One-Loop Radiative Neutrino Mass Model

• One-loop radiative neutrino mass models generate tiny neutrino masses through quantum loop diagrams that introduce new scalar and fermionic fields beyond the Standard Model.
• These models are classified into 'exit' and 'dark-matter' classes, with the latter providing stable candidates due to imposed discrete symmetries like Z2.
• Benchmark scenarios demonstrate that loop suppression, small couplings, and new particle masses naturally satisfy neutrino oscillation data, LFV limits, and dark matter constraints.

A one-loop radiative neutrino mass model is a framework in which tiny neutrino masses are generated quantum mechanically via loop diagrams rather than from renormalizable tree-level couplings. These models typically invoke the introduction of new particles—scalars and/or fermions beyond the Standard Model (SM)—and possibly new symmetries, with lepton number violation (for Majorana neutrino mass) or novel assignment of quantum numbers (for Dirac neutrino mass). This approach addresses the observed smallness of neutrino masses as loop suppression factors and the heaviness of exotic states naturally lead to scales compatible with oscillation data, and often correlates with viable candidates for dark matter and new collider signatures.

1. General Structure and Classification

One-loop radiative models focus on realizing the $d=5$ Weinberg operator,

$$\mathcal{O}_W = \frac{1}{\Lambda}(L^T C \varepsilon H)(L^T C \varepsilon H),$$

through genuine one-loop diagrams with new heavy states in the loop. These models can be classified into two major categories based on cosmological stability and decay of new states (Arbeláez et al., 2022):

• Exit class ("exits"): At least one new field in the loop couples linearly to SM fields and hence can decay, preventing stable charged/colored relics.
• Dark-matter class: All loop fields are odd under a $Z_2$ (either imposed or accidental), ensuring that the lightest neutral state is stable and may serve as a dark-matter WIMP candidate.

Each of these classes encompasses multiple specific model topologies, typically labeled T-I-1, T-I-2, T-I-3, and T-3, denoting different insertions of Yukawa and scalar trilinear couplings in the loop (Arbeláez et al., 2022). The precise field content and representation under $SU(2)_L \times U(1)_Y$ vary, but minimal models generally introduce new scalar multiplets (singlet, doublet, or triplet), vectorlike/lepton partners, and may enforce stabilizing discrete or gauge symmetries.

2. Representative Models and Topologies

2.1 Scalar Triplet Scotogenic Model

A concrete realization is the "scotogenic scalar triplet model," which extends the SM by a hypercharge-zero scalar triplet $\Delta\sim(3,0)$, a charged scalar singlet $h^+ \sim (1,2)$, and vectorlike lepton doublets $\Sigma_{L,R}\sim(2,-1)$. The neutrino mass is generated at one loop through a diagram involving the mixing of $h^+$ and $\Delta^+$, and $\Sigma$ in the internal lines (Brdar et al., 2013). If a discrete $Z_2$ or a gauged $U(1)_D$ is imposed—with all new fields $Z_2$-odd and SM fields $Z_2$-even—the neutral component $\Delta^0$ becomes a stable dark-matter candidate, forbidding tree-level masses.

The analytic neutrino mass expression is: $$(M_\nu)_{ij} = \sum_k \frac{[(g_1)_{ik}(g_2)_{jk} + (g_2)_{ik}(g_1)_{jk}] M_{\Sigma_k} \sin\theta\,\cos\theta}{8\pi^2} \left[\frac{m_{S_1}^2}{m_{S_1}^2 - M_{\Sigma_k}^2} \ln\frac{m_{S_1}^2}{M_{\Sigma_k}^2} - (S_1 \to S_2)\right].$$ Mixing is controlled by the scalar potential parameter $\lambda_7$ through $\sin\theta\cos\theta$ (Brdar et al., 2013).

2.2 Inverse Seesaw Radiative Realization

The minimal radiative inverse seesaw extension consists only of SM singlet fermions ($\nu_R$, $S_L$), without additional scalars or gauge bosons. The radiative mass

$M_{\nu_L}^{1\mbox{-}\rm loop} = \frac{\alpha_W}{16\pi m_W^2} M_D\,\Delta(\mathcal{M}_S)\,M_D^T,$

where $\Delta(\mathcal{M}_S)$ is a loop function over the heavy sector and involves the explicit lepton-number violating soft mass parameter $\mu_R$ in the singlet sector (Dev et al., 2012). Smallness of $m_\nu$ comes from loop suppression, $M_D^2/m_N^2$, and the smallness of $\mu_R$.

2.3 Generic Model Topologies

The full classification includes models where the one-loop diagram is closed by trilinear scalar couplings, with new fermions or scalars running in internal lines. Typical field content includes

• Scalars: singlet $\mathcal{S}$, doublet $\varphi$, triplet $\Xi$, higher $SU(2)_L$ multiplets, with variable hypercharge.
• Fermions: vector-like lepton singlets $E$, doublets, triplets, with quantum numbers ensuring decay to SM leptons or neutrinos (Arbeláez et al., 2022).

3. Loop Mass Generation and Analytic Structure

All such models yield radiative corrections of the schematic form

$$(m_\nu)_{ab} \sim \sum_{\text{internal}} Y_{ai} Y'_{bj} \lambda_{ij} \int \frac{d^4k}{(2\pi)^4} \frac{1}{(k^2 - m_{F_i}^2)\cdots},$$

where $Y$ are Yukawa couplings, $\lambda$ are scalar trilinears, and the loop integral encodes the topology and masses of internal particles (Arbeláez et al., 2022). The suppression of neutrino mass is governed by the loop factor $1/16\pi^2$, small couplings, and the heaviness of exotic particles.

