Calculator Inputs
Use the form below to model vacuum oscillation probabilities for neutrinos or antineutrinos with custom mixing parameters.
Example Data Table
These sample rows illustrate realistic baselines, energies, and output probabilities from typical vacuum calculations.
| Scenario | Initial | Final | L (km) | E (GeV) | Probability |
|---|---|---|---|---|---|
| T2K-like | νμ | νe | 295 | 0.6 | 0.046477 |
| T2K-like | νμ | νμ | 295 | 0.6 | 0.016297 |
| NOvA-like | νμ | νe | 810 | 2.0 | 0.040684 |
| DUNE-like | νμ | νe | 1300 | 2.5 | 0.047191 |
Formula Used
The calculator applies the exact three-flavor vacuum oscillation expression with PMNS matrix elements and mass-squared differences.
P(να → νβ) = δαβ − 4 Σ(i>j) Re[Kαβ,ij] sin²Δij + 2 Σ(i>j) Im[Kαβ,ij] sin(2Δij)Where the phase term is:
Δij = 1.267 × (Δm²ij × L / E)And the interference factor is:
Kαβ,ij = Uαi U*βi U*αj Uβj- L is baseline in kilometers.
- E is neutrino energy in GeV.
- Δm²ij is the mass-squared difference in eV².
- U is the PMNS mixing matrix built from θ12, θ13, θ23, and δCP.
How to Use This Calculator
- Select neutrino or antineutrino mode.
- Choose the initial flavor and desired final flavor.
- Enter baseline length and beam energy.
- Provide mixing angles, CP phase, and mass splittings.
- Press Calculate Probability to view the result above the form.
- Review channel totals, oscillation phases, and PMNS magnitude checks.
- Use the export buttons to save the result as CSV or PDF.
FAQs
1. What does this calculator estimate?
It estimates vacuum oscillation probabilities between electron, muon, and tau neutrino flavors using PMNS mixing inputs, beam energy, and travel distance.
2. Does it include matter effects?
No. This version models oscillation in vacuum. For dense matter paths, effective parameters shift and a dedicated matter-effect solver is more appropriate.
3. Why do probabilities sum close to one?
The sum checks unitarity. Small rounding differences can appear, but the three final-state probabilities should remain extremely close to one.
4. Why can neutrino and antineutrino results differ?
The CP phase changes sign for antineutrinos in vacuum, so interference terms can shift appearance probabilities even when the baseline and energy stay fixed.
5. Which units should I enter?
Use kilometers for baseline, GeV for energy, degrees for mixing angles and CP phase, and eV² for mass-squared differences.
6. What is Δm²32 here?
It is computed internally as Δm²31 minus Δm²21. The calculator shows it because many analyses discuss the 3–2 splitting directly.
7. Can I model survival and appearance channels?
Yes. Choose the same initial and final flavor for survival, or different flavors for appearance probabilities.
8. What do the oscillation length estimates mean?
They show the characteristic vacuum distance associated with each mass-splitting scale. They help compare your chosen baseline against dominant oscillation structure.