In the low-scale type-I seesaw with three right-handed neutrinos in which CP violation originates solely from the low-energy Dirac phase of the PMNS mixing matrix, Granelli, Klarić and Petcov (arXiv:2605.31454, 2026) derived the three CP-conserving non-real Casas-Ibarra forms CP1/CP2/CP3 and showed that the scenario is testable at heavy-neutral-lepton, neutrinoless double-beta decay, and muon charged-lepton-flavour-violation experiments — but only within a set of deliberately restricted regimes. The deferred open problem is to complete the testability and reconstruction program for this Dirac-phase-only hypothesis: extend the heavy-neutrino contribution to neutrinoless double-beta decay beyond the quasi-degenerate regime (mass splittings below 10^-4), recast Big Bang Nucleosynthesis limits point-by-point for the three-HNL flavour structure, project muon charged-lepton-flavour-violation sensitivity for the CP2 and CP3 cases, determine whether realistic measurement combinations can lift the degeneracies among the Dirac phase, the Casas-Ibarra angles, and the lightest neutrino mass, and construct ultraviolet completions in which the low-energy CP structure emerges from approximate CP symmetry.

Background

The type-I seesaw is the minimal extension of the Standard Model that explains why neutrinos are massive: singlet right-handed (RH) neutrinos are added, whose gauge-invariant Majorana mass term and Yukawa couplings to the lepton doublets and the Higgs generate, after electroweak symmetry breaking, one heavy and one light Majorana neutrino per generation, with the light masses seesaw-suppressed by the heavy scale. The light-neutrino mixing is described by the Pontecorvo–Maki–Nakagawa–Sakata (PMNS) matrix, which contains the Dirac CP-violating phase δ\delta (measurable in neutrino oscillations) and two Majorana phases. The neutrino Yukawa couplings can be parameterized through the Casas-Ibarra (CI) formula, in which a complex orthogonal 3×3 matrix RR encodes the mixing between the light and heavy Majorana mass eigenstates.

In the low-scale variant, the heavy neutrinos have masses of roughly 0.1–100 GeV and sizable mixings with the Standard Model leptons, making them experimentally accessible at colliders and beam-dump experiments (heavy neutral leptons, HNLs). This mass range also permits leptogenesis — the dynamical generation of the baryon asymmetry of the Universe (BAU) through CP-violating heavy-neutrino oscillations in the early Universe, later reprocessed by Standard Model sphaleron transitions. A particularly restrictive hypothesis is that the only source of CP violation in the lepton sector is the Dirac phase δ\delta: then baryogenesis and oscillation CP violation share a single measurable origin, and all CP-odd observables vanish for δ=0,π,2π\delta=0,\pi,2\pi.

Granelli, Klarić and Petcov (2026) realized this hypothesis with three RH neutrinos by imposing a CP symmetry on the Yukawa sector while keeping the CI matrix non-real. CP invariance allows only three physically distinct forms of this matrix, called CP1, CP2, and CP3, each parameterized by two real angles xνx_\nu and xNx_N plus one imaginary parameter yy that sets the overall Yukawa size; consistency fixes the Majorana phases to CP-conserving values (for example α21=α31=±π\alpha_{21}=\alpha_{31}=\pm\pi for CP1). The scenario predicts that the squared HNL couplings to the charged flavours, Θe2:Θμ2:Θτ2\Theta^2_e:\Theta^2_\mu:\Theta^2_\tau, occupy only subregions of the flavour (ternary) triangle determined by the neutrino mass ordering and lightest neutrino mass, gives specific forms for the effective Majorana mass mββeffm^{\rm eff}_{\beta\beta} probed in neutrinoless double-beta decay (0νββ0\nu\beta\beta), and supports viable low-scale leptogenesis throughout the experimentally testable region — even for deviations of δ\delta from CP-conserving values as small as O(10−5)\mathcal{O}(10^{-5}), a fact relevant to flavour theories in which CP is realized only approximately.

The analysis, however, deliberately restricted its scope: the 0νββ0\nu\beta\beta treatment of the heavy-neutrino contributions assumed quasi-degenerate heavy masses with splittings ΔM~jk/M~av<10−4\Delta\tilde M_{jk}/\tilde M_{\rm av}<10^{-4}; Big Bang Nucleosynthesis (BBN) limits on long-lived HNLs were not applied because published limits were computed for two HNLs with single-flavour mixings, whereas here the three-HNL flavour structure varies point by point across parameter space; the charged-lepton-flavour-violation (cLFV) projections (processes such as μ→eγ\mu\to e\gamma, μ→eee\mu\to eee, and μ\mu–ee conversion in nuclei, probed by MEG II, Mu3e, Mu2e/COMET) were made only for the CP1 case with normal neutrino mass hierarchy (NH); the parameter degeneracies among δ\delta, xνx_\nu, the lightest neutrino mass, the mass splittings, and xNx_N were described but no reconstruction strategy was quantified; and ultraviolet (UV) completions realizing the required approximate CP symmetry from first principles were left unconstructed. Related but distinct analyses exist in simpler setups: a two-RHN version of Dirac-phase leptogenesis (Granelli, Pascoli, Petcov 2023), a two-RHN study of 0νββ0\nu\beta\beta with the HNL mass splitting as a free parameter (de Vries, Drewes, Georis, Klarić, Plakkot 2024), EFT rate formulae for 0νββ0\nu\beta\beta with sterile neutrinos at general masses (Dekens et al. 2024), flavour-structure-dependent leptogenesis bounds in the two-HNL model (Hernandez, Lopez-Pavon, Rius, Sandner 2022), and community-wide HNL constraint summaries including BBN limits in simplified scenarios (Abdullahi et al. 2022).

