Pith. sign in

REVIEW 4 major objections 4 minor 60 references

In a quark-gluon plasma, the triple-collinear q → q c c̄ splitting remains factorized into sequential 1 → 2 splittings in three strongly ordered limits.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · deepseek-v4-flash

2026-08-01 15:33 UTC pith:RQZWQNWN

load-bearing objection A genuinely useful new formalism for medium-modified 1→3 splitting, with a real factorization claim that is currently conditional on an unquantified small-q window. the 4 major comments →

arxiv 2607.18377 v1 pith:RQZWQNWN submitted 2026-07-20 hep-ph

Factorization of the triple-collinear q to qcbar{c} splitting function at first order in opacity

Shahin Iqbal , Urs Achim Wiedemann This is my paper
classification hep-ph
keywords triple-collinear splitting functionsopacity expansionmedium-modified QCD splittingjet quenchingfactorization of splitting functionsformation timesstrong orderingquark-gluon plasma
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper asks whether the vacuum property that a collinear 1→3 splitting reduces, in the right kinematic limit, to a product of two 1→2 splittings survives when the splitting happens inside dense QCD matter. It computes the fully differential medium-modified splitting for a quark emitting a c c̄ pair at first order in opacity, meaning one scattering of one of the five partons involved, and expresses the result as momentum-shifted vacuum pieces times simple interference factors. It then shows that in three strongly ordered limits, classified by how the inverse formation times of the two branchings compare with the medium length, the medium-modified splitting function factorizes into products of a q→qg and a g→c c̄ splitting, with medium corrections living on one of the two steps. This matters because jet-quenching Monte Carlo generators model in-medium fragmentation as a Markov chain of 1→2 splittings; that assumption is now derived rather than imposed.

Core claim

For the massless channel q → q c c̄ in a finite medium, the first-order-opacity correction is organized as a sum over five attachment points in the amplitude times five in the conjugate amplitude. Each term splits into a color factor, an analytically known longitudinal phase integral, and a Dirac spinor term that is exactly the vacuum spinor structure evaluated with shifted transverse momenta, augmented by shifted invariant masses for longitudinally polarized intermediate gluons. Taking the generalized strong-ordering limit κ2² ≪ κ1², the paper proves that only a subset of phase integrals survives, and those factorize: in the limit C_j L → 0, the result equals a medium-modified q→qg tensor p

What carries the argument

The opacity expansion is rearranged so each contribution is a product of a color factor, a longitudinal phase integral, and a Dirac term. The phase integral compares inverse formation times, expressed through combinations B_i and C_j of the boost-invariant transverse momenta κ1 and κ2, with the medium length L through the interference factor S(C) = 1 − sin(CL)/(CL). The Dirac terms are obtained from the vacuum by momentum shifts κ_i → κ_i + (fraction) q. The phase integrals select which diagrams survive in each strongly ordered limit, and the momentum-shift structure lets the surviving Dirac terms be reassembled as tensor products of the 1→2 splitting tensors, which is what establishes facto

Load-bearing premise

The factorization is proven only while the medium kick q is small enough that the vacuum momentum ordering κ2² ≪ κ1² survives the interaction, meaning B_i ≫ C_j throughout, and the paper gives no quantitative bound on how small |q| must be; the proof also drops all subleading-color terms at leading order in 1/N_c.

What would settle it

Compute the same medium-modified splitting function keeping the subleading-color diagrams (14) and (23) and check whether the strongly ordered limits still hold; alternatively, evaluate the full opacity expression numerically at |q| comparable to κ2 and test whether any non-factorizing interference term survives.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • A jet fragmenting in a quark-gluon plasma can be described, in these limits, as independent sequential 1→2 splittings, with medium effects sitting on one of the two steps and never on both simultaneously at this opacity order.
  • When the c c̄ pair has such a small invariant mass that it forms outside the medium, all charm and anti-charm interactions with the plasma vanish at leading color, and only the q→qg step carries medium modification.
  • When the first splitting forms almost instantaneously, the factorization becomes a sum of two histories: either the vacuum q→qg rescatters and the g→c c̄ remains vacuum, or the q→qg is vacuum with the gluon momentum shifted and the g→c c̄ is medium-modified.
  • In the totally coherent limit, where both branchings occur outside the medium, the first-order opacity correction is zero and the vacuum factorization is recovered.
  • The derivation supplies the missing justification for treating medium-modified 1→2 kernels as a Markov chain in jet-quenching parton-shower event generators.

