In a two-flavour approximation

PHYSICS · Multi-part · Fermi University Physics · Particle Physics & Quantum Fields · Flavour, neutrinos and CP violation · Flavour, neutrinos and CP violation

Symmetry and Flavour, Ch Neutrino masses, mixing and oscillations

Problem

In a two-flavour approximation, the electron and muon flavour states are related to the mass states ∣ν1⟩|\nu_1\rangle, ∣ν2⟩|\nu_2\rangle by a rotation through the solar mixing angle θ=33.7∘\theta = 33.7^\circ: ∣νe⟩=cos⁡θ ∣ν1⟩+sin⁡θ ∣ν2⟩,∣νμ⟩=−sin⁡θ ∣ν1⟩+cos⁡θ ∣ν2⟩.|\nu_e\rangle = \cos\theta\,|\nu_1\rangle + \sin\theta\,|\nu_2\rangle, \qquad |\nu_\mu\rangle = -\sin\theta\,|\nu_1\rangle + \cos\theta\,|\nu_2\rangle. Both mass states propagate coherently once produced. ((a)) Invert these relations to write ∣ν1⟩|\nu_1\rangle and ∣ν2⟩|\nu_2\rangle in terms of ∣νe⟩|\nu_e\rangle and ∣νμ⟩|\nu_\mu\rangle. ((b)) A neutrino is produced by a charged-current interaction as a νe\nu_e. If its mass were measured (in principle), what is the probability the result is ν1\nu_1, and what is the probability it is ν2\nu_2? ((c)) Check that these two probabilities sum to 11, and explain in one sentence why a νe\nu_e does not have a definite mass.

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