19.4. Cluster properties Consider the form factors of a scattering theory based on the functions fx(θ) =

tanh 12

(θ +iπx)

tanh 12

(θ −iπx)

that have the property limθ→∞fx(θ) = 1.

a. Using the Watson equation satisfied by the form factors FOa n (θ1, . . . ,θn) of an operator Oa, prove that taking the limit lim

→∞

FOa n (β1 +, . . . ,βm +,βm+1, . . . ,βn) = FOb m (β1, . . . ,βm)FOc n−m(βm+1, . . . ,βn)

the form factor factorizes in terms of two functions both satisfying the Watson equations. Hence they can be considered the form factors of the operators Ob and Oc. This expresses the cluster property of the form factors.

b. Prove that the form factors of the elementary solutions of the Sinh–Gordon model are self-clustering quantities.