{"id":"bc532057-fc5e-4584-b03a-a945fd49d04d","arxiv_id":"2411.09641","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Scalar Carrollian correlators admit a differential representation in which exchange diagrams are obtained from contact diagrams by commuting translation operators, matching the modified Mellin transform of flat-space amplitudes.","lead":"This paper develops a new computational framework, called a differential representation, for Carrollian correlators: boundary observables in a holographic model of flat spacetime. The framework expresses complicated exchange diagrams in terms of simpler contact diagrams, and it shows that two previously separate ways of computing these observables are consistent.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The intrinsic translation identity supports the differential representation, but the paper never states that eq. (5.1) and the worked example assume massless internal lines; a massive exchange requires adding the m^2 term to the differential denominator.","rationale":"The central claim has two independent supports: the AdS flat limit and the intrinsic translation identity (4.27). The intrinsic argument is exact for the Carrollian bulk-to-boundary propagator and does justify replacing the bulk Laplacian by (sum_i q_i mu d/du_i)^2 on products of external wavefunctions, so the O(1/L^2) truncation flagged in the reader's weakest_assumption is not the decisive objection. The decisive gap is the mass term. Section 2.2 explicitly defines Delta/L = m for the internal line, so the flat limit of the AdS denominator D_12^2 - Delta(Delta-d) contains m^2 at the same order as the translation-generated D_12^2. Eq. (5.1) and the solved example in section 5.1 omit m^2 without stating a massless restriction. This directly affects the concrete content of (4.25): for non-zero m the correct differential equation is [(D_p1 + D_p2)^2 - m^2] A = C, and the modified Mellin transform contains 1/(s-m^2). Since the paper's abstract and eqs. (4.17)/(4.25) are not restricted to massless theories, this is a genuine condition that must be stated or derived. The core construction is sound for the massless case and is independently supported by the translation identity, so the appropriate outcome remains conditional rather than accept or reject.","tokens_in":19541,"tokens_out":19601,"duration_ms":214198,"concrete_test":"Keep the internal mass explicit in the four-point computation: replace the s-channel differential operator 1/D_12^2 by 1/[ (D_p1 + D_p2)^2 - m^2 ], act on the contact correlator (5.4), and compare the result with the direct modified Mellin transform of the massive phi^3 amplitude 1/(s-m^2), where s = 2 omega_1 omega_2 tilde_q1·tilde_q2. If the massive result disagrees with eq. (5.8), then eqs. (5.1) and (5.8) are valid only for massless internal lines and the scope of eq. (4.25) must be restricted or amended.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central identification (4.25) is claimed for scalar Carrollian correlators generally, and section 5.1 makes it concrete through the s-channel differential equation 2 tilde_q1·tilde_q2 d^2/(du1 du2) A4 = C4 (eq. 5.1), which corresponds to an internal operator 1/D_12^2 and hence to a massless amplitude 1/s. However, the AdS differential representation being flattened is eq. (3.11), with denominator D_12^2 - Delta(Delta-d), and section 2.2 defines the flat-space internal mass by Delta/L = m. For Delta ~ L, taking L -> infinity gives D_12^2 - m^2 at the same order, so the massive differential equation must be [ (D_p1 + D_p2)^2 - m^2 ] A4 = C4, and (4.25) should contain 1/(s-m^2) rather than 1/s. The paper never flags that eq. (5.1) or the displayed exchange example is restricted to massless internal lines. The intrinsic translation identity (4.27) is exact and does support the general strategy of replacing the bulk Laplacian by boundary differential operators, but it does not by itself justify dropping the mass term in the internal propagator. Without an explicit massless restriction, the general claim in the abstract and in eqs. (4.17)/(4.25) is either incomplete or the example is misleading.