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Neutrinos · D05 · Mixing and measurement

Neutrino Flavor Propagation

Why does the chance of finding a neutrino in each flavor rise and fall as it travels? Because of phase. A neutrino of a given flavor is a mix of mass states, and each one gains phase at a different rate as it travels. To get the chance of finding a flavor, add the complex amplitudes first and then square. The changing relative phase makes each flavor probability rise and fall with distance, while the probabilities still add up to one.

Follow the phases of the three mass states as a neutrino travels, scan the distance, then sample ideal flavor measurements. Compare three-flavor interference with no mixing, equal masses and the two-flavor limit.

Interactive modelNeutrino Flavor Propagation
Preview distance—\text{—}
Electron-flavor probability—\text{—}
Muon-flavor probability—\text{—}
Tau-flavor probability—\text{—}
Normalization residual—\text{—}

Vacuum prediction with ideal sampling

A probability is not a count

Preparing the model…

Solid cursor: preview. Dashed cursor: recorded trial. The scan uses the recorded parameters; editing controls does not alter it.

N=256N=256

Read the first 32 ideal outcomes

Each trial starts from a freshly prepared state. All three flavors are detected with the same ideal acceptance. The counts leave out interaction thresholds, cross sections, energy resolution and detector efficiency.

Physics tutorial

The phase changes; the mass components stay coherent

BackgroundA weak interaction creates a neutrino with a definite flavor. As it travels, its mass components pick up different phases, and that changes the amplitude for each flavor at the next measurement.

Why it mattersThe same picture links reactor disappearance, atmospheric oscillations and accelerator appearance experiments. This vacuum model shows the interference on its own, before matter and detector physics are added.

Start with the essentials

Focus question
Can the total probability remain one while the probability of the original flavor nearly vanishes?
One-sentence intuition
Add the complex amplitudes first, then square their magnitudes to get probabilities. Each independently prepared neutrino gives one outcome drawn from those probabilities.

Core mathematical model

Amplitude to arrive as each flavor

Aα→β(L)=∑i=13UβiUαi∗e−iϕi,Pαβ=∣Aα→β∣2A_{\alpha\to\beta}(L)=\sum_{i=1}^{3}U_{\beta i}U_{\alpha i}^{*}e^{-i\phi_i},\qquad P_{\alpha\beta}=|A_{\alpha\to\beta}|^2

Rows of the unitary mixing matrix label flavors and columns label mass states. The phase of the first mass state is common to every term and is removed. For antineutrinos the mixing matrix is replaced by its complex conjugate.

Relative phase in practical units

ϕi=2.533865 Δmi12eV2LkmGeVE\phi_i=2.533865\,\frac{\Delta m_{i1}^{2}}{\mathrm{eV}^{2}}\frac{L}{\mathrm{km}}\frac{\mathrm{GeV}}{E}

Distance is in kilometres and energy in GeV; mass-squared differences are written in the usual natural-unit notation. Only differences between the mass squares affect this vacuum calculation.

Two-flavor limit and total probability

Pμτ=sin⁡2(2θ23)sin⁡2 ⁣(Δm312L4E),∑βPαβ=1P_{\mu\tau}=\sin^2(2\theta_{23})\sin^2\!\left(\frac{\Delta m_{31}^2L}{4E}\right),\qquad \sum_\beta P_{\alpha\beta}=1

This expression uses natural units. It holds when the other two mixing angles and the small mass splitting are zero, as in the Two-flavor limit preset; the electron flavor then drops out.

Common difficulties

Three arrows are not three particles

Typical misconceptionA neutrino splits into three classical particles that travel independently.

Better mental modelThe arrows are coherent amplitudes of one quantum state, drawn in a chosen phase convention. Moving the distance preview does not measure the same travelling neutrino again and again.

Flavor probability is not detection efficiency

Typical misconceptionAt the chosen energy, a real detector would see every tau-flavor outcome the model predicts.

Better mental modelThe sampler is an ideal measurement that treats every flavor equally. Identifying a flavor in a real detector depends on interaction thresholds, cross sections, backgrounds and instrument response, and the model leaves all of these out.

Run the experiment

  1. 01

    Find disappearance

    Choose Three flavors, calculate, then scan the distance and watch the three probability bars.

