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Conformal Basis, Optical Theorem, and the Bulk Point Singularity

Ho Tat Lam, Shu-Heng Shao

TL;DR

The paper develops and analyzes the conformal primary basis for wavefunctions in $(d+2)$-dimensional Minkowski space, showing that amplitudes in this basis transform as $d$-dimensional conformal correlators and enabling a conformal-block perspective on flat-space scattering. It translates unitarity into the conformal basis via the optical theorem, deriving a conformal block decomposition over the principal continuous series with three-point OPE data as coefficients, and explicitly verifies this in a $(2+1)$-dimensional scalar model with tree-level exchange. It also clarifies the link between the massless conformal basis and the AdS bulk-point singularity, illustrating how Witten-diagram limits reproduce flat-space results, and discusses a 2d extension that recasts the 1d amplitudes as Lorentzian slices of a higher-dimensional CFT. Together, these results forge connections between flat-space amplitudes, conformal block technology, and holographic locality, with concrete calculations in low dimensions supporting the framework.

Abstract

We study general properties of the conformal basis, the space of wavefunctions in $(d+2)$-dimensional Minkowski space that are primaries of the Lorentz group $SO(1,d+1)$. Scattering amplitudes written in this basis have the same symmetry as $d$-dimensional conformal correlators. We translate the optical theorem, which is a direct consequence of unitarity, into the conformal basis. In the particular case of a tree-level exchange diagram, the optical theorem takes the form of a conformal block decomposition on the principal continuous series, with OPE coefficients being the three-point coupling written in the same basis. We further discuss the relation between the massless conformal basis and the bulk point singularity in AdS/CFT. Some three- and four-point amplitudes in (2+1) dimensions are explicitly computed in this basis to demonstrate these results.

Conformal Basis, Optical Theorem, and the Bulk Point Singularity

TL;DR

The paper develops and analyzes the conformal primary basis for wavefunctions in -dimensional Minkowski space, showing that amplitudes in this basis transform as -dimensional conformal correlators and enabling a conformal-block perspective on flat-space scattering. It translates unitarity into the conformal basis via the optical theorem, deriving a conformal block decomposition over the principal continuous series with three-point OPE data as coefficients, and explicitly verifies this in a -dimensional scalar model with tree-level exchange. It also clarifies the link between the massless conformal basis and the AdS bulk-point singularity, illustrating how Witten-diagram limits reproduce flat-space results, and discusses a 2d extension that recasts the 1d amplitudes as Lorentzian slices of a higher-dimensional CFT. Together, these results forge connections between flat-space amplitudes, conformal block technology, and holographic locality, with concrete calculations in low dimensions supporting the framework.

Abstract

We study general properties of the conformal basis, the space of wavefunctions in -dimensional Minkowski space that are primaries of the Lorentz group . Scattering amplitudes written in this basis have the same symmetry as -dimensional conformal correlators. We translate the optical theorem, which is a direct consequence of unitarity, into the conformal basis. In the particular case of a tree-level exchange diagram, the optical theorem takes the form of a conformal block decomposition on the principal continuous series, with OPE coefficients being the three-point coupling written in the same basis. We further discuss the relation between the massless conformal basis and the bulk point singularity in AdS/CFT. Some three- and four-point amplitudes in (2+1) dimensions are explicitly computed in this basis to demonstrate these results.

Paper Structure

This paper contains 14 sections, 88 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Conformal Primary Bases
  3. One-Dimensional Conformal Correlators
  4. Three-Point Function
  5. Four-Point Function
  6. Crossing Symmetry
  7. Tree-Level Exchange Amplitude
  8. Optical Theorem and Conformal Block Decomposition
  9. Conformal Optical Theorem
  10. Example: Tree-Level Exchange Amplitude
  11. The Bulk Point Singularity in AdS/CFT
  12. Two-Dimensional Crossing Symmetric Four-Point Functions with Positive Block Decompositions
  13. A Two-Dimensional Extension
  14. Positive Conformal Block Decompositions

Figures (3)

  • Figure 1: The analytic structure of the integral \ref{['realline']} for the tree-level exchange four-point function $f_{12\leftrightarrow 34}(z)$. Left: the $s$-channel contribution. Right: the $t$-channel contribution.
  • Figure 2: The bulk point singularity in $AdS_3$. At the bulk point singularity configuration for a four-point function in $AdS_3/CFT_2$, the boundary points are restricted to two constant time slices in $AdS_3$, which are the future and past celestial circles (shown in blue) in the flat space limit to $\mathbb{R}^{1,2}$. The contribution to the Witten diagram is dominated by the integral around the bulk point $Y_0$, which can then be approximated by the flat space scattering process in the massless conformal basis.
  • Figure 3: The two-parameter space for the solution $f_{\Delta_\phi,b}(z,\bar{z})$ to the $2d$ crossing equation. In the shaded region between $0\le b\le 2\Delta_\phi$ we find numerical evidence that the four-point function $f_{\Delta_\phi,b}(z,\bar{z})$ has a non-negative expansion on the $SL(2,\mathbb{C})$ blocks. At the two boundaries $b=0$ and $b=2\Delta_\phi$, $f_{\Delta_\phi,b}(z,\bar{z})$ reduces to the four-point functions in the $2d$ generalized free field theory and that in the free boson theory, respectively.