Fong, Gregoire and Tonero (Phys. Lett. B 816 (2021) 136175) proposed a quasi-Dirac leptogenesis framework in which tiny violation of the symmetry Δtot=(B−L)−(B′−L′)\Delta_{\rm tot}=(B-L)-(B'-L') in a mirror-world seesaw splits both the heavy fermion singlets and the light neutrinos into quasi-Dirac pairs, enabling resonant leptogenesis at the electroweak scale while predicting a light-neutrino mass-squared splitting ε2≡4mνδm\varepsilon^2\equiv 4m_\nu\delta m in the range 10−1210^{-12}–10−6 eV210^{-6}\,\mathrm{eV}^2 that oscillation experiments can probe, via the key relation ∣ϵmax⁡∣≃δm/(2mν)|\epsilon_{\max}|\simeq\delta m/(2m_\nu) linking the CP violation driving the baryon asymmetry to a low-energy observable. The analysis was carried out only for one heavy family under flavor-equilibrated Z2Z_2-symmetric benchmarks, and the paper explicitly flags four open issues: multi-flavor realism, mirror-sector cosmological bounds, collider testability, and experimental coverage of the predicted splitting range, especially ε2≲10−12 eV2\varepsilon^2\lesssim10^{-12}\,\mathrm{eV}^2. The open problem is to determine whether quasi-Dirac leptogenesis survives in a realistic, fully flavor-structured model reproducing the observed neutrino mass differences and mixings, is consistent with cosmological limits on dark radiation and NeffN_{\rm eff}, admits collider-testable realizations, and has its viable parameter space completely mapped onto oscillation, astrophysical-neutrino-flavor, and cosmological probes. Follow-up work since 2024 has substantially developed the astrophysical-flavor and CMB sensitivity direction generically for pseudo-Dirac splittings, but no multi-flavor analysis of this framework, no flavor-model-dependent mapping onto the leptogenesis-viable region, and no collider or mirror-cosmology extension preserving the quasi-Dirac leptogenesis connection has appeared.

Background

The observed baryon asymmetry of the Universe is widely attributed to leptogenesis (Fukugita and Yanagida, 1986): the out-of-equilibrium, CP-violating decays of heavy neutral leptons generate a lepton asymmetry that electroweak sphalerons — the nonperturbative Standard Model processes that violate B+LB+L — partially convert into a baryon asymmetry before they freeze out at T∼132 GeVT\sim132\,\mathrm{GeV}. In the conventional type-I seesaw, where heavy fermion singlets with large Majorana masses generate the light-neutrino masses, the required singlet masses lie far above experimental reach, disconnecting the mechanism from laboratory observables. Resonant leptogenesis (Pilaftsis and Underwood, 2004) evades the usual Davidson–Ibarra lower bound on the singlet mass by exploiting quasi-degenerate singlets whose self-energy corrections enhance the CP asymmetry, allowing singlet masses near the electroweak scale. A neutrino is called pseudo-Dirac or quasi-Dirac when a small lepton-number-violating Majorana mass splits what would be a Dirac fermion into two nearly degenerate Majorana states of opposite CP; for light neutrinos this produces active–sterile pairs — each known weakly interacting ('active') neutrino acquiring an almost gauge-singlet ('sterile') partner — with a small mass-squared splitting εk2\varepsilon_k^2 between the pair members, constrained by solar and atmospheric oscillation data to roughly ε1,22≲10−12\varepsilon_{1,2}^2\lesssim10^{-12}–10−5 eV210^{-5}\,\mathrm{eV}^2 while the splitting of the third eigenstate is only bounded at ε32≲10−5 eV2\varepsilon_3^2\lesssim10^{-5}\,\mathrm{eV}^2 (de Gouvêa, Huang and Jenkins, 2009; Anamiati, Fonseca and Hirsch, 2018). Because ultrahigh-energy astrophysical neutrinos propagate over enormous baselines, their flavor ratios at Earth are sensitive to far smaller splittings than any terrestrial experiment (an idea explored earlier, e.g., in the pseudo-Dirac model of Ahn, Kang and Kim, 2016, where however the large Majorana mass destroyed the connection between leptogenesis and low-energy observables).

