# CP Phases in the PMNS Matrix

I am interested in how to derive the correct number of CP violating phases in the neutrino mixing matrix (or PMNS matrix).

My approach is the following. By hypothesis, the neutrino mixing is given by

$$\nu_{\alpha_L} = \sum_i U_{\alpha i}\nu_{i_L}.$$

So the mixing matrix $U$ have to be a $3 \times 3$ unitary matrix. Hence we have, at most, 9 real parameters. From which we can choose 3 as rotational angles (the equivalent to Euler angles in classical mechanics). Leaving us with 6 phases.

Three of this phases can be absorbed by redefining the $SU(2)$ doublets

$$\pmatrix{\ell \\ \nu_\ell}_L \rightarrow e^{i\phi_\ell} \pmatrix{\ell \\ \nu_\ell}_L.$$

This leaves us with 3 remaining phases, as is expected for Majorana neutrinos.

How does we get rid of the remaining 2 phases for Dirac neutrinos?