Pith. sign in

REVIEW 4 major objections 4 minor 60 references

This paper derives the complete set of leading and subleading massive collinear splitting functions for every Standard Model particle from on-shell massless amplitudes.

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 23:51 UTC pith:HLDJHTC4

load-bearing objection Genuinely useful constructive formalism for massive EW splitting functions, but the abstract's 'complete set' overstates what is actually shown — the paper itself drops the O(v^2/p_T^2) virtuality corrections at the same order. the 4 major comments →

arxiv 2607.15191 v2 pith:HLDJHTC4 submitted 2026-07-16 hep-ph hep-th

Massive On-shell Splitting Functions in Spinor-Helicity Formalism

Yi-Ning Wang , Chao Wu , Jiang-Hao Yu This is my paper
classification hep-ph hep-th
keywords massive splitting functionsspinor-helicity formalismlight-front Galilean symmetrycollinear power countingStandard Model splittingelectroweak showerssubleading mass correctionson-shell amplitude matching
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.

Collinear splitting functions describe how a highly boosted particle branches into nearly parallel daughters; for massive particles the mass corrections are usually added by hand from Feynman diagrams. This paper claims that the entire problem can be reorganized as an on-shell construction: a fixed light-front reference frame makes the hierarchy m < p_T << p_+ explicit, leading terms match massless three-point amplitudes, and mass-suppressed terms are matched by four-point massless amplitudes with an extra scalar leg along the anti-collinear direction. The result is a closed set of leading and subleading splitting functions for all Standard Model particles, together with a dictionary between massless and massive coupling coefficients and a recursive rule for higher-point splittings. A sympathetic reader would care because these functions are the input to parton showers and electroweak resummation, and the construction replaces case-by-case diagrammatic limits with symmetry-dictated amplitudes.

Core claim

The central claim is a constructive equivalence: in the alignment regime m < p_T << p_+, a massive collinear splitting amplitude decomposes into a leading piece that is exactly a massless three-point amplitude built from the large components of collinear spinors, and a subleading piece (order m) that is exactly a massless four-point amplitude containing an extra Higgs leg moving along the anti-collinear direction. The massless three-point amplitude fixes the leading splitting function P^(0)(z); the four-point amplitude, after the pole-to-mass replacement s_{ih} -> m_i^2, fixes P^(1)(z). The paper tabulates these functions for all SM processes, covering fermion, vector, and scalar parents, an

What carries the argument

The engine is the light-front collinear spinor basis: massive spinors are expanded on two fixed lightlike vectors n and n-bar after a longitudinal boost and a transverse Galilean boost. In this basis the two little-group components scale as sqrt(p_+) and p_T/sqrt(p_+), plus one mass-suppressed component m/sqrt(p_+), so the power counting is read off without a second expansion. The subleading component is invisible in three-point matching; the paper probes it by inserting a Higgs momentum along the anti-collinear direction, converting massless four-point poles s_{ih} into m_i^2. That Higgs-insertion rule, together with the massless-to-massive coupling dictionary, is what turns four-point ampl

Load-bearing premise

The load-bearing premise is that adding an extra scalar (Higgs) particle moving opposite the collinear direction, and replacing each resulting pole by the square of the daughter mass, reveals every subleading mass correction; if any spin configuration or coupling channel is missed by this rule, the claimed complete set of subleading splitting functions will be incomplete.

What would settle it

Take one massive splitting process (for example f -> W f) and compute the collinear limit of the full Feynman diagram at relative order m/p_T, keeping all terms up to m^2. If the coefficient of v^2/p_T^4 in the differential rate differs from z z-bar |P^(1)(z)|^2 obtained here, the Higgs-insertion dictionary is incomplete. A cheaper check: verify that each P^(1) entry satisfies the stated relation with the massless four-point amplitude under the subleading spinor replacement for every channel, not just the representative ones.

