  In critical loop models, there exist diagonal fields with arbitrary conformal
dimensions, whose $3$-point functions coincide with those of Liouville theory
at $c\leq 1$. We study their $N$-point functions, which depend on the $2^{N-1}$
weights of combinatorially inequivalent loops on a sphere with $N$ punctures.
Using a numerical conformal bootstrap approach, we find that $4$-point
functions decompose into infinite but discrete linear combinations of conformal
blocks. We conclude that diagonal fields belong to an extension of the $O(n)$
model.
