Neutrino Mass Ordering in Three-Flavor Oscillations

• Neutrino mass ordering is the classification of neutrino mass states, distinguishing between normal (m1 < m2 < m3) and inverted (m3 < m1 < m2) hierarchies.
• Experimental probes employing long-baseline, atmospheric, reactor, and cosmological observations leverage matter effects and spectral distortions to constrain the ordering.
• Global analyses combine diverse data using statistical and Bayesian methods, addressing systematics and offering prospects for decisive future determination.

Neutrino mass ordering refers to the arrangement of the eigenvalues of the three neutrino mass states, denoted $m_1$, $m_2$, and $m_3$, which are mixed by the PMNS matrix to the flavor eigenstates observed in oscillation experiments. The central question is whether $m_3$ is the heaviest (normal ordering, NO: $m_1 < m_2 < m_3$) or the lightest (inverted ordering, IO: $m_3 < m_1 < m_2$). This property is a key open parameter in three-flavor neutrino physics, influencing leptonic flavor models, matter effects in oscillations, neutrinoless double beta decay interpretations, and the cosmological mass sum constraints.

1. Theoretical Framework of Neutrino Mass Ordering

The three-flavor neutrino oscillation paradigm links flavor states $\nu_\alpha$ to mass eigenstates $\nu_i$ via the unitary PMNS mixing matrix $U$, such that $\nu_\alpha = \sum_{i=1}^3 U_{\alpha i}\nu_i$. Oscillation data measure two independent squared-mass differences, typically $\Delta m_{21}^2$ (solar) and $|\Delta m_{31}^2|$ (atmospheric), but leave the sign of $\Delta m_{31}^2$ unresolved. The sign determines the ordering:

• Normal Ordering (NO): $\Delta m_{31}^2 > 0$ (i.e., $m_1 < m_2 < m_3$)
• Inverted Ordering (IO): $\Delta m_{31}^2 < 0$ (i.e., $m_3 < m_1 < m_2$)

The ordering affects the predictions for matter-induced resonance (MSW effect) in long-baseline and atmospheric oscillations, the allowed range for the effective neutrino mass in $0\nu\beta\beta$ decay ($m_{\beta\beta}$), and cosmological observables sensitive to the sum of masses $\Sigma m_i$ (Salas et al., 2018).

Neutrino oscillation probability formulas, especially in the presence of matter, manifest explicit ordering dependence. For example, in the two-flavor, constant-density approximation: $$\sin^2 2\theta_{13}^m = \frac{\sin^2 2\theta_{13}}{[\cos 2\theta_{13} \mp (2\sqrt{2}G_F N_eE)/\Delta m_{31}^2]^2 + \sin^2 2\theta_{13}}$$ with the sign “$\mp$” corresponding to neutrino/antineutrino and the mass ordering (Salas et al., 2018).

2. Experimental Probes of Mass Ordering

A suite of complementary experimental approaches target mass ordering:

• Long-Baseline Accelerator Experiments (T2K, NOvA, DUNE, T2HK) probe $\nu_\mu\to\nu_e$ appearance in matter. Matter effects yield order-dependent resonant enhancements or suppressions, as revealed by the energy and baseline dependence of the appearance probability. The ordering also affects the interplay with the CP phase $\delta$ and $\theta_{23}$ (Salas et al., 2018, Kelly et al., 2020).
• Atmospheric Neutrino Detectors (Super-Kamiokande, IceCube/DeepCore, PINGU, ORCA, INO) measure oscillation patterns over a range of baselines through the Earth, exploiting MSW resonance effects in the multi-GeV regime. High-statistics data and improved reconstruction (energy, angle, topology) are crucial. Statistical approaches in atmospheric neutrino analyses include binned likelihood fits, frequentist and Asimov methods, and careful marginalization of systematics (Aartsen et al., 2019).
• Medium-Baseline Reactor Experiments (JUNO, RENO-50) measure the $\bar\nu_e$ disappearance probability at 50–53 km, exploiting the beat between solar and atmospheric oscillation frequencies as an order-dependent phase shift or spectral distortion. The key ordering-sensitive observable is the effective atmospheric mass-squared difference,

$$\Delta m^2_{ee} = \cos^2\theta_{12}\,\Delta m^2_{31} + \sin^2\theta_{12}\,\Delta m^2_{32}$$

and its spectral imprint (Forero et al., 2021, Parke et al., 12 Apr 2024). JUNO will achieve $\sim 0.3\%$ precision on $\Delta m^2_{ee}$ within a few years.

