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arxiv: 2511.17704 · v3 · submitted 2025-11-21 · ✦ hep-ph

A simple introduction to soft resummation

Stefano Forte , Giovanni Ridolfi This is my paper

Pith reviewed 2026-05-17 19:59 UTC · model grok-4.3

classification ✦ hep-ph
keywords soft resummationthreshold resummationSudakov resummationQCDinfrared factorizationrenormalization groupmass singularities
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The pith

Threshold resummation in QCD follows from a renormalization group argument once infrared singularities cancel.

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

The paper walks through the basic ingredients of soft gluon resummation in QCD in an elementary way. It first recalls how collinear and infrared factorizations separate hard and soft contributions, then shows how infrared singularities cancel between real and virtual diagrams. From that cancellation the authors extract renormalization group equations whose solution organizes all orders of large threshold logarithms into a closed form. This matters because many collider observables receive large corrections from soft emissions that spoil fixed-order perturbation theory, and the resummed expressions restore reliable predictions in those kinematic regions. The presentation also gives equivalent ways to write the resummed cross section and sketches how the same logic applies to transverse momentum resummation.

Core claim

After the cancellation of infrared singularities in QCD, the remaining mass-factorized cross section satisfies a renormalization group equation whose solution yields the all-order threshold-resummed result, with the resummed expression available in several equivalent forms that each exponentiate the large logarithms.

What carries the argument

The renormalization group equation obtained from infrared singularity cancellation, which resums the soft-gluon logarithms that appear near threshold.

If this is right

  • The resummed threshold cross section can be rewritten in several mathematically equivalent forms, each convenient for different applications.
  • The same renormalization group logic extends directly to transverse momentum resummation.
  • Large logarithmic corrections near partonic threshold are organized to all orders without computing each fixed-order diagram separately.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The approach may generalize to other infrared-sensitive observables in collider physics where similar singularity cancellations occur.
  • Numerical implementations of resummed predictions could be simplified by starting from the renormalization group equation rather than diagram-by-diagram exponentiation.

Load-bearing premise

The reader already understands the cancellation of infrared singularities and the factorization of mass singularities in QCD.

What would settle it

An explicit all-order calculation in which the resummed expression obtained from the renormalization group equation fails to reproduce the known threshold logarithms derived by other methods.

Figures

Figures reproduced from arXiv: 2511.17704 by Giovanni Ridolfi, Stefano Forte.

Figure 1
Figure 1. Figure 1: The leading-order Drell-Yan production process (left) and a next-to-leading order real gluon emission correction to it (right). Part I: soft and collinear logarithms Soft resummation in QCD sums to all perturbative orders contributions that are enhanced by powers of the logarithm of the ratio of a soft scale and the hard scale of the process. They arise because contributions coming from real gluon emission… view at source ↗
Figure 2
Figure 2. Figure 2: Radiation of a gluon with momentum k from a quark with momentum p. factorized in terms of universal splitting functions, up to terms with a relative power suppression in the hard scale. The argument of the log is the ratio of the hard scale to the radiator mass, and hence it diverges in the massless limit - a divergence which is usually referred to as collinear or mass singularity. 2.2 Soft radiation Consi… view at source ↗
Figure 3
Figure 3. Figure 3: The leading order Drell-Yan process and the first-order real-emission corrections to it. interested in the form of the phase space integration over the emitted gluon momentum k. To this purpose, it is convenient to use a slightly different version of the Sudakov parametrization, namely the light-cone parametrization, in which instead of p1 and p2 we adopt as basis vectors their linear combinations p ± = p1… view at source ↗
Figure 4
Figure 4. Figure 4: Cancellation of infrared singularities: the interference between the leading-order process (a) and the loop correction (b) cancels the infrared singularity arising due to interference between the two real-emission contributions (c) and (d). A classic result of quantum electrodynamics, the Kinoshita-Lee-Nauenberg (KLN) theorem, guarantees that once one consistently includes loop corrections along with real … view at source ↗
Figure 5
Figure 5. Figure 5: Kinematics of multiple emission. Sect. 6.2, the upper limit of the transverse momentum integral is k 2 t max = Q 2 (1 − z) 2 4z . (41) Substituting this in the expression Eq. (38) of the partonic cross-section, and adding the virtual contribution, one ends up with σˆ(z) = σ0  δ(1 − z) + αs 2π  Pqq(z) ln Q2 µ 2 F +  P r qq(z) ln (1 − z) 2 4z  +  . (42) Hence, the partonic cross-section contains a doub… view at source ↗
Figure 6
Figure 6. Figure 6: Feynman diagrams contributing to the the production of partonic final states up to NLO in QCD. for an electron-positron pair to produce any hadronic final state, normalized to the QED cross￾section for production of a muon-antimuon pair R = σ(e +e − → hadrons) σ(e+e− → µ+µ−) . (49) The numerator receives contributions from any Feynman diagram with partons in the final state, see [PITH_FULL_IMAGE:figures/f… view at source ↗
Figure 7
Figure 7. Figure 7: Graphical representation of the suppression of radiation from internal lines in the soft limit. It is assumed that the line with momentum q is colorless and cannot radiate like in the Drell-Yan process. transform of the highest power of ln(1 − x) is the same power of ln N, with 1 (1−x)+ counting as the first log. However, the Mellin transform also includes lower powers of ln N, all the way down to the cons… view at source ↗
read the original abstract

