$Z_\mathcal{M}$ is given in (\[ZM\]).

### A special point {#pmPET}

The theory shows very interesting behavior if, in addition to the single log limit $\beta=-3b_2/\ell^2-2\sigma\ell^2$, we take the scaling limit $b_2\to\infty$. In order to do so it is advised to rescale the auxiliary field $f_{\mu\nu}$. After rescaling $$\hskip 1cm
 \tilde{f}_{\mu\nu}=\sqrt{2b_2}\,f_{\mu\nu}\,,$$ all fields in the linearized action (\[linaction\]) are normalized “canonically”, i.e. there are no dimensionfull parameters multiplying terms such as $(h/k/f)^{\mu\nu}\mathcal{G}_{\mu\nu}(h/k/f)$. If we now take the limit $b_2\to\infty$, keeping $b_2/\kappa^2=1/\kappa'^2$ finite, the action (\[linaction\]) becomes $$\begin{split}\label{pmaction}\hskip 1cm
 \delta^{(2)} S_{\rm pmPET} = \frac{1}{\kappa'^2}\int{\rm d} ^3x\sqrt{g} \, \Big\{
        &  k^{\mu\nu}\mathcal{G}_{\mu\nu}(h) +\tilde{f}^{\mu\nu}\mathcal{G}_{\mu\nu}(\tilde{f})
         -\frac{1}{2\ell^2}(\tilde{f}^{\mu\nu}\tilde{f}_{\mu\nu}-\tilde{f}^2)\Big\} \,.
\end{split}$$

The theory with the linearized action (\[pmaction\]) has much more gauge symmetry then the original theory (\[linaction\]). First, using the self-adjointness of the tensor operator $\mathcal{G}_{\mu\nu}$ we can write the first term as $h^{\mu\nu}\mathcal{G}_{\mu\nu}(k)$. Thus, the vector modes $v_\mu$ coming from the split of the auxiliary field $k_{\mu\nu}$, see eq. (\[tensorsplit\]), are gauge modes similar to the vector part $\xi_\mu$ of $h_{\mu\nu}$. Therefore we have an additional infinitesimal vector gauge symmetry.

Secondly, we note that the $\tilde{f}_{\mu\nu}$ terms match exactly the Lagrangian of partially massless gravity, a certain parameter limit of NMG [@Bergshoeff:2009aq]. One of the massive modes 