Tracking S-matrix bounds across dimensions

TL;DR

This work performs a non-perturbative S-matrix bootstrap for massive 2→2 identical scalar scattering across dimensions $3\le d\le 11$, treating $d$ as a continuous parameter to bound low-energy observables $\bar{c}_0$ and $\bar{c}_2$ via a primal semidefinite program under ACU. It uncovers two smooth branches of extremal amplitudes separated by sharp kinks at $d=5$ and $d=7$, whose origins lie in the onset of strong threshold behavior and in threshold-induced negativity that alters dispersive representations. By incorporating a detailed threshold ansatz and performing extensive numerics (including a bound-state pole analysis in $d=6$), the study maps the space of allowed S-matrices and reveals structural reorganizations of amplitude data at the kinks. The results illuminate how IR threshold physics constrains UV-like behavior in higher dimensions, discuss implications for UV completion, and outline future directions such as dual bootstrap, large-$d$ analyses, and including inelastic effects.

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.

Figures (5)

• Figure 1: Extrema of $\bar{c}_0$ and $\bar{c}_2$ as functions of $d$. The sharp kinks visible at $d=5$ and $d=7$ coincide with a qualitative change in the threshold structure of the corresponding extremal amplitudes (see Discussion \ref{['sec:discussion']}). Lower error bars corresponds to the best values obtained at finite truncation ($N_{\max}=20$, except in $d=11$ where $N_{\max}=29$), while the upper error bars shows the extrapolated estimates obtained from a linear fit in $(1/N_{\max},\bar{c}_i)$ using the last ten data points. Variations of the fitting procedure lead to qualitatively similar results (see Appendix \ref{['app:convergence']} for more).
• Figure 2: Allowed region in the $(\bar{c}_0,\bar{c}_2)$ plane in $d=6$.
• Figure 3: Bounds on the residue of a bound-state pole in $d=6$ as a function of the bound-state mass $m_b^2$. The two shades of blue indicate $N_{\max}=8$ (lighter) and $N_{\max}=10$ (darker). Dashed black (- - -) shows the renormalised coupling $g_{\text{ren}}$ described in the text, which converges to a finite value at $m_b^2=4$.
• Figure 4: Example of our extrapolation method in $d=8$. Lower dashed line is the best value obtained at finite truncation, here $\max c_0(N_{\max}{=}20)=16.1$, while upper dashed line is the extrapolation of the simple linear fit on last 10 points, here $\max c_0(N_{\max}\rightarrow\infty)=18.1$. We consider these two values respectively as lower and upper values on our best estimate, which we hence define here as $\max c_0=17.1\pm1$ in $d=8$.
• Figure 5: An typical keyhole contour in a dispersion relation, regulating the infrared divergence at $s=4m^2$.