Neutrinos · D05 · Mixing and measurement

Neutrino Flavor Propagation

Rotate the mass-state phases, scan the baseline, then sample an ideal flavor measurement. Compare three-flavor interference with no mixing, equal masses and a two-flavor limit.

Preview distance $\text{—}$

Electron-flavor probability $\text{—}$

Muon-flavor probability $\text{—}$

Tau-flavor probability $\text{—}$

Normalization residual $\text{—}$

Prepare · propagate · sample

Prepare the next trial

AntineutrinoConjugate the mixing matrix; vacuum only.Sampling distance

L

600\,\mathrm{km}

Energy

E

1.2\,\mathrm{GeV}

Prepared flavor

\alpha

\nu_{\mu}

Ideal projection trials

N

256

Random seed

s

17

Mixing angles, phase & masses

Solar mixing angle

\theta_{12}

33^\circ

Reactor mixing angle

\theta_{13}

8.5^\circ

Atmospheric mixing angle

\theta_{23}

45^\circ

CP phase

\delta

-90^\circ

Small mass-squared splitting

\Delta m_{21}^2

7.5\times10^{-5}\,\mathrm{eV}^2

Signed large mass-squared splitting

\Delta m_{31}^2

2.5\times10^{-3}\,\mathrm{eV}^2

Sliders prepare the next record. Calculate to apply them. The distance preview below explores the frozen model without resampling.

Coherent, ultrarelativistic three-flavor propagation in vacuum. Rounded teaching parameters, not a global fit. Ideal projective trials have equal acceptance for all flavors; they are not detector events, especially below charged-lepton production thresholds. No matter effects, energy spread, wave-packet separation, cross sections or detector response. Phase wheels describe components of one state, not separate particle paths.

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

Every trial prepares a new state. All three flavors have ideal equal acceptance here. Counts omit interaction thresholds, cross sections, energy resolution and detector efficiency.

CONTINUE EXPLORING

From flavor to evidence

Propagation predicts a flavor probability. Which parts of the experiment turn an interaction into a usable record?

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. Their squared magnitudes give probabilities; independently prepared trials sample those probabilities.

Core mathematical model

Prepare, propagate, project

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; columns label mass states. The first mass phase is removed as a common phase. Antineutrinos conjugate the mixing matrix.

Units and relative phase

\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 use the conventional natural-unit notation. Only relative mass squares affect this vacuum calculation.

Two-flavor check

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

Natural units in this expression. The Two-flavor preset sets the other mixing angles and the small splitting to zero; the electron flavor then decouples.

Common difficulties

Three arrows are not three particles

Typical misconceptionA neutrino splits into three independently travelling classical particles.

Better mental modelThe arrows are coherent amplitudes of one quantum state, in a chosen phase convention. The preview does not repeatedly measure one travelling neutrino.

Flavor probability is not detection efficiency

Typical misconceptionEvery predicted tau-flavor trial would be visible in a real detector at the chosen energy.

Better mental modelThe sampler is an ideal projection with equal acceptance. Real identification depends on interaction thresholds, cross sections, backgrounds and instrument response, all omitted here.

Run the experiment

01
Find disappearance
Use Three flavors, calculate, then scan the distance. Inspect the three probability bars.
What to observe: The original flavor can become rare without a loss of total probability. The wheel lengths stay constant while their phases rotate.
02
Turn off one ingredient
Try No mixing and Equal masses. Scan again.
What to observe: Both controls remove oscillations, for different reasons. A mass difference alone does not create flavor conversion.
03
Hold the phase scale 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 endpoint probabilities and sampled outcomes agree. The complete distance curves stretch because the energy changed.
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: At a nonzero CP phase some appearance probabilities differ; at zero phase they agree in vacuum. This is a model comparison, not evidence from an experiment.
05
Separate probability from frequency
Keep all physical settings fixed and vary the seed or number of trials.
What to observe: The theoretical curve is unchanged. Finite trial frequencies fluctuate; larger samples reduce typical fluctuation, not necessarily the error of every individual run.

Neutrino Flavor Propagation

A probability is not a count

From flavor to evidence

The phase changes; the mass components stay coherent

Core mathematical model

Prepare, propagate, project

Units and relative phase

Two-flavor check

Common difficulties

Three arrows are not three particles

Flavor probability is not detection efficiency

Run the experiment

Find disappearance

Turn off one ingredient

Hold the phase scale fixed

Compare particle and antiparticle

Separate probability from frequency