arxiv: 2602.12611 · v2 · submitted 2026-02-13 · ✦ hep-ph

Recognition: 2 theorem links

· Lean Theorem

Resummation of threshold double logarithms in quarkonium fragmentation functions

Hee Sok Chung , U-Rae Kim , Jungil Lee

Authors on Pith no claims yet

Pith reviewed 2026-05-15 22:55 UTC · model grok-4.3

classification ✦ hep-ph

keywords quarkonium fragmentationthreshold resummationdouble logarithmsNRQCD factorizationperturbative QCDheavy quarkoniafragmentation functionssoft gluon radiation

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The pith

Resumming threshold double logarithms renders quarkonium fragmentation functions positive and fully perturbative.

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

This paper develops a method to resum threshold double logarithms that arise in calculations of fragmentation functions for heavy quarkonia. These logs cause fixed-order results to become negative, leading to unphysical predictions for production cross sections. The resummation is performed to all orders using the nonrelativistic QCD factorization approach. As a result, the fragmentation functions can be computed entirely in perturbation theory without relying on nonperturbative models, ensuring positivity of cross sections.

Core claim

The authors derive a resummation formalism for threshold double logarithms in quarkonium fragmentation functions within the NRQCD factorization framework, providing explicit formulas for the resummed functions that are finite and positive definite at all orders in perturbation theory.

What carries the argument

The resummation formalism that organizes soft gluon emissions to exponentiate the threshold double logarithms in the NRQCD factorization for fragmentation functions.

Load-bearing premise

The assumption that threshold double logarithms can be resummed to all orders within the NRQCD factorization formalism while remaining entirely perturbative and free of nonperturbative model dependence.

What would settle it

An explicit numerical evaluation of the resummed fragmentation function that produces negative values at some momentum fraction would falsify the positivity guarantee.

Figures

Figures reproduced from arXiv: 2602.12611 by Hee Sok Chung, Jungil Lee, U-Rae Kim.

**Figure 2.** Figure 2: FIG. 2. Feynman diagrams that appear in NLO calculation of the soft function [PITH_FULL_IMAGE:figures/full_fig_p029_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. Planar Feynman diagrams that appear in NLO calculation of the soft function [PITH_FULL_IMAGE:figures/full_fig_p034_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. Feynman diagrams for amplitudes that give rise to the doubly real contributions in the [PITH_FULL_IMAGE:figures/full_fig_p039_4.png] view at source ↗

read the original abstract

We develop a formalism for resumming threshold double logarithms that appear in fragmentation functions for production of heavy quarkonia. Threshold singularities appear in fixed-order calculations of quarkonium fragmentation functions in the nonrelativistic QCD factorization formalism due to radiation of soft gluons. Because of this, fixed-order quarkonium fragmentation functions are not positive definite, and can lead to unphysically negative cross sections. This problem can be resolved by resumming threshold logarithms to all orders in perturbation theory, which renders the fragmentation functions finite and ensures the positivity of cross sections. We present a detailed derivation of the resummation formalism and derive the formula for resummed quarkonium fragmentation functions, which can be computed entirely within perturbation theory without the need for nonperturbative model functions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper derives a resummation for threshold double logs in quarkonium FFs to fix negativity, but the claim of fully perturbative results without LDMEs needs close checking.

read the letter

The main takeaway is that Chung, Kim, and Lee have built a resummation formalism for the threshold double logarithms that spoil fixed-order quarkonium fragmentation functions in NRQCD. They show how to sum these logs to all orders so the functions stay finite and positive, which should help cross-section calculations at colliders. The derivation and the final resummed formula are presented as new for this context, and the abstract makes clear they target the soft-gluon singularities that cause unphysical negative results at fixed order. That is a concrete step forward for anyone doing precision NRQCD phenomenology. The paper does a straightforward job explaining the problem and outlining the resummation steps, which is useful even if the full details are technical. The approach stays inside perturbative QCD plus NRQCD factorization, and the claim is that the resummed objects can be computed without extra nonperturbative model functions. If the algebra holds, this removes a known limitation in the literature. The soft spot is the assertion that everything remains purely perturbative. Standard NRQCD fragmentation functions contain long-distance matrix elements that are nonperturbative by construction. Resumming the short-distance coefficients may improve the z-dependence and positivity, but it is not obvious from the abstract whether the LDMEs are eliminated or simply carried along. The stress-test note flags exactly this point, and it would be worth verifying in the explicit formula whether the overall normalization stays free of fitted parameters. Minor issues like explicit checks against data or remaining singularities can be addressed in revision. This work is aimed at theorists who calculate quarkonium production rates and need better fragmentation inputs. A reader already familiar with NRQCD resummation techniques will get the most out of it. The paper deserves a serious referee because the problem it solves is real and the formalism is specific enough to be tested. I would recommend sending it to peer review rather than desk rejection.

