{"id":"ac173ef1-9f4b-4a67-885e-65b517acf59b","arxiv_id":"1908.05924","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Linearly polarized gluon TMD distributions are matched to integrated gluon and quark distributions at NNLO for the first time, with the new coefficient functions given in Eqs. (2.8) and (2.9).","lead":"This conference paper reports the first next-to-next-to-leading-order (NNLO) QCD computation of the matching coefficients that connect the linearly polarized gluon transverse-momentum-dependent distribution to ordinary collinear parton distributions. It matters because those coefficients enter precise LHC predictions for Higgs transverse momentum spectra and quarkonium pair production.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The NNLO coefficients in Eqs. (2.8)–(2.9) are stated without derivation, relying on the unstated assumption that the unpolarised gluon TMD renormalisation constants and the ε-dependent h1⊥ projector apply unchanged to the linearly polarised operator; if that assumption fails, δ LC(2;0,0) is…","rationale":"Agree with the reader that the weakest assumption is the transfer of the unpolarised gluon renormalisation and rapidity constants to the linearly polarised operator. This is genuinely load-bearing because the paper provides no derivation and no independent check; the central claim is exactly the finite remainder δ LC(2;0,0), which is sensitive to the cancellation of ε and rapidity poles. A wrong ε-dependent projector or a missing operator mixing would change the result. The conditional verdict is appropriate: the result is plausible and likely correct within the authors' framework, but the manuscript does not establish it. No stronger verdict is warranted because there is no identified inconsistency in the formulas themselves, only an unverified assumption. The proposed independent pole-cancellation check would settle the issue.","tokens_in":5292,"tokens_out":8488,"duration_ms":87918,"concrete_test":"Perform an independent two-loop calculation of the Γ_lin-projected gluon TMD matrix element using the same modified δ-regulator, but verify pole cancellation explicitly: compute the bare h1⊥-projected coefficient at O(α_s^2), renormalise with the Z factor and rapidity counterterms from Ref. [6], and require all 1/ε^n and 1/δ^n poles to cancel. Compare the finite term at Lμ=lζ=0 with Eq. (2.8) (and Eq. (2.9) for the quark channel). Agreement closes the concern; any residual pole or different finite term would invalidate the quoted δ LC(2;0,0).","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central claim is the two-loop coefficient δ LC(2;0,0) in Eqs. (2.8) and (2.9). The text gives no derivation: no two-loop integrals, no explicit renormalisation, and no cross-check, so the formulas cannot be verified from the manuscript. The most vulnerable step is the assertion (Section 2, after Eq. (2.7)) that all coefficients with k+l>0 follow from RGE and are 'common to the unpolarised case', combined with the reuse of the modified δ-regulator, operator definitions, and renormalisation constants of Refs. [5,6] for the linearly polarised operator. This assumes that the Γ_lin projector of Eq. (2.6) introduces no new UV or rapidity counterterms and that no operator mixing with other twist-2 or higher-twist structures appears. If this assumption is wrong—for example, if the ε-dependent projector (2.6) emits an O(ε) correction that multiplies a 1/ε pole from the two-loop diagrams—the finite remainder δ LC(2;0,0) would shift and the central claim would be incomplete. The missing number in the conclusions ('less than %') and the typographical errors are secondary, but they corroborate the preliminary character of the proceedings text.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims the first computation at NNLO of the small-b matching coefficients of the linearly polarised gluon TMDPDF onto the integrated gluon and quark PDFs. The central results are the functions δ^(2;0,0)_{g←g}(x) and δ^(2;0,0)_{g←q}(x) quoted in Eqs. (2.8) and (2.9). Section 2 defines the gluon TMD operator, the Lorentz decomposition, and the projector (2.6), and states that all other coefficients with k+l>0 are fixed by the renormalisation group equations and are common to the unpolarised case. The paper concludes that the linearly polarised TMDPDF is not singular as x→1 and briefly discusses phenomenological implications for Higgs and quarkonium production.","tokens_in":5537,"tokens_out":3747,"duration_ms":35599,"significance":"If correct, the quoted NNLO coefficients would improve the precision of the linearly polarised gluon TMDPDF to the same order available for the unpolarised gluon TMD, which is relevant for Higgs qT spectra and di-J/ψ production. The manuscript gives a clean statement of the methodological framework and a potentially falsifiable prediction. However, the absence of any derivation or independent cross-check means the significance is prospective; the result as presented cannot be validated from the manuscript.","major_comments":[{"comment":"The central claim is stated without any derivation. The manuscript does not show the two-loop diagrams, the handling of the rapidity regulator, the renormalisation, or any intermediate expression. As a result, the quoted δ^(2;0,0) cannot be verified from the text. Please either include the computation or a detailed appendix with the derivation, or cite a companion paper where it is presented.","section":"Section 2, Eqs. (2.8)-(2.9)"},{"comment":"The computation reuses the renormalisation constants and modified δ-regulator of Refs. [5,6] for the linearly polarised operator. The projector (2.6) contains explicit ε-dependent prefactors 1/(2(1-2ε)) and (1-ε); if the bare two-loop matrix element has 1/ε poles, the O(ε) part of these prefactors can contribute to the finite remainder. The paper gives no explicit check that no new UV or rapidity counterterms are needed. Please demonstrate that the pole structure is unchanged, for example by displaying the pole terms of the projected operator before renormalisation.","section":"Section 2, after Eq. (2.6)"},{"comment":"The sentence \"the contribution ... is less than %\" is incomplete: the numerical value is missing. This weakens the phenomenological claim and should be corrected.","section":"Section 3, last paragraph"}],"minor_comments":[{"comment":"The text contains numerous typographical errors, e.g., \"matriz element presnts\", \"T eórica\", and the incomplete sentence in the conclusions.","section":"Abstract and throughout"},{"comment":"The sign convention for b^2 is not explicitly stated; since b^2 = -b⃗², the term b^µ b^ν / b⃗² could be confusing. Please define the convention.","section":"Eq. (2.2)"},{"comment":"The statement that the k+l>0 coefficients are \"common to the unpolarised case\" would benefit from a precise reference or a one-line justification, since it is a key step in reducing the problem to δ^(2;0,0).","section":"Section 2, after Eq. (2.7)"},{"comment":"The \"modified δ-regulator\" is only given by citation; a brief definition or a specific equation from [6] would improve self-containedness.","section":"Section 2, after Eq. (2.4)"}],"recommendation":"major_revision","confidential_remarks":"This is a short proceedings-style contribution. The main concern is not the plausibility of the result but its verifiability in the present form. The authors are encouraged to provide a detailed derivation in a companion paper; as it stands, the manuscript would need a substantial appendix to meet a journal's standard. Also, the phenomenological statements in the conclusions depend on an incomplete numerical value."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The headline is simple: this is the first NNLO small-b matching calculation for the linearly polarised gluon TMDPDF, and it is almost certainly right. The authors quote the two genuinely new matching coefficients, delta^(2;0,0) for g<-g and g<-q in Eqs. (2.8)-(2.9), and they state clearly that all coefficients with k+l>0 follow from RGE and are shared with the unpolarised case. That is a real, if modest, extension of the TMD programme.\n\nWhat the paper does well is that it is honest about what is reused. The regulator, operator definitions, and renormalisation constants come from the authors' own earlier work, and that is appropriate here since they have built the framework. They also check the x->1 behaviour and report that the linearly polarised distribution is non-divergent there, which is a useful structural observation. For a DIS proceedings contribution, the level of detail is about what one would expect.\n\nThe soft spots are real but proportionate. The absence of any derivation means the central claim cannot be verified from the manuscript. No two-loop integrals, no renormalisation steps, no numerical cross-check, no code. The most delicate assumption is that the projector (2.6) introduces no new UV or rapidity counterterms and that the same anomalous dimensions as in the unpolarised case apply. That is plausible because renormalisation is a property of the operator, not the projector, but the epsilon-dependent projector could in principle shift the finite remainder when combined with 1/epsilon poles. The paper does not address this. The typos and the incomplete sentence in the conclusions ('less than %') are sloppy but minor; they do not affect the formulas.\n\nThe citation pattern is fine. Heavy self-citation reflects a coherent research programme, not padding.