Neutrino Mass Ordering: Which Neutrino Is Heaviest?

Q: What is the neutrino mass ordering?

The neutrino mass ordering is the unresolved question of how the three neutrino mass states are arranged from lightest to heaviest. Oscillation experiments measure the differences between squared masses rather than the masses themselves. The small solar splitting is known with its sign, so the state labelled two is heavier than the state labelled one. The large atmospheric splitting is known only in magnitude, about 2.5 times 10 to the minus 3 electronvolts squared, which leaves two possibilities. In normal ordering the lone state sits above the close pair, giving m1 smaller than m2 smaller than m3. In inverted ordering the lone state sits below, giving m3 smaller than m1 smaller than m2. Both arrangements reproduce every oscillation measurement made so far. Deciding between them requires matter effects, reactor interference patterns, cosmology, or double beta decay rather than ordinary vacuum oscillation.

Q: Why cannot neutrino oscillations tell us which neutrino is heaviest?

Because the dominant vacuum oscillation probability depends on the square of a sine. For a beam of energy E travelling a distance L, the oscillating term is sin squared of 1.27 times the mass-squared difference in electronvolts squared times L in kilometres divided by E in GeV. Replace a mass-squared difference by minus itself and the sine changes sign, but its square does not, so the measured probability is identical. Vacuum oscillation therefore reads the magnitude of the atmospheric splitting and nothing about its sign. The solar splitting escapes this because solar neutrinos travel through dense plasma, where the sign shows up as a genuine difference in survival probability. The atmospheric splitting needs the same trick on a much larger scale: hundreds or thousands of kilometres of rock, or an interference measurement sensitive to the detailed phase rather than a single oscillation amplitude.

Q: What is the minimum sum of neutrino masses in each ordering?

Setting the lightest state to exactly zero and using the measured splittings gives a hard floor. In normal ordering the masses become 0, 0.0087 and 0.0501 electronvolts, so the sum cannot fall below about 0.059 electronvolts. In inverted ordering the masses become 0.0491, 0.0498 and 0, because the close pair now sits on top, so the sum cannot fall below about 0.099 electronvolts. That factor of 1.7 is what makes cosmology relevant: the sum of neutrino masses suppresses the growth of small-scale structure, so measurements of galaxy clustering and the cosmic microwave background constrain it from above. A cosmological bound tight enough to sit below 0.099 electronvolts would exclude inverted ordering outright, without any accelerator or reactor involved. These are floors, not predictions: if the lightest state turns out to weigh 0.05 electronvolts, both orderings give a sum near 0.16 electronvolts and the cosmological route loses all discriminating power.

Q: Do current experiments prefer normal or inverted ordering?

There is a mild preference for normal ordering, but how mild depends entirely on which datasets you combine. The NuFIT 6.0 global analysis of data through September 2024 finds normal ordering favoured by a chi-squared difference of 6.1, roughly 2.5 standard deviations, once the tabulated Super-Kamiokande atmospheric likelihood is included. Drop that one input and the two orderings fit almost equally well, with inverted ordering formally ahead by a chi-squared difference of 0.6. Long-baseline accelerator data pull in competing directions: T2K and NOvA each favour normal ordering alone, yet agree with one another better under inverted ordering. Cosmological bounds from DESI add pressure against inverted ordering at roughly the three-sigma level, but that result depends on the dark energy model assumed. Nothing here amounts to a measurement, and the field still treats the question as open.

The neutrino mass ordering is still unknown, and four very different kinds of experiment are closing in on it. Here is what each one actually measures.

World of Physics Editorial October 21, 2026

neutrino mass orderingmass hierarchynormal orderinginverted orderingmsw effectjunodesineutrino massmatter effectsparticle physicscosmology

Table of Contents

Two Spacings and One Missing Sign

Here is an awkward gap in our knowledge of matter. Since 1998 we have known that neutrinos have mass, and that the three masses differ, because otherwise neutrino oscillations would not happen. We know one gap between them to better than two per cent. And we still cannot say which neutrino is the heaviest.

