Radiative Neutrino Mass Models

• Radiative neutrino mass models are theoretical frameworks where neutrino masses arise from quantum loop processes involving new scalar or fermionic fields, naturally suppressing mass scales.
• They employ discrete symmetries, such as Z2, to forbid tree-level contributions and stabilize dark matter candidates while allowing for lepton flavor violating signals.
• Key experimental implications include TeV-scale particles accessible at colliders, rare decay observations, and testable predictions in dark matter and neutrinoless double-beta decay searches.

Radiative neutrino mass models constitute a broad class of theoretical frameworks in which the smallness of neutrino masses is attributed to their origin in quantum loop processes, rather than tree-level interactions. These models provide natural explanations for neutrino mass suppression, often correlate the mechanism of neutrino mass generation with the existence and stability of dark matter, and accommodate potential connections to the baryon asymmetry of the Universe. Implementation typically requires the extension of the Standard Model (SM) with additional scalar and/or fermionic fields, precise choices of discrete or continuous symmetries, and controlled breaking of lepton number.

1. Theoretical Foundations of Radiative Neutrino Mass Generation

Radiative neutrino mass models exploit the suppression of mass terms by loop factors, with the canonical seesaw mechanism replaced by higher-order diagrams in which new particles propagate internally. The dimension-5 Weinberg operator, $\mathcal{O}_5 \sim \overline{L^c} L H H/\Lambda$, encapsulates the leading contribution to Majorana neutrino masses in the SM effective field theory, with $\Lambda$ being the scale of new physics. In radiative models, this operator or its higher-dimensional analogues (e.g., $\mathcal{O}_7 = LLHH(H^\dagger H)$ for $d=7$) are realized only at one loop or higher, yielding masses of the form

$$m_\nu \sim \frac{1}{(16\pi^2)^n} \frac{v^2}{M} \cdot (\text{Yukawa and coupling factors}),$$

where $n$ is the loop order, $M$ the mass scale of the mediators, and $v$ the electroweak vacuum expectation value (Cai et al., 2017, Sugiyama, 2015, Cepedello, 2021). The range of viable radiative mechanisms includes one-loop (e.g., Zee, scotogenic), two-loop (Zee–Babu, many "genuine" Weinberg completions), and even three-loop (Krauss-Nasri-Trodden, "cocktail") topologies (Cai et al., 2017).

2. Symmetries and Naturalness

An essential aspect of radiative neutrino mass models is the imposition of discrete (often $Z_2$) or continuous symmetries to forbid tree-level contributions to neutrino masses and sometimes to stabilize potential dark matter candidates (Sugiyama, 2015, Ahriche et al., 2014). The $Z_2$ symmetry ensures that the new states responsible for neutrino mass appear only inside the loops and are stable if they are odd under this symmetry—providing a candidate for dark matter. The possible origin of such stabilizing symmetries includes remnant subgroups from broken gauge symmetries (e.g., $U(1)_x\to Z_2$ through the vacuum expectation value of a scalar field) (Kashiwase et al., 2015), accidental symmetries in the scalar potential (Ahriche et al., 2014), or non-invertible selection rules arising from higher fusion algebras (Kobayashi et al., 20 May 2025).

Crucially, loop suppression allows the new-physics scale $M$ to be substantially lower (TeV scale or below) than in seesaw-type models, making the associated new states potentially accessible to collider searches and indirect probes (Cai et al., 2017).

3. Representative Model Structures

3.1 One-Loop Models

Zee Model—Extension of the SM with a singly-charged scalar $h^+$ and an additional Higgs doublet, generating neutrino masses via one-loop diagrams tied to antisymmetric lepton-number–violating couplings. Various texture zeros and flavor predictions follow from the imposed (possibly family-dependent) discrete symmetries (Sugiyama, 2015, Babu et al., 2013).

Scotogenic ("Ma") Model—Addition of an inert scalar doublet and right-handed neutrinos, all odd under $Z_2$, generating neutrino masses at one loop. Dark matter is realized as the lightest $Z_2$-odd particle, and a small quartic coupling $\lambda_5$ controls the radiative mass term (Kashiwase et al., 2015, Sugiyama, 2015).

3.2 Two-Loop and Multi-Loop Models

Zee–Babu Model and Beyond—Utilizing singly- and doubly-charged scalars to produce neutrino masses at two loops, often with characteristic Yukawa textures and testable LFV predictions (Sugiyama, 2015, Law et al., 2013). Systematic classifications enumerate all topologies for two- and three-loop completions of the Weinberg operator, stressing the role of discrete symmetries in forbidding lower-order contributions (Simoes et al., 2017, Cepedello, 2021).

Extended Topologies—Models may be constructed in which new fields are only singlets or doublets under $SU(2)_L$, maintaining genuine two-loop or higher suppression without tree-level or one-loop diagrams (Simoes et al., 2017).

