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The science
What is simulated
A beam of muon, electron or tau neutrinos (or their antiparticles) travels a straight chord through the Earth. The state is a three-component quantum amplitude ψ = (ψe, ψμ, ψτ) and the flavour probabilities are Pβ = |ψβ|2. Nothing is faked: the amplitude is advanced along the path by the Schrödinger equation i dψ/dL = (2.534/E) [U diag(0, Δm²21, Δm²31) U† + diag(A, 0, 0)] ψ, with L in km, E in GeV and masses in eV², using fourth-order Runge-Kutta with at most 0.05 rad of phase per step.
Mixing and matter
U is the standard PMNS matrix with angles θ12 = 33.4° (fixed), θ13, θ23 and the CP phase δ as sliders; Δm²21 = 7.4×10−5 eV² and |Δm²31| = 2.5×10−3 eV² (normal or inverted ordering). These are rounded values from public global fits and are not the best-fit numbers of any one analysis. Electron neutrinos feel extra forward scattering on electrons in matter: A = 2√2 GF Ne E = 1.526×10−4 eV² (Ye ρ / g cm−3)(E / GeV). For antineutrinos A → −A and U → U*, which is why neutrino and antineutrino curves differ: matter effect and CP phase both contribute to the gap in the lowest graph.
The Earth model
Four constant-density shells (inner core 13.0, outer core 11.1, lower mantle 5.0, crust and upper mantle 3.4 g/cm³) with Ye = 0.468 in the core and 0.495 in the mantle. This is a coarse stand-in for the PREM density profile, accurate to roughly ten percent in density, and the Earth is treated as spherical. The beam is a plane wave of one fixed energy: no decoherence from energy spread, no detector response, no beam divergence.
Simplifications
- θ12 and the solar splitting are fixed; Δm²21 still matters at low energies, where it is included.
- Neutral-current scattering is the same for all flavours and drops out; absorption is negligible below TeV energies and is not modelled.
- The animation shows the probability that a neutrino detected at that distance would have each flavour; there is no single trajectory of a "changing" particle.
Checks
Five boxes: the vacuum evolution against the closed-form PMNS amplitude, unitarity through a 12 000 km chord, the textbook two-flavour constant-density matter formula (switching off all but the e-tau channel), the vacuum CP asymmetry against the Jarlskog expression, and time-reversal symmetry in the symmetric Earth chord.