Where Pith is reading between the lines

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

  • Beyond the paper: the same phase-integral factorization mechanism should extend to the other triple-collinear channels such as q→qgg, g→c c̄ g, and g→ggg; the extra complexity there is combinatorial, involving more vacuum histories, rather than conceptual.
  • Beyond the paper: because the proof restricts the medium kick q to the window where the strong ordering survives, a numerical evaluation of the full non-ordered expression could map how large |q|/κ2 must become before factorization breaks, giving a quantitative diagnostic for overlapping formation times.
  • Beyond the paper: the absence of spin-flip in the collinear high-energy limit means medium-induced spin decorrelation is purely kinematical in this calculation, a feature that could be tested through azimuthal correlations of the c c̄ pair.
  • Beyond the paper: the small-q window is also where an effective kinetic description of jet propagation should agree with this opacity-expansion result; a direct comparison there would test the robustness of factorization outside the strict collinear limit.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The paper develops an opacity-expansion framework for medium-modified collinear splitting functions, based on a reformulation of the Catani-Grazzini compact representation (1.3)-(1.4) in time-ordered perturbation theory. The space-time, color, and Dirac-spinor structures are separated, and the fully differential medium-modified q -> q c cbar splitting function is computed to first order in opacity and leading order in N_c. The central result is the claimed factorization of this splitting function in three strongly ordered collinear limits: Eq. (5.30) (medium-modified q->qg times vacuum g->ccbar), Eq. (5.49) (a sum of two products involving medium-modified q->qg or g->ccbar), and Eq. (5.52) (vanishing medium modification, vacuum factorization). The vacuum limit (4.68) and the g->c cbar medium result (2.67)-(2.69) are checked against the literature.

Significance. If established, this is the first proof that a medium-modified triple-collinear splitting function factorizes into products of 1->2 splitting functions in strongly ordered limits, giving a formal underpinning for Markovian jet-quenching parton showers and for discussions of overlapping formation times. The fully differential result and the compact momentum-shift structure of the Dirac terms (Section 4.3) are potentially useful beyond the specific process. The paper also contains explicit analytic longitudinal phase integrals and consistency checks with existing results, which are strengths. However, the central factorization statement is conditional on a small-momentum-transfer assumption that is not quantified or proven to dominate the q-integral; this must be addressed before the main claim is accepted.