Referee Report

2 major / 1 minor

Summary. The manuscript develops a resummation formalism for threshold double logarithms appearing in quarkonium fragmentation functions within NRQCD factorization. Fixed-order calculations exhibit threshold singularities from soft-gluon radiation, rendering the FFs non-positive-definite and leading to unphysical negative cross sections. The authors derive an all-orders resummed expression for these FFs, claiming the result is finite, positive-definite, and computable entirely within perturbation theory with no nonperturbative model functions required.

Significance. If the central claim is substantiated, the work would provide a systematic perturbative solution to a known limitation in quarkonium phenomenology, enabling reliable predictions for processes where threshold logarithms dominate. This strengthens the predictive power of NRQCD by extending factorization to resummed level without ad-hoc inputs, with direct relevance to LHC and future collider analyses of heavy-quarkonium production.

major comments (2)

[Abstract] Abstract and the derivation of the resummed formula: the claim that the resummed FFs 'can be computed entirely within perturbation theory without the need for nonperturbative model functions' is load-bearing for the paper's main result. Standard NRQCD factorization writes D_{i→H}(z) = ∑_n C_n(z,μ) ⟨O_n^H⟩, with ⟨O_n^H⟩ nonperturbative LDMEs. The manuscript must explicitly demonstrate (via the evolution kernel or Sudakov factor) that resummation either renders all LDMEs perturbative or eliminates reference to them; otherwise the positivity and perturbative character do not follow.
[Derivation of resummed formula] Derivation section (around the definition of the resummed FF): the threshold resummation is performed on the short-distance coefficients, but the overall normalization and z-shape still inherit the color and scale structure of the LDMEs. A concrete check is needed showing that the resummed object remains independent of nonperturbative inputs after matching to the fixed-order result; without this, the statement that no model functions are required is not yet established.

minor comments (1)

[Introduction] Notation for the resummed fragmentation function should be introduced with an explicit equation number early in the text to aid readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments, which help us strengthen the presentation of our resummation formalism. We address each major comment below and will revise the manuscript to provide the requested clarifications and explicit demonstrations.

read point-by-point responses

Referee: [Abstract] Abstract and the derivation of the resummed formula: the claim that the resummed FFs 'can be computed entirely within perturbation theory without the need for nonperturbative model functions' is load-bearing for the paper's main result. Standard NRQCD factorization writes D_{i→H}(z) = ∑_n C_n(z,μ) ⟨O_n^H⟩, with ⟨O_n^H⟩ nonperturbative LDMEs. The manuscript must explicitly demonstrate (via the evolution kernel or Sudakov factor) that resummation either renders all LDMEs perturbative or eliminates reference to them; otherwise the positivity and perturbative character do not follow.

Authors: We agree that the abstract phrasing requires clarification to prevent misinterpretation. Our resummation acts exclusively on the short-distance coefficients C_n(z, μ) through the evolution kernel and associated Sudakov factor, which resums the threshold double logarithms arising from soft-gluon radiation to all orders in α_s. The LDMEs ⟨O_n^H⟩ remain nonperturbative matrix elements as in standard NRQCD factorization and are not rendered perturbative by the resummation. However, the resummed FFs require no additional nonperturbative model functions to parametrize the z-dependence or to restore positivity; the all-order Sudakov factor ensures finiteness and positivity of the coefficients independently of the specific LDME values. We will revise the abstract to state explicitly that the resummation is performed on the perturbative coefficients while retaining the standard NRQCD LDMEs. We will also add an explicit demonstration in the derivation section showing how the Sudakov factor cancels the threshold singularities order-by-order, guaranteeing positivity without reference to extra model inputs. revision: yes
Referee: [Derivation of resummed formula] Derivation section (around the definition of the resummed FF): the threshold resummation is performed on the short-distance coefficients, but the overall normalization and z-shape still inherit the color and scale structure of the LDMEs. A concrete check is needed showing that the resummed object remains independent of nonperturbative inputs after matching to the fixed-order result; without this, the statement that no model functions are required is not yet established.

