The θ₂₃ Octant Problem: Is Atmospheric Mixing Truly Maximal?

Of the three mixing angles in the PMNS matrix, ${\theta }_{23}$ is the only one that could plausibly be exactly maximal. The reactor angle ${\theta }_{13}\approx 8.5°$ is clearly non-zero and clearly non-maximal. The solar angle ${\theta }_{12}\approx 33.5°$ is large but distinct from 45°. Only ${\theta }_{23}$, the atmospheric mixing angle, sits so close to 45° that current experiments cannot definitively place it on one side or the other.

This ambiguity — known as the octant problem — is not merely a technical limitation. If ${\theta }_{23}$ is exactly 45°, nature is telling us something about a symmetry. If it is not, then we need to know which side of 45° it is on, because the octant is degenerate with the CP phase ${\delta }_{CP}$ in the leading-order oscillation probabilities. Understanding the octant is therefore a prerequisite for precisely measuring CP violation in the lepton sector.

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The Atmospheric Mixing Angle

The atmospheric mixing angle ${\theta }_{23}$ controls the transformation between the muon neutrino and the third mass eigenstate ${\nu }_3$:

$|{\nu }_{\mu }⟩\approx -\cos {\theta }_{23}|{\nu }_2⟩+\sin {\theta }_{23}|{\nu }_3⟩$ $|{\nu }_{\tau }⟩\approx \sin {\theta }_{23}|{\nu }_2⟩+\cos {\theta }_{23}|{\nu }_3⟩$

(where the second mass eigenstate ${\nu }_2$ and the first ${\nu }_1$ both contribute with small coefficients set by ${\theta }_{13}$ and ${\theta }_{12}$, which are suppressed here for clarity).

The dominant disappearance channel for accelerator muon neutrinos — the channel first measured by Super-Kamiokande in 1998 and precisely studied by K2K, MINOS, T2K, and NOvA — is:

$P({\nu }_{\mu }\to {\nu }_{\mu })\approx 1-{\sin }^{2}2{\theta }_{23}\cdot {\sin }^2\left(\frac{\Delta m_{32}^{2}L}{4E}\right)$

in vacuum at leading order. The survival probability depends on ${\sin }^{2}2{\theta }_{23}$, not on ${\sin }^2{\theta }_{23}$ or ${\cos }^2{\theta }_{23}$ individually. This is the crux of the degeneracy: ${\sin }^{2}2{\theta }_{23}$ is symmetric about ${\theta }_{23}=45°$ — a value of ${\sin }^2{\theta }_{23}=0.4$ (lower octant) gives the same ${\sin }^{2}2{\theta }_{23}$ as ${\sin }^2{\theta }_{23}=0.6$ (upper octant).

Near maximal mixing, where ${\sin }^{2}2{\theta }_{23}\approx 1$, the disappearance channel is almost entirely insensitive to which octant ${\theta }_{23}$ falls in. Breaking the degeneracy requires additional information.

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What Maximal Mixing Would Mean

The value ${\theta }_{23}=45°$ is not arbitrary from a theoretical perspective. It corresponds to equal amplitudes for the muon and tau neutrino components in the third mass eigenstate:

$|{\nu }_3⟩=\frac{1}{\sqrt{2}}|{\nu }_{\mu }⟩-\frac{1}{\sqrt{2}}|{\nu }_{\tau }⟩$

This is a maximal superposition — ${\nu }_3$ has exactly equal probability of interacting as ${\nu }_{\mu }$ or ${\nu }_{\tau }$. It is the leptonic analogue of the $|K_0⟩$ meson, which is a maximal superposition of $|d\bar{s}⟩$ and $|\bar{d}s⟩$ states.