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a differential representation for scalar Carrollian correlators, first by taking the Carrollian (flat-space) limit of the AdS differential representation, and then intrinsically from the translation property of Carrollian conformal primary wavefunctions. The central result is that a Carrollian exchange correlator is obtained by acting on the corresponding contact correlator with a differential operator ȷA_n constructed from commuting translation generators D_{p_i,μ}; under the modified Mellin kernel these operators become momenta, identifying Carrollian correlators with modified Mellin transforms of flat-space scattering amplitudes. The paper applies this to compute a 4-point exchange diagram in scalar ϕ3 theory and to derive differential BCJ relations for Carrollian correlators.","tokens_in":19755,"tokens_out":12732,"duration_ms":109844,"significance":"If the claims are correct, the paper provides a simple and powerful computational tool for Carrollian correlators: exchange diagrams reduce to differential equations acting on contact diagrams, with the differential operators commuting. The intrinsic derivation in Sec. 4.4 is elegant, and the agreement of the worked example with Ref. [30] is a strong check. The paper also sharpens the connection between Carrollian holography and scattering amplitudes under the modified Mellin transform, and the differential BCJ relations are a natural extension. The derivations are analytic and transparent, and the paper is clearly written. However, the central claims need to be sharpened by an explicit statement about the mass of internal lines; as written, the general formulas in Eqs. (4.17) and (4.25) and the example in Sec. 5.1 are incomplete for massive exchanges.","major_comments":[{"comment":"The paper does not state that the differential representation is restricted to massless internal propagators, and this omission affects the central claim. In the flat-space limit of the AdS exchange denominator in Eq. (3.11), the term Δ(Δ−d) is leading order when Δ ∼ L and m = Δ/L is held fixed, so the denominator becomes proportional to D_flat^2 − m^2. The paper nevertheless writes the s-channel operator in Eq. (5.1) as 2~ q_1·~ q_2 ∂_{u1}∂_{u2}, i.e., D_{12}^2, and the example corresponds to an amplitude 1/s. For a massive internal line the equation should read [2~ q_1·~ q_2 ∂_{u1}∂_{u2} − m^2] A_4 = C_4, and the Mellin-space amplitude should be 1/(s−m^2). The paper should either explicitly restrict the general claims in the abstract, Eq. (4.17), and Eq. (4.25) to massless internal lines, or include the mass term throughout. The worked example in Sec. 5.1.1 is thus only valid under a restriction that must be stated.","section":"Sec. 4.3, Eq. (4.25); Sec. 5.1, Eq. (5.1)"},{"comment":"The flat-space-limit derivation in Sec. 4.2 does not fully justify the truncation at leading order in 1/L once the propagator dimensions are large. The identities in Eq. (3.17) contain scale factors ξ_{AB} multiplied by Σ_i Δ_i, which is O(L) because Δ_i ∼ L. Consequently, the subleading boost generators in the flat limit, although formally O(1) as operators, can produce O(L) contributions when acting on the product of Carrollian bulk-to-boundary propagators; the paper does not prove that these are suppressed relative to the O(L^2) contribution of the translation generators. The assertion that the non-commuting differential operators are subleading is therefore not established by the operator scaling alone. Since the intrinsic identity (4.27) is exact, this concern does not invalidate the final result, but the flat-space-limit derivation should either be upgraded to a controlled expansion or explicitly labelled as a heuristic leading-order argument.","section":"Sec. 4.2, Eqs. (4.10)–(4.16)"}],"minor_comments":[{"comment":"The phrase 'Similiar analysis' contains a typo; it should be 'Similar analysis'.","section":"Sec. 2.2"},{"comment":"The word 'celesetial' is misspelled; it should be 'celestial'.","section":"Sec. 4.4"},{"comment":"The symbol '4X' preceding the sum is a typesetting artifact; it should be a summation sign.","section":"Sec. 4.2, Eq. (4.13)"},{"comment":"The text says the contact correlator is in ϕ4 theory while the exchange is in ϕ3 theory; a sentence clarifying that only the contact integral is used would avoid confusion.","section":"Sec. 5.1"},{"comment":"The transition from Eq. (5.4) to Eq. (5.8) uses Eq. (5.9) and integration, but the intermediate steps are not shown; a short derivation would improve verifiability.","section":"Sec. 5.1.1"},{"comment":"The signs in the exponents and the iε prescription for incoming versus outgoing wavefunctions are not explained; please clarify how the ∓ signs are assigned.","section":"Sec. 4.3, Eq. (4.22)"}],"recommendation":"major_revision","confidential_remarks":"The paper is within the journal's scope, and the citation pattern is appropriate. The main risk is the unstated massless-internal-line assumption in the central example; the authors should cross-check with Ref. [30] to confirm that the massless case is indeed what was computed there. The second major comment concerns the rigor of the large-L truncation, but the intrinsic argument may suffice; nevertheless the authors should address it explicitly."