I'm trying to figure out why a left-handed Majorana mass term is mathematically forbidden by the $SU(2)_L$ symmetry in the context of the seesaw model.
As far as I get it, it is because the left handed neutrino in the Majorana mass term is part of a $SU(2)$ doublet, such that it would be changed in a $SU(2)$ symmetry transformation, e.g. $v_L => e_L$. And for the right handed Majorana mass term it would be allowed, because the right handed neutrino is part of a singlet and not a doublet. Can anybody show me the explicit gauge transformation for that term that breaks symmetry?
Somewhere I've also read that $v_L$ possesses non-zero isospin and hypercharge and that's why the mass term is actually forbidden. That's basically the same like in my interpretation from above, right?
I just want to be sure that I don't get anything wrong here.