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Dispersion Formulas in QFTs, CFTs, and Holography

David Meltzer

TL;DR

The paper establishes momentum-space dispersion formulas for general QFTs and shows their exact relation to conformal dispersion formulas for CFTs, using two independent derivations based on analyticity and the largest-time equation. It demonstrates that the four-point momentum-space dispersion matches the CFT dispersion up to semi-local terms and that Polyakov-Regge expansions in both frameworks are connected by a Fourier transform, with conformal blocks mapped to cut Witten diagrams. It further develops a bulk unitarity method for AdS/CFT correlators in momentum space by combining dispersion with AdS Cutkosky rules, and tests these ideas on scalar and spinning Witten diagrams, including loop diagrams. The results provide a unified on-shell perspective linking S-matrix bootstrap, conformal bootstrap, and holography, with broad implications for computing and constraining AdS/CFT correlators in momentum space.

Abstract

We study momentum space dispersion formulas in general QFTs and their applications for CFT correlation functions. We show, using two independent methods, that QFT dispersion formulas can be written in terms of causal commutators. The first derivation uses analyticity properties of retarded correlators in momentum space. The second derivation uses the largest time equation and the defining properties of the time-ordered product. At four points we show that the momentum space QFT dispersion formula depends on the same causal double-commutators as the CFT dispersion formula. At $n$-points, the QFT dispersion formula depends on a sum of nested advanced commutators. For CFT four-point functions, we show that the momentum space dispersion formula is equivalent to the CFT dispersion formula, up to possible semi-local terms. We also show that the Polyakov-Regge expansions associated to the momentum space and CFT dispersion formulas are related by a Fourier transform. In the process, we prove that the momentum space conformal blocks of the causal double-commutator are equal to cut Witten diagrams. Finally, by combining the momentum space dispersion formulas with the AdS Cutkosky rules, we find a complete, bulk unitarity method for AdS/CFT correlators in momentum space.

Dispersion Formulas in QFTs, CFTs, and Holography

TL;DR

The paper establishes momentum-space dispersion formulas for general QFTs and shows their exact relation to conformal dispersion formulas for CFTs, using two independent derivations based on analyticity and the largest-time equation. It demonstrates that the four-point momentum-space dispersion matches the CFT dispersion up to semi-local terms and that Polyakov-Regge expansions in both frameworks are connected by a Fourier transform, with conformal blocks mapped to cut Witten diagrams. It further develops a bulk unitarity method for AdS/CFT correlators in momentum space by combining dispersion with AdS Cutkosky rules, and tests these ideas on scalar and spinning Witten diagrams, including loop diagrams. The results provide a unified on-shell perspective linking S-matrix bootstrap, conformal bootstrap, and holography, with broad implications for computing and constraining AdS/CFT correlators in momentum space.

Abstract

We study momentum space dispersion formulas in general QFTs and their applications for CFT correlation functions. We show, using two independent methods, that QFT dispersion formulas can be written in terms of causal commutators. The first derivation uses analyticity properties of retarded correlators in momentum space. The second derivation uses the largest time equation and the defining properties of the time-ordered product. At four points we show that the momentum space QFT dispersion formula depends on the same causal double-commutators as the CFT dispersion formula. At -points, the QFT dispersion formula depends on a sum of nested advanced commutators. For CFT four-point functions, we show that the momentum space dispersion formula is equivalent to the CFT dispersion formula, up to possible semi-local terms. We also show that the Polyakov-Regge expansions associated to the momentum space and CFT dispersion formulas are related by a Fourier transform. In the process, we prove that the momentum space conformal blocks of the causal double-commutator are equal to cut Witten diagrams. Finally, by combining the momentum space dispersion formulas with the AdS Cutkosky rules, we find a complete, bulk unitarity method for AdS/CFT correlators in momentum space.

Paper Structure

This paper contains 32 sections, 263 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Summary and Outline
  4. Operator Orderings
  5. Derivation 1: Dispersion from Analyticity
  6. Analyticity in $k$-Space
  7. Two- and Three-Point Functions
  8. Four-Point Function
  9. Five-Point Function
  10. Derivation 2: Dispersion from Largest Time
  11. Largest Time Equations
  12. Four-Point Function
  13. Higher-Point Functions
  14. Relation to Conformal Dispersion
  15. Superbounded Correlators
  16. ...and 17 more sections

Figures (3)

  • Figure 1: The amplitude is first written as a contour integral and then this contour is deformed to wrap the branch cuts on the real line. We have assumed that the arcs at infinity vanish.
  • Figure 2: The figure on the left corresponds to taking the Regge limit in position space. Here we send the four operators to null infinity such that $x_{12}^2$ and $x_{34}^2$ are held fixed and spacelike. The figure on the right gives the same limit in CFT cross-ratio space. In momentum space these limits correspond to taking the large $t$ Regge limit.
  • Figure 3: The $t$-channel Polyakov Regge block is a $t$-channel exchange Witten diagrams plus a finite sum of quartic, contact diagrams.