Problem Statement

Within the low-scale type-I seesaw with three right-handed neutrinos whose CP violation originates solely from the Dirac phase δ\delta of the PMNS matrix (the CP1/CP2/CP3 CP-conserving non-real Casas-Ibarra structure of Granelli–Klarić–Petcov 2026), complete the deferred phenomenological and theoretical program whose common objective is to establish whether the Dirac-phase-only hypothesis is fully testable, falsifiable, and reconstructible across its entire parameter space. Specifically: (i) compute the heavy-neutrino contributions to the effective Majorana mass mββeffm^{\rm eff}_{\beta\beta} in neutrinoless double-beta decay beyond the quasi-degenerate regime, i.e. for relative mass splittings ΔM~jk/M~av\Delta\tilde M_{jk}/\tilde M_{\rm av} not restricted to be below 10−410^{-4}, quantifying the interplay with radiative corrections to the light-neutrino masses; (ii) recast Big Bang Nucleosynthesis limits on long-lived heavy neutral leptons for each viable parameter point, accounting for the point-dependent three-HNL flavour structure rather than single-flavour or two-HNL approximations; (iii) project the reach of muon charged-lepton-flavour-violation experiments (MEG II on μ→eγ\mu\to e\gamma, Mu3e on μ→eee\mu\to eee, Mu2e/COMET and PRISM/PRIME on μ\mu–econversion)fortheCP2andCP3cases,comparablytotheexistingCP1−with−NHanalysis;(iv)determinequantitativelywhetherrealisticcombinationsofmeasurements—heavy−neutrinobranchingratiosandsquaredcouplingse conversion) for the CP2 and CP3 cases, comparably to the existing CP1-with-NH analysis; (iv) determine quantitatively whether realistic combinations of measurements — heavy-neutrino branching ratios and squared couplings |\Theta_{\alpha j}|^2,kinematicresolutionofthethreeheavystatesandtheirsplittings,decay−distributionorright−handed−neutrino−oscillationprobesof, kinematic resolution of the three heavy states and their splittings, decay-distribution or right-handed-neutrino-oscillation probes of x_N,theoscillationvalueof, the oscillation value of \delta,andthelightestneutrinomass—liftthedegeneraciesamong, and the lightest neutrino mass — lift the degeneracies among \delta,, x_\nu,, x_N,, m^{\rm lightest}_\nu,andthemasssplittings,permittingafullparameterreconstructionandaconsistencytestofthescenario;and(v)asthetheoreticalcapstone,constructultravioletcompletionsofthetype−IseesawwithapproximateCPsymmetryinwhichtherequiredlow−energyDirac−phase−onlyCPstructure(includingtheCP−conservingMajorana−phasevaluesandthesmalldeviationsfromCP−conserving, and the mass splittings, permitting a full parameter reconstruction and a consistency test of the scenario; and (v) as the theoretical capstone, construct ultraviolet completions of the type-I seesaw with approximate CP symmetry in which the required low-energy Dirac-phase-only CP structure (including the CP-conserving Majorana-phase values and the small deviations from CP-conserving \delta$ that still allow successful leptogenesis) emerges from first principles rather than being imposed at low energy. All five items (i) through (v) are jointly required: a complete solution must address each of them, and addressing only a proper subset does not resolve the task.

References

  1. A. Granelli, J. Klarić, S. T. Petcov, "Dirac-Phase CP-Violation in the Low-Scale Type-I Seesaw with Three Right-Handed Neutrinos", arXiv:2605.31454 (2026), DOI 10.48550/arxiv.2605.31454, https://arxiv.org/abs/2605.31454
  2. A. Granelli, S. Pascoli, S. T. Petcov, "Low-Scale Leptogenesis with Low-Energy Dirac CP-Violation", Phys. Rev. D 108, L101302 (2023), arXiv:2307.07476, DOI 10.1103/PhysRevD.108.L101302, https://arxiv.org/abs/2307.07476
  3. J. de Vries, M. Drewes, Y. Georis, J. Klarić, V. Plakkot, "Confronting the low-scale seesaw and leptogenesis with neutrinoless double beta decay", JHEP 05 (2025) 090, arXiv:2407.10560, DOI 10.1007/JHEP05(2025)090, https://arxiv.org/abs/2407.10560
  4. W. Dekens, J. de Vries, D. Castillo, J. Menéndez, E. Mereghetti, V. Plakkot, P. Soriano, G. Zhou, "Neutrinoless double beta decay rates in the presence of light sterile neutrinos", JHEP 09 (2024) 201, arXiv:2402.07993, DOI 10.1007/JHEP09(2024)201, https://arxiv.org/abs/2402.07993
  5. P. Hernández, J. López-Pavón, N. Rius, S. Sandner, "Bounds on right-handed neutrino parameters from observable leptogenesis", JHEP 12 (2022) 012, arXiv:2207.01651, DOI 10.1007/JHEP12(2022)012, https://arxiv.org/abs/2207.01651
  6. A. M. Abdullahi et al., "The present and future status of heavy neutral leptons", J. Phys. G 50 (2023) 020501, arXiv:2203.08039, DOI 10.1088/1361-6471/ac98f9, https://arxiv.org/abs/2203.08039