Scattering from production in 2d

TL;DR

The paper develops and tests two numerical implementations of Atkinson's constructive framework to reconstruct 2D S-matrix amplitudes from fixed particle production, enforcing analyticity, crossing, and both elastic and inelastic unitarity. By discretizing and iterating dispersion relations, it shows convergence across broad regions of amplitude space and uncovers a fractal structure of CDD ambiguities, including a fractal-like CDD sector dependence revealed by Newton's method. Fixed-point iteration provides a clear but limited convergence domain, while Newton's method extends this domain and exposes rapid convergence and rich basins of attraction, illustrating a deep link between convergence properties and CDD structure. The results validate the practical viability of Atkinson's approach in 2D and point toward extensions to higher dimensions, where elastic unitarity and more intricate kinematics present additional challenges and opportunities for coupling maximization and amplitude-space exploration.

Abstract

In 1968, Atkinson proved the existence of functions that satisfy all S-matrix axioms in four spacetime dimensions. His proof is constructive and to our knowledge it is the only result of this type. Remarkably, the methods to construct such functions used in the proof were never implemented in practice. In the present paper, we test the applicability of those methods in the simpler setting of two-dimensional S-matrices. We solve the problem of reconstructing the scattering amplitude starting from a given particle production probability. We do this by implementing two numerical iterative schemes (fixed-point iteration and Newton's method), which, by iterating unitarity and dispersion relations, converge to solutions to the S-matrix axioms. We characterize the region in the amplitude-space in which our algorithms converge, and discover a fractal structure connected to the so-called CDD ambiguities which we call "CDD fractal". To our surprise, the question of convergence naturally connects to the recent study of the coupling maximization in the two-dimensional S-matrix bootstrap. The methods exposed here pave the way for applications to higher dimensions, and expose some of the potential challenges that will have to be overcome.

Figures (13)

• Figure 1: Results of the algorithm with no bound state and an arbitrary but regular enough starting point (displayed). Convergence is exponentially fast, so we show only the first five iterations. The inelastic input chosen here is given in \ref{['eq:vi-nobs']} with $H=120$. Linear interpolants, grid given in \ref{['eq:grid-def']}. a) Imaginary part of the S-matrix: iterations (color), analytic (dot-dashed). (b) Real part. (c) Modulus of the S-matrix. 7 seconds for 100 iterations and the grid of \ref{['eq:grid-def']}.
• Figure 2: Convergence of the algorithm is dictated by the spectral radius $\xi$ of the Jacobian. The horizontal axis labels the number of iterations. $||.||_\infty$ represents the maximum of the elements of the discretized S-matrix seen as a vector of the values of the S-matrix, $||S_n-S_{n-1}||_\infty = \mathrm{Max}_{i=0,\dots,N}|S_{n,i}-S_{n-1,i}|$ where $S_{n,i}=S_n(x_i)$.
• Figure 3: Results of the iterations for one bound state and fixed constant at infinity. Obtained for $c_\infty=1.25$, $m_p^2=2.8$. We show only first 5 iterations. The analytical solution is given by \ref{['eq:solution']} with $S_{\text{elastic}} = S_{\text{CDD}}^{\text{pole}} S_{\text{CDD}}^{\text{zero}}$ with $m_z^2 \simeq3.594...$, the value which solves \ref{['eq:constant']} with $v_i=0$. Linear interpolant, grid of eq.\ref{['eq:grid-def']}. A hundred iterations are performed in about 5 seconds.
• Figure 4: Results of the iterations for one bound state and a fixed constant at infinity. Obtained for $c_\infty=-3$,$m_p^2=2.8$, $H=40$. The result of the first four iterations only. The analytic solution is given by \ref{['eq:solution']} with $S_{\text{elastic}} = S_{\text{CDD}}^{\text{pole}} S_{\text{CDD}}^{\text{zero}}$ with $m_z^2=3.225...$, as per eq.\ref{['eq:constant']}. Linear interpolant, grid given in eq. \ref{['eq:grid-def']}.
• Figure 5: Eigenvalues of the Jacobian for a pure CDD-pole amplitude for $m_p^2=2.8$: a) linear interpolation; b) interpolation with Bernstein polynomials.