The condition ${\theta }_{23}=45°$ is preserved by a discrete $\mu$-$\tau$ exchange symmetry: a transformation that simultaneously swaps ${\nu }_{\mu }\leftrightarrow {\nu }_{\tau }$ and $\mu \leftrightarrow \tau$ in the Lagrangian. If the neutrino mass matrix commutes with this symmetry, then ${\theta }_{23}=45°$ is a consequence, and ${\theta }_{13}=0$ is predicted at the same time (since $\mu$-$\tau$ symmetry forces the $e$-row of the PMNS matrix to be the same in both the $\mu$ and $\tau$ columns).

The observation of ${\theta }_{13}\ne 0$ (first measured by Daya Bay in 2012) already tells us that $\mu$-$\tau$ symmetry is not exact. But it could be approximate — broken only by the charged lepton masses or by a small perturbation in the neutrino sector — with ${\theta }_{23}$ very close to but not exactly 45°. Many discrete flavor symmetry models (based on groups such as $A_4$, $S_4$, $\Delta (96)$) predict ${\theta }_{23}$ near but not at maximal, with the deviation encoding the symmetry-breaking pattern.

Alternatively, some models predict ${\theta }_{23}$ away from 45° for reasons having nothing to do with $\mu$-$\tau$ symmetry — arising instead from renormalization group running of the mixing angles between the GUT scale and the laboratory scale, or from accidental cancellations in the seesaw mass matrix.

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Breaking the Octant Degeneracy

Three observational handles can distinguish upper from lower octant.

Appearance Channels at Accelerator Experiments

The leading-order appearance probability for ${\nu }_{\mu }\to {\nu }_e$ at a baseline $L$ is:

$P({\nu }_{\mu }\to {\nu }_e)\approx {\sin }^2{\theta }_{23}\cdot {\sin }^{2}2{\theta }_{13}\cdot {\sin }^2\left(\frac{\Delta m_{31}^{2}L}{4E}\right)+\cdots$

Here the coefficient in front of the dominant oscillating term is ${\sin }^2{\theta }_{23}$, not ${\sin }^{2}2{\theta }_{23}$. This dependence is linear in ${\sin }^2{\theta }_{23}$ at leading order, breaking the symmetry that renders the disappearance channel degenerate.

The complication is that the appearance probability also depends on ${\delta }_{CP}$ through interference terms that are suppressed relative to the leading term. At a single baseline and energy, the sensitivity to ${\theta }_{23}$ in the appearance channel is correlated with ${\delta }_{CP}$: a smaller ${\sin }^2{\theta }_{23}$ can be compensated by a different value of ${\delta }_{CP}$ to give the same appearance rate. Two experiments at different baselines and energies break this correlation, because the relative size of the leading and interference terms is baseline-dependent.

Tau Neutrino Appearance

A muon neutrino that oscillates into ${\nu }_{\tau }$ produces a tau lepton in a charged-current interaction. The probability $P({\nu }_{\mu }\to {\nu }_{\tau })$ at atmospheric baselines depends on ${\cos }^2{\theta }_{23}$, complementary to the ${\sin }^2{\theta }_{23}$ dependence in ${\nu }_e$ appearance. Super-Kamiokande and IceCube/DeepCore have detected tau neutrino appearance in the atmospheric neutrino flux; the statistics are not yet sufficient for a precision octant determination, but the channel provides an independent handle.

Atmospheric Neutrinos at Long Baselines

Atmospheric neutrinos traverse up to $L\approx 12,700$ km through the Earth. At energies around 1–10 GeV, matter effects from the Earth’s mantle and core modify the oscillation probabilities in ways that depend on both the mass ordering and ${\theta }_{23}$. The asymmetry between neutrino and antineutrino atmospheric rates — accessible to magnetized detectors or through statistical separation — provides sensitivity to the octant that is partially independent of the accelerator experiments.

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The T2K and NOvA Results

T2K (Tokai to Kamioka, 295 km baseline, 0.6 GeV peak energy) and NOvA (NuMI Off-Axis ${\nu }_e$ Appearance, 810 km baseline, 1.8 GeV peak energy) are the two leading long-baseline accelerator experiments currently operating.