    What to observe: The original flavor can become rare while the total probability stays at one. The arrows on the wheels keep their length; only their phases turn.
  2. 02

    Turn off one ingredient

    Try No mixing and Equal masses. Scan again.

    What to observe: Both cases stop the oscillations, for different reasons. A mass difference without mixing does not change the flavor.
  3. 03

    Keep distance over energy fixed

    Record at 600 km and 1.2 GeV. Then use 1200 km and 2.4 GeV with the same seed.

    What to observe: The probabilities at the sampling distance and the sampled outcomes match. The full curves against distance are stretched, because the energy changed.
  4. 04

    Compare particle and antiparticle

    Return to Three flavors, record, then enable Antineutrino and record again. Repeat with the CP phase set to zero.

    What to observe: With a nonzero CP phase some appearance probabilities differ between neutrinos and antineutrinos; at zero phase they agree in vacuum. This compares two model settings; it is not experimental evidence.
  5. 05

    Separate probability from frequency

    Keep all physical settings fixed and vary the seed or number of trials.

    What to observe: The predicted curve does not change, but the observed frequencies fluctuate. More trials shrink the typical fluctuation, though not necessarily the error of every single run.

Related experiments and articles

Continue exploring

How flavor change is predicted and tested

Propagation through vacuum and matter predicts a flavor probability. How do solar, atmospheric, reactor and beam experiments test it?

  1. You are hereNeutrino Flavor PropagationStart with coherent vacuum propagation and ideal flavor trials.
  2. LabNeutrino Matter ResonanceFollow a rotating matter Hamiltonian and compare slow with sudden conversion.
  3. LabSolar Neutrino Flavor ObservatorySeparate electron flavor from total active solar flux with three reaction responses.
  4. LabAtmospheric Neutrino Zenith ExperimentTurn Earth-crossing baselines into direction-dependent event counts.
  5. LabReactor Near–Far SpectraFit the same near and far counts while testing calibration and normalization.
  6. LabLong-Baseline CP ObservatoryCompare both accelerator beam polarities on frozen spectra and separate CP phase, matter and response assumptions.

Neutrinos · Reading path

Make a source and test its journey

Articles on this path that use this experiment, followed by the other experiments they suggest.

  1. ArticleHow do you make a beam of particles you cannot steer?
  2. LabLong-Baseline CP Observatory
  3. LabCherenkov Light & Detector Rings
  4. ArticleK2K and MINOS: test flavor change with a distance you choose
  5. ArticleT2K: why look slightly away from the beam axis?
  6. ArticleNOvA: how appearance, disappearance and the Earth fit together
  7. ArticleOPERA: tau appearance and a corrected clock anomaly
  8. ArticleDUNE: what detailed argon images and a long baseline can answer
  9. LabLiquid Argon Time Projection Chamber
  10. ArticleHyper-K: a larger target still needs a precise comparison
  11. ArticleNeutrino CP violation: what comparing neutrinos with antineutrinos can show

Neutrinos · Reading path

Understand flavor change and mass

Articles on this path that use this experiment, followed by the other experiments they suggest.

  1. ArticleBefore anyone saw oscillations, how did the mixing idea take shape?
  2. ArticleWhy can a neutrino change flavor?
  3. ArticleIs finding a missing flavor the same as finding a new one?
  4. ArticleNeutrinos have mass. Why does oscillation not tell us how much?
2 more groups

Neutrinos · Reading path

Weigh the particle and test its nature

Articles on this path that use this experiment, followed by the other experiments they suggest.

  1. ArticleKATRIN: limiting neutrino mass at the end of the beta spectrum
  2. LabBeta Endpoint Mass Observatory
  3. ArticleDirac or Majorana: is the neutrino its own antiparticle?
  4. LabDouble Beta Decay Observatory
  5. ArticleSterile neutrinos: an explanation must pass several tests
  6. ArticleNeutrinos in 2026: which questions remain open?

Neutrinos · Reading path

Read reactors and the Earth

Articles on this path that use this experiment, followed by the other experiments they suggest.

  1. ArticleRENO and Double Chooz: why repeat the mixing-angle measurement?
  2. LabReactor Near–Far Spectra
  3. LabReactor Antineutrino Coincidence
  4. ArticleJUNO: why must a huge detector resolve tiny spectral features?
  5. ArticleShort reactor baselines: look for a change with distance
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