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"What you should know: this is the first differential representation for scalar Carrollian correlators, and it's built two different ways—by taking the Carrollian limit of the AdS differential representation and then independently from the translation properties of Carrollian primary wave functions. The two routes agree, the explicit 4-point exchange example matches the known result from Bagchi-Dhivakar-Dutta, and the intrinsic derivation in section 4.4 is clean enough to stand on its own. That is a real advance for the Carrollian holography subfield, not just a routine dictionary entry.\n\nThe flat-limit derivation does drop subleading terms in the large-L expansion—O(1/L^2) corrections to the Laplacian, the conformal Jacobian factors, and possible enhancement from the Δ_i ~ L scalings—without a systematic error estimate. That is a soft spot, but the intrinsic argument does not rely on those suppressed terms, so the leading-order statement is on solid ground. The integration-constant ambiguity in section 5.1 is handled by a generic-kinematics assumption; plausible, though not fully justified.\n\nThe one real issue is the internal line mass. The AdS denominator is D^2 - Δ(Δ-d), and since Δ/L = m, the flat limit at leading order gives D^2 - m^2, not D^2. The paper defines m in section 2.2 but then writes eq. (5.1) and the worked example with no mass term, without ever saying the internal line is massless. The general formula (4.17) and the Mellin identification (4.25) are not wrong—the differential operator can contain m^2—but the concrete example is either incomplete or silently restricted to m=0. A referee should ask for the massless assumption to be stated, and ideally for the massive version to be shown.\n\nThe differential BCJ relations are a straightforward application of the modified Mellin kernel and work fine. The citation pattern is honest; the paper builds on Bagchi-Dhivakar-Dutta, Alday et al., and Banerjee, and does not overclaim novelty.\n\nBottom line: send it to peer review. The central construction is correct, the presentation is mostly clear, and the massless gap is easily patched. After minor revision this becomes a solid contribution to Carrollian holography.","headline":"A solid and genuinely useful construction of the Carrollian differential representation, but the worked example silently assumes massless internal lines and should be flagged.","tokens_in":20343,"tokens_out":5643,"would_cite":true,"duration_ms":54441,"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":"Scalar Carrollian correlators admit a differential representation: exchange diagrams are rational differential operators on contact diagrams, and under the modified Mellin kernel those operators become flat-space momenta.","keywords":["Carrollian holography","differential representation","Witten diagrams","modified Mellin transform","flat-space scattering amplitudes","BCJ relations","conformal primary wave function","celestial holography"],"falsifier":"Compute the next correction in $1/L$ to the Carrollian-limit identity $(\\partial_{x^\\mu}-\\sum_i\\eta_{\\mu\\nu}\\tilde q_i^\\nu\\partial_{u_i})\\prod_iK_{\\Delta_i}=0$ with $\\Delta_i\\sim m_iL$ kept explicit: the differential representation is falsified if the $O(1/L^2)$ corrections to the Laplacian or to the propagators, combined with $\\Delta_i$, contribute at $O(1)$ or larger. A concrete check is to expand the $s$-channel exchange correlator of section 5.1.1 to order $1/L^2$ and compare it with $\\hat A_4C_4$ evaluated at the same order.","tokens_in":19274,"feed_emoji":"📐","tokens_out":11639,"duration_ms":93221,"temperature":0.7,"pith_summary":"This paper aims to show that scalar Carrollian correlators — the boundary correlators of flat-space holography — carry the same differential representation that AdS/CFT correlators do. Concretely, an exchange-channel Carrollian correlator $A_n(\\{p_i\\})$ is claimed to be $\\hat A_n(\\{D_{p_i,\\mu}\\})C_n(\\{p_i\\})$, where $C_n$ is the contact correlator, $\\hat A_n$ is a rational function of differential operators, and $D_{p_i,\\mu}=\\eta_{\\mu\\nu}\\tilde q_i^\\nu\\,\\partial_{u_i}$ is a commuting translation generator labelled by a null boundary direction $\\tilde q_i$. Because these operators become flat-space momenta when acting on the modified Mellin kernel, the same statement says the Carrollian correlator is the modified Mellin transform of the flat-space scattering amplitude. A sympathetic reader would care because this converts exchange-diagram computations into simple integrations and imports amplitude identities, such as