3.3 Supersymmetric Realizations and Non-Invertible Selection Rules

Supersymmetric frameworks introduce chiral superfields and exploit flat directions in the scalar potential. In Affleck–Dine leptogenesis within radiative schemes, decoupling between the operator responsible for lepton asymmetry ($LH_u$) and the neutrino mass operator allows sufficient baryogenesis at low reheating temperature (Higashi et al., 2011). Non-invertible selection rules, derived from fusion algebras, can be used to systematically forbid tree-level mass operators, with the remnant symmetry ensuring dark matter stability and enabling structured mass textures through family-dependent assignments (Kobayashi et al., 20 May 2025).

4. Phenomenological Implications

Radiative neutrino mass models link the origin of neutrino mass to additional phenomenology:

• Lepton Flavor Violation (LFV): The same set of Yukawa couplings relevant for mass generation often induces rare decays such as $\mu\rightarrow e\gamma$, $\tau \rightarrow 3\mu$, or $\mu-e$ conversion in nuclei. The precise branching ratios are correlated with neutrino mixing angles and are strongly constrained (or on the brink of discovery) by current and future experiments (Sugiyama, 2015, Cai et al., 2017, Babu et al., 2013). LFV constraints impact parameter choices, such as the allowed mass splittings in extended scalar sectors.
• Neutrinoless Double Beta Decay ($0\nu\beta\beta$): Majorana radiative mechanisms generate LNV at the $\Delta L=2$ level, leading to $0\nu\beta\beta$ decay mediated either by the standard mass mechanism or new contributions (e.g., scalar exchange), with effective mass parameter predictions in some cases being sharply constrained by mass textures (Babu et al., 2013, Cai et al., 2017, Cepedello, 2021).
• Dark Matter: In most models possessing a stabilizing symmetry (e.g., $Z_2$), the lightest new particle—be it a scalar or Majorana fermion—serves as a dark matter candidate. The coupling parameters governing neutrino mass generation and dark matter stability are often intertwined, and relic density calculations require both annihilation and semi-annihilation channels to be considered (Simoes et al., 2017, Ahriche et al., 2014, Arhrib et al., 2015, Kobayashi et al., 20 May 2025). Some constructions permit dark matter masses from a few MeV up to several TeV, with direct detection cross sections typified by suppressed rates but within reach of future sensitivity (Ahriche et al., 2014).
• Baryogenesis: Models with low-mass right-handed neutrinos (order 1 TeV) inhibit standard thermal leptogenesis. Mechanisms such as Affleck–Dine leptogenesis utilize flat directions orthogonal to those responsible for neutrino mass, thereby generating adequate lepton asymmetry at lowered reheating temperatures and solving the gravitino problem in supersymmetric contexts (Higashi et al., 2011, Kashiwase et al., 2015).

5. Model Constraints and Predictive Power

Parameter space in radiative neutrino mass models is tightly constrained by the need to:

• Reproduce the observed values of $\Delta m^2$ and mixing angles in the neutrino sector (Hehn et al., 2012, Babu et al., 2013).
• Satisfy LFV limits (e.g., Br($\mu \to e\gamma$) $< 5.7 \times 10^{-13}$) (Jin et al., 2015, Nomura et al., 2016).
• Achieve sufficient dark matter relic abundance and comply with direct and indirect detection limits (Ahriche et al., 2014, Arhrib et al., 2015).
• Avoid tree-level lower-dimensional contributions; achieved by careful symmetry assignments, non-invertible fusion algebra labels, or precise mass and coupling choices to forbid "compressible" subdiagrams (Cepedello, 2021, Kobayashi et al., 20 May 2025).
• Predict or accommodate flavor structures, texture zeros, and mixings consistent with experimental data, sometimes requiring only a minimal set of free parameters owing to the imposed symmetries (Babu et al., 2013, Kobayashi et al., 20 May 2025).

6. Connections to Grand Unification, Flavor, and Topological Classification

Radiative models can often be embedded into grand unified theories (such as $SU(5)$), providing a joint explanation for neutrino mass, unification of gauge couplings, and proton decay constraints (Klein et al., 2019). The flavor structure (including hierarchy and texture zeros) can be "engineered" through family-dependent assignments of fusion algebra elements, symmetries, or field-representation choices (Kobayashi et al., 20 May 2025). The systematic topological classification—enumerating all irreducible diagram structures that give genuine loop-induced neutrino masses—guides model construction and underlines the restrictions imparted by radiative suppression and symmetry criteria (Cepedello, 2021, Simoes et al., 2017).

7. Prospects and Future Directions

The radiative origin of neutrino masses places new physics at or near the TeV scale, making these models testable at colliders and through rare decay processes. Large-scale parameter scans, detailed loop computations, and the use of systematic classification tools continue to expand the list of viable models. Unexplored frontiers include higher-dimensional $\Delta L=2$ operators completed at multi-loop level (Gargalionis et al., 2019), novel symmetry realizations (including non-invertible algebras), and further investigation into the interplay between neutrino mass generation, dark matter phenomenology, and the origin of the matter-antimatter asymmetry. Continuing advances in experimental sensitivity are expected to probe vast swathes of parameter space in the coming years, placing stringent tests on these models’ theoretical underpinnings and phenomenological predictions.