Conformal four-point correlation functions from the operator product expansion

Jean-François Fortin, Valentina Prilepina, Witold Skiba
2020 Journal of High Energy Physics  
We show how to compute conformal blocks of operators in arbitrary Lorentz representations using the formalism described in [1, 2] and present several explicit examples of blocks derived via this method. The procedure for obtaining the blocks has been reduced to (1) determining the relevant group theoretic structures and (2) applying appropriate predetermined substitution rules. The most transparent expressions for the blocks we find are expressed in terms of specific substitutions on the
more » ... tions on the Gegenbauer polynomials. In our examples, we study operators which transform as scalars, symmetric tensors, two-index antisymmetric tensors, as well as mixed representations of the Lorentz group. A different approach for the computation of conformal blocks was recently proposed in [1, 2] . It relies on using the operator product expansion (OPE) in the embedding space [160] [161] [162] [163] . The framework for embedding space OPE was introduced in [164-169], with further developments presented in [170] [171] [172] . This approach can be applied to yield any conformal block in general spacetime dimensions. In this formalism, operators in arbitrary Lorentz representations are uplifted to the embedding space in a uniform manner using products of spinor representations alone. Derivatives naturally occur in the OPE, and hence it is of interest to fully determine their action in order to directly obtain the blocks. These were evaluated explicitly in [1, 2] for any expression that may potentially arise in any M -point function. With the action of derivatives already in hand, computing conformal blocks just requires finding the projection operators for irreducible Lorentz representations and then performing appropriate replacements of terms with the corresponding expressions obtained from derivatives in the OPE. In this work, we derive several four-point conformal blocks using the approach developed in [1, 2] . We have two main goals here. One is to illustrate how the formalism performs in practice. Another is to validate the approach by comparing the results with the existing ones in the literature whenever available. Some of the ingredients needed here, in particular, the projection operators and three-point tensor structures, were studied in detail in [173, 174] ; we rely on those results in this paper. An interesting aspect of the present approach is that all conformal blocks computed here can be expressed in terms of the Gegenbauer polynomials onto which particular substitution rules are then applied. The Gegenbauer polynomials are functions of a variable X, and a set of substitution rules transforms X into the final answer. This paper is organized as follows: we start with an overview of our method and main results in section 2. Section 3 expresses all four-point correlation functions in terms of the conformal blocks. The conformal blocks themselves are obtained by contracting two tensor structures, each originating from the OPE, with the so-called "pre-conformal blocks". These pre-conformal blocks depend primarily on the Lorentz quantum numbers of the exchanged quasi-primary operator. They are computed in two steps using the corresponding hatted projection operators. In the first step, the projection operators are transformed using the three-point tensorial function. In the second step, the result is transformed further by a four-point conformal substitution rule yielding the proper conformal quantity. The resulting pre-conformal blocks are linear combinations of tensorial objects, which involve the generalized Exton G-functions of the conformal cross-ratios. The contractions of the pre-conformal blocks with the two tensor structures can be facilitated with the help of several contiguous relations, leading to the standard conformal blocks. In this work, all pre-conformal blocks and conformal blocks are computed in the s-channel. Section 4 illustrates how the formalism can be applied to derive pre-conformal blocks and conformal blocks in a series of examples. The conformal blocks are all written in terms of appropriate conformal substitutions on the Gegenbauer polynomials. As such, the conformal blocks presented here are the final answers that do not contain any derivatives. Comparison with the existing literature demonstrates the validity of the approach. Finally, section 5 concludes, pointing out the importance of hatted projection operators and tensor structures (d,h,n) 12 can be used to construct higher-point functions for operators in arbitrary Lorentz representations. The resulting expressions generated by the present method are naturally found in closed form with no derivatives or integrals that need to be evaluated. An M -point function is therefore given in terms of D (d,h,n) 12 acting on a function with M − 1 points. However, the upshot is that the action of the OPE differential operator ,
doi:10.1007/jhep08(2020)115 fatcat:7p3sao43xff5zo4bgbsop2qlle