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

If this is right

  • Parton-shower Monte Carlos can adopt the derived P^(0) and P^(1) as branching kernels, giving a consistent treatment of top, W/Z, and Higgs thresholds at next-to-leading logarithmic accuracy.
  • The matching dictionary lets one compute massive coupling coefficients from massless amplitudes, so SM effective-field-theory operators can be imported into the same splitting-function machinery without new Feynman-diagram limits.
  • The recursive substitution rule constructs 1-to-3 and higher splitting amplitudes from 1-to-2 amplitudes, removing the need for case-by-case off-shell collinear limits.
  • In the massless limit the new functions reproduce the known massless splitting kernels, so the massive results are a controlled deformation of existing physics rather than a separate scheme.
  • Because the whole construction is on-shell, it extends to higher perturbative orders by the same amplitude-level matching, potentially simplifying two-loop collinear factorization.

Where Pith is reading between the lines

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

  • If the Higgs-insertion dictionary is as complete as claimed, the same procedure should produce subleading massive splitting functions for any new heavy scalar by substituting its mass and couplings for the Higgs; that is an immediate testable translation the paper does not work out.
  • The Galilean substitution rule has a natural limit: it is exact only when every intermediate state remains in the alignment regime; for p_T ~ m the recursion would need higher-order collinear corrections, which the present framework leaves implicit.
  • The claim that only angle-bracket (or only square-bracket) amplitude forms are needed connects the splitting functions to two-dimensional conformal symmetry; a direct derivation of P^(1)(z) from that symmetry alone would be a clean check of the Higgs-insertion dictionary.
  • One could test the completeness of the 'complete set' by computing the squared subleading amplitude in a single process through two independent channels — direct massive spinor expansion and Higgs-inserted four-point amplitude — and matching term by term; the paper demonstrates this only for representative channels.

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 presents an on-shell, spinor-helicity construction of massive collinear splitting functions. The authors introduce Soper-Weinberg collinear spinors adapted to a light-front Galilean subgroup, decompose massive momenta and spinors onto fixed lightlike reference vectors, and work in the alignment regime m < p_T << p_+. Leading massive three-point amplitudes are matched to massless three-point amplitudes, while subleading mass corrections are obtained by introducing an additional Higgs boson along the anti-collinear direction and matching massless four-point amplitudes to massive amplitudes through a dictionary (|h> -> |i+>, |h] -> |i-], s_ih -> m_i^2). The paper tabulates leading and subleading massive splitting functions for Standard Model particles, reports agreement with earlier diagrammatic results [6,10], and proposes a recursive bootstrap for higher-point splitting functions based on a universal Galilean substitution rule.

Significance. If the completeness claims are justified, this is a useful and systematic contribution: it gives explicit amplitude-level mass-suppressed splitting functions, provides a matching dictionary that avoids some diagrammatic labor, and offers a recursive construction for higher-point kernels. The use of a fixed collinear spinor basis and the emphasis on Galilean symmetry are conceptually clean, and the detailed tables of three-point amplitude matrices and matching relations are valuable reference material. The cross-checks against [6,10] strengthen confidence in the amplitude-induced parts of the results. However, the central claim of a complete set of subleading splitting functions is undercut by the paper's own admission that O(v^2/p_T^2) virtuality corrections are neglected, and the exhaustiveness of the Higgs-insertion dictionary is assumed rather than proved. These issues need to be addressed before the paper can be accepted as a complete derivation.

