obtained simply from Eq. (\[prob\]) by putting the end-to-end vector $n-n'$ to zero there. Equation (\[Z0\]) then reads explicitly $$\label{lnZ0s}
  \ln Z_0 = \frac{N}{2} L^d \int_{-\pi}^{\pi} \frac{\mathrm{d}^d q}{(2
  \pi)^d} \sum_{b=1}^{\infty} \frac{1}{b} \left[ 2 K\sum_\mu \cos
  (q_\mu) \right]^b.$$ Repeating the same steps leading to Eq. (\[discrete\]), one finds $$\begin{aligned}
\label{lnZ0}
  \ln Z_0 &=& \frac{N}{2} L^d \sum_{b=1}^{\infty} \int_{-\pi}^{\pi}
  \frac{\mathrm{d}^d q}{(2 \pi)^d} \frac{1}{b} \, \mathrm{e}^{- (q^2 +
  m^2 a^2) b /2d} \nonumber \\ &=& \frac{N}{2} L^d \sum_{b=1}^{\infty}
  \frac{1}{b} \, \left( \frac{d}{2 \pi b} \right)^{d/2} \mathrm{e}^{-
  m^2 a^2 b/2d} .\end{aligned}$$ It follows that the logarithm of the partition function has the form[@Copeland:1988bh; @Copetal] $$\label{Zell}
\ln Z_0/L^d \sim \sum_b \ell_b,$$ with the loop distribution $$\label{ell}
\ell_b \sim b^{- \tau} \, \mathrm{e}^{- \theta b}.$$ denoting the average number (per lattice site) of times a closed worldline of $b$ steps occurs on the lattice. The parameter $\theta
\propto m^2$ physically determines the line tension of the loops, while the algebraic factor in the loop distribution, characterized by the exponent $\tau=d/2+1$, is an entropy factor, giving a measure of the number of ways a loop of $b$ steps can be embedded in the lattice.

A more familiar expression for $\ln Z_0$ is obtained by again first carrying out 