  Self-avoiding walks (SAWs) and loop-erased random walks (LERWs) are two
ensembles of random paths with numerous applications in mathematics,
statistical physics and quantum field theory. While SAWs are described by the
$n \to 0$ limit of $\phi^4$-theory with $O(n)$-symmetry, LERWs have no obvious
field-theoretic description. We analyse two candidates for a field theory of
LERWs, and discover a connection between the corresponding and a priori
unrelated theories. The first such candidate is the $O(n)$-symmetric $\phi^4$
theory at $n=-2$ whose link to LERWs was known in two dimensions due to
conformal field theory. Here it is established in arbitrary dimension via a
perturbation expansion in the coupling constant. The second candidate is a
field theory for charge-density waves pinned by quenched disorder, whose
relation to LERWs had been conjectured earlier using analogies with Abelian
sandpiles. We explicitly show that both theories yield identical results to
4-loop order and give both a perturbative and a non-perturbative proof of their
equivalence. This allows us to compute the fractal dimension of LERWs to order
$\epsilon^5$ where $\epsilon=4-d$. In particular, in $d=3$ our theory gives
$z_{\rm LERW}(d=3)= 1.6243 \pm 0.001$, in excellent agreement with the estimate
$z = 1.624 00 \pm 0.00005$ of numerical simulations.
