---
title: 'TNMSSM: Triplet-Singlet Extended MSSM'
url: https://www.emergentmind.com/topics/triplet-next-to-minimal-supersymmetric-standard-model-tnmssm
type: topic
---

# TNMSSM: Triplet-Singlet Extended MSSM

Searching arXiv for TNMSSM and related triplet-singlet MSSM papers.
arXiv_search(query="TNMSSM triplet singlet MSSM", max_results=10, sort_by="relevance")
The **Triplet Next-to-Minimal Supersymmetric Standard Model (TNMSSM)** denotes a supersymmetric extension of the MSSM in which the Higgs sector is enlarged by a gauge singlet and triplet superfields. In the usage of [1109.2842], the model contains a singlet superfield \(S\) together with a pair of \(SU(2)_L\) triplet superfields \(T\) and \(\bar T\) carrying hypercharges \(Y=+1\) and \(Y=-1\), all coupled to one another and to the MSSM Higgs doublets. The construction is motivated by combining the NMSSM-type solution to the \(\mu\)-problem with triplet-induced enhancements of the Higgs quartic, thereby addressing the little hierarchy problem and generating both \(\mu\) and \(\mu_T\) dynamically from the singlet vacuum expectation value [1109.2842]. Subsequent work has used the same acronym for closely related triplet-singlet extensions with different triplet hypercharge assignments, including \(Y=0\) constructions [1506.03634], [1512.08651], [1204.6592]. This suggests that “TNMSSM” is best understood as a family of singlet-plus-triplet supersymmetric Higgs-sector extensions rather than a single unique field-content choice.

## 1. Field content and defining interactions

In the hypercharged-triplet realization of [1109.2842], the gauge group remains
\[
SU(3)_c \times SU(2)_L \times U(1)_Y,
\]
and the chiral superfields beyond the MSSM are
\[
S \sim (1,1,0),\qquad T \sim (1,3,+1),\qquad \bar T \sim (1,3,-1).
\]
The triplets are written as
\[
T = \begin{pmatrix} T^+/\sqrt{2} & -T^{++} \\ T^0 & -T^+/\sqrt{2} \end{pmatrix},\qquad
\bar T = \begin{pmatrix} \bar T^-/\sqrt{2} & -\bar T^{0} \\ \bar T^{--} & -\bar T^-/\sqrt{2} \end{pmatrix},
\]
so each triplet contains a neutral component, a singly charged component, and a doubly charged component [1109.2842].

The Higgs-sector superpotential is
\[
W = S \left( \lambda\, H_u \cdot H_d + \lambda_T\, \text{tr}(\bar T T) \right) + \frac{\kappa}{3} S^3 + \chi_u\, H_u \cdot \bar T\, H_u + \chi_d\, H_d \cdot T\, H_d,
\]
supplemented by the usual MSSM Yukawa couplings [1109.2842]. The relevant \(SU(2)_L\) contractions are
\[
H_u \cdot H_d = H_u^+ H_d^- - H_u^0 H_d^0,
\]
\[
H_u \cdot \bar{T} H_u = \sqrt{2}H_u^+H_u^0\bar{T}^- - (H_u^0)^2\bar{T}^0 - (H_u^+)^2\bar{T}^{--},
\]
\[
H_d \cdot T H_d = \sqrt{2}H_d^-H_d^0 T^+ - (H_d^0)^2 T^0 - (H_d^-)^2 T^{++}.
\]
A discrete \(\mathbb{Z}_3\) symmetry forbids bare \(\mu\) and \(B\mu\) terms, so all effective \(\mu\)-like parameters arise from singlet symmetry breaking [1109.2842].

The corresponding soft SUSY-breaking terms include
\[
m_{H_u}^2 |H_u|^2 + m_{H_d}^2 |H_d|^2 + m_S^2 |S|^2 + m_T^2\,\text{tr}(|T|^2) + m_{\bar T}^2\,\text{tr}(|\bar T|^2)
\]
together with trilinears
\[
A\,S\, H_u\cdot H_d,\quad A_T\, S\,\text{tr}(T\bar T),\quad \frac{A_\kappa}{3} S^3,\quad A_u\, H_u\cdot \bar T H_u,\quad A_d\, H_d\cdot T H_d
\]
and the usual MSSM soft terms [1109.2842].

