Evidence for Normal Neutrino Ordering with Combined Atmospheric Neutrino Experiments

This paper investigates whether combining atmospheric-neutrino measurements can determine the neutrino mass ordering and improve measurements of oscillation parameters. The central question is whether the third neutrino mass state is heavier than the other two, known as normal ordering, or lighter, known as inverted ordering. Atmospheric neutrinos provide an independent alternative to accelerator experiments because they span a broad range of energies and travel distances, including paths through Earth’s mantle and core. Matter effects along these paths produce distinct oscillation signatures for the two orderings. The authors jointly analyze Super-Kamiokande data from its five pre-gadolinium operating phases, eight years of IceCube-DeepCore data, and measurements from KM3NeT/ORCA’s initial six-detection-unit configuration, ORCA6. Super-Kamiokande contributes 6,511.3 live days and an exposure of 484.2 kiloton-years, while ORCA6 contributes 5,828 selected events collected over 510 days. The experiments offer complementary strengths: Super-Kamiokande provides detailed event classification and low-energy sensitivity, DeepCore supplies high statistics for measuring atmospheric oscillations, and ORCA6 adds an independent measurement extending to lower energies. The methodological advance is a simultaneous fit of event samples rather than a simple combination of published likelihoods. Atmospheric-flux and neutrino-interaction uncertainties are treated as fully correlated across experiments, while detector-response uncertainties remain independent. This approach allows data from one detector to constrain uncertainties affecting the others. The analysis uses a binned Poisson likelihood, numerical three-flavor neutrino propagation through an Earth-density model, and 143 nuisance parameters. Solar oscillation parameters and the reactor-measured mixing parameter sin²θ₁₃ are fixed. For Super-Kamiokande, the authors develop a high-statistics simulation tuned to released distributions and incorporate corrected energy information supplied by the collaboration. Assuming normal ordering, the combined fit yields an atmospheric mass-squared splitting of 2.44 × 10⁻³ eV², with uncertainties of +0.06 × 10⁻³ and −0.04 × 10⁻³ eV², corresponding to a reported relative precision of 2.2%. The mixing parameter sin²θ₂₃ is 0.52, with uncertainties of +0.03 and −0.04 and a reported precision of 6.1%. Maximal mixing remains allowed, and the preference for the upper octant is negligible, with a χ² difference of only 0.3. Compared with a simple likelihood combination, modeling shared systematic uncertainties narrows the mass-splitting and mixing-parameter intervals by 19% and 9%, respectively. The atmospheric-only constraint on the CP phase is 1.30π, with uncertainties of +0.24π and −0.28π. This sensitivity comes predominantly from Super-Kamiokande’s sub-GeV samples. Values near 0.3π are disfavored by more than 2.7 standard deviations, but the result does not constitute a definitive observation of CP violation. The main result is a χ² difference of 11.0 favoring normal ordering. Because the ordering hypotheses are non-nested, the authors calibrate the significance using simulated pseudoexperiments. Only two of 4,397 inverted-ordering trials produce results as extreme as the observed data, yielding a p-value of 0.000455, equivalent to approximately 3.32 standard deviations. Limited simulation statistics produce a 68% uncertainty range of 3.08–3.60 standard deviations, while the alternative CLs criterion yields a significance of 2.55 standard deviations. Important limitations include tension between experiments, finite Monte Carlo statistics, and the omission of an ORCA6 energy-scale uncertainty. The baseline combined goodness-of-fit p-value is 0.011, improving to approximately 0.12 in a study accounting for finite simulation statistics. The authors conclude that atmospheric data provide evidence, but not definitive proof, for normal ordering. They estimate that adding reactor information could bring the sensitivity close to four standard deviations, while emphasizing the need for a properly calibrated joint analysis.

Atmospheric Neutrinos Offer New Evidence for the Neutrino Mass Order

Neutrinos are among nature’s most elusive particles, yet experiments are steadily uncovering how they behave. One major question remains unresolved: how are their three mass states arranged? In “Evidence for Normal Neutrino Ordering with Combined Atmospheric Neutrino Experiments,” Miaochen Jin, Carlos A. Argüelles, Pablo Fernández-Menéndez, and Iván Martínez-Soler combine observations from three experiments to investigate this puzzle. Their analysis favors normal ordering, while also showing why that evidence should not yet be considered definitive.

What Does Neutrino Mass Ordering Mean?

