Rigorous bounds on the Analytic $S$-matrix

TL;DR

The paper introduces a dual formulation of the S-matrix Bootstrap in $d\ge 3$ that relies solely on analyticity, crossing, and unitarity to produce rigorous bounds, avoiding truncations inherent to primal approaches. It constructs a dual SDP by expanding dual constraints in a spin-bounded functional space and solving with semidefinite programming, anchored by fixed-$t$ dispersion relations. Applied to identical scalar scattering in $d=4$, it yields numerical bounds $g_0\le 2.7272$ and $g_0\ge -8.08$, with rapid convergence for the maximal coupling and slower but informative convergence for the minimal coupling; phase-shift data align with threshold unitarity in the explored regime. The work demonstrates a principled path to non-perturbative, rigorous S-matrix constraints and outlines clear avenues for improving duality gaps and extending the framework to massless theories, EFTs, and multi-particle processes.

Abstract

We consider a dual $S$-matrix Bootstrap approach in $d\geq 3$ space-time dimensions which relies solely on the rigorously proven analyticity, crossing, and unitarity properties of the scattering amplitudes. As a proof of principle, we provide rigorous upper and lower numerical bounds on the quartic coupling for the scattering of identical scalar particles in four dimensions.

Figures (9)

• Figure 1: a) The geometric solution of problem ${\rm max}\,\{x+y\,|\,x^2+y^2 \leq 1\}$. b) The dual function $\bar{\cal O}(\lambda)=\lambda+1/(2\lambda)$ of the same maximization problem. The minimum value is attained for $\lambda=1/\sqrt{2}$ and coincides with the optimal value of the maximization problem.
• Figure 2: Bounds on the quartic coupling $g_0$. On the left, we show in green the region determined by solving the primal problem, in red the one rigorously excluded by solving the dual \ref{['dualproblem']}. The bound on the maximum coupling converges fast both in the primal and dual problem and the gap is relatively small, though non-vanishing -- right-top inset. The bound on the minimum coupling is hard to study using primal: the different green lines denote the numerical coupling for $N_\text{max}=5,8,11,14,17,20$. On the contrary, the red lines for $L=0,{\dots},6$ show that dual convergence is achieved faster -- right-bottom inset. In dashed black we report the best values of Lopez:1976zs.
• Figure 3: The spin-$0$ phase shifts, $\delta_0=\tfrac{1}{2i}\log S_0$, as a function of the center of mass energy $s$ for the maximum (top) and minimum coupling (bottom). We plot in dashed-green and solid-red the phase shifts obtained respectively from the best primal and dual numerics. In the bottom figure, the dashed lines in gray-scale correspond to increasing values of $N_\text{max}$ up to $N_\text{max}=20$ (in green); the solid lines in gray-scale are obtained from the dual for different values of $L$, up to $L=6$ (in red). Although primal and dual results still differ, the physical content they describe is the same, converging one towards the other as the duality gap shrinks.
• Figure 4: Minimum value of the dual objective $D^+(4/3,4/3)$\ref{['dual_objective_appendix']} for the maximum coupling problem as a function of $N_w$ and $L$. Different colors correspond respectively to $L=0$ (green), $L=2$ (blue) and $L=4$ (red). The blue curve is covered by the red one as the bound does improve anymore by increasing $L$. The addition of odd dual variables $w_{2n+1}$ has no effect on the bound. The dashed lines represent the corresponding extrapolations to $L\to \infty$. Our best bound $g_0\leq 2.7272<2.75$ improves the one found in Lopez:1976zs, but is still bigger than the one obtained by the primal Paulos:2018fym and the dual of He:2021eqn.
• Figure 5: In the left panel we show the dual dispersion variables $w_{2n}$ for $n=0,1,2$. On the right, we show the unitarity dual variables. The dual objective $D^+$ is obtained by summing the areas below the $X_J$ curves. We see how small is the contribution to the bound coming from the higher spins. Convergence in $N_w$ is also achieved fast as the various curves tend to overlap by increasing $N_w$.