General Two Higgs Doublet Model (G2HDM)

• The General Two Higgs Doublet Model (G2HDM) is the most general renormalizable extension of the Standard Model with two complex SU(2) doublets, featuring a complete scalar potential and flexible Yukawa sector.
• It incorporates both tree-level flavor-changing neutral currents and multiple CP violation sources, enabling rich phenomenology from collider signals to electroweak baryogenesis.
• The model’s basis independence, unitarity constraints, and effective field theory formulations offer a comprehensive framework for exploring UV completions and experimental tests.

Searching arXiv for the cited G2HDM literature and related terminology. The General Two Higgs Doublet Model (G2HDM) is the most general renormalizable extension of the Standard Model scalar sector based on two complex $SU(2)_L$ doublets, without imposing a discrete $\mathbb{Z}_2$ symmetry and without requiring CP conservation. In that sense it is the unconstrained 2HDM: the scalar potential contains the full set of renormalizable operators, the Yukawa sector contains two independent Yukawa structures for each fermion type, and tree-level flavor-changing neutral scalar interactions and new CP-violating phases are generic unless controlled by alignment, textures, or symmetries. After electroweak symmetry breaking, the spectrum contains five physical Higgs states, conventionally denoted $H^\pm$, $A$, $h$, and $H$, with basis freedom playing a central role in how parameters are represented (Bahl et al., 2022, Sartore et al., 2022, Branco et al., 2011). A recurrent source of ambiguity is nomenclature: in the arXiv literature the acronym G2HDM is also used for the Gauged Two Higgs Doublet Model, which is a distinct framework with an extra $SU(2)_H\times U(1)_X$ gauge sector (Huang et al., 2015, Yang et al., 2024).

1. Scalar-sector definition and basis structure

The scalar potential of the general 2HDM is conventionally written as

$$\begin{aligned} V &= m_{11}^2 \Phi_1^\dagger \Phi_1 + m_{22}^2 \Phi_2^\dagger \Phi_2 - (m_{12}^2 \Phi_1^\dagger \Phi_2 + h.c.) \ &\quad + \frac{\lambda_1}{2} (\Phi_1^\dagger \Phi_1)^2 + \frac{\lambda_2}{2} (\Phi_2^\dagger \Phi_2)^2 + \lambda_3 (\Phi_1^\dagger \Phi_1)(\Phi_2^\dagger \Phi_2) + \lambda_4 (\Phi_1^\dagger \Phi_2)(\Phi_2^\dagger \Phi_1) \ &\quad + \left[\frac{\lambda_5}{2} (\Phi_1^\dagger \Phi_2)^2 + \lambda_6 (\Phi_1^\dagger \Phi_1)(\Phi_1^\dagger \Phi_2) + \lambda_7 (\Phi_2^\dagger \Phi_2)(\Phi_1^\dagger \Phi_2) + h.c.\right]. \end{aligned}$$

Here $m_{11}^2$, $m_{22}^2$, and $\mathbb{Z}_2$0 are real, while $\mathbb{Z}_2$1, $\mathbb{Z}_2$2, $\mathbb{Z}_2$3, and $\mathbb{Z}_2$4 may be complex, giving 14 real parameters in the scalar potential before exploiting basis freedom (Kanemura et al., 2015, Bahl et al., 2022).

A central structural fact is that $\mathbb{Z}_2$5 and $\mathbb{Z}_2$6 are not physical observables. They may be redefined by $\mathbb{Z}_2$7 transformations in Higgs-family space, and many apparently distinct parameter sets therefore describe the same physics. The Higgs basis is particularly useful: only one linear combination acquires a vacuum expectation value, while the orthogonal combination contains the non-SM scalar degrees of freedom. In one standard parametrization,

$\mathbb{Z}_2$8

with $\mathbb{Z}_2$9 and $H^\pm$0 (Kanemura et al., 2015). In the alignment limit, the observed 125 GeV state is dominantly aligned with the vacuum direction, and its couplings approach the SM pattern.

