Webs and posets

  • M. Dukes
  • E. Gardi
  • H. McAslan
  • D. J. Scott
  • C. D. WhiteEmail author
Open Access


The non-Abelian exponentiation theorem has recently been generalised to correlators of multiple Wilson line operators. The perturbative expansions of these correlators exponentiate in terms of sets of diagrams called webs, which together give rise to colour factors corresponding to connected graphs. The colour and kinematic degrees of freedom of individual diagrams in a web are entangled by mixing matrices of purely combinatorial origin. In this paper we relate the combinatorial study of these matrices to properties of partially ordered sets (posets), and hence obtain explicit solutions for certain families of web-mixing matrix, at arbitrary order in perturbation theory. We also provide a general expression for the rank of a general class of mixing matrices, which governs the number of independent colour factors arising from such webs. Finally, we use the poset language to examine a previously conjectured sum rule for the columns of web-mixing matrices which governs the cancellation of the leading subdivergences between diagrams in the web. Our results, when combined with parallel developments in the evaluation of kinematic integrals, offer new insights into the all-order structure of infrared singularities in non-Abelian gauge theories.


QCD Scattering Amplitudes Resummation 


Open Access

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  1. [1]
    I.Y. Arefeva, Quantum contour field equations, Phys. Lett. B 93 (1980) 347 [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  2. [2]
    A.M. Polyakov, Gauge fields as rings of glue, Nucl. Phys. B 164 (1980) 171 [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  3. [3]
    V. Dotsenko and S. Vergeles, Renormalizability of phase factors in the non-Abelian gauge theory, Nucl. Phys. B 169 (1980) 527 [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  4. [4]
    R.A. Brandt, F. Neri and M.-A. Sato, Renormalization of loop functions for all loops, Phys. Rev. D 24 (1981) 879 [INSPIRE].ADSGoogle Scholar
  5. [5]
    G. Korchemsky and A. Radyushkin, Loop space formalism and renormalization group for the infrared asymptotics of QCD, Phys. Lett. B 171 (1986) 459 [INSPIRE].ADSCrossRefGoogle Scholar
  6. [6]
    S. Ivanov, G. Korchemsky and A. Radyushkin, Infrared asymptotics of perturbative QCD: contour gauges, Sov. J. Nucl. Phys. 44 (1986) 145 [Yad. Fiz. 44 (1986) 230] [INSPIRE].Google Scholar
  7. [7]
    G. Korchemsky and A. Radyushkin, Infrared asymptotics of perturbative QCD: renormalization properties of the Wilson loops in higher orders of perturbation theory, Sov. J. Nucl. Phys. 44 (1986) 877 [Yad. Fiz. 44 (1986) 1351] [INSPIRE].Google Scholar
  8. [8]
    G. Korchemsky and A. Radyushkin, Infrared asymptotics of perturbative QCD. Quark and gluon propagators, Sov. J. Nucl. Phys. 45 (1987) 127 [Yad. Fiz. 45 (1987) 198] [INSPIRE].Google Scholar
  9. [9]
    G. Korchemsky and A. Radyushkin, Infrared asymptotics of perturbative QCD. Vertex functions, Sov. J. Nucl. Phys. 45 (1987) 910 [Yad. Fiz. 45 (1987) 1466] [INSPIRE].Google Scholar
  10. [10]
    G. Korchemsky and A. Radyushkin, Renormalization of the Wilson loops beyond the leading order, Nucl. Phys. B 283 (1987) 342 [INSPIRE].ADSCrossRefGoogle Scholar
  11. [11]
    G. Korchemsky, Sudakov form-factor in QCD, Phys. Lett. B 220 (1989) 629 [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  12. [12]
    G. Korchemsky, Asymptotics of the Altarelli-Parisi-Lipatov evolution kernels of parton distributions, Mod. Phys. Lett. A 4 (1989) 1257 [INSPIRE].ADSCrossRefGoogle Scholar
