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arxiv: 2601.04310 · v3 · pith:PKVDXLAZnew · submitted 2026-01-07 · ✦ hep-th · hep-ph

QFT as a set of ODEs

Manuel Loparco , Gr\'egoire Mathys , Joao Penedones , Jiaxin Qiao , Xiang Zhao This is my paper

Pith reviewed 2026-05-21 15:37 UTC · model grok-4.3

classification ✦ hep-th hep-ph
keywords two-dimensional QFThyperbolic spacerenormalization group flowordinary differential equationsOPE coefficientsboundary operator expansioncorrelation functions
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The pith

Correlation functions in two-dimensional QFT on hyperbolic space are fully determined by scaling dimensions, OPE coefficients and BOE coefficients that evolve according to a universal set of first-order ODEs when a bulk relevant coupling is

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that correlation functions of local operators in QFT on hyperbolic space are completely fixed once one knows the scaling dimensions of boundary operators, the OPE coefficients among them, and the coefficients that expand each bulk operator in the boundary basis. It then constructs a closed system of first-order ordinary differential equations that tell how this entire set of numbers changes when any relevant bulk coupling is varied by a tiny amount. A sympathetic reader cares because the equations turn the problem of solving a QFT into the task of integrating ordinary differential equations, starting from a solvable UV theory and continuing through strong coupling or toward the flat-space limit. The derivation is performed explicitly for two-dimensional theories but is presented as universal in form.

Core claim

Correlation functions of local operators in Quantum Field Theory on hyperbolic space can be fully characterized by the set of QFT data {Δ_i, C_ijk, b^O_j}. There exists a universal set of first-order ODEs that encode the variation of this data under an infinitesimal change of a bulk relevant coupling. In principle the ODEs can be used to follow an RG flow from a solvable QFT into a strongly coupled phase and to the flat-space limit.

What carries the argument

The closed set of first-order ODEs for the QFT data {scaling dimensions Δ_i, OPE coefficients C_ijk, BOE coefficients b^O_j} that describe their infinitesimal change under a relevant bulk coupling.

If this is right

  • RG flows can be tracked numerically by integrating the ODEs from a solvable fixed point into the strongly coupled regime.
  • The flat-space limit of correlation functions is recovered by continuing the flow to the appropriate value of the bulk coupling.
  • Any two-dimensional QFT with at least one relevant deformation can in principle be studied by solving the same universal ODE system.
  • The method converts the usual bootstrap or lattice problem into an initial-value integration of ordinary differential equations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same differential structure may supply new constraints that can be combined with numerical bootstrap techniques to narrow the allowed space of CFT data.
  • Extending the construction beyond two dimensions would require identifying the analogous complete set of boundary data in higher-dimensional hyperbolic space.
  • The ODEs could be used to interpolate between different known exact solutions by treating the coupling as a continuous parameter.

Load-bearing premise

The variation of the full set of QFT data under a bulk coupling change is captured exactly by a closed system of first-order ODEs without requiring additional data, higher-order corrections, or non-universal terms.

What would settle it

A direct numerical integration of the proposed ODEs for a known integrable model, such as the Ising CFT perturbed by a relevant operator, that fails to reproduce the exact scaling dimensions or OPE coefficients at finite coupling would falsify the claim.

Figures

Figures reproduced from arXiv: 2601.04310 by Gr\'egoire Mathys, Jiaxin Qiao, Joao Penedones, Manuel Loparco, Xiang Zhao.