major comments (4)
  1. [Section 5, after Eq. (5.7)] The paper explicitly states that it 'neglect[s] the O(v^2/p_T^2) expansion in the propagator virtuality Q^2' and that 'these kinematic corrections must be restored when constructing a splitting function that goes beyond the amplitude-induced mass terms.' This is load-bearing for the central claim. The splitting probability in Eq. (5.2) contains 1/Q^4, and Q_final^2 in Eq. (5.6) is (|p_T|^2 + O(m^2))/(z zbar). Expanding 1/Q^4 multiplies the leading |M^(0)|^2 by O(v^2/p_T^2), producing a contribution at exactly the same v^2/p_T^4 order as the |M^(1)|^2 term. Consequently, P^(1)(z) defined in Eq. (5.11) and used in Eqs. (5.14), (5.54), etc. is not the complete mass-suppressed splitting function, but only the amplitude-induced subset. The abstract's 'complete set' and Section 7's 'full set' are therefore not established as stated. The authors should either restore these kinematic corrections
  2. [Eq. (5.14)] The introductory fermion-scalar example has an algebraic inconsistency. With the amplitude in Eq. (5.13), after setting m_2=m_P=m_f and y=y', one has M = -y (m_f/v)(1/sqrt{zbar} + sqrt{zbar}). From the definition P^(1)= z zbar |M|^2 in Eq. (5.11), the result should be P_Sf^(1) = y^2 (m_f/v)^2 z (1+zbar)^2. The printed result is y^2(1+zbar)^2 z (m_f/v), which is missing one power of m_f/v and does not have the correct mass dimension. Since this is the first worked subleading example and is used to illustrate the method, it must be corrected and the remaining formulas audited for the same issue.
  3. [Section 4.2, Eqs. (4.34)-(4.35)] The Higgs-insertion dictionary is the mechanism by which subleading mass corrections are obtained, but its exhaustiveness is assumed. The matching is demonstrated channel-by-channel for several Standard Model amplitudes, yet no proof or systematic classification is given that every possible subleading spinor structure of the massive three-point amplitude is captured by a massless four-point amplitude with the Higgs momentum along nbar. If there are subleading terms whose pole structure or spinor insertions are not of the form s_ih -> m_i^2, the claimed 'complete set' of subleading splitting functions would be incomplete. A systematic enumeration of the possible I=+ insertions, or a direct comparison with the full diagrammatic expansion including the Q^2 corrections of Eq. (5.6), would close this gap.
  4. [Section 6] The recursive bootstrap is presented as a main result, and Section 7 claims 'Two explicit worked examples (f->fV and V->VV) confirm the validity of the construction.' However, Sections 6.1 and 6.2 contain only the single example f->V fV. No V->VV recursive calculation is shown. Moreover, the universal substitution rule of Eqs. (6.12) and (6.18) is derived for a particular sequential splitting configuration, and its extension to arbitrary final-state multiplicities and to massive intermediate particles is asserted rather than demonstrated. Since this is one of the three pillars advertised in the introduction, the evidence is presently insufficient. Additional worked examples, or a proof of the substitution rule at the level of the Galilean boost action, are needed.
minor comments (4)
  1. [Section 4.3, p. 25] The text says 'Thus the two independent variables are x and p_T' and then repeats 'Thus the two independent variables are z and p_T.' The first should be z.
  2. [Eq. (4.47)] The matrix labels for the S->f f spin components appear to be misaligned: the second row lists 'S->f+ f-' twice and omits the entries corresponding to (f-, f+) and (f+, f+). Please check the labeling against the matrix in Eq. (4.46).
  3. [Throughout] There are several typographical errors: 'requries requires' in Section 3.2, 'genrator' in Appendix A, and inconsistent use of 'x' versus 'z' in Section 4.3. A careful proofreading pass is needed.
  4. [Section 7, Summary] The summary claims two explicit recursive examples, but only one is given in Section 6. Either add the second example or correct the summary.

Circularity Check

0 steps flagged

No significant circularity; central derivation is self-contained and externally checked, with an admitted (non-circular) completeness caveat.

