Singlet–Doublet Majorana Dark Matter

• Singlet–Doublet Majorana Dark Matter is a minimal fermionic model where a neutral Majorana fermion arises from mixing a gauge singlet with an SU(2)L doublet, ensuring stability via a Z₂ symmetry.
• The model employs a well-defined Lagrangian and mass matrix diagonalization that yield Majorana eigenstates, with the mixing angle critically controlling gauge interactions and direct detection rates.
• Features such as blind-spot regions, co-annihilation channels, and compressed spectra make the framework testable, linking dark matter relic density computation with neutrino mass generation.

Singlet-Doublet Majorana Dark Matter designates a class of minimal fermionic dark matter models in which dark matter is realized as a neutral Majorana fermion formed from the quantum admixture of a Standard Model (SM) gauge singlet and a vector-like SU(2)_L doublet. These models are among the simplest and most predictive Weakly Interacting Massive Particle (WIMP) scenarios, tightly constrained by theoretical consistency and experimental data. They generically feature a Z₂ parity ensuring dark matter stability and are frequently motivated by their suitability for simultaneous explanations of thermal dark matter and neutrino mass.

1. Field Content, Symmetries, and Lagrangian Construction

The canonical singlet-doublet Majorana dark matter scenario extends the SM by (i) a vector-like SU(2)_L-doublet Dirac fermion Ψ = (ψ⁰, ψ⁻)ᵗ with hypercharge Y = –1 or –½, and (ii) a gauge singlet Majorana fermion χ (or equivalently, one or more right-handed neutrinos N_R) (Dutta et al., 2021, Dutta et al., 2020). An exact discrete Z₂ symmetry classifies the new dark-sector fields as odd and all SM fields as even, yielding the necessary relic stability.

The minimal Lagrangian, prior to electroweak symmetry breaking (EWSB), is: $\mathcal{L}_\text{DM} \supset \overline{\Psi}(i\slashed{D} - M_D)\Psi - \tfrac12 M_S\,\overline{\chi^c}\chi - (y\,\overline{\Psi}\widetilde{H}\chi + \text{h.c.})$ where H is the SM Higgs doublet and $\widetilde{H} = i \sigma_2 H^*$. After EWSB, the doublet–singlet Yukawa coupling induces a Dirac mass $m_D = y v/\sqrt{2}$. For models motivated by neutrino mass generation, three right-handed neutrinos $N_{R_i}$ are often included, of which just one is Z₂-odd and mixes directly with the doublet (Dutta et al., 2021).

The neutral fermion mass matrix in the $(\chi, \psi^0)$ basis becomes: $$M_\text{neutral} = \begin{pmatrix} M_S & m_D \ m_D & M_D \end{pmatrix}$$ This $2 \times 2$ structure generalizes to larger matrices in models with more Weyl components or extended gauge representations (Lopez-Honorez et al., 2017, Arcadi, 2018).

2. Spectrum, Mass Eigenstates, and Mixing Angles

Diagonalization of the neutral mass matrix yields two or three Majorana eigenstates, with the lightest (commonly labeled $\chi_1$ or $\chi_3$) identified as the dark matter candidate. For the two-state setup: $$m_{\chi_{1,2}} = \frac{M_D + M_S}{2} \mp \frac12 \sqrt{(M_D - M_S)^2 + 4 m_D^2}$$ The mixing angle θ is fixed by

$\tan 2\theta = \frac{2 m_D}{M_D - M_S}$

Admixture of singlet and doublet sectors is parametrized by $\sin^2\theta$, controlling the doublet fraction, and thus the coupling strength to $W^\pm, Z$ and the Higgs.

In generalized scenarios with two doublets or extended scalar sectors (e.g., Two Higgs Doublet Models, scalar-assisted models), the mass matrix increases in size but preserves the qualitative structure: doublet-induced gauge interactions and singlet-induced suppression of direct detection rates (Banik et al., 2018, Arcadi, 2018). The charged mass eigenstates are always Dirac fermions with mass $M_D$.

3. Couplings to Standard Model Bosons and Blind Spots

The Majorana nature of the lightest eigenstate eliminates all tree-level diagonal couplings to the $Z$ boson—a nontrivial and unique feature that sharply distinguishes the Majorana singlet-doublet scenario from its Dirac cousin (Dutta et al., 2021, Cohen et al., 2011). Explicitly, the $Z\chi_1\chi_1$ vertex vanishes; only the $h$ and off-diagonal $Z$ couplings remain relevant at tree level: $$g_{h\chi_1\chi_1} = y\,\sin2\theta = (2\,m_D/v)\sin2\theta$$ Consequently, tree-level elastic spin-independent (SI) and inelastic spin-dependent (SD) nucleon–DM scattering via $H$ and $Z$ exchange, respectively, exhibit sharp dependences on the doublet-singlet mixing.

Blind-spot configurations occur when either $g_{h\chi_1\chi_1}$ or the residual $Z$-exchange coupling vanish through parameter tuning (e.g., $y_2/y_1 = \pm 1$ or a special ratio of $y_2/y_1$ to $M_S/M_D$) (Bhattiprolu et al., 16 May 2025, Cynolter et al., 2015). These regions evade SI or SD direct-detection limits while maintaining appropriate relic density, but restrict allowed parameter combinations.

