Singlet-Doublet fermion origin of dark matter, neutrino mass and inverse first-order electroweak phase transition

Summary

This paper, titled Singlet-Doublet fermion origin of dark matter, neutrino mass and inverse first-order electroweak phase transition, studies a beyond Standard Model (BSM) framework that extends the SM with two generations of SU(2)L singlet and doublet fermions, one singlet scalar, and three right-handed neutrinos (RHNs). The model aims to simultaneously explain dark matter (DM), light Dirac neutrino mass, and an inverse first-order electroweak phase transition (IFOEWPT) with observable gravitational waves (GW).

Model Setup: The authors impose a discrete Z4 symmetry under which the new fermions (χi and Ψi) carry charge-1, the scalar fields φi carry charge i, SM lepton doublets and charged leptons carry charge-i, and the Dirac right-handed neutrinos νR carry charge i. The Z4 symmetry is explicitly broken by a soft term in the scalar potential, which splits the complex scalar singlet φ into two physical scalars with masses Mφ1 and Mφ2, with mass-squared splitting ΔMφ2 = 2μ2φ. After electroweak symmetry breaking (EWSB), the Higgs VEV induces mixing between the singlet fermion and the neutral component of the doublet fermion, leading to two neutral mass eigenstates per generation (χ, ψ for generation 1; χ′, ψ′ for generation 2), with mixing angles sin θ1 and sin θ2 respectively. The lightest state, χ, is the DM candidate.

Radiative Neutrino Mass: The Dirac neutrino mass is generated at one-loop level via diagrams involving the SD fermions and the singlet scalar φ. The one-loop neutrino mass formula is given by:

mναβ = (-iμ2φ/(4π2)) Σi (λΨiα)T ((MY - MX) sin 2θi) F(MX, MY, Mφ1, Mφ2) Iii (λχiβ),

where F is a loop factor. The Yukawa couplings λψ and λχ are parameterized using the PMNS matrix and a projection matrix, with the electron-type coupling of λχ set to zero.

Constraints: The parameter space is constrained by:

• Neutrino oscillation data

• Muon anomalous magnetic moment (g-2)μ, with upper limit Δaμ = 101 × 10−11

• Charged lepton flavor violation (cLFV) bounds: Br(μ→eγ) < 1.5×10−13, Br(τ→eγ) < 3.3×10−8, Br(τ→μγ) < 4.4×10−8

• DM relic density (ΩDM h2)

• DM direct detection constraints from LZ, PANDAX-4T (2024 and 2025), and Darkside

• LEP bounds

• Prompt decay bounds from ATLAS/CMS

• Planck 2018 constraints on ΔNeff

DM Relic Density: The relic density is calculated using coupled Boltzmann equations for two dark sectors (sector 1 containing χ, and sector 2 containing ψ, ψ±, χ′, ψ′, φ1,2). Three regions of parameter space are identified:

1. Region (i): Relic density decided by χχ ↔ νRνR process (ρ1 0.8)

2. Region (ii): Relic density decided by DM annihilation to SM particles via Higgs and gauge processes along with coannihilation (ρ1 > 0.5, ρ2 > 0.8)

3. Region (iii): Relic density depends on decay of heavier generation SD fermions (ρ2 < 0.8)

DM Direct Detection: The spin-independent DM-nucleon scattering cross-section receives contributions from Z-boson mediation (σSI Z ∝ sin4θ1), Higgs mediation (σSI h ∝ sin22θ1), and a loop-level contribution involving νR and φ1,2.

Inverse First-Order Electroweak Phase Transition: The authors study the thermal evolution of the effective potential, which includes contributions from SM particles and two generations of SD fermions. The Universe undergoes a distinctive thermal history:

• At high temperatures, electroweak symmetry is broken through a crossover

• As temperature decreases, the broken vacuum becomes metastable, and the Universe undergoes a first-order phase transition from broken to symmetric phase at Tc1

• Upon further cooling, the symmetric vacuum becomes metastable again, and the Universe undergoes a second first-order phase transition to the broken phase at Tc2

For benchmark point BP1, the critical temperatures are Tc1 = 214.6 GeV and Tc2 = 80.4 GeV, with nucleation temperatures Tn1 = 204.72 GeV and Tn2 = 45.93 GeV. The phase transition parameters (α, β/H) are given in Table 3 for three benchmark points.

Gravitational Waves: The GW spectra from both FOPTs are calculated. For the benchmark points, the GW signals from the inverse FOPT (dashed curves) and the subsequent FOPT (solid curves) are shown in Fig. 12, along with projected sensitivities of future GW observatories (μARES, LISA, DECIGO, UDECIGO, BBO, ET). The BBN constraint on stochastic GW background is also shown.

ΔNeff: The right-handed neutrinos contribute to the effective number of relativistic degrees of freedom. The contribution is given by ΔNeff = NνR × (TνR/TνL)4 = NνR × (gs(TνL dec)/gs(TνR dec))(4/3), where NνR = 2 (since λχ1e = λχ2e = 0). Larger λχ and lighter MDM correspond to larger ΔNeff. The parameter space is constrained by Planck 2018 (Neff = 2.99+0.34-0.33 at 95% CL).

