Conformal four point functions and the operator product expansion

By F. Dolan, H. Osborn. 2000

701 citations recorded by the source corpus.

Various aspects of the four point function for scalar fields in conformally invariant theories are analysed. This depends on an arbitrary function of two conformal invariants u,v. A recurrence relation for the function corresponding to the contribution of an arbitrary spin field in the operator product expansion to the four point function is derived. This is solved explicitly in two and four dimensions in terms of ordinary hypergeometric functions of variables z,x which are simply related to u,v. The operator product expansion analysis is applied to the explicit expressions for the four point function found for free scalar, fermion and vector field theories in four dimensions. The results for four point functions obtained by using the AdS/CFT correspondence are also analysed in terms of functions related to those appearing in the operator product discussion.

Subfields: Astronomy and Astrophysics, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/24f7c91a141f741d4f6446f09ae4820afde47482

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Citations in the corpus

Projectors, shadows, and conformal blocks (2012, methodology): “Unlike the derivation in [21, 22], ours easily generalizes to the case of conformal integrals with tensor indices, which will be needed to compute conformal blocks for operators with spin.”
Conformal Regge theory (2012, background): “(102) This is actually independent of λ because the three point function with the stress-energy tensor is determined by a Ward identity [41, 42]”
Radial Coordinates for Conformal Blocks (2013, methodology): “The expansion coefficients An,j satisfy a three-term recursion relation, also derived earlier from a different point of view by Dolan and Osborn [20].”
Convexity and liberation at large spin (2012, positive): “We mostly follow the conventions of [66]”
Causality constraints in conformal field theory (2015, methodology): “In d = 4 [48] the conformal blocks are known in terms of hypergeometric functions, and reproduced in appendix B.”
Analyticity and the holographic S-matrix (2011, background): “The functional form contributed to a 4-pt function by the exchange of a primary state/operator of dimension ∆ and angular momentum ` along with all of its descendants is called a conformal block [27, 35, 21, 23].”
Eikonal approximation in AdS/CFT: Conformal partial waves and finite N four-point functions (2006, methodology): “1 The d = 2 Case Explicit expressions for the partial waves Th,h̄ and Sh,h̄ exist for d even [9, 10], and are particularly simple in d = 2 where the problem factorizes in left/right equations for z and z̄.”
The ϵ ?> -expansion from conformal field theory (2015, methodology): “They looked at the conformal block of V1 appearing in the decomposition of the four point function 〈V1(x1)V2(x2)V1(x3)V2(x4)〉, using the series representation from [54], Eq.”
Defects in conformal field theory (2016, background): “See [50, 51] for an extensive discussion of conformal blocks in CFT, and [10, 43] for the case of a boundary.”
Superconformal symmetry, correlation functions and the operator product expansion (2001, methodology): “From [13](7) the function G has the expansion G(u, v) = 1 5 ∑”
Central charge bounds in 4D conformal field theory (2010, background): “0)i= Dd (D 1)S D 1 (x2 12 )d 1x2 1 x 2 2 Z Z 1 D Z 2 : (12) These relations determine the coecient c 4;2 appearing in (6) and (10) in terms of the central charge C T and the dimension of ˚[14]: c 4;2 = Dd D 1 1 p C T : (13) 4 Via (6), this crucial relation implies that for large C T, the contrib”
A natural language for AdS/CFT correlators (2011, methodology): “It was used by Dolan and Osborn to find explicit and simple expressions for these contributions, which are the familiar conformal blocks [42, 26].”
Bootstrapping SCFTs with four supercharges (2015, positive): “We follow the normalization of [42], in particular equation (4.”
Correlation functions and massive Kaluza-Klein modes in the AdS//CFT correspondence (2002, background): “6 Operator Product Expansion The operator product expansion of the stress-energy tensor multiplets in N = 4 theory and the AdS5 supergravity is by now a well-developed subject [37]-[41],[26].”
Large spin systematics in CFT (2015, positive): “The four point function in the small u limit is then [10]”
Bootstrapping the O(N ) vector models (2013, methodology): “In [10], special analytic expressions for conformal blocks in even dimensions [39, 40] were used to derive approximations of the form (2.”
A theory of reparameterizations for AdS3 gravity (2018, methodology): “s, then it is a Virasoro block and we notate it as V. In either case, the block is an eigenfunction of the corresponding conformal Casimir. The global blocks are known in closed form and are given by [62,63] F(h i;h;u) = uh 2F 1(h h 1 + h 2;h+ h 3 h 4;2h;u): (6.6) The Virasoro blocks are not known i”