6.4: Neutrino Oscillations

Neutrino oscillation is a phenomenon where a specific flavour of neutrino (electron, muon or tau) is later measured to have different flavour. The probability of measuring a particular flavour varies periodically. The three neutrino states are created by a radioactive decay in a flavour eigenstate as \(|f_1 \rangle \), \(|f_2\rangle \), \(|f_3\rangle\) (electron, muon, tauon). However, these are not eigenstates of energy with a definite mass \(|m_1\rangle \), \(|m_2\rangle \), \(|m_3\rangle \). We can expand the flavour eigenstate using the energy eigenstates as a basis:

\[|f_i\rangle = \sum_j \langle m_j |f_i \rangle |m_j \rangle \nonumber\]

the energy eigenstates show how the wavefunctions behave in time, \(m_j (t) = m_j (0) \text{ exp}(i\omega_j t)\), where \(\omega_j = m_j c^2/\hbar\). \(\omega_{ij} = (m_i − m_j )c^2/\hbar\). Consider an electron neutrino produced by a fusion reaction in the sun, \(\Phi (t = 0) = |f_1\rangle \), its wavefunction then varies as:

\[\Phi (t) = \sum_j |m_j\rangle \langle m_j |f_1\rangle \text{ exp}(i\omega_j t) \nonumber\]

For real neutrinos, the \(\langle m_j |f_1\rangle \) matrix has non-zero, possibly even complex elements everywhere, but here for simplicity we suppose that

\[\langle m_j |f_i \rangle = \begin{vmatrix} a & c & 0 \\ -c & a & 0 \\ 0 & 0 & 1 \end{vmatrix} \nonumber\]

with \(a\) and \(c\) real, time independent and \(a^2 + c^2 = 1\) for normalisation. Our electron neutrono then evolves as \(\Phi (t) = a \text{ exp}(i\omega_1t)|m_1(t) \rangle + c \text{ exp}(i\omega_2t)|m_2(t) \rangle \), so the probability that some time later it is still an electron neutrino is

\[| \langle f_1|\Phi (t) \rangle |^2 = |a^2 \text{ exp}(i\omega_1t) + c^2 \text{ exp}(i\omega_2t)|^2 \\ = a^4 + c^4 + a^2 c^2 (\text{exp}(i\omega_{21}t) + a^2 c^2 \text{ exp} (−i\omega_{21}t) \\ = 1 − 4a^2 c^2 \sin^2 (\omega_{21}t/2) \nonumber\]

which is less than 1: it can somehow “turn into” a muon neutrino. Often, one writes \(a = \sin \theta\) in which case \(4a^2 c^2 = \sin^2 \theta\). \(\theta\) is refered to as a “mixing angle”.

If \(a = c = \sqrt{\frac{1}{2}}\), then with a frequency governed by the difference in masses, the electron neutrino turns completely into a muon neutrino, then back again. With smaller \(c\), there’s always some chance that it will still be an electron neutrino. In reality, it is also possible to oscillate into a tau neutrino. This underlies the “solar neutrino problem”. Detection of solar neutrinos was the subject of the 2002 Nobel prize. Similar oscillation occurs in the kaon system due to a symmetrybreaking effect called “CP violation” subject of the 2008 Nobel prize. Here one of the states is subject to radioactive decay, so a particle not only “turns into” something else, it also disappears when it does so!