Anomalous dimensions at finite conformal spin from OPE inversion

Abstract

We compute anomalous dimensions of higher spin operators in Conformal Field Theory at arbitrary space-time dimension by using the OPE inversion formula of \cite{Caron-Huot:2017vep}, both from the position space representation as well as from the integral (viz. Mellin) representation of the conformal blocks. The Mellin space is advantageous over the position space not only in allowing to write expressions agnostic to the space-time dimension, but also in that it replaces tedious recursion relations in terms of simple sums which are easy to perform. We evaluate the contributions of scalar and spin exchanges in the $t-$channel exactly, in terms of higher order Hypergeometric functions. These relate to a particular exchange of conformal spin $\beta =\Delta +J$ in the $s-$channel through the inversion formula. Our exact results reproduce the special cases for large spin anomalous dimension and OPE coefficients obtained previously in the literature.

37 pages, not figures, some typos corrected, acknowledgement added. Added comments and clarifications regarding finite beta contributions

References in corpus (15)

• The Conformal Bootstrap: Theory, Numerical Techniques, and Applications
• Solving the 3d Ising Model with the Conformal Bootstrap II. c-Minimization and Precise Critical Exponents
• The Lightcone Bootstrap and the Spectrum of the 3d Ising CFT
• Bootstrapping Mixed Correlators in the 3D Ising Model
• Spinning AdS Propagators
• A spacetime derivation of the Lorentzian OPE inversion formula
• Conformal Bootstrap in Mellin Space
• D-independent representation of Conformal Field Theories in D dimensions via transformation to auxiliary Dual Resonance Models. Scalar amplitudes
• Taming the $\epsilon$-expansion with Large Spin Perturbation Theory
• Spinning Mellin Bootstrap: Conformal Partial Waves, Crossing Kernels and Applications
• Loops in Anti-de Sitter Space
• Simplicity in AdS Perturbative Dynamics
• First-order conformal perturbation theory by marginal operators
• OPE inversion in Mellin space
• Mellin Bootstrap for Scalars in Generic Dimension

Cited by in corpus (27)

• The Conformal Bootstrap: Theory, Numerical Techniques, and Applications
• $d$-dimensional SYK, AdS Loops, and $6j$ Symbols
• Quantum Regge Trajectories and the Virasoro Analytic Bootstrap
• On the Polyakov-Mellin bootstrap
• Nonperturbative Mellin Amplitudes: Existence, Properties, Applications
• Recursion Relations in Witten Diagrams and Conformal Partial Waves
• Black Holes, Heavy States, Phase Shift and Anomalous Dimensions
• Crossing symmetry, transcendentality and the Regge behaviour of 1d CFTs
• Distributions in CFT II. Minkowski Space
• Black Holes and Conformal Regge Bootstrap
• Heavy-light Bootstrap from Lorentzian Inversion Formula
• Higher Spin ANEC and the Space of CFTs
• An alternative to diagrams for the critical O(N) model: dimensions and structure constants to order $1/N^2$
• Precision Bootstrap for the $\mathcal{N}=1$ Super-Ising Model
• Resummation at finite conformal spin
• On Genus-one String Amplitudes on $Ad{S}_5\times S^5$
• Closed-form expression for cross-channel conformal blocks near the lightcone
• Lightcone expansions of conformal blocks in closed form
• Late time Wilson lines
• More Analytic Bootstrap: Nonperturbative Effects and Fermions
• The Inversion Formula and 6j Symbol for 3d Fermions
• Factorized lightcone expansion of conformal blocks
• Anomalous Dimensions from Thermal AdS Partition Functions
• Virasoro blocks at large exchange dimension
• Relating diagrammatic expansion with conformal correlator expansion
• Four-point correlation modular bootstrap for OPE densities
• Analytic Studies of Fermions in the Conformal Bootstrap