Mu-Tau Symmetry: Why the PMNS Matrix Looks Almost Symmetric

The PMNS matrix that governs neutrino oscillation has a striking structural feature when written in the standard parameterisation. The atmospheric mixing angle ${\theta }_{23}$ is consistent with $45°$ within current precision, and the reactor mixing angle ${\theta }_{13}$, while no longer consistent with zero (Daya Bay’s 2012 result settled that), is much smaller than the other two angles. The third angle, the solar ${\theta }_{12}$, is moderately large at about $34°$. The overall structure is

${\theta }_{12}\approx 34°,{\theta }_{13}\approx 8.6°,{\theta }_{23}\approx 45°$

Comparing this with the quark mixing matrix, where all three CKM angles are small (1 to 13 degrees), the lepton sector is qualitatively different — two of its three angles are large, and one of them is essentially maximal. The question of why this is the case has occupied theoretical physics since the 1998 Super-Kamiokande discovery of atmospheric oscillation.

One promising answer is μ-τ symmetry: the conjecture that the second and third generations of leptons enter the underlying flavor structure on an equal footing, with the muon-tau interchange leaving the mass matrix invariant. Exact μ-τ symmetry predicts ${\theta }_{23}=45°$ exactly and ${\theta }_{13}=0$ exactly. The observed angles are close to but not exactly these values, so the symmetry can hold approximately but must be broken at some level to accommodate ${\theta }_{13}\ne 0$. The smallness of the symmetry-breaking effects — a few degrees — is the most striking quantitative feature of the lepton mixing pattern.

This post is about what μ-τ symmetry predicts, how the observed angles fit in, and what the symmetry tells us about underlying flavor structure.

The pattern in numbers

The current best-fit values for the three mixing angles, from global oscillation fits of solar, atmospheric, reactor, and accelerator data:

${\theta }_{12}=33.45°\pm 0.75°,{\theta }_{13}=8.62°\pm 0.12°,{\theta }_{23}=49.1°\pm 1.0°$

The atmospheric angle has a small octant ambiguity: ${\sin }^2{\theta }_{23}$ is consistent with values both slightly above and slightly below $0.5$. The latest T2K and NOvA data slightly prefer the upper octant (${\theta }_{23}>45°$), but the lower octant (${\theta }_{23}<45°$) is not excluded.

The reactor angle is now known to 1.4% precision, primarily from the Daya Bay measurement.

Compared to the quark mixing angles — ${\theta }_{12}^{\mathrm{CKM}}\approx 13.0°$, ${\theta }_{13}^{\mathrm{CKM}}\approx 0.2°$, ${\theta }_{23}^{\mathrm{CKM}}\approx 2.4°$ — the lepton angles are systematically larger by factors of three to twenty.

What exact μ-τ symmetry predicts

The PMNS matrix in its standard parameterisation can be written

$U=\left(\begin{array}{ccc}{c}_{12}{c}_{13}& s_{12}{c}_{13}& s_{13}{e}^{-i\delta }\\ -s_{12}{c}_{23}-c_{12}{s}_{23}{s}_{13}{e}^{i\delta }& c_{12}{c}_{23}-s_{12}{s}_{23}{s}_{13}{e}^{i\delta }& s_{23}{c}_{13}\\ s_{12}{s}_{23}-c_{12}{c}_{23}{s}_{13}{e}^{i\delta }& -c_{12}{s}_{23}-s_{12}{c}_{23}{s}_{13}{e}^{i\delta }& c_{23}{c}_{13}\end{array}\right)$

A discrete μ-τ symmetry is the requirement that the neutrino mass matrix in the charged-lepton-mass-eigenstate basis is invariant under the swap ${\nu }_{\mu }\leftrightarrow {\nu }_{\tau }$. In this symmetric limit, the mass matrix takes the form

$M_{\nu }=\left(\begin{array}{ccc}A& B& B\\ B& C& D\\ B& D& C\end{array}\right)$

with three independent complex parameters. Diagonalising this matrix (assuming real, or equivalently CP-conserving, entries) produces

${\theta }_{23}=45°,{\theta }_{13}=0°$

without any additional assumptions. The solar angle ${\theta }_{12}$ is undetermined in pure μ-τ symmetry — it depends on the specific values of $A,B,C,D$ — and could in principle take any value.

