Interactive neutrino physics · Oscillations

How do three neutrino flavours oscillate?

Exact three-flavour oscillation probabilities in vacuum as a function of L/E, with the parameters of the Bari global analysis.

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Interactive figure

Three-flavour oscillation lab

Exact coherent three-flavour vacuum oscillations. The amplitude is summed over the mass eigenstates with the complex PMNS matrix and only then squared, so no curve is an independent sine wave and the three sum to one at every point.

L/E = 500 km/GeV P(ν_e) — P(ν_μ) — P(ν_τ) — Σ = 1.0000000000

Source flavour

L/E — logarithmic

11010²10³10⁴10⁵

Parameters in use

sin²θ₁₂: 0.303
sin²θ₁₃: 0.0223
sin²θ₂₃: 0.473
δ_CP: 1.20 π
δm²: 7.37 × 10⁻⁵ eV²
Δm²: 2.495 × 10⁻³ eV²
Δm²₃₁: —
ordering: normal

Every value above is our own, from Table I of Phys. Rev. D 111, 093006 (arXiv:2503.07752), in this group's conventions: δm² = m₂² − m₁² and Δm² = m₃² − (m₁² + m₂²)/2. They are not NuFIT's Δm²_3ℓ and cannot be compared with it at a glance, which is why Δm²₃₁, the splitting the phases actually need, is derived here rather than quoted: Δm²₃₁ = Δm² + δm²/2. Normal ordering throughout — the same analysis favours it at 2.2σ. Propagation is parameterised by L/E; the scan is a way of reading the curve, not physical time.

About this figure

Three-flavour oscillation lab

Level. Advanced undergraduate: quantum mechanics and the idea of flavour mixing.

The amplitude is summed over the mass eigenstates with the complex PMNS matrix and only then squared, so no curve is an independent sine wave and the three probabilities sum to one at every point. Choose the initial flavour, scan L/E, and read the flavour composition at any point.

Assumptions

Propagation in vacuum: no matter effects.
Normal ordering only — the ordering favoured by the same analysis, at 2.2σ.
Best-fit values of Table I of Phys. Rev. D 111, 093006: sin²θ₁₂ = 0.303, sin²θ₁₃ = 0.0223, sin²θ₂₃ = 0.473, δ/π = 1.20, δm² = 7.37×10⁻⁵ eV², Δm² = 2.495×10⁻³ eV², with Δm² = m₃² − (m₁² + m₂²)/2.

References

Capozzi et al., Phys. Rev. D 111, 093006 (2025)

Where it fits

Cite and reuse

A. Marrone, Three-flavour oscillation lab, interactive figure (2026), https://home.ba.infn.it/~marrone/interactive-oscillations.html.

@misc{Marrone:oscillations,
  author       = {Marrone, Antonio},
  title        = {Three-flavour oscillation lab},
  howpublished = {Interactive figure, \url{https://home.ba.infn.it/~marrone/interactive-oscillations.html}},
  year         = {2026}
}

Text and figures: CC BY 4.0. Code (assets/js/oscillation-lab.js): MIT licence. Use it in lectures and talks, adapt it, redistribute it — with attribution.