Constraints on RH component of fermion triplet under $SU(2)_L$

Question

Consider a fermion $\chi$ whose left-handed part is in a triplet representation of $SU(2)_L$:

$$ \chi_{L} = (\chi^1,\chi^2,\chi^3)_L^{\ \ \text{T}}. $$

The charged current of $\chi_L$ (i.e. its coupling to $W^\pm$) is

$$ - g \left( \overline{\chi^1}\gamma^\mu W_\mu^+\gamma_L \chi^2 + \overline{\chi^2}\gamma^\mu W_\mu^+\gamma_L \chi^3 + h.c.\right).$$

(This can be derived from the term $i\overline{\chi} \gamma^\mu D_\mu \chi$ in a GWS-like Lagrangian with an appropriate Higgs sector).

Question: Are there any constraints on the right-handed part of $\chi$? More specifically, could $\chi_R$ also be in a triplet representation of $SU(2)_L$? (I'm not talking about a $SU(2)_L\times SU(2)_R$ symmetry).

As far as I know, since RH and LH fermions don't interact together, we can use any representation for each part. In the Standard Model, LH particles form doublets while RH particles are singlets under $SU(2)_L$, but this simply comes from experimental data. I guess nothing prevents both $\chi_L$ and $\chi_R$ to be triplets?

A consequence of this is that both representations would have the same quantum numbers (since the electric charge $Q_{EM}=I_{3W}+Y_W/2$ has to be the same for each component). Therefore, their weak coupling would be identical and the fields would be indistinguishable except for their chirality. But I suppose that's not a problem?

Is there a deeper reasoning that would prevent $\chi_R$ to be a weak isospin triplet? (perhaps constraints from the charged/neutral currents, parity, Yukawa coupling, etc. ?)

Answer 1

Well, the triplet representation you wrote is real. There is a term where L and R fermions talk to each other, namely the mass term. For your extreme hypothetical fermions, the mass terms cannot be gauge invariant with the aid of the conventional Higgs doublet, but they can clearly be such for a plain mass term not arising out of a Higgs v.e.v., provided the L and the R are in the same representation, your real triplet one. Nothing inconsistent about this.

So Yukawa couplings are not just unnecessary-- they are problematic. The fermions are then vectorlike, preserving parity. Most model builders sniff at such brutal "just so" mass terms, as they would reflect high energy (GUT?) scales, making them very heavy. And of course no such fermions have been observed, so far.

In fact, if you switched off the EW coupling in your model, you'd realize you have already seen this SU(2) before: it is the SU(2) subgroup of flavor SU(3), the eighfold way of the light quark strong interactions, in the hypothetical limit of equal quark masses for the u,d,s triplet. This SU(3) commutes with $\gamma_5$, so it treats L and R identically.

That is, you may convince yourself that the combination of Gell-Mann matrix generators $$ \frac{\lambda_3 + \sqrt{3}\lambda_8}{2}\equiv S_3 ; \qquad \frac{\lambda_1+\lambda_6}{\sqrt{2}}\equiv S_1; \qquad \frac{\lambda_2+ \lambda_7 }{\sqrt{2}}\equiv S_2 , $$
all hermitean, constitute the conventional spin-1 matrices of SU(2), whose linear combinations may yield the real raising and lowering ones you are using. The common mass term would then be globally SU(2) (and SU(3)) invariant.

Transposing this structure to your model leaving SU(3) of flavor behind, in that fantasy world, and switching on the SU(2) gauge weak interactions, these would couple identically to the L and the R, and preserve parity in that sector. They would also be anomaly free!

The outstanding review by Slansky reviews real and complex representations and their significance and quirks.