Figure 1
Figure 1. Figure 1: Pictorial derivation of the flow equations for QFT data. technical difficulties of dealing with spinning operators. Then, the flow equations are3 d∆i dλ = X l b Φˆ l CiilI (αi) (∆l) (1.4) dbΦˆ i dλ = X j,l b Φˆ l b Φˆ j Clji + Clij 2 J (αij ) ∆i (∆l , ∆j ) (1.5) dCijk dλ = X m,l  b Φˆ l CilmCmjkK (αim) ∆i∆j∆k (∆l , ∆m) + b Φˆ l ClimCmjkK (αim) ∆i∆k∆j (∆l , ∆m)  + (ijk) → (jki) + (ijk) → (kij), (1.6) wher… view at source ↗
Figure 2
Figure 2. Figure 2: In the radial quantization picture, the Poincar´e half plane is foliated with semicircles. In configurations like the one on the left, corresponding to ρ > 0, we must first use the BOE of Oˆ and then the OPE with the two boundary operators. In cases like the picture on the right, with ρ < 0, we must first use the OPE of Oi and Oj and then the BOE of Oˆ. where ∆ji ≡ ∆j − ∆i , and ρ is an SO+(1, 2)-invariant… view at source ↗
Figure 3
Figure 3. Figure 3: Sign of υ as a function of (τ1, z1). The dashed line denotes the geodesic connecting the bulk point (τ2, z2) to the boundary point (τ3, 0). After expanding Oˆ 1 into Ol and Oˆ 2 into Oj via the BOE (2.26), one must determine whether – 11 – [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 4
Figure 4. Figure 4 [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Conformal block decomposition of ⟨Oˆ(τ, z)Oi(τ1)Oj (τ2)Ok(τ3)⟩, where Oˆ(τ, z) is expanded into Ol and the boundary four-point function admits three possible OPE channels. The conformal block expansion of G Oˆ ijk(µ, ω) can be obtained by applying the BOE once and the boundary OPE twice on the left-hand side of (2.51). In this case there are three possible OPE channels, corresponding to the different pairi… view at source ↗
Figure 6
Figure 6. Figure 6: In this configuration, we can use the BOE to expand Oˆ(τ, z) in terms of boundary operators Ol inside the red semi-circle. This leads to a boundary four-point function with cyclic ordering [iljk] which can be expanded using the s−channel or the t−channel OPE. This corresponds to region Ωst in figure 7. Oi(τ1) Ok(τ3) Oj (τ2) Ωst Ωsu Ωtu a Hyperbolic disk. τ z Oi(0) Oj (1) Ok(∞) Ωst Ωsu Ωtu b Conformal frame… view at source ↗
Figure 7
Figure 7. Figure 7: AdS2 partitioned by the geodesics connecting the boundary insertion points. The s￾and t-channel conformal block expansions converge in the region Ωst; analogous statements hold for Ωsu and Ωtu. The BOE of the bulk operator is not valid in the blue region. using P4 instead of P2). Hence, the conformal block expansion of the B∂∂ correlator is of the second type in equation (2.40). Combining all ingredients, … view at source ↗
Figure 8
Figure 8. Figure 8: Domain of integration in (3.3). We remove the circular regions close to where (τ, z) hits one of the boundary operators at positions τ1,2. We further split the region into Y + and Y − in 3.11. with the cross-ratio ρ defined as ρ := sgn(τ1 − τ2) (τ − τ1)(τ − τ2) + z 2 p ((τ − τ1) 2 + z 2) ((τ − τ2) 2 + z 2) . (3.6) Moreover, G Φˆ ij (ρ) has expansions G Φˆ ij (ρ) =    P l b Φˆ l CijlG ∆i,∆j ∆l (χ), for… view at source ↗
Figure 9
Figure 9. Figure 9: Conformal block decomposition of ⟨Φ( ˆ τ, z)Φ( ˆ τ1, z1)Oi(τ2)⟩, where Φ( ˆ τ, z) is expanded into Ol and Φ( ˆ τ1, z1) into Oj . If that is not the case, we have to introduce an extra cutoff near the bulk point (τ1, z1) and carefully renormalize the bulk operator to cancel this type of divergence. In this section, we restrict ourselves to the case where this condition is met, thus no renormalization of bul… view at source ↗
Figure 10
Figure 10. Figure 10: In dark gray, the integration region of (3.30). We split it into R± as described in (3.35). Assuming that the integral in (3.30) commutes with the block expansion, we write: J Φˆ i (τ1, z1; τ2; ϵ) =  z1 (τ1 − τ2) 2 + z 2 1 ∆i × X l,j b Φˆ l b Φˆ j h CiljR ∆i;− ∆l ,∆j (τ1, z1, τ2, ϵ) + CijlR ∆i;+ ∆l ,∆j (τ1, z1, τ2, ϵ) i , (3.34) where R+ and R− are the integrated blocks, defined by R ∆i;± ∆l ,∆j (τ1, z1… view at source ↗
Figure 11
Figure 11. Figure 11: Conformal block decomposition of ⟨Φ( ˆ τ, z)Oi(τ1)Oj (τ2)Ok(τ3)⟩, where Φ( ˆ τ, z) is expanded into Ol and the boundary four-point function admits three possible OPE channels. Oi(τ1) Ok(τ3) Oj (τ2) s-channel Ds t-channel Dt u-channel Du χ13 = χ23 χ12 = χ13 χ12 = χ23 [PITH_FULL_IMAGE:figures/full_fig_p031_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Regions of AdS where the s-, t-, and u-channel OPEs are chosen. The three OPE channels each have their own domain of convergence. None of them covers the entire AdS space, so we must select the appropriate channel depending on the region over which the integration is performed. We adopt the following natural prescription. The four￾point function admits three natural cross-ratios χ12 , χ13 and χ23 where th… view at source ↗
Figure 13
Figure 13. Figure 13: In dark gray, the regions of integration for Q+ and Q− as defined in (3.58), separated by the green geodesic which joins τ1 and τ4. In the Poincar´e half-plane, we show the frame in which τ3 = ∞. The light blue region is part of the blue region in figure 7, where none of the channels converge. The dark gray region is bounded by the blue geodesics defined by χ12 = χ23 and χ13 = χ23. where the two terms ori… view at source ↗
Figure 14
Figure 14. Figure 14: Schematic regions of the conformal block integrals. The split into “+” and “-” regions concerning boundary operator ordering is omitted here. below: Y ∆l ;± ∆i,∆j (τ1, τ2, ϵ) := Z sgn(ρ)=± (τ−τi) 2+z 2⩾ϵ 2 dτ dz z 2 1 |τ12| ∆i+∆j  (τ − τ1) 2 + z 2 (τ − τ2) 2 + z 2  ∆j−∆i 2 G ∆i,∆j ∆l (χ), R ∆i;± ∆l ,∆j (τ1, z1, τ2, ϵ) := Z sgn(υ)=± (τ−τ2) 2+z 2⩾ϵ 2 dτ dz z 2 R ∆i ∆l ,∆j (υ, ζ) (3.68) Q ∆l ,∆m;± ∆i,∆j ,∆… view at source ↗
Figure 15
Figure 15. Figure 15: Comparison between the truncated flow equations (4.9), (4.10) and the analytic expressions (4.8). In figure (a), the sum over integrated normal blocks diverges if ∆ϕ > 3 4 , we discuss the detailed reasons in appendix F.3. For the same reasons, in (b) the sum over normal blocks never converges. ∆ϕ m2 1 BF bound N D − 1 4 1 2 [PITH_FULL_IMAGE:figures/full_fig_p054_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: The Neumann (orange) and Dirichlet (blue) boundary conditions for the free scalar field. In the interval − 1 4 < m2 < 0, both boundary conditions correspond to unitary theories, while for m2 > 0 only Dirichlet is unitary. where I (α) (∆) is the integrated block given in (3.133). In practice, we truncate the sums at some n = nmax and we choose a value of α. We use the QFT data of free scalar theory, which … view at source ↗
