pith. machine review for the scientific record. sign in

arxiv: 2604.18672 · v1 · submitted 2026-04-20 · ✦ hep-th · hep-ph

Recognition: unknown

Tree Amplitudes with Charged Matter in Pure Gauge Theory

Jacob L. Bourjaily , Michael Plesser , Philip Velie
Authors on Pith no claims yet

Pith reviewed 2026-05-10 04:23 UTC · model grok-4.3

classification ✦ hep-th hep-ph
keywords tree amplitudesgauge theoryfermionsdistinct flavorscolor tensorspartial amplitudessupersymmetric Yang-MillsMathematica package
0
0 comments X

The pith

Distinct-flavor tree amplitudes in pure gauge theory reduce to linear combinations of single-flavor supersymmetric Yang-Mills amplitudes.

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

The paper introduces a Mathematica package that computes tree-level scattering amplitudes with any number of gauge bosons and massless fermions carrying arbitrary charges and distinct flavors in non-supersymmetric gauge theory. It establishes that partial amplitudes involving multiple distinct fermion flavors can always be rewritten as linear combinations of simpler single-flavor amplitudes. These single-flavor amplitudes are then obtained directly as component amplitudes extracted from maximally supersymmetric Yang-Mills theory. The approach supplies explicit numeric arrays for all required color tensors once charge generators are chosen, allowing any gauge group including U(1). A reader would care because the reduction turns a hard multi-flavor problem into a set of standard supersymmetric calculations plus straightforward color algebra.

Core claim

Distinct-flavour partial amplitudes are expressed as linear combinations of those involving only a single flavour, which may be evaluated as component amplitudes of maximally supersymmetric Yang-Mills theory. All relevant colour tensors can be realized as explicit, numeric arrays given any choice of charge generators for any gauge theory including U(1). From these arrays all colour contractions needed for cross sections follow directly.

What carries the argument

The linear reduction mapping distinct-flavour fermion partial amplitudes onto single-flavour SYM components, together with the numeric realization of colour tensors for arbitrary charge generators.

If this is right

  • Cross sections follow by contracting the provided numeric color tensors with the reduced amplitudes.
  • The same basis of partial amplitudes and color dressings works for any gauge group and any assignment of fermion charges.
  • Arbitrary numbers of gluons and fermions can be handled once the single-flavor SYM amplitudes are known.
  • The package directly supplies the color-dressed amplitudes needed for phenomenological calculations in pure gauge theories.

Where Pith is reading between the lines

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

  • The same reduction may offer a practical route to multi-flavor amplitudes in effective theories that include additional abelian gauge fields.
  • If analogous reductions exist at loop level, the package could seed numerical or analytic calculations beyond tree order.
  • Phenomenological models with multiple charged matter species under a single gauge group become computationally closer to pure SYM results.

Load-bearing premise

The reduction that converts distinct-flavor fermion amplitudes into single-flavor supersymmetric Yang-Mills components remains valid inside pure non-supersymmetric gauge theory.

What would settle it

Compute a concrete four-fermion two-gluon tree amplitude with two distinct fermion flavors in pure Yang-Mills using Feynman rules and compare the result to the linear combination obtained from the single-flavor SYM components; any mismatch would falsify the claimed reduction.

read the original abstract

We describe the implementation and usage of `fermionic_amplitudes.m', a Mathematica package for the computation of tree amplitudes involving arbitrary numbers of gauge bosons and arbitrarily-charged massless fermions of (possibly) distinct flavours in pure (non-supersymmetric) gauge theory. These are given in terms of a basis of partial amplitudes involving distinct-flavoured fermions dressed by specific colour tensors. Distinct-flavour partial amplitudes are expressed as linear combinations of those involving only a single flavour, which may be evaluated as component amplitudes of (maximally) supersymmetric Yang-Mills theory. All relevant colour tensors can be realized as explicit, numeric arrays given any choice of charge generators (for any gauge theory -- including $u_1$); from these, all colour contractions relevant to cross sections may be readily computed. The complete package and a notebook demonstrating its primary usage and functionality are included in this work's submission's ancillary files on the arXiv.

