Tracking S-matrix bounds across dimensions

TL;DR

This paper explores the bounds of massive 2-to-2 scalar scattering amplitudes across dimensions 3 to 11 using non-perturbative S-matrix bootstrap, revealing a rich structure with sharp transitions at specific dimensions.

Contribution

It introduces a continuous dimension approach in S-matrix bootstrap, uncovering smooth extremal amplitude branches and sharp kinks at critical dimensions, linked to changes in analyticity and positivity constraints.

Findings

Sharp kinks at dimensions 5 and 7 indicating phase transitions.

Smooth extremal amplitude branches across dimensions.

Threshold singularities organize the S-matrix structure.

Abstract

We study massive $2 \to 2$ scattering of identical scalar particles in spacetime dimensions 3 to 11 using non-perturbative S-matrix bootstrap techniques. Treating $d$ as a continuous parameter, we compute two-sided numerical bounds on low-energy observables and find smooth branches of extremal amplitudes separated by sharp kinks at $d=5$ and $d=7$, coinciding with a transition in threshold analyticity and the loss of some well-known dispersive positivity constraints. Our results reveal a rich structure in the space of massive S-matrices across dimensions and identify threshold singularities as a key organizing principle. We comment on numerical limitations at large dimension and on possible implications for ultraviolet completion in higher-dimensional quantum field theory.