In Martin's SUSY Primer, he claims:

For the higgsinos and gauginos, [the ability to have a mass term] follows from the fact that they are fermions in a real representation of the gauge group.

As I understand it, in the Standard Model, explicit fermionic mass terms are prohibited due to the chirality of fermionic fields, i.e. the fact that they transform differently under $SU(2)$ symmetry. I believe this constraint is essentially imposed on the theory by the empirical observation of maximal parity violation of the weak interaction.

So are higgsino and gaugino mass terms allowed in the sense that we have no empirical evidence of a similar constraint such that it's still possible (though not certain) that such mass terms are allowed?

I also don't understand what Martin means by the fact that higgsinos and gauginos are in a "real representation of the gauge group". Why is this not the case for SM fermions?


1 Answer 1


As you point out, maximal parity violation means you need to switch between particles and anti-particles to change the "handedness". Equivalently, the standard model, which involves $SU(N)$ gauge groups, is observed to have left handed fermions in the fundamental representation and right handed fermions in the anti-fundamental representation. The key is that these $N$-dimensional representations are different. I.e. given a set of generators for $N$ or $\bar{N}$, they are not similar to their complex conjugates ($SU(2)$ is a bit special here).

The situation would be different for the fundamental of $SO(N)$ which is real. Then you would be able to make a mass term like \begin{equation} \psi_L^a \psi_L^b \delta_{ab}. \end{equation} The adjoint of $SU(N)$ is also real so you could do something like \begin{equation} \psi^a_{L\bar{a}} \psi^b_{L\bar{b}} \delta_a^{\bar{b}} \delta_b^{\bar{a}}. \end{equation} But if you just want one fundamental index on each fermion, there is no way because $\delta_{ab}$ is not an invariant tensor of $SU(N)$, only $\delta_a^{\bar{a}}$ or equivalently $\epsilon_{a_1 \dots a_N}$. This forces mass terms to involve both representations and therefore both $L$ and $R$. Note that people who talk about fundamental generators of $SU(N)$ with all real entries are really talking about its complexification $SL(N; \mathbb{C})$.


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