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 [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.
Highlights
Conformal field theories (CFTs) are special quantum field theories that enjoy an enhanced symmetry, namely invariance under the conformal group SO(2, d)
Since there are no half-projectors for the exchanged quasi-primary operators in the fourpoint correlation function (2.32), the projection operator to the exchanged representation necessarily appears explicitly in the four-point conformal blocks (2.34)
We have established a set of highly efficient rules for determining all possible four-point conformal blocks in terms of fundamental group theoretic quantities, namely the projection operators of the external and exchanged quasi-primary operators
Summary
Conformal field theories (CFTs) are special quantum field theories that enjoy an enhanced symmetry, namely invariance under the conformal group SO(2, d). As detailed in [75], in the context of this formalism, the procedure for determining a given block comes down to (1) writing down the relevant group theoretic quantities, namely the projection operators and tensor structures (which effectively serve as intertwiners among the respective external and exchanged representations), and (2) identifying the specific linear combination of Gegenbauer polynomials along with the corresponding substitution rules for each piece While this approach is complete and clearly formulated as it stands, it is rather cumbersome to apply in practice for infinite towers of exchanged quasi-primary operators in irreducible representations N m + e1. The rotation matrices entail a careful analysis of the three-point functions, while the conformal blocks necessitate an expansion in Gegenbauer polynomials in a special variable X, coupled with associated substitution rules In these two sections, we consider infinite towers of exchanged quasiprimary operators in some irreducible representations N m + e1.
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