major comments (4)
  1. [Section 5.2, Eqs. (5.11)-(5.12), (4.62)-(4.63), (4.89)] The generalized strong-ordering condition B_i >> C_j is imposed on transverse momenta that are shifted by the medium momentum q. The text after (5.11) states that q must be 'sufficiently small', but no quantitative bound is given, and no argument is provided that the q-integral in (4.89) is dominated by this window. For q ~ kappa1, the shifted C_j in (4.30)-(4.31) become comparable to B_i, and the non-factorizing contributions (12), (13), (24), (34) — whose Dirac terms violate condition (5.10) — are not suppressed by powers of kappa2/kappa1. Since the splitting function is integrated over q with weight |a(q)|^2, the factorization theorems (5.30), (5.49), (5.52) are established only for the integrand in a restricted kinematic region, not for the integrated splitting function. Please derive a power-suppression estimate for the complementary q-region, or else state the result as conditional
  2. [Section 4.3, Eqs. (4.64)-(4.75)] The compact forms of the Dirac structures are central to the factorization analysis, but they are only asserted to have been verified by FeynCalc. No derivation, intermediate expression, or reproducible code/notebook is provided. Since the factorization property (5.10) and the subsequent tensor-product identifications (5.25) rely directly on these forms, the proof is not fully checkable from the manuscript. Please include at least one explicit derivation for a representative contribution, or supply a FeynCalc notebook or ancillary file so the claims can be verified.
  3. [Section 5.2.1-5.2.2, Eqs. (5.14)-(5.22) and (5.34)-(5.46)] The lists of 'one finds' expansions for the longitudinal phase integrals are load-bearing: they determine which contributions survive in the strongly ordered limits and therefore drive the factorizations (5.30) and (5.49). These expansions are presented without derivation or even a generic argument for why certain S(C) combinations vanish or simplify. Please provide the limiting steps, or at least state the generic behavior of S(C) for the relevant C L limits and show how the subleading terms are identified.
  4. [Abstract and Eq. (5.49)] The abstract states that in all three limits the splitting function factorizes 'into a tensor product of q->qg and g->ccbar splitting functions.' In the C_{1/B_i L} limit, however, Eq. (5.49) gives a sum of two tensor products, each with a different medium-modified factor. Moreover, the second product contains a q-dependent shift p5 -> p5+q in the vacuum q->qg factor, which is integrated together with the medium-modified g->ccbar factor. The statement should be softened to 'a sum of factorized products' and the sense in which factorization holds (fixed-q integrand versus integrated product) should be clarified.
minor comments (4)
  1. [Throughout] There are numerous typos and inconsistencies: 'cedium-modified' (Sec. 2.4), 'imaginary contributions' for virtual contributions (Sec. 4.2.2), 'subtelty' (footnote 1), 'propotional' (Sec. 4.3.2), 'rapidtly' (Fig. 8 caption), 'KedMed/JedMed/JetMed' inconsistent naming in the Outlook, and 'bf q' instead of 'q'. A careful proofreading pass is needed.
  2. [Section 2.6] The statement that (2.69) is 'consistent with the terms (A.1) and (A.4) in [55]' would be more useful if the matching were made explicit, e.g., by showing the correspondence of notation for the rescaling and the S factor.
  3. [Section 5.2.1] After Eq. (5.13), the text says the formation time of the second splitting 'becomes arbitrarily large'; more precisely it becomes large compared with the medium length L, since C_j L -> 0. The wording could be misread as an absolute statement.
  4. [Section 6] The limitation to leading order in N_c is stated, but it would help to spell out explicitly that the factorization claims are only established at this order and that subleading-color contributions could in principle modify the factorized form; the conclusion already hints at this but the abstract does not.

Circularity Check

0 steps flagged

No significant circularity: the factorization claims are derived from independently computed opacity-expansion ingredients, and the only same-author citation is a consistency check, not a load-bearing input.

full rationale

The derivation chain is non-circular. The paper starts from the standard Catani-Grazzini representation (Eqs. (1.3)-(1.4)), computes the first-order opacity corrections to 1->2 splittings in Section 2, then builds the q->q c cbar splitting function in Eq. (4.89) from independently determined color factors (4.2)-(4.10), longitudinal phase integrals (4.33)-(4.61), and Dirac terms (4.64)-(4.87). The central factorization results (5.30), (5.49) and (5.52) are obtained by taking explicit limits of these phase integrals and identifying which diagrams survive, e.g. Eqs. (5.14)-(5.22) and (5.34)-(5.46). The 1->2 building blocks such as P^(N=1)_{g->c cbar} in Eq. (2.67) and P^(N=1)_{q->qg} in Eq. (2.71) are derived earlier in the same paper, not assumed from the triple-collinear result. The only noticeable same-group citation is Ref. [55], used at Eq. (2.69) as a consistency check for the rederived medium-modified g->c cbar splitting function; it is not used to derive the triple-collinear factorization and is therefore not load-bearing. No uniqueness theorem or ansatz is imported from the authors' prior work. The proof is conditional on the generalized strong-ordering assumption (5.11)-(5.12), i.e. B_i >> C_j and sufficiently small medium momentum transfer q, and on leading-N_c truncation (4.6), (4.10); the paper does not bound the q-integration window over which factorization holds. This is a correctness/completeness limitation, not circularity: no fitted parameter is renamed as a prediction and no output quantity sets an input assumption.