full rationale

The paper's central chain is not circular. Leading splitting functions are obtained by squaring explicit massive 3-point amplitudes built from SW spinors (e.g., eqs. (4.46), (5.13)-(5.17)), and the massless-to-massive coefficient tables (Tables 1-2) are validated by reproducing the same functions from massless 3-point amplitudes (eqs. (5.33)-(5.43)) and by agreement with the independent diagrammatic results of Refs. [6,10]. The subleading construction uses the Higgs-insertion dictionary, eqs. (4.32)-(4.35): vA(P,1,2,h) -> [M_GE(P,1,2)]_1 with |h> -> |i+>, |h] -> |i-], s_ih -> m_i^2. This is a matching ansatz, not a fit: the massless coefficients are converted to massive ones via the Higgs-mechanism relations vY=m1, vT_s=mP (eq. (4.40)), and the resulting P^(1) is checked against the direct massive-amplitude calculations in section 5.1 and against Refs. [6,10]. No quoted equation reduces to its own input by construction. The self-citations [49-52] are cited only as 'another example' of massless-massive correspondence (footnote 3, p.21) and are not load-bearing; no uniqueness theorem from these works is invoked to force the central result. One non-circular caveat should be weighed: after eq. (5.7) the paper explicitly states it 'neglect[s] the O(v^2/p_T^2) expansion in the propagator virtuality Q^2' and that 'these kinematic corrections must be restored when constructing a splitting function that goes beyond the amplitude-induced mass terms.' This undermines the abstract's wording 'complete set of leading and subleading massive splitting functions' as a completeness claim about all mass corrections, but it is an admitted truncation, not a circular identification. The derivation of the functions actually defined is self-contained; therefore the circularity score is minimal.

Axiom & Free-Parameter Ledger

0 free parameters · 6 axioms · 0 invented entities

The derivation rests on standard spinor-helicity, the Galilean subgroup, and a specially introduced matching dictionary. No new particles or fitted parameters are introduced.

axioms (6)
  • domain assumption Soper-Weinberg spinors form a complete basis for massive spinors in the collinear frame.
    Invoked in §2.3 and used throughout for expanding massive spinors.
  • domain assumption The light-front Galilean subgroup Gal(2) governs collinear kinematics and the power counting in the alignment limit m < p_T << p_+.
    Established in §2.1 and used in §3.3 for the on-shell power counting.
  • standard math Goldstone equivalence gauge removes unitarity violations in longitudinal vector amplitudes.
    Adopted in §4.1 to reorganize the massive V f f amplitude; standard equivalence theorem.
  • ad hoc to paper Massless-massive matching dictionary, including the Higgs-insertion mapping |h>→|i+>, |h]→|i−], and pole-to-mass replacement s_ih→m_i^2, exhaustively captures subleading mass corrections.
    Introduced in §4.2 (eqs. 4.34-4.35); no proof of completeness is given beyond the worked channels.
  • ad hoc to paper The recursive bootstrap universal substitution rule, derived from Galilean symmetry, constructs all higher-point splittings from 1→2 amplitudes.
    Presented in §6 (eq. 6.12); verified only on the f→V f V example.
  • domain assumption Four-point massless amplitudes needed for subleading splittings are constrained by the MHV structure of N=4 SYM.
    Used in Appendix A to justify angle-only forms; not a derivation for SM amplitudes.

pith-pipeline@v1.3.0-alltime-deepseek · 47721 in / 10703 out tokens · 84247 ms · 2026-08-01T23:51:53.191112+00:00 · methodology

0 comments
read the original abstract

Collinear splitting functions govern parton evolution, parton showers, and resummation at high-energy colliders. While on-shell spinor-helicity methods have successfully yielded massless QCD splitting functions, a complete on-shell construction for massive particles, systematically incorporating finite-mass effects, is less developed. We present an on-shell constructive formalism for massive collinear splitting functions based on Soper-Weinberg collinear spinors, whose transformation properties follow from a light-front Galilean subgroup of the Poincar\'e group. Decomposing massive momenta and spinors with respect to fixed lightlike vectors $n$ and $\bar n$ makes the expansion in the alignment regime $m<p_T\ll p_+$ manifest. The leading-order structures are matched to massless three-point amplitudes, while an additional Higgs momentum along $\bar n$ probes the subleading spinor components and relates them to massless four-point amplitudes. We derive the complete set of leading and subleading massive splitting functions for all Standard Model particles and establish a systematic matching dictionary between massless and massive coupling coefficients at both the three- and four-point levels. Higher-point splitting functions are obtained through the recursive bootstrap relation with a universal substitution rule as a consequence of the Galilean symmetry. This constructive framework extends naturally to effective field theory operators and higher perturbative orders, providing a flexible computational tool for precision collider physics and parton shower development.

