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arxiv: 2606.30806 · v1 · pith:TDXCYPSXnew · submitted 2026-06-29 · ✦ hep-th

Thermal Double-Twist Data in Holography

V. Niarchos , C. Papageorgakis , A. Stratoudakis This is my paper

Pith reviewed 2026-07-01 01:18 UTC · model grok-4.3

classification ✦ hep-th
keywords thermal OPE coefficientsdouble-twist operatorsholographic CFTsAdS black branesthermal two-point functionsmomentum space regularizationOPE data extraction
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The pith

A regularization procedure on momentum-space integrals extracts thermal double-twist OPE coefficients from holographic response functions.

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

The paper develops a method to pull thermal OPE coefficients of double-twist operators out of suitably regularized integrals over thermal response functions computed in momentum space. This technique is demonstrated on four-dimensional holographic CFTs whose finite-temperature states are described by an AdS5 black-brane geometry, where the response functions are obtained by numerically solving a radial ODE reduction of the Klein-Gordon equation. The resulting values complete earlier holographic and bootstrap work on thermal two-point functions and include new spin-resolved entries.

Core claim

In four-dimensional holographic CFTs with an Einstein bulk action, the finite-temperature state is captured by an AdS5 black-brane geometry. The thermal response function is computed on the gravitational side by solving numerically a radial ODE reduction of the Klein-Gordon equation. Regularized integrals of these response functions then yield accurate values of the double-twist OPE coefficients, completing previous holographic and bootstrap studies of thermal two-point functions, with some of the reported spin-resolved data being new.

What carries the argument

Regularized integrals of thermal response functions in momentum space, obtained from radial ODE solutions in the AdS black-brane geometry.

If this is right

  • The method supplies accurate numerical values for double-twist data at infinite spatial volume.
  • Some spin-resolved double-twist coefficients become available for the first time.
  • Previous holographic and bootstrap analyses of thermal two-point functions are completed by these extractions.
  • The approach works for any thermal response function that can be computed in momentum space.

Where Pith is reading between the lines

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

  • The same regularization technique could be applied to thermal response functions obtained from other bulk geometries or actions.
  • The new spin-resolved data can serve as input or tests for thermal bootstrap studies beyond the present holographic setting.
  • At large spin the extracted coefficients should approach known universal limits, providing a consistency check on the numerics.

Load-bearing premise

The regularization procedure applied to the momentum-space integrals successfully isolates the double-twist OPE coefficients without introducing uncontrolled errors or missing contributions from other operators.

What would settle it

An independent position-space computation of the same double-twist coefficients in the identical AdS5 black-brane background that produces values differing by more than numerical error.

Figures

Figures reproduced from arXiv: 2606.30806 by A. Stratoudakis, C. Papageorgakis, V. Niarchos.

Figure 1
Figure 1. Figure 1: The thermal response function approaches the zero-temperature result, [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 1
Figure 1. Figure 1: The thermal response function in standard quantization (plot (a)) and its alternative￾quantization counterpart (plot (b)) for ν = 1 2 , as functions of the Matsubara mode n and the spatial momentum k. functional of double-trace deformations. This discussion will in turn serve as preparation for a more complete analysis, with properly regularized integrals, based on the OPE expansion in Section 4. The therm… view at source ↗
Figure 2
Figure 2. Figure 2: The holographic thermal double-twist data a0,0, a1,0, a0,2, and the KMS combination a1,0 + 3a0,2 as functions of the external dimension ∆O across the window ∆O ∈ [1.1, 1.9], at (n ∗ , k∗ ) = (4, 15). All four are smooth and analytic through ∆O = 3 2 . Error bars (often smaller than the markers) are estimated from the k ∗ truncation and grow toward ∆O → 2, where G0 ∼ p −2|ν| falls off ever more slowly. The … view at source ↗
Figure 3
Figure 3. Figure 3: a0,0(∆O) on both quantization branches of the same bulk scalar (m2 ℓ 2 = ∆O(∆O − 4)). The dotted line marks the degenerate point ∆O = 2 (ν = 0). error ∼ 1.4×10−3 , which we quote conservatively as 2×10−3 . This result agrees very well with the zero-separation limit evaluation of [4, 9] (≃ 7.686). The scalar value a0,0 = 1.113079(1) is consistent with the earlier holographic estimate ≃ 1.1 of [7] and improv… view at source ↗
read the original abstract

