Advertisement

Journal of High Energy Physics

, 2017:90 | Cite as

Diagrammatic Hopf algebra of cut Feynman integrals: the one-loop case

  • Samuel Abreu
  • Ruth Britto
  • Claude Duhr
  • Einan Gardi
Open Access
Regular Article - Theoretical Physics

Abstract

We construct a diagrammatic coaction acting on one-loop Feynman graphs and their cuts. The graphs are naturally identified with the corresponding (cut) Feynman integrals in dimensional regularization, whose coefficients of the Laurent expansion in the dimensional regulator are multiple polylogarithms (MPLs). Our main result is the conjecture that this diagrammatic coaction reproduces the combinatorics of the coaction on MPLs order by order in the Laurent expansion. We show that our conjecture holds in a broad range of nontrivial one-loop integrals. We then explore its consequences for the study of discontinuities of Feynman integrals, and the differential equations that they satisfy. In particular, using the diagrammatic coaction along with information from cuts, we explicitly derive differential equations for any one-loop Feynman integral. We also explain how to construct the symbol of any one-loop Feynman integral recursively. Finally, we show that our diagrammatic coaction follows, in the special case of one-loop integrals, from a more general coaction proposed recently, which is constructed by pairing master integrands with corresponding master contours.

Keywords

Scattering Amplitudes Perturbative QCD 

Notes

Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

References

  1. [1]
    A.B. Goncharov, Multiple polylogarithms and mixed Tate motives, math/0103059 [INSPIRE].
  2. [2]
    A.B. Goncharov, Galois symmetries of fundamental groupoids and noncommutative geometry, Duke Math. J. 128 (2005) 209 [math/0208144] [INSPIRE].
  3. [3]
    F. Brown, The massless higher-loop two-point function, Commun. Math. Phys. 287 (2009) 925 [arXiv:0804.1660] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    C. Anastasiou, C. Duhr, F. Dulat and B. Mistlberger, Soft triple-real radiation for Higgs production at N3LO, JHEP 07 (2013) 003 [arXiv:1302.4379] [INSPIRE].ADSCrossRefGoogle Scholar
  5. [5]
    J.M. Henn, Multiloop integrals in dimensional regularization made simple, Phys. Rev. Lett. 110 (2013) 251601 [arXiv:1304.1806] [INSPIRE].ADSCrossRefGoogle Scholar
  6. [6]
    J. Ablinger, J. Blümlein, C. Raab, C. Schneider and F. Wißbrock, Calculating Massive 3-loop Graphs for Operator Matrix Elements by the Method of Hyperlogarithms, Nucl. Phys. B 885 (2014) 409 [arXiv:1403.1137] [INSPIRE].
  7. [7]
    C. Bogner and F. Brown, Feynman integrals and iterated integrals on moduli spaces of curves of genus zero, Commun. Num. Theor. Phys. 09 (2015) 189 [arXiv:1408.1862] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    E. Panzer, Feynman integrals and hyperlogarithms, Ph.D. Thesis, Humboldt University, Berlin, Inst. Math., 2015, [arXiv:1506.07243] [INSPIRE].
  9. [9]
    C. Bogner, MPL — A program for computations with iterated integrals on moduli spaces of curves of genus zero, Comput. Phys. Commun. 203 (2016) 339 [arXiv:1510.04562] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  10. [10]
    A.B. Goncharov, M. Spradlin, C. Vergu and A. Volovich, Classical Polylogarithms for Amplitudes and Wilson Loops, Phys. Rev. Lett. 105 (2010) 151605 [arXiv:1006.5703] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  11. [11]
    M. Caffo, H. Czyz, S. Laporta and E. Remiddi, The master differential equations for the two loop sunrise selfmass amplitudes, Nuovo Cim. A 111 (1998) 365 [hep-th/9805118] [INSPIRE].
