QFT as a set of ODEs: higher dimensions

Abstract

Correlation functions of local operators in Quantum Field Theory (QFT) in Anti-de Sitter space (AdS) are completely fixed by the QFT data: the set of scaling dimensions ${\Delta }_i$ and OPE coefficients $C_{ijk}$ of the boundary operators, and the bulk-boundary (BOE) coefficients $b_i^{\hat{\Phi }}$ encoding how bulk fields decompose into boundary operators. In this work, we generalize the ordinary differential equations (ODEs) that govern the variation of the QFT data under a bulk relevant deformation, originally derived for AdS${}_2$ \cite{Loparco:2026fki}, to the cases of AdS${}_3$ and AdS${}_4$. We demonstrate that these flow equations natively capture the mechanism of merger-annihilation when a boundary operator hits marginality, as well as level repulsion when different ${\Delta }_i$'s approach each other. Furthermore, we address the practical implementation of the framework: we propose substituting the ODE for the OPE coefficients with the crossing equation for greater efficiency, and we observe that Padé approximants dramatically improve the convergence of the sums over boundary operators, at least in free theories. Altogether, these advances lay the groundwork for the future application of the flow equations to the study of strongly coupled QFTs in AdS and their flat space limits.

v1: 58 pages + 5 appendices, 18 figures, one ancillary Mathematica file; v2: added a discussion of the exclusion of the identity operator from the flow equations; revised the section on Padé approximation; corrected typos; v3: ancillary Mathematica file reinserted