---
title: "Neutrino Flavor Propagation Interactive Lab | Sandbox Physics"
description: "Why does the chance of finding a neutrino in each flavor rise and fall as it travels? Because of phase."
canonical_url: "https://sandboxphysics.com/labs/neutrino-flavor-propagation/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/neutrino-flavor-propagation/"
last_modified: "2026-10-27"
---

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 simulation: open this page in a web browser to change the parameters and run the model.

The model propagates a coherent, ultrarelativistic three-flavor state in vacuum. Its parameters are rounded teaching values, not a global fit. The ideal measurements detect every flavor equally well, so they are not detector events, especially below the energy needed to produce the charged lepton. Matter effects, energy spread, wave-packet separation, cross sections and detector response are left out. The phase wheels show components of one state, not separate particle paths.

Vacuum prediction with ideal sampling

## A probability is not a count

Preparing the model…

Preview distance $600\,\mathrm{km}$

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

$N=256$

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.

[Why flavor changes →](https://sandboxphysics.com/stories/neutrinos/neutrino-flavor-oscillations/) [What oscillations say about mass →](https://sandboxphysics.com/stories/neutrinos/oscillations-and-mass/) [PDG · Mixing and oscillations ↗](https://pdg.lbl.gov/2024/reviews/rpp2024-rev-neutrino-mixing.pdf)

Physics tutorial

## The phase changes; the mass components stay coherent

**Background** A 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 matters** The 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

$$
A_{\alpha\to\beta}(L)=\sum_{i=1}^{3}U_{\beta i}U_{\alpha i}^{*}e^{-i\phi_i},\qquad P_{\alpha\beta}=|A_{\alpha\to\beta}|^2
$$

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

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

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

$$
P_{\mu\tau}=\sin^2(2\theta_{23})\sin^2\!\left(\frac{\Delta m_{31}^2L}{4E}\right),\qquad \sum_\beta P_{\alpha\beta}=1
$$

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 misconception** A neutrino splits into three classical particles that travel independently.

**Better mental model** The 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 misconception** At the chosen energy, a real detector would see every tau-flavor outcome the model predicts.

**Better mental model** The 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.

Physical conclusion

Oscillation moves probability between flavors without losing any. Real oscillation measurements add the source, matter, interactions and detector response on top of this coherent evolution.

## 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 here **Neutrino Flavor Propagation** Start with coherent vacuum propagation and ideal flavor trials.](https://sandboxphysics.com/labs/neutrino-flavor-propagation/)
2. [Lab **Neutrino Matter Resonance** Follow a rotating matter Hamiltonian and compare slow with sudden conversion.](https://sandboxphysics.com/labs/neutrino-matter-resonance/)
3. [Lab **Solar Neutrino Flavor Observatory** Separate electron flavor from total active solar flux with three reaction responses.](https://sandboxphysics.com/labs/solar-neutrino-flux/)
4. [Lab **Atmospheric Neutrino Zenith Experiment** Turn Earth-crossing baselines into direction-dependent event counts.](https://sandboxphysics.com/labs/atmospheric-neutrino-zenith/)
5. [Lab **Reactor Near–Far Spectra** Fit the same near and far counts while testing calibration and normalization.](https://sandboxphysics.com/labs/reactor-near-far-spectra/)
6. [Lab **Long-Baseline CP Observatory** Compare both accelerator beam polarities on frozen spectra and separate CP phase, matter and response assumptions.](https://sandboxphysics.com/labs/long-baseline-cp/)

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. [Article **How do you make a beam of particles you cannot steer?**](https://sandboxphysics.com/stories/neutrinos/making-an-accelerator-neutrino-beam/)
2. [Lab **Long-Baseline CP Observatory**](https://sandboxphysics.com/labs/long-baseline-cp/)
3. [Lab **Cherenkov Light & Detector Rings**](https://sandboxphysics.com/labs/cherenkov-rings/)
4. [Article **K2K and MINOS: test flavor change with a distance you choose**](https://sandboxphysics.com/stories/neutrinos/k2k-minos-controlled-baselines/)
5. [Article **T2K: why look slightly away from the beam axis?**](https://sandboxphysics.com/stories/neutrinos/t2k-off-axis-beam-and-appearance/)
6. [Article **NOvA: how appearance, disappearance and the Earth fit together**](https://sandboxphysics.com/stories/neutrinos/nova-appearance-disappearance-and-matter/)
7. [Article **OPERA: tau appearance and a corrected clock anomaly**](https://sandboxphysics.com/stories/neutrinos/opera-tau-appearance-and-timing-correction/)
8. [Article **DUNE: what detailed argon images and a long baseline can answer**](https://sandboxphysics.com/stories/neutrinos/dune-liquid-argon-and-long-baseline/)
9. [Lab **Liquid Argon Time Projection Chamber**](https://sandboxphysics.com/labs/liquid-argon-tpc/)
10. [Article **Hyper-K: a larger target still needs a precise comparison**](https://sandboxphysics.com/stories/neutrinos/hyper-kamiokande-mass-and-systematics/)
11. [Article **Neutrino CP violation: what comparing neutrinos with antineutrinos can show**](https://sandboxphysics.com/stories/neutrinos/neutrino-cp-violation-measurement/)