  12. [12]
    L. Adams, C. Bogner and S. Weinzierl, The two-loop sunrise graph with arbitrary masses, J. Math. Phys. 54 (2013) 052303 [arXiv:1302.7004] [INSPIRE].
  13. [13]
    S. Bloch and P. Vanhove, The elliptic dilogarithm for the sunset graph, J. Number Theor. 148 (2015) 328 [arXiv:1309.5865] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    S. Bloch, M. Kerr and P. Vanhove, A feynman integral via higher normal functions, Compos. Math. 151 (2015) 2329 [arXiv:1406.2664] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    L. Adams, C. Bogner and S. Weinzierl, The two-loop sunrise graph in two space-time dimensions with arbitrary masses in terms of elliptic dilogarithms, J. Math. Phys. 55 (2014) 102301 [arXiv:1405.5640] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    L. Adams, C. Bogner and S. Weinzierl, The two-loop sunrise integral around four space-time dimensions and generalisations of the Clausen and Glaisher functions towards the elliptic case, J. Math. Phys. 56 (2015) 072303 [arXiv:1504.03255] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    L. Adams, C. Bogner and S. Weinzierl, The iterated structure of the all-order result for the two-loop sunrise integral, J. Math. Phys. 57 (2016) 032304 [arXiv:1512.05630] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    S. Bloch, M. Kerr and P. Vanhove, Local mirror symmetry and the sunset Feynman integral, arXiv:1601.08181 [INSPIRE].
  19. [19]
    L. Adams, C. Bogner, A. Schweitzer and S. Weinzierl, The kite integral to all orders in terms of elliptic polylogarithms, J. Math. Phys. 57 (2016) 122302 [arXiv:1607.01571] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    E. Remiddi and L. Tancredi, Differential equations and dispersion relations for Feynman amplitudes. The two-loop massive sunrise and the kite integral, Nucl. Phys. B 907 (2016) 400 [arXiv:1602.01481] [INSPIRE].
  21. [21]
    A. Primo and L. Tancredi, On the maximal cut of Feynman integrals and the solution of their differential equations, Nucl. Phys. B 916 (2017) 94 [arXiv:1610.08397] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    A. Primo and L. Tancredi, Maximal cuts and differential equations for Feynman integrals. An application to the three-loop massive banana graph, Nucl. Phys. B 921 (2017) 316 [arXiv:1704.05465] [INSPIRE].
  23. [23]
    F. Brown, On the decomposition of motivic multiple zeta values, arXiv:1102.1310 [INSPIRE].
  24. [24]
    C. Duhr, Hopf algebras, coproducts and symbols: an application to Higgs boson amplitudes, JHEP 08 (2012) 043 [arXiv:1203.0454] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  25. [25]
    A. Connes and D. Kreimer, Hopf algebras, renormalization and noncommutative geometry, Commun. Math. Phys. 199 (1998) 203 [hep-th/9808042] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    A. Connes and D. Kreimer, Renormalization in quantum field theory and the Riemann-Hilbert problem. 1. The Hopf algebra structure of graphs and the main theorem, Commun. Math. Phys. 210 (2000) 249 [hep-th/9912092] [INSPIRE].
  27. [27]
    D. Kreimer, On the Hopf algebra structure of perturbative quantum field theories, Adv. Theor. Math. Phys. 2 (1998) 303 [q-alg/9707029] [INSPIRE].
  28. [28]
    D. Kreimer and W.D. van Suijlekom, Recursive relations in the core Hopf algebra, Nucl. Phys. B 820 (2009) 682 [arXiv:0903.2849] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    D. Kreimer, The core Hopf algebra, Clay Math. Proc. 11 (2010) 313 [arXiv:0902.1223] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  30. [30]
