# 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!

This page titled 6.4: Neutrino Oscillations is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Graeme Ackland via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.