{"id":"e510bd5c-dda2-48da-8883-0095cc118f8a","arxiv_id":"2502.00105","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"For local quenches in d>2 CFTs, the excess entanglement entropy of radiation grows as ξ^{d/2} at early and late times, obeys an all-time relative-entropy bound, and the holographic model produces a Page-like curve.","lead":"This preprint studies how the entanglement entropy of radiation grows and later decays after a sudden local excitation in a conformal field theory in more than two spacetime dimensions, and derives an all-time upper bound for this entropy. It then checks the same process in a holographic black-hole model, finding a Page-like curve that agrees at early and late times.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Eq. (2.43)'s universal bound is inconsistent with the OPE result (2.30): the displayed leading coefficient is smaller by a factor of π, so the claimed saturation is unsupported.","rationale":"The Reader's verdict is CONDITIONAL, and I agree with that overall assessment. However, the single most load-bearing concern is not the lightest-operator assumption (which the paper states and which only makes the early/late-time estimate conditional), but the internal inconsistency between the universal bound (2.43) and the OPE result (2.30). The displayed bound, when expanded, yields a leading coefficient that is exactly a factor of π smaller than the OPE coefficient, contradicting the paper's claim that (2.44) 'precisely matches' (2.30) and that the stress tensor saturates the bound. The paper's own d=3 and d=2 specializations confirm the discrepancy, since they contain the extra π factor. This is a concrete, checkable error in the central equations, not merely a question of assumptions. It is likely fixable by correcting the prefactor in (2.43), and the independent OPE computation of (2.30) may still be correct, so REJECT would be too strong. The Reader already set the verdict to CONDITIONAL; my concern reinforces that no change is needed.","tokens_in":81890,"tokens_out":7367,"duration_ms":65502,"concrete_test":"Recompute the coefficient in (2.43) by directly evaluating the integral in (2.42) using (2.41) and the Ward identity for c_{OOT}. Equivalently, substitute d=2 and d=3 into the displayed (2.43) and compare the small-ξ expansion with (2.48) and (2.46)/(2.47); if the coefficients differ by π, the prefactor in (2.43) must be corrected. Also check the all-time saturation claim by plotting (2.43) versus (2.30) at fixed small ξ.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central consistency claim connecting the two main CFT results fails as displayed. Expanding the regularized hypergeometric in (2.43) gives, at leading order in ξ, ΔS ≤ [1/(2√π)] Γ(d/2)/Γ((d+3)/2) Δ ξ^{d/2}. Using the duplication formula, this coefficient is exactly (1/π) times the OPE coefficient 2^{d-1} d Γ(d/2)^2 / Γ(d+2) in (2.30)/(2.44). Thus (2.43) does not reduce to (2.44) as claimed; for d=3 it would give ΔS ≤ Δ ξ^{3/2}/8, whereas (2.46) and (2.47) give πΔ ξ^{3/2}/8. The same discrepancy occurs in d=2, where (2.43) predicts 2/(3π) Δ ξ instead of the stated 2/3 Δ ξ in (2.48). Since the paper explicitly uses this reduction to conclude that the stress-tensor OPE saturates the bound ('precisely matches'), the displayed bound and the OPE estimate cannot both be correct as written. The most likely explanation is a missing factor of π in the prefactor of (2.43), but as it stands the central 'universal bound + saturation' claim is internally inconsistent and needs correction before the result can be trusted.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":82133,"tokens_out":9222,"duration_ms":93135,"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":[{"comment":"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.","section":"§2.6, Eqs. (2.43)-(2.44) and (2.46)-(2.48)"},{"comment":"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.","section":"§3.3, Eq. (3.35)"}],"minor_comments":[{"comment":"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.","section":"§2.6, Eq. (2.48)"},{"comment":"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.","section":"§2.5, footnote 4"},{"comment":"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.","section":"Figure 8 caption"},{"comment":"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.","section":"§4.1.1, Eq. (4.15)"}],"recommendation":"major_revision","confidential_remarks":"The factor-\\pi inconsistency in Eq. (2.43) appears to be a local, fixable prefactor error rather than a conceptual failure, but it is load-bearing because it is used to claim saturation of the bound and to identify the holographic perturbative result with the bound. I recommend major revision: the authors should correct the prefactor, re-verify Eq. (2.42) and its hypergeometric evaluation, and re-check the d=2 and d=3 specializations. The BCFT section should also be tightened regarding the undetermined coefficient c_{\\hat O\\hat O D}."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nWhat 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.\n\nSoft 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.\n\nBottom 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.","headline":"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.","tokens_in":82764,"tokens_out":4042,"would_cite":false,"duration_ms":41135,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["conformal field theory","entanglement entropy","local quench","twist operator","conformal defect","relative entropy","holography","Ryu-Takayanagi surface"],"falsifier":"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.","tokens_in":81583,"feed_emoji":"🌀","tokens_out":11467,"duration_ms":101091,"temperature":0.7,"pith_summary":"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.","feed_headline":"Radiation entropy obeys universal d/2 power law after local quench","feed_subtitle":"For any CFT in d>2, early and late decay is set by the quench operator's dimension alone; a bound caps the whole curve.","key_machinery":"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.","core_discovery":"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 $d>2$.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the two-dimensional boundary-quench result and the relative-entropy bound that the paper generalizes to higher dimensions; used for comparison and for the d=2 limit.","marker":"[16]"},{"why":"Provides the higher-dimensional twist operator as a conformal defect, its conformal dimension h_n, and the stress-tensor one-point function in the twist background that carries the OPE coefficient.","marker":"[43]"},{"why":"Gives the general kinematics and OPE of scalar operators with a codimension-two defect, from which the single-cross-ratio structure and defect data are taken.","marker":"[46]"},{"why":"Fixes the OPE coefficient c_{OOT} by a conformal Ward identity, making the early/late-time coefficient depend only on Delta.","marker":"[47]"},{"why":"Introduces the holographic local-quench model (falling massive particle or black hole in AdS4) and the perturbative entanglement-entropy computation that the paper reproduces and compares with its numerics.","marker":"[6]"},{"why":"Provides the vacuum Ryu-Takayanagi minimal-surface solution and the regularized area used as the subtraction in the holographic entropy.","marker":"[14]"},{"why":"Supplies the embedding-space formalism for correlators of two conformal defects, used to show the BCFT correlators depend on one cross-ratio in the constrained kinematics.","marker":"[55]"},{"why":"Gives the local form of the modular Hamiltonian for a spherical entangling region, the input for the relative-entropy bound.","marker":"[53]"}],"fun_headline_variants":["Entropy after local quench follows d/2 power law","Universal entropy growth for quenches in CFTd","Bound and power law for quench radiation entropy","Local quench entropy: universal d/2 exponent"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Entropy after local quench follows d/2 power law","Universal entropy growth for quenches in CFTd","Bound and power law for quench radiation entropy","Local quench entropy: universal d/2 exponent"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000187,"raw_usage":{"total_tokens":1415,"prompt_tokens":1115,"completion_tokens":300,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":731,"completion_tokens_details":{"reasoning_tokens":235}},"tokens_in":731,"tokens_out":300,"duration_ms":3528,"temperature":1.0,"reasoning_tokens":235,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-09T20:10:37.860284+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Radiation, entanglement and islands from a boundary local quench","cited_arxiv_id":"2203.10103","evidence_quote":"Supplies the two-dimensional boundary-quench result and the relative-entropy bound that the paper generalizes to higher dimensions; used for comparison and for the d=2 limit."}],"review_version":1}