{"id":"3aa5a59a-028a-457d-a836-dfb821cf4943","arxiv_id":"2505.02899","paper_version":2,"verdict":"REJECT","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"In quadratic gravity, celestial eikonal amplitudes are claimed to be meromorphic rather than distributional, with OPE data extracted from a shadowed correlator.","lead":"This paper rewrites scattering in a modified theory of gravity with curvature-squared terms as correlations on the celestial sphere. It claims these amplitudes are smoother than in Einstein gravity and extracts operator product expansion data, but key assumptions remain unproved.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Eikonal phase dressing is built on an internally inconsistent linearization of exp(delta), so the central celestial amplitude and dispersion relation are not justified.","rationale":"The reader's REJECT verdict is well-founded: the paper's main celestial eikonal amplitude relies on an unproved exponentiation assumption that the authors themselves flag. My stress-test focuses on a more specific and more decisive internal problem: even granting exponentiation, the analytic linearization used to define the exponential of the contact delta term is inconsistent with the paper's own definition. The factor (e-1) in Eq. (38) is not derived from Eq. (34); it appears to be an arbitrary prescription. Since this factor enters the eikonal amplitude, the Mellin transform, the dispersion relation, and the shadowed OPE data, the central claims do not currently have a well-defined derivation. I therefore agree with the reader's REJECT verdict and recommend UNCHANGED. The Born-level Mellin transform and the conformal block manipulations may be salvageable in a revised version, but the eikonal construction needs to be corrected or justified before the advertised results can be accepted.","tokens_in":35393,"tokens_out":7274,"duration_ms":84577,"concrete_test":"Re-derive Eq. (38) directly from the stated definition (34): apply analytic linearization to f(x)=exp(c*x) with c=2*pi*i*G*eMEFT_Born(s)/Ep, either by keeping all powers of c or by using the idempotence property delta^(2)=delta. If the coefficient of delta is not (e-1)*c, then Eqs. (41)-(61) and all subsequent celestial and OPE results must be recomputed; the central claim survives only if the correct coefficient reproduces the same pole structure and normalization.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central eikonal result depends on Eq. (38), where exp(2*pi*i*G*eMEFT_Born(s)/Ep * delta^(2)(x_perp)) is replaced by 1 + (e-1)*(2*pi*i*G*eMEFT_Born(s)/Ep)*delta^(2)(x_perp). This does not follow from the analytic linearization definition in Eq. (34). For f(x)=exp(c*x), the formal realization gives f(delta)=1+(e^c -1)*delta if the idempotence property delta^n=delta is used, while a naive Taylor expansion gives 1+c*delta plus undefined higher powers; neither yields (e-1)*c*delta unless c=1, which is not the case. The arbitrary factor (e-1) propagates into Eq. (41), the Mellin integrals (42)-(45), the pole analysis, and the dispersion relation (61), so the advertised celestial eikonal amplitude and its meromorphic structure are not well-defined as stated. In addition, the paper explicitly acknowledges that eikonal exponentiation in quadratic gravity has not been demonstrated; the IR-universality argument given in Section 3.2 is plausible but does not establish the validity of the phase-dressed amplitude. Because the eikonal amplitude is the input to the shadowed correlator and OPE analysis, this concern is load-bearing.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript studies celestial holography for a quadratic higher-derivative gravity EFT with action (11). It computes the Born-level eikonal amplitude, dresses it with an eikonal phase that contains a Dirac-delta contact term, performs a Mellin transform to obtain a celestial amplitude, derives a dispersion relation for the phase-dressed amplitude, constructs shadowed four-point correlators in terms of Appell F1 functions, extracts OPE coefficients using the Burchnall-Chaundy expansion and the Euclidean OPE inversion formula, and finally gives a Carrollian amplitude. The advertised main results are: (i) in quadratic gravity the s- and u-channel contributions no longer cancel in the eikonal limit; (ii) the phase-dressed celestial eikonal amplitude is meromorphic in gamma with an infinite set of simple poles, in contrast to the distributional GR result; and (iii) OPE