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 describes the implementation of the Mathematica package fermionic_amplitudes.m for computing tree-level amplitudes involving an arbitrary number of gauge bosons and massless fermions carrying arbitrary charges and (possibly) distinct flavours in pure non-supersymmetric gauge theory. Distinct-flavour partial amplitudes are expressed as linear combinations of single-flavour partial amplitudes that can be evaluated as components of maximally supersymmetric Yang-Mills theory; all relevant colour tensors are supplied as explicit numeric arrays for any choice of charge generators and any gauge group, including U(1). The complete package together with a usage notebook is provided in the arXiv ancillary files.

Significance. If the reduction is correctly implemented, the work supplies a practical, reproducible tool that exploits the tree-level identity between gluon-fermion vertices in pure Yang-Mills and the corresponding components of SYM theory, while handling colour contractions via numeric arrays that remain valid for arbitrary groups. The explicit provision of the package and notebook constitutes a clear strength for verifiability and immediate usability.

minor comments (3)
  1. The manuscript would benefit from a short explicit statement (perhaps in §2 or §3) confirming that the reduction relies only on the tree-level Feynman rules being identical and does not invoke supersymmetry beyond the use of known SYM component amplitudes.
  2. Notation for the colour tensors (e.g., the mapping from charge generators to numeric arrays) should be introduced with one concrete low-point example to aid readers unfamiliar with the numeric-array approach.
  3. The abstract states that the package is included, but the main text should contain a brief table or list of the principal functions and their calling conventions for quick reference.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of our work and for recommending minor revision. No major comments were raised, so there are no specific points requiring response or revision.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper's derivation expresses distinct-flavour partial amplitudes as linear combinations of single-flavour ones evaluated via SYM components, with colour tensors as explicit numeric arrays for arbitrary generators. This follows from the identity of tree-level gluon-fermion vertices and propagators between pure non-supersymmetric Yang-Mills and SYM components—an external, independently verifiable fact not derived or fitted within the paper. Colour handling via numeric arrays is a standard, group-agnostic algebraic technique that preserves all contractions without loss or self-reference. No equations reduce the central claims to fitted inputs, self-definitions, or load-bearing self-citations; the result is self-contained against external benchmarks such as known SYM tree amplitudes and standard colour algebra.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the domain assumption that multi-flavor amplitudes reduce to single-flavor SYM components and that color factors admit numeric array representations for arbitrary gauge groups.

axioms (2)
  • domain assumption Distinct-flavor partial amplitudes can be expressed as linear combinations of single-flavor amplitudes from maximally supersymmetric Yang-Mills theory even in non-supersymmetric pure gauge theory.
    This is the key reduction stated in the abstract as the basis for the package.
  • domain assumption All relevant color tensors for any choice of charge generators can be realized as explicit numeric arrays.
    Stated directly in the abstract as enabling cross-section computations.

pith-pipeline@v0.9.0 · 5461 in / 1313 out tokens · 45977 ms · 2026-05-10T04:23:16.158863+00:00 · methodology

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

49 extracted references · 39 canonical work pages · 1 internal anchor

  1. [1]

    What is the Simplest Quantum Field Theory?

    N. Arkani-Hamed, F. Cachazo, and J. Kaplan, “What is the Simplest Quantum Field Theory?,”JHEP09(2010) 016, arXiv:0808.1446 [hep-th]

    work page Pith review arXiv 2010

  2. [2]

    The All-Loop Integrand For Scattering Amplitudes in PlanarN=4SYM,

    N. Arkani-Hamed, J. L. Bourjaily, F. Cachazo, S. Caron-Huot, and J. Trnka, “The All-Loop Integrand For Scattering Amplitudes in PlanarN=4SYM,”JHEP1101 (2011) 041, arXiv:1008.2958 [hep-th]

    work page arXiv 2011

  3. [3]

    Arkani-Hamed, J

    N. Arkani-Hamed, J. L. Bourjaily, F. Cachazo, A. B. Goncharov, A. Postnikov, and J. Trnka, “Scattering Amplitudes and the Positive Grassmannian,” arXiv:1212.5605 [hep-th]

    work page arXiv

  4. [4]

    Arkani-Hamed and J

    N. Arkani-Hamed and J. Trnka, “The Amplituhedron,”JHEP1410(2014) 30, arXiv:1312.2007 [hep-th]

    work page arXiv 2014

  5. [5]

    Dual-Conformal Regularization of Infrared Loop Divergences and theChiralBox Expansion,