  13. [13]
    J.C. Collins, Sudakov form-factors, Adv. Ser. Direct. High Energy Phys. 5 (1989) 573 [hep-ph/0312336] [INSPIRE].CrossRefGoogle Scholar
  14. [14]
    G. Korchemsky and A. Radyushkin, Infrared factorization, Wilson lines and the heavy quark limit, Phys. Lett. B 279 (1992) 359 [hep-ph/9203222] [INSPIRE].ADSCrossRefGoogle Scholar
  15. [15]
    N. Kidonakis, G. Oderda and G.F. Sterman, Evolution of color exchange in QCD hard scattering, Nucl. Phys. B 531 (1998) 365 [hep-ph/9803241] [INSPIRE].ADSCrossRefGoogle Scholar
  16. [16]
    N. Kidonakis and G.F. Sterman, Resummation for QCD hard scattering, Nucl. Phys. B 505 (1997) 321 [hep-ph/9705234] [INSPIRE].ADSCrossRefGoogle Scholar
  17. [17]
    N. Kidonakis and G.F. Sterman, Subleading logarithms in QCD hard scattering, Phys. Lett. B 387 (1996) 867 [INSPIRE].ADSCrossRefGoogle Scholar
  18. [18]
    N. Kidonakis, Next-to-next-to-leading soft-gluon corrections for the top quark cross section and transverse momentum distribution, Phys. Rev. D 82 (2010) 114030 [arXiv:1009.4935] [INSPIRE].ADSGoogle Scholar
  19. [19]
    J. Drummond, J. Henn, G. Korchemsky and E. Sokatchev, On planar gluon amplitudes/Wilson loops duality, Nucl. Phys. B 795 (2008) 52 [arXiv:0709.2368] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  20. [20]
    B. Basso, G. Korchemsky and J. Kotanski, Cusp anomalous dimension in maximally supersymmetric Yang-Mills theory at strong coupling, Phys. Rev. Lett. 100 (2008) 091601 [arXiv:0708.3933] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  21. [21]
    L.F. Alday and J.M. Maldacena, Gluon scattering amplitudes at strong coupling, JHEP 06 (2007) 064 [arXiv:0705.0303] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  22. [22]
    V. Pestun, Localization of gauge theory on a four-sphere and supersymmetric Wilson loops, Commun. Math. Phys. 313 (2012) 71 [arXiv:0712.2824] [INSPIRE].ADSCrossRefzbMATHMathSciNetGoogle Scholar
  23. [23]
    N. Drukker, Integrable Wilson loops, JHEP 10 (2013) 135 [arXiv:1203.1617] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  24. [24]
    Y.-T. Chien, M.D. Schwartz, D. Simmons-Duffin and I.W. Stewart, Jet physics from static charges in AdS, Phys. Rev. D 85 (2012) 045010 [arXiv:1109.6010] [INSPIRE].ADSGoogle Scholar
  25. [25]
    I. Cherednikov, T. Mertens and F. Van der Veken, Cusped light-like Wilson loops in gauge theories, Phys. Part. Nucl. 44 (2013) 250 [arXiv:1210.1767] [INSPIRE].CrossRefGoogle Scholar
  26. [26]
    I. Cherednikov, T. Mertens and F. Van der Veken, Evolution of cusped light-like Wilson loops and geometry of the loop space, Phys. Rev. D 86 (2012) 085035 [arXiv:1208.1631] [INSPIRE].ADSGoogle Scholar
  27. [27]
    J.M. Henn and T. Huber, The four-loop cusp anomalous dimension in \( \mathcal{N} \) = 4 super Yang-Mills and analytic integration techniques for Wilson line integrals, JHEP 09 (2013) 147 [arXiv:1304.6418] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  28. [28]
    S.G. Naculich and H.J. Schnitzer, Eikonal methods applied to gravitational scattering amplitudes, JHEP 05 (2011) 087 [arXiv:1101.1524] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  29. [29]
    C.D. White, Factorization properties of soft graviton amplitudes, JHEP 05 (2011) 060 [arXiv:1103.2981] [INSPIRE].ADSCrossRefGoogle Scholar
  30. [30]
    R. Akhoury, R. Saotome and G. Sterman, Collinear and soft divergences in perturbative quantum gravity, Phys. Rev. D 84 (2011) 104040 [arXiv:1109.0270] [INSPIRE].ADSGoogle Scholar
