A theoretical model of Dirac neutrinos finds that leptonic CP violation, the absolute neutrino mass scale and the strong-CP phase can be tied together through a set of discrete compatibility branches. In some benchmark regions, the calculation permits sizable—and in one compressed case maximal—leptonic CP while remaining compatible with the stated strong-CP condition. The result is a branch-dependent map of model solutions.
The work, posted as an arXiv preprint version 1 on 26 August 2026, examines a left–right model with generalized parity and a sectorial leptonic reality condition imposed on the Dirac-lepton Yukawa matrices. The study uses analytical reconstruction and numerical scans across mass orderings, leptonic branches and fixed-mass benchmarks.
One condition controls the viable branches
At the heart of the calculation is an exact compatibility test for generic three-generation Yukawa matrices. The parity requirement and the leptonic reality condition can coexist only when one Jarlskog-type CP-odd invariant vanishes. In mixing terms, the allowed left- and right-handed matrices have identical moduli but opposite signs for their Jarlskog invariants.
On those compatible solutions, the left-handed PMNS Jarlskog invariant—the quantity that tracks CP-violating mixing—obeys the identity J_L = −ΔJ_P/2. In plain terms, the compatible leptonic CP signal is fixed by the shift induced in the right-handed sector by parity breaking.
The parity-breaking deformation is denoted by ϵ. The analysis treats the resulting relationships in two limiting regimes. In the ordinary first-order expansion, the response grows with |ϵ|; in the quasi-degenerate limit, where the neutrino masses are more closely packed, the response decreases with |ϵ|. These describe opposite sides of the same nonlinear branch, with the full numerical solution needed between the limits.
The answer changes with ordering and mass scale
The numerical calculation solves the nonlinear compatibility branches separately for normal and inverted neutrino mass ordering and for each leptonic branch. It relates the normalized PMNS Jarlskog invariant to the lightest neutrino mass and to |ϵ|. The benchmarks use NuFIT v6.1 best-fit oscillation inputs, with the lightest mass fixed at 0.01 eV for a light spectrum and at 0.3 eV for a compressed, quasi-degenerate diagnostic.
For the 0.01 eV normal-ordering benchmark, solar-enhanced branches become important first. Atmospheric branches grow at larger values of |ϵ|, while same-sign branches remain weak until close to the physical endpoint. In inverted ordering, the solar-enhanced branches rise faster and approach saturation sooner; atmospheric branches stay weaker and closer to the same-sign family. The sizable CP allowed in the strong-CP-compatible region is concentrated in the solar-enhanced branches.
The higher-mass benchmark changes the picture. At a lightest neutrino mass of 0.3 eV, mixed-sign branches show enhancement at smaller |ϵ|, and the strong-CP-compatible region includes sizable values, including maximal leptonic CP violation. Same-sign branches remain weak. The benchmark was chosen to expose compressed and quasi-degenerate nonlinear structure, rather than to provide a statistical or cosmological fit.
When the calculation is projected onto illustrative intervals of the leptonic phase, compatible solutions extend down to about 1.2 × 10−3 eV for normal ordering and 6 × 10−4 eV for inverted ordering. These regions are unions over discrete branches, so they describe a spread of allowed model solutions rather than one mass value.
A narrow window from the quark sector
The quark reconstruction supplies the other side of the correlation. On matched quark branches, the induced strong-CP phase has a branch-dependent linear response to ϵ at leading order. At the representative parity scale of 10 TeV, inequivalent quark sign classes give coefficient magnitudes from 12.88 to 50.25, so the same deformation maps differently onto the strong-CP phase depending on the branch.
Using the nominal criterion |θ̄| < 10−10 at that 10 TeV scale, all matched quark branches satisfy the bound below |ϵ| ≃ 2 × 10−12. Between that value and about 7.8 × 10−12, only some branches do. Above 7.8 × 10−12, none of the matched branches satisfy the stated criterion. Substantial projected leptonic regions lie below the upper boundary, and the compressed benchmark includes sizable leptonic CP in a strong-CP-compatible region.
A calculation, not a verdict on nature
The curves are tree-level benchmarks evaluated with central oscillation inputs, not statistical confidence regions. The calculation does not include leptonic renormalization-group evolution to the parity scale, an omission that may matter for quasi-degenerate Dirac neutrinos. Outside the stated validity ranges of the truncated analytic expansions, the full nonlinear compatibility solution is required.
The construction is a restricted leptonic benchmark. It does not establish that the sectorial reality condition occurs in nature, provide a measured leptonic CP phase or neutrino mass, demonstrate stability under complete renormalization-group evolution, or explain the origin of the small Dirac-neutrino masses. Its strong-CP interpretation assumes that no Peccei–Quinn relaxation mechanism or cancellation from other CP-odd sources changes the phase map, and the analysis does not propagate hadronic or quark-input uncertainties.
Because the analysis seeks branch-dependent correlations, its output is a family of model solutions rather than a single empirical result. The supplied front matter lists an author affiliation but no funding statement.
Paper data and sources
Original title: Strong CP and the PMNS Phase in Dirac Left-Right Symmetry
Authors: Vladimir Tello
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text