{"id":"2f84eff6-fbca-4ac1-9c39-3d8c7a939e00","arxiv_id":"2506.11766","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Conformal Ward identities fix the leading soft limits of photon, gluon, and graviton correlators in (A)dS, and the subleading soft limit for gravitons.","lead":"This paper derives new mathematical rules for the low-energy limits of photon, gluon, and graviton interactions in curved spacetime called (Anti)-de Sitter space. The rules come from a symmetry principle, conformal invariance, and may simplify calculations of early-universe signals in cosmology.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The soft theorems rest on the Appendix A assertion that an IBP boundary term equals the SCWI inhomogeneous term; this is unproven in the paper and is exactly where the 1/q^2 pole is produced.","rationale":"I agree with the reader that the Appendix A IBP boundary-term identification is the most load-bearing assumption. I considered two other candidate weak points: the restriction to the ε_q^μ component of the SCWI, and the assumption that the remainder R has no 1/q^2 pole. Both are less dangerous: solving the transverse component is sufficient to fix the physical amplitude, and AdS propagators are indeed regular in q→0, so the no-pole assumption is well motivated. By contrast, the Appendix A step is where the derivation converts a symmetry statement into an equation-of-motion statement containing the 1/q^2 pole. Without it, charge conservation (19), the solution for R^{(0)} (21), and all subsequent soft operators do not follow. The paper's checks against explicit amplitudes are genuine evidence, but they are performed inside the same on-shell Mellin-Momentum formalism and therefore do not independently validate the boundary-term identification. The appropriate response is to keep the CONDITIONAL verdict: the central claim is plausible and cross-checked, but it should not be accepted unconditionally until Eq. (13) is derived without invoking [29] Appendix H. If the proposed concrete test passes, the concern is resolved and the paper can be upgraded.","tokens_in":11572,"tokens_out":8609,"duration_ms":87960,"concrete_test":"Compute both sides of Eq. (13) for the simplest nontrivial case: d=3, Δ_ϕ=2, soft photon plus one hard scalar exchange, using the explicit bulk-to-bulk propagator G(k,z,z') in its Mellin-Barnes or Bessel-function representation, and keep all boundary terms from the z-integration by parts. If the boundary term is not exactly the claimed inhomogeneous term of the SCWI, the 1/q^2 pole in (13) shifts and the soft theorem fails. Alternatively, rederive Eq. (13) from the full K^μ action without invoking [29] Appendix H and check whether the discarded boundary term equals the inhomogeneous term for generic kinematics.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central derivation converts the action of K_q+K_h on the soft-exchange diagram into the equation-of-motion operator D_{k_I} in Eq. (59). This step is not derived in the present paper: Appendix A states that after integration by parts the boundary term is 'discarded' and 'identified as the inhomogeneous term' from the spinning SCWI, referring to Appendix H of [29]. That boundary term is exactly the difference between the actual value of (K_q+K_h)(z ε_q·k_h G A_n) and the claimed −(4s_q+d−2)/(2 z q^2) D_{k_I} G ε_q e_h A_n. If it is not precisely equal to the inhomogeneous term, then Eq. (13) is false; the 1/q^2 pole used to enforce charge conservation (19) and to solve for R^{(0)} (21) would be contaminated, and the final soft theorems (26), (37), (50)–(52) do not follow. The manuscript provides no explicit check of this identification for a concrete diagram, and the verification statements in the text are checks of the final formulas against amplitudes constructed with the same formalism, not independent tests of this step.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper derives soft theorems for photon, gluon, and graviton boundary correlators in (Anti)-de Sitter space at tree level. The authors work with on-shell Mellin-momentum amplitudes and impose the special conformal Ward identity (SCWI) order by order in the soft momentum. They claim a leading soft photon theorem, a leading soft gluon theorem, and both leading and subleading soft graviton theorems, together with an infinite partial soft expansion for photons and gluons. The central mechanism is an integration-by-parts step, described in Appendix A, that converts the action of (K_q + K_h) on the soft-exchange diagram into the bulk equation-of-motion operator D_{k_I}, producing the 1/q^2 pole from which the soft theorems are extracted. The paper also discusses modifications due to higher-dimensional operators in Section 6.","tokens_in":11769,"tokens_out":3786,"duration_ms":40998,"significance":"If the central derivation is correct, the paper establishes