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Celestial Optical Theorem

Reiko Liu, Wen-Jie Ma

TL;DR

The work derives a nonperturbative celestial optical theorem from bulk S-matrix unitarity, formulating bootstrap equations for conformal partial wave coefficients in CCFT. It proves a positivity property for the imaginary parts of CPW coefficients and constrains their analytic structure: massless elastic scattering yields CPW poles that are strictly simple, while processes with massive states induce double-trace poles, implying nonperturbative double-trace operators without anomalous dimensions in CCFT. By linking CPW poles to three-point coefficients, the paper provides a concrete mechanism for nonperturbative CCFT constraints and offers a path toward numerical celestial bootstrap. These results deepen the bridge between bulk unitarity and celestial CFT data, and suggest further study of crossing symmetry, conformally soft theorems, and $w_{1+9fty}$-related structures to sharpen the celestial bootstrap program.

Abstract

We establish the nonperturbative celestial optical theorem from the unitarity of $S$-matrix. This theorem provides a set of nonperturbative bootstrap equations of the conformal partial wave (CPW) coefficients. The celestial optical theorem implies that the imaginary part of CPW coefficient with appropriate conformal dimensions is non-negative. By making certain assumptions and using the celestial optical theorem, we derive nonperturbative results concerning the analytic structure of CPW coefficients. We discover that the CPW coefficients of four massless particles must and only have simple poles located at specific positions. The CPW coefficients involving massive particles exhibit double-trace poles, indicating the existence of double-trace operators in nonperturbative CCFT. It is worth noting that, in contrast to AdS/CFT, the conformal dimensions of double-trace operators do not receive anomalous dimensions.

Celestial Optical Theorem

TL;DR

The work derives a nonperturbative celestial optical theorem from bulk S-matrix unitarity, formulating bootstrap equations for conformal partial wave coefficients in CCFT. It proves a positivity property for the imaginary parts of CPW coefficients and constrains their analytic structure: massless elastic scattering yields CPW poles that are strictly simple, while processes with massive states induce double-trace poles, implying nonperturbative double-trace operators without anomalous dimensions in CCFT. By linking CPW poles to three-point coefficients, the paper provides a concrete mechanism for nonperturbative CCFT constraints and offers a path toward numerical celestial bootstrap. These results deepen the bridge between bulk unitarity and celestial CFT data, and suggest further study of crossing symmetry, conformally soft theorems, and -related structures to sharpen the celestial bootstrap program.

Abstract

We establish the nonperturbative celestial optical theorem from the unitarity of -matrix. This theorem provides a set of nonperturbative bootstrap equations of the conformal partial wave (CPW) coefficients. The celestial optical theorem implies that the imaginary part of CPW coefficient with appropriate conformal dimensions is non-negative. By making certain assumptions and using the celestial optical theorem, we derive nonperturbative results concerning the analytic structure of CPW coefficients. We discover that the CPW coefficients of four massless particles must and only have simple poles located at specific positions. The CPW coefficients involving massive particles exhibit double-trace poles, indicating the existence of double-trace operators in nonperturbative CCFT. It is worth noting that, in contrast to AdS/CFT, the conformal dimensions of double-trace operators do not receive anomalous dimensions.
Paper Structure (5 sections, 19 equations, 5 figures)

This paper contains 5 sections, 19 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Celestial Optical Theorem
  4. Applications
  5. Discussion

Figures (5)

  • Figure 1: The relation between scattering amplitude $\mathcal{T}$, celestial amplitude $\mathcal{A}$ and CPW coefficient $\rho$.
  • Figure 2: Comb-channel CPW coefficients. The left is the CPW coefficient $\rho^{\boldsymbol{\Delta},\boldsymbol{J}}_{\boldsymbol{\Delta}',\boldsymbol{J}'}$ in \ref{['eq:conformal_partial_wave_expansion']}. The right is the CPW coefficient $\rho_{IK}$ associated to $\mathcal{A}_{IF}$, and the arrows denote the direction of incoming/outgoing.
  • Figure 3: Illustration of celestial optical theorem \ref{['eq:celestial_optical:S+S=1']}. We use the red color denoting complex conjugation. The pairs of red and blue arrows connected by yellow plaquettes represent the variables that should be integrated out.
  • Figure 4: Dependence of each term in \ref{['eq:celestial_optical:S+S=1']} on external and exchange conformal dimensions and spins. Here $1$ denotes $(\Delta_{1},J_{1})$, $\widehat{3}$ denotes $(\Delta^{*}_{3},-J_{3})$, $1'$ denotes $(\Delta'_{1},J'_{1})$, and similar for others. Notice that $(\boldsymbol{\Delta}'_K, \boldsymbol{J}'_K)$ denotes the collection of exchange conformal weights without the first one $(\Delta'_{1},J'_{1})$.
  • Figure 5: Illustration of positivity.