{"id":"22b30117-c7a6-4cd0-8a2d-8e68e59d6408","arxiv_id":"2508.19226","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"The authors compute the quark jet function at next-to-next-to-leading order in QCD for a class of kT-like resolution variables, giving explicit numerical coefficients for a y23 variant in the E-scheme and WTA scheme.","lead":"This paper computes a new ingredient for precision QCD predictions: the two-loop quark jet function for a class of transverse-momentum-like jet resolution variables in electron-positron collisions. The result is a building block for more accurate next-to-next-to-leading-order calculations of multi-jet rates at colliders.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The NNLO coefficients rest on an unshown zero-bin/soft-function cancellation and an asserted endpoint integral, so Eqs. (52)-(53) are not yet independently verifiable from the text.","rationale":"I read the paper as proposing a semi-numerical method for NNLO quark jet functions of kT-like observables and presenting explicit coefficients for a y23 variant in the E-scheme and WTA scheme. The analytic pole coefficients in Eq. (51) have a plausible structure, and the phase-space parametrizations and regularized kernels in Sec. 2 are laid out coherently. The main numerical ingredients are quoted with integration errors and were obtained with Cuba and Mathematica, but no input code or output data are provided. The two weakest points are exactly the ones the Reader identified: the regulator cancellation depends on an unpublished soft-function paper [68], and the endpoint contribution in Eq. (54) is asserted rather than derived. Both are load-bearing because they feed directly into the finite coefficients that constitute the headline result. I do not see an internal inconsistency that would prove the result wrong, and the method may well be correct, but the evidence presented is not sufficient for full verification. My concern matches the Reader's weakest_assumption, so I agree with the conditional verdict and recommend no change.","tokens_in":16753,"tokens_out":8989,"duration_ms":92306,"concrete_test":"Compute the soft function of Ref. [68] for the same distance variable and explicitly form the sum of the jet-function coefficients in Eqs. (50)-(54) with that soft function; require that all L_N-dependent terms and the epsilon poles cancel. If any residual L_N dependence remains at order epsilon^0, the coefficients in Eqs. (52)-(53) are not the physical jet function and the central claim fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that Eqs. (52)-(53) are the NNLO quark jet function for the considered kT-like variable. This is true only if the zN-regulated collinear integrals combine with the zero-bin/soft-function subtraction of the unpublished companion paper [68] into a regulator-independent physical object. The paper states, after Eq. (10), that the zN prescription makes the soft endpoint non-scaleless and that the resulting zero-bin contribution is 'combined with the soft function [68] to avoid double counting', but the combination is never shown. In addition, the observable-independent endpoint term introduced in Eq. (42) is asserted as the sole source of epsilon poles and is reported in Eq. (54) without a derivation or an explicit integral definition; the text only says it can be evaluated by sector decomposition. These are the two places where an error would change every finite coefficient: an incomplete cancellation of the L_N terms or a mistreated overlapping y-to-0/soft divergence in the endpoint would make Eqs. (52)-(53) scheme-dependent artifacts rather than physical jet-function coefficients. This is not a claim that the calculation is wrong; it is a statement that the central numbers cannot be checked from the paper alone, especially without code or data for the numerical integrations.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper presents a semi-numerical computation of the NNLO quark jet function for a class of kT-like resolution variables in e+e− collisions, following the framework developed at NLO in Refs. [51-53]. Rapidity divergences are regularized with the timelike-vector \"zN\" prescription of Ref. [33]; the 1→3 collinear phase-space integrals over the splitting kernels of Refs. [70-74] are organized by combining the soft-endpoint (zero-bin) contribution with the soft function of the companion paper [68]. For the abelian q→ggq contribution, Eq. (42) separates an observable-independent endpoint term, cut on k⊥, from an observable-dependent subtracted term; the endpoint is evaluated with sector decomposition, the subtracted terms with a dedicated Fortran code using Cuba. The output is the decomposition of