Minimal Three-Higgs-Doublet Models

• Minimal 3HDMs are models with three Higgs doublets that use discrete or continuous symmetries to tightly control the parameter space, enabling phenomena like geometrical CP violation.
• They apply generalized CP constraints, Natural Flavor Conservation, and inert doublet setups to achieve reduced Yukawa textures and a structured approach to dark matter.
• Phenomenological analyses on minimal 3HDMs employ unitarity bounds and collider signatures—such as radiatively enhanced charged-Higgs interactions—to probe their viability over a broad energy range.

Searching arXiv for recent and foundational papers on minimal 3HDMs, alignment, discrete symmetries, CP, dark matter, and flavour-conserving realizations. arXiv search query: "three-Higgs-doublet model minimal 3HDM alignment CP dark matter discrete symmetry generalized CP" Minimal three-Higgs-doublet models (3HDMs) are extensions of the Standard Model scalar sector with three Higgs doublets and a symmetry structure chosen so that the enlarged parameter space remains controlled. In the contemporary literature, the adjective “minimal” does not denote a single canonical model. It can refer to a renormalisable scalar sector with only three doublets and no extra scalar representations, to Natural Flavour Conservation implemented with the smallest discrete symmetry that assigns one doublet to each fermion sector, to generalized-CP constructions with sharply reduced Yukawa parameter counts, or to inert realizations in which one active doublet coexists with two dark doublets stabilized by a residual symmetry (Varzielas, 2018, Das et al., 2019, Bree et al., 2024, Aranda et al., 2019). Across these usages, 3HDMs are repeatedly employed as compact frameworks for structured electroweak symmetry breaking, flavour, CP violation, dark matter, and, more recently, combined neutrino-mass–dark-matter model building (Bento et al., 2022, Bonilla et al., 8 Jul 2026).

1. Meanings of minimality

Across the 3HDM literature, “minimal” is used in several technically distinct senses. The common element is economy of field content or parameterization rather than uniqueness of phenomenological purpose.

Usage of “minimal”	Defining condition	Representative papers
Triplet-based minimality	Three Higgs doublets form one flavour triplet; only renormalisable terms; no extra scalar representations	(Varzielas, 2018)
Flavour-conserving minimality	Three doublets, Natural Flavor Conservation, and a discrete symmetry such as $Z_3$	(Das et al., 2019, Batra et al., 10 Apr 2025)
GCP-constrained minimality	Scalar potential in one of four GCP classes; Yukawa textures with strongly reduced parameter counts	(Bree et al., 2024)
Leptonic minimality	Three doublets with no flavons or exotic leptonic fields beyond right-handed neutrinos or the Weinberg operator	(Dziewit et al., 2023)
Inert/dark minimality	One active doublet plus two inert doublets, with dark stability from $Z_3$ or $U(1)$-based symmetry	(Aranda et al., 2019, Kunčinas et al., 2024)
Non-Abelian neutrino–DM minimality	Three doublets organized as one singlet plus one doublet of $S_3$ or $D_4$	(Bonilla et al., 8 Jul 2026)

In the CP-conserving phenomenological literature, minimal 3HDMs are also described as among the simplest nontrivial extensions of the Standard Model Higgs sector, obtained by adding two more scalar doublets with the same gauge quantum numbers as the Standard Model Higgs (Das et al., 2019). In symmetry-classification work, minimality instead means a scalar potential with the largest realizable symmetry compatible with a finite parameter set, often with the fewest independent quartic couplings in a given class (Ivanov et al., 2012, Bento et al., 2022).

2. Symmetry organization and scalar potentials

The generic renormalisable 3HDM scalar potential can be written in tensor form as

$$V_H = Y_{ij}(\phi_i^\dagger \phi_j) + Z_{ij,kl}(\phi_i^\dagger \phi_j)(\phi_k^\dagger \phi_l),$$

with $Y_{ij}=Y_{ji}^\ast$ and $Z_{ij,kl}=Z_{kl,ij}=Z_{ji,lk}^\ast$ (Bree et al., 2024). Before imposing any additional symmetry, the scalar sector is large: after basis transformations in Higgs space, the general 3HDM retains $30$ independent magnitudes and $16$ independent phases (Bento et al., 2022). The central role of symmetry in 3HDM model building is therefore not optional but structural.

