Singlet-Doublet Mass Splitting in Particle Physics

Updated 20 November 2025

Singlet-doublet mass splitting is the mass difference emerging from the mixing of SM singlets and SU(2) doublets via Yukawa couplings and symmetry breaking.
Analytic formulations using 2×2 mass matrices reveal the dependence on parameters like M_S, M_D, and the Yukawa coupling, distinguishing Majorana and Dirac cases.
Phenomenological implications include altered relic density through co-annihilation, modified direct detection rates, and distinct collider signatures.

A singlet-doublet mass splitting refers generically to the mass difference arising between states that originate as Standard Model singlets and SU(2) doublets, after mixing via Yukawa-type couplings and symmetry breaking effects. This concept is central both in extended Higgs sectors (for instance, in the Higgs singlet-doublet mixing of the NMSSM or its PQ-variant) and in fermionic dark sector models, where the lightest dark matter candidate is an admixture of singlet and doublet components. The precise size and structure of this splitting has direct implications for relic density calculations, direct detection cross sections, collider signatures, and the viability of parameter space in beyond–Standard Model scenarios.

1. Mass Matrices and Mixing Mechanisms

In singlet-doublet models, the physical spectrum after electroweak symmetry breaking is derived from a non-diagonal mass matrix, typically of the form (for the minimal fermion scenario): $\mathcal{M} = \begin{pmatrix} M_S & y v/\sqrt{2} \ y v/\sqrt{2} & M_D \end{pmatrix}$ where $M_S$ is the bare singlet mass, $M_D$ is the doublet mass, $v$ is the SM Higgs vacuum expectation value, and $y$ is a Yukawa coupling. In the scalar (Higgs) sector, as found in the PQ-NMSSM, the active states are also mapped onto a $2\times2$ matrix in the $(h,s)$ basis, with off-diagonal elements set by new couplings and soft parameters (Jeong et al., 2012).

For Dirac fermions, distinct left- and right-handed Yukawa couplings ( $y_1$ , $y_2$ ) can be present, necessitating bi-unitary diagonalization (Yaguna, 2015). For scalar or Majorana fermion cases, symmetric diagonalization suffices (Barman et al., 2019, Dutta et al., 2021).

Diagonalization yields two (or more, in the general case) mass eigenstates, each a superposition of the original singlet and doublet fields, with the mass splitting $\Delta m$ (or $\Delta M$ ) between them controlled by both the diagonal mass difference and the strength of the induced mixing.

2. Analytic Mass Splitting Formulæ

The canonical expression for the singlet-doublet mass splitting for a general $2\times2$ mass matrix is: $\Delta m = \sqrt{(M_D - M_S)^2 + 2 y^2 v^2}$ for Majorana fermions and

$\Delta m = \sqrt{(M_D - M_S)^2 + 4 m_D^2}$

for Dirac fermions, where $m_D = y v/\sqrt{2}$ . Approximations in the small-mixing (weak Yukawa) limit, $y v \ll |M_D - M_S|$ , yield: $\Delta m \simeq |M_D - M_S| + \frac{y^2 v^2}{|M_D - M_S|}$ while in the near-degenerate (maximal mixing) regime, $M_D \approx M_S$ , one finds

$\Delta m \simeq \sqrt{2} y v$

(Barman et al., 2019, Paul et al., 18 Nov 2025, Paul et al., 3 Dec 2024, Yaguna, 2015).

For scalar mass matrices (e.g., in extended Higgs sectors), the structure is analogous, with the mixing controlled by couplings such as $\lambda$ , and soft parameters, and the splitting given via the eigenvalue difference of a $2\times2$ mass-squared matrix (Jeong et al., 2012).

