---
title: '2HDMcT Type-III: Scalar Triplet Seesaw Model'
url: https://www.emergentmind.com/topics/two-higgs-doublet-type-ii-seesaw-model-2hdmct-type-iii
type: topic
---

# 2HDMcT Type-III: Scalar Triplet Seesaw Model

The Two-Higgs-Doublet Type-II Seesaw Model, usually denoted \(2HDMcT\), is the extension of a two-Higgs-doublet model by a complex \(SU(2)_L\) triplet scalar whose vacuum expectation value generates Majorana neutrino masses through the type-II seesaw mechanism. In this literature, the phrase “Type-II” may refer either to the seesaw mechanism or, in some papers, to the Yukawa assignment of the doublet sector; by contrast, the more recent label “2HDMcT Type-III” refers to a **Type-III Yukawa realization of the 2HDM sector inside the scalar-triplet model**, not to a fermionic type-III seesaw. This distinction is essential, because papers on 2HDM-III with fermionic singlets or fermion triplets are structurally different models [1406.6814], [2508.04493], [1812.00244].

## 1. Terminology and model identification

The scalar-triplet literature represented here uses “2HDMcT” for a model with two Higgs doublets and one complex scalar triplet implementing a type-II seesaw. In this usage, the neutrino-mass sector is always scalar-triplet-based. What changes across papers is the treatment of the **2HDM Yukawa sector**: Type-I, Type-II, aligned, and, in one recent study, Type-III [2403.07722], [2104.03275], [2410.11140], [2508.04493].

| Label | Meaning in the cited literature | Representative paper |
|---|---|---|
| \(2HDMcT\) | Two Higgs doublets plus a complex scalar triplet; neutrino masses from type-II seesaw | [1812.07719] |
| “2HDMcT Type-III” | \(2HDMcT\) with a Type-III Yukawa texture assignment in the 2HDM sector | [2508.04493] |
| 2HDM-III with seesaw | Generic 2HDM-III plus fermionic singlets or fermion triplets; **not** a scalar-triplet \(2HDMcT\) | [1812.00244] |

A closely related source of confusion is the word “triplet.” In the genuine \(2HDMcT\), the triplet is a **scalar** and the neutrino-mass mechanism is type-II seesaw. In the 2HDM-III paper “See-Saw Mechanism in the 2HDM through the simultaneous introduction of a singlet and triplet of Majorana,” the triplet is an \(SU(2)_L\) **fermion triplet** used for a type-III seesaw, and there is **no scalar triplet** in the model [1812.00244]. The low-scale type-I-seesaw extension of 2HDM-III with right-handed neutrinos is likewise a different construction [1807.08530].

## 2. Field content, scalar potential, and neutrino-mass operator

The scalar content consists of two electroweak doublets and one complex electroweak triplet. The literature represented here uses both \(Y=1\) and \(Y=2\) conventions for the triplet, while the doublets are the usual two Higgs doublets. A common parametrization is
\[
\Phi_1 = \begin{pmatrix} \phi_1^+ \\ \phi_1^0 \end{pmatrix}, \qquad
\Phi_2 = \begin{pmatrix} \phi_2^+ \\ \phi_2^0 \end{pmatrix}, \qquad
\Delta= \begin{pmatrix} \delta^+/\sqrt{2} & \delta^{++} \\
(v_t+\delta^0+i\eta_0)/\sqrt{2} & -\delta^+/\sqrt{2} \end{pmatrix},
\]
with
\[
\phi_1^0=\frac{1}{\sqrt{2}}(v_1+\psi_1+i\eta_1), \qquad
\phi_2^0=\frac{1}{\sqrt{2}}(v_2+\psi_2+i\eta_2),
\]
and
\[
\sqrt{v_1^2+v_2^2+2v_t^2}=246~{\rm GeV}, \qquad \tan\beta=\frac{v_2}{v_1}.
\]
This enlarged scalar sector yields a doubly charged scalar, additional singly charged states, and triplet-doublet mixing, all absent in an ordinary 2HDM [2403.07722], [2305.08030].

