Seed Conformal Blocks in 4D CFT

Abstract

We compute in closed analytical form the minimal set of "seed" conformal blocks associated to the exchange of generic mixed symmetry spinor/tensor operators in an arbitrary representation (l,\bar l) of the Lorentz group in four dimensional conformal field theories. These blocks arise from 4-point functions involving two scalars, one (0,|l-\bar l|) and one (|l-\bar l|,0) spinors or tensors. We directly solve the set of Casimir equations, that can elegantly be written in a compact form for any (l,\bar l), by using an educated ansatz and reducing the problem to an algebraic linear system. Various details on the form of the ansatz have been deduced by using the so called shadow formalism. The complexity of the conformal blocks depends on the value of p=|l-\bar l | and grows with p, in analogy to what happens to scalar conformal blocks in d even space-time dimensions as d increases. These results open the way to bootstrap 4-point functions involving arbitrary spinor/tensor operators in four dimensional conformal field theories.

38 pages, 4 figures; v2: references added, typo in the final eqs.(5.36) and (5.37) fixed (sorry)

References in corpus (18)

• Bounding scalar operator dimensions in 4D CFT
• Conformal Four Point Functions and the Operator Product Expansion
• Conformal Partial Waves and the Operator Product Expansion
• Solving the 3d Ising Model with the Conformal Bootstrap II. c-Minimization and Precise Critical Exponents
• The Analytic Bootstrap and AdS Superhorizon Locality
• Spinning Conformal Correlators
• Convexity and Liberation at Large Spin
• Projectors, Shadows, and Conformal Blocks
• Bootstrapping the O(N) Vector Models
• Spinning Conformal Blocks
• Witten Diagrams Revisited: The AdS Geometry of Conformal Blocks
• Diagonal Limit for Conformal Blocks in d Dimensions
• Covariant Approaches to Superconformal Blocks
• $\mathcal{N}=1$ Superconformal Blocks for General Scalar Operators
• General Three-Point Functions in 4D CFT
• Scalar-Vector Bootstrap
• Conformal Blocks in the Large D Limit
• Conformal blocks for highly disparate scaling dimensions

Cited by in corpus (42)

• The Conformal Bootstrap: Theory, Numerical Techniques, and Applications
• A Mellin space approach to the conformal bootstrap
• Weight Shifting Operators and Conformal Blocks
• Harmonic Analysis and Mean Field Theory
• Multipoint Conformal Blocks in the Comb Channel
• The 3d Stress-Tensor Bootstrap
• Superintegrability of $d$-dimensional Conformal Blocks
• Spinning Witten Diagrams
• Mellin space bootstrap for global symmetry
• Spinning Mellin Bootstrap: Conformal Partial Waves, Crossing Kernels and Applications
• Harmony of Spinning Conformal Blocks
• Dimensional Reduction for Conformal Blocks
• Casimir recursion relations for general conformal blocks
• From Spinning Conformal Blocks to Matrix Calogero-Sutherland Models
• Simplifying large spin bootstrap in Mellin space
• Integrability of Conformal Blocks I: Calogero-Sutherland Scattering Theory
• Scalar CFTs and Their Large N Limits
• Geodesic Diagrams, Gravitational Interactions & OPE Structures
• Gaudin Models and Multipoint Conformal Blocks: General Theory
• Higher-Point Conformal Blocks in the Comb Channel
• Gaudin Models and Multipoint Conformal Blocks II: Comb channel vertices in 3D and 4D
• A multipoint conformal block chain in $d$ dimensions
• Recursion relations for 5-point conformal blocks
• Spinning Geodesic Witten Diagrams
• Gaudin Models and Multipoint Conformal Blocks III: Comb channel coordinates and OPE factorisation
• New Methods for Conformal Correlation Functions
• Mellin Amplitudes for Fermionic Conformal Correlators
• Towards Feynman rules for conformal blocks
• Towards a Bootstrap approach to higher orders of epsilon expansion
• Charting the space of 3D CFTs with a continuous global symmetry
• Superconformal Blocks: General Theory
• Quantum Integrable Systems from Conformal Blocks
• Efficient Rules for All Conformal Blocks
• Conformal Group Theory of Tensor Structures
• Differential operators for superconformal correlation functions
• Lightcone expansions of conformal blocks in closed form
• Conformal partial waves in momentum space
• Universal Spinning Casimir Equations and Their Solutions
• R-current three-point functions in 4d $\mathcal{N}=1$ superconformal theories
• Towards general super Casimir equations for $4D$ $\mathcal{N}=1$ SCFTs
• Harmonic Analysis in d-dimensional Superconformal Field Theory
• Crossing Symmetry for Long Multiplets in 4D $\mathcal{N}=1$ SCFTs