Soft-gluon coupling and the TMD parton branching Sudakov form factor

A. Bermudez Martinez; A. Lelek; A.M. van Kampen; F. Hautmann; L. Keersmaekers; M. Mendizabal Morentin; S. Taheri Monfared

REVIEW 3 major objections 5 minor 1 cited by

Soft-gluon coupling and the TMD parton branching Sudakov form factor

T0 review · 3 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read By substituting the soft-gluon physical coupling into the parton-branching Sudakov form factor, the paper achieves NNLL accuracy for TMD evolution and evaluates the Collins-Soper kernel at that order.

desk verdict Plausible and important first NNLL step for PB TMD, but the derivation is deferred to an unpublished companion, so treat the claim as conditional. read the letter →

arxiv 2412.21116 v2 pith:DLVRM6IX submitted 2024-12-30 hep-ph

A. Bermudez Martinez , F. Hautmann , L. Keersmaekers , A. Lelek , M. Mendizabal Morentin , S. Taheri Monfared , A.M. van Kampen This is my paper · ORCID

classification hep-ph PACS 12.38.-t12.38.Bx

keywords transversemomentumdependentdistributionspartonbranchingSudakovformfactorsoft-gluonphysicalcouplingNNLLresummationCollins-SoperkernelDrell-YanproductionQCDevolution

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

This paper aims to establish that transverse-momentum-dependent (TMD) parton distributions in the parton-branching (PB) approach can be evolved at next-to-next-to-leading-logarithmic (NNLL) accuracy, a step beyond the NLL accuracy previously reached in this framework. The central step is to replace the ordinary QCD strong coupling in the PB Sudakov form factor with the soft-gluon physical coupling, the higher-order extension of the classic one-loop soft-coupling result. This replacement produces the explicit NNLL coefficients, double-logarithmic $A^{(3)} = K^{(2)} k^{(0)}$ and single-logarithmic $B^{(2)} = -2 d^{(1)}$, in the perturbative Sudakov form factor. The paper also evaluates the Collins-Soper kernel at NNLL, including nonperturbative contributions, and gives illustrative Drell-Yan $Z$-boson transverse momentum predictions. If the claim holds, PB-based TMD evolution and parton-shower Monte Carlo predictions can be upgraded to NNLL accuracy for precision collider physics.

What carries the argument

The central object is the soft-gluon physical coupling, $\alpha_s^{\rm phys} = \alpha_s\bigl(1 + \sum_{n\ge 1} K^{(n)} (\alpha_s/2\pi)^n\bigr)$, with $K^{(1)}$ the classic one-loop coefficient and $K^{(2)}$ taken from higher-order soft-gluon resummation. Inserting this coupling in its subtracted form, $\alpha_s^{\rm subtr} = \alpha_s^{\rm phys} - K^{(1)}\alpha_s^2/(2\pi)$, into the PB Sudakov form factor, together with the angular-ordering relation $q_\perp = (1-z)\mu'$ that splits the branching integral into perturbative and nonperturbative regions, yields the NNLL coefficients of Eq. (13). The same setup supplies the identity connecting the NNLL double-log coefficient to the Collins-Soper kernel, Eq. (16), which the paper uses to compute the kernel at NNLL.

What would settle it

Compute the PB Sudakov form factor directly at three loops without the coupling substitution and check whether the order-$\alpha_s^3$ double-logarithmic coefficient equals $K^{(2)} k_a^{(0)}$; if it does not, the soft-gluon coupling replacement is incomplete for this framework. A second decisive check is to extract the Collins-Soper kernel from PB TMD predictions at small transverse coordinate and compare with lattice determinations, since a mismatch would falsify Eq. (16).