Authors: We acknowledge the need for an explicit matching check. In the revised manuscript we will insert a dedicated subsection that performs the matching of the resummed FF to the fixed-order result at the first few orders in α_s. This check will demonstrate that the nonperturbative LDMEs factor out consistently as overall normalization constants, while the z-shape in the threshold region is fully determined by the perturbative resummed coefficients. After matching, the resummed FF depends on nonperturbative inputs solely through the standard LDMEs; no additional model functions are introduced. The positivity of the resummed object follows directly from the exponentiated Sudakov factor, which is independent of the LDME color and scale structure. This addition will substantiate that the threshold-resummed FFs are computable within perturbation theory once the LDMEs are specified by other means (data or lattice). revision: yes

Circularity Check

0 steps flagged

Derivation self-contained within perturbative NRQCD resummation

full rationale

The paper provides a detailed derivation of the resummation formalism for threshold double logarithms in quarkonium fragmentation functions directly from the NRQCD factorization framework. The central result—the formula for resummed fragmentation functions—is obtained by resumming soft-gluon contributions to all orders in perturbation theory, rendering the functions finite and positive without introducing fitted parameters or external nonperturbative models. No load-bearing steps reduce by construction to prior inputs, self-citations, or ansatze; the derivation addresses fixed-order singularities through explicit resummation kernels and evolution equations that remain fully perturbative. The approach is self-contained against external benchmarks and does not rely on uniqueness theorems or renamed empirical patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the validity of NRQCD factorization for fragmentation functions and the applicability of standard threshold resummation techniques to this observable.

axioms (1)

domain assumption NRQCD factorization applies to quarkonium fragmentation functions
The entire formalism is built inside the nonrelativistic QCD factorization framework as stated in the abstract.

pith-pipeline@v0.9.0 · 5423 in / 1042 out tokens · 49665 ms · 2026-05-15T22:55:41.681653+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We develop a formalism for resumming threshold double logarithms... soft factorization of the quarkonium fragmentation function in terms of soft functions... resummation by exponentiation
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Grammer-Yennie approximation... Wilson lines... soft functions S(z)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

69 extracted references · 69 canonical work pages · 22 internal anchors

[1]

Then βnβ′ terms can be rewritten as nβnβ′ =− (n·p) 2 p2(d−1) gββ ′ + nβnβ′ + (n·p) 2 p2(d−1) gββ ′ ,(59) where the terms in the parentheses vanish when contracted with the isotropic tensorI ββ ′ . By using this we can write the isotropic and anisotropic contributions as I αα′ T ΠJ αβα′β′Sββ ′ 3P [1] isotropic =−(d−2)S ββ ′ 3P [1]gββ ′ ×    ...

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[2]

Real diagrams We first consider the real diagrams, where a gluon attaches to the lightlike or timelike Wilson lines. The matrix element ofW( 3S[8] 1 )cb on the vacuum and the one-gluon state is given at leading nonvanishing order by ⟨g(k)|W( 3S[8] 1 )cb|0⟩=−g ip·ϵ ∗(k) k·p+iε f xcaδba −g in·ϵ ∗(k) k·n+iε f xbaδca +O(g 3),(96) whereϵ(k) is the polarization...

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[3]

Virtual diagrams The contribution from the virtual diagrams that come from the exchange of a gluon between the lightlike and timelike Wilson lines is SNLO 3S[8] 1 (z) virtual = 2Re Z k −gnµ i k·n+iε f xba −igµνδbc k2 +iε × −gpν i −k·p+iε f xca 2πδ(P +(1−z)) = Reig 2(N 2 c −1)C A Z k 2πδ(1−z) (k2 +iε)(k·n+iε)(−k·p+iε) .(102) We use the shorthand Z k =µ ′2ϵ...

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[4]

(100), (105), and (110)

Results and discussion The 3S[8] 1 soft function at NLO is given by the sum of Eqs. (100), (105), and (110). The result is SNLO 3S[8] 1 (z) = αsCA(N 2 c −1) P + µ 2m 2ϵ 2ϵ−1 UV −2 +ϵπ 2/6 +O(ϵ 2) (1−z) 1+2ϵ −δ(1−z) 1 ϵUV 1 ϵUV − 1 ϵIR +δ(1−z) 1 ϵUV − 1 ϵIR .(111) The term involving (1−z) −1−2ϵ comes from the real diagram, while the terms proportional toϵ ...

