Discrete symmetries in the three-Higgs-doublet model

Abstract

N-Higgs-doublet models (NHDM) are among the most popular examples of electroweak symmetry breaking mechanisms beyond the Standard Model. Discrete symmetries imposed on the NHDM scalar potential play a pivotal role in shaping the phenomenology of the model, and various symmetry groups have been studied so far. However, in spite of all efforts, the classification of finite Higgs-family symmetry groups realizable in NHDM for any N>2 is still missing. Here, we solve this problem for the three-Higgs-doublet model by making use of Burnside's theorem and other results from pure finite group theory which are rarely exploited in physics. Our method and results can be also used beyond high-energy physics, for example, in study of possible symmetries in three-band superconductors.Comment: 5 pages; v2: expanded introduction, some minor corrections, matches the published versio

Symmetries of symmetries and geometrical CP violation

“…(4.5). This transformation 11 In case we have two pairs of equal parameters or three (or more) equal parameters the discrete symmetry of the potential is enhanced from (54) to G : 15) or to a continuous group, respectively, in agreement with the maximal realizable symmetry (36) ∼ = G/Z 3 found in [18,19]. 12 As the CP transformations are mapped onto each other under the action of the equivalence transformations, all order two CP transformations are equivalent at the level of the potential.…”

“…In this model one assigns three electroweak Higgs doublets to the three dimensional representation (H 1 , H 2 , H 3 ) = H = 3 of (27). Due to the continuous symmetries and the representation content of the model, the actual discrete symmetry group of the Higgs potential is not (27) but the larger (54) [3,6,18,19]. We will, thus, work with the full discrete symmetry of the potential, G = (54).…”

“…Performing a scan over all subgroups of E while checking for non-trivial solutions of (7.8), we find that, up to conjugation, 17 the largest subgroups of E which allow for a non-trivial are two groups of order 18 (SG (18,4) and SG (18,3)) and a group of order 48 (SG(48, 29)). Of course, also the subgroups of these groups allow for non-trivial solutions to (7.8) which, however, are less restrictive on .…”

“…The group (27) is a normal subgroup of (54) which can be obtained by dropping the generator C. The restriction of the conjugation map conj(C) from (54) to (27) leads to an outer automorphism of (27), which acts as an exchange of all singlet representations with their respective complex conjugates. 18 The real (54)-doublets thus can be obtained from the pairs of mutually complex conjugate one-dimensional representations of (27) as 2 1 = (1 1 , 1 2 ), 2 2 = (1 3 , 1 6 ), 2 3 = (1 4 , 1 8 ), and 2 4 = (1 5 , 1 7 ). The triplet representation is sent to itself under the automorphism with a matrix representation of C acting on the triplet as C, given in (A.3).…”

Higgs decay rate to two photons in a model with two fermiophobic-Higgs doublets

“…The symmetries for the two-Higgs doublet model was obtained in Refs. [6] and in the case of three-Higgs doublet models in [7].…”