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REVIEW 2 major objections 9 minor 55 references

Retarded correlators of holographic CFTs scale anomalously on the light cone, with the power fixed by the curvature of the dual black-hole horizon.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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2026-07-31 20:08 UTC pith:74NQD4SI

load-bearing objection Solid multi-method result on horizon-dependent light-like scaling; planar and hyperbolic are locked down, spherical WKB needs a connection formula the paper never writes. the 2 major comments →

arxiv 2607.24242 v1 pith:74NQD4SI submitted 2026-07-27 hep-th cond-mat.othergr-qc

Light-like retarded correlators and the horizon

Justin R. David , Leonard Schwarze This is my paper
classification hep-th cond-mat.othergr-qc
keywords retarded correlatorslight-like momentaAdS/CFTblack-hole horizonanomalous scalingWKB approximationthermal CFTstress tensor
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

At finite temperature the retarded two-point function of a scalar primary normally grows at large frequency like the zero-temperature power fixed by dimensional analysis. On the light cone that leading contribution vanishes, and the paper shows that the surviving growth is weaker and depends on the horizon topology of the dual black hole. For planar horizons the correlator scales as ω to the power 2(2Δ−d)/(d+2); for spherical or hyperbolic horizons (when frequency squared equals the Casimir on the horizon) it scales as ω to the power (2Δ−d)/2. The claim is established by exact BTZ and hyperbolic-black-hole solutions, numerical Heun analysis in AdS5, WKB matching in general dimension, and large-d checks, and is extended to all three channels of the stress-tensor correlator. A reader cares because the anomalous exponents are universal short-distance data that remain after the identity OPE term drops out and that directly encode the infrared geometry of the dual black hole.

Core claim

In AdS/CFT the large-frequency light-like retarded correlator of a scalar primary of dimension Δ grows as ω^{2(2Δ−d)/(d+2)} for planar black holes and as ω^{(2Δ−d)/2} for spherical or hyperbolic black holes (when ω^{2} equals the horizon Casimir), rather than the naive dimensional-analysis power ω^{2Δ−d} that holds at generic momenta.

What carries the argument

WKB transport of the ingoing horizon solution across the bulk, which carries frequency only as an overall phase; the retarded Green’s function is then fixed solely by the order of the near-boundary Bessel functions whose argument is set by the first non-constant term in the blackening factor.

Load-bearing premise

The WKB solution carries the horizon boundary condition to the boundary only as a phase, with no amplitude mixing from turning points or sub-dominant potential terms.

What would settle it

Evaluate the exact retarded correlator for a holographic CFT whose bulk equation is solvable (BTZ, hyperbolic black hole, or large-d planar black hole) at large light-like frequency and check whether the measured power equals 2(2Δ−d)/(d+2) or (2Δ−d)/2 according to horizon topology.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • On-shell photon and dilepton emission rates at strong coupling acquire anomalous frequency dependence set by horizon curvature.
  • Stress-tensor correlators inherit light-cone exponents 2d/(d+2), 4d/(d+2) and 6d/(d+2) in the scalar, shear and sound channels.
  • Large-d planar correlators reduce exactly to zero-momentum BTZ correlators, giving a dimensional-reduction check of the same exponents.
  • Any CFT dual to an asymptotically AdS black hole of the stated topology must exhibit these reduced powers once momenta equal frequency.

Where Pith is reading between the lines

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

  • The light-cone thermal OPE must contain a specific tower of operators whose coefficients reproduce the reduced powers once the identity contribution cancels; extracting those coefficients would give a pure CFT derivation of the horizon dependence.
  • The same WKB matching applies to charged or higher-derivative black holes whenever the near-boundary blackening expansion is unchanged, predicting charge-independent exponents for d>4.
  • Position-space light-cone singularities previously linked to bulk null geodesics are the natural Fourier dual of these anomalous momentum-space powers.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 9 minor

Summary. The paper studies holographic retarded two-point functions of scalar primaries (and the stress tensor in its three channels) at finite temperature in the light-like limit |k|=ω with ω→∞. The central claim is that the scaling is anomalous and horizon-curvature dependent: for planar AdS black holes G_R∼ω^{2(2Δ−d)/(d+2)} instead of the dimensional-analysis/OPE expectation ω^{2Δ−d} (eq. 1.9), while for spherical and hyperbolic horizons, with ω² set equal to the horizon Casimir, G_R∼ω^{(2Δ−d)/2} (eqs. 1.10–1.11). The planar result is established by an exact BTZ computation matched to an independent CFT Fourier transform via KLT string-integral identities (§2.1), numerics on the Heun connection problem for AdS5 (§2.2, Fig. 1), a general-d WKB analysis (§2.3), and an exact large-d solution (§5). The hyperbolic exponent is confirmed by an exact hypergeometric result (§3.2). The spherical exponent rests entirely on the WKB transport argument of §3.1. Appendix B reproduces the d=2 anomalous term from a Borel-resummed momentum-space OPE.

