Abstract
The chirality-flow formalism, combined with good choices of gauge reference vectors, simplifies tree-level calculations to the extent that it is often possible to write down amplitudes corresponding to Feynman diagrams immediately. It has also proven to give a very sizable speedup in a proof of concept implementation of massless tree-level QED in MadGraph5_aMC@NLO. In the present paper we extend this analysis to QCD, including massive quarks. We define helicity-dependent versions of the gluon vertices, derive constraints on the spinor structure of propagating gluons, and explore the Schouten identity to simplify the four-gluon vertex further. For massive quarks, the chirality-flow formalism sheds light on how to exploit the freedom to measure spin along any direction to shorten the calculations. Overall, this results in a clear speedup for treating the Lorentz structure at high multiplicities.
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T. Gleisberg and S. Hoeche, Comix, a new matrix element generator, JHEP 12 (2008) 039 [arXiv:0808.3674] [INSPIRE].
J. Alwall et al., The automated computation of tree-level and next-to-leading order differential cross sections, and their matching to parton shower simulations, JHEP 07 (2014) 079 [arXiv:1405.0301] [INSPIRE].
T. Sjöstrand et al., An introduction to PYTHIA 8.2, Comput. Phys. Commun. 191 (2015) 159 [arXiv:1410.3012] [INSPIRE].
J. Bellm et al., Herwig 7.0/Herwig++ 3.0 release note, Eur. Phys. J. C 76 (2016) 196 [arXiv:1512.01178] [INSPIRE].
Sherpa collaboration, Event generation with Sherpa 2.2, SciPost Phys. 7 (2019) 034 [arXiv:1905.09127] [INSPIRE].
G.R. Farrar and F. Neri, How to calculate 35640 O(α5) Feynman diagrams in less than an hour, Phys. Lett. B 130 (1983) 109 [INSPIRE].
F.A. Berends and W. Giele, The six gluon process as an example of Weyl-Van Der Waerden spinor calculus, Nucl. Phys. B 294 (1987) 700 [INSPIRE].
F.A. Berends and W.T. Giele, Recursive calculations for processes with n gluons, Nucl. Phys. B 306 (1988) 759 [INSPIRE].
F.A. Berends, W.T. Giele and H. Kuijf, Exact expressions for processes involving a vector boson and up to five partons, Nucl. Phys. B 321 (1989) 39 [INSPIRE].
F.A. Berends and W.T. Giele, Multiple soft gluon radiation in parton processes, Nucl. Phys. B 313 (1989) 595 [INSPIRE].
F.A. Berends, W.T. Giele and H. Kuijf, Exact and approximate expressions for multi-gluon scattering, Nucl. Phys. B 333 (1990) 120 [INSPIRE].
H. Murayama, I. Watanabe and K. Hagiwara, HELAS: HELicity Amplitude Subroutines for Feynman diagram evaluations, KEK-91-11 (1992) [INSPIRE].
S. Dittmaier, Full O(α) radiative corrections to high-energy Compton scattering, Nucl. Phys. B 423 (1994) 384 [hep-ph/9311363] [INSPIRE].
S. Dittmaier, Weyl-van der Waerden formalism for helicity amplitudes of massive particles, Phys. Rev. D 59 (1998) 016007 [hep-ph/9805445] [INSPIRE].
S. Weinzierl, Automated computation of spin- and colour-correlated Born matrix elements, Eur. Phys. J. C 45 (2006) 745 [hep-ph/0510157] [INSPIRE].
S. Keppeler and M. Sjodahl, Orthogonal multiplet bases in SU(Nc) color space, JHEP 09 (2012) 124 [arXiv:1207.0609] [INSPIRE].
M. Sjodahl and J. Thorén, Decomposing color structure into multiplet bases, JHEP 09 (2015) 055 [arXiv:1507.03814] [INSPIRE].
Y.-J. Du, M. Sjödahl and J. Thorén, Recursion in multiplet bases for tree-level MHV gluon amplitudes, JHEP 05 (2015) 119 [arXiv:1503.00530] [INSPIRE].
