{"id":"af544517-c9e4-4e06-a883-9738de752df8","arxiv_id":"2511.23423","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"Adding leading-power quark-mass effects makes standard anti-kT b-jet flavour tagging infrared-collinear-safe through NNLO without changing jet or flavour definitions, up to power corrections.","lead":"This paper shows that heavy-quark jet cross sections at NNLO can be made well-defined with the usual jet algorithms by adding back the quark mass at leading power. The method reproduces a full massive-quark calculation up to small corrections, so experiments need not switch to new flavour-safe jet definitions.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"KLN-based omission of final-state collinear/fragmentation logs is the load-bearing step; it is asserted via one NLO example rather than proved for all singular limits of a generic jet algorithm.","rationale":"The reader identified the KLN-based omission of fragmentation-function matching as the weakest assumption. I agree: the central factorization in eq. (2.1) is asserted by analogy rather than derived, and its correctness hinges on the complete cancellation of final-state collinear (and soft-collinear) mass logarithms in the flavour modulo-2 scheme. The paper's supporting argument is illustrative rather than exhaustive, and the numerical tests, while strong, are consistency checks of the assumed factorization rather than independent verifications of the KLN cancellation for arbitrary QCD configurations. The e+e- test is in QED with a trivial colour structure, so it does not probe multi-parton QCD collinear limits. Thus the concern is real, but it is not demonstrated to be an actual failure; the existing evidence is plausible and the central claim passes initial scrutiny. The verdict CONDITIONAL is appropriate: the method is promising but the KLN premise needs a sharper proof or a dedicated numerical test. I therefore set verdict_should_be to UNCHANGED.","tokens_in":27027,"tokens_out":27347,"duration_ms":302386,"concrete_test":"Perform an independent calculation of the NNLO massless cross section for a simple QCD process (e.g., e+e- -> 2 jets or pp -> Z + jet) in the flavour modulo-2 scheme with anti-kT, including the S-function subtraction but omitting the S-function insertion, and extract the leftover 1/epsilon pole as a function of jet pT and R. Compare its coefficient to the known analytic double-soft bb-pair pole (from Iij(m)-Iij(0) integrals). If the pole matches in all bins and channels and no additional collinear pole appears, the KLN cancellation is verified; if the pole contains an R-dependent or process-dependent piece not attributable to the double-soft limit, the KLN assumption fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim in eq. (2.1) is that the massless cross section in the flavour modulo-2 scheme, after adding the double-soft S functions, reproduces the massive result at leading power. This requires that every final-state collinear (and soft-collinear) mass logarithm cancels under KLN, so that no perturbative fragmentation-function matching is needed. The paper's argument in Section 2 ('The case of collinear final state logs ... can be omitted entirely') only checks two NLO pairs (g->bb vs. gg loop, b->bg vs. b loop) and then states 'By the KLN-theorem, this holds to any order.' This is not a proof for arbitrary jet algorithms and measurements. In particular, a generic jet measurement F is insensitive to the distribution of collinear momentum only if the collinear products are guaranteed to be clustered into a single jet with the same total momentum; this can fail for configurations near the jet boundary or when a soft quark is collinear to a hard parton while the other quark is at wide angle, a configuration not covered by the given NLO examples. If such a configuration produces a non-cancelling collinear (or soft-collinear) mass logarithm, the S-function-only scheme would miss it and the equality in eq. (2.1) would be violated. The paper's numerical checks (pole cancellation, A/B/R independence, analytic-continuation independence) demonstrate consistency of the implementation with the assumed factorization, but they do not independently verify that the only uncancelled pole is the double-soft one; the e+e- test is in QED with a trivial colour structure and does not exercise the potentially problematic QCD multi-parton collinear limits. The factorization itself is asserted by analogy, and the KLN condition is the load-bearing assumption.