    S. Bloch, H. Esnault and D. Kreimer, On motives associated to graph polynomials, Commun. Math. Phys. 267 (2006) 181 [math/0510011] [INSPIRE].
  31. [31]
    F. Brown, Notes on motivic periods, arXiv:1512.06410.
  32. [32]
    E. Panzer and O. Schnetz, The Galois coaction on ϕ 4 periods, Commun. Num. Theor. Phys. 11 (2017) 657 [arXiv:1603.04289] [INSPIRE].
  33. [33]
    F. Brown, Feynman amplitudes, coaction principle and cosmic Galois group, Commun. Num. Theor. Phys. 11 (2017) 453 [arXiv:1512.06409] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  34. [34]
    S. Abreu, R. Britto, C. Duhr and E. Gardi, From multiple unitarity cuts to the coproduct of Feynman integrals, JHEP 10 (2014) 125 [arXiv:1401.3546] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  35. [35]
    S. Abreu, R. Britto and H. Grönqvist, Cuts and coproducts of massive triangle diagrams, JHEP 07 (2015) 111 [arXiv:1504.00206] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  36. [36]
    S. Abreu, R. Britto, C. Duhr and E. Gardi, Algebraic Structure of Cut Feynman Integrals and the Diagrammatic Coaction, Phys. Rev. Lett. 119 (2017) 051601 [arXiv:1703.05064] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  37. [37]
    L. Landau, On analytic properties of vertex parts in quantum field theory, Nucl. Phys. 13 (1959) 181.MathSciNetCrossRefzbMATHGoogle Scholar
  38. [38]
    R.E. Cutkosky, Singularities and discontinuities of Feynman amplitudes, J. Math. Phys. 1 (1960) 429 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  39. [39]
    R. Eden, P. Landshoff, D. Olive and J. Polkinghorne, The Analytic S-Matrix, Cambridge University Press, (1966).Google Scholar
  40. [40]
    G. ’t Hooft and M. Veltman, Diagrammar, NATO Adv. Study Inst. Ser. B Phys. 4 (1974) 177.Google Scholar
  41. [41]
    F. Pham, ed., Singularities of Integrals, Springer, (2005).Google Scholar
  42. [42]
    D. Fotiadi and F. Pham, Analytic study of Some Feynman Graphs by Homological Methods, in Homology and Feynman integrals, R.C. Hwa and V.L. Teplitz, eds., W.A. Benjamin Inc., (1966).Google Scholar
  43. [43]
    R.C. Hwa and V.L. Teplitz, Homology and Feynman Integrals, W.A. Benjamin, Inc., (1966).Google Scholar
  44. [44]
    S. Bloch and D. Kreimer, Cutkosky Rules and Outer Space, arXiv:1512.01705 [INSPIRE].
  45. [45]
    S. Abreu, R. Britto, C. Duhr and E. Gardi, Cuts from residues: the one-loop case, JHEP 06 (2017) 114 [arXiv:1702.03163] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  46. [46]
    F.V. Tkachov, A Theorem on Analytical Calculability of Four Loop Renormalization Group Functions, Phys. Lett. B 100 (1981) 65 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  47. [47]
    K.G. Chetyrkin and F.V. Tkachov, Integration by Parts: The Algorithm to Calculate β-functions in 4 Loops, Nucl. Phys. B 192 (1981) 159 [INSPIRE].ADSCrossRefGoogle Scholar
  48. [48]
    S. Laporta, High precision calculation of multiloop Feynman integrals by difference equations, Int. J. Mod. Phys. A 15 (2000) 5087 [hep-ph/0102033] [INSPIRE].
  49. [49]
    O.V. Tarasov, Connection between Feynman integrals having different values of the space-time dimension, Phys. Rev. D 54 (1996) 6479 [hep-th/9606018] [INSPIRE].
  50. [50]
    Z. Bern, L.J. Dixon and D.A. Kosower, Dimensionally regulated one loop integrals, Phys. Lett. B 302 (1993) 299 [Erratum ibid. B 318 (1993) 649] [hep-ph/9212308] [INSPIRE].
  51. [51]
