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?
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1$\begingroup$ Chern-Simons theory action is not invariant under global gauge transformations with boundary corresponding to nontrivial elements of the homotopy group. However, $e^{i k S / \hbar}$ is invariant for $k$ integer multiples of some minimal value $k_0$. (This is not strictly speaking what you're looking for, because the measure is gauge-invariant, thus not an answer; but thought it might be interesting anyway). $\endgroup$– Prof. LegolasovCommented Feb 23, 2018 at 13:44
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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".