A general renormalizable scalar potential used in modern \(2HDMcT\) analyses is
\[
\begin{aligned}
V(\Phi_1,\Phi_2,\Delta) &= m_{11}^2 \Phi_1^\dagger\Phi_1+m_{22}^2\Phi_2^\dagger\Phi_2-[m_{12}^2\Phi_1^\dagger\Phi_2+{\rm h.c.}] \\
&\quad+\frac{\lambda_1}{2}(\Phi_1^\dagger\Phi_1)^2 +\frac{\lambda_2}{2}(\Phi_2^\dagger\Phi_2)^2 +\lambda_3(\Phi_1^\dagger\Phi_1)(\Phi_2^\dagger\Phi_2) +\lambda_4(\Phi_1^\dagger\Phi_2)(\Phi_2^\dagger\Phi_1) \\
&\quad+\left\{\frac{\lambda_5}{2}(\Phi_1^\dagger\Phi_2)^2 +\big[\beta_1(\Phi_1^\dagger\Phi_1)+\beta_2(\Phi_2^\dagger\Phi_2)\big] \Phi_1^\dagger\Phi_2+{\rm h.c.}\right\} \\
&\quad+\lambda_6\,\Phi_1^\dagger \Phi_1\, {\rm Tr}(\Delta^{\dagger}{\Delta}) +\lambda_7\,\Phi_2^\dagger \Phi_2\, {\rm Tr}(\Delta^{\dagger}{\Delta}) +\lambda_8\,\Phi_1^\dagger{\Delta}\Delta^{\dagger} \Phi_1 +\lambda_9\,\Phi_2^\dagger{\Delta}\Delta^{\dagger} \Phi_2 \\
&\quad+\left\{\mu_1 \Phi_1^T i\sigma^2\Delta^{\dagger}\Phi_1 + \mu_2\Phi_2^T i\sigma^2\Delta^{\dagger}\Phi_2 + \mu_3 \Phi_1^T i\sigma^2\Delta^{\dagger}\Phi_2  + {\rm h.c.}\right\} \\
&\quad +m^2_{\Delta}\, {\rm Tr}(\Delta^{\dagger}{\Delta}) +\bar{\lambda}_8({\rm Tr}\Delta^{\dagger}{\Delta})^2 +\bar{\lambda}_9{\rm Tr}(\Delta^{\dagger}{\Delta})^2 .
\end{aligned}
\]
The dimensionful trilinears \(\mu_1,\mu_2,\mu_3\) are the characteristic doublet-triplet couplings. They connect the doublets to the triplet, induce triplet-doublet mixing after electroweak symmetry breaking, and enter the scalar mass matrices [2403.07722]. In several analyses a \(Z_2\) symmetry is imposed to suppress tree-level FCNCs, with \(\beta_1=\beta_2=0\), softly broken by \(m_{12}^2,\mu_1,\mu_2,\mu_3\) [2403.07722], [2410.11140].

The neutrino-mass operator is the triplet Yukawa interaction
\[
-\mathcal{L}_{\rm Yukawa}\supset -Y_\nu\,L^T\,C\otimes i\sigma^2\,\Delta\,L + {\rm h.c.},
\]
or equivalently
\[
\mathcal{L}_Y^\Delta = \frac{1}{2} y_{ij}\, L_i^T C\, i\tau_2\, \Delta\, L_j + \text{h.c.},
\]
which yields Majorana neutrino masses once the triplet acquires a vev. The cited papers write this either as
\[
(M_\nu)_{ij}=\sqrt{2}\,y_{ij}\,v_\Delta
\]
or
\[
M_\nu=\frac{v_\Delta}{\sqrt2}\,\mathbf h,
\]
with the difference reflecting convention choices [1504.07087], [1406.6814].

## 3. Electroweak symmetry breaking and scalar spectrum

After electroweak symmetry breaking, the \(2HDMcT\) contains **11 physical Higgs states**:
\[
h_1,\ h_2,\ h_3,\qquad A_1,\ A_2,\qquad H_1^\pm,\ H_2^\pm,\qquad H^{\pm\pm}.
\]
This spectrum is one of the main structural differences from an ordinary 2HDM [2403.07722], [2305.08030].

