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)