The entropy of radiation for local quenches in higher dimensions

Andrea Mattiello; Jacopo Sisti; Lorenzo Bianchi

REVIEW 2 major objections 4 minor 74 references

The entropy of radiation for local quenches in higher dimensions

T0 review · 2 major / 4 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read This paper derives a universal $\xi^{d/2}$ law for the excess entanglement entropy of radiation after a local quench in $d>2$ CFTs, plus an all-times bound, assuming the stress tensor is the lightest exchanged operator.

desk verdict Useful higher-dimensional extension of local-quench entanglement entropy, but the advertised universal bound contains a factor-of-π error and needs correction before the saturation claim stands. read the letter →

arxiv 2502.00105 v3 pith:W6BJXN63 submitted 2025-01-31 hep-th

Lorenzo Bianchi , Andrea Mattiello , Jacopo Sisti This is my paper · ORCID

classification hep-th

keywords conformalfieldtheoryentanglemententropylocalquenchtwistoperatordefectrelativeholographyRyu-Takayanagisurface

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The radiation produced by a sudden local excitation of a CFT can be diagnosed by the entanglement entropy across a surrounding sphere. This paper establishes that in any dimension $d>2$ the excess of that entropy over the vacuum behaves at early and late times as a universal power of a single cross-ratio $\xi$, with coefficient set by the quench operator's dimension $\Delta$, as long as the stress tensor is the lightest operator exchanged in the $O\times O$ OPE. It also proves an all-times upper bound on the excess entropy from relative entropy, and shows the stress-tensor contribution saturates that bound in the $\xi\to 0$ limit. The same construction in a boundary CFT yields the same $\xi^{d/2}$ power, but with a coefficient that no Ward identity fixes for $d>2$. The holographic version of the calculation, a heavy particle falling in $\mathrm{AdS}_4$, gives a Page-like curve that agrees with the early/late CFT result, and its perturbative small-mass limit saturates the bound exactly.

What carries the argument

The central object is the higher-dimensional twist operator $\sigma_n$, the extended operator implementing the replica trick; in the orbifold CFT$^n/\mathbb{Z}_n$ it is a codimension-two conformal defect. In the quench kinematics the correlator $\langle \sigma_n O^{\otimes n} O^{\otimes n}\rangle$ collapses to a function of one cross-ratio $\xi$, and the $\xi\to 0$ limit is taken by the bulk OPE of the two $O$ insertions. The leading nontrivial contribution is the one-point function of the symmetrized stress tensor in the twist background, whose coefficient is fixed by a Ward identity; the relative-entropy bound converts the modular Hamiltonian of the sphere, a spatial integral of $T_{tt}$, into an all-times upper bound. For the BCFT the analysis is carried by two-defect kinematics using embedding-space formalism, while the holographic part uses Ryu-Takayanagi surfaces in the black-hole geometry (4.4), with the perturbative area computed from the vacuum embedding.

What would settle it

Compute the leading small-$\xi$ exponent of $\Delta S_{EE}$ in a $d=3$ free scalar CFT: if the composite operator $\phi^2$, of dimension $1$, has a nonzero one-point function around the spherical twist, the leading power should be $\xi^1$ rather than $\xi^{3/2}$, directly testing the assumption that the stress tensor is the lightest exchanged operator.

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Extended reading notes

Core claim

The central claim, in the authors' terms, is that for a CFT in $d>2$ with a local quench by a scalar primary $O$ of dimension $\Delta$, the excess R\'enyi and entanglement entropies across a sphere are controlled by one cross-ratio $\xi$. In the OPE limit $\xi\to 0$, $\Delta S_A^{(n)} \sim \frac{2^{d-2} d \Gamma(d/2)}{(n-1)\pi^{d/2+1}}\frac{h_n \Delta}{C_T} \xi^{d/2}$, whose $n\to 1$ limit gives $\Delta S_{EE} = \frac{2^{d-1} d \Gamma(d/2)^2}{\Gamma(d+2)} \Delta \, \xi^{d/2}+O(\xi^d)$. The factor $h_n$ encodes the twist-operator data, and the coefficient is fixed by a conformal Ward identity. Using relative entropy, the paper derives the all-times bound $\Delta S_{EE} \le \frac{1}{2\sqrt{\pi}} \Delta \, \xi^{d/2} \Gamma(d/2) \, {}_2\tilde{F}_1(1,d/2,(d+3)/2;\xi)$, which reduces to the OPE result at small $\xi$. The authors stress that the universality rests on the assumption that no operator lighter than the stress tensor has a non-vanishing one-point function in the twist background; for the BCFT the analogous assumption involves the displacement operator, and the coefficient $c_{\hat{O}\hat{O}D}$ is not fixed by known Ward identities in $d>2$.