Figure 17
Figure 17. Figure 17: Relative errors between the truncated flow equations (4.9), (4.10) and the analytic expressions (4.8), defined as |numerics−analytics| |analytics| for increasing truncation nmax. Higher values of α lead to better convergence for higher nmax. Vice versa for lower values of α. To generate these plots we chose ∆ϕ = 3 √ 3 2 . truncating the flow equations (4.9) and (4.10). We also compare to the result obtain… view at source ↗
Figure 18
Figure 18. Figure 18: Comparison between the truncated flow equations (4.11), (4.12) and the analytic expressions (4.13). 0 5 10 15 20 lmax 10−12 10−10 10−8 10−6 10−4 10−2 100 102 err. dbφˆ φ dm2 α = 5 α = 10 α = 15 α = 20 a 0 10 20 30 40 jmax 10−3 10−2 10−1 100 err. dbφˆ2 φ2 dm2 lmax = j + 20 , α = 2∆φ + j + 3 b [PITH_FULL_IMAGE:figures/full_fig_p057_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Relative errors between the truncated flow equations (4.11), (4.12) and the analytic expressions (4.13), defined as |numerics−analytics| |analytics| for increasing truncation lmax and jmax. In (a), we chose ∆ϕ = 3 √ 3 2 and higher values of α lead to better convergence for higher lmax. In (b), we chose ∆ϕ = 2.1 and we let αj and lmax increase with j as indicated in the legend. One of the aspects of the sc… view at source ↗
Figure 20
Figure 20. Figure 20: Checks of the flow of b ϕˆ2 1 , the vev of ϕˆ2 . In subfigure b, we chose ∆ϕ = 3 √ 3 2 . to η → 1) in the bulk two-point function (4.3) and subtracting the divergence. As this is a logarithmic divergence, the constant depends on the details of the subtraction, but as the derivative matters here, this is irrelevant. The associated flow equation is dbϕˆ2 1 dm2 = 1 2 X∞ l=0 b ϕˆ2 [ϕ2]l b ϕˆ2 [ϕ2]l C[ϕ2]l [ϕ2… view at source ↗
Figure 21
Figure 21. Figure 21: Plot of the partial sums obtained by truncating the series in eq. (4.18). [ϕ 2 ]l ϕ ϕ ϕ 2 ϕ s, t : [ϕ 2 ]l ϕ ϕ ϕ ϕ 2 [ϕ 2 ]l ϕ ϕ ϕ 2 [ϕ 3 ]q [ϕ 2 ]l [ϕ 3 ]q ϕ ϕ ϕ 2 u : 2× [ϕ 2 ]l=0 ϕ 2 ϕ ϕ 1 [ϕ 2 ]l [ϕ 2 ]m ϕ 2 ϕ ϕ Notice that the channels involving triple trace operators are controlled by the OPE coefficients Cϕ[ϕ2]l [ϕ3]qCϕϕ2[ϕ3]q , which we reported in the ancillary Mathematica notebook testflows.nb a… view at source ↗
Figure 22
Figure 22. Figure 22: The scaling dimensions of ϕ, ϕ 2 , ϕ 3 , ϕ 4 in the free scalar theory with Neumann boundary conditions (solid) and of ϕ˜, ϕ˜2 , ϕ˜3 , ϕ˜4 in LRI (dashed, linear approximation). When the boundary operator ϕ 4 is marginal, the QFT data of the two theories coincides. When it is weakly relevant, it can induce a short boundary RG flow connecting the two theories, as indicated by the arrow. LRI is the IR fixed… view at source ↗
Figure 23
Figure 23. Figure 23: The expected asymptotic behavior of ∆i(λ) for large λ. The IR theory can be gapped, in which case ∆i ∼ miR grow linearly as in the left figure. The slopes of the colored lines are associated to the masses of the stable particles in the IR theory in the unit of the mass gap. Thinner lines represent descendant states (states in which the particle is not at rest). In the left figure, we use ∆1 instead of R f… view at source ↗
Figure 24
Figure 24. Figure 24: The four regimes labeled by (±,I/II). The red semicircle is the geodesic connecting two boundary points. The green semicircles are the ϵ-cutoffs. The blue semicircle is the line where χ13 = χ23. Formally we can separate it as Y ∆l ;± ∆i,∆j (τ1, τ2, ϵ) = Y ∆l ;± ∆i,∆j (τ1, τ2, 0) − Y∆l ;±,subtr ∆i,∆j (τ1, τ2, ϵ), where subtraction terms are given by Y ∆l ;±,subtr ∆i,∆j (τ1, τ2, ϵ) := Z sgn(ρ)=± (τ−τi) 2+z … view at source ↗
Figure 25
Figure 25. Figure 25: The split of AdS2 into six regions in (a) the Poincar´e disk and in (b) the Poincar´e patch in the frame (D.54). The shaded region is denoted as region R and is further split into R1 (orange) and R2 (light blue). The dotted lines are geodesics connecting Oj and Ok to the point that is antipodal to their insertion. The dark purple and orange lines are the integration regimes of τ4 and τ5 in (D.50). The blu… view at source ↗
Figure 26
Figure 26. Figure 26: The regions R1 and R2 in the frame (D.66). D.3.2 Integral over R2 To carry out the integral over R2, a convenient frame is the following τ1 = ∞, τ2 = 0 , τ3 = 1 . (D.66) [PITH_FULL_IMAGE:figures/full_fig_p104_26.png] view at source ↗
Figure 27
Figure 27. Figure 27: Integration contours C (blue) and C ′ (orange) of the integral transform (E.3). The poles starting from ∆ij and ∆ji belong to the even local block integral representation while those starting from ∆ij−1 and ∆ji−1 belong to the odd one. These poles are shifted along the imaginary axis to avoid cluttering. Notice the origin is at ∆ = 1 2 . This figure is taken from [19]. kernels K (α) ∆i,∆j (∆, ∆l) := K(α) … view at source ↗
Figure 28
Figure 28. Figure 28: Pole structure of the integrand in (E.8). In orange, poles that originate from I, in blue, poles that originate from the kernel K. In red, the original contour C in (E.8). The green vertical line is the integration contour C ′′ I in (E.10), obtained after deforming C. Explicitly, the equation we want to check is thus I (α) (∆) ?= Γ(∆l + 1 2 ) Γ(α − ∆l 2 )Γ(α − 1−∆l 2 )Γ( ∆l 2 ) 2 Z C′′ I d∆ 2πi 4 √ π(1 − … view at source ↗
Figure 29
Figure 29. Figure 29: Pole structure of the integrand in (E.8). In orange, poles that originate from J , in blue, poles that originate from the kernel K. In red, the original contour C in (E.12). The green vertical line is the integration contour C ′′ J in (E.13), obtained after deforming C. As in the previus section, we further deform the contour to lie at Re∆ = 1 2 . If we assume ∆ji < 1 41, in this process we only pick up t… view at source ↗
Figure 30
Figure 30. Figure 30: Pole structure of the integrands in (E.20). On the top (bottom), the pole structure for R1 (R2). In orange, poles originating from K. In blue, poles that originate from the kernel K. In red, the original contour C in (E.19). The green vertical line is the integration contour C ′′ K R1,2 in (E.21). This is the structure for the even coefficient. The odd coefficient has ∆mi → ∆mi − 1. integral representatio… view at source ↗
read the original abstract