Axiom & Free-Parameter Ledger

0 free parameters · 6 axioms · 0 invented entities

The calculation has no fitted constants: zeta and z are kinematic variables, n0, L, and |a(q)|^2 are medium-model inputs, and the strong-ordering scales are limits rather than fitted numbers. The load is carried by standard collinear factorization plus the Gyulassy-Wang assumptions and the leading-color/small-q truncations. No new particles, mediators, forces, or conserved quantities are introduced.

axioms (6)
  • standard math Catani-Grazzini dispersive representation of collinear splitting functions (Eqs. (1.3), (1.4)) in axial gauge
    Starting point of the construction; assumed valid as in Ref. [50].
  • domain assumption Gyulassy-Wang model for the medium: A- field with instantaneous scatterers transferring only transverse momentum, homogeneous density n0 theta(y+) theta(L-y+), opacity expansion linear in n0 L |a(q)|^2
    Section 2.3; all medium results inherit this model.
  • domain assumption Time-ordered perturbation theory in mixed representation with instantaneous contributions dropped via the arguments of Sections 2.1 and 3.1-3.2
    Core technical reduction; if instantaneous terms did not cancel, the phase-integral and Dirac factorizations would change.
  • domain assumption Leading-order in N_c truncation; color factors C(14), C(23) and Cbar(23) vanish at this order
    Eqs. (4.6), (4.10); the factorization proof is carried out only at this order.
  • domain assumption Strongly ordered kinematics with O(1) momentum fractions and small medium transfer q such that B_i >> C_j after momentum shifts
    Eqs. (5.11)-(5.12); defines the kinematic window in which factorization is claimed.
  • standard math Known vacuum factorization of spin-correlated 1->2 tensors (5.4) and (5.5)
    Taken from Refs. [50,56]; used to lift scalar splitting functions to the tensor form needed for the factorization proof.

pith-pipeline@v1.3.0-alltime-deepseek · 46887 in / 15810 out tokens · 136560 ms · 2026-08-01T15:33:14.280127+00:00 · methodology

0 comments
read the original abstract

In strongly ordered collinear limits, triple-collinear $1\to 3$ splitting functions factorize into products of $1\to 2$ splitting functions. Here, we investigate whether, and under which conditions, this factorization persists in a finite QCD medium. To this end, we reformulate the opacity expansion as a medium-modification of the compact Catani-Grazzini representation of collinear $1\to n$ splitting functions. We compute the fully differential medium-modified triple-collinear $q\to q c\bar{c}$ splitting function to first order in opacity and leading order in $N_c$, and show that it can be expressed as a sum of momentum-shifted vacuum $q\to q c\bar{c}$ splitting functions multiplied by simple interference factors. In vacuum, this splitting function has a single strongly ordered collinear limit, in which the invariant mass of the outgoing $c\bar{c}$ pair is much smaller than all other invariant masses. The medium-modified splitting function instead exhibits three distinct strongly ordered limits, depending on how the relevant invariant masses -- multiplied by a boost factor and interpretable as inverse formation times -- compare with the medium length. Remarkably, in all three cases the triple-collinear splitting factorizes also in the medium into a tensor product of $q\to qg$ and $g\to c\bar{c}$ splitting functions. In two of these limits, the latter are medium modified to first order in opacity. This first proof shows that factorization extends to medium-modified triple-collinear splitting functions. Lastly, we relate our results to the longstanding problem of overlapping formation times and discuss their implications for jet-quenching parton showers based on medium-modified $1\to 2$ splittings. We also outline further applications of our formalism and note that our results may inform future phenomenological studies of spin (de)correlations in final-state radiation in dense QCD matter.

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

60 extracted references · 57 linked inside Pith

  1. [1]

    Baier, Y

    R. Baier, Y. L. Dokshitzer, A. H. Mueller, S. Peigne, and D. Schiff,Radiative energy loss of high-energy quarks and gluons in a finite volume quark - gluon plasma,Nucl. Phys. B483 (1997) 291–320, [hep-ph/9607355]

    Pith/arXiv arXiv 1997

  2. [2]

    Baier, Y

    R. Baier, Y. L. Dokshitzer, A. H. Mueller, S. Peigne, and D. Schiff,Radiative energy loss and p(T) broadening of high-energy partons in nuclei,Nucl. Phys. B484(1997) 265–282, [hep-ph/9608322]

    Pith/arXiv arXiv 1997

  3. [3]

    B. G. Zakharov,Fully quantum treatment of the Landau-Pomeranchuk-Migdal effect in QED and QCD,JETP Lett.63(1996) 952–957, [hep-ph/9607440]