J^(2)_N,q in Eq. (50): analytic, scheme-independent pole coefficients in Eq. (51), and numerically determined finite coefficients for a y23-like variable defined by the distance in Eq. (44) in the E-scheme (Eq. (52)) and in the WTA scheme (Eq. (53)), together with the endpoint coefficients in Eq. (54). The method is claimed to apply to any observable in the class of Eq. (1).","tokens_in":16993,"tokens_out":23243,"duration_ms":205952,"significance":"If correct, Eqs. (52)-(53) supply the two-loop collinear ingredient that is currently missing for NNLL resummation and qT-like slicing with kT-like observables: jet functions of this class were previously known only at NLO, aside from the small-R inclusive jet function of Ref. [47]. The computation is parameter-free: all coefficients follow from the published splitting kernels and phase-space integrals, with no fit to data, and numerical errors are quoted throughout. The scheme independence of the pole coefficients in Eq. (51), the consistency of the leading poles (B4, D2, A3) with the exponentiation of the zN-regulated NLO result of Eq. (17), and the agreement of D0^WTA with -π²/12 within the quoted error all indicate internal numerical control. The main limitations are that the zero-bin combination with the unpublished soft function [68] is not shown, the endpoint term of Eq. (42) is asserted without derivation, and the distributional expansion of Appendix A is unproved; because each of these steps feeds directly into the finite coefficients of Eqs. (52)-(53), they are examined in the major comments below.","major_comments":[{"comment":"The zero-bin treatment is described but never shown. The text states that the zN prescription makes the soft-endpoint integral non-scaleless and that the resulting zero-bin contribution is \"combined with the soft function [68] to avoid double counting\", but the combination is not displayed anywhere, and Ref. [68] is unpublished. This is load-bearing for the central claim: the cancellation of the overlapping soft-collinear region is what makes the coefficients in Eqs. (52)-(53) scheme-consistent physical quantities rather than artifacts of the zN regulator, and the paper itself notes (Sec. 2) that the freedom in defining the regularized kernels must be \"compensated by different zero-bin contributions\". Please display the zero-bin subtraction explicitly at least at NLO (for Eq. (17), where the q→gq kernel and the phase space are simple), and state unambiguously whether the results in Eqs. (52)-(53) include or exclude the zero-bin.","section":"§2, after Eq. (10)"},{"comment":"The endpoint contribution is asserted without a definition or derivation. The text says that the first term of Eq. (42) is common to all variables in the class and is the only source of ϵ poles, and Eq. (54) lists its coefficients, but the integral itself, its sector decomposition, and any check of the coefficients are absent. Because the endpoint carries all the ϵ poles and contributes sizeable finite terms (D0^EP = -π²/12 C_F², A0^EP = -1.6343868(8) C_F²), an error there would contaminate every coefficient of Eqs. (52)-(53). Please provide the explicit endpoint integral in the variables of Eq. (41) and at least one non-trivial cross-check, such as reproducing the analytic coefficients A3^EP = C_F²/2, A2^EP = 3 C_F²/4, and A1^EP = (1/4 - π²/12) C_F² from an independent evaluation.","section":"§2, Eq. (42), and §3, Eq. (54)"},{"comment":"The distributional expansion in Eq. (55) is stated without proof, citation, or validation, although it is central to the C_F² subtracted contribution: an error in the subtraction terms, or in the limits of Eq. (56), would change the finite coefficients A0 and B0 for both schemes. Please provide a derivation, or a numerical check of Eq. (55) on a set of representative test functions p(z1,z2), or a reference to an established version of this expansion.","section":"Appendix A, Eq. (55)"},{"comment":"The central numerical results cannot be independently checked from the information given. The observable-dependent integrals are described only qualitatively (four-dimensional integrations with a private Fortran code and Cuba), with no integration grids, evaluation counts, or intermediate results, and the endpoint coefficients in Eq. (54) appear without their integral definition; no ancillary files are provided. I ask for at least one non-trivial internal cross-check to be reported, for instance a numerical verification of the scheme independence of the pole coefficients in Eq. (51) computed in both schemes, or a separate listing of the endpoint and subtracted contributions for each colour structure in Eqs. (52)-(53).","section":"§3, Eqs. (52)-(53)"}],"minor_comments":[{"comment":"The