Exotic perturbative anomaly captured only by higher-loop Feynman graphs, but not by any 1-loop Feynman graph?

Question

My question: Are there any perturbative anomaly captured by higher-loop but not by captured at the 1-loop Feynman graph (say, not enough)?

We are familiar with the text book example of a perturbative anomaly captured by 1-loop Feynman graph, such as the ABJ anomaly.

All anomalies in Green-Schwarz mechanism are also perturbative anomaly captured by 1-loop Feynman graphs. I also know global anomalies of Witten SU(2) and global gravitational anomalies not captured by perturbative Feynman graphs.

My question: Are there any perturbative anomaly captured by higher-loop but not by captured at the 1-loop Feynman graph (say, not enough)?

This post imported from StackExchange Physics at 2020-11-24 18:33 (UTC), posted by SE-user annie marie heart

Comments

In a large class of theories, the Adler-Bardeen theorem, which states that if anomalies vanish at one loop, then they vanish to all orders, has been proved to hold.

This post imported from StackExchange Physics at 2020-11-24 18:33 (UTC), posted by SE-user MannyC

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possible dup.? When is an anomaly one-loop exact?.

This post imported from StackExchange Physics at 2020-11-24 18:33 (UTC), posted by SE-user AccidentalFourierTransform

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I am asking beyond 1-loop, thus it is not a duplicated question -- please do not limit the imagination.

This post imported from StackExchange Physics at 2020-11-24 18:33 (UTC), posted by SE-user annie marie heart