The finite Higgs-family symmetries realizable in the scalar sector have been completely classified. Restricting to unitary Higgs-family transformations, the realizable finite groups are

$Z_3$0

(Ivanov et al., 2012). This classification is complemented by the catalogue of continuous Abelian and non-Abelian symmetries used in 3HDM potentials, including $Z_3$1, $Z_3$2, $Z_3$3, $Z_3$4, $Z_3$5, and $Z_3$6 (Keus et al., 2013, Bento et al., 2022). In this sense, the scalar sector of 3HDMs is best understood as a symmetry-stratified space rather than as a single model.

A particularly useful minimal triplet-based construction organizes the three doublets into a faithful triplet of $Z_3$7 or $Z_3$8. For $Z_3$9, the basic $U(1)$0-invariant potential is

$U(1)$1

and subgroup reductions to $U(1)$2 or $U(1)$3 add phase-sensitive quartics (Varzielas, 2018). This realizes minimality in a strict sense: three Higgs doublets arranged as one flavour triplet, no extra scalar representations, and only renormalisable terms.

At the opposite end of the symmetry spectrum, highly constrained scalar sectors can be built from large finite or continuous symmetries. The $U(1)$4-symmetric quartic potential has only two independent quartic parameters, while $U(1)$5, $U(1)$6, and related symmetry classes reduce the quartic sector to a small set of symmetry-adapted couplings (Bento et al., 2022). Such models are “minimal” not because the field content is smaller, but because the orbit of allowed couplings is drastically compressed.

3. Vacuum structure, CP, and alignment

A central organizing principle in 3HDMs is vacuum alignment. Symmetry analyses of the scalar potential repeatedly single out the alignments

$U(1)$7

with $U(1)$8 especially relevant for inert and dark constructions (Keus et al., 2013). In the triplet-based $U(1)$9 potential $S_3$0, the basic vacuum directions are

$S_3$1

up to phases and permutations (Varzielas, 2018). These directions distinguish orbits by the number of nonzero components and already illustrate how discrete family symmetries reduce the minimization problem to a finite list of candidate vacua.

The CP analysis of 3HDMs is substantially subtler than in two-doublet models. In the invariant approach, explicit CP violation is diagnosed by CP-odd basis invariants $S_3$2, while spontaneous CP violation is diagnosed by spontaneous CP-odd invariants $S_3$3 that depend on couplings and vacuum expectation values (Varzielas, 2018). A key example is the “penguin” invariant $S_3$4, whose nonvanishing signals spontaneous CP violation. In the $S_3$5 3HDM with real $S_3$6, the alignment $S_3$7 produces a nonzero spontaneous CP-odd invariant and therefore genuine spontaneous geometrical CP violation, whereas $S_3$8 has complex phases but preserves a generalized CP symmetry and is CP-conserving (Varzielas, 2018). This distinction corrects a common misconception: complex vacuum phases do not by themselves imply CP violation.

Generalized CP (GCP) brings a second layer of structure. In a suitable scalar basis, the GCP matrix can always be written as

$S_3$9

and the scalar potential then falls into exactly four classes: CPa ($D_4$0), CPb ($D_4$1), CPc ($D_4$2), and CPd (generic $D_4$3) (Bree et al., 2024). The appearance of the special $D_4$4 class is intrinsically three-doublet; it has no direct 2HDM analogue. From the earlier finite-group classification, another important theorem follows: the presence of a $D_4$5 Higgs-family symmetry guarantees explicit CP-conservation of the scalar potential (Ivanov et al., 2012).

In phenomenological 3HDMs, one must distinguish vacuum alignment from the Higgs alignment limit. For CP-conserving 3HDMs with three nonzero VEVs, the VEVs are conveniently parameterized by

$D_4$6

The unique CP-even direction coupling to gauge bosons as the Standard Model Higgs is

$D_4$7

and the exact alignment limit is

$D_4$8

so that the light scalar $D_4$9 coincides with $$V_H = Y_{ij}(\phi_i^\dagger \phi_j) + Z_{ij,kl}(\phi_i^\dagger \phi_j)(\phi_k^\dagger \phi_l),$$0 (Das et al., 2019). This gives 3HDM alignment the same analytic structure as in 2HDMs, but with two VEV angles and two alignment conditions instead of one.

