# 2. Masses of Neutrinos¶

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

Fig. 2.1 Source: Mena, O., & Parke, S. (2004). Unified graphical summary of neutrino mixing parameters. Physical Review D, 69(11), 117301. doi:10.1103/PhysRevD.69.117301

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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