Loops in Anti-de Sitter Space.

Igor Bertan,Ivo Sachs

doi:10.1103/physrevlett.121.101601

Igor Bertan, Ivo Sachs

Open Access

https://doi.org/10.1103/physrevlett.121.101601

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Abstract

We obtain analytic results for the four-point amplitude, at one loop, of an interacting scalar field theory in four-dimensional, Euclidean anti-de Sitter space without exerting any conformal field theory knowledge. For the two-point function, we provide analytic expressions up to two loops. In addition, we argue that the critical exponents of correlation functions near the conformal boundary of anti-de Sitter space provide the necessary data for the renormalization conditions, thus replacing the usual on-shell condition.

Highlights

Introduction.—Over the past sixty years, there has been tremendous progress in the calculation of scattering amplitudes in quantum field theory—in particular, concerning higher loop amplitudes in Yang-Mills theory and gravity
Even in de Sitter or anti–de Sitter (AdS) space, which are maximally symmetric, admitting the same number of isometries as Minkowski space, little is known about such amplitudes; see Refs. [4,5,6,7,8,9,10] for recent progress
The reason for this is that, while in Minkowski space, the momentum representation leads to a hierarchy of elementary integrals; in dS or AdS, this is not the case, and the coordinate representation generally leads to integral expressions that are more manageable in (A)dS

Summary

Introduction

Introduction.—Over the past sixty years, there has been tremendous progress in the calculation of scattering amplitudes in quantum field theory—in particular, concerning higher loop amplitudes in Yang-Mills theory and (super) gravity. We compute the two- and four-point functions for λφ4 theory [11] to the second order in the coupling λ on the Poincarepatch of Euclidean AdS4 by explicitly evaluating the corresponding one- and two-loop integrals in coordinate representation. Legs at generic points in AdS, but for insertions on the conformal boundary we are able to get explicit expressions.

Results

Conclusion