Significance. If the results hold, this is a notable and cleanly stated finding: a parameter-free, universal anomalous scaling exponent for light-like thermal correlators at strong coupling, with direct relevance to on-shell emission rates (photon/dilepton, gravitational waves) where light-cone correlators enter. The manuscript's strengths are substantial: the exponents are derived, not fitted; the planar branch is confirmed by four mutually independent methods (exact BTZ plus an independent CFT-side Fourier transform, Heun numerics with error decaying in ω, WKB, exact large-d); the hyperbolic branch has an exact check; and the d=2 OPE/Borel analysis identifies where the anomalous term lives in the momentum-space OPE. The comparisons with generalized free fields, the O(N) model, and the free N=4 glueball correlator (§6) correctly frame the effect as a strong-coupling phenomenon. The stress-tensor channel exponents (1.12) are falsifiable predictions. The spherical branch, however, currently lacks any check independent of the WKB argument, and that argument has a genuine gap (major comments).

major comments (2)
  1. [§3.1.1, eqs. (3.20), (3.26), (3.30)] For κ=+1 the light-like WKB potential has an unanalyzed bulk turning point. Writing a=(R/r*)², the spherical blackening factor in the ρ=r*/r coordinate is f(ρ)=1+aρ²−(1+a)ρ^d, so f(ρ)>1 for 0<ρ<ρ_t with ρ_t=(a/(1+a))^{1/(d−2)}∈(0,1). Hence Q(ρ)=f′(1)²(1−f)/(4f²) (eq. 3.20) is negative in an outer evanescent region and vanishes at ρ_t. The paper's own validity measure (3.26) ∝ |2f′(1)(f−2)f′/(1−f)^{3/2}|/ω diverges at ρ_t, contradicting the claimed validity domain (3.30), ω^{−2/(n+2)}≪ρ≪1. The single-exponential WKB transport (3.16)/(3.22) from horizon to boundary is therefore not justified as written for spherical horizons; an Airy-type connection across ρ_t is required and is nowhere given. Unlike the planar exponent (checked exactly in d=2, numerically in d=4, and at large d) and the hyperbolic exponent (checked exactly in §3.2), the spherical exponent in (1.11) has no independent veri
  2. [§3.1.2, eqs. (3.41)–(3.44)] For κ=+1, f⁽²⁾(0)=2a>0, so the argument z in (3.42) is imaginary and the near-boundary solutions of (3.41) are the modified Bessel functions I_{ν′}(|z|), K_{ν′}(|z|), not the oscillatory J/Y written in (3.42). Correspondingly the large-z asymptotics (3.43) are exponential, not the displayed cos/sin, and cannot match the (evanescent) WKB form (3.22) as written — the displayed matching is internally consistent only for κ=0 and κ=−1. This matters for the exponent, not just the prefactor: the scaling law follows from the ratio of the z^{ν′} and z^{−ν′} branches in (3.44), and if the turning-point connection selected pure I_{ν′} (which has only the z^{ν′} branch), the ρ^{d−Δ} coefficient would vanish or be exponentially suppressed and G_R would be ill-defined. My own estimate is that the in-going wave connects across ρ_t to a K-dominant solution (growing WKB exponential e^{+ω∫κ} maps to e^{−z
minor comments (9)
  1. [Eq. (2.73)] The final equality reads GR ≈ w^{2(2∆−d)/(d+2)}; the RHS should be ω, not w.
  2. [Eq. (4.15)] The sound-channel result is printed as ω^{4d/(d+2)}, but eq. (4.14) gives 2ν=6d/(d+2) and eqs. (1.12), (4.16) also give 6d/(d+2). Presumably a typo, but it appears in a numbered result.
  3. [Eq. (5.30)] Both limits are labeled with the subscript 'shear'; they refer to the sound channel.
  4. [§1 vs §2.3/§3.1.2] The statement after (1.11) that the scaling 'remains the same' for charged black holes is stronger than §2.3 ('our preliminary calculations indicate') and §3.1.2 (restrictions d>4 planar, d≥3 for κ=±1). Please reconcile and state the charge claims with uniform precision.
  5. [Footnote 5; eq. (5.29)] Footnote 5 refers to the domain (2.53); presumably (2.55) is meant. Also eq. (5.29) has 'w⁶' where ω⁶ is intended.
  6. [Fig. 1, §2.2] Please give more detail on the extraction of α (fit range, tolerance of the Heun evaluation, estimated systematic error), and note explicitly in the text that ω≤100 is a numerical-stability cutoff.
  7. [Eq. (2.36)] The term 4πTω_n is noted as differing from the usually cited result of [1]; a sentence clarifying its status (contact term vs physical, and scheme dependence) would help, given the emphasis placed on it.
  8. [§3.1.2 / §6] It would be useful to state explicitly (e.g., in §3.1.2 or the conclusions) that the spherical exponent currently rests only on the WKB argument, and to compare the exponents quantitatively with the known U(1)-current light-like result, eq. (3.19) of [24].
  9. [Throughout] Assorted typos: 'correaltor' (§1, §6), 'mometa' (§5 title), 'Nordstorm' (§3), 'light-ike' (§5.2), 'anstaz' (§3), 'expresssions' (footnote 11), 'form' for 'from' (several places).