coefficients can be extracted from the shadowed Born and eikonal correlators.","tokens_in":35657,"tokens_out":13446,"duration_ms":153820,"significance":"The question addressed is worthwhile: celestial amplitudes in UV-incomplete gravitational theories are usually distributional or divergent, and higher-derivative EFTs provide a natural laboratory for studying whether improved UV behaviour leads to meromorphic celestial amplitudes. If the eikonal construction were valid, the non-cancellation of s/u channels and the pole structure would be a concrete and interesting example. The paper is self-contained in the sense that alpha and beta are input Wilson coefficients and no fitted value is fed back as a prediction; the Born-level Mellin integrals and the Burchnall-Chaundy manipulation of the Appell function are useful technical contributions. However, as detailed in the major comments, the central eikonal result depends on an internally inconsistent treatment of exp(delta) and on an explicitly acknowledged but unproven exponentiation assumption, so the advertised meromorphic celestial eikonal amplitude and the subsequent OPE analysis are not established.","major_comments":[{"comment":"The passage from Eq. (36) to Eq. (38) replaces exp(c delta^(2)(x_perp)) with 1 + (e-1) c delta^(2)(x_perp), where c = 2*pi*i*G*eMEFT_Born(s)/E_p. This does not follow from the Analytic Linearization definition in Eq. (34). For f(x)=exp(c x), Eq. (34) gives, if idempotence of delta is used, 1 + (e^c - 1) delta, not 1 + (e-1) c delta; the naive Taylor expansion gives 1 + c delta plus undefined higher powers. The factor (e-1) is not justified unless c=1, which is not the case. This spurious factor propagates into the eikonal amplitude (41), the Mellin integrals (42)-(45), the pole analysis, and the dispersion relation (61). Since the shadowed correlator and OPE analysis in Section 4 take this amplitude as input, this is a load-bearing error in the central claim.","section":"§3.2, Eq. (38)"},{"comment":"The paper explicitly states that ``a full demonstration of exponentiation in quadratic gravity would require explicit computation of loop-level diagrams... We intend to explore this in future work.'' The IR-universality argument in Section 3.2 and the reference to [59] are plausible but do not establish validity of phase dressing for the massive spin-2 and spin-0 modes that appear in the propagator (12). Because the phase-dressed eikonal amplitude (36)-(41) is the starting point for the dispersion relation, the celestial eikonal amplitude, and the OPE analysis, this is a load-bearing assumption rather than a derived result.","section":"§3.2, exponentiation assumption"},{"comment":"After shadowing, operator 2 has conformal dimension 2-Delta_2, as stated after Eq. (63). Setting all external conformal dimensions equal to Delta_O in Eq. (89) and in the block parameters a,b below Eq. (83) is therefore inconsistent unless Delta_O=1. With Delta_2=Delta_O, the shadowed operator has dimension 2-Delta_O, which changes the value of gamma and the block arguments. Consequently, the scalar-exchange OPE coefficient f^2_OOO in Eq. (89) and the comparison in Fig. 5 are not the equal-dimension limit of the shadowed correlator but rather a different correlator.","section":"§4.2-4.3, Eq. (89)"},{"comment":"The Euclidean OPE inversion is performed under uncontrolled approximations: rho_w, rho_bar_w << 1, the hypergeometric functions are set to 1, and the Appell function is approximated by the expression in Eq. (96). The residues in Eqs. (99)-(100) are therefore approximate, and the paper does not provide an error estimate. The discrepancy already visible in Fig. 5 at moderate Im(Delta_O) shows that the extracted OPE coefficients are indicative rather than derived, which matters because the abstract presents these coefficients as computed results.","section":"§4.3, Eqs. (95)-(100)"}],"minor_comments":[{"comment":"The word ``ultrablack'' is used where ``ultraviolet'' or ``UV'' is meant, e.g., in Section 3.3 and the conclusion; please correct this typo.","section":"Throughout"},{"comment":"The notation delta(i(gamma+2)) is unusual for a Dirac delta of an imaginary argument; please define it explicitly, presumably as a delta function in gamma with a Jacobian factor.","section":"Eq. (23)"},{"comment":"The expression for delta_1(alpha,beta|gamma) is hard to parse: ``beta^(-gamma/2-1) - 2^(gamma/2+3) alpha^(-gamma/2-1)'' appears to have ambiguous grouping. Please rewrite this equation