Consequences of local and global anomaly

Question

Are the physical consequences of anomalies associated with a local symmetry is different from that of a global symmetry? If yes, why? We have global anomaly in the standard model but not local anomaly? Why is that?

Answer 1

If symmetry conserves some important properties of theory (like unitarity) then its explicit breaking makes theory inconsistent. For example, all gauge symmetries must be unbroken due to unitarity of theory. When they become explicitly broken, unphysical states with indefinite metric in Hilbert space arise, and unitarity is lost. That's why we worry about anomalies in gauge theories.

If, however, symmetry isn't gauge one (for example, it may be global chiral symmetry), its explicit breaking doesn't make theory inconsistent. That's why we don't worry about anomaly-induced term of pion-photon interaction: chiral current doesn't interact with EM field, so corresponding symmetry is only global.

Standard model is free from gauge anomalies, because its gauge group is subgroup of SO(10), and there is a fact that a gauge theory with group, which represenations can be constructed as real, is free from anomalies.

If, however, we construct gauge theory (possibly - an extensions of SM) which isn't free from anomalies, we have to think about anomalies cancellation; only in this case theory may be consistent. It can be done by adding new sorts of particles who are charged under gauge group, so they make contribution into anomaly. The charges of particles must be in some sense "tuned" - anomaly coefficient have to be zero.

This is very powefrul method of checking of theories on consistency. Moreover, if some particles are very massive and we work with significantly lower energies, then the integration out of particles doesn't make full decoupling - in an effective action unsuppressed terms of interaction (usually they are called Chern-Simons terms) may remain.