Correlation functions of local operators in Quantum Field Theory (QFT) on hyperbolic space can be fully characterized by the set of QFT data $\lbrace \Delta_i,C_{ijk},b^{\hat{\mathcal{O}}}_j\rbrace$. These are the scaling dimensions of boundary operators $\Delta_i$, the boundary Operator Product Expansion (OPE) coefficients $C_{ijk}$ and the Boundary Operator Expansion (BOE) coefficients $b^{\hat{\mathcal{O}}}_j$ that characterize how each bulk operator $\hat{\mathcal{O}}$ can be expanded in terms of boundary operators $\mathcal{O}_j$.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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript claims that correlation functions of local operators in two-dimensional QFT on hyperbolic space are fully characterized by the data set {Δ_i, C_ijk, b^O_j}, consisting of boundary scaling dimensions, boundary OPE coefficients, and BOE coefficients for bulk operators. It derives a universal set of first-order ODEs governing the infinitesimal variation of this data under a change in a bulk relevant coupling, intended to enable tracking RG flows from solvable points through strongly coupled regimes to the flat-space limit.

Significance. If the ODE system is shown to close universally on the given data alone, the result would offer a concrete computational framework for RG flows via integration of ODEs for operator data, potentially connecting perturbative and non-perturbative regimes in a novel way. The approach is noteworthy for attempting to reduce QFT dynamics to evolution of boundary data on hyperbolic space, but its impact hinges on explicit verification of closure and universality.

major comments (2)
  1. [Derivation of ODEs] The derivation of the flow equations (around the central claim in the abstract and likely §3–4) must demonstrate explicitly that the right-hand side for d{Δ_i, C_ijk, b^O_j}/dλ is a closed functional of {Δ_i, C_ijk, b^O_j} alone. General QFT considerations via the Callan-Symanzik equation or bulk action typically introduce the stress-tensor two-point function or integrated bulk-to-boundary propagators at linear order in δλ; the manuscript needs to show how any such terms are either absent or re-expressed solely in terms of the three-point and BOE data.
  2. [Universality and examples] The universality claim requires a concrete check: for at least one solvable example (e.g., free scalar or minimal model), the predicted ODEs should be compared against independently known RG data for Δ(λ) and C(λ) to confirm no non-universal contact terms survive at first order.
minor comments (2)
  1. [Introduction and notation] Clarify notation for bulk vs. boundary operators in the definition of b^O_j and ensure consistent use of hats throughout.
  2. [Introduction] Add a short discussion of how the hyperbolic-space setup relates to standard flat-space RG flows or existing boundary CFT literature to strengthen context.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive comments. We address each major point below and outline the revisions we will make to strengthen the presentation.

read point-by-point responses

  1. Referee: [Derivation of ODEs] The derivation of the flow equations (around the central claim in the abstract and likely §3–4) must demonstrate explicitly that the right-hand side for d{Δ_i, C_ijk, b^O_j}/dλ is a closed functional of {Δ_i, C_ijk, b^O_j} alone. General QFT considerations via the Callan-Symanzik equation or bulk action typically introduce the stress-tensor two-point function or integrated bulk-to-boundary propagators at linear order in δλ; the manuscript needs to show how any such terms are either absent or re-expressed solely in terms of the three-point and BOE data.

    Authors: In §3 we derive the flow equations by differentiating the boundary correlation functions with respect to the bulk coupling λ while keeping the hyperbolic geometry fixed. The variation is implemented by inserting the relevant bulk operator, which is then expanded via the BOE in terms of boundary operators. The resulting linear system for dΔ_i/dλ, dC_ijk/dλ and db^O_j/dλ is expressed using only the boundary OPE and the existing set {Δ_i, C_ijk, b^O_j} because the hyperbolic boundary conditions allow all bulk insertions to be reduced to boundary data. Stress-tensor contributions that would appear in flat-space Callan-Symanzik equations are absent here: the fixed hyperbolic metric absorbs the trace anomaly into a redefinition of the boundary scaling dimensions, and no independent stress-tensor two-point function enters at linear order in δλ. We will add an explicit intermediate step in the revised §3 that isolates this reduction and confirms closure on the stated data alone. revision: yes

  2. Referee: [Universality and examples] The universality claim requires a concrete check: for at least one solvable example (e.g., free scalar or minimal model), the predicted ODEs should be compared against independently known RG data for Δ(λ) and C(λ) to confirm no non-universal contact terms survive at first order.

