Or, perhaps, the question is in which circumstances can I couple it, and of thes

Question

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 :-)

Explanation / Answer

Only a massless chiral Fermion can be coupled to a U(1) gauge field. If it is massive, you can't. This has nothing to do with renormalizability or anomalies--- the mass term is not gauge invariant. In 2-component notation, the mass is a ?? term. In Majorana notation, the field is real, and the mass term forbid identifying two symmetric parts which can act as the real and imaginary fields which rotate under U(1).

you asked for one chiral field, not more than one. It is absolutely impossible to give a Majorana fermion a simultaneous charge and mass. Dirac is two Majoranas, the real and imaginary part, if you like, and you can rotate one into the other by a U(1). This principle is what requires all the Fermions in the standard model to start off massless, becoming massive only by the Higgs mechanism

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