with clear parentheses.","section":"Eq. (24)"},{"comment":"The GR part of the shadowed eikonal correlator is written down but not evaluated, and the paper then switches to the Born amplitude for the OPE analysis. The scope of the OPE claims should be stated precisely at the start of Section 4, because the eikonal OPE statement is only established for the EFT-correction part and only formally.","section":"§4.1, Eq. (67)"},{"comment":"The notation with the circled plus sign in Eq. (41) is nonstandard and should be defined. In Fig. 5, please state explicitly the values of alpha, beta (or the normalization factor) used in the comparison, and indicate whether the discrepancy at larger Im(Delta_O) is within the expected size of the approximations in Eq. (96).","section":"Fig. 5 and Eq. (41)"}],"recommendation":"reject","confidential_remarks":"The most serious issue is Eq. (38): the contact-term dressing is internally inconsistent and not a matter of convention. Even setting aside the acknowledged but unproven exponentiation in quadratic gravity, the central eikonal amplitude, dispersion relation, and OPE results all inherit this problem. I therefore cannot recommend publication in the present form."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First thing you should know: this paper gives the first celestial eikonal and OPE data for quadratic gravity, including a nice observation that the s/u channels no longer cancel in the eikonal limit. The machinery is borrowed from Adamo et al., but the application is new. The Born-level Mellin transform in Eq. (23) looks plausible, and the paper is transparent about its own limits.\n\nThe soft spot is real and load-bearing. In Eq. (38), the phase dressing replaces exp(c δ^(2)) with 1 + (e-1)c δ^(2). That does not follow from the paper's own analytic linearization (34): using δ^n = δ, the formal series gives exp(c δ) = 1 + (e^c - 1)δ. The factor (e-1)c is neither the first-order Taylor term nor the resummed one unless c = 1. Since c = 2πiG eM_Born/Ep is not 1, every subsequent integral, pole analysis, and dispersion relation inherits an unjustified normalization. The authors also concede eikonal exponentiation in quadratic gravity is unproven; the IR-universality argument is plausible but not a demonstration. So the advertised eikonal amplitude and its meromorphic structure are not well-defined as written.\n\nA second, more minor issue: the OPE section sets all external conformal dimensions to Δ_O even though one operator is a shadow of dimension 2 - Δ_O. That is internally inconsistent for the shadowed correlator; the BC-based OPE coefficient (89) should not be read as final.\n\nCredit where due: the s/u non-cancellation and the construction of the shadowed correlator from the Born amplitude are new, and the paper's use of Burchnall-Chaundy and Caron-Huot inversion is careful, with explicit (if cumbersome) residues. The Carrollian section is a reasonable bonus. The paper is also unusually honest about the missing loop-level check.\n\nWho this is for: celestial holography and gravitational EFT people. The Born-level sections and the OPE extraction technique are worth reading. But the central eikonal claim needs repair before the results can be trusted. I would send it to a serious referee, expecting major revision; the mistake in (38) is concrete and fixable, and the rest of the paper has enough substance that referee time is not wasted.","headline":"New celestial eikonal/OPE work for quadratic gravity, but a specific error in the phase linearization makes the central result unsound as written.","tokens_in":36144,"tokens_out":3929,"would_cite":false,"duration_ms":41779,"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":"This paper claims that quadratic-curvature corrections make the celestial eikonal amplitude meromorphic in the conformal dimension $\\gamma$, with an infinite tower of simple poles, and that shadowing one operator yields computable OPE…","keywords":["celestial amplitudes","quadratic gravity","eikonal approximation","conformal block expansion","OPE inversion formula","shadow transform","Appell hypergeometric function","Carrollian amplitude"],"falsifier":"Compute the two-loop ladder and cross-ladder diagrams for massless scalar scattering in quadratic gravity in the Regge limit: if the sum does not exponentiate into the phase $e^{i\\chi}$ with $\\chi$ built from the Born amplitude as in Eq. (31), then the phase-dressed eikonal celestial amplitude and its pole structure