Decisive unitarity test of the 3×3 lepton (PMNS) mixing matrix via its six leptonic unitarity triangles, and the associated determination of leptonic CP violation

The three-flavour description of neutrino oscillations rests on the assumption that the Pontecorvo–Maki–Nakagawa–Sakata (PMNS) lepton mixing matrix is unitary. Unitarity implies six orthogonality relations among the rows and columns of the matrix, each visualized as a leptonic unitarity triangle whose common area is half the Jarlskog invariant — a direct measure of CP violation in the lepton sector. Whether this unitarity can be tested decisively — that is, whether the matrix elements, including the poorly measured tau-neutrino sector, and the Dirac CP phase can be determined with precision sufficient to close all six triangles at a level that would either establish a deviation or exclude well-motivated non-unitary scenarios, and what precision would be required to establish or exclude leptonic CP violation at discovery-level confidence — remains open. Current global data are consistent with unitarity, but triangle-closure and row/column-normalization precision ranges from a few per mille in the electron sector to order 10% in the tau sector, and leptonic CP violation is not yet established. Projections show the approved next-generation program (Hyper-Kamiokande, DUNE, JUNO) constrains all row and column normalizations only to about the 10% level, so a percent-level closure test of all six triangles requires either new methods, new facilities, or combined oscillation and non-oscillation constraints.

Background

Neutrinos oscillate between the three flavour states because the flavour eigenstates are quantum superpositions of three mass eigenstates, related by the Pontecorvo–Maki–Nakagawa–Sakata (PMNS) mixing matrix $U$. In the standard three-flavour picture $U$ is a $3\times3$ unitary matrix parametrized by three mixing angles $\theta_{12}$, $\theta_{23}$, $\theta_{13}$ and one Dirac CP-violating phase $\delta_{CP}$ (two further Majorana phases do not affect oscillations). Unitarity, $U^{\dagger}U=UU^{\dagger}=\mathbf{1}$, imposes three orthogonality relations among the rows and three among the columns of $U$; each relation, drawn in the complex plane, closes into a triangle. These six leptonic unitarity triangles all have the same area, equal to half the Jarlskog invariant $J=c_{12}s_{12}c_{23}s_{23}c_{13}^{2}s_{13}\sin\delta_{CP}$ (the rephasing-invariant quantity that quantifies CP violation in oscillations), so the triangles simultaneously encode the CP structure and the internal consistency of the matrix. This is the lepton-sector analogue of the unitarity-triangle tests that are a mature precision industry for the quark-sector Cabibbo–Kobayashi–Maskawa (CKM) matrix, where sides and angles are overconstrained at the sub-percent level; the lepton-sector tests are far weaker. They are nonetheless strongly motivated: many mechanisms proposed to explain neutrino mass — heavy neutral leptons in seesaw models (neutrino-mass mechanisms in which the light neutrinos acquire tiny masses through mixing with very heavy neutral leptons), light or eV-scale sterile neutrinos, extra generations — generically make the effective $3\times3$ active-neutrino mixing matrix non-unitary, with deviations that can range from $\mathcal{O}(10^{-3})$ (canonical high-scale seesaw) to percent level and above (eV-scale sterile mixing, still partially allowed by short-baseline and cosmological data). Parke and Ross-Lonergan (2016) performed the first refit of global oscillation data without assuming unitarity, bounding the closure of the six triangles to $\le0.03$ for the $\nu_e\nu_\mu$ triangle and $\le0.1$–$0.2$ for the others, and row/column normalizations to $\le0.2$–$0.4$; Xing and Zhu (2016) plotted the actual shapes of the Dirac and Majorana triangles from the global best-fit parameters. Nguyen Thi Kim Ha et al. (2019) reviewed this triangle picture and illustrated how the uncertainty of the apex position of one triangle — governed dominantly by the uncertainty on $\delta_{CP}$ — shrinks under assumed future Hyper-Kamiokande ($\pm23^\circ$) and combined Hyper-Kamiokande-plus-DUNE ($\pm15^\circ$) phase uncertainties. The empirical landscape: long-baseline accelerator experiments (T2K, NOvA), reactor experiments (Daya Bay, JUNO), solar and atmospheric neutrino data feed global fits in which $\delta_{CP}$ is still comparatively poorly determined and CP violation in the lepton sector is not established; the tau-neutrino row of the matrix is the least well measured part of the Standard Model flavour sector. Next-generation experiments — Hyper-Kamiokande (the T2HK long-baseline program on a 295 km baseline with a megaton-scale water Cherenkov detector), DUNE (a long-baseline experiment with a 1300 km baseline and liquid-argon detectors), and JUNO (a reactor medium-baseline experiment precise on $\theta_{12}$ and the mass-squared differences) — are designed to improve precisely these measurements.

Problem Statement

Can the $3\times3$ lepton (PMNS) mixing matrix be determined completely enough — all nine matrix elements including the tau-neutrino sector, together with the Dirac CP phase $\delta_{CP}$ — to perform a decisive quantitative test of its unitarity? Concretely, the open problem spans three interlinked aspects of this single objective: (1) whether the empirical closure test of all six leptonic unitarity triangles (equivalently, the three row and three column normalization relations $\sum_i|U_{\alpha i}|^2=1$ and the six orthogonality relations $\sum_i U_{\alpha i}U^{*}_{\beta i}=0$) can reach a precision comparable to or better than the deviations predicted by well-motivated non-unitary scenarios — sterile-neutrino mixing at the levels still allowed by current constraints (percent level and above), or seesaw-induced deviations ($\mathcal{O}(10^{-3})$) — so that either a statistically significant deviation from closure would be established or those scenarios would be excluded; (2) what precision on $\delta_{CP}$ and the mixing angles — and on the correlated combinations that form the triangle sides — would be needed to establish or exclude CP violation in the lepton sector ($\delta_{CP}\ne0,\pi$, equivalently a non-zero Jarlskog invariant / non-degenerate triangle area) at discovery-level confidence (conventionally $\ge5\sigma$), the triangles' common area $J/2$ tying their closure to the determination of $\delta_{CP}$; (3) whether the approved next-generation program (Hyper-Kamiokande, DUNE, JUNO, atmospheric $\nu_\tau$ measurements, and non-oscillation constraints), with correlated and systematic uncertainties treated jointly, suffices to reach these precisions, or whether new observables, beam concepts, or facility classes are required. Current data leave all three aspects open: global fits are consistent with unitarity but with triangle-closure and normalization precision spanning from a few per mille (electron sector) to order 10% (tau sector), and leptonic CP violation is not yet established.