abstract contains a typo: \"tranverse-momentum\" should be \"transverse-momentum\".","section":"Abstract"},{"comment":"The phrase \"dubbed kness_T\" appears to be a typesetting or word-formation artifact; the intended terminology (presumably \"kT-ness\") should be written out.","section":"§3, after Eq. (46)"},{"comment":"The definition of ps(t) is ambiguous as typeset: the displayed limit lim_{λ→0} p(λt, λ) depends on two arguments of p, and the same should be clarified for ps(1/t) and for the iterated limits ps1,s2 and ps2,s1.","section":"Appendix A, Eq. (56)"},{"comment":"Refs. [57,58] are cited as independent related work, but no comparison with them is made; a sentence stating how the present results relate to (or differ from) the NNLO jet-function automation of Ref. [57] would help the reader.","section":"References [57,58]"},{"comment":"The coefficients B_EP^1 = 0.06315(3) and B_EP^0 = 0.4688(1) are quoted with relative errors of order 10^-4, much larger than the error on A_EP^0 (relative error ~5×10^-7); a brief comment on the origin of this difference in precision would be informative.","section":"§3, Eq. (54)"},{"comment":"For the non-abelian q→ggq contribution, the text says the additional single-soft-gluon pole \"does not lead to additional complications\" without explaining why; a sentence clarifying that this singularity is a soft, not rapidity, divergence and is therefore handled by dimensional regularization would be useful.","section":"§2, after Eq. (33)"}],"recommendation":"major_revision","confidential_remarks":"The paper's central numbers depend on the unpublished companion Ref. [68], which supplies the soft function used in the zero-bin combination; if [68] is not released alongside this paper, the verification of Eqs. (52)-(53) remains incomplete. The reliance on \"in preparation\" references for a load-bearing step is unusual, and I recommend requiring at least a preprint of [68], or a fully self-contained treatment of the zero-bin combination, before acceptance. The manuscript is otherwise within scope and the results, if verified, would be a useful building block for the resummation and slicing program, but the present draft does not yet allow a reader to distinguish a genuine scheme-consistent calculation from one whose finite terms depend on the unstated zero-bin convention."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. First, this is a genuinely new perturbative QCD result: the first NNLO quark jet function for a kT-like resolution variable, computed for a y23 variant in the E-scheme and the WTA scheme. Second, the numbers are probably right, but the paper as written does not let you verify them. The finite coefficients rest on a cancellation with an unpublished soft function [68] and on an asserted endpoint integral, and neither is shown in enough detail to reproduce the numerics.\n\nWhat the paper does well: the method is clearly explained, the decomposition of the splitting kernels is sensible, and the analytic pole coefficients (Eq. 51) are scheme-independent, which is a good sanity check. The authors quote integration errors on the numerical coefficients, which is more than many papers in this area do. The zN regularization is applied only to the singular terms, and the distributional expansion in Appendix A is a useful technical contribution, though it too is asserted without proof.\n\nThe soft spots are real but not necessarily load-bearing. The main one is the regulator cancellation. The paper states after Eq. (10) that the zero-bin contribution is combined with the soft function of the unpublished paper [68] to avoid double counting, but the combination is never shown. If that cancellation is incomplete, the L_N-dependent terms and the finite coefficients are scheme artifacts. The same applies to the endpoint contribution in Eq. (42), which the text says can be evaluated by sector decomposition and then simply gives the numbers. Neither concern demonstrates an error, and the overall structure of the result looks plausible, but this is a semi-numerical computation and the article does not ship code, data, or enough detail to reproduce the five-digit coefficients. For a result whose main value is those numbers, that is a genuine limitation.