Construction and clustering properties of the 2d non-linear σ -model form factors: O(3)O(3), O(4)O(4), large n examples

Multi-particle form factors of local operators in integrable models in two dimensions seem to have the property that they factorize when one subset of the particles in the external states are boosted by a large rapidity with respect to the others. This remarkable property, which goes under the name of form factor clustering, was first observed by Smirnov in the O(3) non-linear σ  -model and has subsequently found useful applications in integrable models without internal symmetry structure. In this paper we conjecture the nature of form factor clustering for the general O(n)O(n)σ  -model and make some tests in leading orders of the 1/n1/n expansion and for the special cases n=3,4n=3,4.