Dynamical scotogenic generation of the Linear and Inverse seesaws

We propose an economical model in which the tiny active neutrino masses arise from an interplay of linear and inverse seesaw mechanisms. The Standard Model is extended by a local $U(1)’$ gauge symmetry and discrete $\mathbb{Z}{3}\otimes\mathbb{Z}{4}$ symmetries, together with gauge-singlet scalars and neutral leptons. Owing to the preserved discrete symmetries after spontaneous symmetry breaking, the linear and inverse seesaw mechanisms are dynamically generated at the two-loop level, while the same symmetries ensure the stability of both scalar and fermionic dark matter candidates. One of the distinctive features of the model is a fermionic dark matter candidate whose mass is generated at one loop, whereas scalar dark matter masses arise at tree level. The model satisfies current constraints from neutrino oscillation data, dark matter direct detection, invisible Higgs decays, $Z’$ searches, and charged lepton flavor violation, in addition we also discuss predictions for muonium states. The outcome of our analysis is that the inverse seesaw contribution dominates over the linear one, suggesting that atmospheric neutrino mass squared splitting arises from the inverse seesaw mechanism, whereas the solar one is generated from the linear seesaw. Finally, our model offers an explanation of the hierarchy between the atmospheric and solar neutrino mass squared splittings, in addition to the smallness of active neutrino masses, feature not presented in many low-scale seesaw models. In addition, our parameter-space scan shows a slight preference for the normal neutrino mass ordering.

💡 Research Summary

The paper presents a minimal extension of the Standard Model (SM) that simultaneously addresses the origin of tiny active neutrino masses and the existence of dark matter (DM). The authors augment the SM gauge group with a local U(1)′ symmetry and two discrete symmetries, Z₃ ⊗ Z₄. In addition to the SM fields, the model introduces several gauge‑singlet scalars (φ, η, ξ, χ, σ, ρ) and neutral fermions (ν_R, N_R, Ψ_L, Ψ_R, ˜Ψ_R, Ω_L, Ω_R). The scalar potential is constructed to trigger spontaneous breaking of U(1)′ and Z₄ via the vacuum expectation values (VEVs) of σ and ρ at the TeV scale, leaving a residual Z₂ ⊗ Z₃ symmetry. This residual symmetry plays a dual role: it forbids tree‑level contributions to the linear and inverse seesaw (LISS) mechanisms, ensuring that the lepton‑number‑violating (LNV) parameters are generated only at loop level, and it stabilises two DM candidates—a scalar (odd under Z₂) and a fermion (odd under Z₃).

The core novelty lies in the dynamical generation of the LNV parameters. The Majorana mass for Ψ_R, which feeds the inverse seesaw (ISS) sector, arises at one loop through the exchange of the heavy Dirac fermion Ω_R and the real/imaginary components of the singlet scalars ξ and χ. The Dirac mass linking the active neutrinos to the heavy pseudo‑Dirac pair is generated at two loops via a closed scalar line involving σ, ρ, ξ, χ and the trilinear coupling A ϕ η φ. Consequently, the full neutral‑lepton mass matrix takes the LISS form, with the ISS contribution dominating over the linear seesaw (LSS) term. This hierarchy naturally explains why the atmospheric neutrino mass‑squared splitting (Δm²_atm) is set by the ISS while the solar splitting (Δm²_sol) originates from the LSS, providing a dynamical origin for the observed hierarchy without ad‑hoc small parameters.

Phenomenologically, the model is subjected to a comprehensive set of constraints. The Z′ gauge boson is required to satisfy LHC dilepton search limits and to respect bounds from invisible Higgs decays. Dark matter relic density is achieved either through scalar annihilation (via Higgs portal and gauge interactions) or fermionic annihilation (mediated by the Z′). Direct‑detection limits from LUX/XENON place a lower bound on the combination M_{Z′}/g_X for a ~10 GeV fermionic DM candidate. Charged lepton flavour violation (cLFV) processes—μ→eγ, τ→μγ, τ→eγ, ℓ→3ℓ, and μ–e conversion in nuclei—are computed at one loop. The dominant contributions come from box and penguin diagrams involving the scalar sector, and the predicted rates lie close to current experimental sensitivities, making upcoming experiments a decisive test. The model also predicts exotic muonium (Mu) decays and Mu–anti‑Mu oscillations, offering an additional low‑energy probe.

A numerical scan over the parameter space shows a slight preference for the normal ordering of neutrino masses. The viable region respects all experimental bounds while reproducing the observed neutrino oscillation data, the correct DM abundance, and the hierarchy Δm²_atm ≫ Δm²_sol. The authors highlight that the fermionic DM mass is generated radiatively at one loop—a distinctive feature compared to the tree‑level scalar DM masses.

In conclusion, the work delivers a coherent framework where the linear and inverse seesaw mechanisms are realized scotogenically at two loops, the smallness of active neutrino masses is explained without fine‑tuned LNV parameters, and stable dark matter candidates naturally emerge from the same sector. The model makes concrete predictions for cLFV, Z′ searches, and muonium physics, all within reach of forthcoming experiments, thereby offering a testable path toward unraveling the intertwined mysteries of neutrino masses and dark matter.