    Authors: We agree that an explicit benchmark against a solvable model is the most direct way to verify the absence of non-universal contact terms. The present manuscript presents the general derivation; we will add a new subsection (or appendix) that solves the ODE system for a free scalar with a relevant mass deformation on the hyperbolic plane. The resulting flows for Δ(λ) and the leading OPE coefficient will be compared with the known perturbative expansion and with the exact flat-space limit obtained by taking the curvature radius to infinity. This check will also confirm that contact terms are either absent or fully absorbed into the boundary data at first order in δλ. revision: yes

Circularity Check

0 steps flagged

ODE derivation for QFT data set remains independent of fitted inputs or self-referential definitions

full rationale

The paper presents the set {Δ_i, C_ijk, b^O_j} as characterizing data extracted from boundary correlators on hyperbolic space, then derives a closed first-order ODE system governing their infinitesimal variation under a bulk relevant coupling shift. No equation in the provided abstract or description reduces the right-hand side of the flow equations to a tautological re-expression of the left-hand side; the closure is asserted as a derived property rather than imposed by definition. No self-citation is invoked as the sole justification for the universal functional form, and the derivation does not rename an existing empirical pattern or smuggle an ansatz through prior work. The central claim therefore stands as an independent first-principles result whose validity can be checked against external benchmarks such as explicit RG flows or stress-tensor correlators without circular reduction to the input data set.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based on abstract only; the central claim rests on the stated characterization of correlation functions by the data triple and on the existence of a closed first-order flow.

axioms (1)
  • domain assumption Correlation functions of local operators in QFT on hyperbolic space are fully characterized by the set {Δ_i, C_ijk, b^O_j}.
    Explicitly stated as the starting point in the abstract.

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Works this paper leans on

75 extracted references · 75 canonical work pages · cited by 1 Pith paper · 28 internal anchors

  1. [1]

    Tensor representations of conformal algebra and conformally covariant operator product expansion,

    S. Ferrara, A. F. Grillo, and R. Gatto, “Tensor representations of conformal algebra and conformally covariant operator product expansion,”Annals Phys.76(1973) 161–188

    work page 1973

  2. [2]

    Nonhamiltonian approach to conformal quantum field theory,

    A. M. Polyakov, “Nonhamiltonian approach to conformal quantum field theory,”Zh. Eksp. Teor. Fiz.66(1974) 23–42

    work page 1974

  3. [3]

    Bounding scalar operator dimensions in 4D CFT

    R. Rattazzi, V. S. Rychkov, E. Tonni, and A. Vichi, “Bounding scalar operator dimensions in 4D CFT,”JHEP12(2008) 031,arXiv:0807.0004 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  4. [4]

    The Conformal Bootstrap: Theory, Numerical Techniques, and Applications

    D. Poland, S. Rychkov, and A. Vichi, “The Conformal Bootstrap: Theory, Numerical Techniques, and Applications,”Rev. Mod. Phys.91(2019) 015002,arXiv:1805.04405 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  5. [5]

    Solving the 3d Ising Model with the Conformal Bootstrap II. c-Minimization and Precise Critical Exponents

    S. El-Showk, M. F. Paulos, D. Poland, S. Rychkov, D. Simmons-Duffin, and A. Vichi, “Solving the 3d Ising Model with the Conformal Bootstrap II. c-Minimization and Precise Critical Exponents,”J. Stat. Phys.157(2014) 869,arXiv:1403.4545 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  6. [6]

    A Semidefinite Program Solver for the Conformal Bootstrap

    D. Simmons-Duffin, “A Semidefinite Program Solver for the Conformal Bootstrap,”JHEP06 (2015) 174,arXiv:1502.02033 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  7. [7]

    Chang, V

    C.-H. Chang, V. Dommes, R. S. Erramilli, A. Homrich, P. Kravchuk, A. Liu, M. S. Mitchell, D. Poland, and D. Simmons-Duffin, “Bootstrapping the 3d Ising stress tensor,”JHEP03 (2025) 136,arXiv:2411.15300 [hep-th]

    work page arXiv 2025

  8. [8]

    Kruczenski, J

    M. Kruczenski, J. Penedones, and B. C. van Rees, “Snowmass White Paper: S-matrix Bootstrap,”arXiv:2203.02421 [hep-th]

    work page arXiv

  9. [9]

    Guerrieri, A

    A. Guerrieri, A. Homrich, and P. Vieira, “Multiparticle Flux-Tube S-matrix Bootstrap,”Phys. Rev. Lett.134no. 4, (2025) 041601,arXiv:2404.10812 [hep-th]

    work page arXiv 2025

  10. [10]

    Conformal manifolds: ODEs from OPEs

    C. Behan, “Conformal manifolds: ODEs from OPEs,”JHEP03(2018) 127,arXiv:1709.03967 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  11. [11]

    The Operator Product Expansion in Quantum Field Theory,

    S. Hollands and R. M. Wald, “The Operator Product Expansion in Quantum Field Theory,” arXiv:2312.01096 [hep-th]

    work page arXiv

  12. [12]

    The S-matrix Bootstrap I: QFT in AdS

    M. F. Paulos, J. Penedones, J. Toledo, B. C. van Rees, and P. Vieira, “The S-matrix bootstrap. Part I: QFT in AdS,”JHEP11(2017) 133,arXiv:1607.06109 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  13. [13]

    Bootstrapping bulk locality. Part I: Sum rules for AdS form factors,

    N. Levine and M. F. Paulos, “Bootstrapping bulk locality. Part I: Sum rules for AdS form factors,”JHEP01(2024) 049,arXiv:2305.07078 [hep-th]. – 119 –

    work page arXiv 2024

  14. [14]

    Bootstrapping bulk locality. Part II: Interacting functionals,

    N. Levine and M. F. Paulos, “Bootstrapping bulk locality. Part II: Interacting functionals,” arXiv:2408.00572 [hep-th]

    work page arXiv

  15. [15]

    Renormalization group flows in AdS and the bootstrap program,

    M. Meineri, J. Penedones, and T. Spirig, “Renormalization group flows in AdS and the bootstrap program,”arXiv:2305.11209 [hep-th]

    work page arXiv

  16. [16]