The gauge-invariant bilinear formulation sharpens this basis structure. Defining the hermitian matrix

$H^\pm$1

one obtains the basis-independent constraints

$H^\pm$2

In this language, stability, electroweak symmetry breaking, CP properties, scalar masses, trilinear and quartic couplings, gauge interactions, and Yukawa couplings can all be reformulated in terms of gauge-invariant quantities. An especially compact result is the all-order relation

$H^\pm$3

for the charged Higgs mass at a charge-conserving minimum (Sartore et al., 2022).

2. Yukawa structure, flavor violation, and restricted limits

The defining phenomenological feature of the G2HDM is its Yukawa generality. Unlike type-I, type-II, type-X, or type-Y models, both scalar doublets may couple to all fermions. In the Higgs-basis language used in flavor analyses, this introduces, in addition to the SM-like Yukawas $H^\pm$4, a second set of Yukawa matrices $H^\pm$5 for each fermion sector $H^\pm$6. In the alignment regime, the observed Higgs $H^\pm$7 remains approximately SM-like, while the extra neutral and charged scalars $H^\pm$8, $H^\pm$9, and $A$0 retain unsuppressed $A$1-driven interactions (Hou et al., 2023, Teunissen, 2023).

Tree-level flavor-changing neutral currents are then generic, because the two Yukawa structures for a given fermion type are not generically simultaneously diagonalizable. This is the key difference between the general model and the natural-flavor-conserving constructions enforced by discrete symmetries. The standard restricted models are recovered as symmetry limits: type I, type II, lepton-specific, and flipped models impose specific Yukawa assignments so that all fermions of a given charge and chirality couple to only one doublet, thereby eliminating tree-level FCNCs (Branco et al., 2011). In a broader flavor taxonomy, the fully general setup has also been denoted 2HDM-X or, in flavor-centered discussions, associated with type-III 2HDM language (Diaz-Cruz et al., 2010, Herrero-Garcia et al., 2019).

Several control mechanisms have been developed. Yukawa alignment assumes that the two Yukawa matrices for each fermion type are proportional at a high scale, so that tree-level FCNCs vanish there. However, one-loop running spoils exact alignment. In leading-log form, the induced misalignment generates off-diagonal neutral-Higgs couplings proportional to structures such as $A$2, yielding a calculable lower bound on exotic FCNCs in generic aligned 2HDMs barring cancellations and exact discrete symmetries (Braeuninger et al., 2010). Other suppression strategies discussed in the literature include Cheng–Sher-type four-texture ansätze and approximate family symmetries (Diaz-Cruz et al., 2010, Branco et al., 2011).

In recent g2HDM phenomenology, a heuristic flavor pattern has often been adopted in which the extra Yukawas echo the SM hierarchy,

$A$3

This permits $A$4, allows $A$5, and suppresses first-generation couplings. The intent is not to impose natural flavor conservation, but to encode a hierarchical flavor organization that can simultaneously soften FCNC and EDM constraints (Hou et al., 2023, Teunissen, 2023).

3. CP violation and electroweak baryogenesis

The G2HDM admits several qualitatively different sources of CP violation. CP may be violated explicitly or spontaneously in the scalar potential, or generated in the Yukawa sector through complex extra Yukawa matrices. In the language of the Higgs basis, the CP-odd invariants

$A$6

vanish if and only if CP is conserved. A notable feature of the full general model is that exact alignment can coexist with CP violation; this does not occur in the softly broken $A$7-symmetric case, where exact alignment forces CP conservation in the scalar sector (Bahl et al., 2022).

