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REVIEW 4 major objections 6 minor 1 cited by

The Quark Jet Function for $k_T$-like Variables in NNLO QCD

T0 review · 4 major / 6 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read The NNLO quark jet function for kT-like jet-resolution variables is computed for a variant of y23 in two recombination schemes.

desk verdict 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. read the letter →

arxiv 2508.19226 v1 pith:PEDLDNS6 submitted 2025-08-26 hep-ph

Luca Buonocore , Massimiliano Grazzini , Flavio Guadagni , Jürg Haag , Luca Rottoli This is my paper · ORCID
classification hep-ph
keywords jetfunctionsNNLOQCDrapiditydivergenceskT-likeobservablesslicingvariablessoft-collineareffectivetheoryzero-binsubtractione+e-annihilation
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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).

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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 $F$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 6 minor

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).

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 (4)
  1. [§2, after Eq. (10)] 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.
  2. [§2, Eq. (42), and §3, Eq. (54)] 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.
  3. [Appendix A, Eq. (55)] 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.
  4. [§3, Eqs. (52)-(53)] 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).
minor comments (6)
  1. [Abstract] The abstract contains a typo: "tranverse-momentum" should be "transverse-momentum".
  2. [§3, after Eq. (46)] The phrase "dubbed kness_T" appears to be a typesetting or word-formation artifact; the intended terminology (presumably "kT-ness") should be written out.
  3. [Appendix A, Eq. (56)] 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.
  4. [References [57,58]] 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.
  5. [§3, Eq. (54)] 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.
  6. [§2, after Eq. (33)] 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.

Circularity Check

1 steps flagged · score 4.0 of 10

The NNLO collinear integrals are computed from parameter-free splitting kernels, but the physical finite coefficients depend on a zero-bin/soft-function subtraction that is delegated to the unpublished companion paper [68] by coauthor J. Haag, making the central claim partly self-citation load-bearing.

  1. self citation load bearing [Section 2, paragraph following Eq. (10)]
    "Due to the zN prescription, the integral associated with the soft endpoint of the splitting kernel is no longer scaleless. As a result, the procedure generates a non-vanishing zero-bin contribution, which we combine with the soft function [68] to avoid double counting."

    The finite coefficients in Eqs. (52)-(53) are the paper's central result, but their regulator independence is not established inside the paper. The zN-regulated collinear integrals are said to become physical only after combining the zero-bin contribution with the soft function of [68], which is an unpublished manuscript by coauthor J. Haag. No cancellation or explicit soft-function expression is shown, so the quoted numerical coefficients reduce, at the level of the argument, to the collinear integrals computed here minus a subtraction supplied only by this unverified self-citation. If that subtraction is absent or inconsistent, the coefficients are scheme artifacts rather than physical jet-function coefficients.

full rationale

The paper does not fit any parameter to data, and no equation is defined in terms of its own output. The jet function is defined in Eq. (2) directly from QCD splitting kernels and phase-space integrals; the NNLO splitting kernels are taken from standard external references [70-74], and the observable functions for the E-scheme and WTA scheme are derived from the clustering distances, not imposed to reproduce the final coefficients. The zN regularization is an explicit, externally introduced scheme from Ref. [33], and the NLO check in Eq. (17) is an independent cross-check rather than an input to the NNLO calculation. The decomposition in Eq. (42) into an observable-independent endpoint term and a subtracted term is a mathematical identity, and the endpoint term is reported separately in Eq. (54). The main caveat is the zero-bin/soft-function combination: because the zN regulator makes the soft endpoint non-scaleless, the finite jet function requires a subtraction whose explicit form is not given here but is assigned to [68], an unpublished companion paper by one of the authors. This is load-bearing for the regulator independence of Eqs. (52)-(53), and it is not backed by any code, machine-checked proof, or external benchmark in the present text. That makes the result not fully self-contained and earns a moderate circularity score, though it is not a case of fitting a prediction into existence or renaming a known result. The absence of code or data for the numerical integrations is a reproducibility concern, not itself circularity.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The calculation is a fixed-order perturbative computation with no data fitting and no new physical entities. The only introduced object is the regulator vector N, which is a scheme artifact. The main unproven inputs are the zN regularization and the zero-bin and soft-function cancellation in the unpublished Ref. [68].

free parameters (1)
  • timelike auxiliary vector N (zN rapidity regulator) = not fixed (regulator parameter)
    Introduced in Eq. (10) to regulate rapidity divergences. It enters the jet function through LN in Eq. (18) and the coefficients in Eqs. (50)-(53); the paper argues the dependence cancels after combination with the soft function in Ref. [68], but that cancellation is not demonstrated here.
assumptions (5)
  • domain assumption Collinear factorization: the jet function is defined by the collinear limit of QCD matrix elements via Eq. (2) and receives contributions only from collinear splitting kernels.
    The whole calculation assumes the standard SCET and soft-collinear factorization organization; the paper notes this remains useful even where a factorization theorem is not proven (Sec. 1).
  • ad hoc to paper The zN prescription from Ref. [33] regularizes all rapidity divergences, and the zero-bin subtraction with the soft function [68] removes the regulator dependence.
    The procedure is introduced in Sec. 2 after Eq. (10). The paper explicitly allows freedom in defining regularized kernels, compensated by zero-bin contributions, so this is an assumption about the scheme rather than a proven theorem.
  • standard math The one-loop and tree-level splitting kernels from Refs. [70-74] are correct and can be used in the collinear phase-space parametrization of Eq. (25).
    Input ingredients quoted from the literature (Eqs. (19), (21), (33), (35)).
  • standard math The distributional expansion in Eq. (55) of Appendix A is a valid identity in the limit where the transverse momenta k1_perp and k23_perp vanish.
    Stated without proof in Appendix A and used to integrate the product 1/(z1,N z~2,N); a wrong expansion would change the C_F^2 observable-dependent contribution.
  • domain assumption The clustering-based resolution variable q has the collinear approximation q~ given by F(a,b,z,x12) in Eqs. (47)-(49) for the E and WTA schemes.
    The observable function is derived from the recursive recombination algorithm (Eq. (44)); this derivation is not shown in detail.

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Pith. "Pith review of The Quark Jet Function for $k_T$-like Variables in NNLO QCD." pith.science (2026). https://pith.science/paper/PEDLDNS6

@misc{pith2026250819226,
  author       = {Pith},
  title        = {Pith review of: The Quark Jet Function for $k_T$-like Variables in NNLO QCD},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PEDLDNS6}},
  note         = {Machine review of arXiv:2508.19226}
}
abstract

The precise description of jet processes requires observables capable of efficiently capturing the dynamics of the energy flow in hadronic final states. We consider a class of tranverse-momentum like resolution variables that smoothly describe the $n+1$ to $n$ jet transition in multi-jet processes. We discuss a general method for the computation of the corresponding quark jet function at next-to-next-to-leading order in perturbative QCD. Rapidity divergences are regulated by using a time-like auxiliary vector. We present explicit results for a variant of $y_{23}$ in the $E$-scheme and in the WTA scheme.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Dissecting Exclusive Multijet Cross Sections

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    A general leading-power factorization framework for exclusive multijet resolution variables, with a new rapidity regulator and an all-order factorizing k_T-ness variable.

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