• Neutrinoless Double Beta Decay ($0\nu\beta\beta$) probes the Majorana nature of neutrinos and the effective mass $m_{\beta\beta}$: $$m_{\beta\beta} = |\sum_k U_{ek}^2 e^{i\alpha_k} m_k|$$ The IO and NO schemes predict distinct lower bounds for $m_{\beta\beta}$, and high-precision measurements may distinguish the ordering hypothesis provided nuclear matrix element uncertainties and the Majorana/Dirac question are resolved (Ge et al., 2019).
• Cosmology (CMB, LSS surveys): Measurements of $\Sigma m_i$ set upper limits that, when combined with oscillation constraints, can disfavour IO if the cosmological limit falls below $0.1$ eV. Any future detection of $\Sigma m_i \simeq 60$ meV would strongly indicate NO (Hannestad et al., 2016, Salas et al., 2018).
• Supernova Neutrinos: Core-collapse SN events produce flavor– and time–dependent signals sensitive to matter effects inside the SN and thus to mass ordering. Detailed analyses of timing (neutronization burst), channel ratios, and flavor–energy–angle distributions in detectors (e.g., DUNE, Hyper-K, SNO+, HALO) can yield high statistical separation, especially leveraging multi-detector measurements (Brdar et al., 2022, Jesús-Valls, 2022, Saez, 2023).

3. Statistical Methodologies and Global Analyses

The mass ordering determination is a hypothesis-testing problem with discrete (NO/IO) non-nested options:

• Standard $\chi^2$ Difference: $$\Delta\chi^2 = \chi^2_{\mathrm{min},\mathrm{IO}} - \chi^2_{\mathrm{min},\mathrm{NO}}$$ is mapped to a significance under the assumption of Gaussian $\chi^2$ statistics. This method is “median-expected” and does not always reflect the true probability distributions (Stanco, 2016).
• Bayesian Evidence: The Bayesian approach computes evidences $Z_{\rm NO}$ and $Z_{\rm IO}$ from marginalized likelihood integrals. The Bayes factor $B_{\rm NO,IO} = Z_{\rm NO}/Z_{\rm IO}$ quantifies the relative preference. The choice of parameterization (e.g., three masses vs. lightest mass plus splittings) and prior (linear, logarithmic, “ordering-agnostic”) critically affect the inferred strength of evidence. Bayesian posteriors can be mapped to effective Gaussian significances for interpretability (Gariazzo et al., 2018, Gariazzo et al., 2022).
• Novel Test Statistics: Methods inspired by $CL_s$ in Higgs physics or likelihood ratios for Poisson event distributions provide more robust, less “overfitted” significance estimates, explicitly focusing on the rejection of one ordering given data. These techniques better capture event-count fluctuation properties and are particularly effective in channels with low statistics or strong degeneracies (Stanco, 2016).
• Synergies and Global Fits: Combining data from different oscillation channels (e.g., T2HK, JUNO, ICAL), cosmology, and $0\nu\beta\beta$ in coherent fits maximizes sensitivity and robustly separates degeneracies (especially for unfavorable values of $\delta_{\rm CP}$ or poor individual channel resolution). Synergy is strongest when disparate experiments prefer different “best-fit” values of the atmospheric splitting in the wrong ordering, increasing the global $\Delta\chi^2$ beyond the sum of individual contributions (Raikwal et al., 2022, Cabrera et al., 2020).
• Handling Systematics and Priors: The statistical significance depends on nuisance parameters, non-linear detector response (e.g., JUNO's energy calibration), and prior choices. Non–ordering-agnostic priors or inadequate systematic uncertainty treatment can artificially enhance or suppress apparent ordering sensitivity (Gariazzo et al., 2022, Forero et al., 2021).