We provide an elementary pedagogical introduction to some basic concepts and techniques of soft (or Sudakov) resummation, specifically in QCD, paying particular attention to simple but useful tricks of the trade. We briefly review collinear (Altarelli-Parisi) and infrared (eikonal)factorization, cancellation of infrared singularities and factorization of mass singularities. We recall basic concepts on renormalization group invariance and the solution of renormalization group equations. We then show how threshold resummation can be derived from a renormalization group argument following from the cancellation of infrared singularities. We discuss various equivalent forms of the resummed result, and we briefly present transverse momentum resummation.

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.

Referee Report

0 major / 3 minor

Summary. The manuscript provides a pedagogical introduction to soft (Sudakov) resummation in QCD. It briefly reviews collinear (Altarelli-Parisi) and infrared (eikonal) factorization, cancellation of infrared singularities, and factorization of mass singularities. It recalls renormalization group invariance and the solution of RG equations. The central section derives threshold resummation from an RG argument that follows from the cancellation of infrared singularities. Equivalent forms of the resummed result are discussed, and transverse momentum resummation is presented briefly.

Significance. If the derivation and presentation hold, the paper offers a clear, elementary entry point to soft resummation techniques that are widely used in QCD phenomenology to improve fixed-order predictions near thresholds and at small transverse momenta. By explicitly connecting IR cancellation and factorization to an RG equation for the resummed quantity, it reinforces a standard but sometimes opaque route in the literature. The discussion of equivalent forms and simple tricks of the trade adds practical value for readers learning to apply these methods. No new results are claimed, but the focused pedagogical scope could make the work a useful supplement to textbooks or lecture notes.

minor comments (3)
  1. The review of infrared cancellation and mass-singularity factorization is described as brief; adding one or two explicit one-loop examples (with the relevant integrals shown) would help readers who are not already familiar with the cancellation mechanism before the RG step is introduced.
  2. When presenting the various equivalent forms of the resummed threshold result, a short table or side-by-side comparison of the different exponentiations (e.g., in terms of the soft function versus the coefficient function) would improve readability and make the equivalence more transparent.
  3. The brief section on transverse-momentum resummation would benefit from a single sentence noting how the RG argument differs from the threshold case (e.g., the role of the impact-parameter space), even if only at a schematic level.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the positive recommendation to accept it. We are pleased that the pedagogical scope and the explicit connections drawn between infrared cancellation, factorization, and renormalization-group arguments are viewed as providing a useful entry point for readers.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper is a pedagogical introduction that derives threshold resummation from a renormalization group equation following the standard cancellation of infrared singularities and factorization of mass singularities in QCD. This chain relies on established external concepts (Altarelli-Parisi factorization, eikonal approximation, RG invariance) rather than any self-definition, fitted parameters renamed as predictions, or load-bearing self-citations. The derivation is self-contained against standard benchmarks and does not reduce any claimed result to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The presentation rests on standard QCD factorization theorems and renormalization-group invariance, which are treated as background knowledge rather than derived here.

axioms (2)
  • domain assumption Infrared singularities cancel in sufficiently inclusive observables
    Invoked to justify the RG argument for threshold resummation
  • domain assumption Mass singularities factorize and obey renormalization-group equations
    Used to obtain the resummed exponent

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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.

