Abstract

Conformal field theory (CFT) dispersion relations reconstruct correlators in terms of their double discontinuity. When applied to the crossing equation, such dispersive transforms lead to sum rules that suppress the double-twist sector of the spectrum and enjoy positivity properties at large twist. In this paper, we construct dispersive CFT functionals for correlators of unequal scalar operators in position- and Mellin-space. We then evaluate these functionals in the Regge limit to construct mixed correlator holographic CFT functionals which probe scalar particle scattering in Anti-de Sitter spacetime. Finally, we test properties of these dispersive sum rules when applied to the 3D Ising model, and we use truncated sum rules to find approximate solutions to the crossing equation.

Highlights

  • The success of the modern conformal bootstrap stands upon the pillars of high precision numerical methods [1, 2], and deeper understanding of the analytic structure of the bootstrap equations

  • We construct dispersive Conformal field theory (CFT) functionals for correlators of unequal scalar operators in position- and Mellin-space. We evaluate these functionals in the Regge limit to construct mixed correlator holographic CFT functionals which probe scalar particle scattering in Anti-de Sitter spacetime

  • Numerical methods have led to high precision measurements of critical exponents of conformal field theories (CFTs) such as the 3D Ising [5–8] and the O(N ) models [9–11], while analytical methods have proven essential to the study of quantum gravity in Anti-de Sitter (AdS) spacetime [12–18] and perturbative CFTs [19, 20]

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Summary

Introduction

The success of the modern conformal bootstrap stands upon the pillars of high precision numerical methods [1, 2], and deeper understanding of the analytic structure of the bootstrap equations (see [3, 4] and references therein). When evaluated in the Regge limit, these dispersive functionals are further capable of probing bulk AdS physics This insight led to the construction of holographic functionals in [31] where the collinear Bk,v functional served as a seed in their construction. Such holographic functionals are capable of bounding couplings in AdS, and they become dispersion relations in the flat-space limit [36, 37]. Evaluating these Bk,v functionals in position-space can become expensive in certain limits such as for large number of derivatives, or for low twist operators (near the unitarity bound). We use insight gained from this analysis to build a system of truncated sum rules to estimate approximate solutions to crossing for the 3D Ising model

Overview of dispersive CFT sum rules
Generalities
Deriving the position-space Bak,|bmn kernel
Convergence and positivity properties: a plethora of subtraction schemes
Mellin-space functionals
Mellin-space representation of holographic functionals
Applications to the 3D Ising model
Evaluating 3D Ising model sum rules
Approximate solutions to crossing from truncated sum rules
Discussion
C Mack polynomials
D Mixed correlator analytic double-twist functionals
E Varying v for the σσσσ sum rule

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