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New formalism for QCD parton showers

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Stefan Gieseke, Philip Stephens, Bryan Webber

High Energy Physics - Phenomenology · Submitted 7 Oct 2003 · 26 references

Pith's verdict: A careful, explicit formalism paper whose kinematics and soft-limit checks hold up, but whose comparative claims in the abstract are deferred to a companion paper that was still in preparation.

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arXiv IDhep-ph/0310083 (version 2)
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CitationBryan Webber, Philip Stephens, Stefan Gieseke. "New formalism for QCD parton showers." arXiv:hep-ph/0310083 (2003). https://pith.science/paper/HQXS4GTA
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The paper

The paper claims

The authors propose a new ordering variable for QCD parton showers that keeps angular ordering, handles heavy-quark masses, and makes the shower's soft-gluon angular distribution match the exact matrix element in every colour-connection 配置.

The reading

A parton shower is the step that turns the handful of quarks and gluons produced in a hard collision into the hundreds of particles a detector records, and its accuracy depends on how the cascade is ordered. This paper proposes ordering each branching by a variable built from the emission's transverse momentum divided by the momentum fractions, $z^2(1-z)^2$ in the denominator, rather than by virtuality or by the older angular variable used in HERWIG. The authors show that this variable is invariant under boosts along the jet axis, that it can be continued below the heavy-quark mass so the collinear dead cone fills in, and that mass-dependent splitting functions take a compact form in it. The load-bearing step is a bookkeeping rule: for each pair of colour-connected partons the upper limits on the evolution variable are chosen so that the two jet regions just touch in the soft limit, which the authors check reproduces the exact soft-gluon angular distribution in electron-positron annihilation, Drell-Yan, deep inelastic scattering and top decay. If that holds, the shower can be corrected to the exact matrix element everywhere by a simple rejection step, which is what the accompanying implementation does.

What carries the argument

The carrier is a Sudakov-style momentum parametrization: each branching is written as $q_i = alpha_i p + beta_i n + q_{perp i}$, where $p$ is the parent parton momentum and $n$ is a lightlike vector taken along the colour-connected partner. The evolution variable is then $tilde q^2 = p_perp^2/[z^2(1-z)^2] + mu^2/z^2 + Q_g^2/[z(1-z)^2]$ for final-state branching, with $mu$ the larger of the quark mass and the gluon cutoff, so that the same variable measures angle in the massless limit and stays finite inside the dead cone for massive quarks. The second carrier is the jet-boundary rule: the maximum $tilde q$ allowed in each jet is fixed by requiring the two colour-connected jet regions to touch exactly in the soft limit, which is what makes the shower's soft-gluon angular distribution match the exact matrix element rather than merely approximating it.

Extended reading notes

Core claim

The claim is that a shower ordered in $z^2(1-z)^2$ times the new evolution variable $z^2(1-z)^2$ reproduces the correct soft-gluon angular distribution for every configuration of colour connection, provided the upper limits on the evolution variable for the two colour-connected jets are chosen so that the jet regions meet in the soft limit. The authors verify this case by case: for two final-state jets the condition is $(1-b-c+lambda)^2/4$ relating the two limits, for two beam jets the limits multiply to one, for one beam and one final jet the product condition involves the final parton's mass, and for a decaying heavy parton the analogous product condition holds. In each case they compare the shower's soft limit with the exact first-order matrix element and find that the shower always overestimates it, so a rejection step is enough to correct the distribution. They also work out the full kinematics, the argument of the running coupling, and the allowed phase-space boundaries for massive branching, and note that the region of hard, wide-angle gluon emission lies outside both jet regions and must be filled from the matrix element.

Load-bearing premise

The construction assumes that leading-colour coherence is fully captured by choosing the upper limits on the evolution variable so that colour-connected jet regions meet in the soft limit, a modelling choice justified only by leading-logarithmic soft-limit checks rather than derived from the QCD amplitude.

Editorial extensions

If this is right

  • Because the evolution variable is invariant under boosts along the jet axis, the radiation pattern does not depend on which frame along that axis the shower is run in, which simplifies matching to fixed-order calculations.
  • Evolving down to zero transverse momentum with mass-dependent splitting functions removes the sharply defined collinear dead cones of earlier angular-ordered treatments, changing predictions for heavy-quark fragmentation.
  • The shower is claimed to overestimate the exact matrix element in the soft limit in all four colour-connection cases, so matrix-element corrections reduce to a rejection step rather than a subtraction.
  • One formalism covers final-final, initial-initial, initial-final and decay colour connections, so the same machinery serves electron-positron annihilation, hadron colliders, deep inelastic scattering and top decay.
  • The soft region is covered more efficiently, shrinking the dead region that has to be filled by explicit matrix-element events.

