Abstract
We construct a set of Wigner 6𝑗 symbols with gluon lines (adjoint representations) in closed form, expressed in terms of similar 6𝑗 symbols with quark lines (fundamental representations). Together with these Wigner 6𝑗 symbols with quark lines, this gives a set of 6𝑗 symbols sufficient for treating QCD color structure for any number of external particles, in or beyond perturbation theory. This facilitates a complete treatment of QCD color structure in terms of orthogonal multiplet bases, without the need of ever explicitly constructing the corresponding bases. We thereby open up for a completely representation theory based treatment of SU(N) color structure, with the potential of significantly speeding up the color structure treatment.
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References
J.E. Paton and H.-M. Chan, Generalized veneziano model with isospin, Nucl. Phys. B 10 (1969) 516 [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].
M.L. Mangano, S.J. Parke and Z. Xu, Duality and Multi-Gluon Scattering, Nucl. Phys. B 298 (1988) 653 [INSPIRE].
M.L. Mangano, The Color Structure of Gluon Emission, Nucl. Phys. B 309 (1988) 461 [INSPIRE].
D.A. Kosower, Color Factorization for Fermionic Amplitudes, Nucl. Phys. B 315 (1989) 391 [INSPIRE].
Z. Nagy and D.E. Soper, Parton showers with quantum interference, JHEP 09 (2007) 114 [arXiv:0706.0017] [INSPIRE].
M. Sjödahl, Color structure for soft gluon resummation: A General recipe, JHEP 09 (2009) 087 [arXiv:0906.1121] [INSPIRE].
J. Alwall et al., MadGraph 5: Going Beyond, JHEP 06 (2011) 128 [arXiv:1106.0522] [INSPIRE].
M. Sjödahl, ColorFull — a C++ library for calculations in SU(𝑁𝑐) color space, Eur. Phys. J. C 75 (2015) 236 [arXiv:1412.3967] [INSPIRE].
S. Plätzer and M. Sjödahl, Subleading 𝑁𝑐 improved Parton Showers, JHEP 07 (2012) 042 [arXiv:1201.0260] [INSPIRE].
S. Plätzer, M. Sjödahl and J. Thorén, Color matrix element corrections for parton showers, JHEP 11 (2018) 009 [arXiv:1808.00332] [INSPIRE].
R. Frederix and T. Vitos, The colour matrix at next-to-leading-colour accuracy for tree-level multi-parton processes, JHEP 12 (2021) 157 [arXiv:2109.10377] [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].
S. Plätzer, Summing Large-𝑁 Towers in Colour Flow Evolution, Eur. Phys. J. C 74 (2014) 2907 [arXiv:1312.2448] [INSPIRE].
R. Ángeles Martínez et al., Soft gluon evolution and non-global logarithms, JHEP 05 (2018) 044 [arXiv:1802.08531] [INSPIRE].
M. De Angelis, J.R. Forshaw and S. Plätzer, Resummation and Simulation of Soft Gluon Effects beyond Leading Color, Phys. Rev. Lett. 126 (2021) 112001 [arXiv:2007.09648] [INSPIRE].
S. Plätzer and I. Ruffa, Towards Colour Flow Evolution at Two Loops, JHEP 06 (2021) 007 [arXiv:2012.15215] [INSPIRE].
J. Isaacson and S. Prestel, Stochastically sampling color configurations, Phys. Rev. D 99 (2019) 014021 [arXiv:1806.10102] [INSPIRE].
M. Sjödahl and J. Thorén, Decomposing color structure into multiplet bases, JHEP 09 (2015) 055 [arXiv:1507.03814] [INSPIRE].
J.R. Forshaw, J. Holguin and S. Plätzer, Rings and strings: a basis for understanding subleading colour and QCD coherence beyond the two-jet limit, JHEP 05 (2022) 190 [arXiv:2112.13124] [INSPIRE].
A. Kyrieleis and M.H. Seymour, The colour evolution of the process q q → q q g, JHEP 01 (2006) 085 [hep-ph/0510089] [INSPIRE].
