GNMSSM: General Next-to-Minimal Supersymmetric Model

• GNMSSM is a supersymmetric extension of the NMSSM that introduces a gauge-singlet superfield and explicit bilinear and tadpole terms to resolve domain wall and tadpole issues.
• It exhibits rich Higgs and neutralino phenomenology, enabling controlled singlet-doublet mixing to reconcile the 125 GeV SM-like Higgs with a 95 GeV singlet excess and muon g-2 anomalies.
• The model offers viable dark matter scenarios through a singlino-dominated LSP while ensuring a natural, stable vacuum and compliance with cosmological and collider constraints.

The General Next-to-Minimal Supersymmetric Standard Model (GNMSSM) is a supersymmetric extension of the Minimal Supersymmetric Standard Model (MSSM) in which the discrete $\mathbb{Z}_3$ symmetry of the standard Next-to-Minimal Supersymmetric Standard Model (NMSSM) is lifted. The GNMSSM augments the theory with a gauge-singlet chiral superfield $\hat S$ and allows all renormalizable, gauge-invariant, and $R$- and $CP$-conserving superpotential terms. This flexible theoretical structure is constructed to remedy cosmological and ultraviolet problems endemic to the $\mathbb{Z}_3$-NMSSM, enables a rich Higgs and neutralino phenomenology, and naturally admits scenarios reconciling experimental anomalies—such as the muon $g-2$ discrepancy and low-mass scalar excesses—with dark matter, Higgs, and collider constraints (Cao et al., 24 Feb 2024, Li et al., 9 Nov 2025, 0910.1785, Cao et al., 2022, Cao et al., 2023, Meng et al., 11 May 2024, Cao et al., 2022).

1. Model Structure and Lagrangian

The GNMSSM Lagrangian is defined by extending the MSSM to include a gauge-singlet superfield $\hat S$. The general renormalizable superpotential in the Higgs-singlet sector is

$$W_{\rm GNMSSM} = W_{\rm Yukawa} + \lambda\,\hat S\,\hat H_u\cdot\hat H_d + \frac{\kappa}{3}\,\hat S^3 + \mu\,\hat H_u\cdot\hat H_d + \frac{\mu_S}{2}\hat S^2 + \xi_F\,\hat S\,,$$

where:

• $W_{\rm Yukawa}$: MSSM quark and lepton Yukawa couplings,
• $\lambda,\kappa$: dimensionless singlet-doublet and singlet self-couplings,
• $\mu$: supersymmetric Higgsino mass,
• $\mu_S$: supersymmetric singlet mass,
• $\xi_F$: linear singlet (tadpole) term.

The corresponding soft supersymmetry-breaking Lagrangian for the Higgs/singlet sector reads

$$\begin{aligned} -\mathcal{L}_{\rm soft} &= m_{H_u}^2\,|H_u|^2 + m_{H_d}^2\,|H_d|^2 + m_S^2\,|S|^2 \ &+ [ \lambda\,A_\lambda\,H_u\cdot H_d\,S + \frac{\kappa}{3}A_\kappa\,S^3 + B_\mu\,H_u\cdot H_d + \frac{B_S}{2}S^2 + \xi_S\,S + \mathrm{h.c.} ]. \end{aligned}$$

Distinctive to the GNMSSM versus $\mathbb{Z}_3$-NMSSM are the explicit bilinear ($\mu\,H_u H_d$, $\mu_S\,S^2/2$) and linear tadpole ($\xi_F\,S$) terms. These parameters control the Higgsino and singlino mass independently and explicitly break the $\mathbb{Z}_3$ symmetry, resolving both tadpole and cosmological domain wall problems for the singlet.

2. Higgs and Neutralino Sectors

Higgs Sector

After electroweak symmetry breaking, the vacuum expectation values are $$v_u = \langle H_u^0\rangle,\, v_d = \langle H_d^0\rangle,\, v_s = \langle S\rangle$$. The tree-level scalar potential combines $F$-, $D$-, and soft terms, ensuring vacuum stability for generic GNMSSM parameter choices (0910.1785, Cao et al., 2022, Cao et al., 2022). The CP-even mass matrix, in the $(H_{\rm NSM},H_{\rm SM},S)$ basis, is augmented relative to the MSSM via parameters $\lambda,\kappa,\mu_S,\mu,\xi_F,v_s$.

Crucially, the presence of $\mu$ and $\mu_S$ enables decoupling of the Higgsino and singlino masses from the singlet scalar vev $v_s$:

• The Higgsino mass: $\mu_{\rm tot} = \mu+\lambda v_s/\sqrt2$.
• The singlino mass: $m_N = \sqrt2 \kappa v_s + \mu_S$.

The flexibility in these parameters allows for larger singlet-doublet mixing in the CP-even sector without requiring large $\lambda$, increases control over the light singlet-like CP-even Higgs mass $m_{h_s}$, and facilitates the simultaneous realization of a SM-like $h$ at 125 GeV and a predominantly singlet $h_s$ at $\sim 95$ GeV (Cao et al., 24 Feb 2024, Cao et al., 2023).

