For atmospheric neutrinos the relevant two-state mixing is between and

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

For atmospheric neutrinos the relevant two-state mixing is between νμ\nu_\mu and ντ\nu_\tau, described (in a two-flavour approximation) by an effective mass doublet ∣ν2′⟩|\nu_2'\rangle, ∣ν3′⟩|\nu_3'\rangle and a mixing angle θ23\theta_{23}: ∣νμ⟩=cos⁡θ23 ∣ν2′⟩+sin⁡θ23 ∣ν3′⟩,∣ντ⟩=−sin⁡θ23 ∣ν2′⟩+cos⁡θ23 ∣ν3′⟩,|\nu_\mu\rangle = \cos\theta_{23}\,|\nu_2'\rangle + \sin\theta_{23}\,|\nu_3'\rangle, \qquad |\nu_\tau\rangle = -\sin\theta_{23}\,|\nu_2'\rangle + \cos\theta_{23}\,|\nu_3'\rangle, with the near-maximal value θ23=45.0∘\theta_{23} = 45.0^\circ. ((a)) Invert these relations to write the mass state ∣ν3′⟩|\nu_3'\rangle as a combination of ∣νμ⟩|\nu_\mu\rangle and ∣ντ⟩|\nu_\tau\rangle. ((b)) A neutrino is produced purely as the mass state ν3′\nu_3' and later undergoes a charged-current interaction. What is the probability it interacts as a νμ\nu_\mu, and as a ντ\nu_\tau? ((c)) Explain why this even split, rather than a definite flavour, is the physical content of near-maximal θ23\theta_{23} mixing, and contrast this with the CKM matrix, whose mixing angles are all small.

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