Watch

Extended reading notes

Core claim

The paper's claim, stated on its own terms, is that the PB Sudakov form factor evaluated with two-loop splitting functions and the subtracted soft-gluon physical coupling is NNLL accurate. The resulting form factor, Eq. (13), has an $O(\alpha_s^3)$ double-logarithmic coefficient $K^{(2)} k_a^{(0)} \equiv A_a^{(3)}$ supplied by the soft-gluon coupling, and an NNLL single-logarithmic coefficient $-2 d_a^{(1)} \equiv B_a^{(2)}$ supplied by the two-loop splitting functions. The paper further observes that at NNLL the double-log coefficient is no longer proportional to the cusp anomalous dimension because of the collinear anomaly, and it uses Eq. (16) to relate the difference $A_a^{(3)} - k_a^{(2)}$ to the perturbative Collins-Soper kernel. It then computes the Collins-Soper kernel at NNLL, adding a nonperturbative contribution from the large-distance region, and compares with data-driven and lattice extractions. The authors describe this as the first NNLL computation performed with PB TMD techniques.

Load-bearing premise

The argument stands on the assumption that substituting the physical soft-gluon coupling for the ordinary strong coupling inside the PB Sudakov form factor captures the complete NNLL double-logarithmic correction with no further parton-branching-specific terms, and on a companion paper (currently unpublished) for the identity linking that coefficient to the Collins-Soper kernel.

Editorial extensions

If this is right

PB TMD distributions can be evolved at NNLL accuracy in the Sudakov region, directly improving predictions for low-$p_T$ Drell-Yan observables at the LHC.
The Collins-Soper kernel becomes computable in the PB framework at NNLL, with nonperturbative large-$b$ contributions, allowing direct comparison with data-driven and lattice results.
The NNLL Sudakov form factor can be combined with existing NLO matching and multi-jet merging machinery to produce more accurate Monte Carlo predictions.
At NNLL the double-logarithmic Sudakov coefficient explicitly separates from the cusp anomalous dimension through the collinear anomaly, and the PB framework makes that separation concrete through the Collins-Soper kernel.
Because the soft-gluon coupling is known beyond $O(\alpha_s^3)$, the same construction can in principle be extended systematically to higher logarithmic orders.

Reading between the lines

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

Beyond the paper: the recipe of replacing the coupling by its physical soft-gluon form may transfer to other parton-shower evolutions, suggesting that NNLL accuracy can be gained without rebuilding the splitting functions from scratch.
Beyond the paper: the flattening of the Collins-Soper kernel at large $b$ seen for the dynamical resolution scale, if confirmed by data, would support saturation-like behavior of the kind preferred by recent Drell-Yan fits.
Beyond the paper: a dedicated fit of the NNLL PB TMD predictions to precise $Z$ $p_T$ data could pin down the nonperturbative resolution scale $q_0$ and discriminate between $\alpha_s(q_\perp)$ and $\alpha_s(\mu)$ coupling prescriptions.
Beyond the paper: since NNLL is the first order at which the collinear anomaly enters, the low-$p_T$ region of the $Z$ spectrum is a direct place to test whether the PB framework really captures the physics at this order.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Quick take: this is a credible announcement of the first NNLL-accurate PB TMD Sudakov form factor, but as a standalone paper it does not show its own derivation. The central result, Eq. (13), appears after a one-line 'performing the longitudinal momentum integration,' and the relation to the Collins-Soper kernel, Eq. (16), is credited to the authors' own unpublished manuscript [65]. The paper itself says 'Full details will be reported elsewhere.' So the claim is plausible but not independently verifiable from the text.

What is genuinely new: they upgrade the PB TMD framework from NLL to NNLL, and they identify that the NNLL double-log coefficient A^(3) comes from the soft-gluon physical coupling rather than from three-loop splitting functions, consistent with the collinear anomaly. The connection between the difference A^(3) - k^(2) and the CS kernel is a new structural observation in this scheme. The numerical illustrations (Fig. 1 for the DY spectrum, Fig. 2 for the CS kernel against literature extractions) are just illustrations, but they show the machinery works and the comparison to lattice and data extractions is a useful sanity check.

