Zamolodchikov’s conjecture on high-energy elastic vs. inelastic scattering in Ising Field Theory

Prove Zamolodchikov’s conjecture that, in (1+1)-dimensional Ising Field Theory obtained by deforming the critical Ising conformal field theory, the high-energy limit of two-to-two scattering of the lightest particles is purely elastic near the E8 integrable point (η≈0) and completely inelastic near the free-fermion integrable point (η≈∞). Equivalently, determine the high-energy limit of the elastic two-to-two S-matrix element S_{11→11}(E,η) or of the elastic probability P_{11→11}(E,η) as E→∞, establishing that it approaches 1 near η≈0 and 0 near η≈∞.

Background

Ising Field Theory (IFT) in 1+1 dimensions is defined by deforming the critical two-dimensional Ising conformal field theory by its energy and spin operators, and is parametrized by a dimensionless ratio η that interpolates between two integrable limits: the E8 theory at η=0 and massive free fermions at η=∞.

For two-to-two scattering of the lightest particles, the elastic channel probability P_{11→11}(E,η) and its high-energy limit P^∞(η)=lim_{E→∞}P_{11→11}(E,η) encapsulate whether scattering is predominantly elastic or inelastic at high energies. At the two integrable endpoints, the S-matrix is purely elastic, but the behavior for nonzero deformations away from integrability is nontrivial.

Zamolodchikov conjectured a striking dichotomy: in the high-energy limit, scattering remains purely elastic near the E8 point but becomes completely inelastic near the free-fermion point. The paper’s simulations provide evidence consistent with this conjecture, but a proof and a precise characterization of the high-energy asymptotics across η remain open.