Solar neutrinos
Nuclear reactions in the Sun produce electron neutrinos \(\nu_e\). Measurements of the solar neutrino flux proved that nuclear reactions take place in the Sun. The deficit of the measured flux \(\Phi(E)_{\nu_e}\) was explained by Mikheyev, Smirnov and Wolfenstein as a result of neutrino oscillations in the matter of the Sun. This made it possible to measure the mixing angle \(\theta_{12}\) and the difference of the squared neutrino masses \(\Delta m^2_{21}\equiv m_2^2-m_1^2\).
(i) Compute the cross sections of the reactions \(\nu_e + e^- \to e^- + \nu_e\) (\(W\) and \(Z\) exchange) and \(\nu_\mu + e^- \to \nu_\mu + e^-\) (\(Z\) exchange). Plot them as functions of the neutrino energy for \(E_\nu\in(1,10)\) MeV.
(ii) In the Super-Kamiokande experiment (Japan), solar neutrino interactions are observed by detecting the Cherenkov light of the recoil electrons. In a model with two neutrino types, \(\nu_e\) and \(\nu_\mu\), neutrino oscillations redistribute the fluxes, \(\Phi_{\nu_e}\leftrightarrow\Phi_{\nu_\mu}\). How did physicists conclude that there is a deficit of recoil electrons, if electrons appear in both reactions, \(\nu_e + e^- \to e^- + \nu_e\) and \(\nu_\mu + e^- \to \nu_\mu + e^-\)? Support your answer with numerical estimates.
(iii) Compute the expected number of recoil-electron events per year in a water detector of mass 50 kt, assuming a solar neutrino flux \(\Phi=5.7\cdot 10^{6}\) cm\(^{-2}\) s\(^{-1}\) and a mean energy \(\langle E_\nu\rangle =8.5\) MeV.
(iv) The electron neutrino \(|\nu_e\rangle = \cos\theta|\nu_1\rangle+\sin\theta|\nu_2\rangle\) and the muon neutrino \(|\nu_\mu\rangle = -\sin\theta|\nu_1\rangle+\cos\theta|\nu_2\rangle\) are coherent superpositions of the mass states \(|\nu_{1,2}\rangle\). Because of the Mikheyev–Smirnov–Wolfenstein effect, for energies \(E_\nu\ge 5\) MeV what leaves the Sun is neither \(|\nu_e\rangle\) nor \(|\nu_\mu\rangle\) but the pure mass state \(|\nu_2\rangle\). Revisit items (i), (ii), (iii) in the light of this fact and discuss your conclusions.