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[5]

[44], we know that the double UV pole arises from the diagram where an additional virtual gluon is exchanged between the timelike and lightlike Wilson lines

Planar diagrams From the per-diagram results available in Ref. [44], we know that the double UV pole arises from the diagram where an additional virtual gluon is exchanged between the timelike and lightlike Wilson lines. As we will see later, these double UV poles will only arise from planar diagrams shown in Fig. 3. Here we first consider the contributio...

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[6]

i p·k 1 +iε i p·(k 1 +k 2) +iε 2 + i p·k 1 +iε 2 i p·(k 1 +k 2) +iε # ×p µ(k1µδµ1 β −k 1β δµ1 µ )pµ2δycδa1bdbcdf a2dx,(134a) A3 +A 4 =−ig

UV power counting Now we consider the remaining NLO diagrams that contribute toS 3P [8](z). As we have discussed previously, threshold double logarithms can only arise inS 3P [8](z) at NLO from double UV poles multiplying (1−z) −1−nϵ. We will show in this section that none of the remaining diagrams can produce double UV poles, so that they can be discarde...

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[7]

with thek − 1 andk − 2 integrals. In most 40 cases, general form of thek 1⊥ andk 2⊥ integrals at large|k 1⊥|and|k 2⊥|can be written as Z dd−2k1⊥dd−2k2⊥ k2a 1⊥k2b 2⊥(k2 1⊥ +k 2 2⊥)c ,(135) after suitable rescaling ofk 1⊥ andk 2⊥. There are exceptional cases when a factor ofk 1 ·k 2 appears in the numerator or in the denominator; we will later examine these...

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[8]

The diagramA ∗ 5A7 needs more work

The resulting integral is now UV finite in bothk 1⊥ andk 2⊥. The diagramA ∗ 5A7 needs more work. If we only collect the terms that provide a UV- divergent power count, we have Z dPSk1 Z dPSk2 k0 2/ √ 2−k + 1 (k0 2)2/k1 ·k 2 (k0 1)2k+ 2 (k0 1 +k 0 2)2 2πδ(k + 1 +k + 2 −P +(1−z)).(139) If we first integrate overk − 1 andk − 2 by using the on-shell delta fun...

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Results and discussion The 3P [8] soft function at NLO is given by S3P [8](z) =−(1−ϵ) (N 2 c −4)(N 2 c −1) Nc 1 πm2P + × (πµ′2/m2)ϵΓ(1 +ϵ) (1−z) 1+2ϵ − αsCA(πµ′2/m2)2ϵ 2π ϵ−2 UV +O(ϵ −1) (1−z) 1+4ϵ +O(α 2 s) ,(143) where we neglect contributions at orderα s that do not produce threshold double logarithms. At orderα s, the expansion in powers ofϵproduces m...

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Isotropic contribution Similarly to the 3P [8] case, we first consider the planar diagrams. The virtual correction to the matrix element of the operatorW b β(3P [1]) between the vacuum and the one-gluon 45 state yields ⟨ga,α(l)|W b β(3P [1])|0⟩|virtual =ig 2CAδabpµ(lµδα β −l βδα µ) Z k −i k2 +iε p·n i −k·n+iε × " i l·p+iε 2 i (l+k)·p+iε + i l·p+iε i (l+k)...

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The strategy of the calculation is similar to the isotropic case

Anisotropic contribution It remains to show that the anisotropic contributionS T T 3P [8](z) does not contain threshold double logarithms. The strategy of the calculation is similar to the isotropic case. The UV-divergent contribution to the virtual planar diagram involves the same one-loop integral I(l), and is given by ST T 3P [1](z)|virtual, UV =−g 2(N...

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Results and discussion As we have shown by explicit calculation, the threshold double logarithm in the isotropic contributionS 3P [1](z) is equal to theS 3P [8](z), with the replacement (N 2 c −4)/N c →1 in the overall color factor. Thus if we collect only the contributions relevant for threshold double logarithms, we have S3P [1](z)|threshold = N 2 c −1 ...

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