---
title: "Long-Baseline CP Observatory Interactive Lab | Sandbox Physics"
description: "Can the neutrino and antineutrino spectra from an accelerator beam reveal the CP phase? Yes, but not from a simple difference in event counts."
canonical_url: "https://sandboxphysics.com/labs/long-baseline-cp/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/long-baseline-cp/"
last_modified: "2026-10-27"
---

Neutrinos · D08 · Matter & antimatter

# Long-Baseline CP Observatory

**Can the neutrino and antineutrino spectra from an accelerator beam reveal the CP phase?** Yes, but not from a simple difference in event counts. Neutrinos and antineutrinos have different cross-sections, the detector responds to them differently, and matter along the beam affects them differently too. A difference in counts tells you about the CP phase only after the phase, the matter effect and the detector response are pulled apart in one analysis of the same data.

Send a neutrino beam and an antineutrino beam through the Earth and record the electron-flavor events each one produces. Change the CP phase and compare the two mass orderings on the same counts. Then take away the matter and see which part of the asymmetry remains.

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

These are generated teaching spectra, not data or sensitivity forecasts from T2K, NOvA, Hyper-K or DUNE. The model has three active flavors in matter of constant density, fixed mixing parameters, and an illustrative flux, cross-section, background and Gaussian response. Each energy bin is an independent Poisson count, and the signal normalizations are constrained independently. The phase is scanned on a finite grid, so the gaps in the fit objective are only descriptive. They are not confidence intervals, calibrated significance or evidence of real CP violation.

One record from neutrino and antineutrino beams

## The difference needs a model.

Candidate mass ordering Candidate CP phase $0{}^\circ$

Points & whiskers: frozen counts, square-root count scale Purple line: profiled candidate expectation Reveal generated means

### Neutrino appearance

### Antineutrino appearance

### Phase loop · same energy, different medium

Solid: assumed matter. Dashed: vacuum at the same energy and ordering. Each point is a different candidate phase. This is an ideal probability reference, not a reconstruction of detector counts.

### Phase profiles for both mass orderings

Values are relative to the lowest point on the shared grid. Green is normal ordering and gold is inverted. Independent signal normalization constraints are profiled. No confidence thresholds are drawn, and nothing is converted to a significance.

### One journey, three probabilities

Displayed packet polarity Reference energy $2.5\,\mathrm{GeV}$

Position along baseline $0\,\mathrm{km}$

Green: electron. Cyan: muon. Gold: tau. Probabilities sum to one; one coherent packet follows one compressed beam axis. Selected energy and position do not alter or resample the spectra.

### Conditional comparison

- **Best grid phase**: $0$
- **Selected objective gap**: $0$
- **CP conserving grid gap**: $0$
- **Neutrino signal normalization**: $0$
- **Antineutrino signal normalization**: $0$
- **Far electron probability · neutrino**: $0$
- **Far electron probability · antineutrino**: $0$

CP-conserving gap: the smallest objective value at the CP-conserving phases, over both orderings, minus the lowest value on the whole grid. The gap is descriptive; it is not a p-value or a detection claim. Mixing angles, the sizes of the mass splittings, known backgrounds and the assumed response are held fixed. The square-root error bars on counts show the scale of fluctuations; they are not exact Poisson intervals.

Accessible complete spectrum table

Reconstructed energy bins in GeV, including underflow and overflow

| Bin | Neutrino count | Antineutrino count | Candidate neutrino mean | Candidate antineutrino mean |
| --- | --- | --- | --- | --- |

Full phase profile as a table

Five-degree candidate grid; endpoints represent the same phase

| Ordering | Phase · degrees | Objective gap | Neutrino scale | Antineutrino scale |
| --- | --- | --- | --- | --- |

Model, fixed parameters & primary sources

Pure muon-flavor beams evolve with the standard three-flavor complex mixing matrix in constant-density ordinary matter. Electron fraction is one half; mixing squared sines are 0.307, 0.022 and 0.55. Solar splitting is 0.000075 and the magnitude of the atmospheric splitting is 0.0025, both in electronvolt squared. Normal and inverted reverse the latter sign; these are teaching values, not a fitted parameter release. No absorption, sterile flavors or varying density.

Gaussian flux is truncated and normalized over 0.2 to 5 GeV. Illustrative neutrino cross-section is proportional to energy; the antineutrino value is one half. Constant efficiency is 0.8. Fixed flux normalization gives 1800 exposure units before cross-section and efficiency; there is no automatic inverse-square baseline rescaling or real proton-on-target conversion. Changing baseline isolates oscillation at fixed delivered fluence. Gaussian reconstructed energy has a fractional width and a multiplicative scale bias, with every tail retained. Known flat backgrounds have 12 counts per exposure for neutrinos and half as many for antineutrinos, restricted to displayed bins.

Each polarity is independently acquired. Fits use only recorded counts, known backgrounds and public apparatus settings; generated phase, density, ordering and scale truth are excluded. Independent nonnegative signal normalizations have fixed Gaussian constraints centered on one, with no background profiling, energy-dependent shape uncertainties, correlated near-detector fit or auxiliary resampling. The phase grid is finite and can miss a continuous minimum. No confidence coverage or CP discovery calibration.

Geometry, Earth layers, accelerator, near monitor and detector are schematic. The proton beam ends at the target. Charged parents pass through the horns into a schematic decay region and terminate at the dump; neutrinos pass through it and the near monitor into the far volume. Transverse sizes and distances are independently exaggerated. The near monitor contributes no simulated data to this fit. Far tracks are illustrative interactions, not sampled hits. All spectra, probability references and profile charts are outside the apparatus viewport.