exact tree-level soft theorems in (A)dS for spinning correlators, generalizing flat-space soft theorems and showing that conformal Ward identities alone fix the leading (and, for gravitons, subleading) soft behavior. The results are derived without fitting parameters, and the paper includes explicit formulas and checks up to five points, which is a strength. However, the derivation is not self-contained: the key step in Appendix A, the identification of the IBP boundary term with the inhomogeneous term of the SCWI, is deferred to an unpublished appendix of a companion paper [29]. The truncation to only the epsilon_q^mu component of the vector Ward identity is also asserted rather than proved. These omissions are load-bearing, so the significance is currently conditional on the missing derivations being supplied.","major_comments":[{"comment":"The step from Eq. (58) to Eq. (59) is the load-bearing point of the paper: it converts the action of (K_q + K_h) on the soft-exchange diagram into the equation-of-motion operator D_{k_I} acting on the bulk-to-bulk propagator, which produces the 1/q^2 pole used to derive charge conservation and the soft theorems. The text says that one performs integration by parts, discards the boundary term, and identifies it with the inhomogeneous term of the SCWI for spinning correlators, referring to Appendix H of [29]. This identification is not proved in the present paper, and it is exactly the difference between the true value of (K_q + K_h) on the diagram and the claimed expression. Without an explicit derivation or an independent check for a concrete diagram, Eqs. (13), (16), and consequently (26), (37), and (50)-(52) are not established.","section":"Appendix A, Eq. (59)"},{"comment":"The paper restricts to the epsilon_q^mu component of the vector SCWI, stated as 'we will focus solely on the epsilon_q^mu component', but does not prove that solving this single component is sufficient to determine the full amplitude. The final soft theorems are statements about the complete correlator, so one must show that the remaining components of the Ward identity are either automatically satisfied by the proposed solutions or impose no additional constraints. The same issue arises for the gluon and graviton cases in Sections 4 and 5.","section":"Section 3, after Eq. (12)"},{"comment":"There is an inconsistency in the normalization of the delta function for the bulk-to-bulk propagator. Eq. (14) states D_\\Delta G(k,z,z') = \\delta(z-z'), while Appendix A states D_\\Delta G(z,z') = z^{d+1}\\delta(z-z'). These differ by a factor of z^{d+1}. Since this propagator identity is used to eliminate the bulk integral and determine the z-dependence of the soft theorems, the discrepancy must be resolved and the correct normalization stated consistently.","section":"Eq. (14) and Appendix A"},{"comment":"The paper states that formulas are 'verified up to five points' for photons and gluons and for four- and five-graviton amplitudes. However, these verifications use amplitudes constructed within the same Mellin-momentum/on-shell formalism as the derivation, and therefore they do not independently test the contested IBP boundary-term identification in Appendix A or the restriction to the epsilon_q^mu component. An explicit check of Eq. (59) on a simple explicit diagram would be needed to validate the key step.","section":"Sections 3.1, 4.1, 5.1"}],"minor_comments":[{"comment":"There is a typo in the Introduction: 'we perfer to make Lorentz symmetry manifest' should be 'we prefer'.","section":"Abstract/Introduction"},{"comment":"The form of the special conformal generator K^\\mu_A is quoted from prior work [27-29] without derivation or a self-contained definition. Since this operator is central to the paper, a brief derivation or a precise statement of which parts are assumed would improve the presentation.","section":"Section 2, Eq. (6)"},{"comment":"The notation switches between D_{k_n} and D^d_{k_I} without explicit definition; it would be clearer to use a single notation for the bulk-to-bulk propagator inverse throughout.","section":"Section 4, Eq. (36)"},{"comment":"The computation of the phi R^2 three-point amplitude is presented very tersely. Since this example is used to show sensitivity to higher-dimensional operators, a few more intermediate steps or a reference to where the computation is detailed would be helpful.","section":"Section 6, Eq. (54)"}],"recommendation":"major_revision","confidential_remarks":"The paper may well be correct, but as a referee I cannot certify the central derivation because the key IBP boundary-term identification is delegated to an appendix of a companion paper [29] and is not proved here. The editor may wish to consider whether reliance on an unpublished appendix in this manner is acceptable for the journal, and whether the authors should be asked to include a self-contained proof of the identification in the present manuscript."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThis paper derives soft theorems for photon, gluon, and graviton correlators in (A)dS by solving the special conformal Ward identity in the Mellin-Momentum formalism. The genuinely new results are the photon soft theorem (eq. 26) and the subleading graviton soft theorem (eqs. 50–52), neither of which appears in earlier work on soft limits in AdS. The recursive approach—solving the SCWI order by order in the soft momentum to obtain an infinite set of partial soft theorems—is a nice step forward. The paper also cross-checks its formulas against explicit four- and five-point correlators from the authors' prior papers, and those checks give me some confidence that the results are correct.\n\nThe main soft spot is the derivation of the key identity in Appendix A. To get the 1/q^2 pole, the authors need to convert the action of the conformal generator on the soft-exchange diagram into the equation-of-motion operator acting on the bulk-to-bulk propagator. They do this by integrating by parts and discarding a boundary term, which they identify with the inhomogeneous term in the spinning special conformal Ward identity. That identification is asserted, not derived, and the paper refers to Appendix H of the authors' own [29] for the justification. If the identification is wrong, the central soft theorems do not follow. The stress-test note puts the right finger on this: the paper provides no explicit check for a concrete diagram, and the verification statements are checks of final formulas against amplitudes constructed in the same formalism, not independent tests of this step.\n\nThere are two smaller omissions. The restriction to the ε_q^μ component of the Ward identity is asserted rather than justified; presumably the other components follow from other Ward identities, but the paper doesn't say so. Also, the paper leans heavily on the authors' prior work for the on-shell amplitude and the form of the generator; that's not a flaw by itself, but it makes the paper harder to assess in isolation.\n\nOverall, I think the results are likely correct, but the missing proof in Appendix A is a real gap that a referee should ask about. The paper deserves peer review and should be sent out, with a specific request to verify the IBP boundary-term identification. This is for the cosmological bootstrap and holography crowd; they'll find it useful.\n\nRecommendation: send to peer review, and make sure the referee hammers on Appendix A.","headline":"New photon and subleading graviton soft theorems in (A)dS from conformal Ward identities, but the key IBP step in Appendix A is asserted, not proven—worth refereeing, not desk rejection.","tokens_in":12351,"tokens_out":3731,"would_cite":true,"duration_ms":35473,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"The leading soft limit of photon, gluon, and graviton correlators in (A)dS is fixed by the special conformal Ward identity, yielding explicit soft theorems at order $q^0$ (and order $q^1$ for the graviton).","keywords":["soft theorems","conformal Ward identities","AdS/CFT","de Sitter","Witten diagrams","Mellin-Momentum formalism","graviton correlators","gluon correlators"],"falsifier":"Compute the $O(q)$ term of a five-point graviton correlator directly from the Witten-diagram integral without invoking the Ward identity, and compare with the prediction of $S^{(1)}$ in eq. (52); the paper verifies only up to local terms for the four-point case, so a non-local $O(q)$ mismatch at five points would refute the subleading theorem. A more targeted check is to evaluate the boundary term in eqs. (58)--(59) explicitly for a spin-2 external leg and test whether it equals the inhomogeneous SCWI term claimed in Appendix A.","tokens_in":11308,"feed_emoji":"🌌","tokens_out":6278,"duration_ms":54127,"temperature":0.7,"pith_summary":"This paper derives soft theorems—universal statements about what happens to a boundary correlator when one particle's momentum is taken to zero—for photons, gluons, and gravitons in Anti-de Sitter and de Sitter space. The authors show that the special conformal Ward identity, applied to the on-shell part of the Witten diagrams in a Mellin-momentum representation, is enough to fix the leading soft behavior completely (and the subleading soft behavior for gravitons). Unlike flat space, where the soft limit is dominated by diagrams with propagators going on shell and produces a $1/q$ divergence, in (A)dS every diagram contributes and the leading soft theorem sits at order $q^0$. The paper gives explicit soft operators, verifies them up to five-point correlators (photon and gluon) and four- and five-point graviton amplitudes, and shows that higher-dimensional operators alter the graviton soft theorem at leading order. If correct, these results extend the flat-space