The reason sits inside what oscillation experiments measure. The flavour you detect depends on phase differences that go as Δm²L/(4E) for a beam of energy E after a distance L, where Δm² = mᵢ² − mⱼ² is a difference of squared masses. In practical units the oscillating factor is sin²(1.27 Δm²[eV²] × L[km] / E[GeV]).

Two such splittings govern everything we have seen. The small one, from solar and reactor neutrinos, is Δm²₂₁ = (7.49 ± 0.19) × 10⁻⁵ eV² in the NuFIT 6.0 global fit; JUNO has since reported (7.50 ± 0.12) × 10⁻⁵ eV² from 59.1 days of data, a 1.55 per cent uncertainty against KamLAND’s 2.5 per cent. The large splitting is some 35 times bigger: |Δm²₃ℓ| = 2.513 × 10⁻³ eV², known to about 0.8 per cent.

Take square roots and you get spacings, not masses. If the lightest state weighed nothing, the other two would land at 0.0087 eV and 0.0501 eV, and the absolute scale would still be open. A second, stranger freedom hides there too.

Normal, Inverted, and Why Nobody Knows

The sign of the small splitting is known. The sign of the large one is not, and the reason is almost embarrassingly simple: sin²(−x) = sin²(x). Flip Δm²₃₁ to minus itself and the dominant vacuum oscillation probability does not shift by one part in a million. Every accelerator and reactor measurement so far has read the magnitude and been blind to the sign.

Why is Δm²₂₁ different? Solar neutrinos do not travel in vacuum. They are born in dense plasma and climb out through a falling electron density, a journey sensitive to the sign. Combined with θ₁₂ ≈ 33.7° being below 45°, the Sun tells us ν₂ is the heavier of the close pair. Nothing comparable fixes where ν₃ belongs.

So two arrangements remain. In normal ordering the close pair sits at the bottom and ν₃ roughly 0.05 eV above it: m₁ < m₂ < m₃. In inverted ordering the pattern is turned over, ν₃ lightest and the close pair perched on top: m₃ < m₁ < m₂. Both fit every oscillation dataset. The inverted case keeps the labels, so the measured quantity becomes Δm²₃₂ = −2.484 × 10⁻³ eV² — negative, and that minus sign is the whole unsolved problem.

Breaking the Tie with Matter

The way out was found in the 1980s by Wolfenstein, Mikheyev and Smirnov. A neutrino crossing matter scatters coherently forward off the electrons it passes, and because only the electron flavour couples through the charged-current channel, electron neutrinos pick up an extra potential V = √2 GF nₑ that the other two do not. Antineutrinos get −V. It is tiny, but it adds to the mass terms in the oscillation Hamiltonian.

The crucial point is that this term enters as a sum, not a square. Writing A for it, the effective splitting in matter goes as √[(Δm²cos2θ − A)² + (Δm²sin2θ)²], which peaks when A ≈ Δm²cos2θ. That resonance needs A and Δm² to share a sign, and A is positive for neutrinos, negative for antineutrinos. Normal ordering therefore resonates the neutrinos, inverted ordering the antineutrinos: the sign vacuum oscillation discarded returns as a visible asymmetry.

For the atmospheric splitting and mantle density the resonance lands near 6 GeV, and core-crossing paths pick up a further parametric enhancement around 3 to 6 GeV from the density jump at the core-mantle boundary. Hence the interest in atmospheric neutrinos arriving from below, and the design of DUNE and Hyper-Kamiokande: DUNE fires a beam 1300 km through rock to build a measurable matter effect.

JUNO Reads the Interference Fringes

There is a second handle, and it needs no matter at all. A reactor antineutrino disappearance experiment at a medium baseline sees a survival probability containing the slow solar-scale oscillation with fast atmospheric-scale wiggles riding on top. Which of Δm²₃₁ or Δm²₃₂ governs those wiggles depends on the ordering, and the two differ by exactly Δm²₂₁. Over tens of kilometres that shifts the fringes relative to the slow envelope distinguishably.