\n\nThis paper is for the QCD precision community—people who want to use these jet functions in NNLO slicing or resummation calculations. For that audience it is important, and it should go to a serious referee; I would not desk-reject it. But the referee should require the authors to either provide the companion paper, the code, or a much more detailed description of the endpoint and zero-bin treatment before acceptance.","headline":"Genuinely new NNLO coefficients for kT-like jet functions, but the regulator cancellation and endpoint integral are not shown, so the numbers cannot be checked from the paper alone.","tokens_in":17541,"tokens_out":2646,"would_cite":true,"duration_ms":25485,"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 NNLO quark jet function for kT-like jet-resolution variables is computed for a variant of y23 in two recombination schemes.","keywords":["jet functions","NNLO QCD","rapidity divergences","kT-like observables","slicing variables","soft-collinear effective theory","zero-bin subtraction","e+e- annihilation"],"falsifier":"Compute the same jet function with a different rapidity regulator and check that the full combination of jet function, zero-bin, and corresponding soft function reproduces the finite coefficients $D_0$, $A_0$, $B_0$ of Eqs. (52)-(53); any mismatch would expose a regulator-dependent remainder. A more direct check is to insert the published soft function, form the physical combination for $y_{23}$, and verify that every factor of $L_N$ cancels.","tokens_in":1765,"feed_emoji":"📐","tokens_out":2442,"duration_ms":100651,"temperature":0.7,"pith_summary":"The paper establishes a general method for computing the quark jet function at next-to-next-to-leading order (NNLO) for transverse-momentum-like resolution variables that smoothly turn $n+1$ jets into $n$ jets in $e^+e^-$ collisions, and then delivers the explicit two-loop coefficients for a variant of the $y_{23}$ observable. In the E-scheme and in the winner-take-all (WTA) recombination scheme, the full result is Eq. (50), with scheme-independent pole terms in Eq. (51) and numerically evaluated finite coefficients in Eqs. (52) and (53). These numbers are what a resummed calculation or an NNLO slicing calculation needs from the collinear sector for this class of observables. The contribution is the demonstration that the rapidity-divergent integrals can be made finite by a timelike auxiliary vector, the $z_N$ prescription, combined with a soft-function zero-bin subtraction, leaving regulator-independent coefficients.","feed_headline":"Quark jet function computed to NNLO for kT-like variables","feed_subtitle":"Explicit two-loop coefficients for y23 open the way to resummed and slicing predictions for multijet final states.","key_machinery":"The central object is the cumulative jet function $J_q(q_{\\rm cut})$, an integral over collinear phase space of the relevant splitting kernels weighted by a $\\theta$ function that restricts the collinear approximation of the resolution variable to values below $q_{\\rm cut}$. The load-bearing mechanism is the $z_N$ prescription: every singular $1/z$ factor in a splitting kernel is replaced using $z_N = z + N^2 k_\\perp^2 / ((2 p\\cdot N)^2 z)$, with $N$ a timelike auxiliary vector; identity (16) then converts $1/z_N$ into the distribution $(1/z)_+$ plus a logarithm of $N^2 k_\\perp^2 / (2 p\\cdot N)^2$ times $\\delta(z)$. For double-soft configurations the paper uses the uniform distributional expansion (55) for $1/(z_{1,N} \\tilde z_{2,N})$. The endpoint term, common to all variables in the class, is computed once and contains all $\\epsilon$ poles; the observable-dependent subtracted term is finite and evaluated numerically. This machinery converts otherwise divergent rapidity integrals into a sum of explicit poles and the numerical coefficients of Eqs. (52)-(53).","core_discovery":"The paper's central claim is that the NNLO quark jet function for $k_T$-like variables is now available, and for the particular $y_{23}$ variant it is given as an explicit Laurent expansion in $\\epsilon$: scheme-independent pole coefficients in Eq. (51), and numerical finite coefficients for the E-scheme in Eq. (52) and for the WTA scheme in Eq. (53). The jet function is built by integrating the $z_N$-regularised collinear splitting kernels over the collinear phase space: the one-loop $q \\to gq$ kernel and the tree-level $1 \\to 3$ splittings $q \\to \\bar q' q' q$, $q \\to \\bar q q q$, and $q \\to g g q$. The abelian $q \\to g g q$ part is split into strongly-ordered and remainder terms; the endpoint term is evaluated once with sector decomposition, while the subtracted term is finite and integrated numerically. The paper asserts that, together with the soft function of the companion work [68], this gives the physical two-loop jet function for this class of observables.","pith_inferences":["Editorial inference: because the endpoint term is universal, the marginal cost of adding a new observable in this class is one finite four-dimensional numerical integral rather than a new two-loop calculation.","Editorial inference: a decisive cross-check would be to repeat the calculation with an