{"id":"3d5ea3d2-2b90-4aab-b948-3b8da4ce064f","arxiv_id":"2608.09621","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper delivers the first NNLO 0-jettiness soft function for hadronic top-quark pair production and generalises SoftSERVE to massive quarks.","lead":"The paper extends a numerical framework, SoftSERVE, so that two-loop soft corrections can be computed for processes with heavy quarks, and uses it to produce the first 0-jettiness soft function for top-quark pair production at the LHC. This matters because the function is a missing ingredient for precise top-quark predictions at the HL-LHC, needed for modern resummation and subtraction methods.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Tripole cancellation of the unknown O(epsilon^2) imaginary part of I34 is asserted, not demonstrated, and Eq. (4.5) depends on it.","rationale":"The reader identified exactly this assumption, and I agree it is the most fragile point. The paper is internally consistent, with two independent numerical implementations and RG pole checks that strongly support the method claim. However, the finite tripole coefficient - one of the two advertised new results - rests on an O(epsilon^2) contribution that is explicitly stated to be unavailable in the literature and whose cancellation is argued rather than derived. The RG comparison in Fig. 3 checks the pole structure, not the finite O(epsilon^2) piece of the current. Since the paper ships grids rather than the code or an explicit derivation of that piece, the scientific record would be incomplete without either the explicit O(epsilon^2) term of Im I34 or an independent derivation of the tripole cancellation. This does not undermine the method framework, which is validated by the massless-SoftSERVE comparison and the beta_t -> 1 limit, but it should gate acceptance on the addition of that derivation or on a numerical test that demonstrates the finite coefficient is insensitive to the omitted term.","tokens_in":28081,"tokens_out":1561,"duration_ms":13054,"concrete_test":"Recompute the real-virtual tripole sum (2.30) for two independent kinematic points (e.g., beta_t = 0.4, theta = 0.4 pi and beta_t = 0.4, theta = 0.9 pi), omitting the unknown O(epsilon^2) part of Im I34 entirely and then keeping it as an arbitrary constant times the phase-space pole structure. If the extracted c_tri^(2) shifts by more than the quoted grid uncertainties, the Section 2.4.1 cancellation claim is load-bearing and needs an explicit proof or an independent derivation of the O(epsilon^2) term.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central new quantity is the NNLO tripole coefficient c_tri^(2), which enters Eq. (3.17) and is reported in Eq. (4.5)/(4.6) and the numerical grids. Its computation relies on the Section 2.4.1 claim that the O(epsilon^2) piece of Im I34(epsilon) (not present in the literature, as the authors state) is tripole-independent in the soft-collinear limit and cancels in the sum (2.30) after the symmetry combination S(2,Im)(theta) - S(2,Im)(pi-theta). The argument is qualitative: no formula for the O(epsilon^2) coefficient is given, and the claimed independence from the tripole kinematics is only asserted. Because this piece multiplies the 1/epsilon and 1/epsilon^2 phase-space poles, it contributes to the finite tripole coefficient, so an error here would directly change c_tri^(2) (e.g., the 16 pi cos(theta) ln(beta_t)/epsilon and 8 pi cos(theta)(2 ln^2(beta_t) - 4 ln^2 2 + pi^2) terms of Eq. (4.5)/(4.6)). The RG pole cancellation shown in Fig. 3 verifies the 1/epsilon^n structure of the full tripole sum, but it does not pin down the O(epsilon^2) piece of Im I34, because that piece can shift finite terms while leaving the poles unchanged. This is a genuine soft spot in an otherwise strong paper.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper extends the SoftSERVE framework to compute NNLO soft functions for processes with final-state heavy quarks in back-to-back kinematics, focusing on the 0-jettiness soft function for hadronic top-quark pair production. The authors introduce phase-space parameterisations in tailored frame choices for massive and mixed massless-massive dipoles, provide master formulae for the NLO and NNLO (real-virtual, double-real gluon, and double-real quark-antiquark) contributions, and implement the calculation in two independent codes (SoftSERVE and pySecDec). They describe the renormalisation in the combined SCET+HQET framework, verify numerically that the bare poles match RG predictions, and present numerical grids for the dipole and tripole coefficients, including analytic threshold expansions for the tripole sum.","tokens_in":28346,"tokens_out":11949,"duration_ms":90819,"significance":"If correct, the paper is a significant methodological and phenomenological contribution: it generalises a public automated tool to heavy-quark final states, supplies the first NNLO 0-jettiness soft function for top-quark pair production, and provides an essential ingredient for NNLL' resummation and jettiness-slicing applications. The paper is careful in cross-checking the calculation: two independent numerical implementations agree, the pole structure is verified against RG predictions, and analytic threshold behaviour is given for the tripole sum. The main weakness is the not fully demonstrated cancellation