Vector-Like Lepton Doublets Overview

• Vector-like lepton doublets are SU(2)ₗ fermionic fields with hypercharge -1/2, featuring identical left- and right-handed gauge couplings that allow large Dirac masses independent of electroweak symmetry breaking.
• They play a key role in neutrino mass generation through seesaw and loop mechanisms, while also providing stable dark matter candidates via imposed discrete symmetries.
• Their collider phenomenology is characterized by Drell–Yan pair production and multi-lepton plus missing energy signatures, with experimental constraints shaping viable mass and mixing parameters.

A vector-like lepton doublet is a fermionic field that transforms as an SU(2)ₗ doublet with hypercharge Y=–½ (such that its components are a neutral and a singly-charged lepton), and whose left- and right-handed components have identical SU(2)ₗ×U(1)ᵧ gauge quantum numbers. Vector-like lepton doublets are distinguished from chiral leptons of the Standard Model (SM) by the vector-like nature of their gauge couplings, allowing the possibility of large Dirac mass terms without relying on electroweak symmetry breaking. Implementation and phenomenology depend on associated symmetries (e.g., discrete Z₂), additional scalar or fermion representations, and model-building objectives (such as explaining dark matter, neutrino masses, or baryogenesis).

1. Gauge Quantum Numbers and Mass Structure

A canonical vector-like lepton doublet is defined as $L = (N, E)^T$ with gauge transformations under SU(2)ₗ×U(1)ᵧ:

• $L_{L,R} \sim (1,2,-1/2)$
  • $N$: neutral (Q=0)
  • $E$: charged (Q=–1)

A gauge-invariant Dirac mass term is always allowed: $\mathcal{L} \supset -M \bar{L}_L L_R + \text{h.c.}$ as both chiralities transform identically. This is the minimal vector-like pairing. Under a discrete symmetry (e.g., exact Z₂), the doublet can be rendered stable or long-lived, dependent on the model's purpose (e.g., dark matter requires stability (Arina et al., 2012, Bhattacharya et al., 2018)).

2. Key Lagrangian Terms and Interactions

Masses and Yukawa Structure:

By construction, vector-like doublets can have vector-like Dirac masses independent of electroweak symmetry breaking. If direct Yukawa couplings to the Higgs doublet and SM leptons are present, they induce mixing between the vector-like and SM leptons after symmetry breaking. Such terms break chiral symmetry but conserve gauge invariance: $$\mathcal{L} \supset -Y_E \overline{L}_L \phi e_R - y'_e \overline{\ell}_L \phi E_R + \text{h.c.}$$ Here, ϕ denotes the SM Higgs, ℓ_L is the SM lepton doublet, e_R the SM charged-lepton singlet, and E_R the vector-like singlet component.

Majorana Masses:

Additional sources of mass can emerge via the coupling to scalar triplets (Δ) with Y=+2. If a type-II seesaw mechanism is invoked, the neutral component N can acquire a small Majorana mass: $$\mathcal{L} \supset -\tfrac{1}{2} m (\overline{N_{L}^c} N_L + \overline{N_{R}^c} N_R)$$ For example, with an induced Δ vev, $m=\sqrt{2} f_4 \langle\Delta\rangle$, leading to a pseudo-Dirac pair with splitting δ = 2m (Arina et al., 2012).

Additional Interactions:

Vector-like lepton doublets can have couplings to extended scalar sectors (singlets, inert doublets, triplets, or leptoquarks), and to new gauge fields (e.g., Z′), resulting in rich flavor, dark matter, and collider phenomenology (Loualidi et al., 20 Nov 2025, Dehghani et al., 18 Mar 2024, Bigaran et al., 2023, Chakrabarty, 2020).

3. Role in Neutrino Mass, Dark Matter, and Baryogenesis

Neutrino Mass via Seesaw/Loops:

Vector-like doublets participate in neutrino mass generation in several frameworks:

• Type-II seesaw: SM neutrino Majorana masses via triplet vev ($$(M_\nu)_{\alpha\beta} = \sqrt2 (f_L)_{\alpha\beta} \langle\Delta\rangle$$) (Arina et al., 2012).
• Inverse/linear seesaw: In models with additional singlets/triplets, vector-like doublets enable low-scale seesaw realizations leading to small neutrino masses, with the mass matrix structure:

$$\mathcal{M} = \begin{pmatrix} 0 & m_D & 0 \ m_D^T & 0 & M_\psi \ 0 & M_\psi^T & \mu \end{pmatrix}$$

and $$m_\nu \sim m_D (M_\psi^{-1}) \mu (M_\psi^{T-1}) m_D^T$$ (Gu, 2020).

