Classification of finite reparametrization symmetry groups in the three-Higgs-doublet model

Abstract

Symmetries play a crucial role in electroweak symmetry breaking models with non-minimal Higgs content. Within each class of these models, it is desirable to know which symmetry groups can be implemented via the scalar sector. In N-Higgs-doublet models, this classification problem was solved only for N=2 doublets. Very recently, we suggested a method to classify all realizable finite symmetry groups of Higgs-family transformations in the three-Higgs-doublet model (3HDM). Here, we present this classification in all detail together with an introduction to the theory of solvable groups, which play the key role in our derivation. We also consider generalized-CP symmetries, and discuss the interplay between Higgs-family symmetries and CP-conservation. In particular, we prove that presence of the ℤ₄ symmetry guarantees the explicit CP-conservation of the potential. This work completes classification of finite reparametrization symmetry groups in 3HDM.

Appendix: 3HDM potentials with non-abelian Higgs-family symmetry group

Here, for the reader’s convenience, we list once again Higgs potentials with a given symmetry group. We focus here on cases with non-abelian groups from the list (84) because abelian ones were already discussed in detail in [37]. In each case we start from the most general potential compatible with the given realizable group presented in the main text and use the residual reparametrization freedom to simplify the coefficients of the potential (usually, it amounts to rephasing of doublets which makes some of the coefficients real). For each group G, the potential written below faithfully represents all possible Higgs potentials with realizable symmetry group G. In this sense, the symmetry group uniquely defines the phenomenology of the scalar sector of 3HDM, the only exception being D ₆ with its two distinct realizations.

Group $D_{6} \simeq\mathbb{Z}_{3} \rtimes\mathbb{Z}_{2}^{*}$

Consider the most general phase-independent part of the Higgs potential

and the additional terms

(A.1)

For generic λ _i , these terms are symmetric only under the group ℤ₃ generated by

$$ a_3 = \left ( \begin{array}{c@{\quad}c@{\quad}c} \omega& 0 & 0 \\ 0 & \omega^2 & 0 \\ 0 & 0 & 1 \end{array} \right ),\quad \omega= \exp \biggl({2\pi i \over3} \biggr). $$

(A.2)

If it happens that the product λ ₁ λ ₂ λ ₃ is purely real, then by rephasing of doublets one can make all coefficients in (A.1) real. The resulting potential, $V_{0} + V_{\mathbb{Z}_{3}}$, is symmetric under $D_{6} \simeq\mathbb{Z}_{3} \rtimes\mathbb{Z}_{2}^{*}$ generated by a ₃ and the CP-transformation.

Group $D_{8} \simeq\mathbb{Z}_{4} \rtimes\mathbb {Z}_{2}^{*}$

Consider now terms

$$ V_{\mathbb{Z}_4} = \lambda_1 \bigl(\phi_3^\dagger \phi_1\bigr) \bigl(\phi_3^\dagger \phi_2\bigr) + \lambda_2 \bigl(\phi_1^\dagger \phi_2\bigr)^2 + h.c., $$

(A.3)

which are symmetric under the group ℤ₄ generated by

$$ a_4 = \left ( \begin{array}{c@{\quad}c@{\quad}c} i & 0 & 0 \\ 0 & -i & 0 \\ 0 & 0 & 1 \end{array} \right ). $$

(A.4)

It is always possible to compensate the phases of λ ₁ and λ ₂ by an appropriate rephasing of the doublets. Therefore, the potential $V_{0} + V_{\mathbb{Z}_{4}}$ is symmetric under the group $D_{8} \simeq\mathbb{Z}_{4} \rtimes\mathbb{Z}_{2}^{*}$ generated by a ₄ and the CP-transformation.

Group D ₆ of unitary transformations

Let us restrict the coefficients of V ₀ in the way that guarantees the symmetry under ϕ ₁↔ϕ ₂. Then, V ₀ turns into

(A.5)

where all coefficients are real and generic. Imposing the same requirement on $V_{\mathbb{Z}_{3}}$ and performing rephasing, we obtain

(A.6)

where λ ₁ is real and $\sin\psi_{3} \not= 0$. The resulting potential, $V_{1} + V_{D_{6}}$, is symmetric under D ₆ generated by a ₃ and

$$ b = \left ( \begin{array}{c@{\quad}c@{\quad}c} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & -1 \end{array} \right ). $$

(A.7)

There are no other Higgs-family or generalized-CP transformations which leave this potential invariant. Any explicitly CP-violating D ₆-symmetric 3HDM potential can always be brought into this form.

Group $D_{6} \times\mathbb{Z}_{2}^{*}$

If in the previous case we set sinψ ₃=0 in (A.6), then the potential becomes symmetric under $D_{6} \times\mathbb{Z}_{2}^{*}$ generated by a ₃, b, and the generalized CP-transformation b⋅CP.

Group $D_{8} \times\mathbb{Z}_{2}^{*}$

The potential $V_{1} + V_{\mathbb{Z}_{4}}$ is symmetric under the group $D_{8} \times\mathbb{Z}_{2}^{*}$ generated by a ₄, b, and b⋅CP.

Group $A_{4} \rtimes\mathbb{Z}_{2}^{*}$

A potential symmetric under $A_{4} \rtimes\mathbb{Z}_{2}^{*}$ can be brought into the following form:

(A.8)

with complex $\tilde{\lambda}$. Its symmetry group is generated by independent sign flips of the individual doublets, by cyclic permutations of ϕ ₁, ϕ ₂, ϕ ₃, and by the exchange of any pair of doublet together with the CP-transformation. An alternative form of this potential is

(A.9)

Group $S_{4} \times\mathbb{Z}_{2}^{*}$

If the parameter $\tilde{\lambda}$ in (A.8) is real or, equivalently, λ _ReIm=0 in (A.9), the potential becomes symmetric under $S_{4} \times\mathbb{Z}_{2}^{*}$ generated by sign flips, all permutation of the three doublets, and the CP-transformation.

Group (ℤ₃×ℤ₃)⋊ℤ₂≃Δ(54)/ℤ₃

Consider the following potential:

(A.10)

with generic real m ², λ ₀, λ ₁, λ ₂ and complex λ ₃. The symmetry group of this potential is (ℤ₃×ℤ₃)⋊ℤ₂=Δ(54)/ℤ₃. Here, Δ(54) is generated by the same a ₃ and b as before and, in addition, by the cyclic permutation

$$ c = \left ( \begin{array}{c@{\quad}c@{\quad}c} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{array} \right ), $$

(A.11)

while the subgroup ℤ₃ is the center of SU(3).

Group $(\mathbb{Z}_{3}\times\mathbb{Z}_{3})\rtimes (\mathbb{Z}_{2}\times\mathbb{Z}_{2}^{*})$

The potential (A.10) becomes symmetric under a generalized-CP transformation if λ ₃=k⋅π/3 with any integer k. In this case, one can make λ ₃ real by a rephasing transformation. The extra generator then is the CP-transformation.

Group $\varSigma(36)\rtimes\mathbb{Z}_{2}^{*}$

The same potential (A.10) becomes symmetric under the group $\varSigma(36)\rtimes\mathbb{Z}_{2}^{*}$ if, upon rephasing, λ ₃=(3λ ₁−λ ₂)/2. The potential can then be rewritten as

(A.12)

Here I ₀ and I ₁ are the SU(3)-invariants

(A.13)

It is remarkable that this potential has only one “structural” free parameter, and the term containing it reduces the full SU(3) symmetry group to a finite subgroup Σ(36).