In a comment http://motls.blogspot.com.es/2014/07/cms-sees-650-gev-leptoquarks.html#comment-1479399237 to Motl's blog, there are some reference to "chiral leptoquarks". I am guessing that this is an object which is not a Dirac fermion, ie it only exists one of the two chiral components of it. If it is so, can this fermion have QCD and fermion electromagnetic vertices? Because if it is, say, charge -1/3 spin +1/2 and it emits a photon, it should change to spin -1/2 and still have the same charge, shouldn't it?
Why not? The Standard Model, for example, is a chiral gauge theory. Before electroweak symmetry breaking, left- and right-handed quarks are separate, and both have strong and EM interactions. A vector current like ##\bar \psi \gamma^\mu \psi## doesn't change the chirality of the fermion, so there is no problem with a chiral fermion having vector current interactions.
But is it really a vector current? Two of the four components of this ## \psi ## do not exist, they are zero. Or I could imagine they are different of zero but they do not couple to electromagnetism, ie that their EM charge is zero.
Sure; maybe you would prefer to write the vector current out of a two-component Weyl spinor ##\chi## where it would look like ##\chi^{\dagger}_{\dot a} \bar\sigma^{\mu \dot a a} \chi_a## This is a fine vector current; it transforms like a 4-vector and you can write down a theory in which it is coupled to a gauge field: ##\mathcal L = \chi^{\dagger}_{\dot a} \bar\sigma^{\mu \dot a a} (\partial_\mu - i g A_\mu) \chi_a + \cdots## You just have to be careful that any anomalies cancel.
But the Langrangian is not parity invariant anymore, isn't it? Because if I get a electromagnetic lagrangian which is not parity invariant, I feel myself a bit dirty :shy:
Ok I think that the point is that really is not a "vector-like" current, but a "pure V-A" one. This is, while being true that you can build a current V= ##\bar \psi \gamma^\mu \psi##, you can also build an "axial" current A= ##\bar \psi \gamma^\mu \gamma^5 \psi##. In electromagnetism, the combinations V+A and V-A are both of them different of zero. In this theory with a chiral fermion, one of the combinations, say V+A, is zero. Or we could recast the question in terms of representation theory. Usual electromagnetism, as well as color, needs only to use the real representations for quarks and leptons. While there is not an apriori reason for this, one feels a bit uneasy about calling electromagnetism to a theory with complex U(1) representations.