Later analyses of electroweak baryogenesis and Higgs decays use the same hypercharged-triplet field content and superpotential structure, with the same singlet, \(Y=\pm1\) triplets, and couplings \(\lambda,\lambda_T,\kappa,\chi_u,\chi_d\) [2502.11409], [2406.00946]. By contrast, other papers designate as TNMSSM a \(Y=0\) triplet plus singlet extension with superpotential
\[
W_{TS} =  \lambda_T\, \hat H_d \cdot \hat T \hat H_u + \lambda_S\, \hat S\, \hat H_d \cdot \hat H_u + \frac{\kappa}{3}\hat S^3 + \lambda_{TS}\, \hat S\, \mathrm{Tr}[\hat T^2]
\]
or closely related variants [1506.03634], [1512.08651]. This difference in terminology is a recurring feature of the literature represented here.

## 2. Dynamical \(\mu\)-generation and singlet vacuum expectation value

A central feature of the TNMSSM of [1109.2842] is that the singlet vacuum expectation value,
\[
\langle S \rangle \equiv v_s,
\]
generates both the doublet and triplet supersymmetric mass terms:
\[
\mu_{\text{eff}} = \lambda v_s,\qquad \mu^{\text{eff}}_T = \lambda_T v_s.
\]
The same dynamical scale therefore controls the effective Higgsino mass and the tripletino mass [1109.2842]. In the later baryogenesis analysis the same structure is written as
\[
\mu = \lambda v_S,\qquad \mu_T = \lambda_T v_S,
\]
again emphasizing that the singlet VEV solves the MSSM \(\mu\)-problem while simultaneously generating the triplet mass parameter [2502.11409].

For neutral vacuum expectation values
\[
\langle H_u^0 \rangle = v_u,\quad \langle H_d^0 \rangle = v_d,\quad \langle T^0 \rangle = v_T,\quad \langle \bar T^0 \rangle = v_{\bar T},
\]
the neutral Higgs potential contains F-terms, D-terms, soft masses, and trilinears. In the limit of small triplet VEVs, the singlet minimization condition reduces to
\[
(2\kappa^2 v_s^2 + \lambda^2(v_u^2+v_d^2) - \kappa\lambda v_u v_d + A_\kappa v_s + m_S^2)\, v_s = 0.
\]
For \(v_s \neq 0\), a simple approximate condition is that \(m_S^2\) become negative at the electroweak scale [1109.2842].

The radiative mechanism driving \(m_S^2\) negative is one of the distinctive claims of [1109.2842]. Defining
\[
X \equiv \lambda^2 (m_{H_u}^2 + m_{H_d}^2 + m_S^2) + A^2,
\]
\[
X_T \equiv \lambda_T^2 (m_T^2 + m_{\bar T}^2 + m_S^2) + A_T^2,
\]
\[
X_\kappa \equiv 3\kappa^2 m_S^2 + A_\kappa^2,
\]
the singlet soft-mass RGE is
\[
16\pi^2 \frac{d m_S^2}{dt} = 4X + 6X_T + 4X_\kappa.
\]
The singlet-triplet term \(X_T\) is especially important because the triplet masses are not driven tachyonic, owing to large gauge contributions associated with the triplet Casimir, so they can help drive \(m_S^2\) negative while remaining positive themselves [1109.2842]. The paper states that \(m_S^2\) runs from a positive UV value to a negative IR value, whereas \(m_T^2\) and \(m_{\bar T}^2\) remain positive and relatively large, and \(m_{H_u}^2\) behaves as in the MSSM, becoming negative to trigger EWSB [1109.2842].

This radiative pattern is one of the model’s unifying features: the singlet generates \(\mu\) and \(\mu_T\), and the triplet-singlet coupling helps realize the singlet VEV needed to generate them.

