${\gamma }_5$ algebra ambiguities in Feynman amplitudes: Momentum routing invariance and anomalies in $D=4$ and $D=2$

We address the subject of chiral anomalies in two- and four-dimensional theories. Ambiguities associated with the ${\gamma }_5$ algebra within divergent integrals are identified, even though the physical dimension is not altered in the process of regularization. We present a minimal prescription that leads to unique results and apply it to a series of examples. For the particular case of Abelian theories with effective chiral vertices, we show (i) its implication on the way to display the anomalies democratically in the Ward identities, (ii) the possibility to fix an arbitrary surface term in such a way that a momentum routing independent result emerges—this leads to a reinterpretation of the role of momentum routing in the process of choosing the Ward identity to be satisfied in an anomalous process—and (iii) momentum routing invariance is a necessary and sufficient condition to assure vectorial gauge invariance of effective chiral Abelian gauge theories. We also briefly discuss the case of complete chiral theories, using the chiral Schwinger model as an example.

Figure 1

Triangle diagrams which contribute to the ABJ anomaly. We label the internal lines with arbitrary momentum routing.

Figure 2

Diagrams upon which the diagrammatic proof of gauge invariance is constructed.

Figure 3

Pictorial representation of the Ward identities.

Figure 4

Pictorial representation of the Ward identities, showing its connection to momentum routing invariance.

Figure 5

One- and two-point functions needed for the two-loop diagrammatic proof of gauge invariance.

Figure 6

Pictorial representation of two-loop Ward identity for the one-point function.

Figure 7

Pictorial representation of the result of each tadpole insertion.

Figure 8

Pictorial representation of the relation between gauge and MRI for two- and three-point two-loop diagrams (a) and (b), respectively. The external momentum $p$ acts as an arbitrary routing and making the right-hand side zero is the MRI condition while making the left-hand side zero is the gauge invariance condition.