Efficient Rules for All Conformal Blocks

Abstract

We formulate a set of general rules for computing $d$-dimensional four-point global conformal blocks of operators in arbitrary Lorentz representations in the context of the embedding space operator product expansion formalism arXiv:1905.00434. With these rules, the procedure for determining any conformal block of interest is reduced to (1) identifying the relevant projection operators and tensor structures and (2) applying the conformal rules to obtain the blocks. To facilitate the bookkeeping of contributing terms, we introduce a convenient diagrammatic notation. We present several concrete examples to illustrate the general procedure as well as to demonstrate and test the explicit application of the rules. In particular, we consider four-point functions involving scalars $S$ and some specific irreducible representations $R$, namely $⟨SSSS⟩$, $⟨SSSR⟩$, $⟨SRSR⟩$ and $⟨SSRR⟩$ (where, when allowed, $R$ is a vector or a fermion), and determine the corresponding blocks for all possible exchanged representations.

1+61 pages

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Cited by in corpus (6)

• Gaudin Models and Multipoint Conformal Blocks II: Comb channel vertices in 3D and 4D
• Seven-Point Conformal Blocks in the Extended Snowflake Channel and Beyond
• Gaudin Models and Multipoint Conformal Blocks III: Comb channel coordinates and OPE factorisation
• Feynman Rules for Scalar Conformal Blocks
• All Global One- and Two-Dimensional Higher-Point Conformal Blocks
• Harmonic Analysis in d-dimensional Superconformal Field Theory