Neutrinos · D08 · Matter & antimatter

Long-Baseline CP Observatory

Launch two beam polarities through a cutaway landscape. Freeze their electron-flavor spectra, turn the CP phase and compare mass orderings on the same counts. Then remove the matter: which asymmetry survives?

Frozen appearance counts $0$

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

CP conserving objective gap $0$

LAUNCH · FREEZE · COMPARE

Cross the same Earth.

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\,\%

QUESTION 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\,\%

Acquire to apply source settings. Run comparison to apply analysis assumptions. Phase, view and energy selectors reuse the same record.

Generated teaching spectra, not T2K, NOvA, Hyper-K or DUNE data or sensitivity forecasts. Three active flavors in constant density, fixed mixing parameters and illustrative flux, cross-section, background and Gaussian response. Independent Poisson bins and independent constrained signal normalizations. The finite phase grid gives descriptive objective gaps, 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.

Competing phase & ordering profiles

Relative to the common grid minimum. Green: normal ordering. Gold: inverted. Independent signal normalization constraints are profiled. No confidence thresholds or significance conversion.

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 at conserving phases across both orderings, minus the overall grid minimum. This descriptive gap is not a p-value or a detection claim. Mixing angles, splitting magnitudes, known backgrounds and assumed response are held fixed. Count square-root whiskers are a scale guide, 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

Accessible complete phase profile 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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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: Which difference comes from the phase, which from matter, and which from the measurement?
One-sentence intuition: Compare both spectra while accounting for their different cross-sections, matter effects and response. Counts alone are not a CP asymmetry.

Core mathematical model

Three flavors 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]

Constant-density propagation uses the full complex mixing matrix. Antineutrinos conjugate the matrix and reverse the matter potential.

A medium can create asymmetry

a=2\sqrt2\,G_F N_e E

Ordinary matter is not CP symmetric. A beam asymmetry at a conserving phase is not by itself intrinsic CP violation.

Probabilities become 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 response convert a coherent probability into a simulated binned detector expectation. Both overflow bins are retained.

Compare on the same record

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}

At each phase and ordering, profile independent positive signal normalizations. A zero constraint width fixes both to one. Objective differences have no automatic significance calibration.

Common difficulties

A gap is not a discovery

Typical misconceptionA large CP conserving gap proves real CP violation.

Better mental modelThis is a descriptive finite-grid objective comparison with generated counts and simplified constraints. No tail calibration or confidence coverage is provided.

Colored packets are schematic

Typical misconceptionThe beam physically separates into three visible flavor paths.

Better mental modelOne packet follows one beam axis. Its color indicates model flavor probabilities; it is not a measured flavor label or literal particle movie.

Run the experiment

01
Remove the medium
Choose matter asymmetry, select candidate phase zero and inspect the probability loop against its vacuum reference.
What to observe: At a conserving phase, vacuum probabilities agree. Matter can split them; raw counts can differ even in vacuum because the illustrative cross-sections differ.
02
Turn the phase
Select the conjugate preset, or explore candidate phases on the frozen broad-beam record.
What to observe: The probability loop traces neutrino against antineutrino appearance as the phase turns. Dashed vacuum and solid matter loops answer different physical questions.
03
Pay for uncertainty
Compare both polarities with either polarity alone. Change the normalization constraint, then run comparison.
What to observe: The phase profiles and competing ordering minima change. A five-degree grid and fixed mixing parameters limit what these curves establish.
04
Challenge calibration
Acquire the calibration trap. Set the assumed scale to six percent and run comparison without acquiring.
What to observe: The counts stay frozen while predicted spectra and best candidates change. Model assumptions belong in the inference.