Abstract
We compute the two- and four-point holographic correlation functions up to the second order in the coupling constant for a scalar ϕ4 theory in four-dimensional Euclidean anti-de Sitter space. Analytic expressions for the anomalous dimensions of the leading twist operators are found at one loop, both for Neumann and Dirichlet boundary conditions.
Highlights
Since the coupling of the CFT stress tensor to the graviton is present almost universally, one is confronted, when considering the bulk theory beyond the classical level, with the quantization of gravity together with its perturbative pathologies in the ultraviolet
One possible way to get around these problems, in the case of AdS, is to use the conformal bootstrap, in particular, crossing symmetry, to determine the coefficients in the operator product expansion (OPE) and the anomalous dimensions of the corresponding operators in CFT, to make predictions for loop-corrected boundary-to-boundary correlation functions of the dual bulk theory in AdS [10,11,12, 31]
A related approach, followed in ref. [7], is to reduce loop diagrams in global coordinates in AdS to a sum over tree-level diagrams using a discrete Mellin space Kallen-Lehmann representation with weight function inferred from the OPE in the dual CFT.1
Summary
We briefly review the kinematical ingredients that are used throughout the paper. The (d + 1)-dimensional Euclidean anti-de Sitter space, Hd+1, can be embedded into a (d + 2)dimensional flat Minkowski ambient space Md+2. Given two points X and Y on the hyperboloid, there is a simple relation between the geodesic distance ρ and the scalar product of X with Y cosh aρ = −a2X · Y. The flat space limit is obtained by letting a → 0. In order to evaluate the Feynman bulk-to-bulk diagrams we need the bulk-to-bulk propagator Λ(x, y; m) for the scalar field. Bearing in mind the above relation, let us express the propagator by Λ(K; ∆), i.e., as a function of K and ∆. Our model is a conformally coupled scalar field with a quartic self-interaction propagating on a static H4 background, described by the action d4x. Before closing this section let us express schematically the expansions for the two- and the four-point function respectively, φ(x1)φ(x2) =. The reader not interested in the details of the calculation can skip to section 5, where a summary of the obtained results is available
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