Neutralino Sector

The neutralino mass matrix in the $$(-i\tilde B,\,-i\tilde W^0,\,\tilde H_d^0,\,\tilde H_u^0,\,\tilde S)$$ basis becomes

$$M_{\tilde\chi^0} = \begin{pmatrix} M_1 & 0 & -m_Z s_W \cos\beta & +m_Z s_W\sin\beta & 0 \ 0 & M_2 & +m_Z c_W\cos\beta & -m_Z c_W\sin\beta & 0 \ * & * & 0 & -\mu_{\rm tot} & -{\lambda v \over \sqrt2}\sin\!\beta \ * & * & * & 0 & -{\lambda v\over \sqrt2}\cos\!\beta \ * & * & * & * & m_N \end{pmatrix}.$$

A singlino-dominated lightest neutralino ($\tilde\chi_1^0$) is achieved for $m_N \ll \mu_{\rm tot}, M_1, M_2$, with mixing controlled mainly by the $\lambda v/\mu_{\rm tot}$ ratio (Meng et al., 11 May 2024, Li et al., 9 Nov 2025, Cao et al., 2022). This singlet-dominance is the origin of the "secluded" dark sector phenomenology in the GNMSSM.

3. Solution to Cosmological and UV Problems

The explicit $\mathbb{Z}_3$-breaking terms in the superpotential and soft Lagrangian address two long-standing issues of the scale-invariant NMSSM:

• Domain wall problem: The accidental discrete symmetry leads to degenerate vacua and late-time domain walls, which are cosmologically problematic. The explicit breaking terms lift vacuum degeneracy, collapsing walls before nucleosynthesis (0910.1785, Cao et al., 2022).
• Tadpole problem: Planck-suppressed operators in supergravity can generate large singlet tadpoles, destabilizing the weak scale. The GNMSSM allows for appropriate tuning of tadpole and bilinear terms to avoid destabilization and maintain naturalness over a broad parameter region.

Unlike the $\mathbb{Z}_3$-NMSSM, where the effective $\mu$-term is $\mu_{\rm eff}=\lambda v_s/\sqrt 2$, the GNMSSM's $\mu$ and $\mu_S$ ensure that neither fine-tuning nor cosmologically dangerous consequences are forced by discrete symmetries (0910.1785).

4. Collider and Low-Energy Phenomenology

Anomalies and Excesses

The GNMSSM provides unified explanations for:

• Muon anomalous magnetic moment ($g-2$): Light electroweakinos and smuons, with $\tan\beta$ enhanced, yield $\Delta a_\mu\sim 2.5\times 10^{-9}$ predominantly via wino–Higgsino–smuon (WHL) loops (Cao et al., 24 Feb 2024, Cao et al., 2022). Analytic expressions for all leading diagrams—including Bino–Higgsino–(L,R)-slepton and Bino-LR mixing contributions—are given by

$$a_{\mu}^{\rm WHL} \simeq {\alpha_2\over 8\pi} {m_\mu^2\,M_2\,\mu\,\tan\beta \over m_{\tilde\nu}^4} \left[2f_C(...)-{m_{\tilde\nu}^4\over M_{\tilde\mu_L}^4}f_N(...)\right],\,$$

where $f_C, f_N$ are loop functions, and relating parameters of the GNMSSM directly to the measured $a_\mu$.

• Low-mass Higgs signals: Observed diphoton and $b\bar b$ excesses near 95 GeV (LHC, LEP) are naturally interpreted as resonant production of the singlet-dominated CP-even Higgs $h_s$. The couplings to SM states are suppressed but non-negligible due to controlled doublet admixture: $$C_{h_s t\bar t}\simeq V_{h_s}^{\rm SM},\quad C_{h_s b\bar b}\simeq V_{h_s}^{\rm SM}-\tan\beta\,V_{h_s}^{\rm NSM},\quad C_{h_s VV}=V_{h_s}^{\rm SM},$$ with $V_{h_s}^{i}$ as singlet/doublet mixing. Required mixing to match observed strengths: $V_{h_s}^{\rm SM} \sim 0.35$, $V_{h_s}^{\rm NSM}\ \tan\beta \sim 0.07{-}0.11$ (Cao et al., 24 Feb 2024, Cao et al., 2023).

Parameter Space and Experimental Constraints

Global parameter scans with flat priors over $$(\lambda,\,\kappa,\,\tan\beta,\,\mu,\,m_{h_s},\,m_N,\,A_\lambda,\,A_\kappa,\,M_{1,2},\,M_{\tilde\mu_L,\tilde\mu_R})$$ show compatibility with:

• 125 GeV SM-like Higgs mass and couplings (HiggsBounds/HiggsSignals)
• Planck relic density, LZ spin-independent/direct detection bounds,
• B-physics ($B_s\to \mu\mu$, $B\to X_s\gamma$),
• Vacuum stability and perturbative unitarity (Vevacious, SARAH),
• LHC SUSY and extra Higgs searches (CheckMATE, SModelS), requiring, for viable points:
  • $m_h \approx 125$ GeV,
  • $m_{h_s}\simeq 95$ GeV,
  • $\mu_{\rm tot}\gtrsim 210$ GeV,
  • $m_{\tilde\chi_1^0}\gtrsim 140$ GeV,
  • $\tan\beta \gtrsim 18$.

