Analytic Non-Integrability and S-Matrix Factorization

We formulate an equivalence between the 2-dim $\sigma$-model spectrum expanded on a non-trivial massive vacuum and a classical particle Hamiltonian with variable mass and potential. By considering methods of analytic Galoisian non-integrability on appropriate geodesics of the Hamiltonian system we algebraically constrain the particle masses at fixed time, such that integrability is allowed. Through our equivalence this explicitly constrains the masses of the excited spectrum of the dual 2-dim theory in such a way to imply the S-matrix factorization and no particle production. In particular, the integrability of the classical particle system, implies the factorization of the S-matrix in the dual quantum 2-dim theory. Our proposal provides also non-trivial evidence without any assumptions, on the connection between integrability and S-matrix factorization for large class of theories with interactions that break Lorentz invariance.