Three Roads to the Mass Ordering: JUNO, DUNE, and ORCA

The two mass-squared differences that oscillation experiments measure — $\Delta m_{21}^2\approx 7.5\times 10^{-5}$ eV² from solar and reactor data, and $|\Delta m_{31}^2|\approx 2.5\times 10^{-3}$ eV² from atmospheric and accelerator data — pin down the relative spacing of the three neutrino mass eigenstates with great precision. They leave one binary ambiguity unresolved: is the heaviest state ${\nu }_3$, with the two lighter states clustered close together (the normal ordering, NO), or are the two heavier states ${\nu }_1$ and ${\nu }_2$ clustered close together, with the lightest being ${\nu }_3$ (the inverted ordering, IO)?

The question matters everywhere. The interpretation of neutrinoless double beta decay limits depends critically on the ordering — the effective Majorana mass spans a different allowed range in NO versus IO, with IO predicting a higher minimum value within reach of next-generation experiments. The kinematic mass limit from KATRIN constrains a combination of the three masses weighted by the PMNS elements; whether that combination is closer to $\sqrt{|\Delta m_{31}^2|}\approx 50$ meV or larger depends on the ordering. The cosmological mass sum, currently bounded near 100 meV, has different physical implications in the two cases. And essentially every theoretical model of neutrino mass — seesaw, Dirac, anarchy hypotheses — predicts different patterns for the two orderings.

Three completely different experimental strategies are converging on the answer simultaneously. JUNO uses reactor antineutrinos at an intermediate baseline to expose the ordering through interference between the two mass splittings. DUNE uses a long-baseline accelerator beam through the Earth to exploit matter effects that pull neutrinos and antineutrinos differently in the two orderings. ORCA, PINGU and the IceCube Upgrade use atmospheric neutrinos at GeV energies traversing the Earth, where matter-enhanced oscillation differs between the two orderings. Each route attacks the same binary question through wholly different physics. This post is about how each works, what their sensitivities are, and how they complement each other.

Why ordering is hidden in vacuum

In a pure-vacuum two-flavor oscillation, the probability of flavor conversion depends on ${\sin }^2(\Delta m^{2}L/4E)$, which is even under $\Delta m^2\to -\Delta m^2$. The sign is unobservable. Three-flavor mixing complicates this only slightly: the leading-order ${\nu }_{\mu }\to {\nu }_e$ appearance probability still depends mainly on $|\Delta m_{31}^2|$, with sub-leading interference terms that can in principle reveal the sign but require percent-level energy resolution to extract.

The way the sign becomes observable is through interaction. Two routes are available:

The first is matter effects. The Wolfenstein potential adds to the effective Hamiltonian with a sign that depends on whether one considers neutrinos or antineutrinos. When combined with the vacuum oscillation Hamiltonian, the resulting in-matter mass-eigenstate structure depends on whether $\Delta m_{31}^2$ is positive (NO) or negative (IO). A particular oscillation amplitude is enhanced for one sign and suppressed for the other, breaking the vacuum sign degeneracy. The matter-effect contribution grows with baseline, so long-baseline experiments and Earth-traversing atmospheric neutrinos are where matter effects dominate.

The second is interference between $\Delta m_{31}^2$ and $\Delta m_{32}^2$. These two mass-squared differences are not independent: $\Delta m_{31}^2-\Delta m_{32}^2=\Delta m_{21}^2$. Their interference produces a small modulation of the oscillation pattern whose phase depends on the ordering. The effect is subtle but observable in reactor antineutrino disappearance at the kilometer-scale baseline where both splittings contribute oscillation phase. This is the JUNO approach.

