path integral - Symmetries of the theory that are not symmetries of the action nor of the measure?

Can anybody think of an example of any theory in which there is a transformation law that does not leave the action nor the path integral measure invariant, but such that the product of both is invariant so that the transformation is a symmetry?

Answer

The theory of a chiral fermion $\psi$ coupled to a Maxwell field in four dimensions has a famous anomaly in the chiral transformations

$$\psi \rightarrow \psi'=e^{i \gamma_5 \theta} \psi$$ which arises from the non-invariance of the fermion path integral measure $$D\psi' D\bar\psi' = D\psi D\bar\psi \, {\rm\exp}\left( i\frac{\theta}{(4\pi)^2}\int F\wedge F\right)\,.$$ This can be fixed by adding a pseudoscalar (axion) $a(x)$ to the theory, that couples to the Maxwell field as follows $$\mathcal{L}_a = (da + A)\wedge * (da + A) + a \frac{1}{(4\pi)^2}F \wedge F$$ and transforms under the symmetry with a shift $$a \rightarrow a' = a - \theta.$$ The non-invanriace of the classical action cancels the quantum anomaly, giving an example of what you are looking for. This is the toy version of a more general story that goes by the name of "Green-Schwarz anomaly cancellation mechanism".