We explain how to extract thermal OPE coefficients of double-twist operators in scalar two-point functions at infinite spatial volume from suitably regularized integrals of thermal response functions in momentum space. As a specific application, we implement the proposed approach to four-dimensional holographic CFTs with an Einstein bulk action, where the finite-temperature state is captured by an AdS$_5$ black-brane geometry. In that case, the thermal response function can be computed on the gravitational side by solving numerically a radial ODE reduction of the Klein-Gordon equation. We obtain accurate values of double-twist data from this solution, completing previous holographic and bootstrap studies of thermal two-point functions. Some of the reported spin-resolved data that we compute are new.

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

1 major / 0 minor

Summary. The paper explains a method to extract thermal OPE coefficients of double-twist operators from suitably regularized integrals of thermal response functions in momentum space. It applies the approach to 4d holographic CFTs with Einstein gravity by numerically solving a radial ODE reduction of the Klein-Gordon equation on an AdS5 black brane, obtaining accurate double-twist data that completes prior holographic and bootstrap studies, including some new spin-resolved results.

Significance. If the regularization isolates the double-twist contributions without uncontrolled systematics, the work supplies concrete numerical values for thermal double-twist data in a controlled holographic setting, extending existing studies of thermal two-point functions and providing new spin-resolved entries that can be compared with bootstrap predictions.

major comments (1)
  1. [regularization procedure (described after the holographic setup)] The regularization procedure for the momentum-space integrals is load-bearing for the central claim of accurate extraction. The manuscript should supply explicit convergence tests, error estimates, and checks that non-double-twist contributions (including possible continuum effects) are suppressed to the reported precision; without these, the numerical values cannot be fully validated.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their positive assessment of the work and for identifying the need for stronger validation of the regularization procedure. We address the single major comment below and will incorporate the requested material in a revised version of the manuscript.

read point-by-point responses

  1. Referee: [regularization procedure (described after the holographic setup)] The regularization procedure for the momentum-space integrals is load-bearing for the central claim of accurate extraction. The manuscript should supply explicit convergence tests, error estimates, and checks that non-double-twist contributions (including possible continuum effects) are suppressed to the reported precision; without these, the numerical values cannot be fully validated.

    Authors: We agree that the regularization step is central and that the current presentation would benefit from more explicit documentation. In the revised manuscript we will add: (i) convergence plots and tables demonstrating stability of the extracted OPE coefficients under variations of the cutoff and subtraction parameters; (ii) error estimates propagated from the numerical radial ODE solver (including residual tolerances and grid resolution); and (iii) an explicit decomposition showing that the subtracted integrand falls off sufficiently rapidly that continuum and higher-twist contributions lie below the quoted precision. These additions will be placed immediately after the description of the regularization procedure. revision: yes

Circularity Check

0 steps flagged

Minor self-citation of prior holographic work; central numerical extraction independent

full rationale

The paper's derivation starts from the standard holographic dictionary, reduces the Klein-Gordon equation to a radial ODE on the AdS5 black brane, solves it numerically, and extracts double-twist OPE coefficients via regularized momentum-space integrals. This chain contains no self-definitional steps, no fitted parameters renamed as predictions, and no load-bearing uniqueness theorems or ansatze imported via self-citation. The only self-citation is to prior holographic studies of thermal two-point functions, which is not required to justify the present regularization procedure or the reported numerical values. The central claim therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The approach rests on the standard holographic dictionary and the validity of the black-brane geometry for the thermal state; no new free parameters, ad-hoc axioms, or invented entities are introduced beyond conventional AdS/CFT assumptions.

axioms (1)
  • domain assumption Holographic correspondence maps CFT thermal two-point functions to bulk fields in the AdS black-brane geometry
    Invoked when the thermal response function is computed on the gravitational side via the Klein-Gordon equation

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discussion (0)

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