Two Real Scalar Singlet Extension

• The two real scalar singlet extension is a renormalizable framework that introduces two distinct gauge singlet scalars to the Standard Model, enabling rich scalar mixing and multi-Higgs phenomena.
• It modifies the scalar potential to produce complex vacuum dynamics, including one- and two-step electroweak phase transitions capable of generating observable gravitational wave signals.
• The model provides novel dark matter phenomenology via Higgs portal interactions and predicts distinctive collider signatures through resonant and cascade Higgs decays.

The two real scalar singlet extension of the Standard Model (SM) is a renormalizable framework wherein the SM scalar sector is augmented by two distinct real scalar fields, each transforming as a gauge singlet. This construction is motivated by the desire to simultaneously address open questions in particle physics and cosmology, including the origin of dark matter, the nature of the electroweak phase transition, gravitational wave signatures, collider phenomenology, vacuum stability, and naturalness. Distinct $\mathbb{Z}_2$ (or similar discrete) symmetries are typically imposed on the singlet fields to guarantee their stability and to control the coupling structure. The resulting scalar potential generically exhibits mixing among the neutral scalars, providing rich phenomenological consequences accessible at collider and non-collider experiments.

1. General Scalar Potential Structure and Mixing

The most general renormalizable potential for a SM Higgs doublet $H$ and two real singlets $S$ and $X$ (with imposed $\mathbb{Z}_2\otimes\mathbb{Z}'_2$ symmetries such that $S\to -S$, $X\to -X$) is

$$V = \mu_H^2 (H^\dagger H) + \lambda_H (H^\dagger H)^2 + \mu_S^2 S^2 + \lambda_S S^4 + \mu_X^2 X^2 + \lambda_X X^4 + \lambda_{HS} (H^\dagger H) S^2 + \lambda_{HX} (H^\dagger H) X^2 + \lambda_{SX} S^2 X^2,$$

where all terms are even under both singlet discrete symmetries. After electroweak symmetry breaking (EWSB) and spontaneous breaking of the singlet symmetries (unless preserved), $H$, $S$, and $X$ can acquire vacuum expectation values (VEVs), leading to mixing among the CP-even neutral scalars. The neutral scalar mass matrix is generally $3\times3$, and the mass eigenstates ($h_1$, $h_2$, $h_3$) are related to the interaction states by an orthogonal transformation characterized by three mixing angles. One physical eigenstate is identified with the observed 125 GeV SM-like Higgs, while the others are dominantly singlet-like, depending on the mixing pattern (Robens et al., 2019, aali et al., 2020).

Singlet fields with exact or approximate $\mathbb{Z}_2$ symmetries can be dark matter (DM) candidates if they do not mix with the Higgs. In scenarios where both $S$ and $X$ develop VEVs, full mixing occurs, leading to three observable physical scalars with rescaled and novel couplings to SM and non-SM particles, and potentially to each other through cubic and quartic interactions.

2. Electroweak Phase Transition and Gravitational Waves

The addition of two real singlets modifies the finite-temperature Higgs potential, enabling scenarios of strong first-order electroweak phase transitions (EWPT), a prerequisite for electroweak baryogenesis. The field $S_1$ (or $S$) is typically chosen to have a nonzero VEV, altering the shape of the potential, mainly through additional portal and cubic terms; $S_2$ (or $X$) is stabilized by a $\mathbb{Z}_2$ and serves as DM (Tofighi et al., 2015, Shajiee et al., 2018). At finite temperature, the effective potential is extended to include: $$V_{\text{eff}} = V_{\text{tree}} + V_{1-\text{loop}}^{T=0} + V_{1-\text{loop}}^{T\neq 0},$$ with careful treatment of thermal corrections. The dynamics yield two possible sequences:

• Two-step transition: at high temperature, the singlet sector undergoes symmetry breaking first, followed by a first-order phase transition in the Higgs sector, typically assisted by a tree-level barrier stemming from cubic terms or large portal couplings.
• One-step transition: direct transition from the symmetric to the electroweak vacuum.

The order parameter is quantified by $\xi = v_c/T_c$ (the ratio of the broken-phase Higgs VEV to the critical temperature). Scenarios with heavy singlet mass enhance the transition strength even further, with benchmark values $\xi \gtrsim 1-4$ corresponding to a strongly first-order EWPT (Shajiee et al., 2018). Such transitions generate gravitational wave (GW) backgrounds detectable at planned space-based interferometers (eLISA, ALIA, DECIGO, BBO). The wave spectrum depends critically on the latent heat (parameterized by $\alpha$), the inverse duration ($\beta/H$), and the wall velocity, all computable within these models and found to be within reach of future GW experiments for representative benchmark points (Shajiee et al., 2018).

3. Dark Matter Phenomenology

The two singlet setup gives rise to multiple DM candidates due to $\mathbb{Z}_2$ stabilization. The relic abundance is typically set by thermal freeze-out of singlet annihilation to SM particles via the Higgs portal. The interaction terms: $$\lambda_{HS} (H^\dagger H) S^2,\quad \lambda_{HX} (H^\dagger H) X^2,$$ permit $SS\to$ SM or $XX\to$ SM annihilations. If $S$ and $X$ are both stable, their respective abundances sum to match the Planck 2018 measurement ($\Omega_c h^2 = 0.120\pm0.001$). Mutual interactions (e.g., $\lambda_{SX} S^2X^2$) and possible cascade decays (of the heavier singlet to the lighter) further open coannihilation or semi-annihilation channels, altering the DM freeze-out dynamics.