of the O(epsilon^2) imaginary part of the one-loop massive soft current, which affects the finite tripole coefficient.","major_comments":[{"comment":"The cancellation of the O(epsilon^2) piece of Im I_34(epsilon) is asserted rather than demonstrated. The paper states that this piece is not provided in [34,35] and argues that it cancels in the tripole sum (2.30) because it becomes independent of the tripole kinematics in the soft-collinear limit. No explicit formula for this coefficient or a proof of the cancellation is given. Since this piece multiplies the 1/epsilon^2 and 1/epsilon phase-space poles, it contributes directly to the finite tripole coefficient c_tri^(2) in Eq. (3.17) and to the reported threshold result (4.6), e.g. the terms 16 pi cos(theta) ln(beta_t)/epsilon and 8 pi cos(theta) (2 ln^2(beta_t) - 4 ln^2 2 + pi^2). The RG pole check in Fig. 3 does not constrain this contribution because it only verifies the pole structure, which is insensitive to the O(epsilon^2) term. Please provide a derivation of the O(epsilon^2) coefficient of I_34, or a rigorous argument for its cancellation, and quantify any residual contribution to c_tri^(2).","section":"Section 2.4.1 (Eqs. (2.28)-(2.30))"},{"comment":"The threshold expansions for the bare tripole sum and for c_tri^(2) are presented as analytic results, but no derivation is shown. The text says the leading behaviour was 'carefully extracted analytically', yet the reader cannot verify the coefficients of ln^2(beta_t) and ln(beta_t), or the constant pi^2. Given that these expansions are used to control the endpoint region of the grids and are highlighted in the abstract, please include a derivation sketch (or a reference) and state any assumptions about the order of limits (epsilon -> 0 vs beta_t -> 0).","section":"Section 4 (Eqs. (4.5)-(4.6))"}],"minor_comments":[{"comment":"The phrase 'the non-Abelian exponentiation contribution. i.e.' contains a typo (period instead of comma).","section":"Section 2.4.2 (after Eq. (2.31))"},{"comment":"The numerical grids are provided as ancillary files; please ensure the file format and normalisation conventions are documented, and that the files are actually submitted with the arXiv posting.","section":"Section 4"},{"comment":"The notation f_B(y_k,t_k) in the integrand is the result of a remapping from the original f_B(y_k^{-1},t_k); a brief explanatory sentence would avoid confusion.","section":"Section 2.3 (Eq. (2.21))"},{"comment":"The index ranges in the tripole sums, written as k!=l,J and K!=I,j, are not standard; please clarify that k,l run over massless legs and I,J over massive legs, and specify the summation domains.","section":"Section 3.1 (Eq. (3.13))"},{"comment":"The dotted line is described as a small-beta_t approximation for the 1/epsilon coefficient; please state explicitly whether it includes only the ln(beta_t) term or also the constant term, to aid interpretation.","section":"Section 4 (Fig. 3 left)"}],"recommendation":"major_revision","confidential_remarks":"The paper is a strong computational contribution with careful cross-checks. The main concern is the unsupported tripole cancellation in Section 2.4.1, which directly affects the finite tripole coefficient. I would ask the authors to provide a full proof or a direct computation of the O(epsilon^2) term before acceptance. This is fixable within the scope of the manuscript, so I recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: this is a serious, workmanlike paper that delivers the first NNLO 0-jettiness soft function for hadronic top-pair production and a general framework for massive final-state quarks in SoftSERVE. The two independent implementations (SoftSERVE extension and pySecDec) and the numerical pole checks against RG predictions are real evidence; the analytic threshold expansions for the tripole sum are a nice addition. The dipole results are new, the tripole results are new, and the presentation of the renormalisation in the combined SCET+HQET framework is careful and complete.\n\nWhat is actually new: the extension to time-like massive Wilson lines and mixed massless-massive dipoles, the master formulas, and the numerical grids for the coefficient functions. The measured pole cancellation and the consistency relations give me confidence the singular structure is right.\n\nSoft spots: the main one is the Section 2.4.1 claim about the O(epsilon^2) imaginary part of the one-loop massive soft current I34. The paper says this piece is not in the literature, then argues that it becomes tripole-independent in the soft-collinear limit and cancels in the tripole sum (2.30). That argument is qualitative; no formula for that coefficient is given, and the RG pole comparison in Fig. 3 does not pin it down because a constant shift in that O(epsilon^2) piece would change the finite c_tri^(2) without disturbing the poles. I don't think this is a fatal flaw, but it's a genuine gap and a referee should ask the authors to provide an explicit argument or an analytic check. A second softer point: the finite NNLO coefficients are numerical only, and the grids are in ancillary files, so reproducibility is weaker than if the code or analytic results were shipped. That's not a demand, just a proportionality note.