J. Alcock-Zeilinger, S. Keppeler, S. Plätzer and M. Sjodahl, Wigner 6j symbols for SU(N): symbols with at least two quark-lines, J. Math. Phys. 64 (2023) 023504 [arXiv:2209.15013] [INSPIRE].
G. ’t Hooft, A planar diagram theory for strong interactions, Nucl. Phys. B 72 (1974) 461 [INSPIRE].
A. Kanaki and C.G. Papadopoulos, HELAC-PHEGAS: automatic computation of helicity amplitudes and cross-sections, AIP Conf. Proc. 583 (2002) 169 [hep-ph/0012004] [INSPIRE].
F. Maltoni, K. Paul, T. Stelzer and S. Willenbrock, Color flow decomposition of QCD amplitudes, Phys. Rev. D 67 (2003) 014026 [hep-ph/0209271] [INSPIRE].
W. Kilian, T. Ohl, J. Reuter and C. Speckner, QCD in the color-flow representation, JHEP 10 (2012) 022 [arXiv:1206.3700] [INSPIRE].
M. Sjödahl, ColorMath — a package for color summed calculations in SU(Nc), Eur. Phys. J. C 73 (2013) 2310 [arXiv:1211.2099] [INSPIRE].
C. Reuschle and S. Weinzierl, Decomposition of one-loop QCD amplitudes into primitive amplitudes based on shuffle relations, Phys. Rev. D 88 (2013) 105020 [arXiv:1310.0413] [INSPIRE].
M. Sjodahl, ColorFull — a C++ library for calculations in SU(Nc) color space, Eur. Phys. J. C 75 (2015) 236 [arXiv:1412.3967] [INSPIRE].
A. Lifson, C. Reuschle and M. Sjodahl, The chirality-flow formalism, Eur. Phys. J. C 80 (2020) 1006 [arXiv:2003.05877] [INSPIRE].
J. Alnefjord, A. Lifson, C. Reuschle and M. Sjodahl, The chirality-flow formalism for the standard model, Eur. Phys. J. C 81 (2021) 371 [arXiv:2011.10075] [INSPIRE].
A. Lifson, S. Plätzer and M. Sjodahl, One-loop calculations in the chirality-flow formalism, arXiv:2303.02125 [INSPIRE].
A. Lifson, M. Sjodahl and Z. Wettersten, Automating scattering amplitudes with chirality flow, Eur. Phys. J. C 82 (2022) 535 [arXiv:2203.13618] [INSPIRE].
Z. Xu, D.-H. Zhang and L. Chang, Helicity amplitudes for multiple Bremsstrahlung in massless non-Abelian gauge theories, Nucl. Phys. B 291 (1987) 392 [INSPIRE].
M.L. Mangano, S.J. Parke and Z. Xu, Duality and multi-gluon scattering, Nucl. Phys. B 298 (1988) 653 [INSPIRE].
P. De Causmaecker, R. Gastmans, W. Troost and T.T. Wu, Multiple Bremsstrahlung in gauge theories at high-energies. 1. General formalism for quantum electrodynamics, Nucl. Phys. B 206 (1982) 53 [INSPIRE].
F.A. Berends et al., Single Bremsstrahlung processes in gauge theories, Phys. Lett. B 103 (1981) 124 [INSPIRE].
F.A. Berends et al., Multiple Bremsstrahlung in gauge theories at high-energies. 2. Single Bremsstrahlung, Nucl. Phys. B 206 (1982) 61 [INSPIRE].
P. De Causmaecker, R. Gastmans, W. Troost and T.T. Wu, Helicity amplitudes for massless QED, Phys. Lett. B 105 (1981) 215 [INSPIRE].
CALKUL collaboration, Multiple Bremsstrahlung in gauge theories at high-energies. 3. Finite mass effects in collinear photon Bremsstrahlung, Nucl. Phys. B 239 (1984) 382 [INSPIRE].
R. Kleiss, The cross-section for e+e− → e+e−e+e−, Nucl. Phys. B 241 (1984) 61 [INSPIRE].
F.A. Berends, P.H. Daverveldt and R. Kleiss, Complete lowest order calculations for four lepton final states in electron-positron collisions, Nucl. Phys. B 253 (1985) 441 [INSPIRE].
