Zee Model: Radiative Neutrino Mass

• Zee model is a radiative neutrino mass mechanism that extends the Standard Model with a second Higgs doublet and a charged scalar singlet to induce one-loop Majorana mass generation.
• Its framework features an antisymmetric Yukawa coupling and a trilinear scalar interaction that yield a characteristic neutrino mass matrix and predict specific flavor structures.
• The model has evolved into various extensions, including symmetry-controlled variants and the inert Zee model, which incorporates dark matter candidates and addresses flavor-changing constraints.

The Zee model is a radiative Majorana neutrino mass model in which the Standard Model is extended by a second Higgs doublet and a charged scalar singlet, so that neutrino masses arise at one loop without introducing right-handed neutrinos. Its defining ingredients are an antisymmetric coupling of the charged singlet to lepton doublets and a lepton-number-violating trilinear scalar interaction, and it remains the canonical starting point for a large class of Zee-type constructions linking neutrino mass generation to flavor physics, extended scalar sectors, and, in later variants, dark matter (Nomura et al., 2016).

1. Historical origin and defining field content

The Zee model was proposed as an alternative to tree-level mass generation for neutrinos in a framework where Standard Model neutrinos are massless at tree level. Its central idea is that small neutrino masses can be generated radiatively, with the suppression controlled by loop factors such as $1/(16\pi^2)$ and, depending on the realization, by small couplings as well. In its standard electroweak form, the model adds a second Higgs doublet and a charged scalar singlet to the Standard Model scalar sector, while retaining only the Standard Model fermion content (Gaviria et al., 2020).

In a common convention, the new fields are a second Higgs doublet $\Phi_2 \sim (1,2,\tfrac12)$ and a charged scalar singlet $h^+ \sim (1,1,1)$. The key interactions are an antisymmetric Yukawa coupling of the charged singlet to two lepton doublets,

$$f_{ij}\,L_{Li}^T C i\sigma_2 L_{Lj}\, h^+, \qquad f_{ij}=-f_{ji},$$

together with Yukawa couplings of the two Higgs doublets to charged leptons and a trilinear scalar term,

$$\mu\,\Phi_1^T i\sigma_2 \Phi_2\, h^- + \text{h.c.},$$

which mixes the charged scalar states and violates lepton number (Nomura et al., 2016).

These ingredients make the Zee model one of the earliest one-loop Majorana mass models. A recurring source of confusion is the role of hypercharge conventions: equivalent presentations in the literature may assign $Y=\tfrac12$ to the Higgs doublets or use a doubled normalization. This suggests that the invariant content of the model is best identified by its interaction structure rather than by any one normalization convention.

2. One-loop neutrino mass mechanism and flavor structure

The one-loop diagram contains external neutrino legs, an internal charged lepton, and charged scalar states obtained from mixing the doublet and singlet charged scalars. In the simplest formulations, the resulting Majorana mass matrix is controlled by the antisymmetric singlet Yukawa matrix and charged-lepton mass insertions. A modern compact expression for the loop-induced neutrino mass matrix in the charged-lepton mass basis is

$$m^\nu = \kappa \left( F m_\ell Y^\ell + Y^{\ell T} m_\ell F^T \right),$$

where $m_\ell=\mathrm{diag}(m_e,m_\mu,m_\tau)$, $F^T=-F$, $Y^\ell$ denotes the Yukawa couplings of the second Higgs doublet, and

$\Phi_2 \sim (1,2,\tfrac12)$0

encodes the charged-scalar mixing and loop suppression (Chen, 26 Aug 2025).

Because $\Phi_2 \sim (1,2,\tfrac12)$1 is skew-symmetric, it contains only three independent complex entries. This antisymmetry sharply constrains the flavor structure of $\Phi_2 \sim (1,2,\tfrac12)$2. Recent structural work has made this constraint explicit by constructing a pseudovector $\Phi_2 \sim (1,2,\tfrac12)$3 from $\Phi_2 \sim (1,2,\tfrac12)$4 such that $\Phi_2 \sim (1,2,\tfrac12)$5, implying the identity

$\Phi_2 \sim (1,2,\tfrac12)$6

which is independent of $\Phi_2 \sim (1,2,\tfrac12)$7. A concrete consequence is that, for fixed $\Phi_2 \sim (1,2,\tfrac12)$8 and $\Phi_2 \sim (1,2,\tfrac12)$9, only five entries of $h^+ \sim (1,1,1)$0 can be determined from neutrino data, while four remain undetermined by neutrino masses alone (Chen, 26 Aug 2025).