Circularity Check

0 steps flagged

No circularity: anomalous exponents are extracted from bulk ODEs (exact, numerical, WKB) with independent CFT and large-d cross-checks; nothing is forced by definition or self-citation.

full rationale

The central claims (planar exponent 2(2Δ−d)/(d+2); spherical/hyperbolic (2Δ−d)/2; stress-tensor channel exponents) are obtained by solving the bulk radial ODEs with ingoing horizon conditions and reading the ratio of normalisable to non-normalisable boundary coefficients. In d=2 this is done with hypergeometric connection formulae and independently via a KLT/string-integral Fourier transform of the known Euclidean CFT two-point function; in d=4 via numerical Heun connection; in general d via WKB transport of a pure phase plus near-boundary Bessel asymptotics; for hyperbolic horizons via an exact hypergeometric Green’s function; and at large d via exact hypergeometric solutions that recover the d→∞ limit of the planar exponent. No parameter is fitted to data and then re-predicted. Self-citations (e.g. prior large-d one-point work) are peripheral and not load-bearing for the exponents. Possible gaps in the spherical WKB turning-point analysis are correctness/validity issues, not circular reductions of outputs to inputs. The derivation chain is therefore self-contained against its own equations and external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 0 invented entities

The result is a classical bulk calculation inside standard AdS/CFT. It inherits the Son–Starinets retarded prescription, Einstein (or mild higher-derivative) black-hole backgrounds, and the usual scalar/stress-tensor bulk equations. No free parameters are fitted; the only non-standard modeling step is the WKB phase-transport assumption used to read exponents from near-boundary Bessel orders.

axioms (4)
  • domain assumption Son–Starinets prescription: G_R is the ratio of normalizable to non-normalizable boundary coefficients for the bulk solution with infalling horizon conditions.
    Used throughout (eq. 2.7 and analogues); standard in the holographic hydrodynamics literature the paper cites.
  • domain assumption Classical Einstein gravity (or RN/higher-derivative blackening factors with the same near-boundary expansion) dual to a large-N CFT.
    Background metrics in §§2–4; charge/higher-derivative corrections argued not to change the leading near-boundary power for d large enough.
  • ad hoc to paper At large ω the WKB solution carries the horizon condition to the boundary only as a phase, so G_R scaling equals the ratio of powers in the near-boundary Bessel expansion.
    Load-bearing matching step in §§2.3 and 3.1; checked against exact solutions where available but not proved for all channels.
  • domain assumption Light-like kinematics for curved horizons means ω² + p(λ) = 0 with p the Laplacian eigenvalue (Casimir) on the horizon.
    Definition (3.13); natural boundary dispersion but a modeling choice for spherical/hyperbolic cases.

pith-pipeline@v1.2.0-grok45-kimik3 · 40878 in / 2699 out tokens · 79399 ms · 2026-07-31T20:08:52.172250+00:00 · methodology

0 comments
read the original abstract

We show that retarded correlators in conformal field theories of scalar primary operators evaluated at finite temperature using $AdS/CFT$ scale anomalously at large light like momenta. The anomalous scaling depends on the curvature of the horizon. Setting the momenta equal to the frequency $\omega$, the retarded correlator for black holes with planar horizons scales as $\omega^{\frac{2}{d+2} (2\Delta - d) }$ as opposed to the scaling behaviour of $\omega^{2\Delta - d}$ expected by dimensional analysis at generic fixed momenta and large frequencies. $\Delta$ is the dimension of the primary and $d$, the number of space-time dimensions. For black holes with spherical and hyperbolic horizons when the frequency squared equals the Casimir along the horizon, the retarded correlator scales as $\omega^{\frac{2\Delta -d}{2} }$. We establish this using exact results in $d=2$, and a numerical analysis of the Heun equation for the $AdS_5$ planar black hole and finally using the WKB approximation in general $d$. The exact result for black holes with hyperbolic horizon as well as the analysis at large $d$ provides additional checks for the anomalous scaling behaviour. Finally we evaluate the anomalous scaling exponent for the stress tensor correlator in all its 3 channels.

discussion (0)

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