Load-bearing premise

The load-bearing premise is that no operator lighter than the stress tensor (or, in the boundary case, lighter than the displacement operator) appears in the $O\times O$ OPE with a nonzero one-point function around the twist background; if such an operator exists, the leading exponent $d/2$ changes.

Editorial extensions

If this is right

In any CFT$_d$ with $d>2$ satisfying the lightness assumption, the late-time radiation entropy after a local quench decays as $t^{-2d}$ with a coefficient fixed by $\Delta$, $R$, and $\epsilon$: $\Delta S_{EE} \sim \frac{2^{d-1} d \Gamma(d/2)^2}{\Gamma(d+2)} \frac{(2R)^d \epsilon^d}{t^{2d}}$.
The all-times bound (2.43) implies that any operator lighter than the stress tensor must contribute negatively to the excess entropy, so the stress tensor sets the maximal possible early/late growth.
For boundary CFTs, the same exponent $\xi^{d/2}$ governs early/late behavior, but the coefficient is a genuinely new defect datum $c_{\hat{O}\hat{O}D}$; the result saturates the bound of ref. [16].
In $\mathrm{AdS}_4/\mathrm{CFT}_3$ with a heavy local operator, the holographic entanglement entropy follows a Page-like curve symmetric in lightcone time, and the small-mass perturbative result exactly saturates the bound (4.17).
For a tensionless end-of-the-world brane, the homogeneous holographic result is simply rescaled by $1/2$, so the boundary quench entropy is half the bulk one.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Beyond the paper, the same single-cross-ratio mechanism suggests that other sphere-based observables after a local quench, such as mutual information between two spherical regions, will also be governed by the $\xi^{d/2}$ OPE exponent in the same regime.
Because the BCFT coefficient $c_{\hat{O}\hat{O}D}$ is unfixed, one could use conformal-bootstrap constraints to bound it, turning the entropy formula into a numerical test for allowed boundary spectra.
The exact saturation of the bound at small mass hints that the RT surface for the black-hole quench sits at the relative-entropy bound to leading order, a property that may persist at subleading orders in specific large-central-charge limits.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The paper deserves a serious referee, but the headline claim needs fixing. Eq. (2.43), the all-time upper bound, does not reduce to the OPE estimate (2.30) as claimed. Expanding (2.43) at small ξ gives a coefficient that is exactly π times smaller than the OPE coefficient in (2.30)/(2.44). The paper's own d=2 and d=3 expansions in (2.46)–(2.48) agree with the OPE result, not with (2.43). So either (2.43) is missing a factor of π or the rest of the section is inconsistent. Since the paper explicitly uses this reduction to conclude that the stress-tensor contribution saturates the bound, this is load-bearing, not cosmetic. Most likely it is a simple prefactor mistake, and the corrected bound will still work, but the displayed equations must be made mutually consistent and the saturation argument re-verified.

What is genuinely new and good: the extension of the local-quench radiation entropy to d>2 is nontrivial and cleanly executed. The kinematic observation that the Lorentzian time evolution stays on the Euclidean circle in the z-plane is elegant and well explained. The early/late-time OPE estimate (2.30) is a real result, and the d=2 limit correctly reproduces [16]. The BCFT section, with two conformal defects in embedding space, is a solid piece of formal work even though the leading OPE coefficient c_ÔÔD is left undetermined in d>2 — the authors state this openly. The holographic numerical Page-like curve and the perturbative d=3 result (4.15) are a reasonable cross-check, and the tensionless EoW brane case is a nice simple limit, with the λ≠0 obstruction clearly discussed.

Soft spots, in proportion: the bound inconsistency is the main problem. The 'universality' of the OPE estimate is conditional on the stress tensor being the lightest exchanged operator; the paper admits this, but it means the headline should be qualified. The numerics are described without shipped code or data, which is a minor reproducibility gap. The citation pattern and the overlap with [16] are fine — this is a genuine extension, not a repackaging.

