Loops in Anti-de Sitter Space

TL;DR

The paper advances analytic control of quantum corrections for a scalar λφ^4 theory in Euclidean AdS4 by computing the two-point function up to two loops and the four-point function at one loop in coordinate space. It demonstrates that boundary conformal dimensions can serve as renormalization data, effectively replacing the conventional mass-based on-shell condition in AdS, and provides a detailed mapping of bulk loop results to the OPE data of a dual boundary CFT via conformal blocks. The key technical achievements include explicit expressions for the mass-shift structure in two-point functions and a closed-form (up to a finite integral) one-loop correction to the boundary four-point function, encapsulated in the L4 term. Together, these results yield a consistent holographic framework in which bulk amplitudes encode boundary CFT data, with clear predictions for operator dimensions and OPE coefficients and without requiring prior conformal field theory input at tree level.

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.