This enlarged CP structure is one reason the G2HDM has been studied as a framework for electroweak baryogenesis. In one widely studied realization, the Higgs potential is taken CP-conserving and all new CP violation is placed in the Yukawa sector. The baryogenesis source is then dominated by the top sector, especially

$A$8

while a strong first-order electroweak phase transition is supplied by $A$9 Higgs quartics. The favored exotic-scalar spectrum is sub-TeV, with benchmarks such as

$h$0

and the observation that masses around $h$1 GeV would likely make baryogenesis even more efficient (Hou et al., 2023).

Electric dipole moments provide the sharpest low-energy probe of these CP phases. In the g2HDM, the dominant electron EDM contribution is typically the two-loop Barr–Zee diagram, top dominated and sensitive to $h$2 and $h$3. A central result is the flavor-structured cancellation condition

$h$4

with $h$5. This “natural flavor cancellation” can keep the eEDM below present bounds while retaining baryogenesis-scale CP violation. The corresponding phenomenological expectation is that the eEDM may first emerge around $h$6, possibly even around $h$7, while the neutron EDM may appear around $h$8, with $h$9-class experiments probing even cancellation regions (Hou et al., 2023, Teunissen, 2023).

Spontaneous CP violation also arises in UV completions that reduce to the G2HDM at low energies. In the two Higgs bi-doublet left-right symmetric model, the decoupling regime $H$0 yields a general 2HDM with spontaneous CP violation, nontrivial vacuum phases, three light neutral Higgs bosons, and one light charged Higgs pair at the electroweak scale (Liu et al., 2012). This illustrates that the general 2HDM CP structure is not only a bottom-up parametrization but also a natural low-energy limit of more complete theories.

4. Theoretical consistency: unitarity, boundedness from below, and vacuum structure

The G2HDM scalar sector is strongly constrained by perturbative unitarity. At high energy, the relevant amplitudes are two-body scalar scatterings, equivalently including longitudinal gauge bosons through the equivalence theorem. In the most general model, the full problem can be organized by hypercharge and isospin into block-diagonal scattering matrices. One explicit construction yields 14 neutral channels, 8 singly charged channels, and 3 doubly charged channels, reducing the unitarity problem to the eigenvalues of the blocks $H$1, $H$2, $H$3, and the invariant $H$4, with perturbative unitarity imposed on all eigenvalues (Kanemura et al., 2015).

Recent work has supplemented the exact numerical treatment with analytic necessary and sufficient criteria tailored to the fully complex potential. New tools include principal minors, Gershgorin disks, and Frobenius-norm bounds. One practical conclusion is that the crude estimate $H$5 is too loose: unitarity may fail for significantly smaller values, while $H$6 is much safer in numerical scans (Bahl et al., 2022).

Boundedness from below is similarly nontrivial once $H$7 and $H$8 are nonzero and complex. In the restricted $H$9 limit, the familiar necessary and sufficient tree-level conditions are

$SU(2)_H\times U(1)_X$0

but these become only necessary in the general case (Jurciukonis et al., 2018). Exact bounded-from-below conditions for the fully general model have been derived by rewriting the quartic potential as a positivity problem for a quartic polynomial in field directions, and practical fast-scan tests have also been obtained (Bahl et al., 2022).

Vacuum consistency is most transparently described in the bilinear formalism. Writing the potential as

$SU(2)_H\times U(1)_X$1

one obtains the discriminant

$SU(2)_H\times U(1)_X$2

with the criterion

$SU(2)_H\times U(1)_X$3

A physically useful relation is

$SU(2)_H\times U(1)_X$4

which links the neutrality-enforcing Lagrange multiplier directly to the charged Higgs mass. This renders vacuum stability tests interpretable in terms of physical masses (Bahl et al., 2022). The same bilinear machinery also clarifies the distinction between charge-conserving, charge-breaking, and CP-breaking vacua in a basis-independent way (Sartore et al., 2022).