Bottom line: for people working on holographic entanglement and information-paradox toy models, this is worth reading and worth citing once the bound is fixed. Send it to peer review, but ask for a corrected (2.43) and a careful re-derivation of the bound section before acceptance.

Referee Report

2 major / 4 minor

Summary. The paper analyzes the excess entanglement entropy of the radiation emitted by a local quench in a d-dimensional CFT with d>2, using the interpretation of the higher-dimensional twist operator as a conformal defect. It derives an OPE estimate for the early- and late-time behavior of the excess entropy, Eq. (2.30), and proposes an all-time upper bound from relative entropy, Eq. (2.43). The same strategy is extended to a BCFT with two intersecting conformal defects, where the leading OPE contribution is assumed to come from the displacement operator, Eq. (3.35). The paper closes with a holographic analysis in AdS4/CFT3, including numerical Ryu-Takayanagi surfaces in a black hole background, a Page-like curve, and a perturbative result that allegedly saturates the bound.

Significance. If the claims are correct, the paper would give a genuinely higher-dimensional generalization of known two-dimensional local-quench results, with a universal early/late-time coefficient and an all-time bound on radiation entropy. The holographic part provides a nontrivial numerical check and connects the OPE/bound results to a Page-like evolution. The paper is transparent about its main spectral assumption (stress tensor as lightest exchanged operator) and about the fact that the BCFT OPE coefficient c_{\hat O\hat O D} is not fixed by a Ward identity in d>2. The central obstacle is an internal inconsistency in the bound section: Eq. (2.43) as displayed does not reduce to the claimed OPE coefficient and does not match the paper's own specializations.

major comments (2)

[§2.6, Eqs. (2.43)-(2.44) and (2.46)-(2.48)] Eq. (2.43) as written has leading coefficient \Delta/(2\sqrt{\pi})\Gamma(d/2)/\Gamma((d+3)/2), which expands to \Delta/(2\sqrt{\pi})\Gamma(d/2)/\Gamma((d+3)/2)\,\xi^{d/2}. Using the duplication formula this is exactly 1/\pi times the coefficient 2^{d-1}d\Gamma(d/2)^2/\Gamma(d+2) appearing in Eqs. (2.30) and (2.44). Concretely, for d=2 Eq. (2.43) gives 2\Delta\xi/(3\pi), while Eq. (2.48) gives 2\Delta\xi/3; for d=3 it gives \Delta\xi^{3/2}/8, while Eq. (2.47) gives \pi\Delta\xi^{3/2}/8. Thus the displayed Eq. (2.43) is inconsistent with its own claimed expansion (2.44), with its specializations (2.46)-(2.48), and with the holographic saturation statement (4.17), which also contains a factor \pi. The prefactor of Eq. (2.43) must be corrected and the derivation of the integral (2.42) re-examined; until this is fixed, the central claim that the stress-tensor OPE saturates the bound at all times is unsupported.
[§3.3, Eq. (3.35)] The BCFT result (3.35) has the form \Delta S_{EE} \sim [\pi^{(d+1)/2}/(2(d-1)\Gamma((d+3)/2))]\,c_{\hat O\hat O D}\,\xi^{d/2}, and the paper explicitly notes that c_{\hat O\hat O D} is not fixed by any Ward identity in d>2. This means the BCFT prediction is not universal in coefficients but only in the power \xi^{d/2}, and the statement that the result 'perfectly saturates the bound' is conditional on an undetermined OPE coefficient. This limitation should be stated in the abstract and in the concluding discussion, not only in the body of Section 3.3.

minor comments (4)

[§2.6, Eq. (2.48)] The notation '\simeq' after an inequality is slightly misleading; consider writing the asymptotic expansion as an upper bound in the sense '\leq' with the displayed leading term, or explicitly state that the inequality is saturated only at leading order in \xi.
[§2.5, footnote 4] The convention-matching paragraph is helpful but dense; a short table of the dictionary between [16] and the present conventions would make the comparison easier to verify.
[Figure 8 caption] In the caption of Figure 8, the quantity plotted is denoted \Delta S, while the text refers to the entanglement entropy bound; using \Delta S_{EE} consistently in the axis label would avoid confusion.
[§4.1.1, Eq. (4.15)] The perturbative result (4.15) is quoted from [6] with a note on the mass convention; it would be useful to also state the precise dictionary between the cross-ratio \xi and the global-time variable \theta_\infty directly in the main text, since Eq. (4.7) is used later for the comparison with the CFT bound.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the OPE estimate, the relative-entropy bound, and the holographic saturation check are derived from independent conformal and AdS/CFT inputs, and the stated spectrum assumptions are explicit rather than imported from a self-citation.

