Cross-Section Bootstrap: Unveiling the Froissart Amplitude

TL;DR

The paper develops analytic and numerical finite-energy bounds on the integrated total cross-section using non-perturbative S-matrix Bootstrap in arbitrary dimensions, culminating in the Froissart amplitude as the saturating extremal solution. This amplitude features a rising σ_tot, a shrinking diffractive cone, and a dense Regge spectrum including Pomeron-like and singular forward trajectories, all realized within an expanding annular (white-ring) eikonal profile. Through a combination of analytic bounds and a cutting-edge numerical bootstrap, the authors demonstrate that the extremal amplitude maximizes low-energy coefficients and resonances align along Regge trajectories, providing deep insights into soft QCD dynamics beyond perturbation theory. The work also bridges foundational principles with phenomenology, suggesting a principled route to study high-energy hadronic scattering and outlining concrete future directions, including fermionic amplitudes and inelastic channels.

Abstract

We derive a universal bound on the integrated total scattering cross-section at \emph{finite} energies, expressed in terms of a single low-energy coefficient constrained by the non-perturbative S-matrix Bootstrap. At high energies, the bound is compared with proton-proton scattering data; at low energies, with numerical bootstrap results obtained by directly maximizing the cross-section. We conjecture that the amplitude saturating the cross-section at high energies lies at a strongly-coupled corner of the allowed space of low-energy parameters. This universal amplitude exhibits a rising total cross-section, a shrinking elastic differential cross-section with multiple diffractive minima, and a surprisingly rich spectrum of resonances aligning along Regge trajectories, including Pomeron-like and daughter trajectories, as well as unusual ``singular" trajectories in the forward limit which appear deeply interconnected with Froissart growth. Remarkably, the eikonal representation reveals that the scattering is localized within an annular region that slowly expands with energy, challenging the traditional ``disk" diffraction picture. Our results open the door to theoretical and phenomenological studies of \emph{soft} high-energy hadronic scattering via the S-matrix Bootstrap.

Figures (23)

• Figure 1: The integrated cross-section \ref{['eq:integrated_cross_section']} for $pp$ and $p\bar{p}$ scattering is plotted respectively in purple and blue using the phenomenological fits discussed in ParticleDataGroup:2016lqr, and expressed in dimensionless units (setting $1\text{GeV}=1$). The thickness of the curves include $3\sigma$ deviations from the best fit. The red excluded region is the optimal exclusion bound \ref{['eq:sigmatotanalytic']} in the same units.
• Figure 2: $\text{Max}\, c_2$ as a function of the fixed momentum transfer $0<t_0<4$. Different colors correspond to different values of the cutoff $N$. The extrapolation to $N\to\infty$ is in black. The coefficient $c_2(t_0)$ appears to diverge as $t_0 \to 4$ where it reduces to the spin-2 scattering length Correia:2020xtr.
• Figure 3: The dots are obtained by maximizing $\bar{\sigma}_\text{tot}(s)$ for each value of $s=20,30,\dots,200$. Different colors correspond to different values of the ansatz cutoff $N$. The green points with error bars extrapolation to $N\to \infty$ performed with a simple power law fit, which gives an estimate of the optimal numerical bound. In the same figure we show the best analytic bound (red), and the value of $\bar{\sigma}_\text{tot}(s)$ extracted from the Froissart amplitude.
• Figure 4: Profile of $1-{\rm Re\,} S_\ell(s^\prime)$ computed from the amplitude maximizing $\bar{\sigma}_\text{tot}(80)$ for $\ell=6,\dots,18$. This amplitude has resonances arranging into an approximately linear Regge trajectory up to spin $\ell=14$. Above this level, partial waves accumulate close to the boundary of the integration domain resembling a Coon-like behaviour Coon:1969ywFigueroa:2022onw. As we increase $s$, the trajectory extends to larger spins suggesting a regular limit when $s\to \infty$ compatible with the Froissart amplitude proposed in this Letter. The limit however is reached slowly.
• Figure 5: Values of $(c_0, c_2)$ extracted from the amplitudes maximizing $\bar{\sigma}_\text{tot}(s)$ for $s=20,30,\dots,200$. Different colors correspond to different values of $s$. The blue boundary denote the allowed region, and values outside are not permitted.