Direct (e.g., XENON1T (Shajiee et al., 2018)) and indirect (e.g., gamma-ray fluxes, (Drozd et al., 2011)) detection limits provide stringent constraints on the Higgs portal couplings. For example, spin-independent direct detection is mediated by t-channel Higgs exchange and scales as $\sigma_{\text{SI}} \sim \lambda_{HS,X}^2 / m_h^4$. Parameter regions where the invisible Higgs decay $h\to SS$ or $h\to XX$ is kinematically allowed are further constrained by collider measurements of the Higgs invisible width (Robens et al., 2019, Maniatis, 2020).

Additionally, extended mediator models (with a "dark sector" mediator connecting DM to SM) can address structure formation issues. The two-singlet extension, where one field is DM and the other is a light mediator, can achieve velocity-dependent DM self-interactions to alleviate small-scale structure problems, while maintaining consistency with cosmic microwave background and big-bang nucleosynthesis through fast mediator decays to right-handed neutrinos (Maniatis, 2020).

4. Collider Phenomenology and Multi-Scalar Signatures

The two singlet extension predicts a phenomenologically rich scalar sector. After mixing, the three neutral CP-even Higgs states exhibit reduced couplings to SM particles, as the doublet component is diluted among the mass eigenstates. Single production and direct decays of each scalar are uniformly rescaled by the doublet admixture: $$\sigma(pp \to h_a) = \kappa_a^2 \cdot \sigma_{\text{SM}}(M_a),\quad \kappa_a = (\text{doublet admixture of } h_a),\quad \sum_a \kappa_a^2=1,$$ where $a=1,2,3$ denotes the physical states (Robens et al., 2019, aali et al., 2020).

A novel feature is the prevalence of resonant Higgs-to-Higgs decays: heavier scalar mass eigenstates $h_a$ can decay to pairs of lighter scalars ($h_b h_c$), with partial widths given by

$$\Gamma_{a \to b c} = \frac{{\lambda}_{abc}^2}{16\pi M_a^3} \sqrt{\lambda(M_a^2, M_b^2, M_c^2)},$$

where $\lambda$ is the Källén function and ${\lambda}_{abc}$ are the cubic scalar couplings determined by the mixing. These "symmetric decays" ($h_a \to h_b h_b$), "asymmetric decays" ($h_a \to h_b h_c$ with $b\ne c$), and "cascade decays" (successive decays $h_3 \to h_2 h_1$, $h_2 \to h_1 h_1$) lead to multi-Higgs final states (three or four Higgs bosons), giving rise to characteristic LHC signatures such as $6b$, $4b2W$, or $bbbWW$ (Robens et al., 2019). Such multi-boson final states require tailored searches and have generally not been thoroughly explored by collider experiments.

Benchmark scenarios provided in (Robens et al., 2019) fix parameters to maximize various production and cascade rates, offering targets for experimental analyses in upcoming collider runs. The presence of these heavy mixed states and multi-Higgs signatures is a striking prediction of this framework (aali et al., 2020).

5. Vacuum Stability, Naturalness, and Theoretical Constraints

The extended scalar potential's stability conditions are more restrictive than in the single-singlet case due to the increased number of quartic and portal couplings. The positivity condition in the two-singlet scenario demands that, for all field directions: $$\lambda_H > 0,\, \lambda_S > 0,\, \lambda_X > 0,\, \lambda_{HS} + 2\sqrt{\lambda_H \lambda_S}>0,\, \lambda_{HX} + 2\sqrt{\lambda_H \lambda_X}>0,\, \lambda_{SX} + 2\sqrt{\lambda_S \lambda_X}>0,$$ and extensions thereof (Ghorbani, 2021). Absolute vacuum stability requires that the physical EWSB vacuum is the global minimum for all renormalization group (RG) scales up to the Planck scale, imposing further constraints on the parameter space (Ghorbani, 2021).

Naturalness is addressed via the Veltman conditions (VC), which aim to cancel quadratic divergences at one-loop order in the Higgs mass corrections. In the two-singlet model, the VCs generalize to all three CP-even states: $$\delta m_{h_1}^2 \sim 2 m_W^2 + m_Z^2 - 3 t_f m_t^2 + (3\lambda + \lambda_4 + \lambda_5) v^2$$ and analogously for $h_2$ and $h_3$, where the contributions from new quartic couplings can partially cancel the dominant SM top and gauge boson terms (aali et al., 2020). FulfiLLMent of the Veltman conditions reduces the degree of fine-tuning but significantly constrains the viable parameter space and fixes relations among the new quartic couplings.

Tree-level and loop-level (e.g., Coleman–Weinberg) corrections to the effective potential, RG evolution of couplings, unitarity, and perturbativity are all implemented as theoretical constraints to delineate phenomenologically viable regions.

6. Comparative Analysis with Simpler Extensions and Outlook

Relative to single-singlet or single-doublet extensions, the two real scalar singlet extension

• provides broader flexibility in addressing DM, baryogenesis, and collider physics concurrently;
• yields a richer vacuum landscape (multiple critical points, possible multi-step transitions) and more involved stability conditions;
• predicts distinctive collider (multi-Higgs) and GW signatures as in (Robens et al., 2019, Shajiee et al., 2018).

While the increased parameter space can improve compatibility with data (e.g., evading direct detection, solving the little hierarchy problem, enhancing GW signals), the complexity introduces both theoretical and experimental challenges. Constraints from invisible Higgs decays, Higgs coupling measurements, relic density, and direct/indirect DM detection must be applied in conjunction.

Future experiments—precision Higgs measurements, high-luminosity LHC runs, GW observatories, and low-threshold DM searches—are poised to probe the model's viable regions further. Multi-scalar singlet scenarios remain an active research direction due to their minimality, rich phenomenological consequences, and consistent alignment with multiple observations across cosmology, astroparticle physics, and collider experiments.