# 2. Masses of Neutrinos¶

Neutrino masses are still not determined completely. However we have some possible patterns.

One of the questions we have about the masses of neutrinos is the generation of it.

Note

This figure also gives the terms: normal hierarchy (NH) and invertd hierarchy (IH).

Lepton mixing matrix, can be written as the product of three matrices which stands for a rotation in 23, 13(with a CP phase), 12 respectively. This is called the PMNS mixing matrix.

$\begin{split}\mathbf U &= \mathbf {U_{23}} \times \mathbf {U_{13,\delta}} \times \mathbf {U_{12}} \\ & = \begin{pmatrix} 1 & 0 & 0 \\ 0 &\cos\theta_{23} & \sin\theta_{23} \\ 0 -\sin\theta_{23} & \cos\theta_{23} \end{pmatrix} \begin{pmatrix} \cos\theta_{13} & 0 & e^{i\delta} \sin\theta_{13} \\ 0 & 1 & 0 \\ -e^{i\delta}\sin\theta_{13} & 0 & \cos\theta_{13} \end{pmatrix} \begin{pmatrix} \cos\theta_{12} & \sin\theta_{12} & 0 \\ -\sin\theta_{12} & \cos \theta_{12} & 0 \\ 0 & 0 & 1 \end{pmatrix}\end{split}$

Neutrino Mixing vs Quark Mixing

In fact that quarks also has a mixing matrix that mixes the mass eigenstates to produce flavor states. But note that quarks has a mixing matrix of order

$\begin{split}V_{CKM}\sim \begin{pmatrix} 1 & 0.2 & 0.005 \\ 0.2 & 1 & 0.04 \\ 0.005 & 0.04 & 1 \end{pmatrix},\end{split}$

while neutrino has a mixing matrix of order

$\begin{split}U_{PMNS} \sim \begin{pmatrix} 0.8 & 0.5 & ? \\ 0.4 & 0.6 & 0.7 \\ 0.4 & 0.6 & 0.7 \end{pmatrix}.\end{split}$

It is obvious that the mixing in neutrino is much larger than quark mixing.

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