JUNO: reactor vacuum oscillation

The Jiangmen Underground Neutrino Observatory in southern China is a 20-kiloton liquid scintillator detector positioned 53 kilometers from each of two reactor complexes. At this baseline the survival probability for reactor electron antineutrinos shows both the slow solar-scale oscillation and the rapid atmospheric-scale oscillation, interfering in a way that depends on the ordering:

$P({\bar{\nu }}_e\to {\bar{\nu }}_e)=1-{\sin }^{2}2{\theta }_{12}{\cos }^4{\theta }_{13}{\sin }^2\left(\frac{\Delta m_{21}^{2}L}{4E}\right)-{\sin }^{2}2{\theta }_{13}{\sin }^2\left(\frac{\Delta m_{ee}^{2}L}{4E}\right)$

where the effective splitting $\Delta m_{ee}^2$ that the reactor channel measures is a combination of $\Delta m_{31}^2$ and $\Delta m_{32}^2$ weighted by the solar mixing angle, and the resulting fast-oscillation pattern carries a phase shift that depends on the ordering’s sign.

JUNO needs to resolve the fast-oscillation modulation at the level where the NO and IO predictions differ — requiring an unprecedented energy resolution of better than 3 per cent at 1 MeV. Achieving this drove the design: a 35.4-meter-diameter spherical acrylic vessel filled with 20 kilotons of linear-alkylbenzene-based scintillator, viewed by 17,612 large 20-inch photomultipliers plus 25,600 smaller 3-inch PMTs. The total photocathode coverage exceeds 75 per cent — far higher than any previous scintillator experiment.

The expected sensitivity is roughly 3-sigma separation between NO and IO after six years of operation, depending on systematics. JUNO is also the world’s most sensitive measurement of $\Delta m_{21}^2$ and ${\theta }_{12}$, plus a high-precision check on ${\theta }_{13}$ — multiple precision measurements come along with the ordering determination.

DUNE: long-baseline matter effects

The Deep Underground Neutrino Experiment runs a beam from Fermilab to the Sanford Underground Research Facility in South Dakota, a baseline of 1300 kilometers. At GeV energies through the Earth’s crust and mantle, the matter potential is substantial — the matter-effect contribution to the ${\nu }_{\mu }\to {\nu }_e$ appearance probability can shift the rate by ±30% depending on the ordering.

The DUNE far detector is a 40-kiloton liquid-argon TPC arranged in four 10-kiloton modules. The first module is under construction, with the LBNF beamline delivering 1.2-MW initial power, upgradable to 2.4 MW. The on-axis wide-band beam delivers neutrinos and antineutrinos across the first and second oscillation maxima simultaneously.

The matter-effect sign convention is clean: for NO, the ${\nu }_{\mu }\to {\nu }_e$ rate is enhanced relative to vacuum and the ${\bar{\nu }}_{\mu }\to {\bar{\nu }}_e$ rate is suppressed; for IO, the opposite. Compared to the much smaller CP-violation-induced asymmetry that the same experiments are also trying to measure, the matter-effect asymmetry is large and unambiguous given enough statistics.

Expected sensitivity is 5-sigma mass-ordering discovery within 1-2 years of full DUNE operation at modest beam power, with a strong dependence on the value of ${\delta }_{\mathrm{CP}}$ — some combinations of ordering and CP phase produce nearly degenerate observables that take longer to disentangle.

ORCA, PINGU, and atmospheric neutrinos through Earth

The third route is atmospheric neutrinos at GeV energies traversing the Earth. The matter effect through the Earth’s core and mantle produces an MSW-like resonance condition for one sign of $\Delta m_{31}^2$ in ${\nu }_{\mu }$ oscillation and the opposite sign in ${\bar{\nu }}_{\mu }$. The result is a characteristic pattern in the oscillation probability as a function of energy and zenith angle that depends on the ordering: enhanced ${\nu }_{\mu }\to {\nu }_e$ at certain energies for NO, enhanced ${\bar{\nu }}_{\mu }\to {\bar{\nu }}_e$ at the same energies for IO, with the asymmetry visible in the angular and energy distribution of atmospheric events.

Several detectors target this measurement. KM3NeT-ORCA is the Mediterranean-Sea-based oscillation research and cosmology detector that complements ARCA’s astrophysics measurements. ORCA is a dense IceCube-like array of optical sensors in the Mediterranean designed specifically for GeV-scale atmospheric oscillations. The IceCube Upgrade is an extension to IceCube’s deep ice array adding a denser sub-detector aimed at the same energy range. PINGU was an earlier proposed dense in-fill for IceCube; its concept has been folded into the IceCube Upgrade.