JUNO, at 52.5 km under a hill in Guangdong, was built for this. The appeal is what it does not depend on: matter effects at that distance are negligible, and a disappearance channel carries no sensitivity to the CP-violating phase δCP and essentially none to the θ₂₃ octant. Matter-effect measurements must untangle the ordering from both unknowns at once, and those degeneracies are the main reason current data are ambiguous: T2K and NOvA each prefer normal ordering alone, yet fit each other better inverted. JUNO attacks the problem along an orthogonal axis, so its result does not merely add statistics, it breaks that degeneracy. Its projected median sensitivity exceeds 3σ after about six years, and 5σ when combined with KM3NeT/ORCA.

The Cosmological Squeeze

Neutrinos were relativistic in the early universe and streamed out of collapsing overdensities, damping small-scale structure in proportion to Σmν. That makes the relic neutrino background a measurable ingredient of cosmology: galaxy surveys plus the microwave background cap the mass sum.

The orderings have different floors. Set the lightest state to zero and normal ordering gives Σmν ≥ 0.0588 eV, inverted ordering Σmν ≥ 0.0989 eV, so a bound below 0.0989 eV kills inverted ordering outright. DESI DR2 baryon acoustic oscillations with Planck and ACT give Σmν < 0.0642 eV at 95 per cent confidence in ΛCDM, already past that threshold, with tension against the oscillation floors reported at 2.8σ (normal) and 3.3σ (inverted). A Feldman-Cousins treatment at the physical boundary tightens it to 0.053 eV, below normal ordering’s own minimum.

Treat this as live, not decided. The bound is model-dependent: let the dark energy equation of state evolve and it relaxes to about 0.16 eV, which disfavours nothing. A 2024 reanalysis using the Hillipop and Lollipop likelihoods with supernova data weakened the evidence against inverted ordering to moderate or weak. Cosmology is pushing hard in one direction while straining against a floor it cannot cross, and nobody knows yet whether that says something about neutrinos or about dark energy.

What Double Beta Decay Adds

If neutrinos are their own antiparticles, neutrinoless double beta decay can occur at a rate governed by the effective Majorana mass mₑₑ = |Σ Uₑᵢ² mᵢ|. Because those entries carry unknown Majorana phases, the three terms can partly cancel, and how much depends on the ordering.

In inverted ordering the two heavy states dominate with similar weights, so complete cancellation is impossible: mₑₑ stays in roughly the 18 to 48 meV band even with the lightest mass at zero. LEGEND-1000 is designed for a discovery sensitivity of 9 to 21 meV, and nEXO projects 6 to 27 meV, both subject to nuclear matrix elements. Normal ordering offers no such guarantee: near a lightest mass of 0.003 eV the terms cancel almost exactly and mₑₑ can pass through zero. A null result at 10 meV would squeeze inverted ordering hard and leave normal ordering comfortable — logic that runs one way only, and only if the neutrino is Majorana.

Why the Answer Changes Physics

The ordering is one of very few discrete, yes-or-no facts still missing from the neutrino sector, and discrete facts are unusually good at killing models. Flavour-symmetry constructions, seesaw textures and grand unified schemes often predict one ordering, so settling it would clear a large slice of model space in a single stroke, in a field where the basic picture is still being filled in.

It also feeds forward, fixing how deep double beta decay experiments must dig before a null result means anything, and setting the level crossings inside a collapsing star that shape the burst a supernova sends us. It is also the baseline against which anomalies like hints of sterile neutrinos must be judged.

Where It Actually Stands

Mildly in favour of normal ordering, and the word mildly is load-bearing. NuFIT 6.0, analysing data through September 2024, finds normal ordering preferred by Δχ² = 6.1, about 2.5σ, once the tabulated Super-Kamiokande atmospheric likelihood is folded in. Remove that one input and the orderings are effectively tied, inverted ordering formally ahead by Δχ² = 0.6. Present data in principle carry 2.5 to 3σ of discriminating power, which tensions between datasets erode.

