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!