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Journal of High Energy Physics

QFT as a set of ODEs

Manuel Loparco, Grégoire Mathys, João Penedones, Jiaxin Qiao, Xiang Zhao

Published Jul 16, 2026• 1 citation
Publisher

Abstract

A bstract Correlation functions of local operators in Quantum Field Theory (QFT) on hyperbolic space can be fully characterized by the set of QFT data {∆ i , C ijk , $$ {b}_j^{\hat{\mathcal{O}}} $$ b j O ̂ }. These are the scaling dimensions of boundary operators ∆ i , the boundary Operator Product Expansion (OPE) coefficients C ijk and the Boundary Operator Expansion (BOE) coefficients $$ {b}_j^{\hat{\mathcal{O}}} $$ b j O ̂ that characterize how each bulk operator $$ \hat{\mathcal{O}} $$ O ̂ can be expanded in terms of boundary operators $$ {\mathcal{O}}_j $$ O j . For simplicity, we focus on two dimensional QFTs and derive a universal set of first order Ordinary Differential Equations (ODEs) that encode the variation of the QFT data under an infinitesimal change of a bulk relevant coupling. In principle, our ODEs can be used to follow a Renormalization Group (RG) flow starting from a solvable QFT into a strongly coupled phase and to the flat space limit.

DOI
10.1007/jhep07(2026)153

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A bstract Correlation functions of local operators in Quantum Field Theory (QFT) on hyperbolic space can be fully characterized by the set of QFT data {∆ i , C ijk , bjO^{b}_j^{\hat{\mathcal{O}}}bjO^​ b j O ̂ }. These are the scaling dimensions of boundary operators ∆ i , the boundary Operator Product Expansion (OPE) coefficients C ijk and the Boundary Operator Expansion (BOE) coefficients bjO^{b}_j^{\hat{\mathcal{O}}}bjO^​ b j O ̂ that characterize how each bulk operator O^\hat{\mathcal{O}}O^ O ̂ can be expanded in terms of boundary operators Oj{\mathcal{O}}_jOj​ O j . For simplicity, we focus on two dimensional QFTs and derive a universal set of first order Ordinary Differential Equations (ODEs) that encode the variation of the QFT data under an infinitesimal change of a bulk relevant coupling. In principle, our ODEs can be used to follow a Renormalization Group (RG) flow starting from a solvable QFT into a strongly coupled phase and to the flat space limit.
Correlation functions of local operators in Quantum Field Theory (QFT) on hyperbolic space can be fully characterized by the set of QFT data {Δi,Cijk,bjO^}\lbrace Δ_i,C_{ijk},b^{\hat{\mathcal{O}}}_j\rbrace{Δi​,Cijk​,bjO^​}. These are the scaling dimensions of boundary operators ΔiΔ_iΔi​, the boundary Operator Product Expansion (OPE) coefficients CijkC_{ijk}Cijk​ and the Boundary Operator Expansion (BOE) coefficients bjO^b^{\hat{\mathcal{O}}}_jbjO^​ that characterize how each bulk operator O^\hat{\mathcal{O}}O^ can be expanded in terms of boundary operators Oj\mathcal{O}_jOj​.For simplicity, we focus on two dimensional QFTs and derive a universal set of first order Ordinary Differential Equations (ODEs) that encode the variation of the QFT data under an infinitesimal change of a bulk relevant coupling. In principle, our ODEs can be used to follow a Renormalization Group (RG) flow starting from a solvable QFT into a strongly coupled phase and to the flat space limit.
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Abstract We derive a universal asymptotic formula for generic boundary conditions for the average value of the bulk-to-boundary and boundary Operator Product Expansion coefficients of any unitary, compact two-dimensional Boundary CFT (BCFT) with c > 1. The asymptotic limit consists of taking one or more boundary primary operators — which transform under a single copy of the Virasoro algebra — to have parametrically large conformal dimension for fixed central charge. In particular, we find a single universal expression that interpolates between distinct heavy regimes, exactly as in the case of bulk OPE asymptotics [1]. The expression depends universally on the boundary entropy and the central charge, and not on any other details of the theory. We derive these asymptotics by studying crossing symmetry of various correlation functions on higher genus Riemann surfaces with open boundaries. Essential in the derivation is the use of the irrational versions of the crossing kernels that relate holomorphic Virasoro blocks in different channels. Our results strongly suggest an extended version of the Eigenstate Thermalization Hypothesis for boundary OPE coefficients, where the hierarchy between the diagonal and non-diagonal term in the ansatz is further controlled by the boundary entropy. We finally comment on the applications of our results in the context of AdS3/BCFT2, as well as on the recent relation of BCFTs with lower dimensional models of evaporating black holes.
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Numerical fluid dynamics for FRG flow equations: Zero-dimensional QFTs as numerical test cases - Part I: The O(N)O(N)O(N) model
The functional renormalization group (FRG) approach is a powerful tool for studies of a large variety of systems, ranging from statistical physics over the theory of the strong interaction to gravity. The practical application of this approach relies on the derivation of so-called flow equations, which describe the change of the quantum effective action under the variation of a coarse-graining parameter. In the present work, we discuss in detail a novel approach to solve such flow equations. This approach relies on the fact that RG equations can be rewritten such that they exhibit similarities with the conservation laws of fluid dynamics. This observation can be exploited in different ways. First of all, we show that this allows to employ powerful numerical techniques developed in the context of fluid dynamics to solve RG equations. In particular, it allows to reliably treat the emergence of non-analytic behavior in the RG flow of the effective action as it is expected to occur in studies of, e.g., spontaneous symmetry breaking. Second, the analogy between RG equations and fluid dynamics offers the opportunity to gain novel insights into RG flows and their interpretation in general, including the irreversibility of RG flows. We work out this connection in practice by applying it to zero-dimensional quantum-field theoretical models. The generalization to higher-dimensional models is also discussed. Our findings are expected to help improving future FRG studies of quantum field theories in higher dimensions both on a qualitative and quantitative level.
For QFTs in AdS the boundary correlation functions remain conformal even if the bulk theory has a scale. This allows one to constrain RG flows with numerical conformal bootstrap methods. We apply this idea to flows between two-dimensional CFTs, focusing on deformations of the tricritical and ordinary Ising model. We provide non-perturbative constraints for the boundary correlation functions of these flows and compare them with conformal perturbation theory in the vicinity of the fixed points. We also reproduce a completely general constraint on the sign of the TTˉT\bar TTTˉ deformation in two dimensions.