    R.N. Lee, Space-time dimensionality D as complex variable: Calculating loop integrals using dimensional recurrence relation and analytical properties with respect to D, Nucl. Phys. B 830 (2010) 474 [arXiv:0911.0252] [INSPIRE].
  52. [52]
    R.K. Ellis and G. Zanderighi, Scalar one-loop integrals for QCD, JHEP 02 (2008) 002 [arXiv:0712.1851] [INSPIRE].ADSCrossRefGoogle Scholar
  53. [53]
    L.J. Dixon, J.M. Drummond and J.M. Henn, The one-loop six-dimensional hexagon integral and its relation to MHV amplitudes in N = 4 SYM, JHEP 06 (2011) 100 [arXiv:1104.2787] [INSPIRE].
  54. [54]
    V. Del Duca, C. Duhr and V.A. Smirnov, The massless hexagon integral in D = 6 dimensions, Phys. Lett. B 703 (2011) 363 [arXiv:1104.2781] [INSPIRE].
  55. [55]
    V. Del Duca, C. Duhr and V.A. Smirnov, The One-Loop One-Mass Hexagon Integral in D = 6 Dimensions, JHEP 07 (2011) 064 [arXiv:1105.1333] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  56. [56]
    V. Del Duca, L.J. Dixon, J.M. Drummond, C. Duhr, J.M. Henn and V.A. Smirnov, The one-loop six-dimensional hexagon integral with three massive corners, Phys. Rev. D 84 (2011) 045017 [arXiv:1105.2011] [INSPIRE].
  57. [57]
    C.G. Papadopoulos, Simplified differential equations approach for Master Integrals, JHEP 07 (2014) 088 [arXiv:1401.6057] [INSPIRE].ADSCrossRefGoogle Scholar
  58. [58]
    M. Spradlin and A. Volovich, Symbols of One-Loop Integrals From Mixed Tate Motives, JHEP 11 (2011) 084 [arXiv:1105.2024] [INSPIRE].zbMATHGoogle Scholar
  59. [59]
    M.G. Kozlov and R.N. Lee, One-loop pentagon integral in d dimensions from differential equations in ϵ-form, JHEP 02 (2016) 021 [arXiv:1512.01165] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  60. [60]
    D.B. Fairlie, P.V. Landshoff, J. Nuttall and J.C. Polkinghorne, Singularities of the Second Type, J. Math. Phys. 3 (1962) 594.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  61. [61]
    D.B. Fairlie, P.V. Landshoff, J. Nuttall and J.C. Polkinghorne, Physical sheet properties of second type singularities, Phys. Lett. 3 (1962) 55.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  62. [62]
    C. Anastasiou and K. Melnikov, Higgs boson production at hadron colliders in NNLO QCD, Nucl. Phys. B 646 (2002) 220 [hep-ph/0207004] [INSPIRE].
  63. [63]
    C. Anastasiou, L.J. Dixon, K. Melnikov and F. Petriello, Dilepton rapidity distribution in the Drell-Yan process at NNLO in QCD, Phys. Rev. Lett. 91 (2003) 182002 [hep-ph/0306192] [INSPIRE].
  64. [64]
    C. Anastasiou and K. Melnikov, Pseudoscalar Higgs boson production at hadron colliders in NNLO QCD, Phys. Rev. D 67 (2003) 037501 [hep-ph/0208115] [INSPIRE].
  65. [65]
    C. Anastasiou, L.J. Dixon and K. Melnikov, NLO Higgs boson rapidity distributions at hadron colliders, Nucl. Phys. Proc. Suppl. 116 (2003) 193 [hep-ph/0211141] [INSPIRE].
  66. [66]
    C. Anastasiou, L.J. Dixon, K. Melnikov and F. Petriello, High precision QCD at hadron colliders: Electroweak gauge boson rapidity distributions at NNLO, Phys. Rev. D 69 (2004) 094008 [hep-ph/0312266] [INSPIRE].
  67. [67]
    H. Frellesvig and C.G. Papadopoulos, Cuts of Feynman Integrals in Baikov representation, JHEP 04 (2017) 083 [arXiv:1701.07356] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  68. [68]
    M. Zeng, Differential equations on unitarity cut surfaces, JHEP 06 (2017) 121 [arXiv:1702.02355] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  69. [69]
    J. Bosma, M. Sogaard and Y. Zhang, Maximal Cuts in Arbitrary Dimension, JHEP 08 (2017) 051 [arXiv:1704.04255] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  70. [70]