Model Variant	Mass Splitting Expression	Reference
Minimal fermion (Majorana)	$\Delta m = \sqrt{(M_D - M_S)^2 + 2 y^2 v^2}$	(Barman et al., 2019)
Dirac fermion	$\Delta m = \sqrt{(M_D - M_S)^2 + 4 m_D^2}$	(Yaguna, 2015)
2HDM (two-doublet) extension	$\Delta m_{ij} \approx \|m_{D}-m_{S}\| + \mathcal{O}(\lambda^2 v^2/m)$	(Arcadi, 2018)
PQ-NMSSM Higgs sector	See Eq. (28), e.g., $\Delta m = \sqrt{(M^2_{hh}-M^2_{ss})^2 +4(M^2_{hs})^2}/(m_{H_2}+m_{H_1})$	(Jeong et al., 2012)
Scalar singlet-doublet mixing	$\Delta m = \sqrt{\,\sqrt{(m_D^2 - m_S^2)^2 + 4(μ v)^2}\,}$	(Forero et al., 2016)

3. Phenomenological Implications and Parameter Dependencies

The singlet-doublet mass splitting critically determines several aspects of dark sector and Higgs phenomenology:

Co-annihilation Efficiency: For $\Delta m \lesssim O(10~\mathrm{GeV})$ , the heavier doublet-like state is thermally populated at freeze-out, enabling efficient co-annihilation. If $\Delta m \gg T_f \sim m_{\chi_1}/20$ , co-annihilation is Boltzmann-suppressed, and the relic density is set by pure self-annihilation (Paul et al., 3 Dec 2024, Paul et al., 18 Nov 2025, Konar et al., 2020, Arcadi, 2018).
Direct Detection Constraints: Elastic $Z$ -exchange couplings depend on the doublet component and are suppressed when the lightest state is singlet-like and the splitting is small. For Dirac dark matter, severe bounds on vector-coupled scattering restrict both mixing angles and $\Delta m$ , e.g., $\Delta m/m_1 \lesssim 9\%$ (Yaguna, 2015). Majorana cases evade these constraints through suppressed vector couplings (Paul et al., 18 Nov 2025).
Collider Signatures: Charged partners with small $\Delta m$ decay via off-shell $W$ and can yield displaced vertices detectable at the LHC or MATHUSLA if $\sin\theta$ is small and $\Delta m$ resides in the few-GeV regime (Paul et al., 3 Dec 2024).
Relic Density vs. Direct Detection Trade-off: Small $\Delta m$ enhances co-annihilation but may suppress direct detection through mixing angle suppression; conversely, large $\Delta m$ demands stronger mixing to achieve the correct relic density, increasing direct-detection rates (Dutta et al., 2020).

Within the PQ-invariant NMSSM framework, singlet-doublet Higgs mixing can raise the SM-like Higgs mass by several GeV (up to 7 GeV for specific parameters) without requiring large stop mixing, while the magnitude of $\Delta m$ is tightly constrained by LEP and LHC Higgs searches (Jeong et al., 2012).

4. Empirically Allowed Ranges and Benchmark Scenarios

Scans over parameter space in recent studies establish the following numerical regimes for viable singlet-doublet mass splittings, subject to constraints from relic density, direct detection, and collider searches:

Relic Density Allowed Regime: For Majorana dark matter,

$1~\mathrm{GeV} \lesssim \Delta m \lesssim 100~\mathrm{GeV}$

for $\sin\theta$ within $2\times10^{-7} \lesssim \sin\theta \lesssim 0.16$ , with upper $\Delta m$ bound set by perturbativity and unitarity (Paul et al., 18 Nov 2025).

Direct Detection: In the Dirac case, $\Delta m$ is restricted to below $9\%$ of the lightest mass for $m_{\chi_1} \lesssim 750$ GeV, forced by the need to suppress $Z$ -mediated elastic scattering (Yaguna, 2015).
Co-annihilation Window: Co-annihilation is operative for $\Delta m \lesssim (10-20)$ GeV (depending on dark sector mass), as established by Boltzmann suppression analyses (Paul et al., 3 Dec 2024, Konar et al., 2020).
Large Splitting and Higgs Sector: In PQ-NMSSM, $\Delta m$ between two CP-even Higgses can be $3$–$7$ GeV for mixing angles near current experimental limits, contributing to the measured $125$ GeV Higgs mass without excessive stop mixing (Jeong et al., 2012).

