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arxiv: 2605.20859 · v1 · pith:WIPJE5XTnew · submitted 2026-05-20 · ✦ hep-ph · hep-ex

Impact of matter effects on the unitarity test of lepton mixing

Ryuichiro Kitano , Joe Sato , Sho Sugama This is my paper

Pith reviewed 2026-05-21 04:29 UTC · model grok-4.3

classification ✦ hep-ph hep-ex
keywords neutrino oscillationslepton mixingunitarity testmatter effectslong baseline experimentsT2HKneutrino factory
0
0 comments X

The pith

Spectral information from long-baseline neutrino experiments extracts mixing-matrix elements without assuming a parametrization, even after including matter effects, enabling a direct unitarity test.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that the oscillation probability in matter can be written as a sum of terms, each carrying a distinct combination of mixing-matrix elements multiplied by an energy-dependent coefficient. Because these coefficients vary differently with energy, measurements of the full spectrum in appearance channels allow the individual elements to be solved for directly. Once extracted, a particular combination of elements that must vanish in any unitary three-generation model can be checked for consistency with zero. The authors demonstrate that the most sensitive test comes from the CP-conjugate pair at T2HK together with the T-conjugate pair at a neutrino factory, even when standard matter effects are taken into account.

Core claim

By decomposing the matter-affected oscillation probabilities into energy-dependent terms and fitting the observed spectra, the elements of the lepton mixing matrix can be obtained without presupposing any particular parametrization; a linear combination of these elements that is identically zero for a unitary matrix can then be formed and tested for consistency with zero using data from T2HK and a future neutrino factory.

What carries the argument

Decomposition of the oscillation probability into sums of mixing-matrix products each multiplied by a coefficient with a unique energy dependence, allowing the products to be extracted as independent coefficients from spectral data.

If this is right

  • The CP-conjugate appearance channels at T2HK supply the strongest single-experiment constraint on the unitarity-violating quantity.
  • Including the T-conjugate pair at a neutrino factory further tightens the test by providing independent linear combinations of the same matrix elements.
  • In a four-generation extension the same fitting procedure yields a non-vanishing value whose magnitude scales with the size of the extra mixing angles.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The method could be applied to any future long-baseline setup whose beam energy range spans several oscillation maxima, provided the detector can resolve the spectrum finely enough.
  • If the extracted elements are found to satisfy unitarity to high precision, the same data set could be reanalyzed to place limits on non-standard interactions that would otherwise mimic unitarity violation.

Load-bearing premise

The oscillation probabilities remain expressible as linear combinations of a small number of mixing-matrix products whose coefficients have sufficiently different energy dependences even after standard matter effects are included.

What would settle it

A statistically significant non-zero value for the chosen unitarity-violating combination when the full energy spectra of both CP-conjugate and T-conjugate appearance channels are fitted simultaneously.

Figures

Figures reproduced from arXiv: 2605.20859 by Joe Sato, Ryuichiro Kitano, Sho Sugama.

Figure 1
Figure 1. Figure 1: Number of events in the T2HK experiment. The left and right figures show the [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Neutrino flux measured at the Hyper-Kamiokande detector. The anti-muon [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The values of ∆χ 2 max. These results obtained in analyses where events gener￾ated by the three-generation model are fitted using the energy-dependent functions of three￾generation model in matter. The non-linear parameters C5 = Xee 1 , C6 = Xee 2 , and C7 = Xee 3 are varied within their 3σ ranges around the best fit points in [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The values of ∆χ 2 max. These results obtained in analyses where events generated by the three-generation model are fitted using the energy-dependent functions of four-generation model in matter. The non-linear parameters C5 = Xee 1 , C6 = Xee 2 , and C7 = Xee 3 are varied within their 3σ ranges around the best fit points in [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Unitarity test of the three-generation events in matter in terms of [PITH_FULL_IMAGE:figures/full_fig_p018_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Unitarity test of the four-generation events in matter in terms of [PITH_FULL_IMAGE:figures/full_fig_p019_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Unitarity test of the three-generation events in matter in terms of [PITH_FULL_IMAGE:figures/full_fig_p021_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Unitarity test of the four-generation events in matter in terms of [PITH_FULL_IMAGE:figures/full_fig_p022_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Distribution of χ 2 min. These figures show some comparisons of various channel combinations. The figures show the case with Cid = 1.0, Pµ = −1.0, and Nµ = 1022. The black curves represent the probability density functions of the χ 2 distribution for each degree of freedom (see also the blue histogram in [PITH_FULL_IMAGE:figures/full_fig_p026_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Distribution of χ 2 min. These figures show some comparisons of various channel combinations. The figures show the case with Cid = 1.0, Pµ = −1.0, and Nµ = 1022 . The black curves represent the probability density functions of the χ 2 distribution for each degree of freedom. The blue histogram shows the χ 2 min distribution obtained by fitting the three-generation model in matter to the three-generation e… view at source ↗
read the original abstract

