Inelastic Higgs-Portal Complex Singlet

• The inelastic Higgs-portal complex singlet is a framework that introduces a complex gauge-singlet scalar coupling the Standard Model to a hidden sector via the Higgs field.
• It employs kinetic mixing and portal-induced interactions to create distinctive inelastic transitions, resulting in suppressed elastic dark matter scattering and displaced vertex signatures at colliders.
• The model impacts electroweak phase transitions, dark matter phenomenology, and potential gravitational wave production, offering insights for extended supersymmetric and dark sector scenarios.

The inelastic Higgs-portal complex singlet framework generalizes minimal Higgs-portal models by introducing a complex gauge-singlet scalar (or superfield), expanding the phenomenology of hidden sectors that couple to the Standard Model exclusively via the Higgs. This setup generates distinctive inelastic transitions between visible and hidden states, with the complex singlet decomposing into two nearly degenerate real fields—or their superfield generalization—with only off-diagonal or suppressed diagonal couplings to Standard Model states. This mechanism has broad implications for collider, cosmological, and direct detection signatures, and is central to several extensions of the Standard Model, including next-to-minimal supersymmetric scenarios and models of dark matter, dark radiation, and electroweak baryogenesis.

1. Theoretical Structure of the Inelastic Higgs-Portal Complex Singlet

The foundation of the inelastic Higgs-portal complex singlet mechanism is the existence of marginal, UV-induced kinetic mixing between gauge-singlet chiral superfields of the visible sector ($S$) and a hidden sector ($S'$). The most general marginal mixing is realized by the operator: $$\mathcal{L}_\text{kin} = \epsilon \int d^4\theta\, S^\dagger S' + \text{h.c.}$$ where $\epsilon$ is a loop-suppressed coefficient in the range $10^{-5} \lesssim \epsilon \lesssim 10^{-1}$ (Cheung et al., 2010). The visible singlet $S$ is coupled to the two Higgs doublets through a renormalizable superpotential term: $W \supset \lambda S H_u H_d$ In non-supersymmetric contexts, analogous structures appear with a complex singlet scalar $\Phi$ or $S$ coupling to $H^\dagger H$.

When electroweak symmetry breaking occurs, the Higgs doublets acquire VEVs $\langle H_u\rangle = v_u$, $\langle H_d\rangle = v_d$, inducing a shift in $S$ and generating a nonzero $\langle F_S\rangle$; through kinetic mixing this triggers a tadpole (linear term) in the hidden sector superpotential: $W_\text{eff} = -\Lambda_\text{eff}^2 S'$ with $$\Lambda_\text{eff}^2 = \epsilon [ (\lambda/2) v^2 \sin 2\beta + ... ]$$. This drives spontaneous scale generation in the hidden sector, naturally producing hidden-sector masses in the $0.1$–$100$ GeV range, even for small $\epsilon$.

The physical complex singlet decomposes into two real mass eigenstates (often labeled $s_1$, $s_2$), generally with a small mass splitting $\Delta m$ generated by U(1)-breaking portal or mass terms. The kinetic mixing, after suitable field redefinitions, induces small-angle mixing between the visible and hidden singlet scalars and between the singlinos. The typical mixing angle for the fermionic or scalar components is: $\theta \sim \epsilon \frac{m'}{m}$ where $m'$ is the characteristic (hidden-sector) singlet mass scale, $m$ is a weak-scale mass.

2. Portal-Induced Mixing and Inelastic Transitions

The defining phenomenological feature is that the portal-mixed complex singlet supports both diagonal and off-diagonal Higgs portal interactions after mass diagonalization. The Higgs-singlet interactions schematically take the form: $$\mathcal{L} \supset -\frac{1}{2}(f_1 s_1^2 + f_2 s_2^2 + g\, s_1 s_2 )\, h$$ with $f_{1,2}$ and $g$ depending on the mixing angle(s) and the original portal coupling. In the limit where the off-diagonal $g$ term dominates, Higgs-mediated interactions of the dark matter with Standard Model fields become inelastic: dark matter can only scatter into its nearly degenerate partner (e.g., $s_1 \to s_2^{(*)}$) via Higgs exchange (Hooper et al., 30 Jul 2025). If the splitting $\Delta m = m_2 - m_1$ exceeds the kinetic energy of Galactic dark matter ($\sim$ keV), elastic scattering is kinematically forbidden, suppressing direct detection signals.

The generation of the linear term in $S'$ further means that even highly minimal hidden sectors—such as a cubic superpotential $$W_\text{hid} = -\Lambda_\text{eff}^2 S' + (\kappa'/3) (S')^3$$—dynamically produce non-trivial vacua, masses, and self-interactions. The mass spectrum for scalars and singlinos then follows $$m' \sim \mathcal{O}(10 \,\mathrm{MeV} - 100\,\mathrm{GeV})$$ for $\kappa' \sim \mathcal{O}(1)$ (Cheung et al., 2010).

3. Collider and Astrophysical Signatures

One of the haLLMark predictions is the production of Higgs bosons in supersymmetric cascade decays with probability $\mathcal{O}(0.01$–$1)$ when R-parity is conserved, since superpartners produced at the LHC must eventually decay via the portal, necessitating the emission of a Higgs or its VEV (Cheung et al., 2010). Additionally, the small mixing angles ($\theta \sim \epsilon m'/m$) suppress the decay widths of hidden sector scalars back to SM states, leading to displaced vertex signatures: $\gamma c\tau \propto \epsilon^{-2} (m/m')^3$ for hidden scalar $s'$ mass $m' \lesssim 10$ GeV.

