Mass Ordering

Oscillation experiments measure squared-mass differences, not absolute masses. The two independent differences are $\Delta m_{21}^2=m_2^2-m_1^2,\Delta m_{31}^2=m_3^2-m_1^2$ Solar and KamLAND data establish $\Delta m_{21}^2>0$: the ${\nu }_2$ eigenstate is heavier than ${\nu }_1$. The sign of $\Delta m_{31}^2$, however, remains to be determined. Two orderings are compatible with oscillation data alone:

Normal ordering (NO): $m_1<m_2<m_3$, $\Delta m_{31}^2>0$.

Inverted ordering (IO): $m_3<m_1<m_2$, $\Delta m_{31}^2<0$.

Global fits slightly prefer normal ordering, but inverted ordering is not yet excluded.

Why the sign is hard to fix

The vacuum oscillation probability depends on ${\sin }^2(\Delta m^{2}L/4E)$, which is blind to the sign of $\Delta m^2$. The ordering is extracted through matter effects: the forward-scattering of electron neutrinos on electrons in matter shifts effective mass eigenvalues, and the shift has opposite sign for neutrinos and antineutrinos. The resulting difference between $P({\nu }_{\mu }\to {\nu }_e)$ and $P({\bar{\nu }}_{\mu }\to {\bar{\nu }}_e)$ in long-baseline experiments depends on both ${\delta }_{CP}$ and the ordering — a degenerate two-parameter problem.

Three complementary strategies are being pursued.

Strategy 1: atmospheric neutrinos with matter effects

Atmospheric neutrinos travel up to the full Earth diameter. Those that traverse the mantle and outer core encounter substantial matter densities, leading to resonant MSW enhancement at specific energies. The resonance occurs for neutrinos in normal ordering and for antineutrinos in inverted ordering. Detectors capable of distinguishing $\nu$ from $\bar{\nu }$ (or at least measuring the ratio of event rates at different zenith angles and energies) can therefore determine the ordering. IceCube-DeepCore, the PINGU extension, ORCA in the Mediterranean, and the INO facility in India are all designed around this principle.

Strategy 2: long-baseline accelerator experiments

T2K (295 km, Japan) and NOvA (810 km, US) observe ${\nu }_{\mu }\to {\nu }_e$ and ${\bar{\nu }}_{\mu }\to {\bar{\nu }}_e$ appearance. The asymmetry $A=(N_{{\nu }_e}-N_{{\bar{\nu }}_e})/(N_{{\nu }_e}+N_{{\bar{\nu }}_e})$ depends on both ${\delta }_{CP}$ and the mass ordering: $A\sim \sin {\delta }_{CP}+\eta \cdot \text{(matter term)}$ where $\eta =+1$ for normal ordering and $-1$ for inverted. Current NOvA data favor normal ordering at ~2σ; T2K’s asymmetry prefers large CP violation but is statistics-limited. DUNE, with a 1300 km baseline through the mantle, will resolve the ambiguity at high significance in the 2030s.

Strategy 3: reactor neutrinos with precision oscillometry

JUNO in southern China uses 20 kiloton of liquid scintillator at a 53 km baseline from multiple reactor cores. At this distance the ${\bar{\nu }}_e$ spectrum is shaped by both $\Delta m_{21}^2$ (slow oscillation) and $\Delta m_{31}^2$ (rapid oscillation). The interference of the two frequencies produces a distinct spectral pattern whose shape depends on the sign of $\Delta m_{31}^2$. Unlike strategies 1 and 2, this approach does not rely on matter effects and is therefore free of degeneracy with ${\delta }_{CP}$. JUNO reported first oscillation-spectrum measurements in 2025 and is expected to determine the ordering at 3–4σ over roughly six years of data.

Combined global status

As of early 2026, global fits combining all oscillation data give a mild preference for normal ordering. The exact significance depends on which datasets are included and whether cosmological mass bounds are imposed. Within a few years, JUNO and long-baseline experiments are expected to deliver a definitive determination.

Implications

Neutrinoless double beta decay. The effective Majorana mass $⟨m_{\beta \beta }⟩$ has a lower bound set by the lightest mass and the mixing angles. In the inverted ordering, even a lightest mass of zero implies $⟨m_{\beta \beta }⟩\gtrsim 0.015$ eV, within reach of next-generation experiments. Normal ordering allows arbitrarily small $⟨m_{\beta \beta }⟩$ due to possible cancellations among the three mass eigenstates. A null result from LEGEND-1000 or nEXO at the inverted-ordering floor would therefore — combined with a confirmed inverted ordering — strongly disfavor Majorana neutrinos.

Cosmology. In inverted ordering the minimum $\sum m_i$ is about 0.10 eV; in normal ordering, about 0.06 eV. Tightening cosmological bounds below 0.10 eV would disfavor inverted ordering — and indeed recent DESI-era analyses have begun to put pressure on IO.

The two independent routes — oscillation ordering measurement and cosmological mass-sum constraint — therefore triangulate, with consistency checks across both.

Normal versus inverted ordering

Side-by-side comparison of the two allowed orderings. Adjust the lightest neutrino mass to see how the sum Σ m_i changes, with the KATRIN and cosmological bounds indicated for reference.

Frequently asked

Why is mass ordering important beyond particle physics?

The ordering affects the rate of neutrinoless double beta decay (inverted ordering guarantees an observable rate given current experiments; normal ordering does not) and the reach of cosmological neutrino mass bounds. It also influences the expected CP-violation signal magnitude in long-baseline oscillation experiments.