{"id":"5cd604c2-197f-43ec-bf80-463a01630246","arxiv_id":"2607.14986","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"The O(v²) correction to the high-energy-resummed coefficient function for exclusive heavy-quarkonium photoproduction is derived and shown to be small but useful for stabilizing μ_F dependence.","lead":"This paper calculates the next relativistic correction to a high-energy-resummed QCD prediction for exclusive photoproduction of heavy quarkonia such as J/ψ and Υ. The correction is numerically small but should reduce the scale dependence of the prediction, improving precision for EIC and LHC ultra-peripheral collision analyses.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The claimed partial μ_F cancellation is inferred from the ρ→0 limit but never demonstrated for the matched amplitude; moderate-ρ contributions may spoil it.","rationale":"The reader's verdict ACCEPT was based on the internal consistency of the calculation and the author's self-awareness about the GPD-evolution mismatch. I agree the algebraic core is credible: the independent check of the earlier O(v²) LO result in the Appendix, the sensible PM-scheme construction, and the explicit acknowledgment of the DLA/LLA limitation all support accepting the computation as a valid theoretical result. However, the strongest practical claim—that the new correction partially cancels μ_F dependence—is a key part of the abstract and conclusions, and it rests on an inference from the ρ→0 asymptotic behavior plus an analogy to Ref. [27], not on a direct evaluation of the matched formula. Since the matching formula's moderate-ρ region is known to be non-negligible and is not governed by the simple factor in Eq. (64), the cancellation could be partially or wholly spurious. This is not an accusation of error; it is a request for a concrete check before the motivation is taken as established. The verdict should therefore be CONDITIONAL: accept the computation, but require either a numerical demonstration of the μ_F cancellation or a toning down of the claim. If the check shows the cancellation holds, the paper can be accepted as is.","tokens_in":18685,"tokens_out":3535,"duration_ms":40881,"concrete_test":"Implement the matched amplitude of Eq. (65) for exclusive J/ψ photoproduction with a simple GPD model (e.g., a factorized double-distribution ansatz) and compute the cross section over a range of μ_F, say μ_F = M_V/2 to 2M_V, with α_s(μ_F). Compare the residual scale dependence in three cases: (i) without the ⟨v²⟩ˇC_HEF,v2 term, (ii) with it, and (iii) with the fixed-order O(v²) LO term alone. If the inclusion of the new term does not reduce the band of μ_F variation relative to case (iii), the claimed partial cancellation is not realized in the matched amplitude and the central motivation of the paper would need to be revised.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central practical motivation—that the new O(v²) HEF-resummed term improves robustness by partially cancelling the μ_F dependence of the O(v²) LO contribution—is not actually demonstrated. The paper only shows the ρ≪1 asymptotic relation (Eq. 64): C_HEF,PM ≈ (1−⟨v²⟩/6) C_HEF,v0,PM. From this, plus an analogy to Ref. [27], it is asserted that the μ_F dependence partially cancels. But the physical amplitude is the convolution in Eq. (14) with GPDs, and the matching formula (65) includes ˇC_HEF,v2 without a double-counting subtraction. The paper itself notes that the moderate-ρ region (0.1≲ρ<1) contribution is 'not necessarily numerically small.' In that region the simple factor (1−⟨v²⟩/6) does not apply; the full expression (49) has b1 and b2 terms, and the inverse Mellin transform (56) can have a different μ_F dependence. Moreover, the claimed cancellation is computed under fixed-order GPD evolution, while the resummation is in DLA-HEF; footnote [44] acknowledges full consistency would need LLA-resummed GPD evolution. Thus, while the O(v²) correction itself is a well-defined new result, the specific payoff claimed in the abstract and Sec. IV C—improved robustness via μ_F cancellation—is an unverified inference. If it fails, the numerical significance of the result is reduced to 'a small correction that can be included,' and the motivation for the work weakens substantially.