soft theorem program to cosmological and holographic settings where boundary correlators are the observables.","feed_headline":"Conformal symmetry fixes soft limits in (A)dS","feed_subtitle":"Special conformal Ward identities determine leading soft photon, gluon, and graviton correlators to all orders.","key_machinery":"The Mellin-Momentum formalism, in which the boundary correlator is represented as an integral of an on-shell amplitude $A_n$ times bulk-to-boundary propagators; the amplitude $A_n$ is obtained by amputation and obeys an exact special conformal Ward identity $\\sum_a K_a^\\mu A_n = 0$. The generator $K_J^\\mu$ acting on $A_n$ carries a factor $1/k^2$ that, when combined with the soft propagator, produces a $1/q^2$ pole; integrating by parts converts the action on the bulk-to-bulk propagator into the equation-of-motion operator $D_\\Delta$, which cancels the propagator by $D_\\Delta G = \\delta(z-z')$. The paper identifies the integration-by-parts boundary term with the inhomogeneous term on the right-hand side of the special conformal Ward identity for spinning correlators (Appendix A, following Appendix H of the related paper), and this identification is what turns the Ward identity into a recursion that fixes the soft expansion order by order.","core_discovery":"The paper's central claim is that, for tree-level Witten diagrams in (A)dS, the special conformal Ward identity on the on-shell amplitude determines the leading soft behavior of spinning correlators: for photons, $\\langle J(q)\\phi(k_1)\\dots\\rangle = -\\tfrac{1}{2}\\sum_h e_h \\,\\varepsilon_q\\cdot\\partial_{k_h}\\langle\\dots\\rangle + O(q)$; for color-ordered gluons, $\\langle J(q)J(k_1)\\dots J(k_n)\\rangle = \\tfrac{1}{2}(\\varepsilon_q\\cdot\\partial_{k_n} - \\varepsilon_q\\cdot\\partial_{k_1})\\langle J(k_1)\\dots J(k_n)\\rangle + O(q)$; and for gravitons in $d=3$, $\\lim_{q\\to 0}\\langle T(q)T(k_1)\\dots T(k_n)\\rangle = (S^{(0)}+S^{(1)})\\langle T(k_1)\\dots T(k_n)\\rangle$ with $S^{(0)} = -\\tfrac{1}{2}\\sum_a \\varepsilon^{\\mu\\nu} k_{a\\mu}\\partial_{k_a^\\nu}$ and $S^{(1)} = \\tfrac{1}{4}\\sum_a \\varepsilon^{\\mu\\nu}q^\\rho(k_{a\\rho}\\partial_{k_a^\\mu}\\partial_{k_a^\\nu} - 2k_{a\\mu}\\partial_{k_a^\\nu}\\partial_{k_a^\\rho} - 2\\partial_{k_a^\\mu}S_{a\\nu\\rho})$. At order $q^{-2}$ the Ward identity forces charge conservation for photons and universal couplings for gravitons; at higher orders it recursively fixes the remainder function $R$. The paper verifies the photon and gluon theorems up to five points and the graviton theorem up to the four-point correlator, and it shows that a higher-dimensional operator such as $\\phi R^2$ modifies the graviton soft theorem already at leading order.","pith_inferences":["These soft theorems provide boundary consistency conditions that could be imposed on inflationary wavefunction coefficients, giving a cosmological analogue of soft theorems as constraints on amplitudes.","The paper's reliance on a single Ward identity suggests the same recursion may be derivable purely from boundary conformal bootstrap axioms (a direction the authors flag); if so, the soft theorems would follow without bulk diagrammatics.","The undetermined antisymmetric tensors $N$ at higher orders are a testable ambiguity: imposing additional structure, such as color-kinematics duality or parity, could fix them and extend the theorems beyond leading order.","The $\\phi R^2$ example hints that a classification of which effective operators shift the leading soft graviton theorem would map the UV sensitivity of (A)dS soft limits, analogous to soft-theorem analyses in flat-space effective field theory."],"forward_implications":["Charge conservation and universal graviton couplings follow from the $O(q^{-2})$ terms of the special conformal Ward identity, so the soft theorems imply these constraints on any (A)dS boundary theory.","The leading soft photon and gluon theorems hold at order $q^0$ and receive contributions from all diagrams, in contrast to flat space where soft propagators dominate with $1/q$ singularities.","The graviton correlator's leading and subleading soft factors are fully determined: $S^{(0)}$ acts as a momentum-space dilatation-like operator on each hard leg, and $S^{(1)}$ involves the special conformal generators when contracted with $\\eta_{\\mu\\nu}$.","Higher-dimensional operators such as $\\phi R^2$ modify the leading graviton soft theorem, so the (A)dS soft theorems are sensitive to curvature corrections in a way flat-space soft theorems are not.","The recursion yields infinitely many partial soft constraints at higher orders, determined only up to antisymmetric tensors $N$, paralleling the infinite partial soft theorems in flat space."],"supporting_citations":[{"why":"Establishes the exact special conformal Ward identity for on-shell amplitudes and supplies the Appendix H identification