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a leading-power 'massification' prescription to render standard jet flavour definitions IRC-safe at NNLO. Starting from the observation that the only uncancelled singularities for flavour modulo-2 jets are soft bb-pairs, it defines process-independent soft functions Sbb and S∅ (Eqs. (2.1)-(2.6)) that subtract the massless double-soft limit and add the massive one. Final-state collinear/fragmentation mass logs are claimed to cancel by KLN under modulo-2 flavour, so no FF matching is needed. The implementation in Stripper is tested by A/B/R parameter independence, cancellation of leftover ε poles, independence of the d-dimensional continuation, and a comparison of the LP sum with a fully massive e+e-→ttbar-like calculation; phenomenological NNLO results are presented for Z+b-jet and single-inclusive b-jet production. The paper claims that, up to power corrections, the massless calculation plus S functions reproduces the fully massive cross section.","tokens_in":27381,"tokens_out":15703,"duration_ms":163192,"significance":"Should the leading-power factorization hold, this is a significant practical advance: it allows NNLO predictions for b-jets with standard anti-kT clustering and modulo-2 flavour, removing the need for modified jet/flavour definitions or unfolding to theory-specific schemes. The numerical checks in Section 3 are genuinely strong: the cancellation of the naive ε pole, the parameter-independence of the UV subtraction, and the e+e-→ttbar massive-match test are nontrivial and provide real support for the implementation. The phenomenological studies in Section 4 also illustrate important limitations, especially power corrections. However, the central factorization relies on an unproven KLN cancellation of final-state collinear mass logs for generic jet algorithms, so the significance is conditional.","major_comments":[{"comment":"The omission of all perturbative fragmentation-function matching is the load-bearing step of the paper. The argument checks two NLO pairs (Figs. 1 and 3) and then concludes, 'By the KLN-theorem, this holds to any order.' KLN guarantees cancellation only for sums over complete sets of degenerate states; it is not automatic for the flavour-sensitive, jet-clustered observable defined by F in Eq. (2.1). In particular, configurations in which the two soft quarks are each collinear to a different hard parton, so that each jet's modulo-2 flavour changes, are not among the two NLO examples. If such configurations produced a non-cancelling ln(m_b^2), Eq. (2.1) would miss it. The numerical tests in Section 3 are consistency tests of the implementation (pole cancellation, A/B/R independence, analytic-continuation independence); the e+e-→ttbar-like test in Fig. 7 uses Durham, not anti-kT, and the ma","section":"Section 2, 'The case of collinear final state logs ... can be omitted entirely'"},{"comment":"The completeness of the operator set in Eq. (2.1) is assumed rather than derived. Sbb is defined as the massive-minus-massless double-soft limit, and S∅ is fixed by the sum rule Eq. (2.3). This construction ensures that the double-soft singularity is removed by construction, but it does not by itself exclude other leading-power mass-logarithmic corrections at NNLO (e.g. one-loop soft or collinear-soft regions not captured by Sbb/S∅). The numerical checks demonstrate the internal consistency of the implementation and, in Fig. 7, agreement with one fully massive process, but they are not a derivation of the factorization. I would ask the author to state this distinction explicitly and, if possible, to provide a power-counting argument for why the S functions and the standard PDF/α_s decoupling terms exhaust the leading-power mass logarithms through NNLO.","section":"Section 2, Eqs. (2.3)-(2.6) and Section 3"}],"minor_comments":[{"comment":"The phrase 'IRC-safe jet flavour without modifying anything' overstates the change: the method adds a new S-function contribution and uses the flavour modulo-2 scheme. The more precise claim in the body, 'without modifying the jet definition, the flavour definition, or the cross section', is preferable and should be reflected in the title/abstract.","section":"Title and Abstract"},{"comment":"The caption should state explicitly that the orange curves are the sum multiplied by 10, and that the plotted differences are for a single choice of A,B,R, since this affects how the cancellation should be read.","section":"Section 3, Fig. 6"},{"comment":"The paper claims applicability to all common jet algorithms but only presents numerical tests for anti-kT and Durham. A sentence explaining why kT and Cambridge-Aachen are covered by the same KLN and soft-factorization arguments would be useful.","section":"Section 2 and 3"},{"comment":"Ref. [17] has a malformed DOI ('10.1103/b6pf-rj4h'); please correct it.","section":"References"},{"comment":"The arguments of Sbb are written pb, pb without distinguishing the quark and antiquark; using e.g. pb and pbbar would avoid confusion in the phase-space integrations.","section":"Notation in Eq. (2.1)"}],"recommendation":"major_revision","confidential_remarks":"The paper is well-written and the numerical checks are impressive, but the KLN assertion is the main point that prevents me from recommending acceptance. I would be willing to accept after the author provides either a systematic argument for the cancellation of all final-state collinear mass logs under generic jet algorithms, or an additional NNLO anti-kT comparison against a fully massive calculation for a process sensitive to the double-collinear configuration."