Neutrino Oscillation Journey

1 · Two kinds of neutrino states

A neutrino is born and detected as a flavor state — ν_e, ν_μ, ν_τ — labeled by the charged lepton it pairs with at a W-boson vertex. But it travels as a mass state — ν₁, ν₂, ν₃ — the states with definite mass that propagate simply. These are not the same states. Each flavor is a fixed quantum superposition of the three masses:

Nothing exotic happens to the neutrino in flight — this is one state written in two bases, exactly like expressing a spin state along a rotated axis. The change-of-basis matrix U is the PMNS matrix (Pontecorvo–Maki–Nakagawa–Sakata), and it is unitary, so probabilities always total 1.

2 · The mixing matrix — live

Row α says what flavor α is made of. The shares |U_αi|² below are computed from the current parameter values — open the Research Sandbox and drag a slider to watch them respond. Unitarity makes every row sum to 1:

Try it: drag θ₂₃ and the μ and τ rows trade shares. Drag δ_CP and the four middle entries (U_μ1, U_μ2, U_τ1, U_τ2) shift too — their magnitudes contain cos δ.

3 · Three angles and a phase

Any such mixing factors into three plane rotations plus one complex phase that cannot be removed by redefining the fields:

s_ij = sin θ_ij,  c_ij = cos θ_ij.  Current values:  θ₁₂ = θ₂₃ = θ₁₃ = δ_CP =

Each angle owns a visible feature of the curves:

• θ₂₃ (with Δm²₃₁) drives the big, fast ν_μ↔ν_τ wiggle — the atmospheric sector. θ₂₃ ≈ 45° means near-maximal mixing, which is why the swap is so complete.
• θ₁₂ (with Δm²₂₁) drives the slow solar wiggle — visible at long baselines: load KamLAND or JUNO from the experiment menu.
• θ₁₃ is small. It sets how much ν_e appears in a ν_μ beam, and the depth of the reactor ν̄_e dip — load Daya Bay; that dip depth is sin²2θ₁₃.
• δ_CP enters only through interference — section 5.

4 · From phases to wiggles

Each mass state accumulates quantum phase as it travels. Ultra-relativistically E_i ≈ E + m²_i/2E, so after a distance L, states i and j slip out of phase by Δm²_ijL/2E. Detecting flavor β projects the superposition back, and the pieces interfere:

With two flavors this collapses to the one formula worth memorizing:

sin²2θ sets the depth of the wiggle; Δm²L/E sets where it wiggles; 1.267 is nothing but ħ, c, and unit bookkeeping. The full three-flavor expansion — the expression this app evaluates at every pixel of the race track — is

5 · Why δ_CP changes the answer

CP conjugation swaps neutrinos for antineutrinos. In the amplitude, that conjugates every matrix element: U → U^*, i.e. δ → −δ. Look at the formula above: the Re[…] terms don’t care, but the highlighted interference term flips sign. That sign flip is the entire story:

Three consequences you can test in this app:

• Survival is CP-blind. For β = α the product Q_ij is real, so the Im term vanishes identically — no disappearance experiment can measure δ_CP, even in principle. (This is why reactor ν̄_e experiments give such a clean θ₁₃.)
• One number controls all of it. Every CP asymmetry, in every channel, is proportional to the Jarlskog invariant  J = s₁₂c₁₂ s₂₃c₂₃ s₁₃c₁₃² sin δ = . If any angle were 0° or 90°, or δ were 0° or 180°, J = 0 and neutrinos and antineutrinos would oscillate identically. With only two flavors, no phase survives rephasing at all — CP violation needs all three.
• See it live.

Caveat for real experiments: passing through matter also splits ν from ν̄ (the Earth contains electrons, not positrons) — a fake CP asymmetry that must be separated from δ_CP; World 2’s toy matter effect shows the idea. Conventions: this app uses the standard PDG parametrization, and the ordering toggle sets the sign of Δm²₃₁.