    M. Veltman, Diagrammatica: The path to Feynman rules, Cambridge Lect. Notes Phys. 4 (1994) 1.Google Scholar
  71. [71]
    N. Arkani-Hamed, J.L. Bourjaily, F. Cachazo and J. Trnka, Local Integrals for Planar Scattering Amplitudes, JHEP 06 (2012) 125 [arXiv:1012.6032] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  72. [72]
    A.V. Kotikov, Differential equations method: New technique for massive Feynman diagrams calculation, Phys. Lett. B 254 (1991) 158 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  73. [73]
    A.V. Kotikov, Differential equation method: The calculation of N point Feynman diagrams, Phys. Lett. B 267 (1991) 123 [Erratum ibid. B 295 (1992) 409] [INSPIRE].
  74. [74]
    A.V. Kotikov, Differential equations method: The calculation of vertex type Feynman diagrams, Phys. Lett. B 259 (1991) 314 [INSPIRE].
  75. [75]
    T. Gehrmann and E. Remiddi, Differential equations for two loop four point functions, Nucl. Phys. B 580 (2000) 485 [hep-ph/9912329] [INSPIRE].
  76. [76]
    J.A. Lappo-Danilevsky, Théorie algorithmique des corps de Riemann, Rec. Math. Moscou 34 (1927) 113.zbMATHGoogle Scholar
  77. [77]
    A.B. Goncharov, Multiple polylogarithms, cyclotomy and modular complexes, Math. Res. Lett. 5 (1998) 497 [arXiv:1105.2076] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  78. [78]
    T.G. Birthwright, E.W.N. Glover and P. Marquard, Master integrals for massless two-loop vertex diagrams with three offshell legs, JHEP 09 (2004) 042 [hep-ph/0407343] [INSPIRE].
  79. [79]
    F. Chavez and C. Duhr, Three-mass triangle integrals and single-valued polylogarithms, JHEP 11 (2012) 114 [arXiv:1209.2722] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  80. [80]
    D. Gaiotto, J. Maldacena, A. Sever and P. Vieira, Pulling the straps of polygons, JHEP 12 (2011) 011 [arXiv:1102.0062] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  81. [81]
    G.F. Sterman, Partons, factorization and resummation, TASI 95, in QCD and beyond. Proceedings, Theoretical Advanced Study Institute in Elementary Particle Physics, TASI-95, Boulder, U.S.A., June 4-30, 1995, hep-ph/9606312 [INSPIRE].
  82. [82]
    J.M. Drummond, J. Henn, V.A. Smirnov and E. Sokatchev, Magic identities for conformal four-point integrals, JHEP 01 (2007) 064 [hep-th/0607160] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  83. [83]
    J.M. Henn, S.G. Naculich, H.J. Schnitzer and M. Spradlin, More loops and legs in Higgs-regulated N = 4 SYM amplitudes, JHEP 08 (2010) 002 [arXiv:1004.5381] [INSPIRE].
  84. [84]
    J.M. Henn, Dual conformal symmetry at loop level: massive regularization, J. Phys. A 44 (2011) 454011 [arXiv:1103.1016] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  85. [85]
    S. Caron-Huot and J.M. Henn, Iterative structure of finite loop integrals, JHEP 06 (2014) 114 [arXiv:1404.2922] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  86. [86]
    D.J. Broadhurst, Summation of an infinite series of ladder diagrams, Phys. Lett. B 307 (1993) 132 [INSPIRE].
  87. [87]
    A. Goncharov, Volumes of hyperbolic manifolds and mixed Tate motives, alg-geom/9601021.
  88. [88]
    D. Fotiadi, M. Froissart, J. Lascoux and F. Pham, Analytic Properties of Some Integrals over Complex Manifolds, Centre de physique théorique — Ecole polytechnique, Paris France, (1964).Google Scholar
  89. [89]
    D. Fotiadi, M. Froissart, J. Lascoux and F. Pham, Applications of an isotopoy theorem, vol. 4, Pergamon Press, (1965), pp. 159-191.Google Scholar