Fong, Gregoire and Tonero (2021) constructed a minimal framework that ties these threads together. Building on mirror Dirac leptogenesis (Earl, Fong, Gregoire and Tonero, 2020), the Standard Model (SM) is doubled into a mirror sector, and heavy Dirac singlet fermions (NR,NR′)(N_R,N_R') of mass MM couple to both sectors, breaking B−LB-L and B′−L′B'-L' individually while conserving Δtot=(B−L)−(B′−L′)\Delta_{\rm tot}=(B-L)-(B'-L'). Small Δtot\Delta_{\rm tot}-violating Majorana masses m,m′≪Mm,m'\ll M then split both the heavy singlets into quasi-Dirac pairs — resonantly enhancing the CP violation in their decays so that viable leptogenesis occurs at M∼0.5M\sim0.5–10 TeV10\,\mathrm{TeV} — and the light neutrinos into quasi-Dirac active–sterile pairs with splitting δm\delta m. The central result is the relation ∣ϵmax⁡∣≃δm/(2mν)|\epsilon_{\max}|\simeq\delta m/(2m_\nu) between the maximal CP violation available for leptogenesis and the light-neutrino quasi-Dirac splitting, equivalently ε2≡4mνδm\varepsilon^2\equiv4m_\nu\delta m: since successful leptogenesis requires ∣ϵmax⁡∣|\epsilon_{\max}| above a lower bound while oscillation experiments upper-bound ε2\varepsilon^2, the viable parameter space (ε2∼10−12\varepsilon^2\sim10^{-12}–10−6 eV210^{-6}\,\mathrm{eV}^2 for weak-scale MM) overlaps the region probed by solar and atmospheric oscillations, making this a rare leptogenesis model directly testable at low energy; for ε2≲10−12 eV2\varepsilon^2\lesssim10^{-12}\,\mathrm{eV}^2 the flavor content of high-energy astrophysical neutrinos becomes the relevant probe. However, the derivation was performed for a single heavy family (a limiting case of a multi-family model in which resonant leptogenesis proceeds through the lightest quasi-Dirac pair), under the assumption of flavor equilibration in Z2Z_2-symmetric benchmarks (Z2Z_2 denoting the interchange symmetry between the Standard Model and its mirror copy), and the authors explicitly note that explaining the observed neutrino mass differences and mixing requires at least one additional heavy family. They further acknowledge that mirror-sector states can contribute excessive dark radiation (extra relativistic energy density, constrained by the effective neutrino species number NeffN_{\rm eff}), that direct collider production of the heavy singlets is suppressed by the active–sterile mixing yv/M∼mν/Myv/M\sim m_\nu/M so that an explicit extension such as a left–right symmetric gauge sector would be needed for adequate production rates, and that translating existing and future measurements into exclusions or discoveries of the leptogenesis parameter space — particularly below 10−12 eV210^{-12}\,\mathrm{eV}^2 — requires dedicated sensitivity studies.

Problem Statement

Determine whether quasi-Dirac leptogenesis — the mechanism in which small Δtot=(B−L)−(B′−L′)\Delta_{\rm tot}=(B-L)-(B'-L') violation splits the heavy fermion singlets and light neutrinos of a mirror-world seesaw into quasi-Dirac pairs, enabling resonant leptogenesis at the electroweak scale with the light-neutrino splitting linked to the CP violation through ∣ϵmax⁡∣≃δm/(2mν)|\epsilon_{\max}|\simeq\delta m/(2m_\nu) — can be realized as a realistic and fully testable explanation of the cosmic baryon asymmetry. The common objective subsumes four jointly necessary components, and a complete answer must address each and exhibit their interlock: (1) Multi-flavor structure: extend the one-family analysis to a model with full flavor structure (at least one additional heavy family) that reproduces the observed light-neutrino mass differences and mixing angles, and determine whether the CP-splitting relations and the viable leptogenesis parameter regions survive once realistic flavor mixing, flavored washout, and flavor-dependent asymmetry evolution are included. (2) Cosmological consistency: specify how a mirror-sector (or equivalent) realization keeps the abundance of mirror and sterile states within dark-radiation and NeffN_{\rm eff} bounds, including any production-suppression mechanism, and verify it. (3) Collider testability: construct and analyze explicit extensions (for example with left–right symmetric gauge interactions) that provide adequate heavy-singlet production rates while preserving the quasi-Dirac leptogenesis connection between the baryon asymmetry and the light-neutrino splitting. (4) Experimental coverage: quantify which parts of the leptogenesis-viable region in the (M,mν,ε2)(M,m_\nu,\varepsilon^2) parameter space — including which quasi-Dirac-split light-neutrino eigenstate, as fixed by the flavor structure of component (1) — can be excluded or discovered by existing and planned probes, in particular oscillation experiments for ε2≳10−12 eV2\varepsilon^2\gtrsim10^{-12}\,\mathrm{eV}^2 and high-energy astrophysical neutrino flavor measurements and cosmological data for ε2≲10−12 eV2\varepsilon^2\lesssim10^{-12}\,\mathrm{eV}^2. The question is answered either by a demonstrated viable, realistic, and experimentally mapped realization, or by a demonstrated no-go showing that no flavor-consistent, cosmology-consistent realization of the mechanism survives existing data; such a no-go likewise resolves the question and renders components (3) and (4) moot, since no surviving realization remains to be made collider-testable or experimentally mapped.

References

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