## 3. Higgs-sector structure and Higgs-mass enhancement

The CP-even neutral scalar sector contains five real fields,
\[
H_u^0,\ H_d^0,\ T^0,\ \bar T^0,\ S,
\]
leading to a \(5\times5\) mass matrix, while the charged sector contains singly charged states
\[
H_u^+,\ H_d^{-*},\ T^+,\ \bar T^{-*}
\]
and doubly charged states
\[
T^{++},\ \bar T^{--*}.
\]
For small A-terms, an approximate \(U(1)_R\) symmetry produces a light pseudoscalar, often described as an \(R\)-axion [1109.2842].

The tree-level enhancement of the lightest CP-even Higgs mass is summarized by the bound
\[
m_{h^0}^2 \leq M_Z^2 \left( \cos^2 2\beta + \frac{\lambda^2}{\hat g^2} \sin^2 2\beta + \frac{\chi_d^2}{\hat g^2} \cos^4 \beta + \frac{\chi_u^2}{\hat g^2} \sin^4 \beta \right),
\]
with \(\hat g^2 = (g^2 + g'^2)/2\) and \(\tan\beta=v_u/v_d\) [1109.2842]. The \(\lambda\)-dependent term is NMSSM-like and suppressed at large \(\tan\beta\), whereas the triplet-induced \(\chi_u^2 \sin^4\beta\) term is not suppressed in that regime and remains substantial as \(\tan\beta \to \infty\) [1109.2842]. This is the core structural difference between the TNMSSM and the perturbative NMSSM in the large-\(\tan\beta\) regime.

Triplet mixing can also lower the lightest Higgs mass. The leading negative contribution arises from
\[
|F_T|^2 \supset \chi_u \mu_T \bar T^\dagger H_u H_u + \text{h.c.},
\]
giving
\[
\delta m_{h^0}^2 \sim - \frac{ (\chi_u \mu_T v_u)^2 }{ \mu_T^2 + m_T^2 }.
\]
In the non-supersymmetric limit \(m_T \gg \mu_T\), this negative effect is suppressed by the large triplet soft mass while the positive quartic remains unsuppressed [1109.2842]. By contrast, in the opposite limit \(\mu_T \gg m_T\), the triplets can be integrated out supersymmetrically and the residual contribution is smaller,
\[
\delta m_{h^0}^2 \big|_{\text{triplet only} \simeq \frac{\mu}{\mu_T} \chi_u \chi_d\, v^2 \sin 2\beta = \frac{\lambda}{\lambda_T} \chi_u \chi_d\, v^2 \sin 2\beta + \dots
\]
[1109.2842].

Numerically, [1109.2842] reports that with 1-loop stop corrections and modest stops such as \(m_{\tilde t}\approx 300\) GeV with no A-term mixing, large \(\tan\beta\) and large \(m_T/\mu_T\) hierarchies allow the tree-level Higgs mass to exceed \(114\) GeV and go beyond \(120\) GeV, thereby reducing the need for very heavy stops or large stop mixing. This is the paper’s central argument for alleviating the little hierarchy problem [1109.2842].

Related triplet-singlet models with \(Y=0\) triplets exhibit a similar qualitative mechanism, though with different quartic structures and notation. In [1506.03634], for example, the tree-level upper bound becomes
\[
m_{h_1}^2 \le m_Z^2\left( \cos^2 2\beta + \frac{\lambda_T^2}{g_L^2+g_Y^2}\,\sin^2 2\beta + \frac{2\lambda_S^2}{g_L^2+g_Y^2}\,\sin^2 2\beta \right),
\]
and the paper states that for \(\lambda_T,\lambda_S\sim0.8\) the bound can already reach \(\sim125\) GeV at low \(\tan\beta\) [1506.03634]. The common theme across these variants is that triplet and singlet F-terms enlarge the Higgs quartic beyond the MSSM limit.

