# Left-handed Majorana mass term forbidden by $SU(2)$?

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.

The left-handed neutrino is a 2-spinor field $\eta_A$, $A=0,1$, and the Majorana mass term is a bilinear, $\Delta L = \pm 2$ term without the complex conjugation in each term, $$m\cdot \eta_A \eta_B \cdot \epsilon^{AB} + \text{Hermitian conjugate}$$ Note that this Majorana term is the only bilinear term without derivatives that one may construct from a single 2-component spinor (even if the complex conjugate is allowed as well).
However, in the $SU(2)$ gauge-invariant theory, $\eta_A$ is a component of a doublet, so it carries an extra index $i=1,2$ where $i=1$ is the neutrino and $i=2$ is the left-handed electron. So the Majorana term would have to be $$m\cdot \eta_{Aj} \eta_{Bk} \cdot \epsilon^{AB}t^{jk} + \text{Hermitian conjugate}$$ However, the only $SU(2)$-covariant tensor $t^{jk}$ with two indices of the same kind is $\epsilon^{jk}$ which is however antisymmetric, so with this choice of $t$, the term would vanish. Note that $t^{jk}=\delta^{jk}$ is not possible because $\delta$ must have one upper index and one lower index, not two indices of the same kind.
Equivalently, if there were an $SU(2)$-invariant mass term "neutrino . neutrino", by the neutrino-electron symmetry which is an element of $SU(2)$, there would also have to be a term "electron . electron" which would however violate the charge by $\Delta Q = 2$, so it clearly cannot be gauge-invariant.
One may show that the Majorana term would violate the hypercharge, too. Clearly, the neutrino left-handed field has a nonzero hypercharge $Y$, so the bilinear term $\eta\eta$ has a nonzero (double) value of the hypercharge, too, so such a term cannot be gauge-invariant – a faster way to show the same result.
• Dear Michael, 2-component spinors in relativity are the "same" as the non-relativistic 2-component spinors e.g. from the "Pauli equation", one just allows the action of $\sigma^0\sim 1$, the time component, as well. ... A zero term is always invariant under everything. What is non-invariant would be a particular nonzero term. For example, if you choose my 2nd Ansatz and only choose the component $t_{\nu,\nu}$ nonzero, then you reproduce the original Majorana term for neutrinos and nothing else, but a tensor 2x2 whose only component is $t_{22}$ is clearly not invariant under $SU(2)$, right? – Luboš Motl May 27 '16 at 3:42
• But there is no "the" term that we should consider, the $SU(2)$ completion of the neutrino Majorana term could hypothetically be different than one I just described. That's why one must consider all alternatives, which is what I did, and prove that the only solution is zero, which is what I did. What you propose now is simply not a general way to deal with the problem because you haven't specified the "right and only candidate" that should be tested on $SU(2)$ invariance. – Luboš Motl May 27 '16 at 3:44