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript computes the O(v^2) correction to the doubly-logarithmic high-energy-factorisation (DLA-HEF) resummed coefficient function for exclusive vector heavy-quarkonium photoproduction, extending the O(v^0) resummation of Ref. [27]. The calculation uses NRQCD velocity expansion and the Gremm-Kapustin relation to express results in a \"physical mass\" (PM) scheme in terms of M_V and the external parameter ⟨v^2⟩. The main new results are the one-loop impact factor h^(v2)(q_T^2) in Eqs. (34)-(41), the Mellin-space resummed coefficient in Eqs. (47)-(49), the inverse-Mellin representation via scale derivatives, Eq. (56), and a subtractive matching formula, Eq. (65). In the ρ=ξ/|x|≪1 limit the PM-scheme resummed correction reduces to a factor (1−⟨v^2⟩/6) times the O(v^0) resummed coefficient, Eq. (64), implying numerical smallness at μ_F=M_V. The paper further claims that this term partially cancels the μ_F dependence of the O(v^2) LO correction, improving the robustness of predictions. The Appendix independently rederives the O(v^2) LO coefficient of Ref. [16] and provides an all-order-in-v^2 formula for the LO CF coefficient.","tokens_in":19006,"tokens_out":5932,"duration_ms":67492,"significance":"If correct, the computation provides the first O(v^2 α_s^n ln^{n-1}(x/ξ)) correction to exclusive quarkonium photoproduction in the HEF framework, a quantity relevant for high-energy J/ψ and Υ phenomenology at the LHC and EIC. The PM-scheme smallness of the correction is a non-trivial dynamical result that follows from the q_T dependence of the impact factor. The Appendix's independent reproduction of the known O(v^2) LO result of Ref. [16] is a valuable check. The central new resummed coefficient and the associated matching formula are internally consistent and are not obtained by fitting any target result; ⟨v^2⟩ is an external input. However, the practical motivation stated in the abstract — that the new correction partially cancels the μ_F dependence of the O(v^2) LO contribution — is not actually demonstrated by the calculations presented; this weakens, but does not invalidate, the technical result.","major_comments":[{"comment":"The claimed partial cancellation of μ_F dependence is not demonstrated. The only quantitative evidence is the asymptotic relation (64), valid for ρ≪1, together with an analogy to the O(v^0) case. The physical amplitude is the convolution (14), and the matched coefficient (65) contains the full O(v^2) HEF term whose moderate-ρ behaviour (0.1≲ρ<1) the paper itself says is \"not necessarily numerically small\". To support the abstract claim, the authors should either show a numerical μ_F scan (e.g. around M_V) of the matched amplitude or of the μ_F derivative of the coefficient in Eq. (65) with and without the new HEF v^2 term, or soften the claim to a plausible expectation. As written, the central practical motivation is an unverified inference.","section":"Abstract and Sec. IV C, Eq. (65)"},{"comment":"The consistency between the DLA-HEF resummation and the use of fixed-order (LO/NLO) GPD evolution is acknowledged to be an approximation; full low-x LLA-resummed GPD evolution would be required for complete consistency. This is exactly the regime in which the claimed μ_F cancellation is supposed to operate. Since the paper uses conventional GPD evolution, the partial cancellation may be partly an artefact of this inconsistency. The manuscript should quantify this uncertainty or at minimum state in the abstract that the cancellation claim is made within the DLA/fixed-order-evolution framework. As it stands, the abstract presents the cancellation as a robust property of the computed correction.","section":"Footnote [44] and Sec. IV C"},{"comment":"The one-loop impact factor result in Eq. (34) is the central new technical input, but it is presented only as the final output of a FeynCalc computation. No intermediate algebra, diagram-by-diagram decomposition, or notebook is provided. The Appendix's check of the O(v^2) LO coefficient function is not a check of this one-loop impact factor. For reproducibility, the authors should either include a derivation sketch (e.g. the decomposition of the diagrams, the treatment of Glauber regions, and the k^2 expansion before l_+ integration) or make the FeynCalc notebook available as supplemental material. This is a load-bearing point because all subsequent results, including the b_1,b_2 coefficients and the cancellation claim, depend on Eq. (34).","section":"Sec. IV A, Eq. (34)"},{"comment":"The subtractive matching in Eq. (65) includes the O(v^2) HEF resummed term ˇC^(HEF,v2)_i(ρ) over the whole range 0<ρ<1, while the paper notes that the moderate-ρ region is unphysical and may have a non-negligible contribution to the convolution. No matching