In the simplest Zee limit where effectively only one Higgs doublet contributes to charged-lepton Yukawas, the neutrino mass matrix reduces to a form proportional to $h^+ \sim (1,1,1)$1. That limit produces the characteristic Zee texture with vanishing diagonal entries at leading order, a property that shaped much of the model’s early phenomenology (Nomura et al., 2019).

3. Minimal realizations, oscillation data, and the FCNC problem

The original attraction of the Zee model was its economy, but that economy also produced its main phenomenological tension. When the model is restricted so that only one Higgs doublet couples to leptons, the resulting neutrino mass matrix is too constrained to reproduce current oscillation data. This simplest version was shown to be excluded, whereas the general Zee model remains consistent with data once the Yukawa structure is enlarged beyond the minimal texture (He et al., 2011).

A central obstruction is the interplay between neutrino mixing and scalar-mediated flavor physics. If both Higgs doublets couple generically to the same fermion species, neutral scalar exchange produces tree-level Higgs-mediated flavor changing neutral currents. These FCNCs are strongly constrained, so the unconstrained two-Higgs-doublet realization is not phenomenologically acceptable without additional symmetry, alignment, or texture assumptions (Gaviria et al., 2020).

One constrained Zee model based on naturalness considerations illustrates the residual predictive power of the framework. In that analysis, only inverted hierarchy was allowed, $h^+ \sim (1,1,1)$2 had to be non-zero, and the best-fit value was $h^+ \sim (1,1,1)$3. The same study found non-zero CP-violating solutions, with the Jarlskog parameter in the ranges $h^+ \sim (1,1,1)$4, $h^+ \sim (1,1,1)$5, and $h^+ \sim (1,1,1)$6 at $h^+ \sim (1,1,1)$7, $h^+ \sim (1,1,1)$8, and $h^+ \sim (1,1,1)$9, respectively (He et al., 2011). These results should not be read as universal predictions of all Zee realizations; they characterize one specific constrained setup.

A common misconception is therefore that “the Zee model is ruled out.” More precisely, the simplest Zee texture is ruled out, while generalized Zee constructions with additional Yukawa structure remain viable.

4. Symmetry-controlled and flavor-predictive versions

One major line of development replaces ad hoc Yukawa choices by flavor symmetries. A representative example keeps the original Zee particle content but replaces the traditional discrete $$f_{ij}\,L_{Li}^T C i\sigma_2 L_{Lj}\, h^+, \qquad f_{ij}=-f_{ji},$$0 by a flavor-dependent global $$f_{ij}\,L_{Li}^T C i\sigma_2 L_{Lj}\, h^+, \qquad f_{ij}=-f_{ji},$$1. In the successful Class I charge assignment, the quark sector remains effectively Type-I-like, tree-level quark FCNC are absent, and the lepton sector acquires a controlled additional source of flavor violation. In that setup, current neutrino oscillation data can be fitted, while the model predicts characteristic patterns of lepton-flavor-violating decays of the additional Higgs bosons, including correlated $$f_{ij}\,L_{Li}^T C i\sigma_2 L_{Lj}\, h^+, \qquad f_{ij}=-f_{ji},$$2, $$f_{ij}\,L_{Li}^T C i\sigma_2 L_{Lj}\, h^+, \qquad f_{ij}=-f_{ji},$$3, and $$f_{ij}\,L_{Li}^T C i\sigma_2 L_{Lj}\, h^+, \qquad f_{ij}=-f_{ji},$$4 modes (Nomura et al., 2019).