independent rapidity regulator and confirm that the combination of jet function, zero-bin, and soft function reproduces Eqs. (52)-(53); the paper does not contain that comparison.","Editorial inference: until the companion soft function of Ref. [68] is published in matching conventions, external users cannot convert the quoted coefficients into physical cross sections; the result is therefore currently a building block awaiting its counterpart."],"forward_implications":["The coefficients enable NNLL-accurate resummed predictions for this $y_{23}$ variant once the companion soft function is combined.","They supply the collinear ingredient for NNLO slicing calculations of multijet cross sections in $e^+e^-$ collisions that use this variable as the resolution or slicing parameter.","The scheme-independent pole terms in Eq. (51) can be checked against renormalisation and factorisation constraints and against the corresponding gluon jet function.","The endpoint decomposition means every other variable in the class $q \\sim k_t$ uses the same endpoint term; only the subtracted term must be recomputed.","The same approach extends to other recombination schemes and other distance definitions, since the observable enters only through the dimensionless function $F$."],"supporting_citations":[{"why":"Introduces the $z_N$ prescription with a timelike auxiliary vector that the paper uses to regulate rapidity divergences.","marker":"[33]"},{"why":"Companion soft-function calculation that supplies the zero-bin subtraction on which the regulator independence of the result relies.","marker":"[68]"},{"why":"Supplies the $(a,b,z,x_{12})$ phase-space variables in which the triple-collinear integrals and the observable function $F$ are expressed.","marker":"[75]"},{"why":"Supplies sector decomposition used to evaluate the endpoint term that contains all $\\epsilon$ poles.","marker":"[76]"},{"why":"Defines the $e^+e^-$ $k_T$ clustering distance whose variant is the $y_{23}$ observable computed here.","marker":"[77]"},{"why":"Defines the winner-take-all recombination scheme used for the second set of results.","marker":"[56]"},{"why":"Provide the one-loop $q \\to gq$ splitting kernel needed for the real-virtual NNLO contribution.","marker":"[70, 71]"},{"why":"Provide the tree-level $1 \\to 3$ collinear splitting kernels whose regularised versions are integrated.","marker":"[72–74]"},{"why":"Introduces the class of effective transverse momentum variables to which this $y_{23}$ variant belongs.","marker":"[52]"}],"fun_headline_variants":["NNLO quark jet function for kT-like variables","Two-loop quark jet function for kT-like observables","Explicit NNLO quark jet function for y23","NNLO kT-like jet function: explicit y23 results","Quark jet function at NNLO for kT-like variables"],"cache_read_input_tokens":19712,"weakest_assumption_plain":"Everything rests on the assumption that applying the $z_N$ replacement only to the singular terms of the splitting kernels, together with the zero-bin subtraction from the companion soft function, cancels all regulator dependence and yields the same physical coefficients any other rapidity regulator would give; if that cancellation is incomplete, the numbers in Eqs. (52)-(53) are artefacts of the scheme.","fun_headline_variants_meta":{"raw":{"variants":["NNLO quark jet function for kT-like variables","Two-loop quark jet function for kT-like observables","Explicit NNLO quark jet function for y23","NNLO kT-like jet function: explicit y23 results","Quark jet function at NNLO for kT-like variables"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000541,"raw_usage":{"total_tokens":2564,"prompt_tokens":885,"completion_tokens":1679,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":501,"completion_tokens_details":{"reasoning_tokens":1598}},"tokens_in":501,"tokens_out":1679,"duration_ms":12978,"temperature":1.0,"reasoning_tokens":1598,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T16:53:32.240534+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the same jet function with a different rapidity regulator and check that the full combination of jet function, zero-bin, and corresponding soft function reproduces the finite coefficients $D_0$, $A_0$, $B_0$ of Eqs. (52)-(53); any mismatch would expose a regulator-dependent remainder. A more direct check is to insert the published soft function, form the physical combination for $y_{23}$, and verify that every factor of $L_N$ cancels.","supporting_citations":[{"cited_title":"Haag, in preparation","cited_arxiv_id":null,"evidence_quote":"Companion soft-function calculation that supplies the zero-bin subtraction on which the regulator independence of the result relies."}],"review_version":1}