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In two-dimensional critical loop models, including the $O(n)$ and Pottsmodels, the spectrum is exactly known, as are a few structure constants orratios thereof. Using numerical conformal bootstrap methods, we study $235$ ofthe simplest 4-point structure constants. For each structure constant, we findan analytic expression as a product of two factors: 1) a universal function ofconformal dimensions, built from Barnes' double Gamma function, and 2) apolynomial function of loop weights, whose degree obeys a simple upper bound.We conjecture that all structure constants are of this form.  For a few 4-point functions, we build corresponding observables in a latticeloop model. From numerical lattice results, we extract amplitude ratios thatdepend neither on the lattice size nor on the lattice coupling. These ratiosagree with the corresponding ratios of 4-point structure constants.

Critical loop models are exactly solvable