Another major line uses modular flavor symmetry. In a modular $$f_{ij}\,L_{Li}^T C i\sigma_2 L_{Lj}\, h^+, \qquad f_{ij}=-f_{ji},$$5 realization with minimal modular weights, the Zee model fits both charged-lepton masses and neutrino oscillation data while strongly restricting the modulus $$f_{ij}\,L_{Li}^T C i\sigma_2 L_{Lj}\, h^+, \qquad f_{ij}=-f_{ji},$$6. At a benchmark point for normal hierarchy, the model gives $$f_{ij}\,L_{Li}^T C i\sigma_2 L_{Lj}\, h^+, \qquad f_{ij}=-f_{ji},$$7, $$f_{ij}\,L_{Li}^T C i\sigma_2 L_{Lj}\, h^+, \qquad f_{ij}=-f_{ji},$$8 meV, and $$f_{ij}\,L_{Li}^T C i\sigma_2 L_{Lj}\, h^+, \qquad f_{ij}=-f_{ji},$$9 meV, while inverted hierarchy is strongly constrained and effectively disfavored in the minimal-weight setup (Nomura et al., 2021). A non-holomorphic modular $$\mu\,\Phi_1^T i\sigma_2 \Phi_2\, h^- + \text{h.c.},$$0 version is even more economical in the neutrino sector, with two complex free parameters including $$\mu\,\Phi_1^T i\sigma_2 \Phi_2\, h^- + \text{h.c.},$$1, and allows both normal and inverted hierarchy; it predicts $$\mu\,\Phi_1^T i\sigma_2 \Phi_2\, h^- + \text{h.c.},$$2 meV for normal ordering and $$\mu\,\Phi_1^T i\sigma_2 \Phi_2\, h^- + \text{h.c.},$$3 meV for inverted ordering (Nomura et al., 2024).

These symmetry-based constructions indicate that the long-term significance of the Zee model lies not only in its original loop topology, but also in its role as a template for constrained flavor engineering.

5. The inert Zee model and the dark-matter extension of the Zee topology

A particularly influential modernization is the inert Zee model, which imposes an exact $$\mu\,\Phi_1^T i\sigma_2 \Phi_2\, h^- + \text{h.c.},$$4 symmetry under which all new fields are odd and all Standard Model fields are even. In addition to the inert scalar doublet $$\mu\,\Phi_1^T i\sigma_2 \Phi_2\, h^- + \text{h.c.},$$5 and the charged scalar singlet $$\mu\,\Phi_1^T i\sigma_2 \Phi_2\, h^- + \text{h.c.},$$6, it introduces a vector-like charged singlet $$\mu\,\Phi_1^T i\sigma_2 \Phi_2\, h^- + \text{h.c.},$$7 and a vector-like doublet $$\mu\,\Phi_1^T i\sigma_2 \Phi_2\, h^- + \text{h.c.},$$8. The exact $$\mu\,\Phi_1^T i\sigma_2 \Phi_2\, h^- + \text{h.c.},$$9 ensures one-loop neutrino masses, forbids tree-level Higgs-mediated flavor changing neutral currents, and stabilizes the lightest $Y=\tfrac12$0-odd state, thereby supplying a dark matter candidate (Longas et al., 2015).

After electroweak symmetry breaking, the neutral scalar sector contains $Y=\tfrac12$1 and $Y=\tfrac12$2, while the charged singlet and doublet mix into physical charged scalars $Y=\tfrac12$3, and the charged vector-like fermions mix into $Y=\tfrac12$4. The neutrino mass matrix retains the Zee-like bilinear flavor structure,

$Y=\tfrac12$5

so it is rank 2 and predicts one massless neutrino. In the dark sector, $Y=\tfrac12$6 is usually taken as the dark matter candidate. The phenomenology is close to the inert doublet model, with a low-mass regime $Y=\tfrac12$7 and a high-mass regime $Y=\tfrac12$8 (Gaviria et al., 2020).

The distinctive new feature relative to the inert doublet model is the lepton portal. Because of the new vector-like charged fermions, the process

$Y=\tfrac12$9

is opened by $$m^\nu = \kappa \left( F m_\ell Y^\ell + Y^{\ell T} m_\ell F^T \right),$$0-channel exchange. This channel is especially important when the Higgs-portal coupling $$m^\nu = \kappa \left( F m_\ell Y^\ell + Y^{\ell T} m_\ell F^T \right),$$1 is small. In that regime, annihilation into leptons, mainly $$m^\nu = \kappa \left( F m_\ell Y^\ell + Y^{\ell T} m_\ell F^T \right),$$2, can generate the correct relic density even for $$m^\nu = \kappa \left( F m_\ell Y^\ell + Y^{\ell T} m_\ell F^T \right),$$3, and the region $$m^\nu = \kappa \left( F m_\ell Y^\ell + Y^{\ell T} m_\ell F^T \right),$$4 is recovered for $$m^\nu = \kappa \left( F m_\ell Y^\ell + Y^{\ell T} m_\ell F^T \right),$$5 (Gaviria et al., 2020).