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Briefly: this paper makes a strong case that the standard anti-kT jet definition with flavour modulo-2 can be used at NNLO for b-jet cross sections without modifying the jet algorithm, provided one adds a new set of leading-power soft massification terms (the S functions). The argument is plausible and the numerical checks are the most convincing part: the cancellation of leftover ε poles, independence of the UV regulator parameters, independence of the analytical continuation beyond four dimensions, and the e+e- → ttbar-like test in which the LP sum quantitatively reproduces a fully massive NNLO calculation. That test is a direct, nontrivial check of the whole factorisation idea, not just a consistency check of the code. The two LHC applications (Z+b-jet and inclusive b-jet) are also sensible, and the comparison against fully massive NLO results in the small-mass limit supports the claim that the LP approximation is doing what it should.\n\nThe main soft spot is the KLN-based claim that all final-state collinear and soft-collinear mass logarithms cancel when flavour modulo-2 is used, so that perturbative fragmentation-function matching can be omitted entirely. The paper demonstrates this at NLO with two pairs of examples and then invokes KLN to all orders. That is a reasonable physics argument, and I suspect it is correct for any jet algorithm that clusters exactly-collinear partons in the same jet, but it is not a proof for arbitrary measurements. In particular, the case of soft quarks near a jet boundary or double-soft configurations with one quark collinear to a hard parton and the other at wide angle is not explicitly covered. A referee should push on this. If the cancellation is incomplete for some standard algorithm, the massified result would miss collinear mass logarithms and the method would fail in a regime the current tests do not cover.\n\nOther issues are more minor. The S functions are constructed as the difference between the massive and massless double-soft limits, so some of the mass dependence enters by construction; that is fine, but the division between what is new and what is built into the operator should be stated more sharply. The implementation is not public, which limits reproducibility, though the numerical tests mitigate that. The paper also honestly reports that power corrections in the quark mass can be 5–10% at low pT; this does not invalidate the approach but does temper the 'leading power' framing.\n\nBottom line: this is the first practical proposal I know that keeps standard anti-kT and flavour modulo-2 at NNLO without changing the jet or flavour definition and with no experimental changes. It deserves serious peer review. I would send it to referees with a specific request to check the KLN cancellation in collinear configurations of a generic jet algorithm.","headline":"A promising method that makes standard anti-kT plus flavour modulo-2 usable at NNLO by adding soft massification terms; strong numerical checks, but the KLN-based cancellation of collinear logs is argued rather than proven and deserves referee scrutiny.","tokens_in":27883,"tokens_out":3413,"would_cite":true,"duration_ms":39354,"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":"The paper claims that standard flavoured-jet cross sections can be made infrared-collinear safe at NNLO without changing the jet or flavour definitions, by adding back quark-mass effects through process-independent soft functions.","keywords":["jet flavour","infrared-collinear safety","NNLO QCD","heavy-quark mass","massification","soft logarithms","flavour modulo-2 scheme","anti-kT jets"],"falsifier":"A fully massive NNLO computation of single-inclusive b-jet production at LHC energies, compared bin-by-bin with the massless-plus-S prediction: any difference larger than expected power corrections (order m_b/pT) would falsify the claim. A lighter-weight check is to verify explicitly, for a triple-collinear configuration involving a gluon and a soft b-quark pair, that the collinear divergence cancels locally in the flavour modulo-2 scheme without an S-function insertion.","tokens_in":26926,"feed_emoji":"⚛️","tokens_out":7176,"duration_ms":63836,"temperature":0.7,"pith_summary":"Standard b-jet tagging, with jets clustered by the usual anti-kT algorithm and flavour assigned by whether a jet contains an odd number of b quarks, stops being infrared-collinear safe at NNLO when b quarks are treated as massless. The paper argues this is an artefact of dropping the quark mass, and that adding the leading-power quark-mass dependence back through process-independent soft functions restores the safe, fully massive cross section up to power corrections. The payoff is that neither theory nor experiment changes its working definitions: massless NNLO calculations plus a soft correction match measurements made with standard jet clustering and the usual flavour unfolding. The paper demonstrates the mechanism for b-jets in lepton collisions, single-inclusive b-jet production at the LHC, and Z-plus-b-jet production, and finds the mass logarithms are small enough not to break perturbative convergence through NNLO, while power corrections can be larger than those logarithms.","feed_headline":"Massless-quark jet cross sections restored to massive accuracy at NNLO","feed_subtitle":"Adding quark-mass soft logs to massless NNLO results reproduces the massive b-jet cross section without new jet definitions.","key_machinery":"The load-bearing object is the set of soft massification functions S, which factorise the leading-power quark-mass dependence