Bulk renormalization and the AdS/CFT correspondence

TL;DR

This paper develops a systematic renormalization method for quantum field theory in anti-de Sitter space, enabling consistent calculations of correlation functions and their dual conformal field theories, including IR and UV divergence handling.

Contribution

It introduces a scheme-dependent renormalization procedure respecting AdS symmetries, deriving Feynman rules for Witten diagrams from first principles, and applies it to scalar a^4 theory.

Findings

Renormalized 2-point function computed to 2-loops.

Renormalized 4-point function computed to 1-loop.

Method applicable to operators of any dimension in up to 6+1 dimensions.

Abstract

We develop a systematic renormalization procedure for QFT in anti-de Sitter spacetime. UV infinities are regulated using a geodesic point-splitting method, which respects AdS isometries, while IR infinities are regulated by cutting off the radial direction (as in holographic renormalization). The renormalized theory is defined by introducing $Z$ factors for all parameters in the Lagrangian and the boundary conditions of bulk fields (sources of dual operators), and a boundary counterterm action, $S_{\rm ct}$, such that the limit of removing the UV and IR regulators exists. The results are in general scheme dependent (mirroring the analogous result in flat space) and require renormalization conditions. These may be provided by the dual CFT (or by string theory in AdS). Our analysis amounts also to a first principles derivation of the Feynman rules regarding Witten diagrams. The presence and treatment of IR divergences is essential for correctly accounting for anomalous dimensions of dual operators. We apply the method to scalar $\Phi^4$ theory and obtain the renormalized 2-point function of the dual operator to 2-loops, and the renormalized 4-point function to 1-loop order, for operators of any dimension $\Delta$ and bulk spacetime dimension up to $d+1=7$.