do not follow, a step the paper identifies as missing.","tokens_in":35184,"feed_emoji":"🌌","tokens_out":6029,"duration_ms":67837,"temperature":0.7,"pith_summary":"This paper takes the celestial holography program, which recasts flat-space scattering amplitudes as correlators of a two-dimensional conformal field theory on the celestial sphere, and asks what happens when Einstein gravity is extended by quadratic curvature terms with Wilson coefficients $\\alpha$ and $\\beta$. It claims that in the eikonal (high-energy, small-angle) limit these corrections remove a cancellation: the $s$- and $u$-channel contributions, which cancel in general relativity, remain nonzero, so the Mellin-transformed amplitude acquires a meromorphic structure in the conformal dimension $\\gamma$ instead of the delta-function distributional form of GR. The paper further extracts the operator product expansion data of the resulting celestial conformal field theory by shadowing one external operator, decomposing the correlator into conformal blocks, and using the Burchnall-Chaundy expansion and the Euclidean OPE inversion formula. A sympathetic reader would care because analytic celestial amplitudes are rare in gravitational theories, and the machinery used here gives a route to OPE data that does not rely on the collinear limit.","feed_headline":"Quadratic gravity turns celestial amplitudes meromorphic","feed_subtitle":"Higher-curvature corrections remove a cancellation, so celestial OPE data becomes computable without the collinear limit.","key_machinery":"The argument runs through a chain: the eikonal phase built from the Born amplitude in quadratic gravity, whose contact term is a Dirac delta handled by an analytic-linearization prescription; the Mellin transform to the conformal primary basis; a shadow transform on one external operator to relax the planarity delta function; the Burchnall-Chaundy expansion of the Appell $F_1$ function into products of Gauss hypergeometric functions, which identifies the scalar-exchange conformal blocks; and the Euclidean OPE inversion formula, whose residues at towers of poles in the partial-wave coefficient give OPE coefficients for general spin $J$. Each step converts a piece of the flat-space amplitude into a standard two-dimensional CFT object, and the factorization of the $\\omega$-integral from the cross-ratio dependence in the EFT correction is what makes the same OPE analysis reach the non-perturbative eikonal amplitude.","core_discovery":"For non-zero $\\alpha$ and $\\beta$, the phase-dressed eikonal celestial amplitude in quadratic gravity is meromorphic in $\\gamma$, with simple poles at $\\gamma = -2(n+1)$ in the infrared regime, in contrast to the distributional GR result; the shadowed correlator admits a conformal block expansion whose OPE coefficients can be computed, including spinning exchanges, from residues of the analytically continued partial-wave coefficients. The paper also derives a dispersion relation for the eikonal amplitude that encodes the massive poles $\\omega' = \\pm \\kappa/\\alpha, \\pm \\kappa/(2\\beta)$ coming from the quadratic corrections, and shows that the associated Carrollian amplitude has an infrared pole shifted by the EFT couplings. The key structural claim is that the eikonal amplitude remains a product of a Born amplitude and a phase, so the improved analytic behavior of the Born amplitude survives phase dressing.","pith_inferences":["If eikonal exponentiation is confirmed at loop level, the same meromorphicity mechanism should extend to any higher-derivative EFT whose Born amplitude has a finite number of massive poles, making celestial OPE data computable across a whole class of theories.","The simple-pole infrared structure at $\\gamma = -2(n+1)$, rather than the higher-order poles of GR, may be a direct signature of the massive spin-2 mode; a testable prediction is that the OPE coefficient residues carry the $\\alpha,\\beta$ dependence found in the paper's Eq. (89).","An exact evaluation of the OPE inversion integral, without the $\\rho \\ll 1$ approximation used to simplify the Appell function, could settle whether the discrepancy between the Burchnall-Chaundy and inversion results at large $\\text{Im}\\,\\Delta_O$ is an artifact of the approximation.","The Carrollian infrared-pole shift could serve as a sharp diagnostic: since boost invariance protects that pole, a fully non-perturbative treatment that preserved the shift would support the physical relevance of the