\bigg]^2 \,.$$ Here we use the notation of [@Giombi:2008vd], where $q=\exp(i\tau)$ with $\tau=\tau_1+i\tau_2$ and $\tau_1$ and $\tau_2$ being related to the angular momentum $\theta$ and the inverse temperature $\beta$.\
This can be compared to the partition function of single-particle excitations in a parity even rank-three LCFT.\
*Single log* Now we consider the critical line in parameter space where one of the massive modes degenerates with the massless graviton, $\beta=-3b_2/\ell^2-2\sigma\ell^2$, which still implies $\bar{\sigma}=0$. An interesting scaling limit is $b_2\to\infty$ which we will treat separately in section \[pmPET\].

The calculation of the transverse-traceless part is simply done and yields $$\label{ZlogTT}\hskip 1cm
 Z_{\rm log}^{TT}= \big[\det(-\nabla^2-\frac{2}{\ell^2})_2^{TT}\big]^{-1}
                   \big[\det(-\nabla^2-\frac{3}{\ell^2}-\frac{2\sigma\ell^2}{b_2})_2^{TT}\big]^{-1/2} \,.$$ For the vector part we integrate over $v^T$ to obtain a delta function for $u^T$ and find the same result we obtained earlier, see eq. (\[ZT\]). For the scalar part we redefine $\varepsilon=\alpha\varepsilon'$ and $\delta'=\delta+\varepsilon'/2+\bar{F}\ell^2/6$ with $\alpha=b_2/\ell^2+2\sigma\ell^2$. These redefinitions do not produce further ghost determinants. The point where $\alpha$ becomes zero is the tricritical point, which we already covered in the previous subsection. After integration over $h$, $\bar{K}$, $\varepsilon'$, $\delta'$ and $\bar{F}$ we find the result (\[ZS\]). Collecting all contributions the result is $$\begin{split}\label{Zlogpetdet}\hskip 1cm
 Z_{\rm log~PET}^{\rm 1-loop}= Z_{\rm Ein}\cdot \big[\det(-\nabla^2-\frac{2}{\ell^2})_2^{TT}\big]^{-1/2}
  \big[\det(-\nabla^2-\frac{3}{\ell^2}-\frac{2\sigma\ell^2}{b_2})_2^{TT}\big]^{-1/2}\,.
\end{split}$$ Setting $b_2=-2\sigma\ell^4$ 