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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 :-)

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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 $\psi\psi$ 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).

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So you say positively that I can couple a massless chiral Fermion to U(1) but not a Majorana massive one, is it? Well, that could be part of the answer, thanks. Note that in the question I am interested too in the case of a Dirac massive Fermion where the two chiral parts have different couplings to the U(1) field. – arivero Oct 26 '11 at 16:23
I am still not sure of how to read your answer, the "Only" word puts me off. Is it the case that the coupling of the electron to the electromagnetic field is not gauge invariant due to the mass term, and that only a massless electron does work? – arivero Oct 26 '11 at 16:38
@arivero: 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. – Ron Maimon Oct 26 '11 at 20:59
the part before the edit is the original question, yep, bad wording, but it is clear that I was asking beyond the Majorana case. One of the examples I was thinking was to go to Weyl basis and to couple only one of the two fermions. The same trick with Majorana sounds interesting too, and then your insight could apply! Still, I am not sure if the whole Higgs mechanism is needed here, where the gauge group is Abelian. – arivero Oct 26 '11 at 23:05

Sure you can. However, it's not possible to only couple one charged chiral fermion. Gauge anomalies need to be cancelled for consistency. Summing up the cube of all the charges of left handed chiral fermions should end up with zero. With gravity included, it's also needed for the sum of charges to cancel to zero to get rid of mixed gauge-gravity anomalies.

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