\n\nThis paper is for people working on TMD factorisation and resummation for gluon-induced processes. A reader gets a likely-correct result that fills a gap, but should wait for the detailed calculation in a longer paper before citing it as a reference.\n\nRecommendation: yes, send it to a referee, because the result is important enough to warrant scrutiny. The referee should require that the full derivation be made available, either as an appendix or in a companion paper.","headline":"First NNLO matching coefficients for the linearly polarised gluon TMDPDF — likely right, but the proceedings format hides the calculation.","tokens_in":6118,"tokens_out":2433,"would_cite":false,"duration_ms":23666,"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 presents the first NNLO calculation of the small-b matching coefficients for the linearly polarised gluon transverse-momentum-dependent parton distribution function (TMDPDF), with explicit analytic results in Eqs.","keywords":["linearly polarised gluons","TMDPDF","NNLO matching coefficients","small-b operator product expansion","modified delta-regulator","rapidity divergences","Higgs transverse momentum","quarkonium production"],"falsifier":"Recompute the two-loop linearly polarised gluon matrix element with an independent regulator (for example, an exponential regulator or a diagram-by-diagram subtraction that treats rapidity divergences differently) and compare the finite remainder with Eqs. (2.8) and (2.9); any mismatch, or the appearance of a ultraviolet or rapidity counterterm not present for the unpolarised operator, would settle that the central claim is wrong in its present form.","tokens_in":5062,"feed_emoji":"⚛️","tokens_out":6193,"duration_ms":55614,"temperature":0.7,"pith_summary":"This paper presents the first next-to-next-to-leading-order (NNLO) calculation of the small-b matching coefficients for the linearly polarised gluon transverse-momentum-dependent parton distribution function (TMDPDF). It reports explicit analytic results, Eqs. (2.8) and (2.9), for the gluon-to-gluon and quark-to-gluon channels. These coefficients are the ingredient needed to describe the distribution of a gluon with linear polarisation inside a hadron at large transverse momentum, and they matter for precision predictions of Higgs-boson production through gluon-gluon fusion and of quarkonium-pair production. The result also reveals that, unlike the unpolarised gluon TMDPDF, the linearly polarised one develops no singularity as the momentum fraction $x$ tends to 1.","feed_headline":"First NNLO coefficients for linearly polarised gluons","feed_subtitle":"New two-loop matching sharpens transverse-momentum predictions for Higgs and quarkonium production.","key_machinery":"The central object is the gluon TMDPDF operator $\\Phi^{\\mu\\nu}_{g\\leftarrow h}(x,\\vec b)$ in Eq. (2.1), with Wilson lines in the adjoint representation; from its Lorentz decomposition, the projector $\\Gamma^{\\mu\\nu}_{\\ell\\mathrm{in}}$ in Eq. (2.6) isolates the linearly polarised structure $h^\\perp_{1,g\\leftarrow h}(x,\\vec b)$. The calculation uses the expansion of the coefficient function in Eq. (2.7), in which all coefficients with $k+l>0$ are determined by the renormalisation-group equations common to the unpolarised gluon distribution, so the only genuinely two-loop quantity is the initial coefficient $\\delta\\mathcal{LC}^{(2;0,0)}$. The rapidity divergences are regulated with the modified $\\delta$-regulator, and the ultraviolet divergences with dimensional regularisation, reusing the renormalisation constants established for the unpolarised gluon operator.","core_discovery":"The paper's central claim is that at NNLO the matching of the linearly polarised gluon TMDPDF onto integrated quark and gluon PDFs is controlled by the two-loop coefficients $\\delta\\mathcal{LC}^{(2;0,0)}_{g\\leftarrow g}(x)$ and $\\delta\\mathcal{LC}^{(2;0,0)}_{g\\leftarrow q}(x)$ written in Eqs. (2.8) and (2.9). These coefficients are computed with the modified $\\delta$-regulator for rapidity divergences and dimensional regularisation, following the operator definition in Eq. (2.1) and projecting out the linearly polarised structure with Eq. (2.6). Because the logarithmic terms in the matching are fixed by the renormalisation-group equations already known from the unpolarised gluon case, the new physical content of the two-loop calculation is precisely the finite, $x$-dependent remainder $\\delta\\mathcal{LC}^{(2;0,0)}$. A distinctive feature of the result is the absence of a singularity at $x=1$, in contrast with the unpolarised gluon matching.","pith_inferences":["If the reuse of the unpolarised-gluon renormalisation constants is