Three Higgs Doublets Models (3HDM)

• Three-Higgs-doublet models (3HDM) extend the Standard Model by incorporating three scalar SU(2)ₗ doublets, enabling diverse symmetry and CP-violation structures.
• They employ finite group theory to classify discrete symmetries—such as Z₂, Z₃, D₆, and S₄—which constrain the scalar potential and suppress flavor-changing neutral currents.
• 3HDM offers predictive mass relations and novel collider signatures with applications in flavor physics and dark matter, guiding new physics searches beyond the Standard Model.

A three-Higgs-doublet model (3HDM) is an extension of the Standard Model (SM) in which the scalar sector contains three SU(2)ₗ doublet fields with hypercharge +½. These models introduce a multi-component Higgs field structure enabling a diverse range of symmetry, mass, and coupling patterns in both the electroweak and flavor sectors. They serve as templates for rich symmetry and CP-violating phenomena, novel mechanisms for flavor physics, dark matter candidates, and collider signatures. The classification, construction, and phenomenological consequences of symmetry groups that can be imposed on the scalar sector of 3HDMs were established using finite group theory methods, with particular attention to which finite Higgs-family symmetries are consistent with a renormalizable, gauge-invariant scalar potential that does not realize accidental continuous symmetries (Ivanov et al., 2012, Ivanov et al., 2012).

1. Structure of the 3HDM Scalar Potential and Symmetry Principles

The renormalizable, gauge-invariant scalar potential for the three-Higgs-doublet model is

$$V = Y_{ij} (\phi_i)^\dagger \phi_j + Z_{ijkl} (\phi_i)^\dagger \phi_j (\phi_k)^\dagger \phi_l$$

where $\phi_{i=1,2,3}$ are the Higgs doublets, $Y_{ij}$ is a Hermitian $3 \times 3$ matrix, and $Z_{ijkl}$ is a rank-4 tensor symmetric under appropriate index exchanges. Discrete (finite) symmetries—termed "Higgs-family symmetries"—act as transformations among the doublets, modulo the SM gauge redundancy, and are subgroups of $PSU(3) = SU(3)/\mathbb{Z}_3$.

Discrete symmetry imposition restricts the tensor structure, relating the coefficients (e.g., mass parameters and quartic couplings) and leading to predictive scalar spectra. The physical motivation for discrete symmetries in 3HDMs includes suppression of Higgs-mediated flavor-changing neutral currents (FCNCs), the realization of CP violation, and the prevention of unwanted Goldstone bosons.

2. Classification of Discrete Symmetries: Finite Group Theory Approach

The central achievement is the classification of all realizable finite symmetry groups in the scalar potential of 3HDM. Key insights are:

• Available Abelian Building Blocks: Only $Z_2$, $Z_3$, $Z_4$, $Z_2 \times Z_2$, and $Z_3 \times Z_3$ can be imposed without promoting the scalar potential symmetry to a continuous group.
• Burnside’s Theorem and Group Structure: Any finite symmetry group $G$ that can appear must have order $|G| = 2^a 3^b$ with $a, b \geq 0$ (since any prime divisor of $|G|$ must correspond to one of the abelian symmetries above). Burnside’s theorem then implies that $G$ is solvable, i.e., possesses a normal abelian subgroup $A$.
• Self-centralizing Normal Abelian Subgroup: For each allowed abelian $A$, one seeks group extensions $G = A \rtimes K$ where $K \subset \mathrm{Aut}(A)$ and $A = C_G(A)$ (the centralizer condition ensures $K$ acts nontrivially).

Explicit extension construction yields the full list of non-accidentally-continuous finite symmetry groups that can be realized in the scalar potential. These groups and their extensions are as follows:

Abelian Group $A$	Extensions $G$	Realizability in 3HDM
$Z_2$	$Z_2$	always
$Z_3$	$Z_3$, $D_6 = Z_3 \rtimes Z_2$	both realizable
$Z_4$	$Z_4$, $D_8 = Z_4 \rtimes Z_2$	both; $Q_8$ not realizable
$Z_2\times Z_2$	$Z_2\times Z_2$, $D_8$, $A_4$ ($T$), $S_4$ ($O$)	all realizable
$Z_3\times Z_3$	$(Z_3\times Z_3)\rtimes Z_2$, $(Z_3\times Z_3)\rtimes Z_4$	both realizable

Non-abelian groups arise as nontrivial semi-direct products with nontrivial $K$, yielding $D_6$, $D_8$, $A_4$, $S_4$, $A(54)/Z_3$, and $\Sigma(36)$. The explicit construction rules out certain groups, e.g., $Q_8$, because imposing $Q_8$ necessarily restores accidental continuous symmetry in the potential (Ivanov et al., 2012).