- [PDG · Mixing, propagation in matter & accelerator oscillations](https://pdg.lbl.gov/2025/reviews/rpp2025-rev-neutrino-mixing.pdf)
- [T2K · Beam, near detectors & 295 km far detector](https://t2k-experiment.org/t2k/)
- [Fermilab · The 1300 km LBNF/DUNE journey](https://lbnf-dune.fnal.gov/how-it-works/)

[Explore matter resonance →](https://sandboxphysics.com/labs/neutrino-matter-resonance/) [Explore the worldwide history →](https://sandboxphysics.com/stories/neutrinos/)

Physics tutorial

## An asymmetry has more than one cause

**Background** The beam starts as pure muon flavor. The model follows it through uniform matter as one coherent state, then predicts how many electron-flavor neutrinos appear. The mixing angles and the sizes of the mass splittings stay fixed.

**Why it matters** Neutrinos and antineutrinos can travel the same path and still arrive with different probabilities and event rates.

Start with the essentials

- **Focus question**: Which difference comes from the phase, which from matter, and which from the measurement?
- **One-sentence intuition**: A difference in counts is not by itself a CP asymmetry. Compare the two spectra only after accounting for their different cross-sections, matter effects and response.

### Core mathematical model

#### Three-flavor Hamiltonian in matter

$$
H_\nu=\frac{1}{2E}\left[U\,\mathrm{diag}(0,\Delta m^2_{21},\Delta m^2_{31})U^\dagger+\mathrm{diag}(a,0,0)\right]
$$

Propagation through matter of constant density uses the full complex mixing matrix. For antineutrinos the matrix is complex-conjugated and the matter potential changes sign.

#### Matter potential, a second source of asymmetry

$$
a=2\sqrt2\,G_F N_e E
$$

Ordinary matter is not CP symmetric. So a difference between the beams at a CP-conserving phase is not, on its own, intrinsic CP violation.

#### From probability to expected counts

$$
\mu_{pj}=a_p\int\Phi_p(E)\sigma_p(E)\epsilon(E)P_p(E)R_j(E)\,dE+B_{pj}
$$

The teaching flux, cross-section, efficiency and detector response turn the coherent oscillation probability into a simulated expected count in each energy bin. The overflow bins at both ends are kept.

#### Fit objective on the same counts

$$
D=2\sum_{p,j}\left[\mu_{pj}-n_{pj}+n_{pj}\ln\frac{n_{pj}}{\mu_{pj}}\right]+\sum_p\frac{(a_p-1)^2}{s_a^2}
$$

For each phase and mass ordering, the two positive signal normalizations are fitted independently. Setting the constraint width to zero fixes both at one. A difference in this objective does not come with a calibrated significance.

### Common difficulties

#### A large gap is not a discovery

**Typical misconception** If the CP-conserving fit is much worse, real CP violation is proven.

**Better mental model** The gap only describes how the fit objective compares on a finite grid, using generated counts and simplified constraints. It comes with no calibrated tail probability or confidence coverage.

#### The colored packets are a sketch

**Typical misconception** The beam physically splits into three visible paths, one for each flavor.

**Better mental model** A single packet moves along one beam axis. Its color shows the flavor probabilities from the model; it is not a measured flavor or a film of real particles.

### Run the experiment

1. 01

   #### Take away the matter

   Choose Matter asymmetry, set the candidate phase to zero and compare the probability loop with its vacuum reference.

   **What to observe:** At a CP-conserving phase, the vacuum probabilities for neutrinos and antineutrinos are equal, and matter can pull them apart. The raw counts can differ even in vacuum, because the illustrative cross-sections differ.

2. 02

   #### Turn the phase

   Choose Reverse CP phase, or try different candidate phases on the recorded broad-beam data.

   **What to observe:** As the phase turns, the loop traces neutrino appearance against antineutrino appearance. The dashed loop is for vacuum and the solid loop for matter; they answer different physical questions.

3. 03

   #### Add normalization uncertainty

   Compare Both polarities with each polarity alone. Change Signal normalization constraint, then press Run comparison.

   **What to observe:** The phase profiles and the best fits for the two orderings shift. With a five-degree grid and fixed mixing parameters, these curves can only show so much.

4. 04

   #### Test the calibration

   Choose Calibration trap to acquire its data. Set Assumed energy scale bias to six percent and press Run comparison without acquiring again.

   **What to observe:** The counts stay the same while the predicted spectra and the best-fitting candidates change. The model assumptions are part of the inference.

Physical conclusion

Record both spectra once, then separate what the phase, the matter and the measurement each contribute, and keep the assumptions in view.

## 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. [Lab **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. [You are here **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 **Neutrino Flavor Propagation**](https://sandboxphysics.com/labs/neutrino-flavor-propagation/)
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 **DUNE: what detailed argon images and a long baseline can answer**](https://sandboxphysics.com/stories/neutrinos/dune-liquid-argon-and-long-baseline/)
8. [Lab **Liquid Argon Time Projection Chamber**](https://sandboxphysics.com/labs/liquid-argon-tpc/)
9. [Article **Hyper-K: a larger target still needs a precise comparison**](https://sandboxphysics.com/stories/neutrinos/hyper-kamiokande-mass-and-systematics/)
10. [Article **Neutrino CP violation: what comparing neutrinos with antineutrinos can show**](https://sandboxphysics.com/stories/neutrinos/neutrino-cp-violation-measurement/)

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)