In critical loop models, there exist diagonal fields with arbitrary conformal
dimensions, whose 3-point functions coincide with those of Liouville theory
at cβ€1. We study their N-point functions, which depend on the 2Nβ1
weights of topologically 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.Comment: 7 page