Two specific symmetric limits have been historically important:

The tri-bimaximal mixing pattern, proposed by Harrison, Perkins and Scott in 2002, additionally fixes ${\theta }_{12}$ to satisfy ${\sin }^2{\theta }_{12}=1/3$, giving ${\theta }_{12}\approx 35.3°$. Tri-bimaximal mixing is the most predictive symmetric scenario and matched the data well through the late 2000s.

The bimaximal mixing pattern, alternatively, fixes ${\sin }^2{\theta }_{12}=1/2$, giving ${\theta }_{12}=45°$. This was always less well-matched to the data than tri-bimaximal.

Where the data is

Tri-bimaximal mixing predicts ${\theta }_{13}=0$. The Daya Bay, RENO, and Double Chooz reactor measurements decisively measured ${\theta }_{13}\ne 0$ at the level of $8.6°$ in 2012. Pure tri-bimaximal mixing is therefore excluded by data.

What survives is approximate μ-τ symmetry — the conjecture that the symmetric limit holds approximately, with small symmetry-breaking corrections of order ${\theta }_{13}$ producing the observed deviations. In this picture:

• ${\theta }_{23}$ is close to but not exactly maximal, with small deviations of order ${\theta }_{13}^2\approx (0.15{)}^2\approx 0.02$ — consistent with the observed near-maximal value.
• ${\theta }_{13}$ is small and non-zero, set by the symmetry-breaking parameter.
• ${\theta }_{12}$ is constrained by the additional structure of the flavor symmetry, with tri-bimaximal predicting $35.3°$ and the data showing $33.5°$ — a difference of $1.8°$ that the symmetry-breaking corrections can plausibly account for.

The structural pattern — two large angles, one moderate, one small — fits an approximately symmetric lepton sector with breaking effects at the level of ${\theta }_{13}$. The question is then what specific flavor symmetry produces this pattern, and what other predictions does the symmetry make beyond the mixing angles themselves.

Models implementing the symmetry

Several discrete flavor groups have been proposed as the underlying flavor symmetry that produces approximately μ-τ-symmetric mixing.

$A_4$ symmetry is the smallest non-abelian group with three irreducible representations, matching the three lepton generations. In $A_4$ models, the neutrino mass matrix takes the tri-bimaximal form at leading order, with the additional structure of ${\theta }_{12}=35.3°$ predicted automatically. Symmetry-breaking corrections produce the observed deviations, with the specific pattern depending on the $A_4$-breaking scalar field content.

$S_4$ symmetry, the next larger discrete group containing $A_4$ as a subgroup, produces a broader family of mixing patterns that include tri-bimaximal as a sub-case. $S_4$ models have more flexibility in predicting ${\theta }_{12}$, ${\theta }_{13}$, and ${\delta }_{\mathrm{CP}}$, but also fewer specific predictions.

$Z_2\times Z_2$ reflection symmetry is a simpler discrete symmetry approach that imposes the μ-τ interchange directly. The predictions are similar to the tri-bimaximal case but with ${\theta }_{12}$ undetermined by the symmetry alone.

Modular flavor symmetries are a more recent approach that uses the discrete modular group $SL(2,\mathbb{Z})$ to generate the flavor structure from a single complex modulus. The predictions depend on the modulus value but produce specific PMNS patterns that include approximate μ-τ symmetry as a special case.

Each approach makes specific predictions for the Dirac CP phase ${\delta }_{\mathrm{CP}}$ and for the relationship between ${\theta }_{23}$ and ${\theta }_{13}$ that can be tested by current and future oscillation experiments.

The CP-phase prediction

A common feature of approximately μ-τ-symmetric models is a specific prediction for the Dirac CP phase ${\delta }_{\mathrm{CP}}$. Several models predict ${\delta }_{\mathrm{CP}}$ to be close to $\pm \pi /2$ — corresponding to maximal CP violation in the leptonic direction. Others predict CP-conserving values ${\delta }_{\mathrm{CP}}=0$ or $\pi$, depending on whether the symmetry breaking is purely CP-conserving or includes a CP-violating spurion.

