Et Tu, Anomaly ?
For more than 20 years of my research careers in field theory, this is the first time that I was very depressed. It turns out that the chiral anomaly equation is simply a wrong equation. In the 2nd edition of the textbook of "Symmetry and Its Breaking in Quantum Field Theory", I included some of the explanation of the anomaly equation in two dimensional field theory. At the time when the anomaly problem became clear, the revised edition of the textbook was about to appear. Neutrino
What is then the anomaly ? The original discussion started from the decay of neutral pion into two gamma rays. This decay rate is quite large in comparison with the decay rate due to the weak interaction process such as charged pion decay. This is, of course, clear since the neutral pion can decay via electromagnetic interactions in terms of the triangle Feynman diagrams. The evaluation of the T-matrix of the neutral poin decay rate can be calculated, and it is finite. The calculation is explained quite in detail in the textbook of 'Fields and Particles' by Nishijima, and it is indeed consistent with the observed value.
In this sense, there is no anomaly in the triangle diagrams of the neutral pion decay process. However, if one calculates the triangle diagram, then one finds an apparent linear divergence, though its divergence disappears at the evaluation of the Trace. For the neutral pion decay, there is no ambiguity since its decay rate is observed experimentally. ThereforeCpeople started to discuss the triangle diagrams with the axialvector coupling to the fermion pair. By regularizing the T-matrix of this Feynman diagram, they obtained the anomaly equation.
Now a question is as to whether the T-matrix of the triangle diagrams with the axialvector coupling should have the divergences or not. Surprisingly, as one can easily calculate by oneself the triangle diagrams with the axialvector coupling have neither linear divergence nor logarithmic divergence. This can be proved before the momentum integration since the Trace in the T-matrix evaluation clearly shows that the T-matrix vanishes to zero exactly. Furthermore, one can calculate the Z boson decay rate into two photons. I believe that the branching ratio of the Z boson decay should be observed near future even though the predicted value is smaller than the experimental upper limit by three orders of magnitude.
We should understand nature by experiments, and this is always clear and must be kept true. Recent CERN experiments suggest that there should be no Higgs particles at 95 % confidence level. This problem is well discussed in my textbookCand therefore I will not go into details here. But it is clear by now that we should reconsider the standard model of the weak interaction.
Higgs Mechanism
If you want to understand more in detail what is then the anomaly equation in physics, you can read the following papers.
Anomaly