5. Flavor, EDM, and collider phenomenology

Low-energy flavor observables probe the extra Yukawa sector with exceptional sensitivity. In kaon physics, charged-Higgs loops driven by $SU(2)_H\times U(1)_X$5 couplings generate contributions to $SU(2)_H\times U(1)_X$6, $SU(2)_H\times U(1)_X$7, and rare decays such as $SU(2)_H\times U(1)_X$8 and $SU(2)_H\times U(1)_X$9. Among these, $$\begin{aligned} V &= m_{11}^2 \Phi_1^\dagger \Phi_1 + m_{22}^2 \Phi_2^\dagger \Phi_2 - (m_{12}^2 \Phi_1^\dagger \Phi_2 + h.c.) \ &\quad + \frac{\lambda_1}{2} (\Phi_1^\dagger \Phi_1)^2 + \frac{\lambda_2}{2} (\Phi_2^\dagger \Phi_2)^2 + \lambda_3 (\Phi_1^\dagger \Phi_1)(\Phi_2^\dagger \Phi_2) + \lambda_4 (\Phi_1^\dagger \Phi_2)(\Phi_2^\dagger \Phi_1) \ &\quad + \left[\frac{\lambda_5}{2} (\Phi_1^\dagger \Phi_2)^2 + \lambda_6 (\Phi_1^\dagger \Phi_1)(\Phi_1^\dagger \Phi_2) + \lambda_7 (\Phi_2^\dagger \Phi_2)(\Phi_1^\dagger \Phi_2) + h.c.\right]. \end{aligned}$$0 is one of the strongest constraints on $$\begin{aligned} V &= m_{11}^2 \Phi_1^\dagger \Phi_1 + m_{22}^2 \Phi_2^\dagger \Phi_2 - (m_{12}^2 \Phi_1^\dagger \Phi_2 + h.c.) \ &\quad + \frac{\lambda_1}{2} (\Phi_1^\dagger \Phi_1)^2 + \frac{\lambda_2}{2} (\Phi_2^\dagger \Phi_2)^2 + \lambda_3 (\Phi_1^\dagger \Phi_1)(\Phi_2^\dagger \Phi_2) + \lambda_4 (\Phi_1^\dagger \Phi_2)(\Phi_2^\dagger \Phi_1) \ &\quad + \left[\frac{\lambda_5}{2} (\Phi_1^\dagger \Phi_2)^2 + \lambda_6 (\Phi_1^\dagger \Phi_1)(\Phi_1^\dagger \Phi_2) + \lambda_7 (\Phi_2^\dagger \Phi_2)(\Phi_1^\dagger \Phi_2) + h.c.\right]. \end{aligned}$$1, often stronger than the corresponding $$\begin{aligned} V &= m_{11}^2 \Phi_1^\dagger \Phi_1 + m_{22}^2 \Phi_2^\dagger \Phi_2 - (m_{12}^2 \Phi_1^\dagger \Phi_2 + h.c.) \ &\quad + \frac{\lambda_1}{2} (\Phi_1^\dagger \Phi_1)^2 + \frac{\lambda_2}{2} (\Phi_2^\dagger \Phi_2)^2 + \lambda_3 (\Phi_1^\dagger \Phi_1)(\Phi_2^\dagger \Phi_2) + \lambda_4 (\Phi_1^\dagger \Phi_2)(\Phi_2^\dagger \Phi_1) \ &\quad + \left[\frac{\lambda_5}{2} (\Phi_1^\dagger \Phi_2)^2 + \lambda_6 (\Phi_1^\dagger \Phi_1)(\Phi_1^\dagger \Phi_2) + \lambda_7 (\Phi_2^\dagger \Phi_2)(\Phi_1^\dagger \Phi_2) + h.c.