    Pith/arXiv arXiv 1996

  4. [4]

    U. A. Wiedemann,Gluon radiation off hard quarks in a nuclear environment: Opacity expansion,Nucl. Phys. B588(2000) 303–344, [hep-ph/0005129]

    Pith/arXiv arXiv 2000

  5. [5]

    Gyulassy, P

    M. Gyulassy, P. Levai, and I. Vitev,Reaction operator approach to nonAbelian energy loss, Nucl. Phys. B594(2001) 371–419, [nucl-th/0006010]

    Pith/arXiv arXiv 2001

  6. [6]

    Gyulassy and X.-n

    M. Gyulassy and X.-n. Wang,Multiple collisions and induced gluon Bremsstrahlung in QCD, Nucl. Phys. B420(1994) 583–614, [nucl-th/9306003]

    Pith/arXiv arXiv 1994

  7. [7]

    Wang and X.-f

    X.-N. Wang and X.-f. Guo,Multiple parton scattering in nuclei: Parton energy loss,Nucl. Phys. A696(2001) 788–832, [hep-ph/0102230]

    Pith/arXiv arXiv 2001

  8. [8]

    P. B. Arnold, G. D. Moore, and L. G. Yaffe,Effective kinetic theory for high temperature gauge theories,JHEP01(2003) 030, [hep-ph/0209353]. [9]PHENIXCollaboration, K. Adcox et al.,Formation of dense partonic matter in relativistic nucleus-nucleus collisions at RHIC: Experimental evaluation by the PHENIX collaboration, Nucl. Phys. A757(2005) 184–283, [nucl...

    Pith/arXiv arXiv 2003

  9. [13]

    Wang and U

    X.-N. Wang and U. A. Wiedemann,QGP@50: More than Four Decades of Jet Quenching, 8, 2025.arXiv:2508.18794. – 54 –

    Pith/arXiv arXiv 2025

  10. [14]

    J. R. Forshaw, J. Holguin, and S. Plätzer,Building a consistent parton shower,JHEP09(2020) 014, [arXiv:2003.06400]

    Pith/arXiv arXiv 2020

  11. [15]

    Nagy and D

    Z. Nagy and D. E. Soper,Summations of large logarithms by parton showers,Phys. Rev. D104 (2021), no. 5 054049, [arXiv:2011.04773]

    Pith/arXiv arXiv 2021

  12. [16]

    Herren, S

    F. Herren, S. Höche, F. Krauss, D. Reichelt, and M. Schoenherr,A new approach to color-coherent parton evolution,JHEP10(2023) 091, [arXiv:2208.06057]

    Pith/arXiv arXiv 2023

  13. [17]

    C. T. Preuss,A partitioned dipole-antenna shower with improved transverse recoil,JHEP07 (2024) 161, [arXiv:2403.19452]

    Pith/arXiv arXiv 2024

  14. [18]

    van Beekveld et al.,New Standard for the Logarithmic Accuracy of Parton Showers,Phys

    M. van Beekveld et al.,New Standard for the Logarithmic Accuracy of Parton Showers,Phys. Rev. Lett.134(2025), no. 1 011901, [arXiv:2406.02661]

    Pith/arXiv arXiv 2025

  15. [19]

    K. Zapp, J. Stachel, and U. A. Wiedemann,A Local Monte Carlo implementation of the non-abelian Landau-Pomerantschuk-Migdal effect,Phys. Rev. Lett.103(2009) 152302, [arXiv:0812.3888]

    Pith/arXiv arXiv 2009

  16. [20]

    Caucal, E

    P. Caucal, E. Iancu, A. H. Mueller, and G. Soyez,Vacuum-like jet fragmentation in a dense QCD medium,Phys. Rev. Lett.120(2018) 232001, [arXiv:1801.09703]

    Pith/arXiv arXiv 2018

  17. [21]

    Caucal, E

    P. Caucal, E. Iancu, and G. Soyez,Deciphering thezg distribution in ultrarelativistic heavy ion collisions,JHEP10(2019) 273, [arXiv:1907.04866]

    Pith/arXiv arXiv 2019

  18. [22]