  1. Threshold resummation of rapidity distributions at fixed partonic rapidity

    hep-ph 2026-01 accept novelty 6.0

    A general all-order expression for threshold resummation of rapidity distributions at fixed partonic rapidity is derived for colorless final states, with NNLL coefficients determined from NNLO Drell-Yan results and sh...

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Works this paper leans on

19 extracted references · 19 canonical work pages · cited by 1 Pith paper · 4 internal anchors

  1. [1]

    V. B. Berestetskii, E. M. Lifshitz, and L. P. Pitaevskii,Quantum Electrodynamics, vol. 4 of Course of Theoretical Physics. Pergamon Press, Oxford, 1982

    work page 1982

  2. [2]

    G. F. Sterman,An Introduction to quantum field theory. Cambridge University Press, 8, 1993

    work page 1993

  3. [3]

    C. D. White,An Introduction to Webs,J. Phys. G43(2016), no. 3 033002, [arXiv:1507.02167]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  4. [4]

    Becher, A

    T. Becher, A. Broggio, and A. Ferroglia,Introduction to Soft-Collinear Effective Theory, vol. 896. Springer, 2015. 33

    work page 2015

  5. [5]

    G. F. Sterman,Summation of Large Corrections to Short Distance Hadronic Cross-Sections,Nucl. Phys. B281(1987) 310–364

    work page 1987

  6. [6]

    Catani and L

    S. Catani and L. Trentadue,Resummation of the QCD Perturbative Series for Hard Processes,Nucl. Phys. B327(1989) 323–352

    work page 1989

  7. [7]

    M. E. Peskin and D. V. Schroeder,An Introduction to quantum field theory. Addison-Wesley, Reading, USA, 1995

    work page 1995

  8. [8]

    Sudakov Factorization and Resummation

    H. Contopanagos, E. Laenen, and G. F. Sterman,Sudakov factorization and resummation, Nucl. Phys. B484(1997) 303–330, [hep-ph/9604313]

    work page internal anchor Pith review Pith/arXiv arXiv 1997

  9. [9]

    Renormalization group approach to soft gluon resummation

    S. Forte and G. Ridolfi,Renormalization group approach to soft gluon resummation,Nucl. Phys. B650(2003) 229–270, [hep-ph/0209154]

    work page internal anchor Pith review Pith/arXiv arXiv 2003

  10. [10]

    C. Duhr, E. Gardi, S. Jaskiewicz, J. Lübken, and L. Vernazza,Infrared singularities and the collinear limits of multi-leg scattering amplitudes,arXiv:2507.21854

    work page arXiv

  11. [11]

    R. K. Ellis, W. J. Stirling, and B. R. Webber,QCD and collider physics, vol. 8. Cambridge University Press, 2, 2011

    work page 2011

  12. [12]

    Collins,Foundations of Perturbative QCD, vol

    J. Collins,Foundations of Perturbative QCD, vol. 32. Cambridge University Press, 2011

    work page 2011

  13. [13]

    Weinberg,The Quantum theory of fields

    S. Weinberg,The Quantum theory of fields. Vol. 1: Foundations. Cambridge University Press, 6, 2005

    work page 2005

  14. [14]

    D. R. Yennie, S. C. Frautschi, and H. Suura,The infrared divergence phenomena and high-energy processes,Annals Phys.13(1961) 379–452

    work page 1961

  15. [15]

    QCD resummation for hadronic final states

    G. Luisoni and S. Marzani,QCD resummation for hadronic final states,J. Phys. G42 (2015), no. 10 103101, [arXiv:1505.04084]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  16. [16]

    Marzani, G

    S. Marzani, G. Soyez, and M. Spannowsky, Looking inside jets: an introduction to jet substructure and boosted-object phenomenology, vol. 958. Springer, 2019

    work page 2019

  17. [17]

    R. V. Harlander, S. Y. Klein, and M. Lipp,FeynGame,Comput. Phys. Commun.256 (2020) 107465, [arXiv:2003.00896]

    work page arXiv 2020

  18. [18]

    Harlander, S

    R. Harlander, S. Y. Klein, and M. C. Schaaf,FeynGame-2.1 – Feynman diagrams made easy, PoSEPS-HEP2023(2024) 657, [arXiv:2401.12778]

    work page arXiv 2024

  19. [19]

    B¨ undgen, R

    L. Bündgen, R. V. Harlander, S. Y. Klein, and M. C. Schaaf,FeynGame 3.0,Comput. Phys. Commun.314(2025) 109662, [arXiv:2501.04651]. 34

    work page arXiv 2025