The S-matrix bootstrap with neural optimizers I: zero double discontinuity

TL;DR

This work develops a neural-optimizer framework to solve the Atkinson-Mandelstam S-matrix bootstrap for a toy model with zero double discontinuity, enabling unsupervised learning of nonperturbative scattering amplitudes. By pairing neural-parametrized discontinuities with dispersion relations, the authors obtain bounds on the first two low-energy Taylor coefficients and map the corresponding allowed region, cross-validating with traditional primal/dual bootstrap and the rho-bootstrap. A detailed treatment of UV-IR interplay shows high-energy unitarization constraints feed back into low-energy bounds, and the approach yields a practical tool for exploring the space of consistent S-matrices. The results indicate the necessity of double discontinuity to unitarize higher-spin partial waves in regions outside the zero-double-discontinuity almond, providing new insights into the structure of nonperturbative amplitudes and opening avenues for extending to the full Mandelstam problem. Overall, neural optimization offers a flexible, scalable method to navigate nonlinear bootstrap equations and to scan physical parameter spaces that were previously challenging with iterative methods.

Abstract

In this work, we develop machine learning techniques to study nonperturbative scattering amplitudes. We focus on the two-to-two scattering amplitude of identical scalar particles, setting the double discontinuity to zero as a simplifying assumption. Neural networks provide an efficient parameterization for scattering amplitudes, offering a flexible toolkit to describe their fine nonperturbative structure. Combined with the bootstrap approach based on the dispersive representation of the amplitude and machine learning's gradient descent algorithms, they offer a new method to explore the space of consistent S-matrices. We derive bounds on the values of the first two low-energy Taylor coefficients of the amplitude and characterize the resulting amplitudes that populate the allowed region. Crucially, we parallel our neural network analysis with the standard S-matrix bootstrap, both primal and dual, and observe perfect agreement across all approaches.

Figures (12)

• Figure 1: Architecture of a neural network that we use. The output, $\rho(x)$, is the discontinuity of the amplitude. Two sub-networks allow us to describe the high and intermediate energies in finer detail.
• Figure 2: Plots of allowed S-matrices in the space of their Taylor coefficients $c_0$ and $c_2$ defined in \ref{['eq:c0c2def']}. Left.Purple region: space of allowed S-matrices with zero double discontinuity studied in this paper: "the single-disc almond". All four methods (neural optimizer/standard bootstrap; primal/dual) produce identical results; hence, we show a unique shape. The red and orange lines represent, respectively, the fixed-point iteration and Newton's method convergence range along the lower boundary, illustrating the superiority of the gradient-descent neural optimizer. Right. Comparison with the full space of S-matrices as worked out in Chen:2022nymEliasMiro:2022xaa, or, equivalently, "the full almond". Our results shed new light on the role of the double discontinuity for the amplitudes obtained in these papers. In particular, in the complement of our region, surprisingly, the double discontinuity is required to unitarize even the S-wave. Characterizing this double discontinuity is an open problem.
• Figure 3: Plot of the activation function $\mathrm{CELU}$ defined in \ref{['eq:CELU_function']}.
• Figure 4: Example of a neural network with 2 layer-blocks with widths $n$, $m$ and the final layer (linear layer) returning a single output (width $1$). In these graphical conventions, the last layer is always of width $1$. The top and bottom picture represent the same network: the top picture is a condensed notation, which we use in Figure \ref{['fig:NN']} to describe our full architecture, and the bottom is the "definition" of the condensed picture. The function $C$ is the activation function, and the $w$'s and $b$'s are the weights and biases. They have a superscript, which represents the layer, an index, which represents the row number, and arrows when they are vectorial quantities. These indices are given in this picture for the sake of definiteness. Lastly, in this work, $C(\cdot)\equiv\rm{CELU}(\cdot)$, defined in Eq. \ref{['eq:CELU_function']}.
• Figure 5: Full architecture of the neural network for the primal approach, with 8 layer-blocks, implementing skip connections. With the skip connections, layer blocks depend on several layers preceding them simultaneously (the altitude of the arrow in the box does not represent anything, all the variables from all relevant layers enter in a similar way). Intuitively each part of the network learns a specific physical regime: Regge/threshold.