Low-energy theorems and linearity breaking in anomalous amplitudes

This study seeks a better comprehension of anomalies by exploring $(n+1)$-point perturbative amplitudes in a $2n$-dimensional framework. The involved structures combine axial and vector vertices into odd tensors. This configuration enables diverse expressions, considered identities at the integrand level. However, connecting them is not automatic after loop integration, as the divergent nature of amplitudes links to surface terms. The background to this subject is the conflict between the linearity of integration and the translational invariance observed in the context of anomalies. That prohibits the simultaneous satisfaction of all symmetry and linearity properties, constraints that arise through Ward identities and relations among Green functions. Using the method known as Implicit Regularization, we show that trace choices are a means to select the amount of anomaly contributions appearing in each symmetry relation. Such an idea appeared through recipes to take traces in recent works, but we introduce a more complete view. We also emphasize low-energy theorems of finite amplitudes as the source of these violations, proving that the total amount of anomaly remains fixed regardless of any choices.