\n\nOverall: this is a paper that deserves a serious referee. If the tripole cancellation is confirmed, the results stand and will be used for NNLL' resummation and slicing. I'd send it to review.","headline":"First NNLO 0-jettiness soft function for top pairs, with a solid framework and one tripole-cancellation claim that deserves a closer look.","tokens_in":28914,"tokens_out":1908,"would_cite":true,"duration_ms":17553,"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":"This paper generalises the SoftSERVE numerical framework to heavy-quark final states and computes the first NNLO 0-jettiness soft function for hadronic top-quark pair production as grids in top-quark velocity and scattering angle.","keywords":["NNLO soft functions","heavy-quark pair production","0-jettiness","top-quark pair production","SoftSERVE","non-Abelian exponentiation","tripole colour correlations","SCET"],"falsifier":"Evaluate the $O(\\epsilon^2)$ contribution to the imaginary part $I_{34}(\\epsilon)$ of the massive one-loop soft current and insert it into the tripole sum (2.30); if the result is not independent of that term, the NNLO tripole coefficient $c_{\\mathrm{tri}}^{(2)}$ in (3.17) differs from the reported value. A cheaper check is to recompute the bare tripole poles with an independent implementation that keeps the one-loop current without truncating at $O(\\epsilon)$ and compare the finite part against the published grids.","tokens_in":27847,"feed_emoji":"⚛️","tokens_out":12540,"duration_ms":99211,"temperature":0.7,"pith_summary":"Soft functions are the low-energy pieces of QCD factorisation theorems, and until now their NNLO automated computation has been largely confined to processes with only massless partons. This paper extends the SoftSERVE numerical framework to final-state heavy quarks in back-to-back kinematics and applies it to hadronic top-quark pair production, producing the first NNLO 0-jettiness soft function for that process as grids over the top-quark velocity and scattering angle. The paper verifies that the poles of the bare soft function match renormalisation-group predictions, and it derives the threshold behaviour of the tripole colour correlations analytically. These results are the missing ingredient for NNLL′ resummation of 0-jettiness in top-pair production and for NNLO jettiness-slicing calculations at the LHC.","feed_headline":"Top-pair production gets its NNLO soft function","feed_subtitle":"Numerical grids for 0-jettiness enable NNLL′ resummation and NNLO slicing at the LHC.","key_machinery":"The carrying object is a set of dipole-frame phase-space master formulae. For each of the three dipole types -- massless-massless, massive-massive, and mixed massless-massive -- the paper chooses or boosts to a frame in which both emitting directions sit on the z-axis and parametrises the soft momenta by light-cone components and a single transverse angular variable; for the quark-antiquark channel the mixed and massive kernels are the massless kernel times velocity-dependent prefactors, while the observable-dependent measurement function absorbs the remaining angular dependence. The tripole machinery is the colour-sum identity that reduces the six independent tripoles to one structure, together with a Baker-Campbell-Hausdorff-based renormalisation that turns the bare poles into finite coefficients. Non-Abelian exponentiation supplies the key simplification: quadrupole correlations and some tripoles are not computed directly but follow from exponentiating the NLO soft function.","core_discovery":"This paper claims that the SoftSERVE numerical approach to NNLO soft functions, previously limited to massless partons, carries over to final-state heavy quarks in back-to-back kinematics, provided each emitting dipole is treated in a coordinate frame aligned with its two directions. In that frame the squared matrix elements reduce to compact kernels and all non-trivial angular dependence is pushed into the measurement function, so the phase-space integrals stay manageable. Working in Soft-Collinear Effective Theory and assuming non-Abelian exponentiation, the authors compute the renormalised 0-jettiness soft function for hadronic top-quark pair production at NNLO for the first time, delivering the independent dipole and tripole colour coefficients as two-dimensional grids in the top-quark velocity and scattering angle. They further show that the tripole colour sum, which cannot be renormalised dipole by dipole, has only a logarithmic divergence as the top-quark velocity tends to zero because the Coulomb-like $1/\\beta_t$ singularities cancel in the sum.","pith_inferences":["The asserted cancellation of the $O(\\epsilon^2)$ imaginary part of the massive one-loop current in the tripole sum, if correct, suggests a general economy: the same cancellation might remove uncalculated current high-order terms for other heavy-quark dipole configurations, worth checking explicitly.","The numerical grids could be turned into a fast parametrisation by matching the derived small-$\\beta_t$ logarithms, which would make the soft function easy to embed in Monte Carlo event generation.","A direct offshoot test would be to evaluate $I_{34}(\\epsilon)$ to $O(\\epsilon^2)$ numerically and compare the full tripole sum with and without that term; agreement would confirm the paper's completeness argument beyond the pole-cancellation checks."],"forward_implications":["The 0-jettiness soft function for