4. Yukawa textures, flavour, and flavour-conserving realizations

Minimality in the Yukawa sector is highly model dependent. In 3HDMs with generalized CP extended to the Yukawa sector, the quark Yukawa Lagrangian is

$$V_H = Y_{ij}(\phi_i^\dagger \phi_j) + Z_{ij,kl}(\phi_i^\dagger \phi_j)(\phi_k^\dagger \phi_l),$$1

and imposing the same GCP on scalars and fermions yields $$V_H = Y_{ij}(\phi_i^\dagger \phi_j) + Z_{ij,kl}(\phi_i^\dagger \phi_j)(\phi_k^\dagger \phi_l),$$2 candidate assignments $$V_H = Y_{ij}(\phi_i^\dagger \phi_j) + Z_{ij,kl}(\phi_i^\dagger \phi_j)(\phi_k^\dagger \phi_l),$$3, reduced to $$V_H = Y_{ij}(\phi_i^\dagger \phi_j) + Z_{ij,kl}(\phi_i^\dagger \phi_j)(\phi_k^\dagger \phi_l),$$4 physically viable Yukawa textures after excluding degenerate masses, block-diagonal CKM structure, and vanishing Jarlskog invariant (Bree et al., 2024). Several of these textures have only $$V_H = Y_{ij}(\phi_i^\dagger \phi_j) + Z_{ij,kl}(\phi_i^\dagger \phi_j)(\phi_k^\dagger \phi_l),$$5 real Yukawa parameters per charge sector. This is a stronger parameter reduction than in the GCP-constrained 2HDM, where the corresponding numbers are $$V_H = Y_{ij}(\phi_i^\dagger \phi_j) + Z_{ij,kl}(\phi_i^\dagger \phi_j)(\phi_k^\dagger \phi_l),$$6 and $$V_H = Y_{ij}(\phi_i^\dagger \phi_j) + Z_{ij,kl}(\phi_i^\dagger \phi_j)(\phi_k^\dagger \phi_l),$$7 real Yukawa parameters (Bree et al., 2024). In this literature, a “minimal 3HDM” is therefore often one with a GCP-constrained scalar potential and one of the lowest-parameter viable Yukawa textures.

A different sense of minimality is realized by Natural Flavour Conservation. In flavour-conserving 3HDMs, each fermion type couples to only one doublet. The democratic, or Type-Z, assignment is

$$V_H = Y_{ij}(\phi_i^\dagger \phi_j) + Z_{ij,kl}(\phi_i^\dagger \phi_j)(\phi_k^\dagger \phi_l),$$8

so that $$V_H = Y_{ij}(\phi_i^\dagger \phi_j) + Z_{ij,kl}(\phi_i^\dagger \phi_j)(\phi_k^\dagger \phi_l),$$9 couples only to charged leptons, $Y_{ij}=Y_{ji}^\ast$0 only to down-type quarks, and $Y_{ij}=Y_{ji}^\ast$1 only to up-type quarks (Batra et al., 10 Apr 2025). A closely related $Y_{ij}=Y_{ji}^\ast$2-symmetric realization uses the assignments “up-type quarks couple only to $Y_{ij}=Y_{ji}^\ast$3, down-type quarks couple only to $Y_{ij}=Y_{ji}^\ast$4, charged leptons couple only to $Y_{ij}=Y_{ji}^\ast$5” (Das et al., 2019). These are minimal in the Glashow–Weinberg sense: tree-level FCNCs are absent without enlarging the fermion content.

The leptonic sector is more restrictive. A systematic scan of non-Abelian flavour symmetries with order $Y_{ij}=Y_{ji}^\ast$6, using three Higgs doublets and no flavons, found that if one enforces the VEV alignment dictated by a minimal flavour-symmetric scalar potential, nontrivial flavour groups produce severe mass degeneracies and no realistic PMNS mixing (Dziewit et al., 2023). Relaxing the scalar-potential alignment and treating the VEV ratios $Y_{ij}=Y_{ji}^\ast$7 and $Y_{ij}=Y_{ji}^\ast$8 as free parameters does generate mass splittings, but still fails to fit charged-lepton masses, neutrino data, and nontrivial PMNS mixing simultaneously; in all viable cases the PMNS matrix remains monomial (Dziewit et al., 2023). This no-go result sharply delimits what purely Yukawa-sector flavour symmetries can achieve in genuinely minimal leptonic 3HDMs.