full rationale

The CFT early/late-time result (2.30) follows from the defect OPE of the twist-operator correlator with cOOT fixed by the conformal Ward identity (2.27) and the stress-tensor one-point function taken from the independent twist-operator literature [43]; the stress-tensor-lightest assumption is stated as an assumption and its domain of validity is discussed. The all-time bound (2.43) is computed from relative entropy using the modular Hamiltonian (2.39) and the conformally fixed OOT three-point function; [16] is used only as a template for the argument, and the present calculation is self-contained. The BCFT displacement-operator result is derived in Appendix A from the n-dependence of the one-point function and the modular-Hamiltonian two-point function, not assumed from [16]. The holographic numerical evolution and the perturbative HEE result (4.15) are independent gravity-side computations, and the identification Delta=ML is standard AdS/CFT; calling the matching 'saturation' is a comparison, not a fit. I therefore find no fitted parameter renamed as a prediction and no load-bearing self-citation chain. Separately, as a correctness note rather than a circularity finding, the expansion of (2.43) displayed in (2.44) is off by a factor of pi for general d, and the d=3 and d=2 specializations in (2.46) and (2.48) are not literally consequences of (2.43) as written; this internal prefactor inconsistency should be corrected, but it is not an equivalence of a result to its input by construction.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

No new particles or forces are introduced. The holographic black hole and the EoW brane are standard constructions. The central results rest on three physical assumptions: the analytic-continuation step, the lightest-operator conditions, and the Euclidean-contour claim.

free parameters (1)

c_{\hat O\hat O D} (BCFT boundary OPE coefficient)
The BCFT leading entropy in Eq. (3.35) is proportional to this undetermined coefficient; no Ward identity fixes it in d>2, so the estimate depends on an unknown CFT datum.

assumptions (4)

standard math Analytic continuation of Rényi entropies from integer n to real n is assumed.
Used throughout to define entanglement entropy via the n→1 limit; footnote in Sec. 2.2 acknowledges the standard non-uniqueness and states the paper adds nothing to this issue.
domain assumption The stress tensor is the lightest exchanged operator in the bulk OPE of O×O with a non-vanishing one-point function in the presence of the twist operator.
Used to extract Eq. (2.26); the paper discusses conditions such as Δ_L > d/2 or holographic Δ > d/4, but the final 'universal' formula (2.30) depends on this assumption.
domain assumption In the BCFT, the displacement operator D is the lightest exchanged single-copy boundary operator in the OPE of the two boundary scalars.
Used in Sec. 3.3 to obtain Eq. (3.33); alternative lighter boundary operators would change the leading power of ξ.
domain assumption The Lorentzian time evolution stays on a Euclidean contour for the bulk two-point function, so no singularities are crossed and the OPE converges at late time.
Sec. 2.4 establishes that w is a phase; this justifies applying the ξ→0 OPE to the real-time evolution.

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Pith. "Pith review of The entropy of radiation for local quenches in higher dimensions." pith.science (2026). https://pith.science/paper/W6BJXN63

@misc{pith2026250200105,
  author       = {Pith},
  title        = {Pith review of: The entropy of radiation for local quenches in higher dimensions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/W6BJXN63}},
  note         = {Machine review of arXiv:2502.00105}
}

abstract

We investigate the real time dynamics of the radiation produced by a local quench in a $d$-dimensional conformal field theory (CFT) with $d>2$. Using the interpretation of the higher-dimensional twist operator as a conformal defect, we study the time evolution of the entanglement entropy of the radiation across a spherical entangling surface. We provide an analytic estimate for the early- and late-time behavior of the entanglement entropy and derive an upper bound valid at all times. We extend our analysis to the case of a boundary CFT (BCFT) and derive similar results through a detailed discussion of the setup with two conformal defects (the boundary and the twist operator). We conclude with a holographic analysis of the process, computing the time evolution of the holographic entanglement entropy (HEE) as the area of the Ryu-Takayanagi surface in a backreacted geometry. This gives a Page-like curve in agreement with the early- and late-time results obtained with CFT methods. The extension to a holographic BCFT setup is generically hard and we consider the case of a tensionless end-of-the-world brane.

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