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

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

Problem Statement
This work addresses the non-perturbative construction of $2 \to 2$ scattering amplitudes in four-dimensional quantum field theory that include particle production. The authors focus on a specific class of amplitudes defined by "two-particle-recursively-reducible" (2PRR) graphs. These are diagrams that can be reduced to a single vertex by repeatedly cutting two internal lines. While perturbation theory defines these via Feynman diagrams, the authors seek a non-perturbative solution that satisfies unitarity and crossing symmetry without explicitly summing the diagrams.

The core challenge is solving the coupled non-linear integral equations (the Atkinson–Mandelstam bootstrap) that relate the single and double spectral densities of the amplitude. Previous approaches, such as iterative fixed-point methods, often fail to converge at strong coupling or fail to capture the correct high-energy (Regge) behavior. Furthermore, there is a tension in the S-matrix bootstrap literature between extremal solutions (which tend to be elastic) and the Aks theorem, which states that non-trivial scattering in $d > 2$ necessitates particle production.

Methodology
The authors reformulate the bootstrap problem as a physics-informed optimization task using neural networks (PINNs).

1. Formalism: The amplitude $T(s,t)$ is represented via a once-subtracted Mandelstam representation, characterized by a subtraction constant $c_0$, a single spectral density $\rho(s)$, and a double spectral density $\rho(s,t)$.

   • 2PRR Definition: The authors define 2PRR amplitudes by setting the multi-particle double discontinuity $\rho_{MP}(s,t)$ and the associated single spectral inelasticity $\eta_{MP}(s)$ to zero. The full double discontinuity is then $\rho(s,t) = \rho_{el}(s,t) + \rho_{el}(t,s)$, where $\rho_{el}$ satisfies the Mandelstam equation (enforcing elastic unitarity for $J>0$).
   • J=0 Unitarity: A separate condition enforces unitarity for the spin-0 partial wave, linking the single spectral density to the inelasticity $\eta_0$.

2. Neural Network Architecture:

   • Two networks, $NN_\rho$ and $NN_{DDisc}$, parametrize the single and double spectral densities, respectively.
   • Input Representation: To handle the wide range of energy scales (from threshold to the UV), inputs are provided in both linear ($x, y$) and logarithmic ($\log x, \log y$) forms.
   • Structure: The networks utilize parallel subnetworks for different kinematic regimes (low-energy, mixed-energy, asymptotic) which are merged via intertwining layers.
   • Constraints: The architecture hardwires known physical properties, such as the support of $\rho(s,t)$ above the leading Landau curves and the positivity of $\rho(s,t)$ in the Mahoux–Martin region.

3. Optimization Strategy:

   • The unitarity equations are converted into a loss functional $\mathcal{L}$, which is a weighted sum of penalties for violating the Mandelstam equation, spin-0 unitarity, and inelastic unitarity ($|S_J| \le 1$).
   • Gradient Mixing: To handle the different numerical scales of the constraints, the authors employ a gradient-mixing strategy where updates to the single and double spectral networks are driven primarily by their respective dominant loss terms, with controlled cross-coupling.
   • Training: The process involves a staged approach, starting with sparse grids and low energy cutoffs ($10^8 m^2$) and progressively refining with denser grids and higher cutoffs ($10^{16} m^2$) to ensure UV completeness.

Key Contributions and Results

1. Numerical Construction of 2PRR Amplitudes:

   • The authors successfully construct a one-parameter family of 2PRR amplitudes for couplings $0 \le c_0 \lesssim 44\pi$.
   • Emergent Regge Behavior: The solutions exhibit a stable, non-perturbative Regge limit. In the Regge limit ($s \to \infty, t$ fixed), the amplitude approaches a constant, consistent with a $J=0$ Regge pole.
   • Fixed-Angle Decay: At high energies and fixed angles, the amplitude decays as $1/\log s$, consistent with the running of a negative coupling in $\phi^4$ theory.
   • Transparency: The partial waves $S_J(s)$ approach unity as $s \to \infty$, indicating that the scattering becomes transparent at high energies, despite significant inelasticity at intermediate energies.

2. Particle Production and Inelasticity:

   • The 2PRR amplitudes naturally exhibit sizeable particle production, particularly near the four-particle threshold ($s=16m^2$).
   • The inelasticity grows with the coupling $c_0$. The authors observe a power-law growth in the maximal inelasticity, suggesting a critical coupling where the $S$-wave becomes purely inelastic.

3. Aks Screening:

   • The authors extend the framework to treat multi-particle data ($\rho_{MP}$) as a dynamical variable rather than a fixed input.
   • They demonstrate that by tuning $\rho_{MP}$, it is possible to suppress particle production in low-spin partial waves (e.g., enforcing $|S_0|=1$ or $|S_2|=1$). They term this phenomenon "Aks screening."
   • Cost of Screening: Suppressing inelasticity drives the amplitude toward the extremal values found in semidefinite bootstrap studies. However, this requires $\rho_{MP}$ to become large and highly oscillatory. The authors provide bounds showing that the magnitude of $\rho_{MP}$ must grow exponentially with the number of screened spins ($J_{max}$).

4. Consistency with Semidefinite Bootstrap:

   • By injecting the inelasticity profiles extracted from the 2PRR solutions into a standard semidefinite bootstrap (primal) setup, the authors show that the bootstrap recovers the same low-energy observables and Regge behavior, validating the consistency of the two approaches.

Significance and Claims
The paper claims to provide the first numerical solution to the Atkinson–Mandelstam bootstrap for amplitudes with particle production that are fully UV-complete and exhibit emergent Regge behavior.

• Beyond Perturbation: The results reveal non-perturbative features (such as the specific logarithmic decay and Reggeization) that are invisible in fixed-order perturbation theory or approaches focusing solely on low-energy observables.
• Variational Multi-particle Data: A conceptual novelty is treating multi-particle data as a dynamical degree of freedom. This allows the exploration of the full space of consistent amplitudes and offers a mechanism (Aks screening) to reconcile the existence of particle production with the "elastic" nature of extremal bootstrap solutions.
• Limitations: The authors note that for couplings $c_0 \gtrsim 44\pi$, the numerical optimization fails to converge to a Reggeized solution, and for $c_0 > 56\pi$, no converged solutions were found. They interpret this as an empirical observation of the current numerical limits or a physical boundary of the 2PRR class, rather than a definitive exclusion.

The work establishes a bridge between the detailed analytic structure of Feynman diagrams (via the 2PRR class) and the axiomatic constraints of the S-matrix bootstrap, offering a new tool to explore the space of consistent quantum field theories.