Non-local field redefinition and effects on path-integral measure Consider the partition function
$$
Z[0] = \int \left[\mathcal{D}A_\mu\right]\left[\mathcal{D}\pi\right] e^{-i \int d^4x \left(-\frac{1}{2}(\partial\pi)^2-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+ \frac{a}{M^2} \pi^2\left(F_{\mu\nu}F^{\mu\nu}\right)\right)}.
$$
where $a$ is $O(1)$ coefficient, suppressed by some mass-scale $M$.
The action is gauge invariant. Is the path-integral measure invariant under the following field redefinition?
$$
A_\mu \rightarrow A_\mu + \frac{\partial_\mu \pi}{\square}
$$
Or it gives an anomalous contribution to the action, being a non-local transformation?
 A: So, I would start by giving a precise definition of $\frac{\partial_\mu \pi}{\square}$, which is
\begin{align}
\frac{\partial_\mu \pi(x)}{\square}&=\square^{-1}_x\int d^4y\,\delta^{(4)}(x-y)\partial^y_\mu \pi(y)=\int d^4y\ \Delta(x-y)\,\partial^y_\mu \pi(y)\\
&=-\int d^4y\ \partial^y_\mu \Delta(x-y)\,\pi(y)=\int d^4y\ \partial^x_\mu \Delta(x-y)\,\pi(y)
\end{align}
where $\Delta(x-y)$ is defined as
$$\square\, \Delta(x-y)=\delta^{(4)}(x-y)$$
the fields transformations are the following
\begin{align}
&A'_\mu = A_\mu+\frac{\partial_\mu \pi}{\square}=A_\mu+\int d^4y\ \partial^x_\mu \Delta(x-y)\,\pi(y)\\
&\pi'=\pi
\end{align}
the action is invariant under this transformation.
It is now possible to write the Jacobian matrix
$$J^i_j(x,x') = \frac{\delta \phi'_i(x)}{\delta \phi_j(x')}=\left[ \begin{array}{cc} \delta^{(4)}(x-x') & \partial_\mu\Delta(x-x') \\ 0 & \delta^{(4)}(x-x')\end{array} \right]$$
remember that functional derivatives generate distributions and must be evaluated at different points. Furthermore the Jacobian can be considered a $2\times 2$ matrix where in each entry there is a matrix with continuos indices $x,x'$. Once the theory is regularized the determinant can be evaluated.
The formula for the determinant of a matrix of matrices is
$$\det\left[ \begin{array}{cc} A & B \\ 0 & D\end{array} \right]=\det(A)\det(D)$$
since in our case we have that $C = 0$ and so
$$\det J = \det(\delta^{(4)}(x-x'))^2 = \exp\left(2\,tr\log \delta^{(4)}(x-x')\right)$$
this determinant is clearly an highly singular object and the theory must be regularized in order to evaluate it, we can choose 
$$\delta^{(4)}(0)=\lim_{\Lambda\rightarrow\infty} \int \frac{d^4p}{(2\pi)^4}\, \exp\left({-\frac{D^\mu D_\mu}{\Lambda}}\right)$$
where $D_\mu=\partial_\mu+i A_\mu$ and $D^\mu D_\mu$ is the gauge invariant laplacian. You see that
$$\det J=\exp\left( 2\,\int d^4x\,\delta^{4}(0)\, tr\, \log 1\right)=1$$
where $1$ is the trivial transformation $\pi'=\pi$ and $A'_\mu =A_\mu$. So you see that the $tr\log 1=0$ and the transformation is not anomalous since the regularizating function does not contribute to the trace. The chiral anomaly has a nontrivial trace of the log with the regularized delta $\int d^4x\,tr\,\delta^{4}(0) \log M\neq0$ and therefore you get the anomaly etc, in this case it is zero.
Another way of seeing that I think is that the $\pi$ field can be exactly integrated out
$$Z=\int \mathcal{D} A\, \det\left(\square-\frac{2a}{M^2} F^{\mu\nu}F_{\mu\nu}\right)^{-\frac{1}{2}}\, e^{i\int d^4x\, \frac{1}{4}F^{\mu\nu}F_{\mu\nu}}$$
and so the field transformation becomes a simple shift in the field which leaves the path integral measure invariant. Furthermore a gauge anomaly would change the gauge invariance of the effective action
$$\Gamma[A] = -\frac{i}{2}\log\det\left(\square-\frac{2a}{M^2} F^{\mu\nu}F_{\mu\nu}\right)$$
but it is gauge invariant since the field $\pi$ is left unchanged by the tranformation.
Fun fact: if you calculate the determinant and integrate away the the field modes between $M$ and $\Lambda<M$ in the Wilsonian way you get from the expansion of the determinant
$$\det\left(\square-\frac{2a}{M^2} F^{\mu\nu}F_{\mu\nu}\right) = \det(\square)\,\exp\left(\frac{1}{2}\sum_{n=1}^{\infty}\frac{1}{n}(\frac{2a}{M^2})^n\,tr (\square^{-1}F^{\mu\nu}F_{\mu\nu})^n\right)$$
which becomes up to third order of the effective action (numerical factors not guaranteed)
\begin{align}
i\Gamma[A]=&\frac{a}{16\pi^2M^2}(M^2-\Lambda^2)\int d^4x\, F^{\mu\nu}F_{\mu\nu}-\frac{a^2}{16\pi^2M^4}\log\frac{\Lambda}{M}\int d^4x\, (F^{\mu\nu}F_{\mu\nu})^2\\
&+\frac{a^2}{48\pi^2M^4}\left(\frac{1}{\Lambda^2}-\frac{1}{M^2}\right)\int d^4x\,F^{\mu\nu}F_{\mu\nu}\,\square F^{\mu\nu}F_{\mu\nu}\\
&+\frac{a^2}{160\pi^2M^4}\left(\frac{1}{\Lambda^4}-\frac{1}{M^4}\right)\int d^4x\, \square^2 ( F^{\mu\nu}F_{\mu\nu}\, F^{\mu\nu}F_{\mu\nu})\\
&+\frac{a^3}{12\pi^2M^6}\left(\frac{1}{\Lambda^2}-\frac{1}{M^2}\right)\int d^4x\, (F^{\mu\nu}F_{\mu\nu})^3+\ldots
\end{align}
