Conformal Bootstrap in Mellin Space

Abstract

We propose a new approach towards analytically solving for the dynamical content of Conformal Field Theories (CFTs) using the bootstrap philosophy. This combines the original bootstrap idea of Polyakov with the modern technology of the Mellin representation of CFT amplitudes. We employ exchange Witten diagrams with built in crossing symmetry as our basic building blocks rather than the conventional conformal blocks in a particular channel. Demanding consistency with the operator product expansion (OPE) implies an infinite set of constraints on operator dimensions and OPE coefficients. We illustrate the power of this method in the epsilon expansion of the Wilson-Fisher fixed point by reproducing anomalous dimensions and, strikingly, obtaining OPE coefficients to higher orders in epsilon than currently available using other analytic techniques (including Feynman diagram calculations). Our results enable us to get a somewhat better agreement of certain observables in the 3d Ising model, with the precise numerical values that have been recently obtained.

5 pages+appendices. v2:Added--matching of HS current anomalous dimension to O(ε^3); references; minor changes, v3: comparison with numerics updated, typos fixed. v4: Published version

References in corpus (5)

• Solving the 3d Ising Model with the Conformal Bootstrap II. c-Minimization and Precise Critical Exponents
• Writing CFT correlation functions as AdS scattering amplitudes
• Spinning AdS Propagators
• Critical $O(N)$ Models in $6-\epsilon$ Dimensions
• Three Loop Analysis of the Critical $O(N)$ Models in $6-\epsilon$ Dimensions

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• A CFT Perspective on Gravitational Dressing and Bulk Locality
• The nuts and bolts of the BMS Bootstrap
• Momentum space conformal three-point functions of conserved currents and a general spinning operator
• A Crossing-Symmetric OPE Inversion Formula
• Scaling dimensions in QED${}_3$ from the $\epsilon$-expansion
• Inverse Bootstrapping Conformal Field Theories
• On the Exchange Interactions in Holographic p-adic CFT
• Multi-critical multi-field models: a CFT approach to the leading order