## 4. Electroweak symmetry breaking and precision constraints

In the hypercharged-triplet TNMSSM, the electroweak scale is modified to
\[
v^2 = v_u^2 + v_d^2 + 4 v_T^2 + 4 v_{\bar T}^2 \approx (174\ \text{GeV})^2,
\]
while the \(Z\)-mass relation retains the standard form
\[
M_Z^2 = \frac{g'^2 + g^2}{2} v^2 \equiv \hat g^2 v^2
\]
[1109.2842]. The minimization conditions for \(v_u\) and \(v_d\) imply
\[
\frac{1}{2}\sin 2\beta = \frac{v_s(\lambda \kappa v_s + A)}{2(\chi_u^2 v_u^2 + \chi_d^2 v_d^2) + \lambda^2(2v_s^2 + v_u^2 + v_d^2) + m_{H_u}^2 + m_{H_d}^2},
\]
so nonzero EWSB requires
\[
\lambda \kappa v_s + A \neq 0
\]
[1109.2842].

The main tree-level precision constraint comes from the triplet VEVs. For the \(Y=-1\) triplet, the contribution to the Peskin–Takeuchi \(T\) parameter is
\[
\delta T = -\frac{1}{\alpha}\frac{2 v_T^2}{v^2},
\]
and requiring \(T \gtrsim -0.1\) yields
\[
v_T^2 \lesssim (4\ \text{GeV})^2
\]
[1109.2842]. The triplet tadpole estimate
\[
\langle T \rangle \sim \frac{\chi_u \mu_T v_u^2}{m_T^2 + \mu_T^2}
\]
shows that values such as
\[
m_T \sim 600\ \text{GeV},\quad \mu_T \sim 100\text{--}200\ \text{GeV},\quad \chi_u \sim 0.4
\]
naturally give \(v_T \lesssim 4\) GeV without fine-tuning [1109.2842].

The one-loop electroweak precision analysis of [1109.2842] further finds that triplet fermions and higgsinos contribute positively to \(T\), partially compensating the negative tree-level scalar-triplet-VEV effect, and that a significant region of parameter space yields negative \(S\) and modest \(T\). The favorable region is characterized by heavy triplet scalars, light triplet fermions, and small \(\mu_T\) [1109.2842].

More recent work imposes an even stronger \(\rho\)-parameter bound in the hypercharged-triplet model. In [2502.11409], with
\[
\rho = 1 - \frac{2 v_T^2}{v^2},
\]
the experimental value
\[
\rho_{\text{exp}} = 1.00038 \pm 0.00040 \quad (2\sigma)
\]
is quoted as requiring
\[
v_T^2 \lesssim 1~\text{GeV}^2
\]
[2502.11409]. A later dark-matter study likewise states, for the two-triplet singlet model, that
\[
\rho = 1 - \frac{2 v_{T\bar T}^2}{v_{ud}^2}
\]
with \(\rho_{\rm exp}=1.00038\pm0.00020\) implies
\[
v_{T\bar T} \lesssim 1~\mathrm{GeV}
\]
[2410.13659]. These later bounds reflect a more restrictive contemporary treatment of precision data.

By contrast, \(Y=0\) triplet versions of the TNMSSM modify only \(m_W\) at tree level,
\[
m_W^2 = \frac{1}{4}g_L^2\left(v^2 + 4v_T^2\right),\qquad
m_Z^2 = \frac{1}{4}(g_L^2 + g_Y^2)\, v^2,
\]
leading to
\[
\rho = 1 + 4\frac{v_T^2}{v^2}
\]
and bounds such as \(v_T \lesssim 5\) GeV or the benchmark choice \(v_T=3\) GeV [1506.03634], [1512.08651]. This difference in custodial breaking is one of the clearest structural distinctions between the \(Y=\pm1\) and \(Y=0\) versions of the acronym.