  31. [31]
    D. Miller and C. White, The gravitational cusp anomalous dimension from AdS space, Phys. Rev. D 85 (2012) 104034 [arXiv:1201.2358] [INSPIRE].ADSGoogle Scholar
  32. [32]
    M. Beneke and G. Kirilin, Soft-collinear gravity, JHEP 09 (2012) 066 [arXiv:1207.4926] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  33. [33]
    E. Gardi and L. Magnea, Infrared singularities in QCD amplitudes, Nuovo Cim. C 32 (2009) 137 [arXiv:0908.3273] [INSPIRE].Google Scholar
  34. [34]
    J.C. Collins, D.E. Soper and G.F. Sterman, Factorization of hard processes in QCD, Adv. Ser. Direct. High Energy Phys. 5 (1988) 1 [hep-ph/0409313] [INSPIRE].CrossRefGoogle Scholar
  35. [35]
    G. Korchemsky and G. Marchesini, Structure function for large x and renormalization of Wilson loop, Nucl. Phys. B 406 (1993) 225 [hep-ph/9210281] [INSPIRE].ADSCrossRefGoogle Scholar
  36. [36]
    G. Korchemsky and G. Marchesini, Resummation of large infrared corrections using Wilson loops, Phys. Lett. B 313 (1993) 433 [INSPIRE].ADSCrossRefGoogle Scholar
  37. [37]
    S. Catani, M.L. Mangano, P. Nason and L. Trentadue, The resummation of soft gluons in hadronic collisions, Nucl. Phys. B 478 (1996) 273 [hep-ph/9604351] [INSPIRE].ADSCrossRefGoogle Scholar
  38. [38]
    G. Oderda, Dijet rapidity gaps in photoproduction from perturbative QCD, Phys. Rev. D 61 (2000) 014004 [hep-ph/9903240] [INSPIRE].ADSGoogle Scholar
  39. [39]
    R. Bonciani, S. Catani, M.L. Mangano and P. Nason, NLL resummation of the heavy quark hadroproduction cross-section, Nucl. Phys. B 529 (1998) 424 [Erratum ibid. B 803 (2008) 234] [hep-ph/9801375] [INSPIRE].ADSCrossRefGoogle Scholar
  40. [40]
    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].ADSCrossRefGoogle Scholar
  41. [41]
    M. Beneke, M. Czakon, P. Falgari, A. Mitov and C. Schwinn, Threshold expansion of the \( gg\left( {q\overline{q}} \right)\to Q\overline{Q}+X \) cross section at \( \mathcal{O}\left( {\alpha_s^4} \right) \), Phys. Lett. B 690 (2010) 483 [arXiv:0911.5166] [INSPIRE].ADSCrossRefGoogle Scholar
  42. [42]
    V. Ahrens, A. Ferroglia, M. Neubert, B.D. Pecjak and L.L. Yang, Renormalization-group improved predictions for top-quark pair production at hadron colliders, JHEP 09 (2010) 097 [arXiv:1003.5827] [INSPIRE].ADSCrossRefGoogle Scholar
  43. [43]
    M. Czakon, P. Fiedler and A. Mitov, The total top quark pair production cross-section at hadron colliders through \( \mathcal{O}\left( {\alpha_S^4} \right) \), Phys. Rev. Lett. 110 (2013) 252004 [arXiv:1303.6254] [INSPIRE].ADSCrossRefGoogle Scholar
  44. [44]
    C.W. Bauer, S. Fleming and M.E. Luke, Summing Sudakov logarithms in B → X(sγ) in effective field theory, Phys. Rev. D 63 (2000) 014006 [hep-ph/0005275] [INSPIRE].ADSGoogle Scholar
  45. [45]
    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. D 63 (2001) 114020 [hep-ph/0011336] [INSPIRE].ADSGoogle Scholar
  46. [46]
    C.W. Bauer and I.W. Stewart, Invariant operators in collinear effective theory, Phys. Lett. B 516 (2001) 134 [hep-ph/0107001] [INSPIRE].ADSCrossRefGoogle Scholar
  47. [47]
    C.W. Bauer, D. Pirjol and I.W. Stewart, Soft collinear factorization in effective field theory, Phys. Rev. D 65 (2002) 054022 [hep-ph/0109045] [INSPIRE].ADSGoogle Scholar
  48. [48]
    C.W. Bauer, S. Fleming, D. Pirjol, I.Z. Rothstein and I.W. Stewart, Hard scattering factorization from effective field theory, Phys. Rev. D 66 (2002) 014017 [hep-ph/0202088] [INSPIRE].ADSGoogle Scholar
  49. [49]
    T. Becher and M. Neubert, Threshold resummation in momentum space from effective field theory, Phys. Rev. Lett. 97 (2006) 082001 [hep-ph/0605050] [INSPIRE].ADSCrossRefGoogle Scholar