• Radiative mechanisms: Three-loop neutrino masses with asymmetric Yukawa connections between vector-like doublets and inert scalars (Loualidi et al., 20 Nov 2025).

Dark Matter Phenomenology:

Models with an exact Z₂ symmetry can stabilize the lightest neutral state, making it a viable dark matter (DM) candidate. In the presence of a scalar triplet vev, a small (∼100 keV) Majorana splitting between the neutral components yields inelastic DM with suppressed elastic Z-exchange, satisfying stringent direct-detection limits (Arina et al., 2012, Bhattacharya et al., 2018).

• For relic density and direct-detection compatibility, parameters require co-annihilation (small mass splitting Δm between E and N), small mixing ($\sin\theta \lesssim 0.05$), or loop-induced splitting ($\delta m > 100\,\text{keV}$) (Bhattacharya et al., 2018).
• Extended Higgs sectors enable viable inert scalar dark matter, with VLLs affecting annihilation and coannihilation rates, but in some scenarios their influence is negligible if they are sufficiently heavy (Loualidi et al., 20 Nov 2025).

Baryogenesis via Leptogenesis:

If the scalar triplet involved in seesaw generation decays out-of-equilibrium with CP violation, both leptonic and vector-like dark sector asymmetries can be created simultaneously. The baryon asymmetry and dark matter abundance can thus be linked via ratios set by model parameters, e.g.,

$$\frac{\Omega_{DM}}{\Omega_B} = \frac{1}{0.55} \frac{m_{N_4}}{m_p} \frac{\epsilon_{L_4}}{\epsilon_L} \frac{\eta_{L_4}}{\eta_L}$$

for appropriate decay asymmetries and washout factors (Arina et al., 2012).

4. Collider Phenomenology and Experimental Constraints

Production Mechanisms:

Vector-like doublets are dominantly produced in pairs via Drell–Yan processes at hadron colliders: $$q\bar{q} \to Z^*,\gamma^* \to E^+E^-, \quad q\bar{q}' \to W^* \to N E$$ Lepton colliders (ILC, FCC-ee) offer clean environments for threshold scans of pair production, with cross sections sensitive to electroweak charges and mixing angles (Bahrami et al., 2016, Chakraborty et al., 2022, Dinh et al., 28 Aug 2025).

Decay Modes and Multilepton Final States:

Decays are governed by mixing with SM leptons and interactions with additional scalars/gauge bosons:

• If forbidding mixing (Z₂), E→N W^–; N stable (missing energy) (Arina et al., 2012, Bhattacharya et al., 2018).
• With small mixing, branching fractions are typically BR(E→Wν)≈0.5, BR(E→Ze)≈BR(E→He)≈0.25 (Baspehlivan et al., 2022).
• Three-body decays via off-shell intermediates (e.g., with leptoquark mediation or loop splitting) are possible in symmetry-protected or lepton-number–assigned cases, producing τ + jets or multi-b final states (Bigaran et al., 2023).
• Final states such as 2ℓ + MET, 4ℓ + MET, 1ℓ + 2j + MET, and 4j + MET are key probes at both LHC and ILC, with cut-based and multivariate discrimination (Chakraborty et al., 2022).

Experimental Limits and Projections:

• LEP: m_E > 101.9 GeV, m_N > 45 GeV (invisible Z width constraint) (Arina et al., 2012).
• LHC: 2ℓ + MET searches constrain VLL doublets to m_E ≳ 800–1050 GeV with flavor-mixing decays. However, in scenarios with no SM mixing or alternative decay topologies (cascade via leptoquarks, Z₂-odd state), these bounds can be relaxed substantially (Bigaran et al., 2023, Bißmann et al., 2020).
• HL-LHC: Discovery reach extends up to 1.5 TeV for iso-doublet VLLs in multi-lepton channels (Baspehlivan et al., 2022). Future colliders (FCC-hh, CLIC, Muon Collider) can probe up to 6 TeV (Baspehlivan et al., 2022).
• ILC/FCC-ee: Clean lepton colliders can detect VLLs up to the kinematic limit √s/2, with per-mill precision on e⁺e⁻ → ℓ⁺ℓ⁻ cross sections offering powerful indirect constraints (Chakraborty et al., 2022, Dinh et al., 28 Aug 2025).