5. Dark Matter Phenomenology

The GNMSSM realizes a "secluded" WIMP dark matter scenario via a singlino-dominated $\tilde\chi_1^0$ annihilating into singlet-sector scalars: $$\tilde\chi_1^0\tilde\chi_1^0 \to\, h_sA_s\,\text{(s-wave)},\quad h_sh_s,\;A_sA_s\,\text{(p-wave)},$$ with $h_s$, $A_s$ singlet-dominated CP-even/odd Higgses. The annihilation cross sections are approximately

$$\langle\sigma v\rangle_{h_s h_s} \simeq {3 v_F^2\,\kappa^4\over 16\pi m_{\tilde \chi}^2},\qquad \langle\sigma v\rangle_{A_sA_s} \simeq {v_F^2\,\kappa^4\over 48\pi m_{\tilde \chi}^2},\qquad \langle\sigma v\rangle_{h_sA_s} \simeq {\kappa^4\over 4\pi m_{\tilde\chi}^2}.$$

The relic density is achieved for $\kappa\sim 0.1$–$0.7$ at $m_{\tilde\chi_1^0}\sim 40$–$400$ GeV (Meng et al., 11 May 2024, Li et al., 9 Nov 2025, Cao et al., 2022).

Direct detection cross sections scale as $\sigma_{\rm SI}\propto \lambda^2\kappa^2$ (with moderate singlet-doublet Higgs mixing) and $\sigma_{\rm SI}\propto\lambda^4$ if $h_s$ is heavy, ensuring compliance with the LZ bound for $\lambda \lesssim 0.05$–$0.1$.

Nested-sampling and Bayesian analyses favor a singlino-dominated LSP in $\gtrsim 65\%$–$99\%$ of the parameter space, with annihilation typically dominated by $\tilde\chi_1^0\tilde\chi_1^0\to h_sA_s$ ($58\%$), $A_sA_s$ ($35\%$), and $h_sh_s$ ($\sim 2\%$) (Meng et al., 11 May 2024, Li et al., 9 Nov 2025, Cao et al., 2022).

Characteristic mass hierarchies:

• $\tilde S < \tilde B < \tilde H$: light Bino, $\mu_{\rm tot}\sim200$ GeV, annihilation via $A_sA_s$ or $h_sA_s$, mild tuning.
• $\tilde S < \tilde H < \tilde B$: heavy Bino, $\mu_{\rm tot}\gtrsim900$ GeV, requires small $\lambda$ for direct detection, large tuning.

Benchmarks in the literature exemplify points yielding correct $a_\mu$, $m_h$, $m_{h_s}$, $\Omega h^2$, and direct detection rates, for both Bino- and singlino-dominated scenarios (Cao et al., 24 Feb 2024, Cao et al., 2023).

6. Experimental and Future Tests

A comprehensive suite of collider, dark matter, and low-energy measurements constrain the GNMSSM, but large portions of parameter space remain viable:

• High-Luminosity LHC ($3\,\text{ab}^{-1}$): Can probe compressed electroweakino spectra ($$pp\to\tilde\chi\tilde\chi\to\ell\ell+E_T^{\rm miss}$$ down to $\Delta m\sim O(10)$ GeV) and direct slepton production ($m_{\tilde\ell}\lesssim 1$ TeV) (Cao et al., 24 Feb 2024).
• Future $e^+e^-$ colliders (ILC, CLIC, FCC-ee): Expected sensitivity to $g_{h_sVV}$ couplings at a few percent and improved $95$ GeV Higgs mass resolution, facilitating precision studies of singlet-like Higgs states.
• Direct detection: LZ 2024 and future multi-ton experiments will test $\sigma_{\rm SI}$ down to $10^{-49}$ cm$^2$; future improvements by a factor of 5 would strongly impact the allowed parameter space (Meng et al., 11 May 2024, Li et al., 9 Nov 2025).
• Muon $g-2$ (FNAL/J-PARC): Ongoing improvements will further challenge or confirm the surviving corners of GNMSSM parameter space.
• Higgs property measurements: Precision determinations of the 125 GeV Higgs couplings to the percent level will critically test the singlet-doublet mixing structure required for low-mass excesses (Cao et al., 2023).
• Dedicated LHC searches: Targeted analyses for extended decay chains with soft leptons and multi-step cascades will be essential for probing the fully-realized GNMSSM scenario (Li et al., 9 Nov 2025, Cao et al., 2022).

The broad decoupling and flexible parameter structure of the GNMSSM ensure its continued empirical testability and its capacity to synthesize diverse anomalies within a natural, UV-complete, and cosmologically-viable supersymmetric framework.