5. Dark sectors and neutrino-mass realizations

A major branch of minimal 3HDM model building uses two inert doublets and one active doublet. The $Y_{ij}=Y_{ji}^\ast$9-symmetric $Z_{ij,kl}=Z_{kl,ij}=Z_{ji,lk}^\ast$0HDM takes

$Z_{ij,kl}=Z_{kl,ij}=Z_{ji,lk}^\ast$1

with $Z_{ij,kl}=Z_{kl,ij}=Z_{ji,lk}^\ast$2 active and $Z_{ij,kl}=Z_{kl,ij}=Z_{ji,lk}^\ast$3 inert, while all Standard Model fermions and gauge bosons are $Z_{ij,kl}=Z_{kl,ij}=Z_{ji,lk}^\ast$4-neutral (Aranda et al., 2019). The inert vacuum

$Z_{ij,kl}=Z_{kl,ij}=Z_{ji,lk}^\ast$5

keeps $Z_{ij,kl}=Z_{kl,ij}=Z_{ji,lk}^\ast$6 exact after electroweak symmetry breaking. The neutral inert sector then exhibits pairwise mass degeneracy,

$Z_{ij,kl}=Z_{kl,ij}=Z_{ji,lk}^\ast$7

and the lightest pair $Z_{ij,kl}=Z_{kl,ij}=Z_{ji,lk}^\ast$8 forms what the paper calls “Hermaphrodite DM”: two mass-degenerate states of opposite CP parity that contribute equally to the relic density (Aranda et al., 2019).

Continuous symmetries lead to a distinct class of minimal dark 3HDMs. In $Z_{ij,kl}=Z_{kl,ij}=Z_{ji,lk}^\ast$9-based models, dark stability is provided by an unbroken continuous symmetry rather than by $30$0. In the $30$1-symmetric 3HDM with vacuum $30$2, each inert doublet yields an exactly degenerate neutral pair,

$30$3

and the model therefore contains a multi-component dark sector with two independent mass scales (Kunčinas et al., 2024). After imposing theoretical consistency, Higgs invisible-width constraints, HiggsTools, direct detection, and relic-density bounds, viable solutions were found over a broad dark-matter mass range $30$4 GeV, with the upper limit set by the scan cutoff (Kunčinas et al., 2024). In CP-conserving realizations, these degenerate pairs can be viewed either as CP-even/CP-odd partners or as states with opposite $30$5 charges.

The latest extension of this logic combines minimal non-Abelian symmetry, neutrino masses, and dark-matter stability. In 3HDMs based on global $30$6 or $30$7, the scalar sector is organized as one singlet doublet $30$8 plus one flavour doublet $30$9, and a residual $16$0 parity left by spontaneous symmetry breaking stabilizes the dark matter candidate (Bonilla et al., 8 Jul 2026). The same dark field runs in a one-loop neutrino-mass diagram, alongside a tree-level type-I seesaw contribution. The paper identifies $16$1 and $16$2 as the smallest non-Abelian groups realizing this structure and emphasizes that the resulting minimal models conserve CP in both the Yukawa and scalar sectors (Bonilla et al., 8 Jul 2026). This places 3HDMs in direct continuity with scotogenic and residual-symmetry approaches, but with the scalar content fixed at three doublets.

6. Constraints, viable parameter space, and collider signatures

Theoretical control of minimal 3HDMs depends heavily on perturbative unitarity. For the generic quartic tensor $16$3, a set of symmetry-independent necessary conditions in 3HDMs is

$16$4

and the full set of symmetry-constrained models can be treated efficiently with a principal-minors method that avoids diagonalization of large scattering matrices (Bento et al., 2022). This work also provides the complete catalogue of unitarity bounds for finite and continuous symmetry classes in 3HDMs, including $16$5, $16$6, $16$7, $16$8, $16$9, $Z_3$00, $Z_3$01, $Z_3$02, $Z_3$03, $Z_3$04, and $Z_3$05 (Bento et al., 2022).

Phenomenological viability near the Higgs alignment limit has now been mapped in detail for the democratic $Z_3$06-symmetric 3HDM. After imposing bounded-from-below conditions, perturbativity, unitarity, HiggsBounds, HiggsSignals, $Z_3$07, and oblique-parameter constraints, three CP-even mass orderings were analyzed (Batra et al., 10 Apr 2025).