out of the cross section. Their key element is the double-soft function for a quark-antiquark pair, built from the universal soft-gluon factors attaching the pair's radiation to the hard partons; it controls how an infinitesimally soft b-quark pair is deposited into the final state and how the jet-clustering then counts flavour. The no-pair function S∅ is not computed directly but obtained by the unitarity condition that all S functions add to one. Final-state collinear mass logarithms are argued to cancel by the standard theorem for cancellation of mass singularities","core_discovery":"The central claim is captured by a factorisation of leading-power quark-mass effects: the massive-quark cross section equals the massless-quark cross section convoluted with a sequence of soft functions S∅, Sbb, Sbbbb, … that insert infinitesimally soft b-quark pairs into the final state. At NNLO only the no-insertion function S∅ and the one-pair function Sbb are needed; Sbb is determined by the universal double-soft limit of a massive quark-antiquark pair, and S∅ is fixed by the requirement that all the soft functions sum to the identity, so that flavour-blind cross sections are mass-blind. Adding these terms cancels the leftover divergence that standard flavour tagging acquires at NNLO, tu","pith_inferences":["Inference: if the cancellation of final-state collinear mass logarithms were ever found incomplete for some configuration of a standard jet algorithm, the scheme would miss collinear mass logarithms and the advertised equality with the massive cross section would fail; this is the assumption most worth testing.","Inference: the same soft-function language suggests a path to resummation of the non-global flavour-changing logarithms by iterating strict soft limits in a parton-shower style, which the paper sketches but does not develop.","Inference: extending the construction to strange quarks would require non-perturbative soft functions fitted to data, a possibility the paper names but leaves unbuilt; this would turn the method from a perturbative fix into a hadronisation-level description.","Inference: the paper's finding that power corrections can exceed the mass logarithms implies that at low jet pT or high precision the practical endgame is to compute massive cross sections anyway, with the massless-plus-S approach serving mainly as the high-pT or higher-order shortcut."],"forward_implications":["A massless NNLO calculation for flavoured jets, supplemented by the S-function contribution, becomes a leading-power replacement for a fully massive calculation up to power corrections.","Standard anti-kT jets with flavour modulo-2 tagging become IRC-safe at NNLO without changing the jet definition, the flavour definition, or the cross section.","Experimental analyses keep standard clustering and the usual unfolding from any-flavour to flavour modulo-2; no new algorithm or parameter needs to be adopted.","The soft mass logarithms at NNLO scale as αs² ln(m_b²/Q²), are numerically small, and do not spoil perturbative convergence through NNLO.","The construction extends beyond NNLO; at N3LO only one-loop integrations in the soft functions are needed, a minor cost compared to full massive amplitudes."],"fun_headline_variants":["Soft logs turn massless b-jet NNLO into massive accuracy","Massless to massive: NNLO b-jets via soft quark logs","NNLO b-jets: add soft logs, avoid new jet definitions","Leading-power mass effects restore massive b-jet at NNLO","Without new jets: soft logs give massive b-jet NNLO"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The construction assumes that, in the flavour modulo-2 scheme with a standard jet algorithm, all final-state collinear and soft-collinear mass logarithms cancel by the standard cancellation theorem for mass singularities, so that only the soft S functions need to be added; the paper asserts this on the basis of an argument rather than proving it.","fun_headline_variants_meta":{"raw":{"variants":["Soft logs turn massless b-jet NNLO into massive accuracy","Massless to massive: NNLO b-jets via soft quark logs","NNLO b-jets: add soft logs, avoid new jet definitions","Leading-power mass effects restore massive b-jet at NNLO","Without new jets: soft logs give massive b-jet NNLO"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000673,"raw_usage":{"total_tokens":2942,"prompt_tokens":826,"completion_tokens":2116,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":570,"completion_tokens_details":{"reasoning_tokens":2024}},"tokens_in":570,"tokens_out":2116,"duration_ms":12874,"temperature":1.0,"reasoning_tokens":2024,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T19:30:59.808541+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A fully massive NNLO computation of single-inclusive b-jet production at LHC energies, compared bin-by-bin with the massless-plus-S prediction: any difference larger than expected power corrections (order m_b/pT) would falsify the claim. A lighter-weight check is to verify explicitly, for a triple-collinear configuration involving a gluon and a soft b-quark pair, that the collinear divergence cancels locally in the flavour modulo-2 scheme without an S-function insertion.","supporting_citations":[],"review_version":1}