    K. C. Zapp, F. Krauss, and U. A. Wiedemann,A perturbative framework for jet quenching, JHEP03(2013) 080, [arXiv:1212.1599]

    Pith/arXiv arXiv 2013

  19. [23]

    K. C. Zapp,JEWEL 2.0.0: directions for use,Eur. Phys. J. C74(2014), no. 2 2762, [arXiv:1311.0048]

    Pith/arXiv arXiv 2014

  20. [24]

    Schenke, C

    B. Schenke, C. Gale, and S. Jeon,MARTINI: An Event generator for relativistic heavy-ion collisions,Phys. Rev. C80(2009) 054913, [arXiv:0909.2037]

    Pith/arXiv arXiv 2009

  21. [25]

    Y. He, T. Luo, X.-N. Wang, and Y. Zhu,Linear Boltzmann Transport for Jet Propagation in the Quark-Gluon Plasma: Elastic Processes and Medium Recoil,Phys. Rev. C91(2015) 054908, [arXiv:1503.03313]. [Erratum: Phys.Rev.C 97, 019902 (2018)]

    Pith/arXiv arXiv 2015

  22. [26]

    T. Luo, Y. He, S. Cao, and X.-N. Wang,Linear Boltzmann transport for jet propagation in the quark-gluon plasma: Inelastic processes and jet modification,Phys. Rev. C109(2024), no. 3 034919, [arXiv:2306.13742]

    Pith/arXiv arXiv 2024

  23. [27]

    J. H. Putschke et al.,The JETSCAPE framework,arXiv:1903.07706

    Pith/arXiv arXiv 1903

  24. [28]

    Caucal, E

    P. Caucal, E. Iancu, A. H. Mueller, and G. Soyez,Nuclear modification factors for jet fragmentation,JHEP10(2020) 204, [arXiv:2005.05852]

    Pith/arXiv arXiv 2020

  25. [29]

    Caucal, E

    P. Caucal, E. Iancu, and G. Soyez,Jet radiation in a longitudinally expanding medium,JHEP 04(2021) 209, [arXiv:2012.01457]

    Pith/arXiv arXiv 2021

  26. [30]

    Casalderrey-Solana, D

    J. Casalderrey-Solana, D. C. Gulhan, J. G. Milhano, D. Pablos, and K. Rajagopal,A Hybrid Strong/Weak Coupling Approach to Jet Quenching,JHEP10(2014) 019, [arXiv:1405.3864]. [Erratum: JHEP 09, 175 (2015)]

    Pith/arXiv arXiv 2014

  27. [31]

    Armesto, H

    N. Armesto, H. Ma, Y. Mehtar-Tani, C. A. Salgado, and K. Tywoniuk,Coherence effects and broadening in medium-induced QCD radiation off a massiveq¯qantenna,JHEP01(2012) 109, [arXiv:1110.4343]. – 55 –

    Pith/arXiv arXiv 2012

  28. [32]

    Casalderrey-Solana and E

    J. Casalderrey-Solana and E. Iancu,Interference effects in medium-induced gluon radiation, JHEP08(2011) 015, [arXiv:1105.1760]

    Pith/arXiv arXiv 2011

  29. [33]

    Mehtar-Tani, C

    Y. Mehtar-Tani, C. A. Salgado, and K. Tywoniuk,Anti-angular ordering of gluon radiation in QCD media,Phys. Rev. Lett.106(2011) 122002, [arXiv:1009.2965]

    Pith/arXiv arXiv 2011

  30. [34]

    Mehtar-Tani, C

    Y. Mehtar-Tani, C. A. Salgado, and K. Tywoniuk,Jets in QCD Media: From Color Coherence to Decoherence,Phys. Lett. B707(2012) 156–159, [arXiv:1102.4317]

    Pith/arXiv arXiv 2012

  31. [35]

    Mehtar-Tani, C

    Y. Mehtar-Tani, C. A. Salgado, and K. Tywoniuk,The radiation pattern of a QCD antenna in a dilute medium,JHEP04(2012) 064, [arXiv:1112.5031]

    Pith/arXiv arXiv 2012

  32. [36]

    Mehtar-Tani, C

    Y. Mehtar-Tani, C. A. Salgado, and K. Tywoniuk,The Radiation pattern of a QCD antenna in a dense medium,JHEP10(2012) 197, [arXiv:1205.5739]