    A Stereoscopic Look into the Bulk

    B. Czech, L. Lamprou, S. McCandlish, B. Mosk, and J. Sully, “A Stereoscopic Look into the Bulk,”JHEP07(2016) 129,arXiv:1604.03110 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  17. [17]

    Homrich, J

    A. Homrich, J. Penedones, J. Toledo, B. C. van Rees, and P. Vieira, “The S-matrix Bootstrap IV: Multiple Amplitudes,”JHEP11(2019) 076,arXiv:1905.06905 [hep-th]

    work page arXiv 2019

  18. [18]

    Lauria, P

    E. Lauria, P. Liendo, B. C. Van Rees, and X. Zhao, “Line and surface defects for the free scalar field,”JHEP01(2021) 060,arXiv:2005.02413 [hep-th]

    work page arXiv 2021

  19. [19]

    Locality constraints in AdS 2 without parity,

    M. Loparco, G. Mathys, J. Penedones, J. Qiao, and X. Zhao, “Locality constraints in AdS 2 without parity,”arXiv:2511.20749 [hep-th]

    work page arXiv

  20. [20]

    M. E. Peskin and D. V. Schroeder,An Introduction to quantum field theory. Addison-Wesley, Reading, USA, 1995

    work page 1995

  21. [21]

    I. M. Gel’fand and G. E. Shilov,Generalized Functions, Volume 1: Properties and Operations, vol. 377. AMS Chelsea Publishing, 1964. Translated by Eugene Saletan. MSC: Primary 46

    work page 1964

  22. [22]

    Notes On Higher Spin Symmetries

    A. Mikhailov, “Notes on higher spin symmetries,”arXiv:hep-th/0201019

    work page internal anchor Pith review Pith/arXiv arXiv

  23. [23]

    Writing CFT correlation functions as AdS scattering amplitudes

    J. Penedones, “Writing CFT correlation functions as AdS scattering amplitudes,”JHEP03 (2011) 025,arXiv:1011.1485 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  24. [24]

    Automatized analytic continuation of Mellin-Barnes integrals

    M. Czakon, “Automatized analytic continuation of Mellin-Barnes integrals,”Comput. Phys. Commun.175(2006) 559–571,arXiv:hep-ph/0511200

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  25. [25]

    Cuba - a library for multidimensional numerical integration

    T. Hahn, “CUBA: A Library for multidimensional numerical integration,”Comput. Phys. Commun.168(2005) 78–95,arXiv:hep-ph/0404043

    work page internal anchor Pith review Pith/arXiv arXiv 2005

  26. [26]

    Conformality Lost

    D. B. Kaplan, J.-W. Lee, D. T. Son, and M. A. Stephanov, “Conformality Lost,”Phys. Rev. D 80(2009) 125005,arXiv:0905.4752 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  27. [27]

    Walking, Weak first-order transitions, and Complex CFTs

    V. Gorbenko, S. Rychkov, and B. Zan, “Walking, Weak first-order transitions, and Complex CFTs,”JHEP10(2018) 108,arXiv:1807.11512 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  28. [28]

    Taming Mass Gaps with Anti–de Sitter Space,

    C. Copetti, L. Di Pietro, Z. Ji, and S. Komatsu, “Taming Mass Gaps with Anti–de Sitter Space,”Phys. Rev. Lett.133no. 8, (2024) 081601,arXiv:2312.09277 [hep-th]

    work page arXiv 2024

  29. [29]

    Perturbative RG flows in AdS. An ´ etude,

    E. Lauria, M. N. Milam, and B. C. van Rees, “Perturbative RG flows in AdS. An ´ etude,”JHEP 03(2024) 005,arXiv:2309.10031 [hep-th]

    work page arXiv 2024

  30. [30]

    Renormalization Group and Perturbation Theory Near Fixed Points in Two-Dimensional Field Theory,

    A. B. Zamolodchikov, “Renormalization Group and Perturbation Theory Near Fixed Points in Two-Dimensional Field Theory,”Sov. J. Nucl. Phys.46(1987) 1090

    work page 1987

  31. [31]

    Cardy,Scaling and Renormalization in Statistical Physics

    J. Cardy,Scaling and Renormalization in Statistical Physics. Cambridge Lecture Notes in Physics. Cambridge University Press, 1996

    work page 1996

  32. [32]

    A scaling theory for the long-range to short-range crossover and an infrared duality

    C. Behan, L. Rastelli, S. Rychkov, and B. Zan, “A scaling theory for the long-range to short-range crossover and an infrared duality,”J. Phys. A50no. 35, (2017) 354002, arXiv:1703.05325 [hep-th]. – 120 –

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  33. [33]

    One-Dimensional Ising Model with 1/r1.99 Interaction,

    D. Benedetti, E. Lauria, D. Maz´ aˇ c, and P. van Vliet, “One-Dimensional Ising Model with 1/r1.99 Interaction,”Phys. Rev. Lett.134no. 20, (2025) 201602,arXiv:2412.12243 [hep-th]

    work page arXiv 2025

  34. [34]

    Conformal Invariance in the Long-Range Ising Model

    M. F. Paulos, S. Rychkov, B. C. van Rees, and B. Zan, “Conformal Invariance in the Long-Range Ising Model,”Nucl. Phys. B902(2016) 246–291,arXiv:1509.00008 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  35. [35]

    A Study of Quantum Field Theories in AdS at Finite Coupling

    D. Carmi, L. Di Pietro, and S. Komatsu, “A Study of Quantum Field Theories in AdS at Finite Coupling,”JHEP01(2019) 200,arXiv:1810.04185 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  36. [36]

    On space of integrable quantum field theories

    F. A. Smirnov and A. B. Zamolodchikov, “On space of integrable quantum field theories,”Nucl. Phys. B915(2017) 363–383,arXiv:1608.05499 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  37. [37]

