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, $S_{\mathrm{ct}}$, 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 ${\mathrm{\Phi }}^4$ theory and obtain the renormalized two-point function of the dual operator to two loops, and the renormalized four-point function to one-loop order, for operators of any dimension $\mathrm{\Delta }$ and bulk spacetime dimension up to $d+1=7$.

• Received 26 August 2022
• Accepted 6 January 2023

DOI:https://doi.org/10.1103/PhysRevD.107.L021901

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Published by the American Physical Society