Figures

Figures reproduced from arXiv: 2607.15191 by Chao Wu, Jiang-Hao Yu, Yi-Ning Wang.

Figure 1
Figure 1. Figure 1: Decomposition of a massive momentum (blue arrow) into two massless momenta (black arrows) [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Decomposition of a massive momentum (blue arrow) into massless momenta on the light cone. In [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Configuration of the four massless particles in momentum space: three particles (black arrows) [PITH_FULL_IMAGE:figures/full_fig_p023_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Factorization of the cross section into the hard part [PITH_FULL_IMAGE:figures/full_fig_p032_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Illustration of a Galilean boost acting on a massive particle (blue arrow) and the two massless [PITH_FULL_IMAGE:figures/full_fig_p034_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Leading splitting function matching involving fermions. The left column corresponds to massless [PITH_FULL_IMAGE:figures/full_fig_p038_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Leading splitting function matching with bosons only. The left column corresponds to massless [PITH_FULL_IMAGE:figures/full_fig_p039_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Subleading splitting function matching involving fermions. The left column corresponds to mass [PITH_FULL_IMAGE:figures/full_fig_p042_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Subleading splitting function matching with boson only. The left column corresponds to massless [PITH_FULL_IMAGE:figures/full_fig_p043_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Typical decomposition of a 1 → 3 process. The parent particle P first undergoes a splitting into particle 1 and an intermediate state q, which subsequently decays into particles 2 and 3. To address this, we perform a standard boost Lp that brings the second 1 → 2 process into the standard splitting parametrization, where the transverse momentum of the intermediate particle vanishes: A(q; 2, 3) Lp −−→ A(P;… view at source ↗
Figure 11
Figure 11. Figure 11: Typical massless amplitude corresponding to the subleading contributions in a sequential 1 [PITH_FULL_IMAGE:figures/full_fig_p048_11.png] view at source ↗

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 · 36 linked inside Pith

  1. [1]

    Altarelli and G

    G. Altarelli and G. Parisi,Asymptotic Freedom in Parton Language,Nucl. Phys. B126(1977) 298–318

    1977

  2. [2]

    Y. L. Dokshitzer,Calculation of the Structure Functions for Deep Inelastic Scattering and e+ e- Annihilation by Perturbation Theory in Quantum Chromodynamics,Sov. Phys. JETP46(1977) 641–653

    1977

  3. [3]

    V. N. Gribov and L. N. Lipatov,Deep inelastic e p scattering in perturbation theory,Sov. J. Nucl. Phys.15(1972) 438–450

    1972

  4. [4]

    Ciafaloni, P

    M. Ciafaloni, P. Ciafaloni and D. Comelli,Towards collinear evolution equations in electroweak theory, Phys. Rev. Lett.88(2002) 102001, [hep-ph/0111109]

    Pith/arXiv arXiv 2002

  5. [5]

    J.-y. Chiu, A. Fuhrer, R. Kelley and A. V. Manohar,Soft and Collinear Functions for the Standard Model,Phys. Rev. D81(2010) 014023, [0909.0947]

    Pith/arXiv arXiv 2010

  6. [6]

    J. Chen, T. Han and B. Tweedie,Electroweak Splitting Functions and High Energy Showering,JHEP 11(2017) 093, [1611.00788]

    Pith/arXiv arXiv 2017

  7. [7]

    C. W. Bauer, N. Ferland and B. R. Webber,Standard Model Parton Distributions at Very High Energies,JHEP08(2017) 036, [1703.08562]

    Pith/arXiv arXiv 2017

  8. [8]

    Cuomo, L

    G. Cuomo, L. Vecchi and A. Wulzer,Goldstone Equivalence and High Energy Electroweak Physics, SciPost Phys.8(2020) 078, [1911.12366]

    Pith/arXiv arXiv 2020

  9. [9]

    T. Han, Y. Ma and K. Xie,High energy leptonic collisions and electroweak parton distribution functions,Phys. Rev. D103(2021) L031301, [2007.14300]

    Pith/arXiv arXiv 2021

  10. [10]