Lepton-flavor observables remain crucial. In the inert Zee model, $$m^\nu = \kappa \left( F m_\ell Y^\ell + Y^{\ell T} m_\ell F^T \right),$$6, $$m^\nu = \kappa \left( F m_\ell Y^\ell + Y^{\ell T} m_\ell F^T \right),$$7, $$m^\nu = \kappa \left( F m_\ell Y^\ell + Y^{\ell T} m_\ell F^T \right),$$8-$$m^\nu = \kappa \left( F m_\ell Y^\ell + Y^{\ell T} m_\ell F^T \right),$$9 conversion, and electric dipole moments are generated at one loop by the same Yukawa couplings that control neutrino masses. A dedicated analysis found that $m_\ell=\mathrm{diag}(m_e,m_\mu,m_\tau)$0-$m_\ell=\mathrm{diag}(m_e,m_\mu,m_\tau)$1 conversion is the most constraining and promising observable for testing the model, while future electron EDM measurements can probe CP-violating parameter regions beyond the reach of charged-lepton-flavor-violation searches (Longas, 2018).

6. Zee-type descendants, broader legacy, and current perspective

The Zee model has also generated a family of descendants in which the basic radiative idea is preserved but the loop order, gauge quantum numbers, or ultraviolet setting are modified. The following summary captures representative directions.

Construction	Added structure	Representative feature
Zee–Babu model	$m_\ell=\mathrm{diag}(m_e,m_\mu,m_\tau)$2, $m_\ell=\mathrm{diag}(m_e,m_\mu,m_\tau)$3, and $m_\ell=\mathrm{diag}(m_e,m_\mu,m_\tau)$4	two-loop mass matrix $m_\ell=\mathrm{diag}(m_e,m_\mu,m_\tau)$5 (Nomura et al., 2016)
Colored Zee–Babu model	leptoquark plus diquark	short-range new-physics contribution can cancel light-neutrino exchange, yielding “hidden” $m_\ell=\mathrm{diag}(m_e,m_\mu,m_\tau)$6 (Chen et al., 2022)
5D split-fermion Zee model	$m_\ell=\mathrm{diag}(m_e,m_\mu,m_\tau)$7 with localized fermions	effective 4D Yukawas arise from wavefunction overlap; explicit configurations fit lepton data and predict nonzero $m_\ell=\mathrm{diag}(m_e,m_\mu,m_\tau)$8 (Chang et al., 2010)
Minimal 3-3-1 realization	no extra fields beyond m331 multiplets	modified Zee and Zee–Babu mechanisms are automatically implemented (Machado et al., 2018)

The Zee–Babu descendant is conceptually the closest relative. It promotes the one-loop Zee idea to a two-loop mechanism by adding a doubly charged scalar and a cubic $m_\ell=\mathrm{diag}(m_e,m_\mu,m_\tau)$9 term, thereby replacing the one-loop charged-scalar–charged-lepton diagram by a two-loop structure with stronger suppression and richer charged-scalar phenomenology (Vien et al., 2014). Colored variants further connect the radiative neutrino mechanism to leptoquark and diquark sectors, while extra-dimensional and gauge-extended realizations show that the Zee topology can emerge from broader model frameworks without being inserted by hand (Chen et al., 2022).

In current usage, it is often useful to distinguish the strict “Zee model” from the wider class of “Zee-type” mechanisms. The former denotes the original one-loop two-Higgs-doublet-plus-charged-singlet construction. The latter includes inert, modular, colored, higher-dimensional, and gauge-extended descendants. A plausible implication is that the enduring value of the Zee model lies less in any one minimal texture than in the loop topology it introduced: a radiative, symmetry-sensitive origin of Majorana neutrino mass that continues to organize model building across flavor physics, dark matter, collider phenomenology, and neutrinoless double beta decay.