Neutrino Oscillations
Why electron neutrinos turn into muon ones, and the proof that neutrinos have mass.
Flavor eigenstates $|\nu_\alpha\rangle$ ($\alpha = e, \mu, \tau$) — what is produced/detected in weak interactions — are not mass eigenstates $|\nu_i\rangle$ ($i = 1, 2, 3$). They are related by a unitary PMNS matrix:
$$|\nu_\alpha\rangle = \sum_i U_{\alpha i} |\nu_i\rangle.$$Mass eigenstates propagate with phases $e^{-iE_i t}$. For two-flavor mixing with angle $\theta$ and squared-mass difference $\Delta m^2 = m_2^2 - m_1^2$, the survival probability for ultra-relativistic neutrinos with energy $E$ travelling distance $L$ is
$$P(\nu_e \to \nu_e) = 1 - \sin^2(2\theta) \sin^2\!\left(\frac{\Delta m^2\, L}{4 E}\right).$$(Convenient: $\Delta m^2$ in eV², $L$ in km, $E$ in GeV, prefactor $1.27$.)
Oscillations were established by atmospheric ($\nu_\mu$, Super-K 1998) and solar ($\nu_e$, SNO 2001) experiments — confirming nonzero neutrino mass (Nobel 2015). Mass hierarchy and CP violation in the neutrino sector are still active research.