  90. [90]
    J. Drummond, C. Duhr, B. Eden, P. Heslop, J. Pennington and V.A. Smirnov, Leading singularities and off-shell conformal integrals, JHEP 08 (2013) 133 [arXiv:1303.6909] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  91. [91]
    J.M. Drummond, Generalised ladders and single-valued polylogarithms, JHEP 02 (2013) 092 [arXiv:1207.3824] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  92. [92]
    L.J. Dixon, J.M. Drummond, M. von Hippel and J. Pennington, Hexagon functions and the three-loop remainder function, JHEP 12 (2013) 049 [arXiv:1308.2276] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  93. [93]
    L.J. Dixon, J.M. Drummond, C. Duhr and J. Pennington, The four-loop remainder function and multi-Regge behavior at NNLLA in planar N = 4 super-Yang-Mills theory, JHEP 06 (2014) 116 [arXiv:1402.3300] [INSPIRE].
  94. [94]
    L.J. Dixon, C. Duhr and J. Pennington, Single-valued harmonic polylogarithms and the multi-Regge limit, JHEP 10 (2012) 074 [arXiv:1207.0186] [INSPIRE].ADSCrossRefGoogle Scholar
  95. [95]
    T. Dennen, M. Spradlin and A. Volovich, Landau Singularities and Symbology: One- and Two-loop MHV Amplitudes in SYM Theory, JHEP 03 (2016) 069 [arXiv:1512.07909] [INSPIRE].ADSCrossRefGoogle Scholar
  96. [96]
    K.T. Chen, Iterated path integrals, Bull. Amer. Math. Soc. 83 (1977) 831.MathSciNetCrossRefzbMATHGoogle Scholar
  97. [97]
    A.B. Goncharov, A simple construction of Grassmannian polylogarithms, arXiv:0908.2238 [INSPIRE].
  98. [98]
    F.C.S. Brown, Multiple zeta values and periods of moduli spaces0,n, Annales Sci. Ecole Norm. Sup. 42 (2009) 371 [math/0606419] [INSPIRE].
  99. [99]
    C. Duhr, H. Gangl and J.R. Rhodes, From polygons and symbols to polylogarithmic functions, JHEP 10 (2012) 075 [arXiv:1110.0458] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  100. [100]
    A. Brandhuber, B. Spence and G. Travaglini, From trees to loops and back, JHEP 01 (2006) 142 [hep-th/0510253] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  101. [101]
    Z. Bern, L.J. Dixon and D.A. Kosower, Dimensionally regulated pentagon integrals, Nucl. Phys. B 412 (1994) 751 [hep-ph/9306240] [INSPIRE].
  102. [102]
    S.A. Joni and G.C. Rota, Coalgebras and Bialgebras in Combinatorics, Stud. Appl. Math. 61 (1979) 93.Google Scholar
  103. [103]
    W.R. Schmitt, Incidence Hopf Algebras, J. Pure Appl. Algebra 96 (1994) 299.MathSciNetCrossRefzbMATHGoogle Scholar
  104. [104]
    W.R. Schmitt, Antipodes and Incidence Coalgebras, J. Combin. Theory Ser. A 46 (1987) 264.MathSciNetCrossRefzbMATHGoogle Scholar
  105. [105]
    D. Simmons-Duffin, Projectors, Shadows and Conformal Blocks, JHEP 04 (2014) 146 [arXiv:1204.3894] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Author(s) 2017

Authors and Affiliations

  1. 1.Physikalisches InstitutAlbert-Ludwigs-Universität FreiburgFreiburgGermany
  2. 2.School of MathematicsTrinity CollegeDublin 2Ireland
  3. 3.Hamilton Mathematics InstituteTrinity CollegeDublin 2Ireland
  4. 4.Institut de Physique ThéoriqueUniversité Paris Saclay, CEA, CNRSGif-sur-Yvette cedexFrance
  5. 5.Kavli Institute for Theoretical PhysicsUniversity of CaliforniaSanta BarbaraU.S.A.
  6. 6.Theoretical Physics Department, CERNGenevaSwitzerland
  7. 7.Center for Cosmology, Particle Physics and Phenomenology (CP3)Université Catholique de LouvainLouvain-La-NeuveBelgium
  8. 8.Higgs Centre for Theoretical Physics, School of Physics and AstronomyThe University of EdinburghEdinburghU.K.

Personalised recommendations