\right]. \end{aligned}$$2-physics limits, especially when the charged Higgs is heavy. A distinctive result is the special sensitivity of $$\begin{aligned} V &= m_{11}^2 \Phi_1^\dagger \Phi_1 + m_{22}^2 \Phi_2^\dagger \Phi_2 - (m_{12}^2 \Phi_1^\dagger \Phi_2 + h.c.) \ &\quad + \frac{\lambda_1}{2} (\Phi_1^\dagger \Phi_1)^2 + \frac{\lambda_2}{2} (\Phi_2^\dagger \Phi_2)^2 + \lambda_3 (\Phi_1^\dagger \Phi_1)(\Phi_2^\dagger \Phi_2) + \lambda_4 (\Phi_1^\dagger \Phi_2)(\Phi_2^\dagger \Phi_1) \ &\quad + \left[\frac{\lambda_5}{2} (\Phi_1^\dagger \Phi_2)^2 + \lambda_6 (\Phi_1^\dagger \Phi_1)(\Phi_1^\dagger \Phi_2) + \lambda_7 (\Phi_2^\dagger \Phi_2)(\Phi_1^\dagger \Phi_2) + h.c.\right]. \end{aligned}$$3 to a TeV-scale charged Higgs: the $$\begin{aligned} V &= m_{11}^2 \Phi_1^\dagger \Phi_1 + m_{22}^2 \Phi_2^\dagger \Phi_2 - (m_{12}^2 \Phi_1^\dagger \Phi_2 + h.c.) \ &\quad + \frac{\lambda_1}{2} (\Phi_1^\dagger \Phi_1)^2 + \frac{\lambda_2}{2} (\Phi_2^\dagger \Phi_2)^2 + \lambda_3 (\Phi_1^\dagger \Phi_1)(\Phi_2^\dagger \Phi_2) + \lambda_4 (\Phi_1^\dagger \Phi_2)(\Phi_2^\dagger \Phi_1) \ &\quad + \left[\frac{\lambda_5}{2} (\Phi_1^\dagger \Phi_2)^2 + \lambda_6 (\Phi_1^\dagger \Phi_1)(\Phi_1^\dagger \Phi_2) + \lambda_7 (\Phi_2^\dagger \Phi_2)(\Phi_1^\dagger \Phi_2) + h.c.\right]. \end{aligned}$$4 amplitude contains a double CKM enhancement of the $$\begin{aligned} V &= m_{11}^2 \Phi_1^\dagger \Phi_1 + m_{22}^2 \Phi_2^\dagger \Phi_2 - (m_{12}^2 \Phi_1^\dagger \Phi_2 + h.c.) \ &\quad + \frac{\lambda_1}{2} (\Phi_1^\dagger \Phi_1)^2 + \frac{\lambda_2}{2} (\Phi_2^\dagger \Phi_2)^2 + \lambda_3 (\Phi_1^\dagger \Phi_1)(\Phi_2^\dagger \Phi_2) + \lambda_4 (\Phi_1^\dagger \Phi_2)(\Phi_2^\dagger \Phi_1) \ &\quad + \left[\frac{\lambda_5}{2} (\Phi_1^\dagger \Phi_2)^2 + \lambda_6 (\Phi_1^\dagger \Phi_1)(\Phi_1^\dagger \Phi_2) + \lambda_7 (\Phi_2^\dagger \Phi_2)(\Phi_1^\dagger \Phi_2) + h.c.\right]. \end{aligned}$$5 term, making this decay far more powerful than naive heavy-mass suppression would suggest (Hou et al., 2022).