    Pith/arXiv arXiv 2012

  33. [37]

    Barata, F

    J. Barata, F. Domínguez, C. A. Salgado, and V. Vila,A modified in-medium evolution equation with color coherence,JHEP05(2021) 148, [arXiv:2101.12135]

    Pith/arXiv arXiv 2021

  34. [38]

    Abreu, X

    S. Abreu, X. Mayo López, G. Milhano, and A. Soto-Ontoso,A generalized picture of colour decoherence in dense QCD media,JHEP03(2025) 216, [arXiv:2410.24135]

    Pith/arXiv arXiv 2025

  35. [39]

    T. Liou, A. H. Mueller, and B. Wu,Radiativep⊥-broadening of high-energy quarks and gluons in QCD matter,Nucl. Phys. A916(2013) 102–125, [arXiv:1304.7677]

    Pith/arXiv arXiv 2013

  36. [40]

    Blaizot and Y

    J.-P. Blaizot and Y. Mehtar-Tani,Renormalization of the jet-quenching parameter,Nucl. Phys. A929(2014) 202–229, [arXiv:1403.2323]

    Pith/arXiv arXiv 2014

  37. [41]

    Wu,Radiative energy loss and radiativep⊥-broadening of high-energy partons in QCD matter,JHEP12(2014) 081, [arXiv:1408.5459]

    B. Wu,Radiative energy loss and radiativep⊥-broadening of high-energy partons in QCD matter,JHEP12(2014) 081, [arXiv:1408.5459]

    Pith/arXiv arXiv 2014

  38. [42]

    Casalderrey-Solana, D

    J. Casalderrey-Solana, D. Pablos, and K. Tywoniuk,Two-gluon emission and interference in a thin QCD medium: insights into jet formation,JHEP11(2016) 174, [arXiv:1512.07561]

    Pith/arXiv arXiv 2016

  39. [43]

    Arnold and S

    P. Arnold and S. Iqbal,The LPM effect in sequential bremsstrahlung,JHEP04(2015) 070, [arXiv:1501.04964]. [Erratum: JHEP 09, 072 (2016)]

    Pith/arXiv arXiv 2015

  40. [44]

    Arnold, H.-C

    P. Arnold, H.-C. Chang, and S. Iqbal,The LPM effect in sequential bremsstrahlung 2: factorization,JHEP09(2016) 078, [arXiv:1605.07624]

    arXiv 2016

  41. [45]

    Arnold and S

    P. Arnold and S. Iqbal,In-medium loop corrections and longitudinally polarized gauge bosons in high-energy showers,JHEP12(2018) 120, [arXiv:1806.08796]. [Erratum: JHEP 12, 098 (2023), Erratum: JHEP 09, 169 (2024)]

    Pith/arXiv arXiv 2018

  42. [46]

    Arnold, T

    P. Arnold, T. Gorda, and S. Iqbal,The LPM effect in sequential bremsstrahlung: nearly complete results for QCD,JHEP11(2020) 053, [arXiv:2007.15018]. [Erratum: JHEP 05, 114 (2022)]

    Pith/arXiv arXiv 2020

  43. [47]

    Arnold, T

    P. Arnold, T. Gorda, and S. Iqbal,The LPM effect in sequential bremsstrahlung: analytic results for sub-leading (single) logarithms,JHEP04(2022) 085, [arXiv:2112.05161]

    Pith/arXiv arXiv 2022

  44. [48]

    Arnold, O

    P. Arnold, O. Elgedawy, and S. Iqbal,Landau-Pomeranchuk-Migdal effect in sequential bremsstrahlung: Gluon shower development,Phys. Rev. D108(2023), no. 7 074015, [arXiv:2302.10215]

    Pith/arXiv arXiv 2023

  45. [49]

    Fickinger, G

    M. Fickinger, G. Ovanesyan, and I. Vitev,Angular distributions of higher order splitting functions in the vacuum and in dense QCD matter,JHEP07(2013) 059, [arXiv:1304.3497]. – 56 –

    Pith/arXiv arXiv 2013

  46. [50]