    Nardi, L

    F. Nardi, L. Ricci and A. Wulzer,Low-virtuality splitting in the Standard Model,JHEP10(2024) 215, [2405.08220]

    Pith/arXiv arXiv 2024

  11. [11]

    Dittmaier and M

    S. Dittmaier and M. Reyer,Electroweak splitting functions in the Standard Model and beyond,JHEP 01(2026) 119, [2507.06568]

    arXiv 2026

  12. [12]

    Ciafaloni and D

    P. Ciafaloni and D. Comelli,Sudakov enhancement of electroweak corrections,Phys. Lett. B446 (1999) 278–284, [hep-ph/9809321]. 54

    Pith/arXiv arXiv 1999

  13. [13]

    V. S. Fadin, L. N. Lipatov, A. D. Martin and M. Melles,Resummation of double logarithms in electroweak high-energy processes,Phys. Rev. D61(2000) 094002, [hep-ph/9910338]

    Pith/arXiv arXiv 2000

  14. [14]

    Ciafaloni and D

    M. Ciafaloni and D. Comelli,Electroweak Sudakov form-factors and nonfactorizable soft QED effects at high-energies,Phys. Lett. B476(2000) 49–57, [hep-ph/0003142]

    Pith/arXiv arXiv 2000

  15. [15]

    J.-y. Chiu, F. Golf, R. Kelley and A. V. Manohar,Electroweak Sudakov corrections using effective field theory,Phys. Rev. Lett.100(2008) 021802, [0709.2377]

    Pith/arXiv arXiv 2008

  16. [16]

    A. V. Manohar and W. J. Waalewijn,Electroweak Logarithms in Inclusive Cross Sections,JHEP08 (2018) 137, [1802.08687]

    Pith/arXiv arXiv 2018

  17. [17]

    Pagani and M

    D. Pagani and M. Zaro,One-loop electroweak Sudakov logarithms: a revisitation and automation, JHEP02(2022) 161, [2110.03714]

    Pith/arXiv arXiv 2022

  18. [18]

    L. J. Dixon,A brief introduction to modern amplitude methods, inTheoretical Advanced Study Institute in Elementary Particle Physics: Particle Physics: The Higgs Boson and Beyond, pp. 31–67, 2014.1310.5353. DOI

    arXiv 2014

  19. [19]

    Elvang and Y.-t

    H. Elvang and Y.-t. Huang,Scattering Amplitudes in Gauge Theory and Gravity. Cambridge University Press, 4, 2015

    2015

  20. [20]

    Cheung,TASI lectures on scattering amplitudes., inTheoretical Advanced Study Institute in Elementary Particle Physics: Anticipating the Next Discoveries in Particle Physics, pp

    C. Cheung,TASI lectures on scattering amplitudes., inTheoretical Advanced Study Institute in Elementary Particle Physics: Anticipating the Next Discoveries in Particle Physics, pp. 571–623, 2018.1708.03872. DOI

    arXiv 2018

  21. [21]

    Badger, J

    S. Badger, J. Henn, J. C. Plefka and S. Zoia,Scattering Amplitudes in Quantum Field Theory,Lect. Notes Phys.1021(2024) pp., [2306.05976]

    Pith/arXiv arXiv 2024

  22. [22]

    L. J. Dixon,Calculating scattering amplitudes efficiently, inTheoretical Advanced Study Institute in Elementary Particle Physics (TASI 95): QCD and Beyond, pp. 539–584, 1, 1996.hep-ph/9601359

    Pith/arXiv arXiv 1996

  23. [23]

    Z. Bern, L. J. Dixon, D. C. Dunbar and D. A. Kosower,One loop n point gauge theory amplitudes, unitarity and collinear limits,Nucl. Phys. B425(1994) 217–260, [hep-ph/9403226]

    Pith/arXiv arXiv 1994

  24. [24]

    Z. Bern, L. J. Dixon and D. A. Kosower,Two-loop g —>gg splitting amplitudes in QCD,JHEP08 (2004) 012, [hep-ph/0404293]

    Pith/arXiv arXiv 2004

  25. [25]

