An Analytical Toolkit for the S-matrix Bootstrap

TL;DR

This work develops an analytical toolkit for constraining nonperturbative S-matrices in $d\ge 3$ using extended analyticity and elastic unitarity. By deriving threshold and large-$J$ expansions and connecting them through the Froissart-Gribov inversion, it links low-energy data to finite-energy, finite-spin scattering, while highlighting positivity and production constraints via the double spectral density. The authors also outline concrete improvements to modern numerical bootstrap programs to incorporate elastic-unity and Landau-curve structures, and discuss connections to conformal bootstrap. Together, these results provide a structured approach to constraining physically allowed S-matrices and guiding numerical explorations toward bootstrap-solvable theories. The framework offers practical pathways to bound inelasticity and to refine numerical schemes that aim to solve or approximate higher-dimensional scattering in quantum field theories.

Abstract

We revisit analytical methods for constraining the nonperturbative $S$-matrix of unitary, relativistic, gapped theories in $d \geq 3$ spacetime dimensions. We assume extended analyticity of the two-to-two scattering amplitude and use it together with elastic unitarity to develop two natural expansions of the amplitude. One is the threshold (non-relativistic) expansion and the other is the large spin expansion. The two are related by the Froissart-Gribov inversion formula. When combined with crossing and a local bound on the discontinuity of the amplitude, this allows us to constrain scattering at finite energy and spin in terms of the low-energy parameters measured in the experiment. Finally, we discuss the modern numerical approach to the $S$-matrix bootstrap and how it can be improved based on the results of our analysis.

Figures (14)

• Figure 1: In the elastic strip $4m^2<s<16m^2$ the discontinuity of the amplitude comes from two intermediate particle exchange only. This result in the exact elastic unitarity equation (\ref{['eq:elauni1']}). The corresponding phase space integration kernel ${\cal P}_d(\cos\theta,\cos\theta',\cos\theta")$ in (\ref{['eq:dlips']}) is proportional to a step/delta function which has a simple geometrical origin. In the center of mass frame we have three $(d-1)$-dimensional vectors, $\vec{p}_1=-\vec{p}_2$, $\vec{p}_3=-\vec{p}_4$, and $\vec{q'}=-\vec{q"}$. The geometrical angles between these three vectors, $\{\theta,\theta',\theta"\}$ are therefore restricted to the range $\theta_1+\theta_2\ge\theta_3$, where $\theta_{1,2,3}$ are any permutation of $\{\theta,\theta',\theta"\}$.
• Figure 2: a. The partial wave projection integral (\ref{['eq:fJNeumann']}) is a contour integral (in blue) that circles around the cut of the $Q$-function, between $t=0$ and $t=-(s-4m^2)$, (in red). b. We partially open up the contour. Sometimes this representation for partial waves is called the truncated Froissart-Gribov formula. The advantage of this representation is that we only use a finite amount of extended analyticity that has not been rigorously proven. c. We open the contour all the way to infinity and arrive at the usual Froissart-Gribov formula (\ref{['FG1']}) with two integrations of the discontinuity of the amplitude along the $t$-channel and $u$-channel cuts (in black).
• Figure 3: The region of integration in equation (\ref{['eq:doubleSD']}). As $s$ or $t$ approaches the Landau curve from above, the integration region shrinks to zero. As a result, the double spectral density vanishes below the Landau curve $z=2 z_1^2-1$.
• Figure 4: The double discontinuity of the amplitude $\rho(s,t)$ in the real $(s,t)$ plane. In gray is the Steinmann shadow region where $\rho$ vanishes. This region extends inside the elastic bands, $\{(4m^2<s<16m^2,t),(s,4m^2<t<16m^2)\}$, and is bounded by the Landau curve $t_1(s)={16m^2 s\over s-4m^2}$ (in red) and its crossed curve $t_1^\text{cross}(s)={4m^2 s\over s-16m^2}$ (in blue). These two curves extend out of the elastic bands, where there are additional multi-particle contributions. They cross at $s=t=20m^2$.
• Figure 5: a. The Landau diagrams that contribute to the double spectral density in the elastic strip $4m^2<s<16m^2$ and $4m^2<t$. In this kinematical regime there can be only two particles in the $s$-channel. Hence, the corresponding Landau diagrams have a simple structure of iterative two-particle exchange in the $s$-channel and in between, any number of particles exchange in the $t$-channel. b. Analogously to figure \ref{['stepfunction']}, the corresponding Landau curves originate from a simple geometric constraint on physical kinematics, see discussion after (\ref{['thetacurve']}).