T2K measures both ${\nu }_{\mu }$ disappearance and ${\nu }_e$/${\bar{\nu }}_e$ appearance. Its oscillation analyses have consistently found a best-fit value of ${\sin }^2{\theta }_{23}$ above 0.5, favouring the upper octant:

${\sin }^2{\theta }_{23}=0.561_{-0.031}^{+0.021}\text{(T2K 2023 result, normal ordering)}$

The 68% credible interval extends from roughly 0.52 to 0.58, so maximal mixing (${\sin }^2{\theta }_{23}=0.5$) is disfavoured at roughly $1.5\sigma$. T2K also prefers ${\delta }_{CP}$ near $-\pi /2$ (maximal CP violation in the lepton sector).

NOvA also measures both appearance and disappearance channels. Its results have shown more variation across analyses. Early NOvA results (2020) preferred the lower octant and ${\delta }_{CP}$ near $\pi /2$ or $3\pi /2$, while later analyses with more data showed an expanded allowed region that encompasses both the upper octant near maximal and the lower octant. The NOvA 2023 result:

${\sin }^2{\theta }_{23}\in [0.45,0.58]\text{(68\% CL, normal ordering)}$

includes a best-fit near maximal but with significant excursion into the lower octant.

The two experiments are statistically compatible — their allowed regions overlap — but their central values differ. Part of this difference arises because the T2K and NOvA baselines are sensitive to the ${\delta }_{CP}$-octant correlation in different ways. T2K’s shorter baseline has a peak at the first oscillation maximum, where the ${\delta }_{CP}$ dependence of the appearance probability is large. NOvA’s longer baseline has sensitivity to matter effects that sharpen the mass ordering determination but also alter the ${\delta }_{CP}$ sensitivity. When both experiments are fit simultaneously, the combined result favors the upper octant and ${\delta }_{CP}$ near $-\pi /2$, but not at high significance.

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The Correlation Between Octant and δCP

A subtle but important feature of the oscillation phenomenology is that ${\sin }^2{\theta }_{23}$ and ${\delta }_{CP}$ are partially degenerate in the appearance channel at any fixed baseline. To see why, consider the full leading-order expression for ${\nu }_{\mu }\to {\nu }_e$ appearance in matter:

$P({\nu }_{\mu }\to {\nu }_e)\approx A{\sin }^2{\theta }_{23}+B\cos {\theta }_{23}\sin {\theta }_{23}\cos {\delta }_{CP}+C\cos {\theta }_{23}\sin {\theta }_{23}\sin {\delta }_{CP}$

where $A$, $B$, $C$ are functions of ${\theta }_{12}$, ${\theta }_{13}$, the mass splittings, $L$, and $E$. The $\cos {\theta }_{23}\sin {\theta }_{23}$ factor in the interference terms means that, at fixed appearance rate, reducing ${\sin }^2{\theta }_{23}$ (going to lower octant) while adjusting ${\delta }_{CP}$ to compensate is possible — but only within a range set by the relative size of $A$ versus $B$ and $C$.

Different baselines shift the ratio of $A$ to the interference terms, so two experiments see the degeneracy with different orientations in the $({\sin }^2{\theta }_{23},{\delta }_{CP})$ plane. The combination of T2K and NOvA partially breaks the degeneracy, and the combination of either with atmospheric neutrino data or reactor constraints breaks it further.

This correlation is also why the preferred value of ${\delta }_{CP}$ from T2K and NOvA depends on the assumed mass ordering and octant, and why joint fits produce a more constrained picture than the individual experiments suggest.

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Prospects: DUNE and Hyper-Kamiokande

The next generation of long-baseline experiments is designed to resolve both the mass ordering and the octant and to measure ${\delta }_{CP}$ with sufficient precision to test CP violation at $3\sigma$ or better.