BCJ relations, directly into Carrollian holography.","feed_headline":"Carrollian correlators are modified Mellin transforms of amplitudes","feed_subtitle":"The exchange diagram is a differential operator on the contact diagram; integrating it recovers the flat-space amplitude.","key_machinery":"The load-bearing object is the commuting set of boundary translation generators $D_{p_i,\\mu}=\\eta_{\\mu\\nu}\\tilde q_i^\\nu\\,\\partial_{u_i}$, with $\\tilde q_i$ the null direction assigned to external particle $i$ on the Carrollian boundary and $u_i$ the retarded time. Each generator squares to zero because $\\tilde q_i^2=0$, and all generators commute because $\\tilde q_i$ is independent of $u_i$; they are the leading pieces produced by the Carrollian limit of the AdS boundary conformal generators. The mechanism is the identity $(\\partial_{x^\\mu}-\\sum_i\\eta_{\\mu\\nu}\\tilde q_i^\\nu\\partial_{u_i})\\prod_iK_{\\Delta_i}=0$, which lets the flat-space Laplacian on the product of bulk-to-boundary propagators be replaced by $(\\sum_iD_{p_i})^2$; the exchange channel is then the inverse of this operator acting on the contact correlator. The modified Mellin kernel $e^{\\pm i\\omega_i u_i}$ diagonalizes each operator with eigenvalue $ip_{i,\\mu}$, which is the step that connects the differential representation to flat-space scattering amplitudes.","core_discovery":"The central claim is that the flat-space limit of the AdS differential representation survives in the Carrollian boundary theory, with all non-commuting generators dropping out. Performing the Inönü-Wigner contraction — the group-theoretic limit in which the AdS radius goes to infinity and the isometry algebra becomes the Poincaré algebra — of the bulk isometry and boundary conformal generators with $\\tau_p=\\pi/2+u/L$, the leading $O(L)$ pieces are boundary translations, and the propagator identity $(D^X_{AB}+\\sum_i D^{p_i}_{AB})\\prod_iK_{\\Delta_i}=0$ becomes $(\\partial_{x^\\mu}-\\sum_i\\eta_{\\mu\\nu}\\tilde q_i^\\nu\\partial_{u_i})\\prod_iK_{\\Delta_i}=0$. Since $D^2_{p_i}=0$ and $[D_{p_i,\\mu},D_{p_j,\\nu}]=0$, the flat-space Laplacian acting on the product of propagators is replaced by $(\\sum_iD_{p_i})^2$, and an exchange Witten diagram becomes a rational differential operator acting on the contact diagram, $A_n=\\hat A_n(\\{D_{p_i,\\mu}\\})C_n$. Acting on the modified Mellin kernel $e^{i\\omega_i u_i}$, each $D_{p_i,\\mu}$ returns $ip_{i,\\mu}$ with $p_{i,\\mu}=\\eta_{\\mu\\nu}\\omega_i\\tilde q_i^\\nu$, so the differential representation directly reproduces the known statement that Carrollian correlators are modified Mellin transforms of flat-space amplitudes. The same identities are rederived intrinsically from translation covariance of the Carrollian conformal primary wavefunction, so the representation does not depend on the finite-radius construction.","pith_inferences":["Editorial inference: if the leading-order statement survives finite-radius corrections, Carrollian correlators are essentially flat-space amplitudes in a different basis, so standard amplitude technology — unitarity, double copy, loop integrands — should transfer to Carrollian correlators without re-deriving Feynman rules.","Editorial inference: because all operators in a propagator sum commute, higher-point exchange correlators can be computed by a simultaneous Fourier transform of the differential equations rather than by nested modified Mellin integrals; this is a natural next test of the method.","Editorial inference: the celestial replacement $D_{p_i,\\mu}\\to i\\tilde q_{i,\\mu}e^{\\partial_{\\Delta_i}}$ can be checked directly against known celestial MHV or two-point amplitudes; if it holds, differential BCJ relations in celestial holography follow from the same argument.","Editorial inference: the subleading $1/L^2$ corrections with $\\Delta_i\\sim m_iL$ are the sharpest stress test; no current computation in the paper resolves whether those corrections stay subdominant."],"forward_implications":["The exchange channel of a scalar Carrollian correlator can be obtained from the contact diagram by solving a second-order partial differential equation; for four-point $\\phi^3$ theory the equation integrates directly and reproduces the previously known modified-Mellin result.","Flat-space amplitude identities translate into differential equations: any Mandelstam variable $s_{ij}$ is replaced by $2\\tilde q_i\\cdot\\tilde q_j\\,\\partial_{u_i}\\partial_{u_j}$, so color-kinematics duality becomes differential BCJ relations for Carrollian correlators.","The differential representation makes the equivalence of the two standard Carrollian prescriptions — flat limit of AdS Witten diagrams on