Reading between the lines

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

  • The rule that colour-connected jet regions touch in the soft limit is a partition of phase space chosen for convenience, not something derived from the QCD amplitude; the checks in the paper are all leading-logarithmic soft-limit comparisons, so a different true coherence pattern at finite angle would not show up in them.
  • The claim that the shower overestimates the exact result in the soft limit suggests a clean numerical test: plot the ratio of shower to exact matrix element across the whole jet region, not just its soft corner, and check whether it stays above one everywhere.
  • The authors themselves note that in the hard-gluon region of top decay the shower underestimates the matrix element at the highest gluon energies, so the overestimation property is not global and a practical implementation needs to know where that boundary lies.
  • The boost invariance along the jet axis looks like a natural language for matching showers to next-to-leading-order calculations, which the authors flag as future work rather than carry out here.
Original abstract

We present a new formalism for parton shower simulation of QCD jets, which incorporates the following features: invariance under boosts along jet axes, improved treatment of heavy quark fragmentation, angular-ordered evolution with soft gluon coherence, more accurate soft gluon angular distributions, and better coverage of phase space. It is implemented in the new HERWIG++ event generator.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 2,301 input tokens, $0.0001 reported.

What the abstract itself states, as probabilities: claims first 0.06 · states limitation 0.05 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.03 · compares with prior results 0.18 · mentions code or data availability 0.27.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; .

We present a new formalism for parton shower simulation of QCD jets, which incorporates the following features: invariance under boosts along jet axes, improved treatment of heavy quark fragmentation, angular-ordered evolution with soft gluon coherence, more accurate soft gluon angular distributions, and better coverage of phase space. It is implemented in the new HERWIG++ event generator.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We present a new formalism for parton shower simulation of QCD jets, which incorporates the following features: invariance under boosts along jet axes, improved treatment of heavy quark fragmentation, angular-ordered evolution with soft gluon coherence, more accurate soft gluon angular distributions, and better coverage of phase space. contribution 0.99 0.92 0.57 asserted 1.00
s1 It is implemented in the new HERWIG++ event generator. method 0.61 0.77 0.06 asserted 1.00

What kind of paper the classifier reports this is: method or tool 1.00 · evidence from simulation or computation 0.75.

