  We discuss a correlation function factorization, which relates a three-point
function to the square root of three two-point functions. This factorization is
known to hold for certain scaling operators at the two-dimensional percolation
point and in a few other cases. The correlation functions are evaluated in the
upper half-plane (or any conformally equivalent region) with operators at two
arbitrary points on the real axis, and a third arbitrary point on either the
real axis or in the interior. This type of result is of interest because it is
both exact and universal, relates higher-order correlation functions to
lower-order ones, and has a simple interpretation in terms of cluster or loop
probabilities in several statistical models. This motivated us to use the
techniques of conformal field theory to determine the general conditions for
its validity.
  Here, we discover a correlation function which factorizes in this way for any
central charge c, generalizing previous results. In particular, the
factorization holds for either FK (Fortuin-Kasteleyn) or spin clusters in the
Q-state Potts models; it also applies to either the dense or dilute phases of
the O(n) loop models. Further, only one other non-trivial set of highest-weight
operators (in an irreducible Verma module) factorizes in this way. In this case
the operators have negative dimension (for c < 1) and do not seem to have a
physical realization.