Testing the unitarity of the lepton mixing matrix, in a manner analogous to the unitarity tests of the CKM matrix in the quark sector, is an important step toward probing physics beyond the standard three-generation framework. In long baseline neutrino oscillation experiments, the formula of the oscillation probabilities can be written as a sum of terms with various combinations of the mixing-matrix elements, and their coefficients depend differently on energy. By observing the spectral information of long baseline experiments such as T2HK and a future neutrino factory at J-PARC with a $\nu_e$ beam, the elements of the mixing matrix can be extracted without assuming a specific parametrization of the mixing matrix. We investigate how such an extraction method can be applied to neutrino oscillations by taking into account matter effects, and discuss how one can test unitarity of the mixing matrix in future long baseline experiments. As a concrete example, we examine the unitarity test by using a four-generation model, where we look at a quantity which should be vanishing in a unitary model. Among possible combinations of measurements, the most powerful test can be provided from the energy spectra of the CP-conjugate appearance channels $\nu_\mu \to \nu_e$ and $\bar{\nu}_\mu \to \bar{\nu}_e$ at T2HK, as well as from the T-conjugate pair $\nu_\mu \to \nu_e$ and $\nu_e \to \nu_\mu$ available at neutrino factories.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The manuscript investigates extracting the elements of the lepton mixing matrix from energy spectra in long-baseline experiments (T2HK and a neutrino factory with a νe beam) without assuming a specific parametrization, while incorporating matter effects. It then tests unitarity via a quantity that vanishes for unitary mixing, using a four-generation model as an example. The authors conclude that the most powerful tests come from the CP-conjugate appearance channels (νμ → νe and ν̄μ → ν̄e) at T2HK and the T-conjugate pair (νμ → νe and νe → νμ) at neutrino factories.

Significance. If the extraction procedure remains valid once matter potentials are included, the work supplies a concrete, experiment-specific route to model-independent unitarity tests of the PMNS matrix. This is directly analogous to CKM unitarity tests and would be valuable for constraining or discovering additional neutrino generations or other new physics in upcoming facilities.

major comments (1)
  1. [Section deriving the oscillation probability formula with matter effects] The central extraction method relies on writing the oscillation probability as a linear combination of terms whose coefficients (functions of the vacuum Uαi) multiply distinct energy-dependent factors. In constant-density matter the effective Hamiltonian is H = (1/(2E))U diag(0,Δm²₂₁,Δm²₃₁,…)U† + diag(V,0,0,…); its eigenvalues λk(E) and eigenvectors become E-dependent. Please show explicitly (in the section deriving the probability formula with matter effects) that the probability can still be decomposed into independent coefficients times separable E-dependent functions, or provide numerical evidence that any resulting degeneracies do not prevent unique extraction of the U elements from realistic spectra.
minor comments (1)
  1. [Abstract] The abstract states that the formula 'can be written as a sum of terms with various combinations of the mixing-matrix elements' but does not indicate whether this decomposition is performed in vacuum or in matter; a brief clarifying sentence would help readers.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive evaluation of our work's significance and for the constructive major comment. We address the point raised below and will strengthen the manuscript accordingly.

read point-by-point responses

  1. Referee: [Section deriving the oscillation probability formula with matter effects] The central extraction method relies on writing the oscillation probability as a linear combination of terms whose coefficients (functions of the vacuum Uαi) multiply distinct energy-dependent factors. In constant-density matter the effective Hamiltonian is H = (1/(2E))U diag(0,Δm²₂₁,Δm²₃₁,…)U† + diag(V,0,0,…); its eigenvalues λk(E) and eigenvectors become E-dependent. Please show explicitly (in the section deriving the probability formula with matter effects) that the probability can still be decomposed into independent coefficients times separable E-dependent functions, or provide numerical evidence that any resulting degeneracies do not prevent unique extraction of the U elements from realistic spectra.

    Authors: We thank the referee for highlighting this subtlety. In the presence of constant-density matter, the effective eigenvalues and eigenvectors are indeed energy-dependent, so the probability does not factorize in exactly the same manner as in vacuum. In our analysis we incorporated matter effects by numerically evaluating the full oscillation probabilities (via the effective Hamiltonian) for each energy bin while treating the vacuum mixing-matrix elements Uαi as the fit parameters. To confirm that this procedure still permits unique extraction, we will add to the revised manuscript a dedicated numerical study. Using simulated spectra for T2HK and the neutrino factory (including realistic statistics and the matter potential), we will demonstrate that the fitted vacuum U elements recover the input values with no significant degeneracies. This study will be placed immediately after the probability formula section. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation remains self-contained

full rationale

The paper's central method decomposes oscillation probabilities into energy-dependent coefficients to extract mixing-matrix elements without parametrization, then constructs a unitarity test from a vanishing quantity in a four-generation extension. This relies on standard Hamiltonian diagonalization in matter and independent channel combinations (e.g., CP- and T-conjugate spectra), without reducing any load-bearing step to a self-definition, fitted input renamed as prediction, or self-citation chain. The extraction and test are presented as falsifiable against external data and benchmarks, with no quoted equations showing the vanishing quantity defined circularly from the same fit. The skeptic concern about E-dependent frequencies affects correctness but does not create a definitional loop within the paper's logic.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, the analysis rests on the standard decomposition of neutrino oscillation probabilities into mixing-matrix terms whose energy dependence differs; no free parameters, invented entities, or non-standard axioms are mentioned.

axioms (1)
  • domain assumption Neutrino oscillation probabilities in matter can be written as sums of terms involving products of mixing-matrix elements with distinct energy-dependent coefficients.
    Invoked when stating that spectral information allows extraction of matrix elements.

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Reference graph

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