    J. L. Bourjaily, S. Caron-Huot, and J. Trnka, “Dual-Conformal Regularization of Infrared Loop Divergences and theChiralBox Expansion,”JHEP1501(2015) 001, arXiv:1303.4734 [hep-th]

    work page arXiv 2015

  6. [6]

    Local Integrand Representations of All Two-Loop Amplitudes in Planar SYM,

    J. L. Bourjaily and J. Trnka, “Local Integrand Representations of All Two-Loop Amplitudes in Planar SYM,”JHEP08(2015) 119, arXiv:1505.05886 [hep-th]

    work page arXiv 2015

  7. [7]

    Prescriptive Unitarity,

    J. L. Bourjaily, E. Herrmann, and J. Trnka, “Prescriptive Unitarity,”JHEP06 (2017) 059, arXiv:1704.05460 [hep-th]

    work page arXiv 2017

  8. [8]

    Loops from Cuts,

    J. L. Bourjaily and S. Caron-Huot, “Loops from Cuts,”Phys. Rev. D108(2023) no. 2, 025008, arXiv:2305.01673 [hep-th]

    work page arXiv 2023

  9. [9]

    Bourjaily,Computational Tools for Trees in Gauge Theory and Gravity, 2312.17745

    J. L. Bourjaily, “Computational Tools for Trees in Gauge Theory and Gravity,” arXiv:2312.17745 [hep-th]

    work page arXiv

  10. [10]

    All Tree-Level Amplitudes in Massless QCD,

    L. J. Dixon, J. M. Henn, J. Plefka, and T. Schuster, “All Tree-Level Amplitudes in Massless QCD,”JHEP1101(2011) 035, arXiv:1010.3991 [hep-ph]

    work page arXiv 2011

  11. [11]

    Bourjaily,Efficient Tree-Amplitudes in N=4: Automatic BCFW Recursion in Mathematica,1011.2447

    J. L. Bourjaily, “Efficient Tree-Amplitudes inN=4: Automatic BCFW Recursion in Mathematica,” arXiv:1011.2447 [hep-ph]

    work page arXiv

  12. [12]

    FEYN CALC: Computer Algebraic Calculation of Feynman Amplitudes,

    R. Mertig, M. Bohm, and A. Denner, “FEYN CALC: Computer Algebraic Calculation of Feynman Amplitudes,”Comput. Phys. Commun.64(1991) 345–359

    1991

  13. [13]

    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 [hep-ph]

    work page arXiv 2020

  14. [14]

    Gleisberg, S

    T. Gleisberg, S. Hoeche, F. Krauss, A. Schalicke, S. Schumann, and J.-C. Winter, “SHERPA 1.α: A Proof of Concept Version,”JHEP02(2004) 056, arXiv:hep-ph/0311263

    work page arXiv 2004

  15. [15]

    The automated computation of tree-level and next-to-leading order differential cross sections, and their matching to parton shower simulations

    J. Alwall, R. Frederix, S. Frixione, V. Hirschi, F. Maltoni, O. Mattelaer, H. S. Shao, T. Stelzer, P. Torrielli, and M. Zaro, “The Automated Computation of Tree-Level and Next-to-Leading Order Differential Cross Sections, and their Matching to Parton Shower Simulations,”JHEP07(2014) 079, arXiv:1405.0301 [hep-ph]. – 45 –

    work page internal anchor Pith review arXiv 2014

  16. [16]

    Melia, Dyck words and multiquark primitive ampli- tudes, Phys

    T. Melia, “Dyck Words and Multiquark Primitive Amplitudes,”Phys. Rev. D88 (2013) no. 1, 014020, arXiv:1304.7809 [hep-ph]

    work page arXiv 2013

  17. [17]

    Getting More Flavor out of One-Flavor QCD,

    T. Melia, “Getting More Flavor out of One-Flavor QCD,”Phys. Rev. D89(2014) no. 7, 074012, arXiv:1312.0599 [hep-ph]

    work page arXiv 2014

  18. [18]

    Johansson and A

    H. Johansson and A. Ochirov, “Color-Kinematics Duality for QCD Amplitudes,” JHEP01(2016) 170, arXiv:1507.00332 [hep-ph]

    work page arXiv 2016

  19. [19]

    On Tree Amplitudes in Gauge Theory and Gravity,

    N. Arkani-Hamed and J. Kaplan, “On Tree Amplitudes in Gauge Theory and Gravity,”JHEP0804(2008) 076, arXiv:0801.2385 [hep-th]

    work page arXiv 2008

  20. [20]