Y.L. Dokshitzer and G. Marchesini, Soft gluons at large angles in hadron collisions, JHEP 01 (2006) 007 [hep-ph/0509078] [INSPIRE].
M. Sjödahl, Color evolution of 2 → 3 processes, JHEP 12 (2008) 083 [arXiv:0807.0555] [INSPIRE].
M. Beneke, P. Falgari and C. Schwinn, Soft radiation in heavy-particle pair production: All-order colour structure and two-loop anomalous dimension, Nucl. Phys. B 828 (2010) 69 [arXiv:0907.1443] [INSPIRE].
S. Keppeler and M. Sjödahl, Orthogonal multiplet bases in SU(𝑁𝑐) color space, JHEP 09 (2012) 124 [arXiv:1207.0609] [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].
S. Keppeler and M. Sjödahl, Hermitian Young Operators, J. Math. Phys. 55 (2014) 021702 [arXiv:1307.6147] [INSPIRE].
J. Alcock-Zeilinger and H. Weigert, Simplification Rules for Birdtrack Operators, J. Math. Phys. 58 (2017) 051701 [arXiv:1610.08801] [INSPIRE].
J. Alcock-Zeilinger and H. Weigert, Compact Hermitian Young Projection Operators, J. Math. Phys. 58 (2017) 051702 [arXiv:1610.10088] [INSPIRE].
J. Alcock-Zeilinger and H. Weigert, Transition Operators, J. Math. Phys. 58 (2017) 051703 [arXiv:1610.08802] [INSPIRE].
M. Sjödahl and J. Thorén, QCD multiplet bases with arbitrary parton ordering, JHEP 11 (2018) 198 [arXiv:1809.05002] [INSPIRE].
A. Alex, M. Kalus, A. Huckleberry and J. von Delft, A numerical algorithm for the explicit calculation of SU(𝑁) and SL(𝑁, 𝐶) Clebsch-Gordan coefficients, J. Math. Phys. 52 (2011) 023507 [arXiv:1009.0437] [INSPIRE].
T. Dytrych et al., SU3lib: A C++ library for accurate computation of Wigner and Racah coefficients of SU(3), Comput. Phys. Commun. 269 (2021) 108137 [INSPIRE].
H.T. Johansson and C. Forssén, Fast and Accurate Evaluation of Wigner 3𝑗, 6𝑗, and 9𝑗 Symbols Using Prime Factorization and Multiword Integer Arithmetic, SIAM J. Sci. Comput. 38 (2016) A376 [arXiv:1504.08329] [INSPIRE].
J. Alcock-Zeilinger, S. Keppeler, S. Plätzer and M. Sjödahl, Wigner 6j symbols for SU(N): Symbols with at least two quark-lines, J. Math. Phys. 64 (2023) 023504 [arXiv:2209.15013] [INSPIRE].
P. Cvitanović, Group Theory: Birdtracks, Lie’s, and Exceptional Groups, Princeton University Press (2020) [INSPIRE].
S. Keppeler, Birdtracks for SU(𝑁), SciPost Phys. Lect. Notes 3 (2018) 1 [arXiv:1707.07280] [INSPIRE].
R.C. King, Generalized Young Tableaux and the General Linear Group, J. Math. Phys. 11 (1970) 280.
S. Plätzer, Colour evolution and infrared physics, JHEP 07 (2023) 126 [arXiv:2204.06956] [INSPIRE].
Acknowledgments
We are thankful to Judith Alcock-Zeilinger for useful discussions. We also thank the Erwin Schrödinger International Institute for Mathematics and Physics (ESI) in Vienna, for hospitality, discussions, and support, both via the Research in Teams programme “Amplitude Level Evolution II: Cracking down on color bases” (RIT0521), where this work was initiated, and via the ESI-QFT 2023 workshop, where the scientific part was concluded. MS acknowledges 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).
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Keppeler, S., Plätzer, S. & Sjodahl, M. Wigner 6j symbols with gluon lines: completing the set of 6j symbols required for color decomposition. J. High Energ. Phys. 2024, 51 (2024). https://doi.org/10.1007/JHEP05(2024)051
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DOI: https://doi.org/10.1007/JHEP05(2024)051