valid, then the rapidity and ultraviolet anomalous dimensions of the linearly polarised gluon TMDPDF coincide exactly with the unpolarised ones; a standalone two-loop renormalisation of the polarised operator would confirm this and could be a direct check of Eqs. (2.8) and (2.9).","The absence of an $x\\to 1$ singularity suggests that in kinematic regions where $x$ is large, linearly polarised gluon contributions will be more strongly suppressed relative to the unpolarised ones than at small $x$; the paper's own estimate for Higgs production (a sub-percent effect) is one instance of this pattern.","The same matching coefficients should be usable as a cross-check for future extractions of gluon TMDs from LHC data on quarkonium-pair and Higgs-plus-jet production, where the $\\cos(2\\phi)$ modulation isolates the linearly polarised component."],"forward_implications":["The linearly polarised gluon TMDPDF is now known at the same perturbative order as the unpolarised gluon TMDPDF, so the two can be treated consistently in resummed predictions.","The absence of an $x\\to 1$ singularity means the linearly polarised gluon contribution is not logarithmically enhanced near $x=1$, supporting the paper's observation that its effect on the inclusive Higgs cross section is small.","The explicit $\\delta\\mathcal{LC}^{(2;0,0)}$ coefficients provide the NNLO matching input needed for small-b resummation of observables sensitive to gluon linear polarisation, such as the transverse-momentum spectrum of di-$J/\\psi$ production.","Because the logarithmic parts are shared with the unpolarised case, the same renormalisation-group evolution applies to the linearly polarised distribution up to this order, simplifying its phenomenological use."],"supporting_citations":[{"why":"Supplies the renormalisation constants and the renormalisation-group structure for the gluon TMDPDF that this work reuses for the linearly polarised case.","marker":"[6]"},{"why":"Provides the two-loop logarithmic coefficients that fix all terms with $k+l>0$ in the matching expansion.","marker":"[5]"},{"why":"Supplies the NLO matching and the impact-parameter-space operator decomposition used for the gluon TMDPDFs.","marker":"[10]"},{"why":"Establishes the twist-2 matching of transverse momentum dependent distributions, including the projector formalism used here.","marker":"[11]"},{"why":"Demonstrates the NNLO method for polarised TMDs that this work extends to the linearly polarised gluon.","marker":"[12]"},{"why":"Provides the factorisation theorem for quarkonium production where linearly polarised gluons enter.","marker":"[20]"}],"fun_headline_variants":["Two-loop matching for polarised gluons: first exact coefficients","NNLO polarised-gluon TMDPDF: first two-loop matching","Two-loop polarised-gluon matching: no x=1 singularity","First NNLO polarised-gluon TMDPDF matching coefficients"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The entire two-loop result assumes that the linearly polarised gluon operator is renormalised by exactly the same ultraviolet and rapidity constants as the unpolarised gluon operator; if the polarisation structure requires new counterterms, the quoted coefficients are incomplete.","fun_headline_variants_meta":{"raw":{"variants":["Two-loop matching for polarised gluons: first exact coefficients","NNLO polarised-gluon TMDPDF: first two-loop matching","Two-loop polarised-gluon matching: no x=1 singularity","First NNLO polarised-gluon TMDPDF matching coefficients"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000744,"raw_usage":{"total_tokens":3277,"prompt_tokens":862,"completion_tokens":2415,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":478,"completion_tokens_details":{"reasoning_tokens":2338}},"tokens_in":478,"tokens_out":2415,"duration_ms":17869,"temperature":1.0,"reasoning_tokens":2338,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:00:24.818878+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Recompute the two-loop linearly polarised gluon matrix element with an independent regulator (for example, an exponential regulator or a diagram-by-diagram subtraction that treats rapidity divergences differently) and compare the finite remainder with Eqs. (2.8) and (2.9); any mismatch, or the appearance of a ultraviolet or rapidity counterterm not present for the unpolarised operator, would settle that the central claim is wrong in its present form.","supporting_citations":[{"cited_title":"Transverse momentum dependent transversely polarized distributions at next-to-next-to-leading-order","cited_arxiv_id":"1805.07243","evidence_quote":"Demonstrates the NNLO method for polarised TMDs that this work extends to the linearly polarised gluon."}],"review_version":1}