Dispersion Relation for CFT Four-Point Functions

Abstract

We present a dispersion relation in conformal field theory which expresses the four point function as an integral over its single discontinuity. Exploiting the analytic properties of the OPE and crossing symmetry of the correlator, we show that in perturbative settings the correlator depends only on the spectrum of the theory, as well as the OPE coefficients of certain low twist operators, and can be reconstructed unambiguously. In contrast to the Lorentzian inversion formula, the validity of the dispersion relation does not assume Regge behavior and is not restricted to the exchange of spinning operators. As an application, the correlator $⟨\varphi \varphi \varphi \varphi ⟩$ in ${\varphi }^4$ theory at the Wilson-Fisher fixed point is computed in closed form to order ${\epsilon }^2$ in the $\epsilon$ expansion.

19 pages, 2 figures

References in corpus (24)

• Conformal Four Point Functions and the Operator Product Expansion
• The Analytic Bootstrap and AdS Superhorizon Locality
• Analyticity in Spin in Conformal Theories
• Convexity and Liberation at Large Spin
• Algorithms for the symbolic integration of hyperlogarithms with applications to Feynman integrals
• HypExp, a Mathematica package for expanding hypergeometric functions around integer-valued parameters
• OPE Convergence in Conformal Field Theory
• HypExp 2, Expanding Hypergeometric Functions about Half-Integer Parameters
• Large Spin Perturbation Theory
• Unitarity and the Holographic S-Matrix
• A Mellin space approach to the conformal bootstrap
• Analytic Bounds and Emergence of ${\text{AdS}}_2$ Physics from the Conformal Bootstrap
• All Tree-Level Correlators in AdS${}_5\times$S${}_5$ Supergravity: Hidden Ten-Dimensional Conformal Symmetry
• Four-point Functions of Lowest Weight CPOs in N=4 SYM_4 in Supergravity Approximation
• The Analytic Functional Bootstrap I: 1D CFTs and 2D S-Matrices
• The Analytic Functional Bootstrap II: Natural Bases for the Crossing Equation
• Solving CFTs with Weakly Broken Higher Spin Symmetry
• Universality at large transverse spin in defect CFT
• Taming the $\epsilon$-expansion with Large Spin Perturbation Theory
• More $\mathcal{N}=4$ superconformal bootstrap
• On the Polyakov-Mellin bootstrap
• Analytic Bootstrap for Boundary CFT
• Conjectures for Large N Superconformal N=4 Chiral Primary Four Point Functions
• Large spin systematics in CFT

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• Crossing Symmetric Dispersion Relations for Mellin Amplitudes
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• Aspects of CFTs on Real Projective Space
• Holographic Correlators at Finite Temperature
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• Bootstrapping Monodromy Defects in the Wess-Zumino Model
• Froissart bound for/from CFT Mellin amplitudes
• Conformal dispersion relations for defects and boundaries
• Subleading Bounds on Chaos
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• The perturbative CFT optical theorem and high-energy string scattering in AdS at one loop
• Interacting conformal scalar in a wedge
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• Large $N$ analytical functional bootstrap I: 1D CFTs and total positivity
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