# Seesaw mass matrix and neutrino masses

## Main Question or Discussion Point

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?

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Orodruin
Staff Emeritus
Homework Helper
Gold Member
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.