## 5. Renormalization-group behavior, perturbativity, and naturalness

The hypercharged-triplet TNMSSM introduces a nontrivial new RGE system for \(\lambda,\kappa,\lambda_T,\chi_u,\chi_d\):
\[
\begin{aligned}
16\pi^2 \frac{d\lambda}{dt} &= \lambda\left[ 3h_t^2 + 3h_b^2 + h_\tau^2 + 4\lambda^2 + 6\chi_d^2 + 6\chi_u^2 + 3\lambda_T^2 + 2\kappa^2 - 3g_2^2 - \frac{3}{5}g_1^2 \right],\\
16\pi^2 \frac{d\kappa}{dt} &= \kappa\left[ 6\lambda^2 + 9\lambda_T^2 + 6\kappa^2 \right],\\
16\pi^2 \frac{d\lambda_T}{dt} &= \lambda_T\left[ 2(\chi_d^2 + \chi_u^2 + \lambda^2 + \kappa^2) + 5\lambda_T^2 - 8g_2^2 - \frac{12}{5}g_1^2 \right],\\
16\pi^2 \frac{d\chi_u}{dt} &= \chi_u\left[ 6h_t^2 + 2\lambda^2 + 14\chi_u^2 + \lambda_T^2 - 7g_2^2 - \frac{9}{5}g_1^2 \right],\\
16\pi^2 \frac{d\chi_d}{dt} &= \chi_d\left[ 6h_b^2 + 2h_\tau^2 + 2\lambda^2 + 14\chi_d^2 + \lambda_T^2 - 7g_2^2 - \frac{9}{5}g_1^2 \right].
\end{aligned}
\]
The paper emphasizes that the triplet couplings are better controlled in the UV because of the larger \(SU(2)_L\) Casimir, and that with \(\chi_{u,d}\sim0.2\text{–}0.5\) and \(\lambda,\kappa\sim0.3\) at the electroweak scale, all couplings remain perturbative up to \(\sim10^{16}\) GeV [1109.2842].

The Higgs-sector soft masses obey
\[
\begin{aligned}
16\pi^2 \frac{d m_{H_u}^2}{dt} &= 6X_t + 2X + 12X_u - 6g_2^2 M_2^2 - \frac{6}{5}g_1^2 M_1^2 + \frac{3}{5}g_1^2 S,\\
16\pi^2 \frac{d m_{H_d}^2}{dt} &= 6X_b + 2X_\tau + 2X + 12X_d - 6g_2^2 M_2^2 - \frac{6}{5}g_1^2 M_1^2 - \frac{3}{5}g_1^2 S,\\
16\pi^2 \frac{d m_T^2}{dt} &= 4X_d + 2X_T - 16g_2^2 M_2^2 - \frac{24}{5}g_1^2 M_1^2 + \frac{6}{5}g_1^2 S,\\
16\pi^2 \frac{d m_{\bar T}^2}{dt} &= 4X_u + 2X_T - 16g_2^2 M_2^2 - \frac{24}{5}g_1^2 M_1^2 - \frac{6}{5}g_1^2 S,\\
16\pi^2 \frac{d m_S^2}{dt} &= 4X + 6X_T + 4X_\kappa,
\end{aligned}
\]
and [1109.2842] states that \(m_{H_u}^2\) and \(m_S^2\) run from positive values at the GUT scale to negative values at the electroweak scale, while \(m_T^2\) and \(m_{\bar T}^2\) remain positive and large [1109.2842].

The naturalness argument is qualitative rather than based on a specific fine-tuning measure in [1109.2842]. The claimed improvements are threefold: the Higgs mass receives a large tree-level enhancement from triplet quartics, \(\mu\) and \(\mu_T\) arise naturally from \(v_s\), and the singlet VEV is generated radiatively by the triplet-singlet coupling [1109.2842]. The paper explicitly states that the model offers a substantially improved tuning compared to the MSSM and perturbative NMSSM [1109.2842].

The \(Y=0\) literature develops this point further. In [1506.03634], one-loop Higgs corrections are computed with a Coleman–Weinberg effective potential, and the paper states that electroweak/Higgs-sector corrections can be as large or larger than the usual top/stop corrections, allowing a \(125\) GeV Higgs with light stops and reduced fine-tuning [1506.03634]. In [1204.6592], the singlet-plus-\(Y=0\)-triplet model yields a tree-level Higgs mass of \(119\)–\(120\) GeV, with radiative corrections raising it to \(125\) GeV, and the paper states that no significant contributions from stop loops are needed, alleviating the fine-tuning problem [1204.6592].