3. Explicit Group Extensions and Physical Realization

For each abelian "building block," the explicit structure of the possible nonabelian extensions is dictated by automorphism considerations:

• For $Z_3$, $\mathrm{Aut}(Z_3) \cong Z_2$, allowing $D_6 = Z_3 \rtimes Z_2$.
• For $Z_4$, both $D_8$ and $Q_8$ are a priori possible, but only $D_8$ is physically realizable in 3HDM since $Q_8$ requires coefficient vanishing that accidentally enhances the symmetry to continuous.
• For $Z_2\times Z_2$ (with $\mathrm{Aut}(Z_2\times Z_2)=S_3$), one finds $A_4 = (Z_2\times Z_2)\rtimes Z_3$ and $S_4 = (Z_2\times Z_2)\rtimes S_3$ among the allowed nonabelian symmetries.

In the $Z_3\times Z_3$ case, only specific automorphism subgroups (especially those contained in $Sp(2,3)\simeq SL(2,3)$) are compatible with the physical realization, leading to discrete groups such as $A(54)/Z_3$ and $\Sigma(36)$. In each case, the imposition of the group restricts the form of the potential via symmetry-motivated equalities and relations among coefficients.

4. Phenomenological and Theoretical Implications

• Scalar Mass Spectra: Imposing a discrete symmetry typically enforces relations among mass parameters, generating characteristic mass degeneracies or mass relations. For example, an $S_3$ or $D_6$-symmetric potential naturally leads to mass degeneracy among subsets of the Higgs bosons, mimicking aspects of the two-Higgs-doublet model (2HDM).
• CP Violation: Some finite groups, such as those with a $Z_4$ factor, can enforce explicit CP conservation in the scalar sector. Notably, the presence of a $Z_4$ suffices to guarantee explicit CP conservation, whereas other groups such as $D_6$ admit both CP-violating and CP-conserving realizations depending on the embedding of generalized CP (Ivanov et al., 2012).
• Absence of Extra Goldstones: Unlike continuous symmetries, discrete finite symmetries do not lead to unwanted Goldstone bosons if spontaneously broken, preventing an overabundance of massless scalar states.
• Implications for Vacuum Structure: The breakdown of the full imposed symmetry by the Higgs vacuum expectation values determines the pattern of residual symmetries, affecting possible dark matter stabilization and flavor structures (Ivanov et al., 2014).

5. Beyond the Higgs Sector: Applications and Methodological Impact

The methods used—especially the application of Burnside’s theorem and finite group extension techniques—are broadly applicable in any context involving multiple order parameters or complex symmetry patterns. In particular, analogous symmetry classification plays a role in the study of phase patterns in three-band superconductors and more generally in any model with three coupled scalar fields or order parameters (Ivanov et al., 2012). Identification of all possible finite symmetry realizations in 3HDM thus has cross-disciplinary utility.

The formalism described here underscores the importance of systematic group-theoretic classification, not only for phenomenological model-building but also for ensuring completeness and consistency in the space of possible discrete-symmetry-extended scalar sectors.

6. Summary Table: Allowed Symmetry Groups in 3HDM

Type	Symmetry Groups (and isomorphisms)
Abelian	$Z_2$, $Z_3$, $Z_4$, $Z_2 \times Z_2$, $Z_3 \times Z_3$
Nonabelian	$D_6$, $D_8$, $A_4 (T)$, $S_4 (O)$, $A(54)/Z_3$, $\Sigma(36)$

These results establish that the only possible finite (non-continuous) Higgs-family symmetry groups in the three-Higgs-doublet model scalar sector are: $$Z_2,\ Z_3,\ Z_4,\ Z_2 \times Z_2,\ D_6,\ D_8,\ T \cong A_4,\ O\cong S_4,\ (Z_3 \times Z_3) \rtimes Z_2,\ (Z_3 \times Z_3) \rtimes Z_4$$ All are realized as appropriate extensions of one of the main abelian building blocks via a nontrivial automorphism group action (Ivanov et al., 2012, Ivanov et al., 2012).

7. Role of These Results in Model Building and Interdisciplinarity

The finite group classification provides a comprehensive toolkit for model builders specifying which scalar potentials can be consistently endowed with discrete symmetries. This has direct impact for:

• Controlling Higgs-mediated FCNC: By selecting admissible symmetry groups, one can systematically avoid large tree-level FCNCs.
• Enabling Predictive Mass Patterns: Enforced degeneracies and textures translate into distinctive collider signals and decay hierarchies.
• Geometric CP Violation: The possibility of calculable, symmetry-protected CP-violating vacuum phases—referred to as "geometric CP violation"—emerges for certain symmetry groups.
• Interdisciplinary Transfer: The methodology applies with minimal modification to other multi-component systems with order parameters transforming under finite groups, such as in condensed matter physics.

In conclusion, the classification of finite discrete symmetries in the 3HDM scalar sector provides a rigorous foundation for the construction and analysis of multi-Higgs doublet models, establishing the allowed landscape of symmetry-motivated potentials and guiding phenomenological exploration well beyond the Standard Model. The approach exemplifies the power of finite group theory in constraining and organizing scalar sector model building (Ivanov et al., 2012, Ivanov et al., 2012).