# Vector-like Representation of fermions

In the literature, they often extend the Standard Model by adding a so-called vector-like fermion which is a multiplet invariant under $$SU(2)_L\times U(1)$$. The left- and right-handed components of vector-like multiplet are supposed to transform the same way, under the symmetry group. As a result, their mass terms, are not forbidden by the symmetry. See for an example see these slides.

I have problem understanding the gauge transformation. If the multiplet $$\Psi$$ is invariant under $$SU(2)_L$$, only the left-handed particles $$\Psi_L =P_L \Psi$$ should be charged under the gauge symmetry. The right-handed component $$\Psi_R$$ should not be charged under $$SU(2)_L$$.

So my understanding is that vector-like multiplet should be in a representation of $$SU(2)$$ that is different from the usual chiral representation of the SM fermion doublets.

If this is correct, then how is that vector-like representation of fermions related to the SM chiral representation? Is there a linear transformation?

Any helps or comments would be appreciated.

• Minor comment to the post (v2): Please consider to mention explicitly author, title, etc. of link, so it is possible to reconstruct link in case of link rot. – Qmechanic Feb 13 at 11:38