4. Present Status and Combined Constraints

Recent analyses report:

• With current data, the global preference for NO is typically 2.5–3.5$\sigma$, with a Bayesian odds ratio varying strongly (few:1 up to $\sim$40:1) depending on the choice of mass parameterization and prior (Gariazzo et al., 2018, Gariazzo et al., 2022, Salas et al., 2018). The statistical significance is mainly driven by oscillation data; cosmological and $0\nu\beta\beta$ constraints enhance it modestly.
• The “sum rule” formalism, exploiting the precise measurement of both $\Delta m_{ee}^2$ in reactor (JUNO) and $\Delta m_{\mu\mu}^2$ in long-baseline (T2K/NOvA), can independently determine the mass ordering at 3$\sigma$ level soon after JUNO's startup—with discrimination at 5$\sigma$ anticipated with DUNE-scale accelerator experiments (Parke et al., 12 Apr 2024).
• Any claim of “decisive” evidence is highly prior- and method-dependent; ordering-agnostic frameworks (sampling over lightest mass and splittings, with minimally informative priors) yield more robust, data-driven results (Gariazzo et al., 2022).
• The status is sensitive to statistical fluctuations in appearance/disappearance data, as recently seen in T2K/NOvA where the preference for NO diminished when new data were included (Kelly et al., 2020).

5. Future Prospects and Experimental Sensitivities

Next-generation experiments promise to definitively resolve the ordering:

• JUNO/RENO-50: Expected to reach $<1\%$ uncertainty on $\Delta m^2_{ee}$, providing %%%%52$m_3$53%%%% sensitivity on its own, rising to $>5\sigma$ when combined with accelerator disappearance data, assuming central values remain stable (Forero et al., 2021, Parke et al., 12 Apr 2024).
• DUNE/T2HK/Hyper-K/INO/ORCA/PINGU: Individually expected to reach $>5\sigma$ determination within a decade by leveraging long baseline matter effects and high statistical power (Salas et al., 2018, Raikwal et al., 2022).
• Combined Global Fits: Simultaneous analysis of reactor, accelerator, atmospheric, and cosmological data will be essential for unambiguous resolution, exploiting differences in best-fit atmospheric splittings and the sum rule consistency (Cabrera et al., 2020, Salas et al., 2018).
• Supernova and Relic Neutrinos: Next galactic supernovae, with multi-channel and timing signatures (especially in DUNE and Hyper-Kamiokande), as well as large-scale detection of the diffuse supernova neutrino background (DSNB), may provide independent ordering evidence and even probe dynamical mass ordering scenarios (Jesús-Valls, 2022, Brdar et al., 2022, Perez-Gonzalez et al., 27 Jan 2025).

6. Phenomenological and Model-Building Implications

Determining the mass ordering is foundational for flavor model construction, constraining seesaw scenarios, and interpreting CP violation effects. The normal ordering is favored by indirect empirical relations between lepton mixing and mass ratios in dispersive analyses, which associate observed large lepton mixing with the $m_2^2/m_3^2 \gg m_s^2/m_b^2$ inequality, and with internal SM consistency requirements through cancellation conditions in box diagrams (Li, 2023). The choice of ordering informs $0\nu\beta\beta$ strategies—as in NO, the funnel region and sensitivity to Majorana phases is distinct, while in IO the “floor” for $m_{\beta\beta}$ is higher, providing a different experimental target (Ge et al., 2019).

The synergetic approach of leveraging both vacuum and matter-enhanced oscillations also functions as a precision consistency test of the three-neutrino paradigm: any discrepancy between inferred mass ordering from pure-vacuum (JUNO) and matter-dominated (long-baseline) measurements would motivate beyond–Standard Model interpretations, such as non-standard interactions (Cabrera et al., 2020, Capozzi et al., 2020).

7. Challenges, Open Issues, and Ongoing Developments

While the global evidence supports normal ordering, the strength of this conclusion is highly method- and prior-dependent, as well as subject to statistical fluctuations in the datasets. The presence of non-standard neutrino interactions (NSI), particularly in the $e$–$\tau$ sector, can obscure or erase the ordering sensitivity in accelerator appearance data, increasing the parameter degeneracies and necessitating careful accounting in global fits (Capozzi et al., 2020).

Ongoing and future research must continue to refine statistical frameworks (accounting for discrete hypothesis testing and full likelihood coverage), minimize systematic uncertainties (especially in spectral calibration and flux predictions), and maintain a consistent approach to prior selection across combined analyses. The integration of cosmological, terrestrial, and astrophysical probes—each sensitive to different aspects of the mass spectrum—affords a comprehensive path to settling the question of neutrino mass ordering and realizing its full implications in particle physics and cosmology.