What was examined, and what was not

sources read
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not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
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  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. Final-state parton branching. The blob represents the hard subprocess. The momentum fraction and relative transverse momentum are now defined as zi = αi αi−1 , p⊥i = q⊥i − ziq⊥i−1 . (2.3) Then we have q 2 i−1 = q 2 i zi + k 2 i 1 − zi + p⊥i 2 zi(1 − zi) . (2.4) 2.1.2 Running coupling To find the optimal argument of αS, we consider the branching of a quark of virtuality q 2 into an on-shell quark and an off-shell gluon of virtuality k 2 [15]. From eq. (2.4), the propagator denominator is q 2 − m2 = 1 − z z m2 + k 2 1 − z + p⊥ 2 z(1 − z) = 1 1 − z
Figure 2. Initial-state parton branching. The blob represents the hard subprocess. Then q 2 i−1 = ziq 2 i − zi 1 − zi k 2 i − p⊥i 2 1 − zi . (2.18) We assume a massless variable-flavour-number evolution scheme [19,20] for constituent parton branching, setting m = 0 and putting all emitted gluons at the minimum virtuality, k 2 i = Q2 g . The angular evolution variable now relates only to the angle of the emitted gluon and therefore we choose q˜ 2 i = p⊥i 2 + ziQ2 g (1 − zi) 2 , (2.19) with ordering condition simply ˜qi+1 < q˜i . Correspondingly, for the argument of the
Figure 3. Phase space for e +e − → qqg¯ for mq = 5 GeV, Q2 = m2 Z, with symmetric definition of quark and antiquark jets. For the maximal quark jet we get from eq. (3.24) κ˜q = 4(1 − 2 √ ρ) , (3.25) as shown in fig. 4 together with the complementary antiquark jet region given by eq. (3.22). – 11 –
Figure 4. Phase space for e +e − → qqg¯ for mq = 5 GeV, Q2 = m2 Z , with maximal region for the quark jet. 3.3.1 Exact matrix element The e +e − → V → qqg¯ differential cross section, where V represents a vector current such as a virtual photon, is given to first order in αS by [22, 23] 1 σV d 2σV dxq dxq¯ = αS 2π CF v  (xq + 2ρ) 2 + (xq¯ + 2ρ) 2 + ζV (1 + 2ρ)(1 − xq)(1 − xq¯) − 2ρ (1 − xq) 2 − 2ρ (1 − xq¯) 2  (3.26) where ζV = −8ρ(1 + 2ρ) (3.27) and σV = σ0 (1 + 2ρ) v (3.28) is the Born cross section for heavy quark production by a vector current, σ0 being the
Figure 5. The function f(˜κ) giving the gluon angular distribution in the soft limit, for m = 5 GeV, Q2 = m2 Z . The exact result eq. (3.33), solid curve, and shower approximation (3.32), dashed, are not distinguishable on this scale. into a region whose width vanishes quadratically as xq, xq¯ → 1, as illustrated in fig. 7. The mapping shown is xq → x ′ q = 1 −  1 4 − (1 − xq)  = 7 4 − xq , xq¯ → x ′ q¯ = 1 − 2(1 − x ′ q )  3 4 − (1 − xq)  = 5 8 + 1 2 xq + 3 2 xq¯ − 2xqxq¯ (3.37) when xq > xq¯ > 3 4 . Within the mapped region, the integrand then has an extr
Figure 6. The function F D(˜κq) giving the contribution of the dead region to the cross section, for m = 5 GeV, Q2 = m2 Z. Solid: vector current. Dashed: axial current. For emission of a gluon g from b we write qi = αipb + βipc + q⊥i (4.2) where q⊥g = k⊥, q⊥a = −k⊥, q⊥b = q⊥c = 0. Notice that in this case the recoil transverse momentum is taken by the colour singlet a so we cannot preserve its 4-momentum. We choose to preserve its mass and rapidity, so that αa = βa = √ 1 + κ , (4.3) where as before κ ≡ k⊥ 2 /Q2 . Now we have βb = αc = 0 , αgβg = κ , αa = αb − αg , βa = βc − βg , (4.4)
Figure 7. The soft region, with jet boundaries (solid) and mapped region (dashed), for m = 5 GeV, Q2 = m2 Z . Thus we find αa = βa = p 1 + (1 − z) 2κ , ˜ αb = 1 z p 1 + (1 − z) 2κ , ˜ βc = 1 + (1 − z)˜κ p 1 + (1 − z) 2κ˜ . (4.6) 4.1 Phase space variables It is convenient to express the kinematics in terms of the “reduced” Mandelstam invariants: s¯ = (qb + qc) 2 /Q2 , t¯= (qb − qg) 2 /Q2 , u¯ = (qc − qg) 2 /Q2 . (4.7) The phase space limits are 1 < s < S/Q ¯ 2 , 1 − s <¯ t <¯ 0 , u¯ = 1 − s¯− t¯ (4.8) where S is the beam-beam c.m. energy squared. In terms of the show
Figure 8. Beam jets (B,C) and dead region (D) in initial-state branching. Thus curves of constant ˜κ in the (¯s,t¯) plane are given by t¯= κ˜(1 − s¯) κ˜ + ¯s (4.10) and the Jacobian factor for conversion of the shower variables to the Mandelstam invariants is ∂(¯s,t¯) ∂(z, κ˜) = 1 − z z s . ¯ (4.11) For the other beam jet c we have t¯↔ u¯ and thus t¯= s¯(1 − s¯) κ˜ + ¯s . (4.12) We see that in order for the jet regions to touch without overlapping in the soft limit ¯s → 1, t¯→ 0, we need ˜κ < κ˜b in jet b and ˜κ < κ˜c in jet c, where ˜κc = 1/κ˜b. The most symmetrical choice is ˜κc
Figure 9. Beam jet (B), outgoing jet (C) and dead region (D) in initial-final state branching: c = 0.25, ˜κb = 1.25, ˜κc = 1.5
Figure 10. Beam jet (B), outgoing jet (C) and dead region (D) in initial-final state branching: c = 0.25, ˜κb = 10, ˜κc = 0.40625. 5.4 Example: deep inelastic scattering Consider deep inelastic scattering on a hadron of momentum P µ by exchange of a virtual photon of momentum q µ . If the contribution to the Born cross section from scattering on a quark of momentum fraction xB = Q2/2P · q is represented by σ0 (a function of xB and Q2 ), then the correction due to single gluon emission is given by 1 σ0 d 2σ dxp dzp = CF αS 2π D(xB/xp) D(xB) 1 + (xp + zp − 1)2 xp(1 − xp)(1 − zp) . (
Figure 11. Phase space for decay t → W bg, with symmetric choice of emission regions for the b (B) and the t (T1,T2), and the dead region (D). The exact t → W bg differential decay rate to first order in αS is given in [26]: 1 Γ0 d 2Γ dxW dxg = αS π CF (1 + a − xW )x 2 g ( xg − (1 + a − xW )(1 − xg) + x 2 g 1 − a +xg (xW + xg − 1 − a) 2 2(1 − a) 2 + 2a(1 + a − xW )x 2 g (1 − a) 2(1 + 2a) ) (6.31) where Γ0 is the lowest-order decay rate. In the soft region xW → 1 + a, xg → 0 this becomes 1 Γ0 d 2Γ dxW dxg ∼ αS π CF xg  1 1 + a − xW − 1 (1 − a)xg  . (
Figure 12. Phase space for decay t → W bg, with maximal region (B) for emission from the b, together with complementary regions of emission from the t (T1,T2) and the dead region (D). where ft(˜κ) = κ˜ − 1 (1 − a)˜κ2 . (6.35) In the same region the parton shower approximation (2.12) gives d 2P dxgdxW ∼ 1 (1 − a)ǫ d 2P dz dκ˜ ∼ αs π CF (1 − a)ǫ 2κ˜  1 − 1 κ˜  = αs π CF ǫ 2 f(˜κ) . (6.36) Thus we see that, for emission from the top quark, the shower approximation is exact in the soft limit. At higher gluon energies, inside the region T1 the parton shower overestimates

