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

Abstract

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.

Figures (34)

• Figure 1: A series of two-particle-recursively-reducible (2PRR) graphs that impose two-particle unitarity. Each of the graphs can be reduced to a single vertex by performing two-particle cuts. Each graph comes with its crossing images. We call the amplitudes defined via this series 2PRR scattering amplitudes.
• Figure 2: We characterize the space of amplitudes using the multi-particle data. In perturbation theory, it is defined by the sum of all two-particle-irreducible graphs.
• Figure 3: Typical neural network used to parametrize the amplitude. For better expressivity, we design the network using connected subnetworks that deal with different regions.
• Figure 4: Structure of the double discontinuity $\rho(s,t)$ in the $(s,t)$-plane. Red: leading $s$-channel Landau curve (eq.\ref{['eq:LeadingLC-s']}), Blue: leading $t$-channel Landau curve (Eq. \ref{['eq:LeadingLC-t']}) Black: conjectured leading Landau curve for the onset of the inelastic contribution $\rho_\text{MP}$Correia:2021etg (see Eq. \ref{['eq:PlanarCrossLandauCurve']}). The purple region indicates the domain where both elastic contributions overlap. In the $(x=4/s,y=4/t)$-plane, the red and blue LC become straight lines, see e.g.Tourkine:2023xtu.
• Figure 5: The Mahoux--Martin region in the $(s,t)$ plane, shown as a hatched area. In this region the double spectral function $\rho(s,t)$ is strictly positive.