# Higher point anomalies cancellation from trace

I had asked this question within this one before, but having also made other 2 independent enough questions there, decided to ask this one by itself here.

So, it is a well known fact that the cancellation of the triangle anomalies is enough to guarantee all higher order anomalies also vanish (and Adler and Bardeen proved that no radiation corrections can make the anomalies vanish).

The easier way I have found to understand the first statement is the following: if the triangle anomaly vanish, we must have $$\text{tr}\{t_A\{t_B,t_C\}\}=0$$ (where the $$t$$ are the generators appearing in the vertex). In turn, taking the box anomaly as an example (with an odd number of axial vertices), is proportional to

$$\text{tr}(t_A\{t_B,[t_C,t_D]\})=f_{CDZ}\text{tr}(t_A\{t_B,t_Z\})$$

which equals zero if the triangle does.

The same kind of thing happens with the pentagon graph, that is, a triangle trace can be factored out.

What I don't understand is how the minus sign in the commutator in the above equation appears. More explicitly: the anticommutator in the triangle case is simply a way of condensing the sum of both Bose-related diagrams. But how could a minus sign appear (which happens in the box and penthagon)?

It is easy to conceptually track this down to the momentum reorderings in the loop coming from the permutation of vertices positions, but writing the expression down I couldn't figure it out precisely. If that is the case, what is the technique to relabel those momenta and get the signs right?

If this is too much calculation and figure to be written here I would be extremely happy just with the pointing to a reference which does this calculations.

• Please stop making trivial edits to bump the question into the front page. Thank you. May 26, 2019 at 17:18
• @AccidentalFourierTransform This time it was not the case. It just seemed that the curly brackets of the trace made the expression hard to read because of the anticommutator. In any case, sorry about that. May 26, 2019 at 18:15