quantum field theory - Can I couple a chiral fermion to electrodynamics?

Or, perhaps, the question is in which circumstances can I couple it, and of these, which are the simplest.

For instance, I think that you can not have a massive Dirac fermion and just couple the left part of it to the electromagnetic field: you trigger some vector-axial current and then trigger the anomaly and spoil renormalizability, do you?

And, is the problem different if the fermion is massless, or if we just use a Weyl left fermion without ever adding the right handed counterpart?

(EDIT: this last paragraph could be a source of confusion, I am afraid... Of course in the massive case I still should have both left and right Weyl fermions, but with different coupling to the abelian field, and even one of the couplings could be zero. I am interested on answers for both cases, massive and massless fermions. Pure Majorana mass is of minor importance, but it is fine for completeness :-)