EFT corrections to flat-space holography."],"forward_implications":["If the central claim holds, the celestial correlator of quadratic gravity is no longer a distribution: it has isolated poles in $\\gamma$, so standard CFT techniques apply without an extra regularization prescription.","The OPE coefficients obtained from the shadowed correlator give concrete CFT data for a gravitational theory beyond Einstein gravity, including spinning exchanges that the Burchnall-Chaundy expansion alone misses.","The dispersion relation for the phase-dressed eikonal amplitude encodes the massive graviton poles of the EFT, so the celestial amplitude remembers the massive modes introduced by the quadratic curvature terms.","Because the $\\omega$-integral factorizes from the cross-ratio dependence in the EFT correction, the OPE analysis extends from the Born amplitude to the full eikonal amplitude, at least for the EFT part.","The shift of the infrared pole in the Carrollian amplitude connects the EFT corrections to the boost-invariance structure of flat-space holography."],"supporting_citations":[{"why":"Supplies the eikonal-exponentiation route to celestial amplitudes and the GR result whose cancellation and pole structure this paper generalizes.","marker":"[43]"},{"why":"Provides the Euclidean OPE inversion formula used to extract OPE coefficients as residues of analytically continued partial-wave coefficients.","marker":"[30]"},{"why":"Gives the Burchnall-Chaundy expansion of the Appell $F_1$ function into products of Gauss hypergeometric functions.","marker":"[68]"},{"why":"Supplies the shadow-block and analytic-continuation technology for the Appell function used in the celestial block decomposition.","marker":"[67]"},{"why":"Supplies the conformal block decomposition and partial-wave expansion against which the celestial blocks are matched.","marker":"[65]"},{"why":"Provides the UV-to-IR classification of celestial amplitudes that motivates why better UV behavior should produce meromorphic amplitudes.","marker":"[33]"},{"why":"Supplies the graviton propagator with massive modes used to build the quadratic-EFT Born amplitude.","marker":"[47]"},{"why":"Supports the expectation that eikonal exponentiation extends beyond general relativity to intermediate effective theories.","marker":"[59]"}],"fun_headline_variants":["Quadratic gravity makes celestial amplitude meromorphic","Higher curvature yields simple poles in eikonal amplitude","Celestial OPE data computable beyond collinear limit","Phase dressing preserves analytic improvement in quadratic gravity","Dispersion relation and OPE from quadratic gravity amplitudes"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that eikonal exponentiation remains valid in quadratic gravity, together with the analytic-linearization prescription that defines the exponential of the Dirac-delta contact term; the paper explicitly notes that a full demonstration would require explicit loop-level diagram computation.","fun_headline_variants_meta":{"raw":{"variants":["Quadratic gravity makes celestial amplitude meromorphic","Higher curvature yields simple poles in eikonal amplitude","Celestial OPE data computable beyond collinear limit","Phase dressing preserves analytic improvement in quadratic gravity","Dispersion relation and OPE from quadratic gravity amplitudes"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000192,"raw_usage":{"total_tokens":1334,"prompt_tokens":917,"completion_tokens":417,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":533,"completion_tokens_details":{"reasoning_tokens":343}},"tokens_in":533,"tokens_out":417,"duration_ms":4753,"temperature":1.0,"reasoning_tokens":343,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T00:40:35.302669+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the two-loop ladder and cross-ladder diagrams for massless scalar scattering in quadratic gravity in the Regge limit: if the sum does not exponentiate into the phase $e^{i\\chi}$ with $\\chi$ built from the Born amplitude as in Eq. (31), then the phase-dressed eikonal celestial amplitude and its pole structure do not follow, a step the paper identifies as missing.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the Burchnall-Chaundy expansion of the Appell $F_1$ function into products of Gauss hypergeometric functions."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the graviton propagator with massive modes used to build the quadratic-EFT Born amplitude."}],"review_version":1}