I would like to know whether the next calculations are true or not.
1) First, in quantum chromodynamics, a violation of the axial group UA(1) leads to a nonconservation of the axial current:
∂μJ5,μ=2iˉqˆmqγ5q+Nfg28π2ϵμναβtr(GμνGαβ) ,
where Gμν - gluon field strength tensor. The violation of the axial group is connected with the fact that the vacuum of quantum chromodynamics has a complex topological structure, and this eventually leads to an additional term in the Lagrangian:
Lθ=θg216π2ϵμναβtr(GμνGαβ)
2) Second, in addition to the "internal", anomaly of chromodynamics written above, there are external anomalies in the chromodynamics of external currents, the simplest of which corresponds to the process π0→γγ:
∂μJem5,μ=2m(ˉqγ5τ3q)+e216π2ϵμναβFμνFαβ ,
where q - quark field, Fμν - electromagnetic field strength. The corresponding Lagrangian for the anomaly has the form (ˉqγ5τ3q=fπm2π0):
Lem=−Nce296π2fπϵμναβFμνFαβπ0
Isn't this violation related to the topological properties of the theory?
In addition to this anomaly, there is a huge number of others, for example an anomaly corresponding to the process γ→πππ. In order to describe all the anomalies, the Wess-Zumino-Witten action is used. This is possible due to the following statement: any non-Abelian anomaly in four-dimensionality can be represented through the action of Wess-Zumino-Witten in five-dimension (Chern-Simons term) (for further information please refer to https://physics.stackexchange.com/questions/193568/can-the-effective-vertex-for-gamma-to3-pi-be-derived-directly-from-the-anomal and https://physics.stackexchange.com/questions/43317/chiral-anomaly-in-odd-spacetime-dimensions).
W=−iNc96π2∫10dx5∫d4xϵμνσλρTr[−j−μFLνσFLλρ−j+μFRνσFRλρ−12j+μFLνσU(x5)FRλρU†(x5)−12j+μFRνσU†(x5)FLλρU(x5)+iFLμνj−σj−λj−ρ+iFRμνj+σj+λj+ρ+25j−μj−νj−σj−λj−ρ]
My questions:
1) So, do I understand everything correctly?
2) Which books clearly explain the distinction between an axial anomaly in QCD (Theta Vacuum: axion -> 2 gluons) and an axial anomaly in QCD of current (Chern–Simons term: pion->two photons, photon->three pions, ...)?