Symmetries in quantum field theory and anomalies
Suppose we have a lagrangian quantum field theory, thus a theory where we can write an action in the form \begin{equation} S = \displaystyle \int d^4 x \; \mathcal L \, \left( \partial_{\mu} \phi , \phi \right) \ . \end{equation} In classical field theory, we define a symmetry as an action on the fields and/or on spacetime such that the lagrangian changes by a total divergence, which does not change the action $S$ (in this premise I made some simplifying assumptions: I take the lagrangian to be time independent and spacetime to be <PERSON> $\mathcal M_4$). Suppose we consider a quantum field theory with the same lagrangian as above. The symmetry found before (as I know) is said to be a classical symmetry of the system. However, at the quantum level, this symmetry may be spoiled. I know two mechanisms that may spoil this symmetry.
1. If we treat the QFT perturbatively, by means of renormalization loop diagrams may add additional terms to our lagrangian which may spoil the symmetry. 669
This may happen also if under the flow of renormalization, some couplings of the initial lagrangian changes and break the symmetry.
2. If we look at the QFT via the partition function, it may happen that the functional measure is not invariant, and it changes by the determinant of the Jacobian of the transformation. This is usually called anomaly. This gives an additional terms to the lagrangian, using the fact that \begin{equation} \text{det}(A) = e^{\text{tr}(\text{log}(A))} \ . \end{equation} Notice also that the logarithm is not a local term in the lagrangian, and thus the anomaly introduce a non-local term.
My first question is whether there are more mechanisms which can spoil a symmetry at the quantum level.
The second question is whether these two that I described are somehow related. For example, it may happen that expanding the log in the formula for the anomaly, we can recast the series of local terms that come from renormalization. Personally, I think they are independent, because I think we may have an anomalous symmetry even in a renormalizable theory, where e.g. the form of the local terms is fixed and only couplings change.
Last, to connect all the dots, is there a systematic approach to the question whether a quantum field theory spoils a symmetry or not? I am really interested in the subject but I find it confusing, probably because of this lack of systematicity (at least in my mind).
Thanks a lot anyone in advance for any comment! 669