Anomaly cancellation and fermion number violation

Question

In the standard model, an axial $SU(3)$ currents has anomaly which after quantization leads to the fermion number violation. However, taking all the fermions into account we note that the anomalies cancel. Does the cancellation imply that the net fermion number violation is zero?

Answer 1

I'm not sure what this SU(3) is supposed to mean--certainly not the vectorlike color group. You are probably talking about nonperturbative EW SU(2) anomalies where baryon B and lepton L numbers are violated individually---but their difference B-L is preserved, so baryon creation is accompanied by lepton creation. (These are not perturbative triangle anomalies, which all cancel for the standard gauged generators' currents: they'd better, for consistency of the theory.)

In the Standard Model the parity-violating SU(2) electroweak interaction induces an axial anomaly contribution in the (vector!) baryon number current. Through electroweak instantons, sphalerons, whatever, this leads to the possibilty of baryon and lepton number violation through quantum tunneling processes in the θ-vacuum for the Standard Model fields. This baryon number violation never appears in perturbative calculations, but is generated through nonperturbative transitions between different vacuum states.

Each transition violates baryon number (and lepton number) by ∆B=∆L= ± 3n_f where n_f is the number of families (fermion generations). If you counted antileptons as fermions, you might think of B-L as some type of fermion number, but what's the point?