The rate for such a decay is: $\Gamma \propto \epsilon^2 y^2 (m'/v)^2 m'$ where $y$ is a representative SM Yukawa coupling. The visible decay products typically include the heaviest kinematically accessible SM pairs—e.g., $b\bar{b}$, $\tau^+\tau^-$, photons.

More generally, inelastic Higgs-portal models predict missing energy signatures or displaced decays at hadron colliders, with modified Higgs signal strengths if the diagonal mixing is appreciable. For dark matter, direct detection cross sections via exchanged singlet scalars are suppressed by small mixing angles: $$\sigma_T = \frac{\mu_T^2}{2\pi} \frac{\lambda'^2}{m'^4} [ Z g_{s'p p} + (A-Z) g_{s'n n} ]^2$$ where $g_{s'NN}$ is proportional to nucleon–Higgs matrix elements and the mixing angles (Cheung et al., 2010).

If the mass splitting $\Delta m$ is of order or greater than the nuclear recoil scale, direct detection is inelastic and strongly suppressed; loop-induced elastic scattering provides only a subdominant contribution. These models therefore can accommodate null results in dark matter direct searches, while remaining compatible with observed relic densities and producing observable effects in cosmological or collider experiments (Hooper et al., 30 Jul 2025).

Astrophysically, the scenario can explain features such as the Galactic Center gamma-ray excess for appropriate mass and portal coupling, since dark matter annihilation via the Higgs portal produces a $hh$ final state whose decays yield a characteristic photon spectrum.

4. Cosmological and Electroweak Implications

The inelastic Higgs-portal complex singlet scenario substantially modifies the Higgs potential and electroweak symmetry breaking structure. The hidden sector, via the induced linear term, receives a mass scale inherited from electroweak symmetry breaking, transmitting weak-scale physics to the hidden sector without direct gauge interactions. This can induce a strong first-order electroweak phase transition, as the effective potential receives extra scalar contributions, enabling the conditions necessary for electroweak baryogenesis (Hooper et al., 30 Jul 2025).

Bubble nucleation during such a first-order transition sources a stochastic gravitational wave background. The strength, frequency, and duration of the gravitational wave signal depend on model parameters such as vacuum energy difference, bubble wall velocity, and latent heat released, with formulas for the gravitational wave spectrum shape and amplitude given in (Hooper et al., 30 Jul 2025). Predicted signals may fall within the sensitivity reach of planned space-based detectors such as LISA, Taiji, or DECIGO.

In more elaborate models with extended Higgs (e.g., two-doublet) or neutrino sectors, the singlet portal can connect to seesaw right-handed neutrinos, yielding complementary hidden sector–visible sector mixing structures and further cosmological/phenomenological consequences (Cheung et al., 2010, Baek et al., 2013).

5. Dark Matter and Inelastic Scattering Phenomenology

If the lightest mass eigenstate of the complex singlet is stable (protected by a discrete symmetry or accidental stability), it serves as a viable dark matter candidate. The inelasticity of the portal suppresses elastic scattering off nuclei; the leading observable channels in direct detection require upscattering to the heavier state, which is typically forbidden kinematically in conventional detectors due to small mass splitting. The resulting spin-independent cross sections are safely below current experimental bounds (Hooper et al., 30 Jul 2025, Cheung et al., 2010).

Thermal relic abundance is set by annihilation via Higgs-mediated $s$-channel processes. For complex singlets, if the inelastic transition is dominant, the effective amplitude is parameterically: $$\mathcal{A}(D M~D M \to h^* \to \text{SM}) \sim \frac{g\, v}{\langle h \rangle^2}$$ where $g$ is the off-diagonal coupling.

Parameter regions consistent with relic abundance, collider bounds, and direct detection null results occur naturally for singlet masses in the $100$–$150$ GeV range and small mixing angles, with the off-diagonal portal coupling $g$ typically $\mathcal{O}(0.1)$ or less (Hooper et al., 30 Jul 2025).

6. Model Extensions and Broader Implications

Variants appear in supersymmetric scenarios (next-to-minimal SUSY, singlet-extended MSSM), models where the portal is mediated by higher-dimensional operators, and settings where the singlet is part of a larger dark or neutrino sector. Kinetic mixing between right-handed neutrinos and hidden sector singlets, as detailed in (Cheung et al., 2010), induces displaced visible-lepton vertices (e.g., $n' \rightarrow \ell W$) without a corresponding effective tadpole, distinguishing this scenario from scalar-only portals.

Additional fields or higher symmetries (e.g., local versus global, U(1) versus discrete) can modify the vacuum structure, dark radiation content ($\Delta N_\text{eff}$), and stability properties. A typical scenario predicts local or global minima coexisting in the scalar potential, imposing constraints from vacuum stability and cosmological history.

The inelastic Higgs-portal complex singlet offers a predictive dark sector signature suite: missing energy and displaced vertex events at colliders, null or inelastic signals in direct detection, potential explanation of gamma-ray astrophysical excesses, and possible gravitational wave backgrounds from modified electroweak phase transitions. All are linked through the portal-induced inelastic mixing and the dynamically transmitted weak-scale mass to the hidden sector—a direct result of electroweak symmetry breaking propagated through singlet kinetic mixing (Cheung et al., 2010, Hooper et al., 30 Jul 2025).