Tailoring three-point functions and Integrability III. Classical tunneling

We compute three-point functions between one large classical operator and two large BPS operators at weak coupling. We consider operators made out of the scalars of N = 4 SYM, dual to strings moving in the sphere. The three-point function exponentiates and can be thought of as a classical tunneling process in which the classical string-like operator decays into two classical BPS states. From an Integrability/Condensed Matter point of view, we simplified inner products of spin chain Bethe states in a classical limit corresponding to long wavelength excitations above the ferromagnetic vacuum. As a byproduct we solved a new long-range Ising model in the thermodynamic limit.