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 $\langle SSSS\rangle$, $\langle SSSR\rangle$, $\langle SRSR\rangle$ and $\langle SSRR\rangle$ (where, when allowed, $R$ is a vector or a fermion), and determine the corresponding blocks for all possible exchanged representations.