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This Bardeen counterterm is an elusive beast, I must say. Yet I will share what I have found and understand: Define a $\mathrm{U}(1)$ gauge theory by writing its action in left- and right-handed chiral spinors as $$S_{\mathrm{chiral}}[A] = \int \bar \psi_L (\mathrm{i} {\not \hspace{-4px} \partial} - \not \hspace{-5px} A)\psi_L + \bar \psi_R (\mathrm{i} ... 1 A pretty exhaustive summary in the context of Standard Model already exists in the following source: ''Dynamics of the Standard Model'' - Donoghue, Golowich, Holstein, Chapter 3 - Symmetries and Anomalies A limited preview can be found here. (Embarrassingly though, the very first page of the chapter is excluded from Google's preview!) But here's the ... 3 I'd say that there is not a systematic summary of the status of symmetries on particle physics, but if any, it should be spread all over the PDG review. However, I'd like to comment on a few points. So far Lorentz symmetry is exact on all sectors.{}^\dagger Scaling (part of the conformal transformations) is broken once an energy scale is introduced in ... 6 Anomalies (not anamolies) are a whole subject whose basics are covered by one or several chapters of almost any good enough quantum field theory textbook so it's counterproductive to retype this whole chapter here. But generally, in quantum field theory, anomalies are quantum mechanical effects breaking symmetries that exist in the classical theory – ... 1 Alright, I will try to answer why we need Dirac eigenstates in this procedure, but I am not sure if it is anything more that the tautology that the Fujikawa method is precisely defined by using the Dirac eigenstates. Let me briefly recap what the idea is: (All of this is for a Euclidean theory.) We consider the infinitesimal local transformaton$$\psi(x) ...