The calculation is performed in Feynman parameterization; e.g., for four propagators,

$$\mathcal{I}_4(m_1, m_2, m_3, m_4) = \frac{i}{16\pi^2}\int_0^1 dx_1 \cdots dx_4\,\delta(\sum x_i-1)\,\frac{1}{[x_1 m_1^2 + ... + x_4 m_4^2]^2}.$$

4. Phenomenology: Dark Matter, Collider, and Lepton Flavor Signatures

Phenomenological signatures strongly depend on the symmetry structure and quantum numbers:

• Dark matter: If the lightest new $Z_2$-odd state is electrically neutral, it can be a viable WIMP candidate. For the triplet scalar, the freeze-out abundance consistent with $\Omega h^2 \simeq 0.12$ is obtained for $m_{\Delta^0}\sim2.5$ TeV (real, $Z_2$-odd) or $\sim 1.8$ TeV (complex, $U(1)_D$-charged), with dominant annihilation into electroweak bosons (Brdar et al., 2013).
• LFV and $\mu\to e\gamma$: The Yukawa couplings responsible for neutrino mass also mediate lepton-flavor violation. Given the required smallness of the couplings to match oscillation data, induced LFV rates are securely below current bounds for most viable benchmarks (Brdar et al., 2013).
• Collider signals: New scalar triplet/doublet and vectorlike leptons can be produced via Drell–Yan, yielding multi-lepton and missing energy final states. The $h \to \gamma\gamma$ decay receives loop corrections from charged scalars, with the signal strength $R_{\gamma\gamma}$ constraining quartic couplings $\lambda_4$, $\lambda_5\lesssim\mathcal{O}(1)$ for $m_S\sim200$ GeV (Brdar et al., 2013). If $\Delta^0$ is multi-TeV, direct production is suppressed, but models with lighter DM or broken $U(1)_D$ can manifest via additional signatures from dark photons or dark Higgs bosons.
• Direct and indirect detection: For $Y=0$, tree-level $Z$-mediated DM scattering is absent, while loop-induced cross-sections remain well below current direct-detection sensitivities. Indirect searches for WIMP annihilation into gauge bosons are promising for next-generation Cherenkov experiments especially if Sommerfeld enhancement is relevant (Brdar et al., 2013).

5. Constraints, Model Statistics, and High-Scale Consistency

A comprehensive paper enumerates a total of hundreds of possible 1-loop completions of the Weinberg operator, after consistent imposition of cosmological and phenomenological constraints (Arbeláez et al., 2022):

• Exit class: 406 models, of which 38 use SM internal lines; 368 use only BSM fields but feature "exit" particles ensuring no stable charged relics.
• Dark-matter class: 318 models (115 require explicit $Z_2$, 203 have an accidental $Z_2$ due to quantum numbers).

The extended field content can cause Landau poles in gauge couplings if large $SU(2)_L$ or $SU(3)$ multiplets are present, potentially constraining the cutoff of the theory. Only a select few models remain perturbative to the GUT scale; a unique "exit" model achieves one-loop unification at $m_G\simeq 10^{17}$ GeV (Arbeláez et al., 2022).

6. Benchmark Scenarios

The following benchmarks exemplify the viable parameter space (Brdar et al., 2013):

Benchmark	Particle Masses	Key Couplings	$m_\nu$	DM Type	Phenomenology
A (weak-scale)	$M_\Sigma = m_{h^+} = m_{\Delta^+}=200$ GeV	$g_1=g_2=\lambda_7=10^{-4}$	$0.1$ eV	No DM (no symmetry)	$h\to\gamma\gamma$ enhancement
B ($Z_2$)	$m_{\Delta^0}=2.5$ TeV, $M_\Sigma=2$ TeV	$g_1=g_2=\lambda_7=10^{-3}$	$0.05$ eV	Real triplet WIMP	LHC out of reach, LFV suppressed
C ($U(1)_D$)	$m_{\Delta^0}=1.8$ TeV, $g_D=0.1$, $m_{\gamma_D}=100$ MeV, $m_{\zeta_R}=200$ MeV	$\epsilon_\mu\sim10^{-4}$	varies	Complex triplet + dark sector	Astrophysical signatures, LHC+DM

These points illustrate the interplay between accessible parameter space for neutrino masses and parallel implications for dark matter searches and collider probes.

7. Outlook and Broader Implications

One-loop radiative neutrino mass models substantially enlarge the landscape of neutrino model building. They achieve natural suppression of $m_\nu$ via loop factors and potentially small new couplings, robustly address cosmological constraints by classifying decay and stability of novel states, and generically provide correlated phenomenology linking neutrino oscillation, dark matter, flavor violation, and collider physics. The explicit catalogue and statistical paper demonstrate that hundreds of valid 1-loop completions exist, in contrast to only three tree-level seesaw types, but various requirements (dark matter viability, absence of charged relics, collider and LFV limits, high-scale perturbativity) serve to prune this space to a more constrained set of viable models (Arbeláez et al., 2022, Brdar et al., 2013).