of the integration-by-parts boundary term with the inhomogeneous SCWI term, the load-bearing step of the derivation.","marker":"[29]"},{"why":"Provides the on-shell bootstrap for $n$-gluon and graviton scattering in (A)dS and the factorization used to build the soft diagram (a).","marker":"[28]"},{"why":"Supplies the Mellin-Momentum amplitude bootstrap and the four-point graviton amplitude used for verification.","marker":"[27]"},{"why":"Derives soft limits of gluon and graviton correlators in AdS and provides the soft expansion and integration of the bulk-to-bulk propagator that the authors reuse.","marker":"[20]"},{"why":"Gives the decomposition of spinning correlators into transverse-traceless and longitudinal parts and the three-point amplitudes used for the $\\phi R^2$ example.","marker":"[32]"},{"why":"Provides the de Sitter graviton four-point function used to verify the graviton soft theorem.","marker":"[33]"},{"why":"Provides the graviton trispectrum from gluons used in the verification of the graviton soft theorem.","marker":"[34]"},{"why":"Justifies the on-shell amputation of external states via generalized LSZ, the basis for focusing on the on-shell amplitude $A_n$.","marker":"[26]"}],"fun_headline_variants":["Conformal Ward identities dictate soft limits in (A)dS","Soft photon, gluon, graviton theorems from conformal symmetry","Leading soft behavior in (A)dS fixed by conformal invariance","Conformal symmetry pins down soft correlators in (A)dS","Ward identities yield soft theorems for photons, gluons, gravitons"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the integration-by-parts boundary term produced when the special conformal generator acts on a Witten diagram is exactly the inhomogeneous term on the right-hand side of the special conformal Ward identity for spinning correlators; if that identification fails, the $1/q^2$ pole and the soft theorems built from it do not follow.","fun_headline_variants_meta":{"raw":{"variants":["Conformal Ward identities dictate soft limits in (A)dS","Soft photon, gluon, graviton theorems from conformal symmetry","Leading soft behavior in (A)dS fixed by conformal invariance","Conformal symmetry pins down soft correlators in (A)dS","Ward identities yield soft theorems for photons, gluons, gravitons"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000714,"raw_usage":{"total_tokens":3232,"prompt_tokens":987,"completion_tokens":2245,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":603,"completion_tokens_details":{"reasoning_tokens":2153}},"tokens_in":603,"tokens_out":2245,"duration_ms":16036,"temperature":1.0,"reasoning_tokens":2153,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T04:04:03.781049+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the $O(q)$ term of a five-point graviton correlator directly from the Witten-diagram integral without invoking the Ward identity, and compare with the prediction of $S^{(1)}$ in eq. (52); the paper verifies only up to local terms for the four-point case, so a non-local $O(q)$ mismatch at five points would refute the subleading theorem. A more targeted check is to evaluate the boundary term in eqs. (58)--(59) explicitly for a spin-2 external leg and test whether it equals the inhomogeneous SCWI term claimed in Appendix A.","supporting_citations":[{"cited_title":"Mei and Y","cited_arxiv_id":null,"evidence_quote":"Establishes the exact special conformal Ward identity for on-shell amplitudes and supplies the Appendix H identification of the integration-by-parts boundary term with the inhomogeneous SCWI term, the load-bearing step of the derivation."},{"cited_title":"Mei and Y","cited_arxiv_id":null,"evidence_quote":"Provides the on-shell bootstrap for $n$-gluon and graviton scattering in (A)dS and the factorization used to build the soft diagram (a)."},{"cited_title":"Mei, Amplitude Bootstrap in (Anti) de Sitter Space And The Four-Point Graviton from Double Copy , arXiv:2305.1389","cited_arxiv_id":null,"evidence_quote":"Supplies the Mellin-Momentum amplitude bootstrap and the four-point graviton amplitude used for verification."},{"cited_title":"Chowdhury, A","cited_arxiv_id":null,"evidence_quote":"Derives soft limits of gluon and graviton correlators in AdS and provides the soft expansion and integration of the bulk-to-bulk propagator that the authors reuse."},{"cited_title":"Bonifacio, H","cited_arxiv_id":null,"evidence_quote":"Provides the de Sitter graviton four-point function used to verify the graviton soft theorem."},{"cited_title":"Armstrong, H","cited_arxiv_id":null,"evidence_quote":"Provides the graviton trispectrum from gluons used in the verification of the graviton soft theorem."},{"cited_title":"Cheung, J","cited_arxiv_id":null,"evidence_quote":"Justifies the on-shell amputation of external states via generalized LSZ, the basis for focusing on the on-shell amplitude $A_n$."}],"review_version":1}