Calculation
The chiral anomaly can be calculated exactly by one-loop Feynman diagrams, e.g. the famous "triangle diagram", contributing to the pion decays, and, namely .
Here, a wavy line signifies a photon, and the right-going (left-going) bold lines signify an electron and a positron, alternatingly. The fact that by encircling the triangle an odd number of times, a line which originally may be seen as a positron, returns as an electron, is actually already a signature for the anomalous behavior of the diagram.
This can also be calculated directly from the change in the measure of the fermionic fields under the chiral transformation.
Wess and Zumino developed a set of conditions on how the partition function ought to behave under gauge transformations called the Wess-Zumino consistency conditions.
Fujikawa derived this anomaly using the correspondence between functional determinants and the partition function using the Atiyah-Singer index theorem. See Fujikawa's method.
Read more about this topic: Chiral Anomaly
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