    On-Shell Constructibility of Tree Amplitudes in General Field Theories,

    T. Cohen, H. Elvang, and M. Kiermaier, “On-Shell Constructibility of Tree Amplitudes in General Field Theories,”JHEP1104(2011) 053, arXiv:1010.0257 [hep-th]

    work page arXiv 2011

  21. [21]

    The Supersymmetry of Cuts in Pure Gauge Theory and Gravity,

    J. L. Bourjaily, “The Supersymmetry of Cuts in Pure Gauge Theory and Gravity,” arXiv:2512.24984 [hep-th]

    work page arXiv

  22. [22]

    Consistency Conditions on theS-Matrix of Massless Particles,

    P. Benincasa and F. Cachazo, “Consistency Conditions on theS-Matrix of Massless Particles,” arXiv:0705.4305 [hep-th]

    work page arXiv

  23. [23]

    Proof of a New Colour Decomposition for QCD Amplitudes,

    T. Melia, “Proof of a New Colour Decomposition for QCD Amplitudes,”JHEP12 (2015) 107, arXiv:1509.03297 [hep-ph]

    work page arXiv 2015

  24. [24]

    Maltoni, K

    F. Maltoni, K. Paul, T. Stelzer, and S. Willenbrock, “Color Flow Decomposition of QCD Amplitudes,”Phys. Rev. D67(2003) 014026, arXiv:hep-ph/0209271

    work page arXiv 2003

  25. [25]

    Multi-Gluon Cross-Sections and Five Jet Production at Hadron Colliders,

    R. Kleiss and H. Kuijf, “Multi-Gluon Cross-Sections and Five Jet Production at Hadron Colliders,”Nucl. Phys.B312(1989) 616

    1989

  26. [26]

    Bern, J.J.M

    Z. Bern, J. Carrasco, and H. Johansson, “New Relations for Gauge-Theory Amplitudes,”Phys. Rev.D78(2008) 085011, arXiv:0805.3993 [hep-ph]

    work page arXiv 2008

  27. [27]

    Perturbative Quantum Gravity as a Double Copy of Gauge Theory,

    Z. Bern, J. J. M. Carrasco, and H. Johansson, “Perturbative Quantum Gravity as a Double Copy of Gauge Theory,”Phys. Rev. Lett.105(2010) 061602, arXiv:1004.0476 [hep-th]

    work page arXiv 2010

  28. [28]

    Bern, J.J

    Z. Bern, J. J. M. Carrasco, M. Chiodaroli, H. Johansson, and R. Roiban, “The Duality Between Color and Kinematics and its Applications,” arXiv:1909.01358 [hep-th]

    work page arXiv 1909

  29. [29]

    Snowmass white paper: the double copy and its applications,

    T. Adamo, J. J. M. Carrasco, M. Carrillo-González, M. Chiodaroli, H. Elvang, H. Johansson, D. O’Connell, R. Roiban, and O. Schlotterer, “Snowmass White Paper: the Double Copy and its Applications,” inSnowmass 2021. 4, 2022. arXiv:2204.06547 [hep-th]

    work page arXiv 2021

  30. [30]

    Supergravity Amplitudes, the Double Copy and Ultraviolet Behavior,

    Z. Bern, J. J. M. Carrasco, M. Chiodaroli, H. Johansson, and R. Roiban, “Supergravity Amplitudes, the Double Copy and Ultraviolet Behavior,” arXiv:2304.07392 [hep-th]

    work page arXiv

  31. [31]

    Group Theory for Feynman Diagrams in non-Abelian Gauge Theories,

    P. Cvitanovic, “Group Theory for Feynman Diagrams in non-Abelian Gauge Theories,”Phys. Rev. D14(1976) 1536–1553. – 46 –

    1976

  32. [32]

    The Six Gluon Process as an Example of Weyl-Van Der Waerden Spinor Calculus,

    F. A. Berends and W. Giele, “The Six Gluon Process as an Example of Weyl-Van Der Waerden Spinor Calculus,”Nucl. Phys. B294(1987) 700–732

    1987

  33. [33]

    Duality and Multi-Gluon Scattering,

    M. L. Mangano, S. J. Parke, and Z. Xu, “Duality and Multi-Gluon Scattering,” Nucl. Phys.B298(1988) 653–672

    1988

  34. [34]