  50. [50]
    T. Becher, M. Neubert and B.D. Pecjak, Factorization and momentum-space resummation in deep-inelastic scattering, JHEP 01 (2007) 076 [hep-ph/0607228] [INSPIRE].ADSCrossRefGoogle Scholar
  51. [51]
    T. Becher, M. Neubert and G. Xu, Dynamical threshold enhancement and resummation in Drell-Yan production, JHEP 07 (2008) 030 [arXiv:0710.0680] [INSPIRE].ADSCrossRefGoogle Scholar
  52. [52]
    D. Yennie, S.C. Frautschi and H. Suura, The infrared divergence phenomena and high-energy processes, Annals Phys. 13 (1961) 379 [INSPIRE].ADSCrossRefGoogle Scholar
  53. [53]
    G.F. Sterman, Summation of large corrections to short distance hadronic cross-sections, Nucl. Phys. B 281 (1987) 310 [INSPIRE].ADSCrossRefGoogle Scholar
  54. [54]
    S. Catani and L. Trentadue, Resummation of the QCD perturbative series for hard processes, Nucl. Phys. B 327 (1989) 323 [INSPIRE].ADSCrossRefGoogle Scholar
  55. [55]
    E. Laenen, G. Stavenga and C.D. White, Path integral approach to eikonal and next-to-eikonal exponentiation, JHEP 03 (2009) 054 [arXiv:0811.2067] [INSPIRE].ADSCrossRefGoogle Scholar
  56. [56]
    M.G. Sotiropoulos and G.F. Sterman, Color exchange in near forward hard elastic scattering, Nucl. Phys. B 419 (1994) 59 [hep-ph/9310279] [INSPIRE].ADSCrossRefGoogle Scholar
  57. [57]
    G.P. Korchemsky, On near forward high-energy scattering in QCD, Phys. Lett. B 325 (1994) 459 [hep-ph/9311294] [INSPIRE].ADSCrossRefGoogle Scholar
  58. [58]
    I. Korchemskaya and G. Korchemsky, High-energy scattering in QCD and cross singularities of Wilson loops, Nucl. Phys. B 437 (1995) 127 [hep-ph/9409446] [INSPIRE].ADSCrossRefGoogle Scholar
  59. [59]
    I. Korchemskaya and G. Korchemsky, Evolution equation for gluon Regge trajectory, Phys. Lett. B 387 (1996) 346 [hep-ph/9607229] [INSPIRE].ADSCrossRefGoogle Scholar
  60. [60]
    I. Balitsky, Operator expansion for high-energy scattering, Nucl. Phys. B 463 (1996) 99 [hep-ph/9509348] [INSPIRE].ADSCrossRefGoogle Scholar
  61. [61]
    Y.V. Kovchegov, Non-Abelian Weizsacker-Williams field and a two-dimensional effective color charge density for a very large nucleus, Phys. Rev. D 54 (1996) 5463 [hep-ph/9605446] [INSPIRE].ADSGoogle Scholar
  62. [62]
    I. Balitsky, High-energy QCD and Wilson lines, hep-ph/0101042 [INSPIRE].
  63. [63]
    I. Balitsky and G.A. Chirilli, High-energy amplitudes in N = 4 SYM in the next-to-leading order, Phys. Lett. B 687 (2010) 204 [arXiv:0911.5192] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  64. [64]
    J. Jalilian-Marian, A. Kovner, L.D. McLerran and H. Weigert, The intrinsic glue distribution at very small x, Phys. Rev. D 55 (1997) 5414 [hep-ph/9606337] [INSPIRE].ADSGoogle Scholar
  65. [65]
    E. Gardi, J. Kuokkanen, K. Rummukainen and H. Weigert, Running coupling and power corrections in nonlinear evolution at the high-energy limit, Nucl. Phys. A 784 (2007) 282 [hep-ph/0609087] [INSPIRE].ADSCrossRefGoogle Scholar
  66. [66]
    V. Del Duca, C. Duhr, E. Gardi, L. Magnea and C.D. White, An infrared approach to Reggeization, Phys. Rev. D 85 (2012) 071104 [arXiv:1108.5947] [INSPIRE].ADSGoogle Scholar
  67. [67]
    V. Del Duca, C. Duhr, E. Gardi, L. Magnea and C.D. White, The infrared structure of gauge theory amplitudes in the high-energy limit, JHEP 12 (2011) 021 [arXiv:1109.3581] [INSPIRE].ADSCrossRefGoogle Scholar
  68. [68]
    Y.V. Kovchegov and E. Levin, Quantum chromodynamics at high energy, Cambridge Univ. Pr., Cambridge U.K. (2012) [INSPIRE].