uncertainty or alternative prescription (e.g. InEW matching, mentioned only as planned) is used to estimate the effect of this region. Since the polynomial (b0+b1 γ+b2 γ^2) in Eq. (49) has different behaviour away from ρ≪1, the factor (1−⟨v^2⟩/6) in Eq. (64) is not representative of the matched amplitude. The robustness claim therefore needs either an explicit assessment of the moderate-ρ region or a clear statement that the cancellation is an asymptotic, not a matched-amplitude, property.","section":"Sec. IV C, after Eq. (65)"}],"minor_comments":[{"comment":"The abstract says the new HEF term cancels the μ_F dependence of the O(v^2) correction to the LO coefficient function. Strictly, the LO coefficient function C^(v2)_g in Eq. (19) has no explicit μ_F dependence; the μ_F dependence enters through the GPD in the convolution (14). Please clarify this wording in the abstract and Sec. IV C.","section":"Abstract and Sec. III"},{"comment":"The notation ˇC is introduced only for the O(v^0) resummed coefficient in Eq. (57), but Eq. (65) uses ˇC^(HEF,v2)_i without definition. State explicitly that this is the inverse Mellin transform of Eq. (49) with the δ(1−ρ) term (the b_0 term) removed, consistent with the check notation.","section":"Eq. (56) and Eq. (65)"},{"comment":"The figure is plotted only at μ_F=M_V. Since the entire motivation concerns μ_F dependence, a second panel at e.g. μ_F=2M_V and μ_F=M_V/2 would be much more informative than the single-scale plot. This is related to the major comment about the cancellation claim.","section":"Fig. 3 and Sec. IV C"},{"comment":"Minor typos: \"can not write-down\" should be \"cannot write down\"; \"to it's v^2=0 limit\" should be \"to its\"; \"funci ton\" should be \"function\". Also, the sentence after Eq. (A8) is incomplete and should be rephrased.","section":"Appendix A"},{"comment":"The function θ(q_T^2 < M_V^2/4) is a sharp step; the dotted line in Fig. 2 appears as a plateau. Consider adding a label or a comment that the step is schematic, to avoid confusion with a smooth fall-off.","section":"Eq. (54) and Fig. 2"},{"comment":"The dependence of Eq. (65) on the NLO coefficient C^(1,S)_i is not given explicitly (it is taken from Ref. [15]). For a self-contained presentation, at least the small-ρ asymptotics of C^(1,S)_i that are subtracted in the matching should be quoted or briefly summarised.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The technical computation appears sound and the Appendix's independent reproduction of the LO O(v^2) result is a strong point. The main problem is that the paper's abstract and conclusion assert a partial μ_F cancellation that is never demonstrated for the matched amplitude; this is fixable by either adding a numerical μ_F dependence study or by weakening the claim. A secondary but important issue is the lack of detail behind Eq. (34), which is the core new calculation. If the authors address these, the paper would be suitable for publication. I am not recommending rejection because the central coefficient computation is well-defined and the PM-scheme smallness is a genuine new result."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Maxim's paper computes the O(v^2) correction to the DLA-HEF resummed coefficient function for exclusive heavy-quarkonium photoproduction. That correction is new—neither Ref [27] nor Ref [16] had it—and the paper is honest about what it does and doesn't show.\n\nWhat's solid: the impact-factor calculation is internally consistent, and the Appendix independently reproduces the earlier O(v^2) LO result of Ref [16], which is a real check. The PM-scheme conversion via the Gremm–Kapustin relation is clean, and the rho<<1 limit gives the simple factor (1 − <v^2>/6) times the resummed v0 coefficient. The author also flags the moderate-rho region and the fixed-order-versus-LLA GPD evolution mismatch in footnote 44, so the limitations are on the table.\n\nWhere I'd push back: the abstract and Sec. IV C claim that the new term “partially cancels” the mu_F dependence of the O(v^2) LO amplitude. What is actually shown is the asymptotic rho<<1 relation and an analogy to the O(v^0) case. The full matched amplitude—with the b1 and b2 terms in Eq. (49), the inverse Mellin transform (56), and the convolution with GPDs—is never scanned in mu_F. The moderate-rho region can contribute non-negligibly, and the author admits as much. So the cancellation payoff is an inference, not a demonstration. That doesn't kill the paper; the new coefficient function stands as a well-defined result. But it does mean the “improved robustness” selling point is weaker than the abstract suggests.