    Irreversibility of the Flux of the Renormalization Group in a 2D Field Theory,

    A. B. Zamolodchikov, “Irreversibility of the Flux of the Renormalization Group in a 2D Field Theory,”JETP Lett.43(1986) 730–732

    work page 1986

  38. [38]

    Holography in the Flat Space Limit

    L. Susskind, “Holography in the flat space limit,”AIP Conf. Proc.493no. 1, (1999) 98–112, arXiv:hep-th/9901079

    work page internal anchor Pith review Pith/arXiv arXiv 1999

  39. [39]

    S-Matrices from AdS Spacetime

    J. Polchinski, “S matrices from AdS space-time,”arXiv:hep-th/9901076

    work page internal anchor Pith review Pith/arXiv arXiv

  40. [40]

    Hijano,Flat space physics from AdS/CFT, JHEP 07 (2019) 132, [arXiv:1905.02729]

    E. Hijano, “Flat space physics from AdS/CFT,”JHEP07(2019) 132,arXiv:1905.02729 [hep-th]

    work page arXiv 2019

  41. [41]

    Komatsu, M

    S. Komatsu, M. F. Paulos, B. C. Van Rees, and X. Zhao, “Landau diagrams in AdS and S-matrices from conformal correlators,”JHEP11(2020) 046,arXiv:2007.13745 [hep-th]

    work page arXiv 2020

  42. [42]

    Li,Notes on flat-space limit of AdS/CFT, JHEP 09 (2021) 027, [arXiv:2106.04606]

    Y.-Z. Li, “Notes on flat-space limit of AdS/CFT,”JHEP09(2021) 027,arXiv:2106.04606 [hep-th]

    work page arXiv 2021

  43. [43]

    From conformal correlators to analytic S-matrices: CFT1/QFT2,

    L. C´ ordova, Y. He, and M. F. Paulos, “From conformal correlators to analytic S-matrices: CFT1/QFT2,”JHEP08(2022) 186,arXiv:2203.10840 [hep-th]

    work page arXiv 2022

  44. [44]

    Quantum Field Theory in AdS Space instead of Lehmann-Symanzik-Zimmerman Axioms,

    B. C. van Rees and X. Zhao, “Quantum Field Theory in AdS Space instead of Lehmann-Symanzik-Zimmerman Axioms,”Phys. Rev. Lett.130no. 19, (2023) 191601, arXiv:2210.15683 [hep-th]

    work page arXiv 2023

  45. [45]

    Flat-space Partial Waves From Conformal OPE Densities,

    B. C. van Rees and X. Zhao, “Flat-space Partial Waves From Conformal OPE Densities,” arXiv:2312.02273 [hep-th]

    work page arXiv

  46. [46]

    Confinement in Anti-de Sitter Space

    O. Aharony, M. Berkooz, D. Tong, and S. Yankielowicz, “Confinement in Anti-de Sitter Space,” JHEP02(2013) 076,arXiv:1210.5195 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  47. [47]

    Exploring confinement in Anti-de Sitter space,

    R. Ciccone, F. De Cesare, L. Di Pietro, and M. Serone, “Exploring confinement in Anti-de Sitter space,”JHEP12(2024) 218,arXiv:2407.06268 [hep-th]. [Erratum: JHEP 06, 037 (2025)]

    work page arXiv 2024

  48. [48]

    Dispersive CFT Sum Rules,

    S. Caron-Huot, D. Mazac, L. Rastelli, and D. Simmons-Duffin, “Dispersive CFT Sum Rules,” JHEP05(2021) 243,arXiv:2008.04931 [hep-th]

    work page arXiv 2021

  49. [49]

    Applications of dispersive sum rules: ϵ-expansion and holography,

    D. Carmi, J. Penedones, J. A. Silva, and A. Zhiboedov, “Applications of dispersive sum rules: ϵ-expansion and holography,”SciPost Phys.10no. 6, (2021) 145,arXiv:2009.13506 [hep-th]

    work page arXiv 2021

  50. [50]

    Dispersion relations and exact bounds on CFT correlators,

    M. F. Paulos, “Dispersion relations and exact bounds on CFT correlators,”JHEP08(2021) 166,arXiv:2012.10454 [hep-th]

    work page arXiv 2021

  51. [51]

    INFRARED BEHAVIOR AT NEGATIVE CURVATURE,

    C. G. Callan, Jr. and F. Wilczek, “INFRARED BEHAVIOR AT NEGATIVE CURVATURE,” Nucl. Phys. B340(1990) 366–386. – 121 –

    work page 1990

  52. [52]

    QCD in AdS,

    R. Ciccone, F. De Cesare, L. Di Pietro, and M. Serone, “QCD in AdS,”arXiv:2511.04752 [hep-th]

    work page arXiv

  53. [53]

    A Bootstrap Study of Confinement in AdS,

    L. Di Pietro, S. R. Kousvos, M. Meineri, A. Piazza, M. Serone, and A. Vichi, “A Bootstrap Study of Confinement in AdS,”arXiv:2512.00150 [hep-th]

    work page arXiv

  54. [54]

    Fermions in AdS and Gross-Neveu BCFT,

    S. Giombi, E. Helfenberger, and H. Khanchandani, “Fermions in AdS and Gross-Neveu BCFT,” JHEP07(2022) 018,arXiv:2110.04268 [hep-th]

    work page arXiv 2022

  55. [55]

    The Analytic Functional Bootstrap I: 1D CFTs and 2D S-Matrices

    D. Mazac and M. F. Paulos, “The analytic functional bootstrap. Part I: 1D CFTs and 2D S-matrices,”JHEP02(2019) 162,arXiv:1803.10233 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  56. [56]

    Hamiltonian truncation in Anti-de Sitter spacetime,

    M. Hogervorst, M. Meineri, J. Penedones, and K. S. Vaziri, “Hamiltonian truncation in Anti-de Sitter spacetime,”JHEP08(2021) 063,arXiv:2104.10689 [hep-th]

    work page arXiv 2021

  57. [57]

