Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Proof of the Conformal Collider Bounds

Diego M. Hofman, Daliang Li, David Meltzer, David Poland, Fernando Rejon-Barrera

TL;DR

This work delivers a full field-theory proof that the conformal collider bounds of Hofman and Maldacena hold in any unitary parity-preserving CFT with a unique stress tensor in $d>2$, using bootstrap-derived analyticity, crossing symmetry, and reflection positivity. The authors derive sharp bounds on the three-point functions $\langle JJT\rangle$ and $\langle TTT\rangle$, relate them to central-charge ratios like $a/c$ in 4d (with stronger constraints in supersymmetric theories), and compute large-spin anomalous dimensions showing they are negative, implying universal gravitational attraction in AdS. They extend the analysis to non-conserved currents and connect to deep inelastic scattering results, while clarifying saturation by free field theories and discussing implications for the conformal bootstrap program and holography. The results provide a principled, first-principles confirmation that causality/positivity constraints tightly bound CFT data and have broad consequences for holographic duals and higher-spin consistency. This strengthens the bootstrap’s role in constraining quantum field theories and deepens our understanding of universal energy positivity in conformal dynamics.

Abstract

In this paper, we prove that the "conformal collider bounds" originally proposed by Hofman and Maldacena hold for any unitary parity-preserving conformal field theory (CFT) with a unique stress tensor in spacetime dimensions larger than 2. In particular this implies that the ratio of central charges for a unitary 4d CFT lies in the interval $\frac{31}{18} \geq \frac{a}{c} \geq \frac{1}{3}$. For superconformal theories this is further reduced to $\frac{3}{2} \geq \frac{a}{c} \geq \frac{1}{2}$. The proof relies only on CFT first principles - in particular, bootstrap methods - and thus constitutes the first complete field theory proof of these bounds. We further elaborate on similar bounds for non-conserved currents and relate them to results obtained recently from deep inelastic scattering.

A Proof of the Conformal Collider Bounds

TL;DR

This work delivers a full field-theory proof that the conformal collider bounds of Hofman and Maldacena hold in any unitary parity-preserving CFT with a unique stress tensor in , using bootstrap-derived analyticity, crossing symmetry, and reflection positivity. The authors derive sharp bounds on the three-point functions and , relate them to central-charge ratios like in 4d (with stronger constraints in supersymmetric theories), and compute large-spin anomalous dimensions showing they are negative, implying universal gravitational attraction in AdS. They extend the analysis to non-conserved currents and connect to deep inelastic scattering results, while clarifying saturation by free field theories and discussing implications for the conformal bootstrap program and holography. The results provide a principled, first-principles confirmation that causality/positivity constraints tightly bound CFT data and have broad consequences for holographic duals and higher-spin consistency. This strengthens the bootstrap’s role in constraining quantum field theories and deepens our understanding of universal energy positivity in conformal dynamics.

Abstract

In this paper, we prove that the "conformal collider bounds" originally proposed by Hofman and Maldacena hold for any unitary parity-preserving conformal field theory (CFT) with a unique stress tensor in spacetime dimensions larger than 2. In particular this implies that the ratio of central charges for a unitary 4d CFT lies in the interval . For superconformal theories this is further reduced to . The proof relies only on CFT first principles - in particular, bootstrap methods - and thus constitutes the first complete field theory proof of these bounds. We further elaborate on similar bounds for non-conserved currents and relate them to results obtained recently from deep inelastic scattering.

Paper Structure

This paper contains 33 sections, 133 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Overview
  3. Scalar Correlators
  4. Bounds on $\langle JJT \rangle$
  5. Crossing Symmetry
  6. Reflection Positivity
  7. $\langle J^+ \phi\phi J^+ \rangle$
  8. $\langle J^t \phi\phi J^t \rangle$
  9. Anomalous Dimensions
  10. Bounds on $\langle TTT \rangle$
  11. Crossing Symmetry
  12. $\langle T^{++} \phi\phi T^{++} \rangle$
  13. $\langle T^{+t} \phi\phi T^{+t} \rangle$
  14. $\langle T^{tt} \phi\phi T^{tt} \rangle$
  15. Anomalous Dimensions
  16. ...and 18 more sections

Figures (2)

  • Figure 1: The region $D$ is a small half disc on the complex $\sigma$ plane above the origin (which is excluded). Here $0<\eta\ll R\ll 1.$
  • Figure 2: Conformal collider bounds in several dimensions $d$. Filled regions correspond to the allowed values of the parameters $t_{2}$, $t_{4}$.