5. Flavor, Precision, and Cosmological Constraints

Lepton Flavor Violation and Precision Electroweak:

Mixing with SM leptons induces lepton flavor violation (LFV) and changes Z and W couplings, bounded by:

• Z→ℓ^+ℓ^- width measurements (e.g., Y_{ij}^e contributions in SU(5) embedding) (Dorsner et al., 2014).
• CLFV processes: μ→eγ, τ→μγ, μ→e conversion, τ→3μ, Z→μτ (Hernández et al., 2021, Lee et al., 2021).
• Electroweak oblique parameters S and T constrain allowed mass splittings and mixing angles, with scalar extensions providing mechanisms for cancellation or relaxation of the T parameter (Garg et al., 2013).

Dark Matter Direct/Indirect Detection:

• Z-mediated elastic scattering of N is strongly constrained by XENON1T, requiring suppressed mixing or splitting-induced (δ > 100 keV) inelasticity (Bhattacharya et al., 2018).
• Co-annihilation and Majorana-splitting–induced inelastic dark matter allow survival against direct exclusion. Scalar dark matter scenarios, especially with heavy VLLs, likewise satisfy current direct-detection and Planck abundance bounds (Loualidi et al., 20 Nov 2025).

Cosmological Implications:

Vector-like doublets can participate in the generation of the baryon asymmetry via decays of associated heavy states (scalar triplets, singlets, or triplet fermions), providing simultaneous solutions to dark matter and baryogenesis requirements at high scales (Arina et al., 2012, Gu, 2020).

6. Model Building Contexts and Theoretical Motivations

Theoretical Motivations:

• Flavor Democracy motivates the inclusion of vector-like doublets as a means to lift degeneracies and produce realistic lepton mass spectra (Baspehlivan et al., 2022).
• GUTs (E₆, SU(5), SO(10)) predict light vector-like doublets, often embedded in fundamental or antisymmetric representations (Dorsner et al., 2014, Baspehlivan et al., 2022).
• Asymptotically safe UV completions and Higgs vacuum stability scenarios motivate the inclusion of VLLs with extended scalar sectors (Bißmann et al., 2020, Chakrabarty, 2020).

Extensions and Variants:

• Multiple generations, S₃ or flavor symmetries, or lepton number assignments (vector-like “antileptons”) give rise to exotic phenomenology, including novel decay chains and almost background-free multilepton signatures (Chakraborty et al., 2022, Bigaran et al., 2023).
• Couplings to additional gauge bosons (e.g., Z′), extended Higgs or scalar sectors, or leptoquarks further enrich the phenomenological profile.

Parameter Space Summary Table

Observable	Typical Bounds / Features	Reference
m_E (LEP limit)	m_{E} > 101.9 GeV	(Arina et al., 2012)
m_N (LEP limit)	m_{N} > 45 GeV	(Arina et al., 2012)
LHC multilepton	m_{E} ≳ 800–1050 GeV	(Bißmann et al., 2020)
Direct detection	Suppressed if δ > 100 keV	(Bhattacharya et al., 2018)
LFV (e.g. μ→eγ)	Mixing angles ≪ 1	(Hernández et al., 2021)
HL-LHC reach	m_{VLL} ≲ 1.5 TeV	(Baspehlivan et al., 2022)
FCC-hh reach	m_{VLL} ≲ 6 TeV	(Baspehlivan et al., 2022)

7. Outlook and Open Directions

Vector-like lepton doublets remain among the most theoretically motivated and phenomenologically rich extensions of the SM. They provide simultaneous portals to explain dark matter, baryogenesis, and neutrino masses, and serve as sensitive probes of flavor and precision physics. The minimal setup is highly constrained by direct and indirect searches, but extended models featuring new scalar, fermionic, or gauge sectors introduce novel decay channels and signatures—particularly in multi-lepton final states and inelastic dark matter paradigms.

Future lepton colliders (ILC, FCC-ee) and high-luminosity hadron machines (HL-LHC, FCC-hh) are projected to continue probing the viable parameter space well into the multi-TeV regime, with dedicated strategies required for scenarios with suppressed or nonstandard decay chains (Dinh et al., 28 Aug 2025, Chakraborty et al., 2022, Bigaran et al., 2023). In UV model-building, vector-like lepton doublets maintain a central role within GUTs and flavor frameworks, with ongoing work aiming to integrate their phenomenological requirements with gauge coupling unification and vacuum stability constraints (Dorsner et al., 2014, Chakrabarty, 2020, Garg et al., 2013).