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.

Frozen appearance counts $0$

Best phase on fixed grid $0{}^\circ$

CP conserving objective gap $0$

Compare neutrinos and antineutrinos over one baseline

Send both beams through the same rock

Baseline

L

1300\,\mathrm{km}

Beam flux center

E_0

2.5\,\mathrm{GeV}

Generated CP phase

\delta_{\mathrm{CP}}

-90\,{}^\circ

Generated constant density

\rho

2.8\,\frac{\mathrm g}{\mathrm{cm}^3}

Exposure per polarity

t

3\,\mathrm{arb.}

Generated mass ordering

Beam width & detector response

Beam flux width

w

0.8\,\mathrm{GeV}

Fractional energy resolution

r

15\,\%

Generated energy scale bias

b

0\,\%

Test candidate models on the same counts

Polarities used in objectiveAssumed constant density

\rho_{\mathrm{fit}}

2.8\,\frac{\mathrm g}{\mathrm{cm}^3}

Assumed energy scale bias

b_{\mathrm{fit}}

0\,\%

Signal normalization constraint

s_a

5\,\%

Source settings take effect when you acquire. Analysis assumptions take effect when you run the comparison. The phase, view and energy selectors all reuse the same record.

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.

Candidate mass orderingCandidate CP phase

0{}^\circ

Points & whiskers: frozen counts, square-root count scalePurple line: profiled candidate expectationReveal 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 polarityReference 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.

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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 misconceptionIf the CP-conserving fit is much worse, real CP violation is proven.

Better mental modelThe 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 misconceptionThe beam physically splits into three visible paths, one for each flavor.

Better mental modelA 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

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.
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.
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.
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.

Long-Baseline CP Observatory

The difference needs a model.

Neutrino appearance

Antineutrino appearance

Phase loop · same energy, different medium

Phase profiles for both mass orderings

One journey, three probabilities

Conditional comparison

An asymmetry has more than one cause

Core mathematical model

Three-flavor Hamiltonian in matter

Matter potential, a second source of asymmetry

From probability to expected counts

Fit objective on the same counts

Common difficulties

A large gap is not a discovery

The colored packets are a sketch

Run the experiment

Take away the matter

Turn the phase

Add normalization uncertainty

Test the calibration

Related experiments and articles

How flavor change is predicted and tested

Make a source and test its journey