Collider Signatures: The charged doublet fermion ψ1± can be produced via gauge interactions and decay into DM and charged leptons. The decay length depends on the mixing angle sin θ1. For small mixing, long-lived charged particles give displaced vertex signatures (probed by LHC DV and MATHUSLA); for larger mixing, prompt decays give multilepton final states (pp → 3l + ETmiss).

Final Allowed Parameter Space: The final allowed parameter space (red points in Fig. 16) satisfies all constraints. The cyan region (Mχ′ > 2215 GeV) does not allow IFOEWPT. Black points are ruled out by ΔNeff constraints, maroon points by prompt decay bounds, and gray shaded region by LEP data. Three benchmark points (BP1, BP2, BP3) are given in Table 2, with parameters: Mχ = 40.96, 294.84, 724.59 GeV; ΔM1 = 284.46, 61.70, 14.82 GeV; Mχ′ = 400, 600, 800 GeV; sin θ1 = 0.00407, 0.01264, 0.00972; Mψ′ = 1661, 1965, 2225 GeV; y2 = 2.763, 2.991, 3.123; Mφ = 189.10, 594.12, 783.01 GeV.

RGE Running: The singlet-doublet Yukawa coupling yχ exceeds the perturbative limit at μ 4.85 TeV and develops a Landau pole at μ 26 TeV for BP1, indicating the possibility of new UV completions or strong dynamics at a few TeV scale.

Based on this paper, I can improve AI systems in the following specific ways:

1. Multi-constraint parameter space optimizer: Build an AI system that automatically scans the full 10+ dimensional parameter space (masses, mixing angles, Yukawa couplings) while simultaneously satisfying all experimental constraints (neutrino oscillation data, DM relic density, direct detection limits, cLFV bounds, muon g-2, LEP/collider bounds, ΔNeff). The AI would use Bayesian optimization or reinforcement learning to efficiently find viable regions, rather than random scanning.

2. Phase transition thermal history predictor: Train a neural network to predict the critical temperatures (Tc1, Tc2), nucleation temperatures, and phase transition strength parameters (α, β/H) directly from model parameters, replacing expensive finite-temperature effective potential calculations. This would enable rapid exploration of the IFOEWPT parameter space.

3. Gravitational wave signal classifier: Develop a deep learning model that takes model parameters as input and predicts the resulting GW spectra (peak frequency, amplitude, shape) for both phase transitions, then automatically assesses detectability against LISA, DECIGO, BBO, ET, and μARES sensitivities.

4. Yukawa coupling structure solver: Create an AI that inverts the one-loop neutrino mass formula to find Yukawa coupling matrices (λψ, λχ) that reproduce measured neutrino masses and mixing angles, using the PMNS matrix and projection matrix parameterization, while respecting the λχ1e = λχ2e = 0 constraint.

5. DM relic density region classifier: Build a classifier that automatically identifies which of the three relic density regions (annihilation to νR, SM annihilation/coannihilation, or decay-dominated) a given parameter point falls into, based on the ρ1 and ρ2 parameters, enabling faster identification of viable DM scenarios.

6. Collider signature predictor: Train a model to predict whether a given parameter point yields long-lived charged particles (displaced vertices for MATHUSLA/LHC DV searches) or prompt decays (multilepton final states), based on the mixing angle sin θ1 and mass splittings, optimizing experimental search strategies.

7. UV completion advisor: Use the RGE running information (Landau pole at 26 TeV for BP1) to train a system that suggests possible UV completions (e.g., strong dynamics, extra gauge symmetries) when perturbativity is violated, aiding model-building.

8. Benchmark point generator: Create a generative model (e.g., VAE or normalizing flow) trained on the final allowed parameter space (red points in Fig. 16) to produce new benchmark points that satisfy all constraints, useful for phenomenology studies and experimental collaborations.

9. ΔNeff constraint optimizer: Build an AI that minimizes ΔNeff while maintaining correct DM relic density, by optimizing the trade-off between λχ couplings and DM mass, ensuring Planck 2018 compatibility.

10. Automated model validation pipeline: Develop an end-to-end AI system that takes any BSM model file as input, automatically derives the mass spectra, computes all relevant observables (neutrino masses, DM properties, phase transition parameters, GW signals, collider signatures), and outputs a comprehensive viability report against all current experimental bounds.

We study the possibility of an inverse first-order electroweak phase transition (IFOEWPT) and observable gravitational waves (GW) in a radiative neutrino mass model of scotogenic type where singlet-doublet (SD) fermions, the lightest of whom is the dark matter (DM) candidate, generate the necessary seesaw at one-loop level. Considering the possibility of light neutrinos being Dirac for simplicity and additional detection prospects, we extend the standard model (SM) with two generations of SU(2) L singlet and doublet fermions, one singlet scalar, and three right-handed neutrinos (RHNs). While RHNs provide the right chiral parts of light Dirac neutrinos, the SD fermions and the scalar singlet facilitate the one-loop neutrino mass diagram. The neutral component of the lighter SD fermion, stabilized under a residual Z 2 symmetry plays the role of DM while the heavier SD fermions strongly couple to the Higgs leading to an IFOEWPT where the Universe undergoes two different first-order phase transition as it goes from the symmetric to the final broken Higgs phase. We constrain the parameter space from the requirements of generating the correct neutrino mass, DM relic as well as IFOEWPT while incorporating the existing constraints from different experiments. The final allowed parameter space of the model can be probed at collider, direct-detection, GW and cosmic microwave background (CMB) experiments in near future.