4. Dark Matter Relic Density and Dynamics

Relic abundance of singlet-doublet Majorana dark matter is determined by standard thermal freeze-out, where annihilation and (for small mass splittings) co-annihilation channels dominate: $$\Omega_\chi h^2 \approx \frac{1.07 \times 10^9\,\text{GeV}^{-1}}{\sqrt{g_*} M_\text{Pl} \int_{x_f}^\infty \langle \sigma v \rangle x^{-2} dx}$$ The dominant annihilation processes include

• $\chi_1\chi_1 \to W^+W^-$, $ZZ$, $hh$ (via t-channel and s-channel diagrams)
• $f\bar{f}$ via Higgs exchange (controlled by $g_{h\chi_1\chi_1}$, thus by $\sin2\theta$)
• Co-annihilation with heavier doublet and singlet states when $\Delta M \lesssim 20$–$50$ GeV.

The precise competition between direct annihilation, co-annihilation, and conversion-driven processes further enlarges the viable parameter space—especially for small mixing angles, where conversion processes maintain quasi-equilibrium down to freeze-out (Paul et al., 18 Nov 2025). The allowed mass range, as constrained by relic density and current direct-detection, extends from as low as 1 GeV up to $\sim$1.75 TeV for mixing angles $2\times10^{-7}\lesssim \sin\theta \lesssim 0.16$ (Paul et al., 18 Nov 2025).

5. Direct and Indirect Detection Constraints

The primary direct detection signal arises from Higgs-mediated SI scattering,

$$\sigma_\text{SI} = \frac{f_N^2}{\pi} \frac{\mu_{\chi N}^2}{m_h^4} (y\,\sin2\theta)^2$$

where $\mu_{\chi N}$ is the DM-nucleon reduced mass and $f_N \approx 0.3$. For $m_\chi \sim 100$ GeV, the null results from XENON1T/LZ require $y\,\sin2\theta \lesssim O(10^{-2})$ (Dutta et al., 2021, Paul et al., 18 Nov 2025).

Spin-dependent rates are further suppressed by the absence of a tree-level diagonal $Z$ coupling and the predominantly inelastic nature of possible transitions. This implies that, outside special resonance regions (e.g., $m_{\chi_1} \sim m_h/2$ or $m_Z/2$) or small mass-splitting co-annihilation corridors, SI direct detection experiments provide the leading constraint.

Highlighting "blind-spot" regions, parameter combinations can yield vanishing SI or SD scattering, evading experimental limits while reproducing the observed relic abundance (Cynolter et al., 2015, Bhattiprolu et al., 16 May 2025). These remain consistent with heavy DM masses and compressed spectra.

6. Collider Phenomenology and Theoretical Consistency

Collider signatures are set by the compressed spectrum: doublet-like charged states ($\psi^\pm$) and heavier neutral states, with splittings $\Delta M \sim 1$–$50$ GeV. Production proceeds via Drell-Yan mechanisms, and decay products typically feature soft leptons or pions plus missing energy—implying challenges for LHC searches, but viable prospects for future colliders or dedicated searches for long-lived charged tracks in the very small-mixing regime (Bhattiprolu et al., 16 May 2025, Paul et al., 18 Nov 2025).

Perturbative unitarity restricts the Yukawa couplings $|y_i| \lesssim 4\sqrt{\pi}$, compatible with the parameter space selected by relic, direct detection, and collider constraints (Cynolter et al., 2015). Absolute vacuum stability, especially in scalar-extended variants, further restricts the singlet-doublet mixing angle $\sin\theta$ to a narrow band (Banik et al., 2018).

A prominent theoretical feature is the "RG focus" effect: one-loop renormalization group evolution can naturally drive generic UV initial conditions to infrared parameter values very close to the direct-detection blind spots, suggesting improved UV robustness and predictivity of such models (Bhattiprolu et al., 16 May 2025).

7. Connections to Neutrino Mass, Electroweak Physics, and Extended Frameworks

Motivated by the minimality and predictive power, singlet-doublet Majorana DM models often accommodate neutrino mass generation. In the benchmark setup, two Z₂-even right-handed neutrinos induce a classic Type-I seesaw, giving

$m_\nu = -m_D^{(\nu)} M_R^{-1} (m_D^{(\nu)})^T$

fitting solar and atmospheric neutrino data, while the Z₂-odd state participates in DM mixing (Dutta et al., 2021, Dutta et al., 2020). Radiative neutrino mass may also be realized at one loop in extensions with additional Z₂-odd scalars, which connect dark matter and neutrino phenomenology and may induce observable lepton flavor violation (Dey et al., 24 Nov 2025).

The framework is compatible with a range of electroweak phenomena: corrections to the $W$-boson mass via oblique parameters (Borah et al., 2022), stabilization of the electroweak vacuum in scalar-extended models (Banik et al., 2018), or first-order electroweak phase transitions leading to gravitational wave signals within reach of next-generation detectors (Dey et al., 24 Nov 2025).

The parameter space—after imposing all constraints—consists of the following canonical regions:

• $m_\chi \gtrsim 100$ GeV, typically up to a few TeV,
• mass splitting $\Delta M \sim 10$–$200$ GeV,
• mixing angle $\sin\theta$ in the range $10^{-7}$–$0.5$, with smaller values favored by direct-detection null results,
• exceptional survival of "Higgs funnel" ($m_\chi \simeq m_h/2$) or co-annihilation strips (Dutta et al., 2021, Bhattiprolu et al., 16 May 2025, Paul et al., 18 Nov 2025).

Singlet-doublet Majorana models supply not only a UV-complete, minimal extension for WIMP dark matter, but also a rich and testable framework for linking dark matter with neutrino physics, flavor, electroweak precision, and collider searches (Dutta et al., 2021, Dey et al., 24 Nov 2025, Dutta et al., 2020, Bhattiprolu et al., 16 May 2025).