    Cohen and M

    T. Cohen and M. Riembau,Recursion for Wilson-line form factors,JHEP10(2024) 132, [2406.03540]

    Pith/arXiv arXiv 2024

  26. [26]

    Kleiss and W

    R. Kleiss and W. J. Stirling,Spinor Techniques for Calculating p anti-p —>W+- / Z0 + Jets,Nucl. Phys. B262(1985) 235–262

    1985

  27. [27]

    Kleiss and W

    R. Kleiss and W. J. Stirling,Cross-sections for the Production of an Arbitrary Number of Photons in Electron - Positron Annihilation,Phys. Lett. B179(1986) 159–163

    1986

  28. [28]

    Dittmaier,Weyl-van der Waerden formalism for helicity amplitudes of massive particles,Phys

    S. Dittmaier,Weyl-van der Waerden formalism for helicity amplitudes of massive particles,Phys. Rev. D59(1998) 016007, [hep-ph/9805445]

    Pith/arXiv arXiv 1998

  29. [29]

    Schwinn and S

    C. Schwinn and S. Weinzierl,On-shell recursion relations for all Born QCD amplitudes,JHEP04 (2007) 072, [hep-ph/0703021]. 55

    Pith/arXiv arXiv 2007

  30. [30]

    R. H. Boels and C. Schwinn,On-shell supersymmetry for massive multiplets,Phys. Rev. D84(2011) 065006, [1104.2280]

    Pith/arXiv arXiv 2011

  31. [31]

    A. J. Larkoski and M. E. Peskin,Antenna Splitting Functions for Massive Particles,Phys. Rev. D84 (2011) 034034, [1106.2182]

    Pith/arXiv arXiv 2011

  32. [32]

    Kleiss and R

    R. Kleiss and R. Verheyen,Collinear electroweak radiation in antenna parton showers,Eur. Phys. J. C80(2020) 980, [2002.09248]

    Pith/arXiv arXiv 2020

  33. [33]

    Brooks, P

    H. Brooks, P. Skands and R. Verheyen,Interleaved resonance decays and electroweak radiation in the Vincia parton shower,SciPost Phys.12(2022) 101, [2108.10786]

    Pith/arXiv arXiv 2022

  34. [34]

    Arkani-Hamed, T.-C

    N. Arkani-Hamed, T.-C. Huang and Y.-t. Huang,Scattering Amplitudes for All Masses and Spins, JHEP11(2021) 070, [1709.04891]

    Pith/arXiv arXiv 2021

  35. [35]

    Weinberg,Dynamics at Infinite Momentum,Phys

    S. Weinberg,Dynamics at Infinite Momentum,Phys. Rev.150(1966) 1313–1318

    1966

  36. [36]

    D. E. Soper,Field Theories in the Infinite Momentum Frame,Phys. Rev. D4(1971) 1620–1634

    1971

  37. [37]

    E. P. Wigner,On Unitary Representations of the Inhomogeneous Lorentz Group,Annals Math.40 (1939) 149–204

    1939

  38. [38]

    Susskind,Model of selfinduced strong interactions,Phys

    L. Susskind,Model of selfinduced strong interactions,Phys. Rev.165(1968) 1535–1546

    1968

  39. [39]

    Chang and S.-K

    S.-J. Chang and S.-K. Ma,Feynman rules and quantum electrodynamics at infinite momentum,Phys. Rev.180(1969) 1506–1513

    1969

  40. [40]

    Bardakci and M

    K. Bardakci and M. B. Halpern,Theories at infinite momentum,Phys. Rev.176(1968) 1686–1699

    1968

  41. [41]

    J. B. Kogut and D. E. Soper,Quantum Electrodynamics in the Infinite Momentum Frame,Phys. Rev. D1(1970) 2901–2913

    1970

  42. [42]

    Bargmann,On Unitary ray representations of continuous groups,Annals Math.59(1954) 1–46

    V. Bargmann,On Unitary ray representations of continuous groups,Annals Math.59(1954) 1–46

    1954

  43. [43]

