibility issues, i.e., (i) how to (...truncated)