Neutrinos change flavor as they travel, a phenomenon called oscillation. These changes depend on differences between the squared masses of three neutrino states, along with mixing parameters that describe how the states combine. Researchers know that two mass states form a relatively close pair. The unresolved question is whether the third state sits above that pair, called normal ordering, or below it, called inverted ordering.

Atmospheric neutrinos offer a natural way to investigate. Produced when cosmic rays interact with Earth’s atmosphere, they arrive with a wide range of energies and travel distances. Some reach detectors from overhead; others pass through Earth’s mantle or core. Interactions with matter along these routes modify oscillations differently for the two orderings, leaving potentially distinguishable patterns in the data.

Three Experiments, Complementary Strengths

The study brings together Super-Kamiokande’s five pre-gadolinium operating phases, eight years of IceCube-DeepCore observations, and data from the initial six-detection-unit configuration of KM3NeT/ORCA, known as ORCA6. Each contributes a different perspective:

• Super-Kamiokande supplies detailed event classification and sensitivity at low energies, with 6,511.3 live days and an exposure of 484.2 kiloton-years.
• DeepCore contributes high statistics that help pin down atmospheric oscillation parameters.
• ORCA6 adds an independent measurement extending to lower energies, using 5,828 selected events collected over 510 days.

The value of combining them goes beyond simply increasing the event count. Their different capabilities can help distinguish genuine oscillation patterns from uncertainties in neutrino production, interactions, and detector behavior.

A Joint Fit, Not Just Added Results

The methodological centerpiece is a simultaneous fit of event samples, rather than a simple combination of published likelihoods. Atmospheric-flux and neutrino-interaction uncertainties are treated as fully correlated across experiments, while detector-response uncertainties remain independent. This lets observations in one detector constrain uncertainties that also affect the others.

The analysis uses a binned Poisson likelihood, numerical three-flavor propagation through an Earth-density model, and 143 nuisance parameters. Solar oscillation parameters and the reactor-measured mixing parameter sin²θ₁₃ are fixed. For Super-Kamiokande, the researchers develop a high-statistics simulation tuned to released distributions and incorporate corrected energy information supplied by the collaboration.

Sharper Measurements of Neutrino Oscillations

Assuming normal ordering, the combined fit places the atmospheric mass-squared splitting at 2.44 × 10⁻³ eV², with uncertainties of +0.06 × 10⁻³ and −0.04 × 10⁻³ eV². The reported relative precision is 2.2%. The mixing parameter sin²θ₂₃ is 0.52, with uncertainties of +0.03 and −0.04, corresponding to a reported precision of 6.1%.

Maximal mixing remains allowed. The preference for the upper octant—values of θ₂₃ above 45 degrees—is negligible, with a χ² difference of just 0.3. Importantly, accounting for shared systematic uncertainties narrows the mass-splitting and mixing-parameter intervals by 19% and 9%, respectively, compared with a simple likelihood combination.

A Limited Window on CP Violation

The atmospheric-only fit also constrains the CP phase to 1.30π, with uncertainties of +0.24π and −0.28π. This sensitivity comes predominantly from Super-Kamiokande’s sub-GeV samples. Values near 0.3π are disfavored by more than 2.7 standard deviations, but this result is not a definitive observation of CP violation.

How Strong Is the Ordering Evidence?

The main result is a χ² difference of 11.0 favoring normal ordering. Because normal and inverted ordering are non-nested hypotheses, the researchers calibrate its significance using simulated pseudoexperiments rather than relying on a straightforward χ² conversion.

Only two of 4,397 inverted-ordering trials produce results as extreme as the observed data. That gives a p-value of 0.000455, equivalent to approximately 3.32 standard deviations. Limited simulation statistics leave a 68% uncertainty range of 3.08–3.60 standard deviations. An alternative criterion, CLs, gives a more conservative significance of 2.55 standard deviations.

Promising Evidence, With Important Caveats

The analysis has limitations, including tension between experiments, finite Monte Carlo statistics, and an omitted ORCA6 energy-scale uncertainty. The baseline combined goodness-of-fit p-value is 0.011, improving to approximately 0.12 in a study accounting for finite simulation statistics. These checks matter when judging how confidently the combined model describes the observations.

The conclusion is therefore measured: atmospheric neutrinos provide evidence for normal ordering, not definitive proof. The authors estimate that adding reactor information could bring sensitivity close to four standard deviations, but emphasize the need for a properly calibrated joint analysis. The study demonstrates how combining complementary detectors—and carefully connecting their shared uncertainties—can extract more information from neutrinos passing through Earth.