  69. [69]
    A.H. Mueller, Soft gluons in the infinite momentum wave function and the BFKL Pomeron, Nucl. Phys. B 415 (1994) 373 [INSPIRE].ADSCrossRefGoogle Scholar
  70. [70]
    S. Melville, S. Naculich, H. Schnitzer and C. White, Wilson line approach to gravity in the high energy limit, arXiv:1306.6019 [INSPIRE].
  71. [71]
    R. Akhoury, R. Saotome and G. Sterman, High energy scattering in perturbative quantum gravity at next to leading power, arXiv:1308.5204 [INSPIRE].
  72. [72]
    J. Ware, R. Saotome and R. Akhoury, Construction of an asymptotic S matrix for perturbative quantum gravity, JHEP 10 (2013) 159 [arXiv:1308.6285] [INSPIRE].ADSCrossRefGoogle Scholar
  73. [73]
    J. Gatheral, Exponentiation of eikonal cross-sections in non-Abelian gauge theories, Phys. Lett. B 133 (1983) 90 [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  74. [74]
    J. Frenkel and J. Taylor, Non-Abelian eikonal exponentiation, Nucl. Phys. B 246 (1984) 231 [INSPIRE].ADSCrossRefGoogle Scholar
  75. [75]
    G.F. Sterman, Infrared divergences in perturbative QCD (talk), AIP Conf. Proc. 74 (1981) 22 [INSPIRE].ADSCrossRefGoogle Scholar
  76. [76]
    S.M. Aybat, L.J. Dixon and G.F. Sterman, The two-loop anomalous dimension matrix for soft gluon exchange, Phys. Rev. Lett. 97 (2006) 072001 [hep-ph/0606254] [INSPIRE].ADSCrossRefGoogle Scholar
  77. [77]
    S.M. Aybat, L.J. Dixon and G.F. Sterman, The two-loop soft anomalous dimension matrix and resummation at next-to-next-to leading pole, Phys. Rev. D 74 (2006) 074004 [hep-ph/0607309] [INSPIRE].ADSGoogle Scholar
  78. [78]
    A. Ferroglia, M. Neubert, B.D. Pecjak and L.L. Yang, Two-loop divergences of scattering amplitudes with massive partons, Phys. Rev. Lett. 103 (2009) 201601 [arXiv:0907.4791] [INSPIRE].ADSCrossRefGoogle Scholar
  79. [79]
    A. Ferroglia, M. Neubert, B.D. Pecjak and L.L. Yang, Two-loop divergences of massive scattering amplitudes in non-Abelian gauge theories, JHEP 11 (2009) 062[arXiv:0908.3676] [INSPIRE].ADSCrossRefGoogle Scholar
  80. [80]
    A. Mitov, G.F. Sterman and I. Sung, Computation of the soft anomalous dimension matrix in coordinate space, Phys. Rev. D 82 (2010) 034020 [arXiv:1005.4646] [INSPIRE].ADSGoogle Scholar
  81. [81]
    T. Becher and M. Neubert, Infrared singularities of scattering amplitudes in perturbative QCD, Phys. Rev. Lett. 102 (2009) 162001 [arXiv:0901.0722] [INSPIRE].ADSCrossRefGoogle Scholar
  82. [82]
    T. Becher and M. Neubert, On the structure of infrared singularities of gauge-theory amplitudes, JHEP 06 (2009) 081 [Erratum ibid. 11 (2013) 024] [arXiv:0903.1126] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  83. [83]
    E. Gardi and L. Magnea, Factorization constraints for soft anomalous dimensions in QCD scattering amplitudes, JHEP 03 (2009) 079 [arXiv:0901.1091] [INSPIRE].ADSCrossRefGoogle Scholar
  84. [84]
    L.J. Dixon, E. Gardi and L. Magnea, On soft singularities at three loops and beyond, JHEP 02 (2010) 081 [arXiv:0910.3653] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  85. [85]
    V. Ahrens, M. Neubert and L. Vernazza, Structure of infrared singularities of gauge-theory amplitudes at three and four loops, JHEP 09 (2012) 138 [arXiv:1208.4847] [INSPIRE].ADSCrossRefGoogle Scholar
  86. [86]
    S.G. Naculich, H. Nastase and H.J. Schnitzer, All-loop infrared-divergent behavior of most-subleading-color gauge-theory amplitudes, JHEP 04 (2013) 114 [arXiv:1301.2234] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  87. [87]