References

1. Ha Nguyen Thi Kim, Van Nguyen Thi Hong, Son Cao Van, 'Unitarity of neutrino mixing matrix', EPJ Web of Conferences 206 (2019) 09009, DOI: 10.1051/epjconf/201920609009, https://doi.org/10.1051/epjconf/201920609009
2. Stephen Parke, Mark Ross-Lonergan, 'Unitarity and the three flavor neutrino mixing matrix', Physical Review D 93 (2016) 113009, arXiv:1508.05095, DOI: 10.1103/PhysRevD.93.113009, https://arxiv.org/abs/1508.05095
3. Zhi-zhong Xing, Jing-yu Zhu, 'Leptonic unitarity triangles and effective mass triangles of the Majorana neutrinos', Nuclear Physics B 908 (2016) 302-317, arXiv:1511.00450, DOI: 10.1016/j.nuclphysb.2016.03.031, https://arxiv.org/abs/1511.00450
4. Sebastian A.R. Ellis, Kevin J. Kelly, Shirley Weishi Li, 'Leptonic Unitarity Triangles', Physical Review D 102 (2020) 115027, arXiv:2004.13719, DOI: 10.1103/PhysRevD.102.115027, https://arxiv.org/abs/2004.13719
5. Sebastian A.R. Ellis, Kevin J. Kelly, Shirley Weishi Li, 'Current and Future Neutrino Oscillation Constraints on Leptonic Unitarity', Journal of High Energy Physics 12 (2020) 068, arXiv:2008.01088, DOI: 10.1007/JHEP12(2020)068, https://arxiv.org/abs/2008.01088
6. Zhuojun Hu, Jiajie Ling, Jian Tang, TseChun Wang, 'Global oscillation data analysis on the 3ν mixing without unitarity', Journal of High Energy Physics 01 (2021) 124, arXiv:2008.09730, DOI: 10.1007/JHEP01(2021)124, https://arxiv.org/abs/2008.09730
7. Peter B. Denton, Julia Gehrlein, 'New oscillation and scattering constraints on the tau row matrix elements without assuming unitarity', Journal of High Energy Physics 06 (2022) 135, arXiv:2109.14575, DOI: 10.1007/JHEP06(2022)135, https://arxiv.org/abs/2109.14575
8. Mehedi Masud, Poonam Mehta, Christoph A. Ternes, Mariam Tórtola, 'Non-standard neutrino oscillations: perspective from unitarity triangles', Journal of High Energy Physics 05 (2021) 171, arXiv:2103.11143, DOI: 10.1007/JHEP05(2021)171, https://arxiv.org/abs/2103.11143
9. T2K and NOvA Collaborations (S. Abubakar et al.), 'Joint neutrino oscillation analysis from the T2K and NOvA experiments', Nature 646 (2025) 818-824, DOI: 10.1038/s41586-025-09599-3, arXiv:2510.19888, https://arxiv.org/abs/2510.19888
10. Ana Maria Garcia Trzeciak, Hiroshi Nunokawa, Alexander A. Quiroga, 'Impact of unitarity violation on sensitivity of the leptonic CP phase at Hyper-Kamiokande and DUNE', Journal of High Energy Physics 11 (2025) 059, arXiv:2502.10873, DOI: 10.1007/JHEP11(2025)059, https://arxiv.org/abs/2502.10873
11. Ryuichiro Kitano, Joe Sato, Sho Sugama, 'Unitarity test of lepton mixing via energy dependence of neutrino oscillation', Journal of High Energy Physics 04 (2026) 091, arXiv:2508.20389, DOI: 10.1007/JHEP04(2026)091, https://arxiv.org/abs/2508.20389
12. Ryuichiro Kitano, Joe Sato, Sho Sugama, 'Impact of matter effects on the unitarity test of lepton mixing', arXiv preprint (2026), arXiv:2605.20859, https://arxiv.org/abs/2605.20859
13. Mathieu Guigue, Lorenzo Restrepo, 'Geometrical Constraints On Leptonic Unitarity Triangles', arXiv preprint (2026), arXiv:2601.18601, https://arxiv.org/abs/2601.18601
14. Shu Luo, 'Rescaled leptonic unitarity triangles and rephasing invariants', Nuclear Physics B 1024 (2026) 117340, arXiv:2511.16942, DOI: 10.1016/j.nuclphysb.2026.117340, https://arxiv.org/abs/2511.16942
15. Zhi-zhong Xing, 'Leptonic first-row correlation and unitarity waiting for further JUNO tests', Science Bulletin 71 (2026) 1899-1902, arXiv:2510.17583, DOI: 10.1016/j.scib.2026.03.043, https://arxiv.org/abs/2510.17583
16. Hyper-Kamiokande Proto-Collaboration (K. Abe et al.), 'Hyper-Kamiokande Design Report', arXiv preprint (2018), arXiv:1805.04163, https://arxiv.org/abs/1805.04163