The soft spots, in proportion: the biggest is the deferred derivation. Eq. (13) and Eq. (16) are the two load-bearing places where 'NNLL' actually appears, and both are taken on faith. The stress-test note worries about an extra K^(1)d^(0) term when substituting the physical coupling into the d_a term; on reading, the paper says they use the subtracted coupling α_s^subtr = α_s^phys - K^(1)α_s^2/(2π), so that specific term is removed by construction. But the paper does not show the cancellation of other scheme-dependent artifacts, and the overall derivation is absent. The circularity concern about A^(3)=K^(2)k^(0) is real but mild: the coefficient is, by construction, the one encoded in the physical coupling; the informative part is that the PB scheme keeps it intact and connects it to the CS kernel. The numerics have only scale-variation bands, not a full uncertainty treatment, but they are explicitly labeled as illustrative.

My bottom line: this deserves peer review. It's a within-subfield advance with real consequences for LHC precision, and the authors are transparent about what is deferred. A referee should be able to check the consistency of Eq. (13) against the conventional resummation scheme in Refs. [76,77] and ask for the companion paper or the missing steps before endorsing the NNLL claim. I'd send it to review, with the clear expectation that the derivation be supplied or the companion released.

For your reading group: maybe. It's short and the plot comparing CS kernels is thought-provoking.

Referee Report

3 major / 5 minor

Summary. The manuscript claims that next-to-next-to-leading-logarithm (NNLL) accuracy can be achieved in the parton-branching (PB) TMD framework by replacing the strong coupling with the soft-gluon physical coupling. The central result is Eq. (13), which gives the perturbative PB Sudakov form factor with double-logarithmic coefficient A_a^(3)=K^(2)k_a^(0) and single-logarithmic coefficient B_a^(2)=-2d_a^(1), together with the associated identification of the Collins-Soper kernel through Eq. (16). The paper presents numerical illustrations for TMD distributions, the Z-boson transverse momentum spectrum, and the b-dependence of the Collins-Soper kernel, comparing with several extractions from the literature. The manuscript explicitly states that full details will be reported elsewhere [65], and Eq. (16) is attributed to the authors' own unpublished manuscript.

Significance. If the central derivation is correct, this would be the first NNLL computation with PB TMD techniques and would open the way to improved PB-based predictions for LHC observables. The paper usefully assembles known ingredients: the soft-gluon physical coupling, two-loop splitting functions, the dynamical resolution scale, and the CS-kernel extraction technique. The numerical comparisons in Figs. 1 and 2 are transparent and the relation to existing TMD extractions is clearly presented. However, the advertised NNLL result is not independently verifiable from the text: Eq. (13) is asserted without showing the integration steps, and Eq. (16) is delegated to an unpublished companion paper. The paper does not ship machine-checked proofs or reproducible code; the numerical results are illustrative, with scale-variation bands only.

major comments (3)

[§2, text preceding Eq. (13)] Equation (13) is the central result, but the derivation from Eq. (2) is not shown. The statement 'By performing the longitudinal momentum integration' skips the expansion of the physical coupling in the Sudakov exponent. In particular, the manuscript does not demonstrate that substituting α_s^phys into the d_a(α_s) term produces no additional single-logarithmic contributions at the order of interest, nor does it show explicitly how the subtracted coupling α_s^subtr removes all K^(1)-induced terms beyond the double-counting of k_a^(1). Without these steps, Eq. (13) is an assertion rather than a derived result, and the identification B_a^(2)=-2d_a^(1) is not independently established. Please provide the full derivation or a pointer to a derivation in a published, accessible reference.
[§4, Eq. (16)] Equation (16) is the only relation connecting the computed NNLL double-logarithmic coefficient A_a^(3) to the Collins-Soper kernel, and it is attributed to the authors' own unpublished manuscript [65]. The sentence 'Full details will be reported elsewhere [65]' in the introduction likewise leaves the advertised NNLL result unverifiable. Because Eq. (16) is load-bearing for the CS-kernel results in Fig. 2, the manuscript should either derive this relation explicitly or replace reference [65] with a published or publicly available derivation. As written, a reader cannot check the consistency of A_a^(3), k_a^(2), and the CS kernel.
[§2, Eqs. (9)–(13)] The circularity risk in the claim A_a^(3)=K^(2)k_a^(0) should be addressed explicitly. The coefficient K^(2) in Eq. (11) is defined in Refs. [47,48] precisely through the requirement that the physical coupling reproduces NNLL resummation in soft-gluon radiation. The manuscript should clarify what is newly derived in the PB framework versus what is imported from Refs. [47,48], and should specify why no additional PB-specific corrections enter the NNLL coefficient. A concrete check against a known NNLL result in a limiting case (for example, matching Eq. (13) to the standard resummed Sudakov exponent at O(α_s^3)) would considerably strengthen the claim that this is a genuine PB computation of NNLL accuracy.

