Is the PMNS Matrix Really Unitary? Tests of the Three-Flavor Closure

The PMNS matrix is usually drawn in the simplest possible way: a 3×3 array of complex numbers that connects the three neutrino flavor states with the three light mass eigenstates. Its unitarity — the property that its rows and columns are orthogonal and unit-normalized — is what guarantees the cleanness of three-flavor oscillation analysis. Three mixing angles, one Dirac phase, two Majorana phases: nine real numbers, no more, no less.

But this picture rests on an assumption that nothing the field has yet measured directly proves. If there exist additional heavy neutral leptons that mix with the three active ones — as the seesaw mechanism requires, and as low-scale variants might bring within experimental reach — then the matrix we measure at low energies is only the upper-left 3×3 corner of a larger object. That larger object is unitary; the 3×3 corner is not. Its rows and columns satisfy modified normalization conditions, and the standard oscillation formulas pick up small but specific correction terms.

Whether the 3×3 closure actually holds is a separate experimental question from any of the three-flavor measurements. Asking it requires combining precision oscillation data with information from charged-lepton processes, from electroweak precision tests, and from direct heavy-neutral-lepton searches. The current limits are at the few per cent level for most matrix entries; the next generation of experiments will sharpen them by an order of magnitude. This post is about what unitarity means here, how it gets tested, and why the assumption has been quietly built into every neutrino oscillation analysis for the past three decades.

The structure of a non-unitary submatrix

In a world with three light neutrinos and $n$ heavy ones, the full mixing matrix is a $(3+n)\times (3+n)$ unitary object $\mathcal{U}$ that we can partition as

$\mathcal{U}=\left(\begin{array}{cc}N& K\\ R& S\end{array}\right)$

The upper-left $3\times 3$ block $N$ relates the three light flavor states ${\nu }_e,{\nu }_{\mu },{\nu }_{\tau }$ to the three light mass eigenstates ${\nu }_1,{\nu }_2,{\nu }_3$. The upper-right block $K$ relates the three light flavors to the $n$ heavy mass eigenstates. The lower blocks describe the corresponding mappings on the heavy-flavor side. Full unitarity of $\mathcal{U}$ implies

$N{N}^{†}+K{K}^{†}=1_{3\times 3}$

so $N$ alone is unitary only if $K=0$ — that is, only if there is no mixing between light and heavy sectors. Any heavy-light mixing makes $N$ deviate from unitarity by

$N{N}^{†}=1-\eta ,\eta =K{K}^{†}$

The matrix $\eta$ is Hermitian and positive-definite, with entries that quantify how much each row or column of $N$ “leaks” into the heavy sector. Bounding $\eta$ is what unitarity tests do. The diagonal entries ${\eta }_{\alpha \alpha }$ tell us how much the flavor $\alpha$ couples to the heavy sector through its mass-eigenstate composition; the off-diagonal entries ${\eta }_{\alpha \beta }$ encode flavor-changing combinations of the same mixings.

For the natural seesaw scenarios at the grand-unification scale, $\eta$ is exponentially small — $K\sim m_D/M_R\sim 10^{-12}$, and $\eta \sim 10^{-24}$, far below any experimental reach. For low-scale seesaw variants with heavy neutrinos in the GeV-to-TeV range, $\eta$ can sit at the few-per-cent level and is testable. The unitarity test is therefore a window onto the low-scale corner of the seesaw parameter space.

Where non-unitarity shows up

Several distinct physical processes are sensitive to deviations from PMNS unitarity, and they probe different combinations of the $\eta$ matrix entries.

The first is short-baseline oscillation. If $N$ is not unitary, then at very short distances — where standard oscillation has not had time to develop — the flavor transition probabilities are not zero but proportional to $|{\eta }_{\alpha \beta }{|}^2$. A muon neutrino produced at a source could be detected as an electron neutrino at essentially zero distance, with probability set by $|{\eta }_{e\mu }{|}^2$. Experiments such as MicroBooNE, NOMAD, and the various short-baseline reactor experiments look for exactly this kind of zero-distance conversion and constrain the off-diagonal entries.

The second is survival probability normalisation. At any baseline, the survival probability $P({\nu }_{\alpha }\to {\nu }_{\alpha })$ in a non-unitary scenario is multiplied by an overall factor $(1-2{\eta }_{\alpha \alpha })$, reflecting the loss of probability to the heavy sector. This is what reactor antineutrino disappearance experiments and accelerator muon-neutrino disappearance measurements actually measure. The expected number of events in a perfectly known flux is reduced relative to the unitary prediction by ${\eta }_{\alpha \alpha }$, so precision flux predictions translate into bounds on the diagonal entries.

The third is electroweak precision tests at the Z pole. The effective Z coupling to a light flavor neutrino is rescaled by the same factor of $(1-{\eta }_{\alpha \alpha })$. The total invisible Z width measured at LEP, which determines the effective number of light neutrino species, would be reduced from three to $3-{\sum }_{\alpha }{\eta }_{\alpha \alpha }$ in a non-unitary scenario. Comparing the LEP measurement of $N_{\nu }=2.984\pm 0.008$ with the Standard-Model prediction of $3.000$ already implies ${\sum }_{\alpha }{\eta }_{\alpha \alpha }\lesssim 0.02$ — a global bound on the trace of $\eta$.

The fourth, and tightest in some directions, is charged-lepton flavor violation. The same heavy-light mixing that produces ${\eta }_{e\mu }$ also induces $\mu \to e\gamma$ and $\mu \to e$ conversion through loop diagrams. The non-observation of these processes places very stringent indirect bounds: $|{\eta }_{e\mu }|\lesssim 10^{-5}$ or better, depending on the assumed heavy-neutrino mass spectrum.