The triplet vev is small. In the notation \(v_\Delta\), a widely used approximate expression is
\[
v_\Delta \approx \frac{1}{\sqrt{2}}\,
\frac{\mu_1 v_1^2+\mu_2 v_2^2+\mu_3 v_1 v_2}
{m_\Delta^2+\dfrac{\lambda_6+\lambda_7}{2}v_1^2+\dfrac{\bar\lambda_6+\bar\lambda_7}{2}v_2^2},
\]
and one convenient relation adopted in early analyses is
\[
\mu_3 \sim -\frac{\mu_1 v_1^2+\mu_2 v_2^2}{v_1 v_2},
\]
which preserves \(v_\Delta\ll v\) by cancellation [1406.6814], [1504.07087]. A more explicit tadpole formulation in the notation of [1812.07719] is
\[
m_\Delta^2= \frac{\sqrt{2}(\mu_1 v_1^2+v_2(\mu_3 v_1+\mu_2 v_2)) -v_t(\lambda_{68}^+ v_1^2+2\bar\lambda_{89}^+ v_t^2+\lambda_{79}^+ v_2^2)} {2v_t}.
\]

The CP-even neutral mass matrix is \(3\times 3\). One commonly used form is
\[
{\mathcal{M}_{\mathcal{CP}_{even}^2=
\begin{pmatrix}
m_{\rho_1\rho_1}^2 & m_{\rho_2\rho_1}^2 & m_{\rho_0\rho_1}^2 \\
m_{\rho_1\rho_2}^2 & m_{\rho_2\rho_2}^2 & m_{\rho_0\rho_2}^2 \\
m_{\rho_1\rho_0}^2 & m_{\rho_2\rho_0}^2 & m_{\rho_0\rho_0}^2
\end{pmatrix},
\]
with, for example,
\[
m_{\rho_1\rho_1}^2=\lambda_1 v_1^2+\frac{v_2(\sqrt{2}m_3^2+\mu_3 v_t)}{\sqrt{2}v_1},
\qquad
m_{\rho_2\rho_2}^2=\lambda_2 v_2^2+\frac{v_1(\sqrt{2}m_3^2+\mu_3 v_t)}{\sqrt{2}v_2},
\]
\[
m_{\rho_0\rho_0}^2=
\frac{4(\bar\lambda_8+\bar\lambda_9)v_t^3+\sqrt{2}(\mu_1 v_1^2+\mu_3 v_2v_1+\mu_2 v_2^2)}{2v_t},
\]
and off-diagonal entries
\[
m_{\rho_0\rho_1}^2=
\frac{1}{\sqrt{2}}\left(\sqrt{2}v_1v_t(\lambda_6+\lambda_8)-(2\mu_1 v_1+\mu_3 v_2)\right),
\]
\[
m_{\rho_0\rho_2}^2=
\frac{1}{\sqrt{2}}\left(\sqrt{2}v_2v_t(\lambda_7+\lambda_9)-(2\mu_2 v_2+\mu_3 v_1)\right).
\]
Diagonalization is performed by an orthogonal matrix \(\mathcal E\),
\[
\mathcal E\,\mathcal M_{\mathcal{CP}_{even}^2\,\mathcal E^T
= {\rm diag}(m_{h_1}^2,m_{h_2}^2,m_{h_3}^2),
\]
parameterized by \(\alpha_1,\alpha_2,\alpha_3\) [2403.07722], [2410.11140].

The singly charged sector is the most distinctive structural novelty. In the reduced \((H^\pm,\delta^\pm)\) basis, the physical states are
\[
\begin{pmatrix} H_1^\pm\\ H_2^\pm \end{pmatrix}
=
\begin{pmatrix} \cos\theta_\pm & \sin\theta_\pm\\ -\sin\theta_\pm & \cos\theta_\pm \end{pmatrix}
\begin{pmatrix} H^\pm\\ \delta^\pm \end{pmatrix},
\]
with
\[
(m_{H_{1,2}^\pm})^2
=
\frac12(m_{\delta^\pm}^2+m_{H^\pm}^2)
\mp
\frac12\sqrt{(m_{\delta^\pm}^2-m_{H^\pm}^2)^2+4(m_{H^-\delta^+}^2)^2},
\]
\[
\tan2\theta_\pm = -\frac{2m_{H^-\delta^+}^2}{m_{\delta^\pm}^2-m_{H^\pm}^2}.
\]
The crucial point is that \(m_{H^-\delta^+}^2\) is proportional to the cubic \(\mu\)-parameters and can therefore be large, producing a sizable \(\theta_\pm\) [1504.07087], [1406.6814].

