The S-matrix Bootstrap I: QFT in AdS

TL;DR

This work develops a program to study massive QFT through conformal bootstrap by embedding the theory in hyperbolic space and focusing on boundary correlators, which obey crossing in one lower dimension. In the flat-space limit ($R\to\infty$ with $\Delta_i\sim m_i R$), these boundary data map to bulk S-matrix observables via a Mellin-space limit and a direct phase-shift relation, both yielding unitarity-compatible constraints. The authors implement a 1D conformal bootstrap for two-dimensional QFTs to obtain nonperturbative upper bounds on cubic couplings, finding perfect agreement with analytic S-matrix bounds from a companion paper, thereby providing strong evidence for the boundary CT–S-matrix correspondence. This approach opens pathways to nonperturbative insights into higher-dimensional massive QFTs and suggests avenues for refining the S-matrix program through boundary conformal data and multi-correlator bootstrap. Overall, the work establishes a concrete and highly nontrivial bridge between conformal bootstrap techniques and flat-space scattering theory in a controlled AdS setting, with clear implications for future explorations.

Abstract

We propose a strategy to study massive Quantum Field Theory (QFT) using conformal bootstrap methods. The idea is to consider QFT in hyperbolic space and study correlation functions of its boundary operators. We show that these are solutions of the crossing equations in one lower dimension. By sending the curvature radius of the background hyperbolic space to infinity we expect to recover flat-space physics. We explain that this regime corresponds to large scaling dimensions of the boundary operators, and discuss how to obtain the flat-space scattering amplitudes from the corresponding limit of the boundary correlators. We implement this strategy to obtain universal bounds on the strength of cubic couplings in 2D flat-space QFTs using 1D conformal bootstrap techniques. Our numerical results match precisely the analytic bounds obtained in our companion paper using S-matrix bootstrap techniques.

Figures (14)

• Figure 1: Hyperbolic space in Poincaré coordinates (left) and global coordinates (right). Surfaces of constant $\tau$ correspond to hemispheres of radius $\sqrt{z^{2}+r^{2}}=e^{\tau}$ in the right picture. The boundary point $B$ corresponds to $\tau=-\infty$ in global coordinates.
• Figure 2: Three-point Witten diagram.
• Figure 3: On the left, we show the euclidean preparation of the two particle scattering state by the insertion of $\mathcal{O}_1$ and $\mathcal{O}_2$ inside the unit sphere. Projecting onto primaries of spin $l$ and dimension $\Delta \in ]E-1,E+1]$ and taking the limit $x^2\to 1$ we obtain the state $|\Psi_l(E)\rangle$. On the right, we depict the Lorentzian evolution of this state starting from $t=0$. The blue lines indicate timelike geodesics that represent the classical evolution of two massive particles in AdS in the center of mass frame. The periodicity of these geodesics leads to a scattering event for each time interval $\Delta t =\pi$.
• Figure 4: In scenario I we vary $m_2$ and find an upper bound on $g_{112}$. In scenario II we vary $m_b$ and find an upper bound on $g_{111}$. In both scenarios the mass of the scattered particle is $m_1$.
• Figure 5: Numerical bounds for scenario I in the specific case with $\Delta_2 = 1.2 \Delta_1$. The orange line is the extrapolation of the numerical results to $N = \infty$ and for large $\Delta_1$ it accurately matches the expected flat-space slope as indicated by the dashed line.