Current global bounds

Combining all the inputs in a global fit produces a constraint matrix on $\eta$ that looks, very roughly,

$|\eta |\lesssim \left(\begin{array}{ccc}2.5\times 10^{-3}& 6\times 10^{-5}& 3\times 10^{-3}\\ \cdot & 5\times 10^{-4}& 3\times 10^{-3}\\ \cdot & \cdot & 6\times 10^{-3}\end{array}\right)$

where the diagonal entries are bounded by survival-probability normalisations and Z-pole data, the off-diagonal entries are bounded by short-baseline searches and charged-lepton flavor violation, and the lower triangle is implied by Hermiticity. The ${\eta }_{e\mu }$ element is by far the most tightly constrained, because the $\mu \to e$ processes provide such powerful indirect leverage. The ${\nu }_{\tau }$-sector entries are the loosest, because direct measurements of tau-flavor flux are hardest.

The picture sits comfortably with all data: no entry of $\eta$ has been measured to be different from zero, and the upper bounds are consistent with no heavy-light mixing at all. The bounds rule out a substantial portion of low-scale seesaw parameter space and limit how strongly any heavy neutral lepton accessible at colliders can couple to the active sector.

Why precision oscillation experiments care

The next generation of long-baseline oscillation experiments — DUNE, Hyper-Kamiokande, JUNO — aims to measure the standard PMNS parameters at the per-cent level. At that precision the unitarity assumption is no longer free: a per-cent-level non-unitarity would shift the extracted values of ${\theta }_{23}$, ${\delta }_{\mathrm{CP}}$, and the mass ordering by amounts comparable to the experimental error.

Several recent analyses have begun to fold unitarity tests directly into the oscillation fits, allowing $\eta$ to vary and extracting the standard parameters and the unitarity violation simultaneously. The results so far are statistically consistent with unitarity, but the joint fits sometimes shift the central value of ${\delta }_{\mathrm{CP}}$ by a noticeable amount, hinting that the unitarity assumption may eventually need to be relaxed as precision improves. The cleanest test is to combine appearance and disappearance data with reactor measurements of ${\theta }_{13}$ — which is essentially independent of the unitarity question — and look for any inconsistency in the closure relations.

A related consequence is that the CP-violation discovery depends on unitarity. The phase ${\delta }_{\mathrm{CP}}$ that drives the oscillation asymmetry is, in a unitary three-flavor framework, the only source. In a non-unitary scenario, additional CP-violating phases enter through the heavy-light mixing and contribute to the measured asymmetry. Disentangling the two requires combining channels with different sensitivities to the standard versus non-unitary contributions.

What might break it

A clear non-unitarity signal would most likely appear in one of three places.

The first is in flux-versus-rate comparisons at high-precision reactor experiments such as JUNO. If the measured event rate falls short of the predicted rate by an amount that cannot be absorbed into reactor flux uncertainties, the deficit could be a unitarity violation — equivalently, oscillations into a sterile sector. The challenge is that reactor-flux predictions have their own per-cent-level uncertainties, so disentangling the two interpretations needs independent input from flux models or near-detector measurements.

The second is in the tau-flavor sector. Constraints on the third row and column of $\eta$ are the loosest because direct tau-neutrino measurements are statistics-limited. DUNE’s tau-appearance channel and IceCube’s astrophysical tau-neutrino flavor measurement will both improve the bounds substantially.

The third is at a direct discovery of a heavy neutral lepton at the LHC or at fixed-target experiments such as SHiP, which would simultaneously break unitarity at a measurable level. The mixing strength inferred from the discovery would predict specific deviations from PMNS unitarity that the oscillation experiments could then confirm.

Why this is foundational

PMNS unitarity is in some sense the boring foundation of three-flavor oscillation analysis — the assumption everyone makes and nobody usually questions. Testing it explicitly is part of the maturation of the field, in the same way that CKM unitarity tests in the quark sector have become a major part of flavor physics. Just as a measured deviation in the Cabibbo angle versus inclusive semileptonic decay rates would point to new physics in the quark sector, a measured deviation in PMNS unitarity would point unambiguously to heavy neutral leptons.

The current status — bounds at the per-cent level, consistent with unitarity, with substantial improvements expected in the next decade — places the field in a position to either confirm or refute the three-flavor closure with confidence. The answer will determine how the next generation of oscillation experiments interpret their data and whether the seesaw mechanism has a low-scale realisation within experimental reach.

Summary

The 3×3 PMNS matrix that appears in three-flavor oscillation analyses is only the upper-left block of a larger unitary matrix if heavy neutral leptons exist and mix with the three active neutrinos. The deviation from unitarity, parameterised by the matrix $\eta =K{K}^{†}$, is bounded by a combination of short-baseline oscillation, reactor antineutrino normalisations, Z-pole precision measurements, and charged-lepton flavor violation searches. The current global limits put the diagonal entries of $\eta$ at the level of $10^{-3}$ to $10^{-4}$ and the most-constrained off-diagonal at $10^{-5}$. Next-generation experiments push the precision into territory where the unitarity assumption can no longer be silently imposed: DUNE, Hyper-Kamiokande and JUNO will either confirm three-flavor closure to per-cent precision or detect the first hint of a heavier neutral lepton through a small but coherent pattern of deviations. Either result will reshape the interpretation of the standard mixing parameters.