PMNS mixing and flavour oscillation

The weak charged current couples νₑ to e, ν_μ to μ, ν_τ to τ. That vertex is what a source and a detector actually measure, so it defines the flavour basis. Insert the mass-basis resolution of the identity, Σᵢ |νᵢ⟩⟨νᵢ| = 1, and each flavour state is expanded with coefficients ⟨νᵢ|ν_α⟩ = U*_αi. The PMNS matrix is nothing more than that table of inner products between two orthonormal bases, which is why it is unitary, and it differs from the identity because the neutrino mass matrix, whatever mechanism generates it, is not diagonal in the basis where the charged-lepton masses are. With two flavours the table is a real rotation, |νₑ⟩ = cos θ |ν₁⟩ + sin θ |ν₂⟩ and |ν_μ⟩ = −sin θ |ν₁⟩ + cos θ |ν₂⟩, and one angle is the whole story. Note what propagation does not change: the mass content of the travelling state |ψ(L)⟩, |⟨νᵢ|ψ(L)⟩|² = |U_αi|², is the same at every L. Only the relative phases of the components evolve, and that moves the state relative to the flavour basis your detector is built to ask about.

In the mass basis the propagator is diagonal, exp(−iEᵢt/ħ), which is exactly why you rotate there. Neutrinos in every terrestrial experiment are ultrarelativistic — mᵢ ≲ 0.1 eV against E ≳ 1 MeV — so Eᵢ = √(p² + mᵢ²) ≈ p + mᵢ²/2E, and with t ≈ L the phase is Eᵢt ≈ pL + mᵢ²L/2E. The pL piece is identical for all three components and factors out. Restoring units, the relative phase between components i and j is Δm²ᵢⱼc⁴L/(2ħcE), and half of it, Δm²c⁴L/(4ħcE), is what sits inside the sin². In the units experiments use — Δm² in eV², L in km, E in GeV — the numerical factor is 10³/(4 × 10⁹ × 1.973×10⁻⁷) = 1.267. That is all the famous 1.27 is: a unit conversion. Whether you assume the three components share a momentum or share an energy shifts the phase only at relative order m²/E² ≲ 10⁻¹⁴, which no experiment can see.

Write the amplitude and factor out the first phase: A(α→β) = exp(−i m₁²L/2E) Σᵢ U*_αi exp(−i Δm²ᵢ₁ L/2E) U_βi. The prefactor is a global phase, and P = |A|² is blind to it. So the observables are the two independent splittings Δm²₂₁ and Δm²₃₁, with Δm²₃₂ = Δm²₃₁ − Δm²₂₁ fixed by them — never the masses. Expanding the square gives P(α→β) = δ_αβ − 4 Σ_(i>j) Re(U*_αi U_βi U_αj U*_βj) sin²(Δm²ᵢⱼL/4E) + 2 Σ_(i>j) Im(U*_αi U_βi U_αj U*_βj) sin(Δm²ᵢⱼL/2E). The Re term is symmetric under α ↔ β; the Im term is antisymmetric and flips sign for antineutrinos. That second term is CP violation, and it survives only if all three splittings and a non-zero δ are present, which is why two-flavour physics can never show it.

Collapse to two flavours and P(α→β) = sin²2θ sin²(1.27 Δm² L/E). The mixing angle appears only in the prefactor and the splitting only inside the sine, so the two are fitted from different features of the same data: the depth of the dip gives sin²2θ, the L/E at which the dip sits gives Δm². Two real experiments show how far apart that can put you. Daya Bay watches reactor ν̄ₑ at E ≈ 4 MeV over L ≈ 1.65 km, which is L/E ≈ 410 km GeV⁻¹, and reads sin²2θ₁₃ = 0.086 — a floor only 8.6% below one. T2K watches ν_μ at E = 0.60 GeV over L = 295 km, L/E ≈ 492 km GeV⁻¹, and reads sin²2θ₂₃ ≈ 0.99 — a dip that removes 99% of the beam. Nearly the same L/E, nearly the same Δm², utterly different depths, because the two channels read different entries of the third column of U: |Uₑ₃|² ≈ 0.02 for νₑ, |U_μ₃|² ≈ 0.54 for ν_μ.

With Δm²₃₁/Δm²₂₁ = 2.5×10⁻³/7.5×10⁻⁵ = 33, the two phases in Pₑₑ run at very different rates, and which term you can see is a design choice. At Daya Bay's L/E = 412 km GeV⁻¹ the atmospheric phase is 1.31 rad and the solar phase is 0.039 rad, so sin²Φ₃₁ = 0.933 against sin²Φ₂₁ = 1.5×10⁻³, a factor 605. Folding in cos⁴θ₁₃ sin²2θ₁₂ = 0.814 and sin²2θ₁₃ = 0.086 gives a solar contribution of 0.0013 against an atmospheric 0.080 — still a factor 64, and still too big to throw away at 0.2% systematics. KamLAND, at L/E ≈ 5×10⁴ km GeV⁻¹, has the opposite problem: Φ₃₁ ≈ 159 rad oscillates far faster than its energy resolution, so sin²Φ₃₁ averages to ½ and the fast term becomes a constant offset rather than a fringe.

Two approximations were smuggled in. Treating each mass eigenstate as a plane wave gives it infinite extent, so the three components never stop overlapping; real wave packets separate after a coherence length, beyond which the interference terms damp and P settles at its average. And vacuum propagation is not what solar or long-baseline neutrinos do: coherent forward scattering off electrons adds V = √2 G_F nₑ to the νₑ diagonal of the effective Hamiltonian only, and that term flips sign between ν and ν̄, so the ordering of the states — the sign of Δm²₃₁, invisible to a two-flavour vacuum fit because sin² is even — becomes measurable in matter. What no oscillation experiment can reach is the absolute scale: m₁ cancelled as a global phase, and the Majorana phases cancel because they multiply U on the right and leave every product U*_αi U_βi untouched.