We study a class of Wigner 6j symbols involving two fundamental representations and derive explicit formulas for all 6j symbols in this class. Our formulas express the 6j symbols in terms of the dimensions of the involved representations, and they are thereby functions of N. We view these explicit formulas as a first step toward efficiently decomposing color structures in terms of group invariants.
REFERENCES
1.
J. E.
Paton
and C.
Hong-Mo
, “Generalized Veneziano model with isospin
,” Nucl. Phys. B
10
, 516
–520
(1969
).2.
F. A.
Berends
and W.
Giele
, “The six-gluon process as an example of Weyl-van der Waerden spinor calculus
,” Nucl. Phys. B
294
, 700
(1987
).3.
M.
Mangano
, S.
Parke
, and Z.
Xu
, “Duality and multi-gluon scattering
,” Nucl. Phys. B
298
, 653
(1988
).4.
M.
Mangano
, “The color structure of gluon emission
,” Nucl. Phys. B
309
, 461
(1988
).5.
D. A.
Kosower
, “Color factorization for fermionic amplitudes
,” Nucl. Phys. B
315
, 391
–418
(1989
).6.
Z.
Nagy
and D. E.
Soper
, “Parton showers with quantum interference
,” J. High Energy Phys.
2007
(9
), 114
; arXiv:0706.0017 [hep-ph].7.
M.
Sjodahl
, “Color structure for soft gluon resummation—A general recipe
,” J. High Energy Phys.
2009
(9
), 87
; arXiv:0906.1121 [hep-ph].8.
J.
Alwall
, M.
Herquet
, F.
Maltoni
, O.
Mattelaer
, and T.
Stelzer
, “MadGraph 5: Going beyond
,” J. High Energy Phys.
2011
(6
), 128
; arXiv:1106.0522 [hep-ph].9.
M.
Sjödahl
, “ColorFull: A C++ library for calculations in SU(Nc) color space
,” Eur. Phys. J. C
75
, 236
(2015
); arXiv:1412.3967 [hep-ph].10.
S.
Plätzer
and M.
Sjodahl
, “Subleading Nc improved parton showers
,” J. High Energy Phys.
2012
(7
), 42
; arXiv:1201.0260 [hep-ph].11.
S.
Plätzer
, M.
Sjodahl
, and J.
Thorén
, “Color matrix element corrections for parton showers
,” J. High Energy Phys.
2018
(11
), 9
; arXiv:1808.00332 [hep-ph].12.
G.
’t Hooft
, “A planar diagram theory for strong interactions
,” Nucl. Phys. B
72
, 461
(1974
).13.
A.
Kanaki
and C. G.
Papadopoulos
, “HELAC-PHEGAS: Automatic computation of helicity amplitudes and cross-sections
,” AIP Conf. Proc.
583
, 169
(2002
); arXiv:hep-ph/0012004.14.
F.
Maltoni
, K.
Paul
, T.
Stelzer
, and S.
Willenbrock
, “Color flow decomposition of QCD amplitudes
,” Phys. Rev. D
67
, 014026
(2003
); arXiv:hep-ph/0209271 [hep-ph].15.
S.
Plätzer
, “Summing large-N towers in colour flow evolution
,” Eur. Phys. J. C
74
, 2907
(2014
); arXiv:1312.2448 [hep-ph].16.
R.
Ángeles Martínez
, M.
De Angelis
, J. R.
Forshaw
, S.
Plätzer
, and M. H.
Seymour
, “Soft gluon evolution and non-global logarithms
,” J. High Energy Phys.
2018
(5
), 44
; arXiv:1802.08531 [hep-ph].17.
M.
De Angelis
, J. R.
Forshaw
, and S.
Plätzer
, “Resummation and simulation of soft gluon effects beyond leading color
,” Phys. Rev. Lett.
126
, 112001
(2021
); arXiv:2007.09648 [hep-ph].18.
S.
Plätzer
and I.
