Neutrinos · D04 · A planet-sized baseline

Atmospheric Neutrino Zenith Experiment

Follow a neutrino through Earth, acquire a sky sample, then compare its direction counts with two predictions. Change energy bands to see which travel distances matter.

Travel distance $\text{—}$

Muon survival probability $\text{—}$

Selected simulated events $\text{—}$

Up / down count ratio $\text{—}$

Teaching simulation, not Super-Kamiokande data. Two-flavor vacuum muon disappearance; spherical Earth, surface detector and one production height. A declared energy spectrum, angular flux and acceptance generate the same events used in every count. Perfect energy and direction reconstruction. No matter effects, electron or tau event samples, geomagnetic asymmetry, altitude distribution, backgrounds or real detector response. Marker sizes are enlarged; the ray geometry is to scale.

Preparing the sky sample…

Arrival zenith cosine

\cos\theta_z

-0.8

Preview energy

3\,\mathrm{GeV}

Negative cosine: upward-going, through Earth. Positive: downward-going, from the nearby atmosphere. The two sliders inspect a single hypothetical ray; they do not select or resample the sky record.

Sampled countsOscillation predictionNo-oscillation prediction

Whiskers show the square root of the observed count, a scale for counting fluctuations, not a confidence interval. Predictions integrate the same source and acceptance used to generate events. The up/down ratio excludes the two central horizon bins.

Read the ten direction bins

Energy bands use perfect true energy and direction. These simplified muon-flavor event counts omit backgrounds, lepton scattering angles and resolution; they cannot reproduce the experiment’s fitted significance.

CONTINUE EXPLORING

From flavor to evidence

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

Physics tutorial

The source is all around us. The baseline is not.

BackgroundCosmic-ray cascades produce neutrinos in the atmosphere. A surface detector can receive a short downward-going path or a long upward-going path through Earth. The 1998 Super-Kamiokande analysis used direction-dependent event deficits as evidence for oscillation.

Why it mattersA flavor probability alone is not an event prediction. This Lab combines propagation with an explicit teaching source spectrum, exposure and direction-dependent acceptance, then compares the generated counts with integrated expectations.

Start with the essentials

Focus question: Does the upward deficit remain when source and acceptance are symmetric between the two hemispheres?
One-sentence intuition: Compare each direction with its own no-oscillation expectation. An uneven histogram can come from flux or acceptance; a long-baseline flavor deficit adds a different dependence on energy and direction.

Core mathematical model

A ray intersects the production shell

\begin{aligned}c&=\cos\theta_z\\L(c,h)&=\sqrt{R^2c^2+2Rh+h^2}-Rc\end{aligned}

The detector is at the surface of a sphere of radius 6371 km. Positive cosine is downward-going; negative is upward-going. The source is at one fixed height above the sphere. The formula follows from intersecting a ray with that shell.

Muon-flavor survival

\begin{aligned}P_{\mu\mu}&=1-\sin^2(2\theta)\sin^2\phi\\\phi&=1.266933\,\frac{\Delta m^2}{\mathrm{eV}^2}\frac{L}{\mathrm{km}}\frac{\mathrm{GeV}}{E}\end{aligned}

Two-flavor ultrarelativistic vacuum propagation. A smaller survival probability removes events from this simplified muon-flavor sample; it does not mean that Earth absorbed the neutrino.

The declared source and acceptance

\begin{aligned}f(c)&=1+a(1-c^2)\\\epsilon(c)&=1-b(1-c^2)\end{aligned}

The first factor enhances horizontal flux; the second reduces horizontal acceptance. Both are symmetric between upward and downward directions. They are illustrative functions, not a fitted atmospheric flux or detector calibration.

From spectrum to expected counts

\begin{aligned}\lambda_B&=\frac{N_0}{2I}\int_B q\,dE\,dc\\q&=E^{-1.7}f(c)\epsilon(c)P_{\mu\mu}\end{aligned}

Energy is the numerical value in GeV, restricted to 1–20 GeV; I is the integral of its power-law weight over that range. The event weight combines a teaching flux with power −2.7 and an interaction weight proportional to energy. The reference exposure is the expected total without oscillation when both angular adjustments are zero.

A check on the complete no-oscillation sample

\frac{N_{\mathrm{null}}}{N_0}=1+\frac{2}{3}(a-b)-\frac{8}{15}ab

Integrating the even angular polynomial gives this independent normalization check. Increasing the acceptance loss lowers the expected total. Turning off both angular adjustments gives a flat no-oscillation histogram.

Common difficulties

Upward does not mean produced underground

Typical misconceptionThe upward sample comes from a source inside Earth.

Better mental modelIts source is atmospheric on the far side. The arrow points from production to the same surface detector. Earth provides a longer path here, not a neutrino source.

A detector measures outgoing particles

Typical misconceptionThe plotted neutrino energy and direction are directly measured without uncertainty.

Better mental modelThis Lab assumes perfect reconstruction. Actual muon directions differ from their parent neutrinos, and energy, efficiency, backgrounds and nuclear interactions enter a real analysis. The simplified bins are not the collaboration’s published bins.

Run the experiment

01
Turn a direction into a distance
Use Through Earth, Horizon and Overhead. Inspect Path detail for the short downward ray.
What to observe: Overhead distance equals the production height. The upward vertical path adds Earth’s diameter. Even a horizontal path is longer than the height because Earth is curved.
02
Find the upward deficit
Acquire Oscillating sky. Compare the sampled points with the green prediction and dashed no-oscillation line.
What to observe: The two predictions use the same flux, acceptance and exposure. Their separation comes from survival probability, while points fluctuate because they are sampled events.
03
Use energy as a second check
Select each recorded energy band, then change only the preview energy.
What to observe: Band buttons filter the same recorded events and integrate both predictions over matching energy bounds. Preview energy changes one hypothetical ray only. Higher energy changes the oscillation phase at a given distance.
04
Make a null control
Try No oscillation, then Uniform control. Change the horizontal flux and acceptance in the advanced controls and acquire again.
What to observe: An angular shape can exist without oscillation. These even source and acceptance functions preserve an expected up/down ratio of one in the null model; finite samples need not have equal counts.
05
Accumulate evidence without moving the goalposts
Try Long exposure and vary the random seed. Export the record and sum the individual event rows into the ten bins.
What to observe: The CSV event rows reproduce the displayed counts. Each bin also includes its two integrated predictions. Larger exposure reduces relative counting fluctuations; this teaching model does not report a discovery significance.