Figures reproduced from arXiv:hep-ph/0310083 by the authors.

References
How later papers used it

94 later papers cite it

Evidence: substantial. 84 of the citing papers share no author with it, and 32 of those carry a confident label from Pith's citation classifier.

Pith read 23 of the 153 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

Uses its method or tool 4 of 37 papers with this label · 1 by the same authors

By the same authors

How the literature uses it

86 arXiv papers cite this in 144 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 84 sentences without a confident label

Uses its method or tool 35 papers
  • In this paper we will use a similar approach to that of Ref. [ref] in order to analyse the behaviour of the improved angular-ordered shower of Ref. [ref] .

    Logarithmic Accuracy of Angular-Ordered Parton Showers · Introduction · 0.92

  • We therefore combine our fixed-order NLO results (supplemented with loop-induced and EFT contributions) with the angular-ordered parton shower [ref] through the subtractive (i.e. MC@NLO-like) matching algorithm based on [ref] .[The loop-induced and EFT contributions are treated as LO QCD processes in that regard.] Uncertainties in the shower are mainly quantified by varying the hard shower scale $μ_Q$ which provides a reliable estimate of missing logarithmic orders as well as the impact of large-angle, hard and thus unreliably modelled emissions.

    Anomalous coupling, top-mass and parton-shower effects in ${W^+W^-}$ production · Parton-shower effects · 0.99

  • To avoid this pole, the region $x_b,x_b̅>3/4$ is mapped into a region which includes $D$ but whose width vanishes quadratically as $x_b,x_b̅→1$ [ref] .

    MC@NLO for the hadronic decay of Higgs bosons in associated production with vector bosons · Divergences in dead region · 0.96

Compares with it 1 paper
  • A sample of eight million events produced with Herwig 7.1.4 [ref] is also used as an example of an angular ordered shower for comparison with Sherpa's $k_T$ ordered shower.

    Deep Learning as a Parton Shower · Jet Distributions Predicted by the CNN · 0.82

Cites it as support 1 paper
  • The second property comes from the tendency of parton shower to emit softer parton at earlier stage with larger opening angle [ref] .

    The Georgi Algorithms of Jet Clustering · Clustering with Massive Subjets · 0.96

Cites it as background 12 papers
  • In this paper we present a unified analysis of the two main approaches to formulating parton showers: dipole showers [ref] and angular ordered showers [ref] .

    Building a consistent parton shower · Introduction · 0.91

  • This restricts the integral over $q̃^2$ and $z$ to a region where [equation] We note that also other choices are in principle possible and have been discussed in the context of radiation within the 'dead cone' for massive quarks [ref] .

    On the Cutoff Dependence of the Quark Mass Parameter in Angular Ordered Parton Showers · Massless case · 0.85

  • Examples include 's angular-ordered showers [ref] , the Manchester-Vienna dipole shower [ref] , [ref] , and the [ref] , [ref] , and [ref] showers.

    Timelike showers with jet recoils · Introduction · 0.83

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