DrDu said:
Maybe someone can give me an idea of this anomaly cancellation argument.
The anomalies in 4-dim. quantum field theory are usually contained in so-called triangle graphs with three external gauge boson lines and three inner fermion lines forming a triangle. This graph is divergent and has to be renormalized. It contributes to two different continuity equations (in quantum field theory replaced by so-called Ward identities). Now one finds that in order to keep one current conservation law one must violate the other one; the renormalization is not able to protect both conservations laws which means that one current becomes anomalous.
Usually one choses the renormalization such that the gauge current (e.g. the electromagnetic current) derived from a local gauge symmetry remains conserved whereas the other current (axial current) derived from a global symmetry becomes anomalous (the two currents are due to the fact that one can project to left- or right-handed fermions; therefore instead of calling in axial anomaly sometimes one refers to it as chiral anomaly). The reason for gauge current conservation is renormalizibility, i.e. consistency of the theory. The anomaly itself has physical effects which can be seene.g. in pion decay and the mass of the eta-prime meson.
Now in electroweak interactions the left and the right handed currents become gauge currents which are conserved separately in classical field theory. But due to the above arguments that means that one can no longer protect both gauge symmetries in the current conservation b/c one must necessarily break gauge invariance either in the left or in the right handed sector.
That would mean that the theory becomes inconsistent, but there is one way to protect both gauge symmetries in the left- and in the right-handed sector. Roughly speaking each fermion species comes with its own triangle anomaly. But the external gauge bosons do not carry any fermion information which means that in order to calculate the total contribution of the triangle graphs to the current conservation one has to sum over all triangle graphs. Each triangle comes with a pre-factor that is related to the (electroweak) charges of the inner fermion in that graph. So the sum over all graphs vanishes iff the sum over these pre-factors vanishes which results in a constraint for the electroweak charges of the fermions.
In the SM the anomaly has to cancel in each generation, which essentially means that given the electric charge of the fermions (up, down, e, e-neutrino) and the multiplicity of the fermions in the graph (e.g. counting different colors) the electric charges must fulfill certain consistency conditions. In addition it means that one generation has to be complete. That was one reason for the existence of the top quark: an incomplete 3rd generation (., bottom, tau,tau-neutrino) would cause the gauge current to become anomalous whereas a complete 3rd generation (top, bottom, tau,tau-neutrino) saves the consistency.
So once one knows the electric charge of the electron the charges of up and down are essentially fixed: afaik one can derive both Q(d) = -Q(u)/2 and Q(d) = Q(e)/3; the last equation is related to the fact that there are three colors - and the different colors are counted individually in the triangle diagrams.
But using these equations one automatically finds that Q(proton) = 1 and Q(neutron) = 0. In addition one finds Q(proton) = - Q(electron).
I'lltry to find some references where all this is derived rigorously.