Singlet Scalar Dark Matter Candidate

• Singlet Scalar Dark Matter Candidate is a neutral field stabilized by a Z2 symmetry, interacting with Standard Model particles primarily through the Higgs portal.
• Thermal freeze-out and freeze-in mechanisms determine its relic density, with distinct viable regions identified in low-mass, resonance, and high-mass domains.
• Direct and indirect detection strategies leverage Higgs-mediated nuclear scattering and gamma-ray searches to reconcile theoretical predictions with experimental constraints.

A singlet scalar dark matter candidate refers to a neutral, Standard Model (SM) gauge singlet scalar field that is stabilized—typically by a discrete symmetry such as $\mathbb{Z}_2$—and interacts with SM fields primarily via the Higgs portal. This candidate represents one of the simplest and most studied minimal extensions of the SM for addressing the origin and phenomenology of cold dark matter (CDM). Such a construction permits analytical control, sharp correlations between model parameters and observable quantities, and compatibility with current theoretical and experimental constraints.

1. Theoretical Construction and Stabilizing Symmetry

The core theoretical framework introduces a real scalar field $S$ that is a singlet under the SM gauge group. The most general renormalizable Lagrangian involving $S$ is: $$\mathcal{L} = \mathcal{L}_\mathrm{SM} + \frac{1}{2} (\partial_\mu S)^2 - \frac{1}{2} m_0^2 S^2 - \frac{\lambda_S}{4} S^4 - \lambda S^2 H^\dagger H$$ where $H$ is the SM Higgs doublet and $\lambda$ is the dimensionless Higgs–singlet coupling (the "Higgs portal" interaction).

To ensure cosmological stability, a discrete $\mathbb{Z}_2$ symmetry is imposed: $S \to -S$, all SM fields invariant. This forbids all odd-power terms in $S$—including $S$ and $S^3$ terms—and prevents $S$ from mixing with the Higgs or decaying into SM particles. As a result, $S$ remains stable on cosmological timescales, making it a dark matter candidate that couples to the visible sector exclusively via Higgs portal interactions (Biswas et al., 2011).

2. Thermal Production, Relic Density, and the Boltzmann Equation

In the standard WIMP (Weakly Interacting Massive Particle) paradigm, the relic abundance of $S$ is set by thermal freeze-out. The evolution of the number density $n$ of $S$ is governed by the Boltzmann equation: $$\frac{dn}{dt} + 3Hn = - \langle \sigma v \rangle (n^2 - n_\mathrm{eq}^2)$$ where $H$ is the Hubble parameter, $\langle \sigma v \rangle$ is the thermally averaged annihilation cross-section into SM final states, and $n_\mathrm{eq}$ is the equilibrium number density. After changing variables to the yield $Y = n/s$ (with $s$ the entropy density) and $x = m_S/T$, the equation reads: $$\frac{dY}{dx} = -\sqrt{\frac{45}{\pi G}} \frac{g_*^{1/2} m_S}{x^2} \langle \sigma v \rangle (Y^2 - Y_\mathrm{eq}^2)$$ This equation, integrated from high to low temperature, yields the present-day yield $Y_0$, which sets the relic density via

$$\Omega h^2 = 2.755 \times 10^8 \left( \frac{m_S}{\mathrm{GeV}} \right) Y_0$$

For the model to match the cosmological dark matter abundance measured by Planck, the freeze-out temperature is typically $T_f \sim m_S/20$ (Biswas et al., 2011, Collaboration et al., 2017).

In contrast, in the FIMP (Feebly Interacting Massive Particle) regime, where $\lambda$ is extremely small ($\sim 10^{-11} - 10^{-12}$), $S$ never attains equilibrium and is produced via freeze-in: $$\frac{dY}{dT} = \sqrt{\frac{\pi g_*(T)}{45}} M_P \langle \sigma v \rangle Y_\mathrm{eq}^2$$ Here, $Y \ll Y_\mathrm{eq}$ throughout cosmic history, leading to a suppressed abundance and, crucially, no detectable signals in current direct or indirect detection experiments (Yaguna, 2011).

3. Allowed Parameter Space and Experimental Constraints

The phenomenologically viable parameter space is delineated by requiring that the predicted relic density matches the range from cosmological observations (e.g., $0.099 \leq \Omega h^2 \leq 0.123$ from WMAP/Planck) and that the direct detection cross-section does not exceed exclusion bounds from experiments such as LUX, Xenon1T, PandaX, CDMS-II, CoGeNT, DAMA, and EDELWEISS-II.

