Anomalies At Different Energy Scales
Quantum anomalies were discovered via the process of renormalization, when some divergent integrals cannot be regularized in such a way that all the symmetries are preserved simultaneously. This is related to the physics at the UV (i.e. at high energies). However, due to Gerard 't Hooft's anomaly matching condition, any chiral anomaly can be described either by the UV degrees of freedom (those relevant at high energies) or by the IR degrees of freedom (those relevant at low energies). Thus one cannot cancel the anomaly by some UV completion of the theory — an anomalous symmetry is simply not a symmetry of the theory, even though classically it looks as if it is.
Read more about this topic: Anomaly (physics)
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