## 6. Spectrum, collider signatures, dark matter, and baryogenesis

The hypercharged-triplet TNMSSM has an extended electroweak spectrum comprising five CP-even neutral scalars, several CP-odd scalars including a light \(R\)-axion in small-A regimes, singly charged Higgs states, doubly charged Higgs states, a singlino, singly charged tripletinos, and doubly charged tripletinos [1109.2842]. In the non-supersymmetric limit \(m_T \gg \mu_T\), triplet scalars are heavy while triplet fermions remain at the electroweak scale, including a light doubly charged fermion [1109.2842]. Representative spectra in [1109.2842] with \(M_1\sim200\) GeV and \(M_2\sim220\) GeV contain a weak-scale neutralino, a weak-scale singly charged chargino, and a doubly charged fermion in the non-SUSY limit [1109.2842].

Collider implications emphasized in [1109.2842] include light doubly charged fermions and possibly scalars, same-sign dilepton or multi-lepton signatures, and compressed electroweak spectra that can reduce visible energy in SUSY decays and weaken standard missing-energy searches. The same paper notes that baryon-number violating R-parity violation can be allowed by an appropriate \(\mathbb{Z}_6\) choice while forbidding the dangerous \(W=\xi\,L\cdot T L\) neutrino-mass operator, thereby weakening proton-decay issues and changing collider signatures toward multi-jet plus low-MET final states [1109.2842].

A later precision-Higgs study of exclusive decays \(h\to MZ\) in the hypercharged-triplet TNMSSM finds that the indirect contributions through the effective \(h\gamma Z\) vertex dominate over direct contributions and that the signal strengths \(\mu_{MZ}^{ggF}\) are generically enhanced by about \(10\%\)–\(25\%\), correlated with a moderately enhanced \(h\to Z\gamma\) rate [2406.00946]. In that analysis, \(\mu_{Z\gamma}^{ggF}\) ranges from \(1.164\) to \(1.248\), while representative \(\mu_{MZ}^{ggF}\) values lie between about \(1.07\) and \(1.26\) depending on the meson and parameter choice [2406.00946]. The paper attributes these effects to loop contributions from singly and doubly charged Higgs bosons, singly and doubly charged charginos, and sfermions [2406.00946].

Dark-matter analyses highlight the enlarged neutralino sector. In the hypercharged-triplet plus singlet model there are seven neutralinos, and [2410.13659] studies a bino-like LSP with coannihilation mediated by triplinos. After imposing the Higgs mass, B-physics, \((g-2)_\mu\), LHC, and LZ constraints, the paper finds that a bino-like neutralino with mass in the region
\[
[100, 450]~\rm{GeV}
\]
can account for the correct relic density, with the viable coannihilation window characterized by
\[
\mu_T/M_1 \approx 1.18\text{–}1.23
\]
[2410.13659]. Most of this viable parameter space is stated to be testable by Xenon-nT or the LHC in the near future [2410.13659].

Electroweak baryogenesis in the hypercharged-triplet TNMSSM has also been investigated. Using a finite-temperature effective potential
\[
V_{\text{eff}}(h,\eta,S) = M(T)^2 h^2 + m_\eta^2 \eta^2 + m_S^2 S^2 + \frac{1}{8}(g_1^2+g_2^2)(h^2 \cos 2\beta + 2\eta^2 \cos 2\beta')^2 + \frac{2}{3} A_\kappa S^3,
\]
[2502.11409] reports that the tree-level cubic term \(\frac{2}{3}A_\kappa S^3\) generates a barrier sufficient for a strong first-order phase transition with
\[
\frac{v_c}{T_c} \gtrsim 1.3.
\]
The baryon asymmetry is approximated as
\[
Y_B = F_1 \sin\theta_\mu + F_2 \sin(\theta_\mu + \theta_T),
\]
and the paper states that the TNMSSM can account for the observed baryon asymmetry while satisfying EDM bounds when different contributions to the electron EDM cancel each other [2502.11409]. A characteristic benchmark requires \(|\theta_\mu|\approx0.013\) near the resonance \(|\mu|\approx|M_2|\), with cancellations between \(\theta_\mu\) and \(\theta_{M_2}\) controlling \(d_e\) [2502.11409].

