Skip to main content
SpringerLink
Log in

Hopf algebras, coproducts and symbols: an application to Higgs boson amplitudes

  • Published:
Journal of High Energy Physics Aims and scope Submit manuscript

Abstract

We show how the Hopf algebra structure of multiple polylogarithms can be used to simplify complicated expressions for multi-loop amplitudes in perturbative quantum field theory and we argue that, unlike the recently popularized symbol-based approach, the coproduct incorporates information about the ζ values. We illustrate our approach by rewriting the two-loop helicity amplitudes for a Higgs boson plus three gluons in a simplified and compact form involving only classical polylogarithms.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. N. Nielsen, Der Eulersche Dilogarithmus und seine Verallgemeinerungen, Nova Acta Leopoldina 90 (1909) 123.

    Google Scholar 

  2. E. Remiddi and J. Vermaseren, Harmonic polylogarithms, Int. J. Mod. Phys. A 15 (2000) 725 [hep-ph/9905237] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  3. T. Gehrmann and E. Remiddi, Two loop master integrals for γ ∗ → 3 jets: the planar topologies, Nucl. Phys. B 601 (2001) 248 [hep-ph/0008287] [INSPIRE].

    Article  ADS  Google Scholar 

  4. J. Ablinger, J. Blumlein and C. Schneider, Harmonic sums and polylogarithms generated by cyclotomic polynomials, J. Math. Phys. 52 (2011) 102301 [arXiv:1105.6063] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  5. J. Vermaseren, A. Vogt and S. Moch, The third-order QCD corrections to deep-inelastic scattering by photon exchange, Nucl. Phys. B 724 (2005) 3 [hep-ph/0504242] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  6. S. Moch, J. Vermaseren and A. Vogt, The longitudinal structure function at the third order, Phys. Lett. B 606 (2005) 123 [hep-ph/0411112] [INSPIRE].

    Article  ADS  Google Scholar 

  7. A. Vogt, S. Moch and J. Vermaseren, The three-loop splitting functions in QCD: the singlet case, Nucl. Phys. B 691 (2004) 129 [hep-ph/0404111] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  8. S. Moch, J. Vermaseren and A. Vogt, The three loop splitting functions in QCD: the nonsinglet case, Nucl. Phys. B 688 (2004) 101 [hep-ph/0403192] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  9. R. Bonciani, P. Mastrolia and E. Remiddi, Master integrals for the two loop QCD virtual corrections to the forward backward asymmetry, Nucl. Phys. B 690 (2004) 138 [hep-ph/0311145] [INSPIRE].

    Article  ADS  Google Scholar 

  10. W. Bernreuther et al., Two-loop QCD corrections to the heavy quark form-factors: the vector contributions, Nucl. Phys. B 706 (2005) 245 [hep-ph/0406046] [INSPIRE].

    Article  ADS  Google Scholar 

  11. W. Bernreuther et al., Two-loop QCD corrections to the heavy quark form-factors: axial vector contributions, Nucl. Phys. B 712 (2005) 229 [hep-ph/0412259] [INSPIRE].

    Article  ADS  Google Scholar 

  12. W. Bernreuther et al., Two-loop QCD corrections to the heavy quark form-factors: anomaly contributions, Nucl. Phys. B 723 (2005) 91 [hep-ph/0504190] [INSPIRE].

    Article  ADS  Google Scholar 

  13. P. Mastrolia and E. Remiddi, Two loop form-factors in QED, Nucl. Phys. B 664 (2003) 341 [hep-ph/0302162] [INSPIRE].

    Article  ADS  Google Scholar 

  14. R. Bonciani, A. Ferroglia, P. Mastrolia, E. Remiddi and J. van der Bij, Two-loop N f = 1 QED Bhabha scattering: soft emission and numerical evaluation of the differential cross-section, Nucl. Phys. B 716 (2005) 280 [hep-ph/0411321] [INSPIRE].

    Article  ADS  Google Scholar 

  15. R. Bonciani, A. Ferroglia, P. Mastrolia, E. Remiddi and J. van der Bij, Two-loop N f = 1 QED Bhabha scattering differential cross section, Nucl. Phys. B 701 (2004) 121 [hep-ph/0405275] [INSPIRE].

    Article  ADS  Google Scholar 

  16. M. Czakon, J. Gluza and T. Riemann, Master integrals for massive two-loop Bhabha scattering in QED, Phys. Rev. D 71 (2005) 073009 [hep-ph/0412164] [INSPIRE].

