Neural S-matrix bootstrap II: solvable 4d amplitudes with particle production

We study a model for nonperturbative unitarization of the four-point contact scalar amplitude in four dimensions. It is defined through an infinite sum of planar diagrams, constructed using two-particle unitarity and crossing symmetry. We reformulate the problem in terms of a set of nonlinear integral equations obeyed by the single and double discontinuities of the amplitude. We then solve them using a neural-network ansatz trained by minimizing a physics-informed loss functional. We obtain a one-parameter family of amplitudes, which exhibit rich structure: sizeable particle production, nontrivial emergent Regge behavior, Landau curves, a logarithmic decay at high energy and fixed angle. Finally, we go beyond the two-particle-reducible setup by treating the multi-particle data—supported above the multi-particle Landau curves due to multi-particle unitarity—as a dynamical variable. We demonstrate that it can be tuned to suppress low-spin particle production—a phenomenon we call Aks screening—at the cost of generating larger and oscillatory double spectral density in the multi-particle region.