## 7. Relation to MSSM, NMSSM, triplet extensions, and custodial variants

Relative to the MSSM, the TNMSSM replaces the hard \(\mu\)-parameter by a singlet-induced \(\mu\), enlarges the Higgs quartic through triplet and singlet F-terms, and thereby reduces the need for large stop-sector radiative corrections [1109.2842]. Relative to the NMSSM, it preserves the NMSSM solution to the \(\mu\)-problem while avoiding the suppression of the extra quartic at large \(\tan\beta\), because the triplet-induced \(\chi_u^2 \sin^4\beta\) term remains large precisely where the MSSM tree-level contribution is already maximized [1109.2842]. Relative to a triplet-extended MSSM without a singlet, it solves the additional \(\mu_T\)-problem through \(\mu_T=\lambda_T v_s\) and uses the singlet-triplet coupling to facilitate the radiative generation of \(v_s\) [1109.2842].

The literature also contains a custodial triplet extension with \(Y=(0,\pm1)\) triplets and an explicit global \(SU(2)_L\otimes SU(2)_R\) symmetry, described as a supersymmetric Georgi–Machacek construction [1308.4025]. That model is not a TNMSSM in the singlet-plus-triplet sense, because it does not include a gauge singlet, but it provides an important benchmark for custodial triplet dynamics. Its central property is that custodial symmetry preserves
\[
\rho=1
\]
at tree level even for sizable triplet VEVs, and organizes the scalar sector into degenerate custodial singlets, triplets, and fiveplets [1308.4025]. This suggests a possible conceptual direction for broader triplet-singlet model building: custodial organization can relax precision constraints that otherwise force hypercharged triplet VEVs to be small.

A persistent source of ambiguity in the term “TNMSSM” is therefore the coexistence of at least two model classes in the literature summarized here:

| Usage of TNMSSM | Additional fields | Representative papers |
|---|---|---|
| Hypercharged-triplet TNMSSM | \(S\) and \(T,\bar T\) with \(Y=\pm1\) | [1109.2842], [2502.11409], [2406.00946], [2410.13659] |
| \(Y=0\) triplet-singlet TNMSSM | \(S\) and one \(Y=0\) triplet | [1506.03634], [1512.08651], [1204.6592] |

The two versions share the same broad aims—solving the \(\mu\)-problem, raising the Higgs mass, enriching electroweakino and Higgs phenomenology—but differ substantially in custodial-symmetry breaking, charged-state content, and precision constraints.

Across these variants, the encyclopedic picture is consistent. The TNMSSM is a supersymmetric Higgs-sector extension built from the MSSM doublets plus a singlet and triplet representations, typically arranged to generate effective \(\mu\)-terms dynamically, enlarge the Higgs quartic at tree level, and produce a rich spectrum of neutral, charged, and sometimes doubly charged states. In the hypercharged-triplet realization of [1109.2842], the model is explicitly presented as one in which “the NMSSM and the triplet-extended MSSM can successfully solve problems of one another,” with the singlet supplying the \(\mu\)-solution and the triplets supplying the Higgs-mass solution [1109.2842]. Subsequent work has extended that framework to electroweak baryogenesis [2502.11409], precision Higgs decays [2406.00946], and neutralino dark matter with triplinos [2410.13659], while parallel \(Y=0\) constructions have explored charged Higgs phenomenology [1512.08651], hidden Higgs scenarios and loop Higgs masses [1506.03634], and mixed Higgsino–triplino dark matter around \(100\) GeV [1204.6592].

Source: https://www.emergentmind.com/topics/triplet-next-to-minimal-supersymmetric-standard-model-tnmssm