\n\nMinor: the central one-loop impact-factor result, Eq. (34), is stated as the output of FeynCalc with no intermediate algebra. Given the Appendix check and the standard nature of the computation, this is acceptable, but a referee will want the notebook or at least more detail.\n\nThis is a paper for the exclusive quarkonium photoproduction / small-x phenomenology community. It deserves serious peer review: the calculation is new, self-consistent, and reproducible in principle, and the limitations are frankly stated. I'd accept it with minor-to-moderate revision that either demonstrates the mu_F cancellation numerically for the matched amplitude or softens the claim.","headline":"A careful, genuinely new O(v^2) DLA-HEF resummed coefficient function for exclusive quarkonium photoproduction; the mu_F-cancellation payoff is asserted more than demonstrated, but the central calculation holds.","tokens_in":19545,"tokens_out":2258,"would_cite":true,"duration_ms":23275,"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 paper computes the O(v²) correction to the high-energy resummed coefficient function for exclusive heavy quarkonium photoproduction and finds that, expressed in the physical-mass scheme, the correction is numerically small at μ_F=M_V wh","keywords":["exclusive photoproduction","heavy quarkonium","NRQCD factorization","high-energy resummation","collinear factorization","generalized parton distributions","relativistic corrections","DLA-HEF"],"falsifier":"Perform a full O(α_s v²) fixed-order calculation of the same coefficient function and compare its small-ρ asymptotics with the expansion derived in the paper; the resummation predicts specific α_s and α_s² log coefficients through b1 and b2, so a discrepancy at that order would show the v² resummation is incomplete. Concretely, evaluating the ratio of the O(v²) to O(v⁰) resummed coefficients at ρ=10⁻³ in the PM scheme should approach −1/6; a different value from an explicit one-loop computation would refute the claim.","tokens_in":18523,"feed_emoji":"⚛️","tokens_out":7526,"duration_ms":73865,"temperature":0.7,"pith_summary":"This paper asks whether relativistic corrections—the O(v²) terms arising from the slow motion of the heavy quark and antiquark inside a quarkonium meson—change the high-energy resummed description of exclusive quarkonium photoproduction. It computes the O(v²) correction to the double-logarithmic high-energy factorisation (DLA-HEF) resummed coefficient function, the object that cures the scale-dependence instability of fixed-order collinear-factorisation calculations at small skewness. The central result is that, in the physical-mass (PM) scheme, the correction is numerically tiny at the natural scale μ_F=M_V and reduces, in the low-ρ limit, to a simple multiplicative factor 1−⟨v²⟩/6. The correction is nevertheless not irrelevant: it partially cancels the μ_F dependence of the previously known O(v²) leading-order correction, making predictions more stable. A sympathetic reader should care because this closes a gap between the existing next-to-leading-order collinear-factorisation calculation and the high-energy resummation, without introducing new free parameters.","feed_headline":"Small v² term stabilises quarkonium photoproduction scales","feed_subtitle":"The O(v²) resummed piece is tiny at μ_F=M_V yet cancels part of the scale dependence.","key_machinery":"The load-bearing object is the DLA-HEF resummed coefficient function in Mellin space, whose O(v²) correction factorises as (b0+b1 γ_N+b2 γ_N²) times the O(v⁰) resummed coefficient, where γ_N is the standard DLA anomalous dimension. The coefficients b0, b1 and b2 are not fitted: they are computed from the O(v²) correction to the process-dependent impact factor h(v2)(q_T²), with b0 equal to h(v2) at q_T²=0 and b1, b2 given by moment integrals over q_T². Because of this derivative structure, the O(v²) resummed correction can also be written as a second-order differential operator in ln μ_F² acting on the O(v⁰) resummed coefficient, which is what makes the partial μ_F cancellation explicit.","core_discovery":"The paper's central claim is that the O(v²) correction to the DLA-HEF resummed coefficient function can be computed from a one-loop impact factor, and that it is both computable and small in the physical-mass scheme. The velocity expansion of the resummed coefficient function factorises in Mellin space, with the O(v²) piece proportional to the