Ruffa
, “Towards colour flow evolution at two loops
,” J. High Energy Phys.
2021
(6
), 7
; arXiv:2012.15215 [hep-ph].19.
S.
Keppeler
and M.
Sjödahl
, “Orthogonal multiplet bases in SU(Nc) color space
,” J. High Energy Phys.
2012
(9
), 124
; arXiv:1207.0609 [hep-ph].20.
A.
Kyrieleis
and M. H.
Seymour
, “The colour evolution of the process qq → qqg
,” J. High Energy Phys.
2006
(1
), 85
; arXiv:hep-ph/0510089.21.
Y. L.
Dokshitzer
and G.
Marchesini
, “Soft gluons at large angles in hadron collisions
,” J. High Energy Phys.
2006
(1
), 7
; arXiv:hep-ph/0509078.22.
M.
Sjödahl
, “Color evolution of 2 → 3 processes
,” J. High Energy Phys.
2008
(12
), 83
; arXiv:0807.0555 [hep-ph].23.
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
, 69
–101
(2010
); arXiv:0907.1443 [hep-ph].24.
Y.-J.
Du
, M.
Sjödahl
, and J.
Thorén
, “Recursion in multiplet bases for tree-level MHV gluon amplitudes
,” J. High Energy Phys.
2015
(5
), 119
; arXiv:1503.00530 [hep-ph].25.
M.
Sjodahl
and J.
Thorén
, “Decomposing color structure into multiplet bases
,” J. High Energy Phys.
2015
(9
), 55
; arXiv:1507.03814 [hep-ph].26.
S.
Keppeler
and M.
Sjödahl
, “Hermitian Young operators
,” J. Math. Phys.
55
, 021702
(2014
); arXiv:1307.6147 [math-ph].27.
J.
Alcock-Zeilinger
and H.
Weigert
, “Simplification rules for birdtrack operators
,” J. Math. Phys.
58
, 051701
(2017
); arXiv:1610.08801.28.
J.
Alcock-Zeilinger
and H.
Weigert
, “Compact Hermitian Young projection operators
,” J. Math. Phys.
58
, 051702
(2017
); arXiv:1610.10088.29.
J.
Alcock-Zeilinger
and H.
Weigert
, “Transition operators
,” J. Math. Phys.
58
, 051703
(2017
); arXiv:1610.08802 [math-ph].30.
M.
Sjodahl
and J.
Thorén
, “QCD multiplet bases with arbitrary parton ordering
,” J. High Energy Phys.
2018
(11
), 198
; arXiv:1809.05002 [hep-ph].31.
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
, A376
–A384
(2016
).32.
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
, 023507
(2011
).33.
T.
Dytrych
, D.
Langr
, J. P.
Draayer
, K. D.
Launey
, and D.
Gazda
, “SU3lib: A C++ library for accurate computation of Wigner and Racah coefficients of SU(3)
,” Comput. Phys. Commun.
269
, 108137
(2021
).34.
S.
Gieseke
, P.
Kirchgaeßer
, S.
Plätzer
, and A.
Siodmok
, “Colour reconnection from soft gluon evolution
,” J. High Energy Phys.
2018
(11
), 149
; arXiv:1808.06770 [hep-ph].35.
36.
P.
Cvitanović
, Group Theory: Birdtracks, Lie’s and Exceptional Groups
(Princeton University Press
, Princeton
, 2008
).37.
S.
Keppeler
, “Birdtracks for SU(N)
,” SciPost Phys. Lect. Notes
3
, 1
(2018
); arXiv:1707.07280 [math-ph].38.
39.
B. E.
Sagan
, The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions
, 2nd ed. (Springer
, New York
, 2000
).40.
R. C.
King
, “Generalized Young tableaux and the general linear group
,” J. Math. Phys.
11
, 280
–293
(1970
).© 2023 Author(s). Published under an exclusive license by AIP Publishing.
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