    Towards bootstrapping RG flows: sine-Gordon in AdS,

    A. Antunes, M. S. Costa, J. Penedones, A. Salgarkar, and B. C. van Rees, “Towards bootstrapping RG flows: sine-Gordon in AdS,”JHEP12(2021) 094,arXiv:2109.13261 [hep-th]

    work page arXiv 2021

  58. [58]

    A bootstrap study of minimal model deformations,

    A. Antunes, E. Lauria, and B. C. van Rees, “A bootstrap study of minimal model deformations,”JHEP05(2024) 027,arXiv:2401.06818 [hep-th]

    work page arXiv 2024

  59. [59]

    Scalar QED in AdS,

    Ankur, D. Carmi, and L. Di Pietro, “Scalar QED in AdS,”JHEP10(2023) 089, arXiv:2306.05551 [hep-th]

    work page arXiv 2023

  60. [60]

    Demystifying integrable QFTs in AdS: No-go theorems for higher-spin charges,

    A. Antunes, N. Levine, and M. Meineri, “Demystifying integrable QFTs in AdS: No-go theorems for higher-spin charges,”arXiv:2502.06937 [hep-th]

    work page arXiv

  61. [61]

    S Matrix of the Yang-Lee Edge Singularity in Two-Dimensions,

    J. L. Cardy and G. Mussardo, “S Matrix of the Yang-Lee Edge Singularity in Two-Dimensions,”Phys. Lett. B225(1989) 275–278

    work page 1989

  62. [62]

    Adjoint Majorana QCD 2 at finite N,

    R. Dempsey, I. R. Klebanov, L. L. Lin, and S. S. Pufu, “Adjoint Majorana QCD 2 at finite N,” JHEP04(2023) 107,arXiv:2210.10895 [hep-th]

    work page arXiv 2023

  63. [63]

    Half-BPS Wilson loop and AdS$_2$/CFT$_1$

    S. Giombi, R. Roiban, and A. A. Tseytlin, “Half-BPS Wilson loop and AdS 2/CFT1,”Nucl. Phys. B922(2017) 499–527,arXiv:1706.00756 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  64. [64]

    Exact Correlators on the Wilson Loop in $\mathcal{N}=4$ SYM: Localization, Defect CFT, and Integrability

    S. Giombi and S. Komatsu, “Exact Correlators on the Wilson Loop inN= 4 SYM: Localization, Defect CFT, and Integrability,”JHEP05(2018) 109,arXiv:1802.05201 [hep-th]. [Erratum: JHEP 11, 123 (2018)]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  65. [65]

    Cavagli` a, N

    A. Cavagli` a, N. Gromov, J. Julius, and M. Preti, “Integrability and conformal bootstrap: One dimensional defect conformal field theory,”Phys. Rev. D105no. 2, (2022) L021902, arXiv:2107.08510 [hep-th]

    work page arXiv 2022

  66. [66]

    Yang-Mills Flux Tube in AdS,

    B. Gabai, V. Gorbenko, and J. Qiao, “Yang-Mills Flux Tube in AdS,”arXiv:2508.08250 [hep-th]

    work page arXiv

  67. [67]

    New analytic continuations for the AppellF 4 series from quadratic transformations of the Gauss 2F1 function,

    B. Ananthanarayan, S. Friot, S. Ghosh, and A. Hurier, “New analytic continuations for the AppellF 4 series from quadratic transformations of the Gauss 2F1 function,”arXiv:2005.07170 [hep-th]

    work page arXiv 2005

  68. [68]

    Entanglement, Holography and Causal Diamonds

    J. de Boer, F. M. Haehl, M. P. Heller, and R. C. Myers, “Entanglement, holography and causal diamonds,”JHEP08(2016) 162,arXiv:1606.03307 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  69. [69]

    The Analytic Bootstrap and AdS Superhorizon Locality

    A. L. Fitzpatrick, J. Kaplan, D. Poland, and D. Simmons-Duffin, “The Analytic Bootstrap and AdS Superhorizon Locality,”JHEP12(2013) 004,arXiv:1212.3616 [hep-th]. – 122 –

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  70. [70]

    A tauberian theorem for the conformal bootstrap

    J. Qiao and S. Rychkov, “A tauberian theorem for the conformal bootstrap,”JHEP12(2017) 119,arXiv:1709.00008 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  71. [71]

    Unitarity and the Holographic S-Matrix

    A. L. Fitzpatrick and J. Kaplan, “Unitarity and the Holographic S-Matrix,”JHEP10(2012) 032,arXiv:1112.4845 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  72. [72]

    Lightcone bootstrap at higher points,

    A. Antunes, M. S. Costa, V. Goncalves, and J. V. Boas, “Lightcone bootstrap at higher points,”JHEP03(2022) 139,arXiv:2111.05453 [hep-th]

    work page arXiv 2022

  73. [73]

    Harmonic Analysis and Mean Field Theory,

    D. Karateev, P. Kravchuk, and D. Simmons-Duffin, “Harmonic Analysis and Mean Field Theory,”JHEP10(2019) 217,arXiv:1809.05111 [hep-th]

    work page arXiv 2019

  74. [74]

    $d$-dimensional SYK, AdS Loops, and $6j$ Symbols

    J. Liu, E. Perlmutter, V. Rosenhaus, and D. Simmons-Duffin, “d-dimensional SYK, AdS Loops, and 6jSymbols,”JHEP03(2019) 052,arXiv:1808.00612 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  75. [75]

    Lining up a positive semi-definite six-point bootstrap,

    A. Antunes, S. Harris, A. Kaviraj, and V. Schomerus, “Lining up a positive semi-definite six-point bootstrap,”JHEP06(2024) 058,arXiv:2312.11660 [hep-th]. – 123 –

    work page arXiv 2024