The same scan studies show that $$\begin{aligned} V &= m_{11}^2 \Phi_1^\dagger \Phi_1 + m_{22}^2 \Phi_2^\dagger \Phi_2 - (m_{12}^2 \Phi_1^\dagger \Phi_2 + h.c.) \ &\quad + \frac{\lambda_1}{2} (\Phi_1^\dagger \Phi_1)^2 + \frac{\lambda_2}{2} (\Phi_2^\dagger \Phi_2)^2 + \lambda_3 (\Phi_1^\dagger \Phi_1)(\Phi_2^\dagger \Phi_2) + \lambda_4 (\Phi_1^\dagger \Phi_2)(\Phi_2^\dagger \Phi_1) \ &\quad + \left[\frac{\lambda_5}{2} (\Phi_1^\dagger \Phi_2)^2 + \lambda_6 (\Phi_1^\dagger \Phi_1)(\Phi_1^\dagger \Phi_2) + \lambda_7 (\Phi_2^\dagger \Phi_2)(\Phi_1^\dagger \Phi_2) + h.c.\right]. \end{aligned}$$6 is achievable after imposing $$\begin{aligned} V &= m_{11}^2 \Phi_1^\dagger \Phi_1 + m_{22}^2 \Phi_2^\dagger \Phi_2 - (m_{12}^2 \Phi_1^\dagger \Phi_2 + h.c.) \ &\quad + \frac{\lambda_1}{2} (\Phi_1^\dagger \Phi_1)^2 + \frac{\lambda_2}{2} (\Phi_2^\dagger \Phi_2)^2 + \lambda_3 (\Phi_1^\dagger \Phi_1)(\Phi_2^\dagger \Phi_2) + \lambda_4 (\Phi_1^\dagger \Phi_2)(\Phi_2^\dagger \Phi_1) \ &\quad + \left[\frac{\lambda_5}{2} (\Phi_1^\dagger \Phi_2)^2 + \lambda_6 (\Phi_1^\dagger \Phi_1)(\Phi_1^\dagger \Phi_2) + \lambda_7 (\Phi_2^\dagger \Phi_2)(\Phi_1^\dagger \Phi_2) + h.c.\right]. \end{aligned}$$7-physics and kaon-mixing bounds, while light-$$\begin{aligned} V &= m_{11}^2 \Phi_1^\dagger \Phi_1 + m_{22}^2 \Phi_2^\dagger \Phi_2 - (m_{12}^2 \Phi_1^\dagger \Phi_2 + h.c.) \ &\quad + \frac{\lambda_1}{2} (\Phi_1^\dagger \Phi_1)^2 + \frac{\lambda_2}{2} (\Phi_2^\dagger \Phi_2)^2 + \lambda_3 (\Phi_1^\dagger \Phi_1)(\Phi_2^\dagger \Phi_2) + \lambda_4 (\Phi_1^\dagger \Phi_2)(\Phi_2^\dagger \Phi_1) \ &\quad + \left[\frac{\lambda_5}{2} (\Phi_1^\dagger \Phi_2)^2 + \lambda_6 (\Phi_1^\dagger \Phi_1)(\Phi_1^\dagger \Phi_2) + \lambda_7 (\Phi_2^\dagger \Phi_2)(\Phi_1^\dagger \Phi_2) + h.c.\right]. \end{aligned}$$8 benchmarks can also produce negative values down to about $$\begin{aligned} V &= m_{11}^2 \Phi_1^\dagger \Phi_1 + m_{22}^2 \Phi_2^\dagger \Phi_2 - (m_{12}^2 \Phi_1^\dagger \Phi_2 + h.c.) \ &\quad + \frac{\lambda_1}{2} (\Phi_1^\dagger \Phi_1)^2 + \frac{\lambda_2}{2} (\Phi_2^\dagger \Phi_2)^2 + \lambda_3 (\Phi_1^\dagger \Phi_1)(\Phi_2^\dagger \Phi_2) + \lambda_4 (\Phi_1^\dagger \Phi_2)(\Phi_2^\dagger \Phi_1) \ &\quad + \left[\frac{\lambda_5}{2} (\Phi_1^\dagger \Phi_2)^2 + \lambda_6 (\Phi_1^\dagger \Phi_1)(\Phi_1^\dagger \Phi_2) + \lambda_7 (\Phi_2^\dagger \Phi_2)(\Phi_1^\dagger \Phi_2) + h.c.\right]. \end{aligned}$$9. In contrast, $m_{11}^2$0 are much less informative in the studied parameter space: $m_{11}^2$1 is dominated by long-distance two-photon physics, and $m_{11}^2$2 changes by at most about $m_{11}^2$3 (Hou et al., 2022).