The doubly charged scalar originates directly from the triplet. One explicit formula used in the \(Y=2\) convention is
\[
m_{H^{\pm\pm}^2=
\frac{\sqrt{2}\mu_1 v_1^2 + \sqrt{2}\mu_3 v_1 v_2 + \sqrt{2}\mu_2 v_2^2 - \lambda_8 v_1^2 v_t - \lambda_9 v_2^2 v_t - 2 \bar{\lambda}_9 v_t^3}{2v_t},
\]
while older analyses often worked with approximate hierarchies such as \(m_{\delta^\pm}=m_{\delta^{\pm\pm}}+100~{\rm GeV}\) in collider benchmarks [2403.07722], [1504.07087].

## 4. Theoretical consistency and constrained parameter space

A detailed theory analysis of the \(2HDMcT\) established a complete set of tree-level unitarity constraints on the coupling parameters of the potential and exact tree-level boundedness-from-below constraints for all directions [1812.07719]. The core quartic positivity conditions include
\[
\lambda_1>0,\qquad \lambda_2>0,
\]
\[
\lambda_3+\sqrt{\lambda_1\lambda_2}>0,
\]
\[
\lambda_3+\lambda_4-|\lambda_5|+\sqrt{\lambda_1\lambda_2}>0,
\]
together with triplet-sector requirements such as
\[
\bar\lambda_8+\bar\lambda_9>0,\qquad \bar\lambda_8+\frac12\bar\lambda_9>0,
\]
plus additional mixed doublet-triplet conditions denoted \(\Omega_1,\ldots,\Omega_5\) [1812.07719]. Tree-level unitarity is imposed through scalar-scattering eigenvalues \(a_i\) with
\[
|a_i|\le 0.5,\qquad i=1,\dots,19.
\]

Electroweak precision is central because the triplet vev breaks custodial symmetry. One early tree-level result is
\[
m_W^2=\frac{g^2v^2}{4}\left(1+\frac{2v_\Delta^2}{v^2}\right),\qquad
m_Z^2=\frac{g^2v^2}{4\cos^2\theta_W}\left(1+\frac{4v_\Delta^2}{v^2}\right),
\]
so that
\[
\rho=\frac{1+2v_\Delta^2/v^2}{1+4v_\Delta^2/v^2}.
\]
Using \(\rho=1.0004^{+0.0003}_{-0.0004}\), one paper inferred
\[
v_\Delta<3.4~\text{GeV}
\]
at \(2\sigma\), while later scans typically restricted \(v_t\) to the \(0\)–\(1\) GeV or \(0\)–\(2\) GeV range [1406.6814], [2305.08030], [2508.04493].

The \(2HDMcT\) parameter space has also been revisited under modified Veltman conditions and \(\bar B\to X_s\gamma\). In one detailed study, the one-loop quadratic-divergence combinations were written as
\[
\delta T_{d_1}= \frac{m_W^2}{v^2}\left(2+\frac{1}{c_w^2}\right) +3\lambda_1+2\lambda_3+\lambda_4 +\left(3\lambda_6+\frac{3}{2}\lambda_8\right) -\left(\frac{\sqrt2}{v}\right)^2\frac{m_D^2}{\cos^2\beta},
\]
\[
\delta T_{d_2}= \frac{m_W^2}{v^2}\left(2+\frac{1}{c_w^2}\right) +3\lambda_2+2\lambda_3+\lambda_4 +\left(3\lambda_7+\frac{3}{2}\lambda_9\right) -\left(\frac{\sqrt2}{v}\right)^2\frac{m_U^2}{\sin^2\beta},
\]
\[
\delta T_t= 4\frac{m_W^2}{v^2}\left(1+\frac{1}{c_w^2}\right) +2(\lambda_6+\lambda_7)+\lambda_8+\lambda_9+8\bar\lambda_8+6\bar\lambda_9,
\]
and the surviving sample was significant around \(\delta T\simeq 3.5\) GeV [2305.08030].