    Catani and M

    S. Catani and M. Grazzini,Infrared factorization of tree level QCD amplitudes at the next-to-next-to-leading order and beyond,Nucl. Phys. B570(2000) 287–325, [hep-ph/9908523]

    Pith/arXiv arXiv 2000

  47. [51]

    H. E. Haber,Useful relations among the generators in the defining and adjoint representations of SU(N),SciPost Phys. Lect. Notes21(2021) 1, [arXiv:1912.13302]

    Pith/arXiv arXiv 2021

  48. [52]

    Peigné,Color in QCD: An Introduction Featuring the Birdtrack Pictorial Technique

    S. Peigné,Color in QCD: An Introduction Featuring the Birdtrack Pictorial Technique. SpringerBriefs in Physics. Springer, 5, 2024

    2024

  49. [53]

    Shtabovenko, R

    V. Shtabovenko, R. Mertig, and F. Orellana,New Developments in FeynCalc 9.0,Comput. Phys. Commun.207(2016) 432–444, [arXiv:1601.01167]

    Pith/arXiv arXiv 2016

  50. [54]

    Shtabovenko, R

    V. Shtabovenko, R. Mertig, and F. Orellana,FeynCalc 9.3: New features and improvements, Comput. Phys. Commun.256(2020) 107478, [arXiv:2001.04407]

    Pith/arXiv arXiv 2020

  51. [55]

    Attems, J

    M. Attems, J. Brewer, G. M. Innocenti, A. Mazeliauskas, S. Park, W. van der Schee, and U. A. Wiedemann,The medium-modifiedg→c csplitting function in the BDMPS-Z formalism,JHEP 01(2023) 080, [arXiv:2203.11241]

    Pith/arXiv arXiv 2023

  52. [56]

    Craft, M

    E. Craft, M. Gonzalez, K. Lee, B. Mecaj, and I. Moult,The 1→3 massive splitting functions from QCD factorization and SCET,JHEP07(2024) 080, [arXiv:2310.06736]

    Pith/arXiv arXiv 2024

  53. [57]

    P. K. Dhani, G. Rodrigo, and G. F. R. Sborlini,Triple-collinear splittings with massive particles,JHEP12(2023) 188, [arXiv:2310.05803]

    Pith/arXiv arXiv 2023

  54. [58]

    Attems, J

    M. Attems, J. Brewer, G. M. Innocenti, A. Mazeliauskas, S. Park, W. van der Schee, G. Soyez, and U. A. Wiedemann,Medium-Enhanced cc¯Radiation,Phys. Rev. Lett.132(2024), no. 21 212301, [arXiv:2209.13600]

    Pith/arXiv arXiv 2024

  55. [59]

    Brewer, W

    J. Brewer, W. van der Schee, and U. Wiedemann,Testing the QCD formation time with reconstructed parton splittings,arXiv:2503.11764

    Pith/arXiv arXiv

  56. [60]

    J. M. Silva and A. Soto-Ontoso,Collinear spin correlations of final-state radiation in dense QCD matter,JHEP05(2026) 026, [arXiv:2511.17737]

    arXiv 2026

  57. [61]

    Hauksson and E

    S. Hauksson and E. Iancu,Jet polarisation in an anisotropic medium,JHEP08(2023) 027, [arXiv:2303.03914]

    Pith/arXiv arXiv 2023

  58. [62]

    Barata, C

    J. Barata, C. A. Salgado, and J. M. Silva,Gluon to qq antenna in anisotropic QCD matter: spin-polarized and azimuthal jet observables,JHEP12(2024) 023, [arXiv:2407.04774]

    Pith/arXiv arXiv 2024

  59. [63]

    Bossi, A

    H. Bossi, A. S. Kudinoor, I. Moult, D. Pablos, A. Rai, and K. Rajagopal,Imaging the wakes of jets with energy-energy-energy correlators,JHEP12(2024) 073, [arXiv:2407.13818]

    Pith/arXiv arXiv 2024

  60. [64]

    Barata, I

    J. Barata, I. Moult, A. V. Sadofyev, and J. M. Silva,Dissecting Jet Modification in the QGP with Multi-Point Energy Correlators,arXiv:2503.13603. – 57 –

    Pith/arXiv arXiv