A key result is the emergence of two distinct viable regions in the $(m_S, \lambda)$ parameter space (Biswas et al., 2011, Collaboration et al., 2017): | Mass Region | $m_S$ [GeV] | $\lambda$ or $\delta_2$ | Features | |:---|:---|:---|:---| | Low-mass | 6–16 | 0.7–1.25 | Supported by anomalies/WMAP; high coupling | | Resonance | $\sim$62.5 | $10^{-4}$–$10^{-3}$ | Near $m_h/2$, annihilation via Higgs resonance | | High-mass | 52.5–1000+ | 0.02–0.4 | Satisfies WMAP + direct detection; low coupling |

In the resonance region around $m_S \sim m_h/2$, the annihilation cross-section is resonantly enhanced, allowing even small couplings to be consistent with the observed relic density. Experimental bounds on the invisible Higgs width (e.g., $\text{Br}_\mathrm{inv} < 19\%$ at LHC) further exclude regions with $m_S < m_h/2$ and large coupling. For $m_S > 1$ TeV, heavier dark matter is allowed for moderate to order-one couplings (Collaboration et al., 2017).

4. Direct and Indirect Detection Signatures

Direct detection proceeds via elastic spin-independent scattering of $S$ on nuclei through Higgs exchange. The nucleon cross-section is

$$\sigma_N^{\text{scalar}} = \frac{\delta_2^2 v^2 |\mathcal{A}_N|^2}{4\pi} \frac{m_r^2}{m_S^2 m_h^4}$$

with $m_r$ the $S$-nucleon reduced mass and $v$ the Higgs vacuum expectation value. For a target nucleus with atomic number $A$,

$$\sigma^{\text{scalar}}_{\text{nucleus}} = A^2 \left( \frac{m_r(\text{nucleus}, S)^2}{m_r(\text{nucleon}, S)^2} \right) \sigma_N^{\text{scalar}}$$

Detection rates, including annual modulation (Earth's motion), are predicted and show that for Xe targets and $m_S = 55$–65 GeV, the differential rate is suppressed beyond $\sim$80 keV; for Ge and $m_S =$ 8–10 GeV, the rate drops beyond $\sim$10 keV.

Indirect detection focuses on gamma-ray line searches from $S S \to \gamma\gamma$ annihilations in the Galactic Center. The predicted flux,

$$\frac{d\Phi_\gamma}{dE_\gamma} = \frac{1}{8\pi} \frac{\langle \sigma v \rangle_{S S \to \gamma\gamma}}{m_S^2} \frac{dN_\gamma}{dE_\gamma} r_\odot \rho_\odot^2 J$$

is several orders of magnitude below the Fermi–LAT signal for a 130 GeV scalar, barring a large astrophysical or particle boost factor (Biswas et al., 2011).

5. Theoretical and Experimental Implications

The gauge singlet scalar scenario demonstrates a tightly correlated map between particle theory parameters and cosmological, collider, and direct detection observables. Its virtues as a dark matter candidate include theoretical minimality, stability guaranteed by symmetry, and predictive phenomenology. Results show that:

• Both WIMP (thermal freeze-out) and FIMP (freeze-in) regimes are admitted, with drastically different signal expectations (Yaguna, 2011).
• The viable parameter space is split into resonance (fine-tuned) and high-mass (heavier $S$) regions; frequentist analyses favor both, but Bayesian statistical approaches penalize the resonance on grounds of fine-tuning (Collaboration et al., 2017).
• Portions of parameter space yield rates for direct detection signals (cross-sections $10^{-45}$–$10^{-44}$ cm$^2$) within reach of next-generation experiments; indirect gamma-ray signatures are generically unobservable barring enhancement mechanisms (Biswas et al., 2011).
• For light scalar masses ($m_S < m_h/2$), significant regions are already excluded by Higgs invisible width limits and direct detection; the high-mass region remains a robust candidate.

6. Summary Table: Key Model Ingredients and Constraints

Model Feature	Scalar Singlet DM Scenario
Stability	$\mathbb{Z}_2$ symmetry: $S \to -S$
Portal	Higgs portal: $\lambda S^2 H^\dagger H$
Production	Freeze-out (WIMP), Freeze-in (FIMP)
Detection	Direct (Higgs exchange), Indirect ($\gamma$-ray)
Key Constraints	Relic density, direct searches, Higgs width
Viable $m_S$	6–16 GeV (high coupling), $\sim$62 GeV (res.), 52.5 GeV–1 TeV (low coupling), $>1$ TeV (high-mass)
Disfavored Areas	$m_S < m_h/2$, large coupling (inv. decay); lower masses (direct detection)

7. Outlook and Research Directions

Current and future direct detection experiments, precision Higgs measurements, and indirect searches continue to test the parameter space for singlet scalar dark matter. High-mass regions above the Higgs pole and resonance regions remain under especially close scrutiny. The contrast between WIMP-excludable and FIMP-viable regimes opens a research avenue for interpreting sustained null results. The singlet scalar paradigm thus serves as a template for mapping theoretical constructs to empirical tests in the particle–cosmology interface (Biswas et al., 2011, Yaguna, 2011, Collaboration et al., 2017).