    Multi-Gluon Scattering: A String-Based Calculation,

    D. Kosower, B.-H. Lee, and V. P. Nair, “Multi-Gluon Scattering: A String-Based Calculation,”Phys. Lett. B201(1988) 85–89

    1988

  35. [35]

    Color Factorization for Fermionic Amplitudes,

    D. A. Kosower, “Color Factorization for Fermionic Amplitudes,”Nucl. Phys. B315 (1989) 391–418

    1989

  36. [36]

    Color Decomposition of One Loop Amplitudes in Gauge Theories,

    Z. Bern and D. A. Kosower, “Color Decomposition of One Loop Amplitudes in Gauge Theories,”Nucl. Phys. B362(1991) 389–448

    1991

  37. [37]

    Multiparton amplitudes in gauge theories,

    M. L. Mangano and S. J. Parke, “Multiparton Amplitudes in Gauge Theories,” Phys. Rept.200(1991) 301–367, arXiv:hep-th/0509223

    work page arXiv 1991

  38. [38]

    Pure Gravities via Color-Kinematics Duality for Fundamental Matter,

    H. Johansson and A. Ochirov, “Pure Gravities via Color-Kinematics Duality for Fundamental Matter,”JHEP11(2015) 046, arXiv:1407.4772 [hep-th]

    work page arXiv 2015

  39. [39]

    Cyclic Mario Worlds—Color-Decomposition for One-Loop QCD,

    G. Kälin, “Cyclic Mario Worlds—Color-Decomposition for One-Loop QCD,”JHEP 04(2018) 141, arXiv:1712.03539 [hep-th]

    work page arXiv 2018

  40. [40]

    Ochirov and B

    A. Ochirov and B. Page, “Multi-Quark Colour Decompositions from Unitarity,” JHEP10(2019) 058, arXiv:1908.02695 [hep-ph]

    work page arXiv 2019

  41. [41]

    Calculating scattering amplitudes efficiently,

    L. J. Dixon, “Calculating Scattering Amplitudes Efficiently,” arXiv:hep-ph/9601359

    work page arXiv

  42. [42]

    Bourjaily, M

    J. L. Bourjaily, M. Plesser, and C. Vergu, “The Colour Dependence of Amplitudes,” arXiv:2512.23806 [hep-th]

    work page arXiv

  43. [43]

    The Many Colours of Amplitudes,

    J. L. Bourjaily, M. Plesser, and C. Vergu, “The Many Colours of Amplitudes,” arXiv:2412.21189 [hep-th]

    work page arXiv

  44. [44]

    New color decompositions for gauge amplitudes at tree and loop level,

    V. Del Duca, L. J. Dixon, and F. Maltoni, “New Color Decompositions for Gauge Amplitudes at Tree and Loop Level,”Nucl. Phys.B571(2000) 51–70, arXiv:hep-ph/9910563 [hep-ph]

    work page arXiv 2000

  45. [45]

    Gauge-Invariant Double Copies via Recursion,

    J. L. Bourjaily, N. Kalyanapuram, K. Patatoukos, M. Plesser, and Y. Zhang, “Gauge-Invariant Double Copies via Recursion,”Phys. Rev. Lett.131(2023) no. 19, 191601, arXiv:2307.02542 [hep-th]

    work page arXiv 2023

  46. [46]

    Rational Representations of Simple Lie Algebras

    J. L. Bourjaily, M. Plesser, and P. Velie, “Rational Representations of Simple Lie Algebras.” In preparation

  47. [47]

    Cvitanović,Group Theory

    P. Cvitanović,Group Theory. Princeton University Press, Princeton, NJ, 2008. https://doi.org/10.1515/9781400837670. Birdtracks, Lie Algebras, and Exceptional Groups

    work page doi:10.1515/9781400837670 2008

  48. [48]

    Diagonalization of Color Factors,

    D. Zeppenfeld, “Diagonalization of Color Factors,”Int. J. Mod. Phys. A3(1988) 2175–2179

    1988

  49. [49]

    Scattering amplitudes for all masses and spins,

    N. Arkani-Hamed, T.-C. Huang, and Y.-t. Huang, “Scattering Amplitudes for all Masses and Spins,”JHEP11(2021) 070, arXiv:1709.04891 [hep-th]. – 47 –

    work page arXiv 2021