the sum over the loop length $b$ in Eq. (\[lnZ0s\]): $$\begin{aligned}
 \ln Z_0 &=& - \frac{N}{2} L^d \int_{-\pi}^{\pi} \frac{\mathrm{d}^d
  q}{(2 \pi)^d} \ln \left[1-2 K \sum_\mu \cos (q_\mu) \right] \nonumber
  \\ &=& - \frac{N}{2} \Omega \int_{-\pi/a}^{\pi/a} \frac{\mathrm{d}^d k}{(2
  \pi)^d} \ln \left\{2 \sum_\mu \left[1-\cos \left(k_\mu a\right)\right]
  + m^2 a^2 \right\},\end{aligned}$$ where in the last step an irrelevant additive factor is ignored and $\Omega=(L a)^d$ is the volume of the spacetime box. This is the standard representation of the logarithm of the partition function describing a free field theory on a hypercubic lattice. In the continuum limit $a
\to 0$, the momentum integral can again best be carried out by using the Schwinger proper time representation this time of the logarithm, $$\label{lnS}
\ln(a) = \lim_{z \to 0} \frac{1}{z} - \int_0^\infty
\frac{\mathrm{d}s}{s} \, \mathrm{e}^{-s a},$$ where the first term is an irrelevant diverging contribution. Apart from further irrelevant additive constants this leads to $$\begin{aligned}
\label{Z_0S} 
\ln Z_0 &=& \ln \mathrm{Det}^{-N/2}\left( - \partial_\mu^2 + m^2 \right)
  \nonumber \\ &=& - \frac{N}{2} \Omega \int_{-\infty}^\infty
  \frac{\mathrm{d}^d k}{(2 \pi)^d} \ln \left(k^2 + m^2 \right) \nonumber
  \\ &=& \frac{N}{2} \Omega \int_0^\infty \frac{\mathrm{d}s}{s} \,
  \left(\frac{1}{4 \pi s}\right)^{d/2} \mathrm{e}^{-s m^2} ,\end{aligned}$$ where the operator $- \partial_\mu^2 + m^2$ is the continuum 