That same analysis concluded that naturalness affects drastically the masses of heavy Higgs \(h_3\), \(A_2\), \(H_2^\pm\) and \(H^{\pm\pm}\). For \(|\mu_2|>40\) GeV it found
\[
m_{h_2}\lesssim 880\ \mathrm{GeV},\qquad
m_{A_1}\lesssim 920\ \mathrm{GeV},\qquad
m_{H_1^\pm}\lesssim 918\ \mathrm{GeV},
\]
and, once \(\bar B\to X_s\gamma\) is included, the lower mass limits of nonstandard Higgs bosons are pushed up to higher values located between \(490\) and \(750\) GeV. It also reported that the \(\bar B\to X_s\gamma\) experimental limits can only be accommodated if two conditions are fulfilled:
\[
\alpha_1>0,\qquad m_{h_3}>m_{H_2^\pm}>m_{H^{\pm\pm}}.
\]
The \(\bar B\to X_s\gamma\) interval used there was
\[
[3.11\times 10^{-4},\ 3.87\times 10^{-4}]
\]
at \(95\%\) C.L. [2305.08030].

## 5. Charged-scalar mixing and doubly charged Higgs phenomenology

The phenomenological signature most often emphasized in the \(2HDMcT\) is the large mixing between the singly charged scalar from the doublet sector and the singly charged scalar from the triplet. This changes the usual doubly charged Higgs phenomenology of the minimal type-II seesaw. Early analyses stressed that the new interactions in the scalar potential cause the sizable mixture of charged Higgses in triplet and doublet, opening the new channels
\[
\delta^{++}\to H_1^+W^{+(*)},\qquad \delta^{++}\to H_1^+H_1^+,
\]
with \(H_1^\pm\) the lightest charged Higgs [1406.6814].

This has two immediate consequences. First, the triplet charged Higgs inherits quark couplings through its doublet admixture, so the triplet charged Higgs could couple to quarks in the model [1406.6814]. Second, the doubly charged state no longer decays predominantly through the classic minimal-type-II channels
\[
\delta^{\pm\pm}\to \ell^\pm\ell^\pm,\qquad \delta^{\pm\pm}\to W^\pm W^\pm,
\]
because the cascade channels can dominate even when the triplet vev is at GeV level [1406.6814], [1504.07087].

A typical cascade emphasized in the literature is
\[
\delta^{++}\to W^+ H_1^+ \to W^+ t\bar b \to W^+W^+ b\bar b,
\]
and the corresponding QCD-assisted production channels are
\[
pp \to H_2^+ \bar t b \; (H_2^- t \bar b) \to \delta^{++} W^- \bar t b \; (\delta^{--} W^+ t \bar b),
\]
\[
pp \to H_2^+ \bar t \; (H_2^- t) \to \delta^{++} W^- \bar t \; (\delta^{--} W^+ t).
\]
These channels are absent or negligible in the minimal type-II seesaw because there the triplet states do not couple significantly to quarks [1504.07087].

A focused LHC study of this regime used **MADGRAPH/MADEVENT 5**, **PYTHIA 6**, **PGS 4**, and **CalcHEP 3.6.15** with **CTEQ6L** PDFs, and defined the significance as
\[
S= \sqrt{ 2\left[ (n_s+n_b)\ln\left(1+\frac{n_s}{n_b}\right)-n_s \right] }.
\]
At \(\sqrt{s}=13\) TeV it found that with integrated luminosity \(40~{\rm fb}^{-1}\),
\[
m_{\delta^{\pm\pm}} \lesssim 330~{\rm GeV}
\]
could be discovered at the \(5\sigma\) level, while for \(300~{\rm fb}^{-1}\),
\[
m_{\delta^{\pm\pm}} \lesssim 450~{\rm GeV}
\]
could be discovered at the \(5\sigma\) level [1504.07087].

The triplet also reshapes loop phenomenology. In a study of muon \(g-2\) in a two-Higgs-doublet model with a type-II seesaw mechanism, the dominant effect came from two-loop Barr-Zee diagrams mediated by the Higgs triplet fields, especially the doubly charged scalar. The paper stressed that, unlike the usual two-Higgs-doublet models that require exotic Higgs bosons light in mass, the masses of the corresponding particles in the model are of \({\cal O}(100)\) GeV, and that the doubly-charged Higgs boson presents a different decay pattern from the usual Higgs triplet model, motivating multi-\(\tau\) searches at the LHC [2104.03275].

## 6. Yukawa realizations, loop probes, and the “Type-III” scenario

The scalar-triplet sector does not fix the Yukawa realization of the two-doublet part. The literature represented here includes a Type-II Yukawa assignment, an aligned scheme, a Type-I texture for the \(95\) GeV excess problem, and a recent Type-III texture at a muon collider [2403.07722], [2104.03275], [2410.11140], [2508.04493].