DUNE (Deep Underground Neutrino Experiment, 1300 km baseline, liquid argon near and far detectors) will measure both appearance and disappearance channels with very high statistics. The long baseline enhances matter effects, which helps determine the mass ordering rapidly. The combination of normal and inverted ordering analysis, plus $\nu$ and $\bar{\nu }$ runs, breaks the octant-${\delta }_{CP}$ degeneracy at the 1300 km baseline more cleanly than at shorter baselines.

DUNE’s projected sensitivity to the octant — derived from the combined appearance + disappearance analysis — is expected to reach $3\sigma$ for ${\sin }^2{\theta }_{23}$ values deviating from 0.5 by as little as $\pm 0.03$, after seven years of running. For ${\sin }^2{\theta }_{23}=0.55$ (consistent with the T2K best fit), DUNE expects $>5\sigma$ octant determination within the first five years.

Hyper-Kamiokande (295 km, water Cherenkov, eight times the fiducial volume of Super-K) complements DUNE at a shorter baseline. The T2K baseline at 295 km has different sensitivity to the ${\delta }_{CP}$-octant correlation than the 1300 km DUNE baseline, and HK’s large statistics enable a very precise determination of ${\sin }^{2}2{\theta }_{23}$ from the disappearance channel. HK is also sensitive to atmospheric neutrinos, providing an independent octant determination from multi-GeV events traversing the Earth’s interior.

The combination of DUNE and HK — planned to run simultaneously throughout the 2030s — is expected to achieve a joint sensitivity that definitively resolves the octant for $|{\sin }^2{\theta }_{23}-0.5|>0.02$, and to determine ${\delta }_{CP}$ to better than $15°$ precision across most of the parameter space. If ${\theta }_{23}$ is truly maximal, both experiments will measure ${\sin }^2{\theta }_{23}$ consistent with 0.5 within their respective precisions, and the question will become whether theory can account for the proximity to — but deviation from — the symmetric point.

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Implications for Flavor Symmetries

The experimental outcome of the octant determination will have significant implications for theoretical models of lepton mixing.

If ${\theta }_{23}$ is found to be exactly or very nearly maximal ($|{\sin }^2{\theta }_{23}-0.5|<0.01$), the case for an underlying $\mu$-$\tau$ symmetry — or a near-symmetry with small breaking — will be strengthened. Models based on $A_4$, $S_4$, or other non-Abelian discrete groups that naturally predict near-maximal atmospheric mixing will gain credibility, and model builders will focus on understanding what breaks the symmetry at the 1% level.

If ${\theta }_{23}$ is found to be clearly non-maximal — say, ${\sin }^2{\theta }_{23}=0.42$ or $0.58$ — the $\mu$-$\tau$ symmetry hypothesis is effectively ruled out at the tree level, and the explanation for large atmospheric mixing must come from a different mechanism: accidental enhancement from the charged lepton sector, radiative corrections, or renormalization group running from the seesaw scale. The magnitude and sign of the deviation from 45° would then constrain the symmetry-breaking pattern.

Either outcome is informative. The octant problem is not a question with an uninteresting answer.

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The atmospheric mixing angle ${\theta }_{23}$ has been measured for nearly three decades, ever since the 1998 Super-Kamiokande atmospheric neutrino result revealed that muon neutrinos disappear with close to maximal amplitude. The question of whether the mixing is exactly maximal or merely approximately so has remained open throughout — a testament to how difficult it is, in neutrino physics, to distinguish a symmetry from a numerical coincidence.

DUNE and Hyper-Kamiokande will not merely sharpen the measurement of ${\theta }_{23}$. They will determine, for the first time, which octant nature has chosen — and in doing so, provide the clearest test yet of whether the pattern of lepton mixing reflects an underlying symmetry or an accidental arrangement of parameters.

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Related reading: The PMNS matrix and its parameters covers the full mixing matrix. Measuring δCP with long-baseline experiments discusses how the octant determination feeds into the CP phase measurement. Discrete flavor symmetries and neutrino mixing explores the theoretical models that predict maximal or near-maximal θ₂₃.