    S. Caron-Huot, When does the gluon reggeize?, arXiv:1309.6521 [INSPIRE].
  88. [88]
    E. Gardi, E. Laenen, G. Stavenga and C.D. White, Webs in multiparton scattering using the replica trick, JHEP 11 (2010) 155 [arXiv:1008.0098] [INSPIRE].ADSCrossRefGoogle Scholar
  89. [89]
    E. Gardi and C.D. White, General properties of multiparton webs: proofs from combinatorics, JHEP 03 (2011) 079 [arXiv:1102.0756] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  90. [90]
    E. Gardi, J.M. Smillie and C.D. White, On the renormalization of multiparton webs, JHEP 09 (2011) 114 [arXiv:1108.1357] [INSPIRE].ADSCrossRefGoogle Scholar
  91. [91]
    E. Gardi, J.M. Smillie and C.D. White, The non-Abelian exponentiation theorem for multiple Wilson lines, JHEP 06 (2013) 088 [arXiv:1304.7040] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  92. [92]
    A. Mitov, G. Sterman and I. Sung, Diagrammatic exponentiation for products of Wilson lines, Phys. Rev. D 82 (2010) 096010 [arXiv:1008.0099] [INSPIRE].ADSGoogle Scholar
  93. [93]
    E. Gardi, From webs to polylogarithms, arXiv:1310.5268 [INSPIRE].
  94. [94]
    G. Falcioni, E. Gardi, M. Harley, L. Magnea and C.D. White, Calculating three-loop webs, in preparation.Google Scholar
  95. [95]
    M. Dukes, E. Gardi, E. Steingrimsson and C.D. White, Web worlds, web-colouring matrices and web-mixing matrices, J. Comb. Theory Ser. A 120 (2013) 1012 [arXiv:1301.6576] [INSPIRE].CrossRefMathSciNetGoogle Scholar
  96. [96]
    V. Del Duca, L.J. Dixon and F. Maltoni, New color decompositions for gauge amplitudes at tree and loop level, Nucl. Phys. B 571 (2000) 51 [hep-ph/9910563] [INSPIRE].ADSCrossRefGoogle Scholar
  97. [97]
    D.R. Mazur, Combinatorics: a guided tour, Mathematical Association of America, U.S.A. (2010).Google Scholar
  98. [98]
    G. Brightwell and P. Winkler, Counting linear extensions, Order 8 (1991) 225.CrossRefzbMATHMathSciNetGoogle Scholar
  99. [99]
    B. Sury, T. Wang and F.Z. Zhao, Identities involving reciprocals of binomial coefficients, J. Integer Seq. 7 (2004) 04.2.8.MathSciNetGoogle Scholar

Copyright information

© The Author(s) 2014

Authors and Affiliations

  • M. Dukes
    • 1
  • E. Gardi
    • 2
  • H. McAslan
    • 3
  • D. J. Scott
    • 4
  • C. D. White
    • 5
    Email author
  1. 1.Department of Computer and Information SciencesUniversity of StrathclydeGlasgowU.K
  2. 2.Higgs Centre for Theoretical Physics, School of Physics and AstronomyThe University of EdinburghEdinburghScotland, U.K
  3. 3.Department of Physics and AstronomyUniversity of SussexBrightonU.K
  4. 4.Institute for Particle Physics PhenomenologyUniversity of DurhamDurhamU.K
  5. 5.SUPA, School of Physics and AstronomyUniversity of GlasgowGlasgowScotland, U.K

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