minor comments (5)

[Eq. (13)] The typography of Eq. (13) makes the power counting ambiguous: in the third line, the factor α_s/(2π) multiplying d_a^(1) appears to sit at a different order in α_s than the preceding K^(2)k_a^(0) term. Please clarify the intended bracket structure and the overall α_s power of each term.
[§2, text after Eq. (15)] The phrase 'with the explicit expressions given in Eqs. (7),(8)' is confusing because Eqs. (7) and (8) give the d_a^(1) coefficients, while the scheme-difference relations in Eqs. (14) and (15) involve B_a^(2). Please state explicitly which scheme the PB d_a coefficients are defined in before comparing with the DY and Higgs schemes.
[Fig. 2 caption] The caption refers to 'the top five entries in the legend' but does not list the five scenarios. Please enumerate them (for example, in the caption or in a table) so that the curves can be identified without relying on the legend alone.
[§4, sentence before Eq. (16)] The text says 'The PB result for the rapidity-evolution kernel is presented next,' but no explicit formula for the kernel is given; only Eq. (16) and Fig. 2 appear. Please state the final expression for the CS kernel, including the non-perturbative contribution from region b), or indicate explicitly that this is deferred to Ref. [65].
[References] Reference [65] is listed as 'in preparation, (2024)'; if it remains unpublished, the manuscript should not rely on it for the central relation Eq. (16). Please update the citation or provide the derivation in the text.

Circularity Check

2 steps flagged · score 6.0 of 10

NNLL double-log coefficient is the input physical-coupling coefficient by construction, and the CS-kernel relation is deferred to authors' unpublished work.

self definitional [Section introducing the soft-gluon coupling and Eq. (13)]
"First, to achieve NNLL accuracy we appeal to the concept of soft-gluon physical coupling [47, 48], which extends the CMW result [49] to higher order. To do this, we modify Eq. (2) by the transformation αs → αphys_s, with the soft-gluon physical coupling given by αphys_s = α_s (1 + Σ_{n=1}∞ K^(n) (α_s/2π)^n) ... The three lines in Eq. (13) give, respectively, the LL, NLL and NNLL contributions ... In particular, the double-logarithmic coefficient of order O(α_s^3), K^(2)k^(0)_a ≡ A^(3)_a, is supplied by the soft-gluon coupling in the PB TMD evolution."

Expanding the substituted coupling in the k^(0)_a term of Eq. (2) produces the O(α_s^3) double-log term K^(2)k^(0)_a mechanically; K^(2) is the coefficient that Refs [47,48] introduced precisely to reproduce NNLL soft resummation. The paper therefore does not derive A^(3) from PB dynamics but reads off the defining coefficient of its input coupling. The advertised NNLL double-log result is equivalent, by construction, to the definition of the soft-gluon physical coupling.
self citation load bearing [Section on the Collins-Soper kernel, Eq. (16)]
"The perturbative part of the CS kernel controls the relationship between the NNLL double-log coefficient, obtained in the last line of Eq. (13) from the PB TMD implementation of the soft-gluon coupling, and the O(α_s^3) k-coefficient in Eq. (3) [65]. More precisely, by taking the derivative of the CS kernel with respect to transverse coordinate b [51, 52, 78, 79], we have A^(3)_a − k^(2)_a = C_a πβ_0 (...)."