    Levy-Leblond,Galilei Group and Nonrelativistic Quantum Mechanics,J

    J.-M. Levy-Leblond,Galilei Group and Nonrelativistic Quantum Mechanics,J. Math. Phys.4(1963) 776

    1963

  44. [44]

    Conde and A

    E. Conde and A. Marzolla,Lorentz Constraints on Massive Three-Point Amplitudes,JHEP09(2016) 041, [1601.08113]

    Pith/arXiv arXiv 2016

  45. [45]

    J. M. Cornwall, D. N. Levin and G. Tiktopoulos,Derivation of Gauge Invariance from High-Energy Unitarity Bounds on the S Matrix,Phys. Rev. D10(1974) 1145–1167

    1974

  46. [46]

    M. S. Chanowitz and M. K. Gaillard,The TeV Physics of Strongly Interacting W’s and Z’s,Nucl. Phys. B261(1985) 379–431

    1985

  47. [47]

    Yao and C

    Y.-P. Yao and C. P. Yuan,Modification of the Equivalence Theorem Due to Loop Corrections,Phys. Rev. D38(1988) 2237

    1988

  48. [48]

    Bagger and C

    J. Bagger and C. Schmidt,Equivalence Theorem Redux,Phys. Rev. D41(1990) 264

    1990

  49. [49]

    Ni, Y.-N

    Y.-H. Ni, Y.-N. Wang, C. Wu and J.-H. Yu,Extended Poincare Symmetry Dictates Massive Scattering Amplitudes,2412.03762. 56

    Pith/arXiv arXiv

  50. [50]

    Ni, Y.-N

    Y.-H. Ni, Y.-N. Wang, C. Wu and J.-H. Yu,Massive Helicity-Chirality Spinor Formalism from Massless Amplitudes with On-shell Mass Insertion,2501.09062

    Pith/arXiv arXiv

  51. [51]

    Y.-H. Ni, C. Wu and J.-H. Yu,Massless-Massive Amplitude Correspondence II: Constructive Massive Amplitudes in Standard Model,2601.10622

    arXiv

  52. [52]

    Y.-H. Ni, C. Wu and J.-H. Yu,Massless-Massive Amplitude Correspondence I: Helicity-chirality Matching and On-shell Higgsing,2601.10620

    arXiv

  53. [53]

    Catani and M

    S. Catani and M. Grazzini,Collinear factorization and splitting functions for next-to-next-to-leading order QCD calculations,Phys. Lett. B446(1999) 143–152, [hep-ph/9810389]

    Pith/arXiv arXiv 1999

  54. [54]

    J. M. Campbell and E. W. N. Glover,Double unresolved approximations to multiparton scattering amplitudes,Nucl. Phys. B527(1998) 264–288, [hep-ph/9710255]

    Pith/arXiv arXiv 1998

  55. [55]

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

    Pith/arXiv arXiv 2023

  56. [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, [2310.06736]

    Pith/arXiv arXiv 2024

  57. [57]

    M. R. Masouminia and P. Richardson,Implementation of angularly ordered electroweak parton shower in Herwig 7,JHEP04(2022) 112, [2108.10817]

    Pith/arXiv arXiv 2022

  58. [58]

    Bothmann and D

    E. Bothmann and D. Napoletano,Automated evaluation of electroweak Sudakov logarithms in Sherpa, Eur. Phys. J. C80(2020) 1024, [2006.14635]

    Pith/arXiv arXiv 2020

  59. [59]

    C. W. Bauer, S. Fleming, D. Pirjol and I. W. Stewart,An Effective field theory for collinear and soft gluons: Heavy to light decays,Phys. Rev. D63(2001) 114020, [hep-ph/0011336]

    Pith/arXiv arXiv 2001

  60. [60]

    C. W. Bauer, D. Pirjol and I. W. Stewart,Soft collinear factorization in effective field theory,Phys. Rev. D65(2002) 054022, [hep-ph/0109045]. 57

    Pith/arXiv arXiv 2002