Efficient rules for all conformal blocks

• Published in: The journal of high energy physics Vol. 2021; no. 11; pp. 1 - 61
• Main Authors: Fortin, Jean-François, Ma, Wen-Jie, Prilepina, Valentina, Skiba, Witold
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 09.11.2021; Springer Nature B.V; SpringerOpen
• Subjects: Classical and Quantum Gravitation; Conformal and W Symmetry; Conformal Field Theory; Elementary Particles; Fermions; Field Theories in; High energy physics; Higher Dimensions; Operators; Phase transitions; Physics; Physics and Astronomy; Quantum Field Theories; Quantum Field Theory; Quantum Physics; Regular Article - Theoretical Physics; Relativity Theory; Representations; Scalars; String Theory; Tensors; Field Theories in; Conformal and W Symmetry; Higher Dimensions; Conformal Field Theory
• ISSN: 1029-8479; 1029-8479
• DOI: 10.1007/JHEP11(2021)052

A bstract 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 [1]. 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.