parameter range of $\beta$ and $b_2$ \[see the action (\[linaction\])\], or equivalently the masses $M_+$ and $M_-$ of the two propagating massive gravitons. Subsequently, we specialize to the logarithmic points and subsection \[critloc\] is devoted to the critical loci: the tricritical point, the logarithmic line (single log) where one of the two masses $M_+$ and $M_-$ goes to zero, and the massive logarithmic line (massive log) where $M_+$ and $M_-$ degenerate with each other. Section \[pmPET\] covers a special parameter limit that leads to a partially massless theory. At the critical loci the calculation confirms the conjecture that PET gravity is dual to parity even logarithmic CFTs of rank two and three. This is shown in section \[CFTinterpret\] using the combinatorial counting argument of [@Gaberdiel:2010xv].

The calculation will follow the lead of [@Gaberdiel:2010xv] and all formulas and technicalities that are not explained in detail here can be found in that reference.

Linearized action and ghost determinants
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The semi-classical one-loop contribution to the gravity partition function is given by $$\label{pathint}\hskip 1cm
 Z_{\rm PET}^{\rm 1-loop}=\int\mathcal{D}h_{\mu\nu}\mathcal{D}k_{\mu\nu}\mathcal{D}f_{\mu\nu} \,
                          e^{-\delta^{(2)} S_{\rm PET}} \,.$$ The action in the exponent is the linearized Euclidean action of PET gravity, given by [@Bergshoeff:2012ev] $$\begin{split}\label{linaction}\hskip 1cm
 \delta^{(2)} S_{\rm PET} 