Each of these reaches 3-sigma mass-ordering sensitivity over a few years of operation, with the ultimate precision limited by atmospheric flux modelling uncertainties and by the energy and angular resolution of the detector at the relevant GeV scale.

Why three independent answers matter

Each route has its own dominant systematic and its own kind of degeneracy with other oscillation parameters. JUNO is essentially insensitive to ${\delta }_{\mathrm{CP}}$ but acutely sensitive to the absolute energy scale of its scintillator. DUNE is degenerate between certain combinations of ordering and CP phase, requiring high statistics to disentangle. ORCA’s sensitivity depends on the atmospheric flux model and on detector resolution at GeV energies.

A coherent answer requires either a single experiment reaching the discovery threshold or the combination of multiple consistent results from different experiments. The latter is more likely to happen first: combining a 3-sigma JUNO result, a 3-sigma DUNE result, and a 3-sigma ORCA result over the next 5-7 years produces a combined significance well above 5-sigma even if no single experiment crosses the threshold by itself. The combined picture also addresses concerns that any one experiment’s result might be artificially driven by a misunderstood systematic.

The pattern is reminiscent of how three independent measurements eventually pinned down the Hubble constant — different methods, different systematics, eventually converging on a coherent answer. The mass ordering problem will resolve the same way.

Cosmology and 0νββ as the prize

A confirmed mass ordering would immediately sharpen two other major neutrino-mass measurements.

Cosmology currently constrains the sum of neutrino masses at $\sum m_{\nu }<0.12$ eV from CMB plus large-scale structure. Future surveys (CMB-S4, Simons Observatory, DESI, Euclid) project sensitivity down to $\sim 0.04$ eV at the per-cent level. In the inverted ordering, the minimum sum is about $0.10$ eV — directly testable against future cosmological measurements. A cosmological mass sum below this minimum would force the normal ordering. Combined with a direct mass-ordering measurement from oscillation experiments, this becomes an independent consistency test of the cosmological model itself.

Neutrinoless double beta decay depends on the ordering through the effective Majorana mass $|m_{\beta \beta }|$. In the inverted ordering, $|m_{\beta \beta }|$ has a minimum value around 15 meV — within reach of the next generation of $0\nu \beta \beta$ experiments. In the normal ordering the minimum can vanish, with a possible cancellation among the three mass eigenstates that suppresses the effective Majorana mass arbitrarily. A confirmed inverted ordering would essentially guarantee that next-generation $0\nu \beta \beta$ experiments at multi-ton scale will either observe a signal or rule out the Majorana hypothesis.

A normal ordering, by contrast, would extend the discovery horizon by orders of magnitude and put $0\nu \beta \beta$ in a regime where only multi-decade exposures would access the relevant parameter space.

Summary

The neutrino mass ordering — whether ${\nu }_3$ is the heaviest mass eigenstate (normal) or the lightest (inverted) — is one of the major unsolved questions in neutrino physics, with implications for $0\nu \beta \beta$, cosmology, and BSM model building. Three completely independent experimental strategies are converging on the answer. JUNO exploits the interference between $\Delta m_{31}^2$ and $\Delta m_{32}^2$ in reactor antineutrino disappearance, with 3-sigma sensitivity after six years requiring sub-3% energy resolution. DUNE uses the matter-induced asymmetry between ${\nu }_{\mu }\to {\nu }_e$ and ${\bar{\nu }}_{\mu }\to {\bar{\nu }}_e$ appearance over its 1300-km baseline, with 5-sigma reach in 1-2 years of full operation. ORCA, the IceCube Upgrade, and PINGU exploit the zenith-angle pattern of atmospheric neutrinos at GeV energies through the Earth, with 3-sigma reach over a few years. Combined, these three independent measurements will resolve the ordering well above any single experiment’s individual threshold, with the answer expected within the next 5-7 years.