The current best-fit value of ${\delta }_{\mathrm{CP}}$ from T2K plus NOvA combined fits is around $3\pi /2$ or $-\pi /2$ (the two are equivalent), favouring large CP violation in the direction ${\nu }_e$ enhancement over ${\bar{\nu }}_e$. This is consistent with the predictions of approximately μ-τ-symmetric models that include CP-violating spurions in the breaking sector.

DUNE and Hyper-K’s projected ${\delta }_{\mathrm{CP}}$ measurements will discriminate between maximal CP violation ($\pm \pi /2$) and CP conservation ($0,\pi$) at the $5\sigma$ level for substantial parameter regions. A confirmed ${\delta }_{\mathrm{CP}}\approx -\pi /2$ would strongly favour μ-τ-symmetric models with CP-breaking spurions.

Approximate vs. exact: what the small parameters mean

Exact μ-τ symmetry predicts a zero ${\theta }_{13}$, but the observed value is $8.6°$. The natural question is whether ${\theta }_{13}$ is “small” relative to expectations from random anarchy.

Under the anarchy hypothesis, the lepton mixing angles are random uniform draws from the allowed range. The probability of obtaining ${\theta }_{13}<10°$ by chance is approximately ${\sin }^2(10°)\approx 0.03$ — about 3%, small but not absurdly small. The probability of also obtaining ${\theta }_{23}$ within $5°$ of maximal at the same time is approximately another factor of three smaller. So the combined probability of the observed pattern by anarchic chance is around 1%.

This is not a definitive disproof of anarchy, but it is enough motivation for the field to explore structured-symmetry alternatives, of which approximate μ-τ symmetry is the most natural.

Tests in the next decade

The next generation of oscillation experiments will provide several tests of approximate μ-τ symmetry:

Precision ${\theta }_{23}$ measurement by DUNE and Hyper-K will reduce the uncertainty to below half a degree. Models predict either exactly maximal ${\theta }_{23}=45°$ or specific small deviations at the level of ${\theta }_{13}^2\approx 0.02$. A measured ${\theta }_{23}$ within $0.5°$ of maximal would be consistent with most symmetric models; a measured value farther away would rule out the simplest implementations.

Precision ${\delta }_{\mathrm{CP}}$ measurement will discriminate between symmetric models with different CP-spurion structure. The current data favouring $-\pi /2$ is suggestive but not yet decisive.

Determination of the octant of ${\theta }_{23}$ — whether it is above or below $45°$ — will distinguish models that predict the upper octant from those that predict the lower octant. The two outcomes correspond to different patterns of symmetry breaking and are currently slightly preferred toward the upper octant.

Solar-angle precision from JUNO will tighten ${\theta }_{12}$ to the per-cent level, where the deviation from tri-bimaximal ${\sin }^2{\theta }_{12}=1/3$ can be measured precisely and compared to symmetric-model predictions.

Summary

The PMNS mixing matrix has two large angles (${\theta }_{12}\approx 34°$, ${\theta }_{23}\approx 45°$) and one moderate angle (${\theta }_{13}\approx 8.6°$), a pattern strikingly different from the all-small CKM angles in the quark sector. Approximate μ-τ symmetry, the conjecture that the underlying flavor structure is invariant under interchange of the second and third lepton generations, predicts exact ${\theta }_{23}=45°$ and ${\theta }_{13}=0$ at the symmetric limit, with the small observed deviations arising from symmetry-breaking corrections at the level of ${\theta }_{13}$ itself. Specific discrete flavor symmetries — $A_4$, $S_4$, $Z_2\times Z_2$, modular flavor groups — implement μ-τ symmetry with additional structure that predicts specific values for ${\theta }_{12}$ and ${\delta }_{\mathrm{CP}}$. The current data are consistent with approximate μ-τ symmetry, with the favoured ${\delta }_{\mathrm{CP}}$ value near $-\pi /2$ matching the predictions of symmetric models with CP-violating spurions. DUNE, Hyper-K, and JUNO will deliver per-cent-level precision on all three mixing angles and the CP phase over the next decade, providing definitive tests of the approximate-symmetric picture and discriminating between the competing flavor models. The lepton mixing pattern, if it ultimately confirms approximate μ-τ symmetry, will be one of the cleaner empirical hints about the underlying flavor structure of the Standard Model.