    ADS  Google Scholar 

  17. Z. Bern, M. Czakon, L.J. Dixon, D.A. Kosower and V.A. Smirnov, The four-loop planar amplitude and cusp anomalous dimension in maximally supersymmetric Yang-Mills theory, Phys. Rev. D 75 (2007) 085010 [hep-th/0610248] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  18. G. Heinrich and V.A. Smirnov, Analytical evaluation of dimensionally regularized massive on-shell double boxes, Phys. Lett. B 598 (2004) 55 [hep-ph/0406053] [INSPIRE].

    Article  ADS  Google Scholar 

  19. V.A. Smirnov, Analytical result for dimensionally regularized massive on-shell planar double box, Phys. Lett. B 524 (2002) 129 [hep-ph/0111160] [INSPIRE].

    Article  ADS  Google Scholar 

  20. L. Bork, D. Kazakov and G. Vartanov, On form factors in N = 4 SYM, JHEP 02 (2011) 063 [arXiv:1011.2440] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  21. 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].

    Article  MathSciNet  ADS  Google Scholar 

  22. U. Aglietti, R. Bonciani, G. Degrassi and A. Vicini, Analytic results for virtual QCD corrections to Higgs production and decay, JHEP 01 (2007) 021 [hep-ph/0611266] [INSPIRE].

    Article  ADS  Google Scholar 

  23. U. Aglietti, R. Bonciani, G. Degrassi and A. Vicini, Master integrals for the two-loop light fermion contributions to gg → H and H → γγ, Phys. Lett. B 600 (2004) 57 [hep-ph/0407162] [INSPIRE].

    Article  ADS  Google Scholar 

  24. U. Aglietti, R. Bonciani, G. Degrassi and A. Vicini, Two loop light fermion contribution to Higgs production and decays, Phys. Lett. B 595 (2004) 432 [hep-ph/0404071] [INSPIRE].

    Article  ADS  Google Scholar 

  25. T. Gehrmann and E. Remiddi, Two loop master integrals for γ ∗ → 3 jets: the nonplanar topologies, Nucl. Phys. B 601 (2001) 287 [hep-ph/0101124] [INSPIRE].

    Article  ADS  Google Scholar 

  26. C. Anastasiou, S. Beerli, S. Bucherer, A. Daleo and Z. Kunszt, Two-loop amplitudes and master integrals for the production of a Higgs boson via a massive quark and a scalar-quark loop, JHEP 01 (2007) 082 [hep-ph/0611236] [INSPIRE].

    Article  ADS  Google Scholar 

  27. S. Moch, P. Uwer and S. Weinzierl, Two loop amplitudes with nested sums: fermionic contributions to \( {e^{ + }}{e^{ - }} \to q\bar{q}g \), Phys. Rev. D 66 (2002) 114001 [hep-ph/0207043] [INSPIRE].

    ADS  Google Scholar 

  28. S. Moch, P. Uwer and S. Weinzierl, Two loop amplitudes for \( {e^{ + }}{e^{ - }} \to q\bar{q}g \) : the N f contribution, Acta Phys. Polon. B 33 (2002) 2921 [hep-ph/0207167] [INSPIRE].

    ADS  Google Scholar 

  29. U. Aglietti, V. Del Duca, C. Duhr, G. Somogyi and Z. Trócsányi, Analytic integration of real-virtual counterterms in NNLO jet cross sections. I, JHEP 09 (2008) 107 [arXiv:0807.0514] [INSPIRE].

    Article  ADS  Google Scholar 

  30. V. Del Duca, C. Duhr, E. Nigel Glover and V.A. Smirnov, The one-loop pentagon to higher orders in epsilon, JHEP 01 (2010) 042 [arXiv:0905.0097] [INSPIRE].

    Article  ADS  Google Scholar 

  31. T. Gehrmann and E. Remiddi, Numerical evaluation of harmonic polylogarithms, Comput. Phys. Commun. 141 (2001) 296 [hep-ph/0107173] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  32. D. Maître, HPL, a Mathematica implementation of the harmonic polylogarithms, Comput. Phys. Commun. 174 (2006) 222 [hep-ph/0507152] [INSPIRE].

