General Bootstrap Equations in 4D CFTs

TL;DR

This work addresses the challenge of spinning correlators in 4D CFTs by unifying the covariant embedding formalism with the non-covariant conformal frame approach, enabling generic 2-, 3-, and 4-point bootstrap computations. It develops a complete dictionary of tensor structures, normalization conventions, seed conformal blocks, and Casimir operators, and shows how to map seamlessly between the two formalisms, including treatment of conservation and permutation symmetries. The authors formulate the conformal partial wave decomposition in terms of seed blocks and differential operators, derive and solve bootstrap equations from crossing symmetry, and provide a freely available Mathematica package implementing these methods. The framework is designed to handle operators in arbitrary Lorentz representations, paving the way for systematic analytic and numerical bootstrap studies of spinning 4D CFTs with practical computational tools. Overall, this work delivers a practical, interconnected toolkit for constructing and validating spinning conformal blocks and crossing equations in 4D, with broad potential impact on constraints for QCD-like theories and beyond.

Abstract

We provide a framework for generic 4D conformal bootstrap computations. It is based on the unification of two independent approaches, the covariant (embedding) formalism and the non-covariant (conformal frame) formalism. We construct their main ingredients (tensor structures and differential operators) and establish a precise connection between them. We supplement the discussion by additional details like classification of tensor structures of n-point functions, normalization of 2-point functions and seed conformal blocks, Casimir differential operators and treatment of conserved operators and permutation symmetries. Finally, we implement our framework in a Mathematica package and make it freely available.