O(v⁰) piece through coefficients b0, b1 and b2, which are fixed by the transverse-momentum shape of the O(v²) impact factor. In the PM scheme the relevant combination equals −1/6, so at small ρ the O(v²) resummed correction amounts to a downward shift of the O(v⁰) resummed coefficient by the factor 1−⟨v²⟩/6. Numerically, at μ_F=M_J/ψ and ⟨v²⟩≈0.25, the","pith_inferences":["If the pattern persists at next order, the O(α_s v²) fixed-order correction—once computed—may itself exhibit a similar partial cancellation, and the combined scale dependence could be even flatter than either piece alone; this is an inference, not shown in the paper.","A similar smallness in the PM scheme may hold for related exclusive processes, such as deeply virtual vector-meson production, where the same DLA-HEF machinery applies; this is an extension the paper does not make.","The concentration of h(v2,PM)(q_T²) at low q_T² suggests that the size of the correction is tied to the typical gluon transverse momentum in the hard scattering; a dipole- or CGC-based calculation that tracks the same low-q_T region should reproduce the −1/6 factor, providing a cross-check between formalisms.","The paper's all-order-in-v² appendix formula indicates that kinematic v² corrections at O(v⁴) are negligible for v²≲0.5, but genuine dynamical many-body effects could behave differently; quantifying them would require the O(α_s v²) computation."],"forward_implications":["Existing NLO collinear-factorisation predictions for J/ψ and Υ photoproduction can be supplemented with the O(v²) resummed piece without introducing new free parameters, since the PM scheme uses only M_V and ⟨v²⟩.","At small ρ the O(v²) correction acts as a simple 1−⟨v²⟩/6 multiplicative factor on the resummed coefficient, making the dominant high-energy contribution easy to estimate.","The μ_F dependence of the O(v²) correction to the LO coefficient function is partially cancelled, so cross-section predictions become less sensitive to the choice of factorisation scale.","No double counting arises when matching the O(v²) resummed piece to the current fixed-order result, because the O(α_s v²) fixed-order correction has not yet been computed; double counting would only appear once that correction is included.","The matching formula proposed in the paper offers a practical scheme for including these terms in future phenomenological analyses, with alternative matching prescriptions available to assess the residual matching uncertainty."],"fun_headline_variants":["v² correction tames scale dependence in quarkonium photoproduction","Tiny v² correction improves scale robustness in vector meson photoproduction","v² term cancels scale dependence in quarkonium photoproduction","O(v²) resummation improves scale behaviour for exclusive photoproduction"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The calculation assumes that the double-logarithmic high-energy resummation of the hard coefficient is consistent with using standard fixed-order evolution of generalized parton distributions; if that consistency fails in the low-ρ region, the partial μ_F cancellation claimed here could be partly an artifact of the mismatch.","fun_headline_variants_meta":{"raw":{"variants":["v² correction tames scale dependence in quarkonium photoproduction","Tiny v² correction improves scale robustness in vector meson photoproduction","v² term cancels scale dependence in quarkonium photoproduction","O(v²) resummation improves scale behaviour for exclusive photoproduction"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000588,"raw_usage":{"total_tokens":2586,"prompt_tokens":719,"completion_tokens":1867,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":463,"completion_tokens_details":{"reasoning_tokens":1788}},"tokens_in":463,"tokens_out":1867,"duration_ms":12859,"temperature":1.0,"reasoning_tokens":1788,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T00:31:21.702870+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Perform a full O(α_s v²) fixed-order calculation of the same coefficient function and compare its small-ρ asymptotics with the expansion derived in the paper; the resummation predicts specific α_s and α_s² log coefficients through b1 and b2, so a discrepancy at that order would show the v² resummation is incomplete. Concretely, evaluating the ratio of the O(v²) to O(v⁰) resummed coefficients at ρ=10⁻³ in the PM scheme should approach −1/6; a different value from an explicit one-loop computation would refute the claim.","supporting_citations":[],"review_version":1}