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.

💡 Research Summary

In this work the authors introduce a novel non‑perturbative construction of a four‑dimensional scalar 2→2 scattering amplitude that incorporates both two‑particle unitarity and crossing symmetry in a fully analytic framework. The starting point is a class of planar Feynman‑graph topologies called “two‑particle‑recursively‑reducible” (2PRR) graphs. By definition each 2PRR graph can be split by a two‑particle cut into two smaller graphs of the same class, and all crossing images are included. Summing an infinite series of such graphs yields a well‑defined function of the coupling λ with a finite radius of convergence, providing a concrete model for a non‑perturbative unitarization of the contact λ⁴!ϕ⁴ amplitude.

The authors reformulate the problem using the Atkinson–Mandelstam representation, a once‑subtracted double‑dispersion relation: \