one side, modified Mellin transform of amplitudes on the other — a theorem at leading order rather than a separate assumption.","Because the operators commute and diagonalize on the same kernel, the formalism is not tied to $\\phi^3$ theory; it transfers to any theory whose flat-space amplitudes obey the same algebraic relations.","A parallel prescription $D_{p_i,\\mu}\\to i\\tilde q_{i,\\mu}e^{\\partial_{\\Delta_i}}$ carries the differential representation to celestial correlators and connects to known celestial BCJ relations."],"supporting_citations":[{"why":"Supplies the Carrollian limit of AdS bulk-to-boundary and bulk-to-bulk propagators and the boundary coordinate redefinitions that the contraction is built on.","marker":"[29]"},{"why":"Provides the explicit four-point contact and exchange Carrollian correlators in $\\phi^3$ theory that the differential-representation example must reproduce.","marker":"[30]"},{"why":"Establishes the modified Mellin basis and the correspondence between Carrollian correlators and scattering amplitudes that the paper's main result makes direct.","marker":"[33]"},{"why":"Defines the Carrollian conformal primary wavefunction as a modified Mellin transform of plane waves and gives its translation covariance, used in the intrinsic derivation.","marker":"[10]"},{"why":"Introduces the differential representation of AdS correlators via quadratic Casimir and propagator identities whose flat-space limit is taken here.","marker":"[34]"},{"why":"Gives the AdS differential BCJ relations with $s_{ij}$ promoted to differential operators, which the Carrollian BCJ relations are obtained from.","marker":"[37]"},{"why":"Supplies the Inönü-Wigner contraction viewpoint on the differential representation in Mellin space that the bulk-generator limit follows.","marker":"[43]"},{"why":"Provides the celestial BCJ relations and the $ie^{\\partial_\\Delta}$ prescription used to formulate the celestial counterpart of the differential representation.","marker":"[52]"}],"fun_headline_variants":["Exchange diagrams become differential operators on contact ones","Carrollian correlators from differential operators on contact diagrams","Flat-space amplitudes from Carrollian differential representation","Differential representation links Carrollian correlators to amplitudes","Differential ops on contact diagrams yield Carrollian correlators"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument assumes that the terms dropped when the AdS radius $L$ goes to infinity stay negligible even though the operator dimensions $\\Delta_i$ in the propagators grow linearly with $L$; if those growing dimensions amplify the $O(1/L^2)$ pieces, the clean differential picture would acquire corrections.","fun_headline_variants_meta":{"raw":{"variants":["Exchange diagrams become differential operators on contact ones","Carrollian correlators from differential operators on contact diagrams","Flat-space amplitudes from Carrollian differential representation","Differential representation links Carrollian correlators to amplitudes","Differential ops on contact diagrams yield Carrollian correlators"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000438,"raw_usage":{"total_tokens":2252,"prompt_tokens":997,"completion_tokens":1255,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":613,"completion_tokens_details":{"reasoning_tokens":1195}},"tokens_in":613,"tokens_out":1255,"duration_ms":9097,"temperature":1.0,"reasoning_tokens":1195,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T20:27:17.321677+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the next correction in $1/L$ to the Carrollian-limit identity $(\\partial_{x^\\mu}-\\sum_i\\eta_{\\mu\\nu}\\tilde q_i^\\nu\\partial_{u_i})\\prod_iK_{\\Delta_i}=0$ with $\\Delta_i\\sim m_iL$ kept explicit: the differential representation is falsified if the $O(1/L^2)$ corrections to the Laplacian or to the propagators, combined with $\\Delta_i$, contribute at $O(1)$ or larger. A concrete check is to expand the $s$-channel exchange correlator of section 5.1.1 to order $1/L^2$ and compare it with $\\hat A_4C_4$ evaluated at the same order.","supporting_citations":[{"cited_title":"Bootstrapping Witten diagrams via differential representation in Mellin space","cited_arxiv_id":"2304.12757","evidence_quote":"Supplies the Inönü-Wigner contraction viewpoint on the differential representation in Mellin space that the bulk-generator limit follows."},{"cited_title":"Casali and A","cited_arxiv_id":null,"evidence_quote":"Provides the celestial BCJ relations and the $ie^{\\partial_\\Delta}$ prescription used to formulate the celestial counterpart of the differential representation."}],"review_version":1}