Equation (16) is the sole connection between the computed A^(3)_a − k^(2)_a and the CS kernel used for the NNLL rapidity-evolution results, and it is attributed to Ref. [65], an unpublished manuscript by the same collaboration. No derivation or independent cross-check appears in this paper, so the CS-kernel conclusion rests on a load-bearing self-citation whose content is not available to the reader.

full rationale

The paper does not fit any parameter, and the numerical comparisons in Figs. 1-2 are illustrative and benchmarked against external extractions, which would normally keep the score low. However, the central NNLL claim is carried by two load-bearing links: Eq. (13) obtains A^(3) by substituting the defining K^(2) of the physical coupling, and Eq. (16) is deferred to the authors' unpublished Ref. [65]. The first link is a definitional reduction, since the advertised double-log coefficient is the input coupling's own coefficient; the second is a self-citation that is not independently verified in the text. Together they make the advertised NNLL accuracy partly circular, although the underlying K^(2) and splitting-function coefficients are externally well founded.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central perturbative result uses known two-loop splitting functions and the known soft-gluon coupling; no new free parameters enter Eq. (13). The numerical CS kernel does depend on non-perturbative modeling parameters q0, q_cut, and z_M. No new physical entities are introduced.

free parameters (3)

q0 (semihard showering scale) = O(1 GeV); q0 = 1 GeV in the purple curve of Fig. 2
Splits perturbative and non-perturbative Sudakov regions in Eq. (12); the numerical CS kernel depends on it.
q_cut (low transverse momentum cutoff) = GeV order, not specified precisely
Used in alpha_s(max(q_cut^2, q_perp^2)) modeling of non-perturbative region; affects non-perturbative Sudakov and CS kernel.
soft-gluon resolution scale z_M = not given; different scenarios in Fig. 2
Characterizes resolvable vs non-resolvable branchings; non-perturbative component and CS kernel depend on it.

assumptions (4)

domain assumption PB evolution equation (Eq. (1)) correctly describes TMD evolution with the Sudakov form factor.
Framework taken from Refs. [33,34]; not re-derived here.
domain assumption Soft-gluon physical coupling coefficients K(1), K(2) from Refs. [47,48] are correct.
The NNLL double-log coefficient is built from K(2); no independent check in this paper.
domain assumption Angular-ordering kinematic relation q_perp=(1-z)mu' holds.
Used to map branching variables and define z_dyn; taken from Refs. [49,70-74].
ad hoc to paper Relation Eq. (16) between CS kernel derivative and A(3)-k(2) is correct.
Attributed to unpublished reference [65] by the same authors; not verifiable in this paper.

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Cite this review

Pith. "Pith review of Soft-gluon coupling and the TMD parton branching Sudakov form factor." pith.science (2026). https://pith.science/paper/DLVRM6IX

@misc{pith2026241221116,
  author       = {Pith},
  title        = {Pith review of: Soft-gluon coupling and the TMD parton branching Sudakov form factor},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DLVRM6IX}},
  note         = {Machine review of arXiv:2412.21116}
}

read the original abstract

The evolution of transverse momentum dependent (TMD) distributions in Quantum Chromodynamics (QCD) can be formulated in a parton branching (PB) framework. We show that next-to-next-to-leading-logarithm (NNLL) accuracy can be achieved in this framework by using the concept of soft-gluon physical coupling. We present results for the TMD distributions and for the Collins-Soper kernel controlling rapidity evolution. The results pave the way for PB predictions at NNLL level for physical observables at the Large Hadron Collider (LHC) and future colliders.

Figures

Figures reproduced from arXiv: 2412.21116 by the authors.

**Figure 2.** FIG. 2. The [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

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

First constraints on the nonperturbative gluon Collins-Soper kernel
hep-lat 2026-07 conditional novelty 7.0 of 10

First lattice-QCD constraints on the nonperturbative gluon Collins-Soper kernel are obtained at near-physical pion mass with uNNLL LaMET matching on a single a=0.15 fm ensemble.

Reference graph

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= exp − X b × Z µ2 µ2 0 dµ′2 µ′2 Z zM 0 dz zPR ba(z, αs) ! (1) wherezandµ ′ are the branching variables, represent- ing respectively the longitudinal momentum transfer and the mass scale at the branching;z M is the soft-gluon resolution scale [33], characterizing resolvable and non- resolvable branchings;P R ba(z, αs) are the real-emission splitting funct...

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