A 2.5σ preference that evaporates when one dataset is dropped is not what physics records as an answer. The answer will come from JUNO’s fringes, DUNE’s trip through rock, Hyper-Kamiokande’s atmospheric sample and ORCA’s view through the Earth, probably from several agreeing. Until then the mass ordering stays the rare question with exactly two possible answers, both still on the table after nearly thirty years of looking.

Sources and Further Reading

Frequently Asked Questions

What is the neutrino mass ordering?

The neutrino mass ordering is the unresolved question of how the three neutrino mass states are arranged from lightest to heaviest. Oscillation experiments measure the differences between squared masses rather than the masses themselves. The small solar splitting is known with its sign, so the state labelled two is heavier than the state labelled one. The large atmospheric splitting is known only in magnitude, about 2.5 times 10 to the minus 3 electronvolts squared, which leaves two possibilities. In normal ordering the lone state sits above the close pair, giving m1 smaller than m2 smaller than m3. In inverted ordering the lone state sits below, giving m3 smaller than m1 smaller than m2. Both arrangements reproduce every oscillation measurement made so far. Deciding between them requires matter effects, reactor interference patterns, cosmology, or double beta decay rather than ordinary vacuum oscillation.

Why cannot neutrino oscillations tell us which neutrino is heaviest?

Because the dominant vacuum oscillation probability depends on the square of a sine. For a beam of energy E travelling a distance L, the oscillating term is sin squared of 1.27 times the mass-squared difference in electronvolts squared times L in kilometres divided by E in GeV. Replace a mass-squared difference by minus itself and the sine changes sign, but its square does not, so the measured probability is identical. Vacuum oscillation therefore reads the magnitude of the atmospheric splitting and nothing about its sign. The solar splitting escapes this because solar neutrinos travel through dense plasma, where the sign shows up as a genuine difference in survival probability. The atmospheric splitting needs the same trick on a much larger scale: hundreds or thousands of kilometres of rock, or an interference measurement sensitive to the detailed phase rather than a single oscillation amplitude.

What is the minimum sum of neutrino masses in each ordering?

Setting the lightest state to exactly zero and using the measured splittings gives a hard floor. In normal ordering the masses become 0, 0.0087 and 0.0501 electronvolts, so the sum cannot fall below about 0.059 electronvolts. In inverted ordering the masses become 0.0491, 0.0498 and 0, because the close pair now sits on top, so the sum cannot fall below about 0.099 electronvolts. That factor of 1.7 is what makes cosmology relevant: the sum of neutrino masses suppresses the growth of small-scale structure, so measurements of galaxy clustering and the cosmic microwave background constrain it from above. A cosmological bound tight enough to sit below 0.099 electronvolts would exclude inverted ordering outright, without any accelerator or reactor involved. These are floors, not predictions: if the lightest state turns out to weigh 0.05 electronvolts, both orderings give a sum near 0.16 electronvolts and the cosmological route loses all discriminating power.

Do current experiments prefer normal or inverted ordering?

There is a mild preference for normal ordering, but how mild depends entirely on which datasets you combine. The NuFIT 6.0 global analysis of data through September 2024 finds normal ordering favoured by a chi-squared difference of 6.1, roughly 2.5 standard deviations, once the tabulated Super-Kamiokande atmospheric likelihood is included. Drop that one input and the two orderings fit almost equally well, with inverted ordering formally ahead by a chi-squared difference of 0.6. Long-baseline accelerator data pull in competing directions: T2K and NOvA each favour normal ordering alone, yet agree with one another better under inverted ordering. Cosmological bounds from DESI add pressure against inverted ordering at roughly the three-sigma level, but that result depends on the dark energy model assumed. Nothing here amounts to a measurement, and the field still treats the question as open.