In the Type-II Yukawa assignment used in one \(e^+e^- \to h_1\gamma\) study,
\[
-{\cal L}^{II}_{Yukawa}= -y_u \bar Q_L \tilde\Phi_2 u_R -y_d \bar Q_L \Phi_1 d_R -y_\ell \bar L_L \Phi_1 \ell_R + h.c.,
\]
and the loop-induced process
\[
e^+e^- \to h_1\gamma
\]
was found to be strongly dependent on \(\alpha_1\), \(\lambda_7\), \(\lambda_9\), and the trilinear Higgs couplings. After imposing theoretical constraints and experimental data, including the \(B\to X_s\gamma\) limit at \(95\%\) C.L., the paper found
\[
\sigma (e^+e^- \to h_1\gamma)
\]
can be enhanced up to
\[
8.1\times 10^{-2}\ {\rm fb},
\]
and that \(R_{\gamma h_1}\) is entirely correlated with both the \(h_1\to\gamma\gamma\) and \(h_1\to\gamma Z\) signal strengths [2403.07722].

A distinct phenomenological use of the \(2HDMcT\) appeared in a study of the \(95\) GeV excesses. There the analysis was performed using the **Type-I Yukawa texture**, and the paper explicitly stated that the other Yukawa textures—Types II, III, and IV—do not explain the excesses within the allowed parameter space. In that framework, a light CP-even Higgs boson \(h_1\) around \(95\) GeV could simultaneously account for the \(\gamma\gamma\) and \(b\bar b\) excesses, while the combination of a nearly degenerate \(h_1\) and \(A_1\) gave the best three-channel fit with
\[
\chi^2_{\gamma\gamma+\tau^+\tau^-+b\bar b}=2.25,
\]
corresponding to
\[
0.64\sigma
\]
in the paper’s convention [2410.11140].

The label “\(2HDMcT\) Type-III” appears explicitly in a later charged-Higgs study at future \(\mu^+\mu^-\) colliders. In that paper, “Type-III” refers to the **Type-III Yukawa texture assignment of the 2HDM sector embedded in the \(2HDMcT\)**, not to a fermionic type-III seesaw [2508.04493]. Its central point is that, in Type-II and Type-III, the neutral-Higgs couplings to muons are approximately proportional to
\[
\frac{1}{\cos\beta}\approx \tan\beta
\]
at large \(\tan\beta\), but the Type-III realization permits much larger \(\tan\beta\) values than the Type-II realization under the imposed bounds [2508.04493].

For a \(3\) TeV muon collider, that study found
\[
\sigma(\mu^+\mu^- \to H_1^+ H_1^-)\lesssim 3.15~{\rm fb},
\]
\[
\sigma(\mu^+\mu^- \to H_1^\pm H_2^\mp)\lesssim 0.17~{\rm fb},
\]
with enhancement up to
\[
0.62~{\rm fb}
\]
if electroweak precision observables constraints are relaxed, and
\[
\sigma(\mu^+\mu^- \to H_1^\pm W^\mp)\lesssim 4.12~{\rm fb}.
\]
The paper emphasized that the \(\mu^+\mu^- \to H_1^+ W^-\) cross section is significantly enhanced due to the large \(\tan\beta\) amplification characteristic of the Type-III scenario, and it presented discovery \((5\sigma)\) regions at a \(3\) TeV muon collider for \(\mu^+\mu^- \to H_1^+ H_1^-\), \(\mu^+\mu^- \to H_1^+ H_2^-\), and \(\mu^+\mu^- \to W^\pm H_1^\mp\) [2508.04493].

Taken together, these studies show that the \(2HDMcT\) is best understood as a **scalar type-II-seesaw extension of the 2HDM** whose phenomenology is highly sensitive to the Yukawa realization chosen for the doublet sector. A plausible implication is that the phrase “2HDMcT Type-III” should be read with care: in the current literature it designates a Yukawa texture inside a scalar-triplet type-II-seesaw model, not a fermionic type-III seesaw.

Source: https://www.emergentmind.com/topics/two-higgs-doublet-type-ii-seesaw-model-2hdmct-type-iii