In the 'neutrino option', the Standard Model Higgs mass parameter is generated radiatively by the one-loop threshold corrections of PeV-scale right-handed neutrinos that simultaneously produce the observed light-neutrino masses through the type-I seesaw. Successful baryogenesis in this scenario requires resonant leptogenesis, which in turn requires the two right-handed neutrinos to form a quasi-degenerate pseudo-Dirac pair: Brdar, Helmboldt, Iwamoto and Schmitz (2019) bound the relative mass splitting δM=ΔM/M1\delta M=\Delta M/M_1 to δM∼10−6\delta M\sim10^{-6}–10−410^{-4} for normal light-neutrino ordering and δM∼10−8\delta M\sim10^{-8}–10−610^{-6} for inverted ordering, with the companion leptogenesis study confirming splittings as small as ∼10−8\sim10^{-8} near the lower viable mass bound. No symmetry or ultraviolet mechanism is known that produces and stabilizes such a tiny splitting without fine tuning while preserving all other constraints of the scenario; constructing one is the open problem.

Background

The type-I seesaw extends the Standard Model (SM) by heavy neutral fermions NIN_I (right-handed neutrinos, RHNs) with Majorana mass terms MIM_I — gauge-invariant mass terms coupling each NIN_I only to itself, allowed because NIN_I carries no SM charge — and Yukawa couplings yIαy_{I\alpha} to the SM lepton doublets; light neutrinos acquire masses mν≃−mDM−1mDTm_\nu\simeq -m_D M^{-1} m_D^T with mD=yvm_D=yv, vv being the Higgs vacuum expectation value. The PMNS matrix is the lepton mixing matrix measured in neutrino oscillations. Brivio and Trott (2017) proposed the 'neutrino option': if the SM Higgs potential obeys classical scale invariance in the ultraviolet (UV; i.e. no tree-level Higgs mass term at high energies), integrating out PeV-scale RHNs generates the Higgs mass parameter μ02≃Tr(yy†)M12/16π2\mu_0^2\simeq\mathrm{Tr}(yy^\dagger)M_1^2/16\pi^2 at one loop through Coleman–Weinberg threshold corrections (radiative corrections to the effective potential from heavy fields), so that the electroweak scale and the neutrino masses have a common origin. Brivio and Trott (2019) computed the one-loop corrections to this matching relation and verified its consistency with the observed neutrino and Standard-Model masses. Brdar, Helmboldt, Iwamoto and Schmitz (2019) showed that the scenario is viable for RHN masses M1≃106.5M_1\simeq10^{6.5}–107 GeV10^{7}\,\mathrm{GeV} and compatible with baryogenesis through resonant leptogenesis, the mechanism in which the CP asymmetry in RHN decays is enhanced when two RHNs are nearly degenerate so that their self-energy diagrams interfere; this requires the two RHNs to form a pseudo-Dirac pair (two Majorana fermions arranged so that lepton number is only weakly violated, mimicking a Dirac fermion) with relative mass splitting δM=ΔM/M1\delta M=\Delta M/M_1 of order 10−610^{-6}–10−410^{-4} for normal light-neutrino ordering and 10−810^{-8}–10−610^{-6} for inverted ordering, with splittings as small as 10−810^{-8} required near the lower end of the viable mass range, as the companion analysis of Brivio, Moffat, Pascoli, Petcov and Turner (2019) established. Such extreme quasi-degeneracy is not protected by any symmetry in the minimal model and looks tuned unless a mechanism enforces it. The question is sharpened by two subsequent developments: Brivio, Talbert and Trott (2021) proved no-go constraints showing that the PeV Majorana mass scale itself cannot be generated perturbatively from a deeper scale in minimal setups because of global symmetries of the seesaw Lagrangian, while non-perturbative routes (string D-brane instantons, non-perturbative string-theory effects that can generate operators forbidden in perturbation theory) or framework-changing routes (inverse-seesaw and related lepton-number-protected extensions, which add singlet fermions whose small masses are protected by approximate lepton-number conservation) either address only the mass scale or abandon the minimal two-RHN type-I setting.

Problem Statement

Construct and analyze a concrete, phenomenologically viable symmetry-based mechanism — a flavor symmetry, a spontaneous-breaking pattern, or ultraviolet dynamics — that, within the type-I seesaw realization of the neutrino option (two Majorana right-handed neutrinos with masses M1≃106.5M_1\simeq10^{6.5}–107 GeV10^{7}\,\mathrm{GeV}, classically scale-invariant boundary conditions for the SM Higgs potential whose mass parameter is generated by the one-loop right-handed-neutrino threshold corrections), automatically produces and stabilizes the quasi-degenerate pseudo-Dirac spectrum with relative mass splitting δM=ΔM/M1\delta M=\Delta M/M_1 in the range required for successful resonant leptogenesis (of order 10−610^{-6}–10−410^{-4} for normal and 10−810^{-8}–10−610^{-6} for inverted light-neutrino ordering, with values as small as ∼10−8\sim10^{-8} near the lower viable mass bound), without ad hoc fine tuning of the degeneracy. Here 'without ad hoc fine tuning' means that the smallness of δM\delta M must follow from the symmetry-breaking order parameter — δM\delta M is suppressed by it (or by the equivalent small parameter of an ultraviolet-dynamics mechanism) and vanishes in the symmetry-restored limit — rather than arising from a cancellation among symmetry-unrelated parameters; an O(1)\mathcal{O}(1) coefficient in that suppression is acceptable. Staying within this setting means that the seesaw sector itself remains the minimal two-RHN type-I form, while the mechanism may add symmetry-breaking structure or invoke non-perturbative ultraviolet dynamics. The mechanism must simultaneously preserve: (i) the neutrino-option matching μ02≃Tr(yy†)M12/16π2\mu_0^2\simeq\mathrm{Tr}(yy^\dagger)M_1^2/16\pi^2 with the correct sign and magnitude of the radiatively generated Higgs mass; (ii) the observed light-neutrino masses and PMNS mixing parameters; and (iii) generation of the observed baryon asymmetry via resonant leptogenesis, including renormalization-group (RG) stability of the degeneracy between the symmetry-breaking scale and the leptogenesis scale. The analysis should derive the resulting RHN mass matrix and verify these conditions.

References

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