Blind Spots: The parameter region where the $h\chi_1\chi_1$ or $Z\chi_1\chi_1$ coupling vanishes (“blind spots”) coincides with specific $\Delta m$ values determined by destructive interference between singlet and doublet components (Calibbi et al., 2015, Cynolter et al., 2015). In such regions, elastic cross sections plummet despite substantial mixing.
Scalar Higgs Mixings: In extended Higgs sectors, as in the PQ–NMSSM, the splitting between singlet- and doublet-like Higgs states enters as an explicit function of soft terms ( $m_S^2,~A_\lambda,~B,~\mu$ ) and quartic couplings. Experimental bounds on the Higgs sector place nontrivial constraints on the level of allowed mixing and hence on the splitting (Jeong et al., 2012).
Conversion-driven Freeze-out (Co-scattering): For very small mixing angle and low splittings, conversion-driven processes (e.g., $N_1$ SM $\rightarrow N_2$ SM, with $N_2$ subsequently annihilating) become essential for depleting the relic density. This pushes the viable parameter space into regimes testable by displaced-vertex searches, a phenomenology recently recognized in (Paul et al., 3 Dec 2024, Paul et al., 18 Nov 2025).

Constraint/Regime	$\Delta m$ (typ.)	Mixing $\sin\theta$	Reference
Co-annihilation	$1$–$20$ GeV	$10^{-4}$ – $10^{-2}$	(Paul et al., 3 Dec 2024)
Annihilation-dominated	$>100$ GeV	$0.05$–$0.16$	(Paul et al., 18 Nov 2025)
Direct-detection (Dirac)	$<9\%~m_1$	$\ll1$	(Yaguna, 2015)
PQ-NMSSM Higgs	$3$–$7$ GeV (CP-even)	up to $0.25$	(Jeong et al., 2012)
Scalar doublet-singlet NSI	$80$ GeV to $10$ TeV	$\sim0.3$	(Forero et al., 2016)

6. Unitarity and Theoretical Limits

The Yukawa couplings that set the off-diagonal splitting contributions are restricted by perturbative unitarity. Explicit bounds include $|y| \leq 4\sqrt{\pi}\approx 7.1$ , giving a maximum off-diagonal entry $y v/\sqrt{2} \lesssim 1.23$ TeV (Cynolter et al., 2015). These theoretical limits, together with stability and vacuum constraints in more elaborate models (e.g., $B\!-\!3L_\tau$ extensions, scalar–assisted setups), ensure that splittings in excess of several hundred GeV cannot be realized in weakly-coupled regimes (Barman et al., 2019, Banik et al., 2018).

7. Summary and Thematic Perspective

Singlet-doublet mass splitting is a robust, model-independent signature of extensions involving singlet and doublet fields coupled via new Yukawa or soft terms. It is universally expressible as a simple function of mass parameters and couplings, with analytic structure determined by the diagonalization of a $2\times2$ (or, in extended Higgs or fermion sectors, $3\times3$ ) mass matrix. The physical consequences of the splitting are wide-ranging:

It defines the thermal history of the dark sector (annihilation, co-annihilation, co-scattering).
It sets the scale for direct-detection rates, being directly correlated with mixing and $Z$ -mediated couplings.
It shapes collider phenomenology, determining the lifetimes and decay topologies of next-to-lightest states.
In Higgs sectors, it governs the ability of models like the PQ-invariant NMSSM to naturally realize a 125 GeV Higgs without large radiative corrections.

Comprehensive analyses confirm that, modulo theoretical and experimental constraints, allowed singlet-doublet splittings reside between a few GeV and a few hundred GeV, with key “corridors” determined by the interplay of relic density, direct-detection, and collider bounds (Paul et al., 18 Nov 2025, Paul et al., 3 Dec 2024, Konar et al., 2020, Jeong et al., 2012). The analytic and phenomenological tools applied in this context are now standard framework elements in model building and phenomenological studies across particle and astroparticle physics.