The Pontecorvo Matrix

A field note on two-flavor mixing

Before you tackle the full three-flavor PMNS formalism, you strip it down to its skeleton. Pontecorvo — Bruno Pontecorvo, who in 1957 proposed that neutrinos might oscillate even before anyone knew whether they had mass — showed that the essential physics lives in two dimensions.

The two-flavor model

Take two flavors: νₑ and ν_μ. Take two mass eigenstates: ν₁ and ν₂, with masses m₁ and m₂. The flavor states are related to the mass states by a single rotation parameterized by one mixing angle θ:

|νₑ⟩ = cosθ |ν₁⟩ + sinθ |ν₂⟩ |νμ⟩ = −sinθ |ν₁⟩ + cosθ |ν₂⟩

This is the Pontecorvo matrix — a 2×2 rotation matrix, the simplest unitary mixing matrix you can write. It is the prototype for everything that follows.

Time evolution and the survival probability

An electron neutrino produced at t = 0 is a superposition of mass eigenstates. Each evolves as e^(−iEᵢt), so at time t the state is:

|ν(t)⟩ = cosθ · e^(−iE₁t) |ν₁⟩ + sinθ · e^(−iE₂t) |ν₂⟩

The probability of detecting it still as an electron neutrino is |⟨νₑ|ν(t)⟩|². Working through the algebra:

P(νₑ → νₑ) = 1 − sin²(2θ) sin²(Δm² L / 4E)

where Δm² = m₂² − m₁², we've used the ultra-relativistic approximation Eᵢ ≈ E + mᵢ²/2E, and replaced t with the propagation distance L.

What the formula tells you

The amplitude of oscillation is controlled entirely by sin²(2θ). If θ = 45° (maximal mixing), sin²(2θ) = 1 and the neutrino oscillates between flavors completely — at maximum it can be found as the other flavor. If θ is small, the oscillation is suppressed.

The oscillation term sin²(Δm²L/4E) creates an interference pattern in the (L, E) plane. Nodes — where no oscillation occurs — line up along curves of fixed L/E. This is the defining experimental signature: plot the survival probability against L/E and you should see a sinusoidal pattern.

For the oscillation to be observable, the coherence length must exceed the detector distance. The coherence length L_coh ≈ 4√2 E² / (Δm² σ_x), where σ_x is the wavepacket width of the produced neutrino. In practice, for laboratory energies and terrestrial distances, this condition is trivially satisfied.

Why two flavors is enough (for now)

Many real experiments can be well approximated in the two-flavor limit. Solar neutrinos at short baselines are dominated by the Δm²₂₁ (solar) scale with θ₁₂ as the relevant angle. Atmospheric neutrinos at high energy are dominated by Δm²₃₁ with θ₂₃. Reactor experiments at short distances probe Δm²₃₁ and θ₁₃.

The Pontecorvo approximation reduces the three-flavor oscillation probability to manageable analytical form while preserving the essential physics: different mass eigenstates accumulate different phases, and flavor detection is a projection onto a misaligned basis.

Historical note

Pontecorvo made his proposal in the context of solar neutrinos, two years before the neutrino was even conclusively detected as a distinct particle. He reasoned — correctly — that if neutrinos had mass and mixings existed, the Sun's electron neutrinos would not arrive at Earth as pure flavor states. The experimental confirmation came forty years later, through Super-Kamiokande and SNO.

References

• Pontecorvo, B. (1957). "Inverse beta-process and non-conservation of lepton charge." Zh. Eksp. Teor. Fiz.
• Bilenky, S. M., & Pontecorvo, B. (1978). "Lepton mixing and neutrino oscillations." Physics Letters B.