Higgs-mediated quark flavor violation is another characteristic signature. In full scans of the general 2HDM consistent with perturbativity, vacuum stability, unitarity, oblique parameters, Higgs signal strengths, $m_{11}^2$4 mixing, $m_{11}^2$5, and direct $m_{11}^2$6 limits, the allowed rates remain substantial. At $m_{11}^2$7,

$m_{11}^2$8

while at $m_{11}^2$9,

$m_{22}^2$0

If the mild $m_{22}^2$1-mass-splitting discrepancy were confirmed and dominated by tree-level scalar exchange, the model would predict

$m_{22}^2$2

This phenomenology is difficult to reproduce in vector-like-quark-only completions because correlated $m_{22}^2$3-flavor violation suppresses the corresponding Higgs flavor-violating rates much more strongly (Herrero-Garcia et al., 2019).

The Higgs self-interactions are also unusually flexible. In a Higgs-basis analysis imposing tree-level unitarity, boundedness from below, the $m_{22}^2$4-parameter constraint, and $m_{22}^2$5, the cubic self-coupling $m_{22}^2$6 of the observed Higgs may even flip sign, while the quartic coupling satisfies

$m_{22}^2$7

The same analysis finds that the extra scalar masses are typically below about $m_{22}^2$8 GeV for $m_{22}^2$9 and below about $\mathbb{Z}_2$00 GeV for $\mathbb{Z}_2$01, unless alignment is approached very closely (Jurciukonis et al., 2018). This suggests a phenomenology in which alignment does not imply decoupling, but often coexists with a relatively light extended scalar spectrum.

6. Effective-field-theory form, UV origins, and conceptual boundaries

The general 2HDM is frequently treated as an effective field theory. The 2HDM EFT supplements the renormalizable Lagrangian with higher-dimensional operators and retains both doublets explicitly in the low-energy theory: $\mathbb{Z}_2$02 A complete Warsaw-like construction yields 228 dimension-six operators, together with three dimension-five neutrino-mass operators. Rewriting the EFT in the Higgs basis separates operators that modify masses and standard-model-like couplings from operators that contribute only to scattering amplitudes, and makes correlations among multi-Higgs processes more transparent. The same framework also provides specific operator subsets for the type-I, II, X, and Y symmetry-restricted models, with 76 operators common to all four types (Dermisek et al., 2024).

Several UV completions illuminate how the G2HDM can emerge nontrivially. The two Higgs bi-doublet left-right symmetric model has a genuine decoupling regime in which, for $\mathbb{Z}_2$03, the theory reduces to a general 2HDM with spontaneous CP violation and a light electroweak-scale two-doublet sector (Liu et al., 2012). A composite near-conformal construction based on dilaton EFT yields instead a highly constrained inert 2HDM corner, with

$\mathbb{Z}_2$04

and a specific hierarchy among quartic couplings, showing how strong dynamics can populate only a narrow subspace of the general parameter space (Appelquist et al., 2022). In 6D gauge-Higgs unification, two Higgs doublets can arise from extra-dimensional gauge fields, but the quartic potential is then a restricted subset of the full 2HDM potential, often of the form

$\mathbb{Z}_2$05

rather than the fully general scalar interaction (Chang et al., 2012).

These examples clarify an important conceptual point. The G2HDM is best viewed not as one model but as the maximal renormalizable 2HDM parameter space consistent with gauge symmetry. Symmetry-restricted 2HDMs, aligned models, BGL constructions, EFT deformations, and UV completions all occupy identifiable submanifolds inside that space. Equally important, the general 2HDM should not be conflated with the Gauged Two Higgs Doublet Model, which embeds the two doublets into an $\mathbb{Z}_2$06 gauge multiplet and uses the same acronym in a different sense (Huang et al., 2015, Yang et al., 2024). The former is a general low-energy field-theoretic framework; the latter is a distinct gauge extension.