Neutrinos · Reading path

### Understand flavor change and mass

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

1. [Article **Before anyone saw oscillations, how did the mixing idea take shape?**](https://sandboxphysics.com/stories/neutrinos/pontecorvo-maki-nakagawa-sakata-mixing/)
2. [Article **Why can a neutrino change flavor?**](https://sandboxphysics.com/stories/neutrinos/neutrino-flavor-oscillations/)
3. [Article **Is finding a missing flavor the same as finding a new one?**](https://sandboxphysics.com/stories/neutrinos/neutrino-appearance-and-disappearance/)
4. [Article **Neutrinos have mass. Why does oscillation not tell us how much?**](https://sandboxphysics.com/stories/neutrinos/oscillations-and-mass/)

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. [Article **KATRIN: limiting neutrino mass at the end of the beta spectrum**](https://sandboxphysics.com/stories/neutrinos/katrin-beta-endpoint-neutrino-mass/)
2. [Lab **Beta Endpoint Mass Observatory**](https://sandboxphysics.com/labs/beta-endpoint-mass/)
3. [Article **Dirac or Majorana: is the neutrino its own antiparticle?**](https://sandboxphysics.com/stories/neutrinos/dirac-majorana-neutrino-nature/)
4. [Lab **Double Beta Decay Observatory**](https://sandboxphysics.com/labs/neutrinoless-double-beta/)
5. [Article **Sterile neutrinos: an explanation must pass several tests**](https://sandboxphysics.com/stories/neutrinos/sterile-neutrinos-anomalies-and-tests/)
6. [Article **Neutrinos in 2026: which questions remain open?**](https://sandboxphysics.com/stories/neutrinos/open-neutrino-questions-2026/)

Neutrinos · Reading path

### Read reactors and the Earth

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

1. [Article **RENO and Double Chooz: why repeat the mixing-angle measurement?**](https://sandboxphysics.com/stories/neutrinos/reno-double-chooz-independent-mixing-tests/)
2. [Lab **Reactor Near–Far Spectra**](https://sandboxphysics.com/labs/reactor-near-far-spectra/)
3. [Lab **Reactor Antineutrino Coincidence**](https://sandboxphysics.com/labs/reactor-coincidence/)
4. [Article **JUNO: why must a huge detector resolve tiny spectral features?**](https://sandboxphysics.com/stories/neutrinos/juno-fine-spectrum-and-mass-ordering/)
5. [Article **Short reactor baselines: look for a change with distance**](https://sandboxphysics.com/stories/neutrinos/short-baseline-reactor-sterile-tests/)

Fields & matter · Starter path

## From a field to an electron microscope

Map an electric field and steer a charged particle. Then find the electron's wavelength and use electron beams to see a crystal and a surface.

[Start here → **Make an Electric Field Visible** Move the probe around a dipole and check that the field always points downhill on the potential landscape. Observe · 8 min](https://sandboxphysics.com/labs/electric-field/)

1. [1 Make an Electric Field Visible](https://sandboxphysics.com/labs/electric-field/)
2. [2 A Charge Bending through Fields](https://sandboxphysics.com/labs/charged-particle/)
3. [3 Electron Wavelength & Voltage](https://sandboxphysics.com/labs/electron-wavelength/)
4. [4 Electron Diffraction & Reciprocal Space](https://sandboxphysics.com/labs/electron-diffraction/)
5. [5 TEM Bright Field, Dark Field & SAED](https://sandboxphysics.com/labs/tem-mode-selector/)
6. [6 SEM Secondary-Electron Topography](https://sandboxphysics.com/labs/sem-topography/)

[View the full path →](https://sandboxphysics.com/topics/fields-and-matter/#starter-path)