arbitrary shape and length.[@Samuel] Being a noninteracting theory, the closed worldlines can freely intersect themselves or other worldlines and even share bonds. On the spacetime lattice, the loop gas (\[loopgas\]) can be expressed as a sum over all possible tangles of closed paths $$\label{phantom} 
Z_0 = \sum_\mathrm{loops} K^b N^l,$$ with $b$ and $l$ denoting the total number of occupied bonds and separate loops forming a given tangle. In Eq. (\[phantom\]), the loops are no longer considered to be rooted, i.e., they can start and end at any spacetime coordinate in the system. In comparison with rooted closed paths, i.e., closed paths all starting at a given spacetime coordinate, this yields a spacetime volume factor $\Omega$. In the continuum, Eq. (\[phantom\]) translates into the path integral representation $$\label{explicit}
Z_0 = \sum_{l=0}^{\infty} \frac{1}{l!}  \left(\frac{N}{2}\right)^l
\prod_{r=1}^l \left[ \Omega \int_0^\infty \frac{\mathrm{d} s_r}{s_r}  \oint
\mathrm{D} x_r(s'_r) \right] \mathrm{e}^{-W^{(l)}_0},$$ as follows from Eq. (\[lnZ0pi\]). Here, $W^{(l)}_0$ denotes the extension of the single particle action (\[W0\]) to $l$ particles: $$\label{Wl0}
W^{(l)}_0 = \sum_{r=1}^l \int_0^{s_r} \mathrm{d} s'_r \,
\left[\tfrac{1}{4} \dot{x}_r^2 (s'_r) + m^2 \right].$$

### $|\phi|^4$ Field Theory {#sec:phi4}

We next perturbatively include the interaction in the theory (\[Sor\]). At the level of the partition function (\[Z\]), this 