J.F. Gunion and Z. Kunszt, Four jet processes: gluon-gluon scattering to nonidentical quark-anti-quark pairs, Phys. Lett. B 159 (1985) 167 [INSPIRE].
J.F. Gunion and Z. Kunszt, Improved analytic techniques for tree graph calculations and the \( Ggq\overline{q} \) lepton anti-lepton subprocess, Phys. Lett. B 161 (1985) 333 [INSPIRE].
R. Kleiss and W.J. Stirling, Cross-sections for the production of an arbitrary number of photons in electron-positron annihilation, Phys. Lett. B 179 (1986) 159 [INSPIRE].
K. Hagiwara and D. Zeppenfeld, Helicity amplitudes for heavy lepton production in e+e− annihilation, Nucl. Phys. B 274 (1986) 1 [INSPIRE].
R. Kleiss and W.J. Stirling, Spinor techniques for calculating \( p\overline{p} \) → W±/Z0 + jets, Nucl. Phys. B 262 (1985) 235 [INSPIRE].
R. Kleiss, Hard Bremsstrahlung amplitudes for e+e− collisions with polarized beams at LEP/SLC energies, Z. Phys. C 33 (1987) 433 [INSPIRE].
CALKUL collaboration, New techniques and results in gauge theory calculations, in the proceedings of the Europhysics study conference: electroweak effects at high energies, (1987), p. 599 [INSPIRE].
C. Schwinn and S. Weinzierl, Scalar diagrammatic rules for Born amplitudes in QCD, JHEP 05 (2005) 006 [hep-th/0503015] [INSPIRE].
R. Gastmans and T.T. Wu, The ubiquitous photon: helicity method for QED and QCD, Int. Ser. Monogr. Phys. 80 (1990) 1 [INSPIRE].
M.E. Peskin and D.V. Schroeder, An introduction to quantum field theory, Addison-Wesley, Reading, MA, U.S.A. (1995).
P. de Aquino et al., ALOHA: Automatic Libraries Of Helicity Amplitudes for Feynman diagram computations, Comput. Phys. Commun. 183 (2012) 2254 [arXiv:1108.2041] [INSPIRE].
T. Ohlsson, Relativistic quantum physics: from advanced quantum mechanics to introductory quantum field theory, Cambridge University Press, Cambridge, U.K. (2012) [INSPIRE].
R. Kleiss, W.J. Stirling and S.D. Ellis, A new Monte Carlo treatment of multiparticle phase space at high-energies, Comput. Phys. Commun. 40 (1986) 359 [INSPIRE].
O. Mattelaer and K. Ostrolenk, Speeding up MadGraph5_aMC@NLO, Eur. Phys. J. C 81 (2021) 435 [arXiv:2102.00773] [INSPIRE].
N. Nethercote and J. Seward, Valgrind: a framework for heavyweight dynamic binary instrumentation, in PLDI ’07, (2007).
J. Weidendorfer, M. Kowarschik and C. Trinitis, A tool suite for simulation based analysis of memory access behavior, in Lecture notes in computer science, Springer, Berlin, Heidelberg, Germany (2004), p. 440 [https://doi.org/10.1007/978-3-540-24688-6_58].
Acknowledgments
We thank Olivier Mattelaer for productive discussions on the MadGraph5_aMC@NLO implementation. We also thank Rikkert Frederix and Olivier Mattelaer for constructive feedback on the manuscript. ZW thanks Stefan Roiser and Robert Schöfbeck for enabling pursuit of this work. AL and MS acknowledge support by the Swedish Research Council (contract number 2016-05996, as well as the European Union’s Horizon 2020 research and innovation programme (grant agreement No 668679). The authors have in part also been supported by the European Union’s Horizon 2020 research and innovation programme as part of the Marie Sklodowska-Curie Innovative Training Network MCnetITN3 (grant agreement no. 722104).
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Boman, E., Lifson, A., Sjodahl, M. et al. Simplifying QCD event generation with chirality flow, reference vectors and spin directions. J. High Energ. Phys. 2024, 5 (2024). https://doi.org/10.1007/JHEP02(2024)005
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DOI: https://doi.org/10.1007/JHEP02(2024)005