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Restrictions can be imposed on the anomalous terms from general considerations even without fully solving the Feynman diagrams. In fact, the restrictions on the Schwinger term in the commutator given in the question are explicitly described in detail by Roman Jackiw in his review article: Field theoretic investigations in current algebra (section 2.2.). His ...


It seems OP's main question is how to understand the representation of the matter fields of YM theory. The matter fields can in principle transform in any representation $\rho:G\to {\rm End}(V)$ of the local gauge group $G=SU(N)$, e.g. the fundamental, or adjoint representation. Here ${\rm End}(V)$ denotes the algebra of endomorphisms on the vector space $...


Note that, for example, \begin{align} [A_\mu,\partial_\nu]f&=A_\mu\partial_\nu f-\partial_\nu(A_\mu f)\\ &=A_\mu\partial_\nu f-\partial_\nu(A_\mu)f-A_\mu\partial_\nu f\\ &=-f\partial_\nu A_\mu\,. \end{align} So you don't get terms like $A_\mu\partial_\nu$.

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