Seesaw mass matrix and neutrino masses

[rruben]

Hi

Since a few days I've been confused about the seesaw mass matrix explaining neutrino masses. It is the following matrix:
$\begin{pmatrix} 0 & m\\ m & M \\ \end{pmatrix}$.

As can easily be checked it has two eigenvalues which are given by $M$ and $-m^2/M$ in the limit $M>>m$ (the limit doesn't really matter one is always negative when M and m are positive). It seems really weird to me that you would have a negative mass.

As a lot of papers on the subject (Y. Chikashige, R.N. Mohapatra and R.D. Peccei, Phys. Lett. B98 (1981) 265 and others) will tell you the "mass eigenstates" have masses $M$ and $m^2/M$ without the minus sign. This makes me feel like I'm missing something that makes the sign irrelevant. Could anyone help me with this?

Thanks in advance!

 

[Orodruin]

The sign is not irrelevant, it is related to the Majorana phase of the eigenstate. The mass of the particle is the absolute value of the eigenvalue.

Edit: You will notice that also the Dirac mass matrix
$$
\begin{pmatrix}
0 & m \\
m & 0
\end{pmatrix}
$$
has one positive and one negative eigenvalue.

 

[rruben]

Thanks for your reply. I will look into it a bit more.