Hierarchy scenario	125 GeV state	Main outcome
Regular	$Z_3$08 lightest	$Z_3$09 typically $Z_3$10 GeV; viable
Medial	$Z_3$11 intermediate	one lighter CP-even scalar allowed, $Z_3$12 GeV
Inverted	$Z_3$13 heaviest	two lighter CP-even states excluded by $Z_3$14

The central conclusion is sharp: a single lighter CP-even Higgs below $Z_3$15 GeV remains allowed, but the presence of two lighter CP-even Higgs bosons is ruled out in this minimal Type-Z setup (Batra et al., 10 Apr 2025). The same analysis used active learning to navigate the $Z_3$16-dimensional parameter space, reflecting the practical complexity of even symmetry-reduced 3HDMs.

Naturalness considerations point in the same direction: symmetry helps, but does not trivialize the hierarchy problem. In the $Z_3$17 3HDM, the strict Veltman condition for the Standard-Model-like scalar cannot be imposed while maintaining phenomenological viability, but the minimum achievable ratio $Z_3$18 is about $Z_3$19 for $Z_3$20 TeV (Chakrabarty et al., 2018). In the active $Z_3$21 3HDM, by contrast, it is possible to set $Z_3$22 and tune several nonstandard-scalar Veltman coefficients down to $Z_3$23, although other scalars remain more finely tuned (Chakrabarty et al., 2018). Minimal 3HDMs therefore ameliorate the Higgs mass fine-tuning problem in a model-dependent way, rather than eliminating it universally.

CP-violating flavour-conserving 3HDMs exhibit a distinctive charged-Higgs phenomenon absent in 2HDMs. When CP violation is isolated in the charged Higgs sector, the two physical charged Higgs bosons generate electron and neutron EDMs through Barr–Zee and Weinberg-operator contributions, but a new cancellation mechanism appears: the cancellation becomes exact when the charged Higgs masses are degenerate, and mass degeneracies at the $Z_3$24 level are sufficient to evade current EDM bounds in viable regions (Logan et al., 2020). The same study found allowed parameter space with both charged Higgs bosons lighter than $Z_3$25 GeV and large CP-violating phases, while satisfying perturbativity, direct-search bounds, and $Z_3$26 (Logan et al., 2020). This is one of the cleanest examples of a genuinely 3HDM effect tied directly to the presence of two charged Higgs eigenstates.

A complementary collider discriminator is the loop-induced $Z_3$27 interaction in flavour-conserving 3HDMs. In the alignment limit, these vertices are purely radiative, and their one-loop amplitudes can be organized into UV-finite and gauge-invariant subsets (Chakrabarty et al., 25 Dec 2025). After imposing theoretical consistency, $Z_3$28, and $Z_3$29, the corresponding form factors show a sizeable increment of about $Z_3$30 over the analogous 2HDM quantities (Chakrabarty et al., 25 Dec 2025). At the $Z_3$31 TeV LHC, vector-boson-fusion production of $Z_3$32 followed by the cascade

$Z_3$33

can yield $Z_3$34 values in the $Z_3$35 range (Chakrabarty et al., 25 Dec 2025). Such a signal would simultaneously probe the radiative $Z_3$36 coupling and establish the existence of two charged Higgs states, making it a particularly direct signature of a flavour-conserving 3HDM rather than a 2HDM.

Taken together, these results show that minimal 3HDMs are not a single model class but a family of tightly constrained constructions. Their scalar potentials are best understood through realizable discrete and continuous symmetries; their vacuum structure supports both CP-conserving and spontaneously CP-violating phases; their Yukawa sectors range from highly predictive generalized-CP textures to flavour-conserving democratic assignments; and their inert or residual-symmetry realizations naturally accommodate multi-component dark matter and, in the newest non-Abelian constructions, neutrino masses. The common theme is that three doublets are already sufficient to generate phenomena—geometrical CP violation, exact charged-Higgs EDM cancellations, multi-component dark sectors, and radiatively enhanced $Z_3$37 vertices—that do not arise, or arise less flexibly, in two-doublet models (Varzielas, 2018, Bree et al., 2024, Logan et al., 2020, Chakrabarty et al., 25 Dec 2025).