hadronic top-pair production is now available at NNLO, so NNLL′ resummation and NNLO slicing for this process have the missing ingredient.","The framework applies to any Soft-Collinear Effective Theory I observable that obeys non-Abelian exponentiation, not just 0-jettiness, so other event-shape soft functions with heavy quarks become accessible.","Because colour structures were kept generic rather than tied to a specific colour basis, a subset of the dipole results also serve top-pair production at lepton colliders and single-top production.","The analytic small-velocity expansion of the tripole sum lets users interpolate reliably between the grid points and the threshold endpoint."],"supporting_citations":[{"why":"Establishes the two-loop anomalous dimensions of generic dijet soft functions on which the SoftSERVE renormalisation machinery is built.","marker":"[10]"},{"why":"Supplies the correlated-emission master formulae and measurement-function conventions used throughout the double-real and real-virtual phase-space integrals.","marker":"[11]"},{"why":"Provides the generic massless-dipole parameterisation and NNLO soft-function master formulae that the paper adapts to massive emitters by frame choice.","marker":"[13]"},{"why":"Derives the 0-jettiness factorisation and resummation for top-pair production and gives the NLO soft function that anchors the colour conventions.","marker":"[31]"},{"why":"Gives the simplified one-loop soft-gluon current with massive fermions whose real and imaginary parts enter the real-virtual tripole integrals.","marker":"[34]"},{"why":"Provides the singular expansion of the massive one-loop soft current, including the O(epsilon^2) imaginary part whose cancellation in the tripole sum is argued in Section 2.4.1.","marker":"[35]"},{"why":"Supplies the tree-level double-real QCD matrix elements whose kernels enter the quark-antiquark and gluon phase-space master formula.","marker":"[36]"},{"why":"Provides the double-real radiation matrix elements for massive-quark pairs used for the gluonic channel kernels with non-trivial mass corrections.","marker":"[37]"},{"why":"Provides the two-loop hard anomalous dimension for massive partons from which the soft anomalous dimension is extracted via consistency relations.","marker":"[39]"},{"why":"Defines the function g(beta_IJ) and the massive cusp anomalous dimension that govern the genuine two-loop tripole contribution to the soft anomalous dimension.","marker":"[40]"}],"fun_headline_variants":["First NNLO soft function for top-quark pairs","Top pairs get NNLO soft function grids","SoftSERVE goes heavy: NNLO soft functions","Heavy-quark NNLO soft function from SoftSERVE","NNLO soft functions for hadronic top pairs"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole NNLO tripole result depends on the claim, made in Section 2.4.1 without an explicit formula, that the uncomputed $O(\\epsilon^2)$ piece of the imaginary part of the one-loop massive soft current becomes independent of the tripole's kinematics in the soft-collinear limit and therefore cancels in the tripole sum; if that cancellation fails, the finite tripole coefficient changes.","fun_headline_variants_meta":{"raw":{"variants":["First NNLO soft function for top-quark pairs","Top pairs get NNLO soft function grids","SoftSERVE goes heavy: NNLO soft functions","Heavy-quark NNLO soft function from SoftSERVE","NNLO soft functions for hadronic top pairs"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00097,"raw_usage":{"total_tokens":4086,"prompt_tokens":869,"completion_tokens":3217,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":485,"completion_tokens_details":{"reasoning_tokens":3140}},"tokens_in":485,"tokens_out":3217,"duration_ms":21576,"temperature":1.0,"reasoning_tokens":3140,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T13:56:58.246714+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Evaluate the $O(\\epsilon^2)$ contribution to the imaginary part $I_{34}(\\epsilon)$ of the massive one-loop soft current and insert it into the tripole sum (2.30); if the result is not independent of that term, the NNLO tripole coefficient $c_{\\mathrm{tri}}^{(2)}$ in (3.17) differs from the reported value. A cheaper check is to recompute the bare tripole poles with an independent implementation that keeps the one-loop current without truncating at $O(\\epsilon)$ and compare the finite part against the published grids.","supporting_citations":[{"cited_title":"Two-loop anomalous dimensions of generic dijet soft functions","cited_arxiv_id":"1805.12414","evidence_quote":"Establishes the two-loop anomalous dimensions of generic dijet soft functions on which the SoftSERVE renormalisation machinery is built."},{"cited_title":"A simplified expression for the one-loop soft-gluon current with massive fermions","cited_arxiv_id":"1804.02069","evidence_quote":"Gives the simplified one-loop soft-gluon current with massive fermions whose real and imaginary parts enter the real-virtual tripole integrals."},{"cited_title":"The singular behavior of one-loop massive QCD amplitudes with one external soft gluon","cited_arxiv_id":"1107.4384","evidence_quote":"Provides the singular expansion of the massive one-loop soft current, including the O(epsilon^2) imaginary part whose cancellation in the tripole sum is argued in Section 2.4.1."}],"review_version":1}