    Article  ADS  Google Scholar 

  33. D. Maître, Extension of HPL to complex arguments, Comput. Phys. Commun. 183 (2012) 846 [hep-ph/0703052] [INSPIRE].

    Article  ADS  Google Scholar 

  34. J. Vollinga and S. Weinzierl, Numerical evaluation of multiple polylogarithms, Comput. Phys. Commun. 167 (2005) 177 [hep-ph/0410259] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  35. A.I. Davydychev and M.Y. Kalmykov, New results for the ǫ-expansion of certain one, two and three loop Feynman diagrams, Nucl. Phys. B 605 (2001) 266 [hep-th/0012189] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  36. A.I. Davydychev and M.Y. Kalmykov, Massive Feynman diagrams and inverse binomial sums, Nucl. Phys. B 699 (2004) 3 [hep-th/0303162] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  37. A.B. Goncharov, Multiple polylogarithms, cyclotomy and modular complexes, Math. Res. Lett. 5 (1998) 497 [arXiv:1105.2076].

    Article  MathSciNet  MATH  Google Scholar 

  38. A.B. Goncharov, Multiple polylogarithms and mixed Tate motives, math/0103059.

  39. S. Laporta and E. Remiddi, Analytic treatment of the two loop equal mass sunrise graph, Nucl. Phys. B 704 (2005) 349 [hep-ph/0406160] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  40. 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].

    Article  MathSciNet  ADS  Google Scholar 

  41. V. Del Duca, C. Duhr and V.A. Smirnov, An analytic result for the two-loop hexagon Wilson loop in N = 4 SYM, JHEP 03 (2010) 099 [arXiv:0911.5332] [INSPIRE].

    Article  ADS  Google Scholar 

  42. V. Del Duca, C. Duhr and V.A. Smirnov, The two-loop hexagon Wilson loop in N = 4 SYM, JHEP 05 (2010) 084 [arXiv:1003.1702] [INSPIRE].

    Article  ADS  Google Scholar 

  43. A.B. Goncharov, A simple construction of Grassmannian polylogarithms, arXiv:0908.2238.

  44. L.F. Alday, Some analytic results for two-loop scattering amplitudes, JHEP 07 (2011) 080 [arXiv:1009.1110] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  45. D. Gaiotto, J. Maldacena, A. Sever and P. Vieira, Pulling the straps of polygons, JHEP 12 (2011) 011 [arXiv:1102.0062] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  46. P. Heslop and V.V. Khoze, Wilson loops @ 3-loops in special kinematics, JHEP 11 (2011) 152 [arXiv:1109.0058] [INSPIRE].

    Article  ADS  Google Scholar 

  47. M. Spradlin and A. Volovich, Symbols of one-loop integrals from mixed Tate motives, JHEP 11 (2011) 084 [arXiv:1105.2024] [INSPIRE].

    Google Scholar 

  48. 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].

    Article  MathSciNet  ADS  Google Scholar 

  49. 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].

    Article  ADS  Google Scholar 

  50. 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].

    Article  ADS  Google Scholar 

  51. V. Del Duca et al., The one-loop six-dimensional hexagon integral with three massive corners, Phys. Rev. D 84 (2011) 045017 [arXiv:1105.2011] [INSPIRE].

    ADS  Google Scholar 

  52. S. Buehler and C. Duhr, CHAPLIN — Complex harmonic polylogarithms in Fortran, arXiv:1106.5739 [INSPIRE].

  53. A. Brandhuber, G. Travaglini and G. Yang, Analytic two-loop form factors in N = 4 SYM, JHEP 05 (2012) 082 [arXiv:1201.4170] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  54. S. Caron-Huot, Superconformal symmetry and two-loop amplitudes in planar N = 4 super Yang-Mills, JHEP 12 (2011) 066 [arXiv:1105.5606] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  55. L.J. Dixon, J.M. Drummond and J.M. Henn, Analytic result for the two-loop six-point NMHV amplitude in N = 4 super Yang-Mills theory, JHEP 01 (2012) 024 [arXiv:1111.1704] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  56. L.J. Dixon, J.M. Drummond and J.M. Henn, Bootstrapping the three-loop hexagon, JHEP 11 (2011) 023 [arXiv:1108.4461] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  57. C. Duhr, H. Gangl and J.R. Rhodes, From polygons and symbols to polylogarithmic functions, arXiv:1110.0458 [INSPIRE].

  58. A.B. Goncharov, Galois symmetries of fundamental groupoids and noncommutative geometry, Duke Math. J. 128 (2005) 209 [math/0208144].

    Article  MathSciNet  MATH  Google Scholar 

  59. F. Brown, On the decomposition of motivic multiple zeta values, arXiv:1102.1310 [INSPIRE].

  60. A. Koukoutsakis, Higgs bosons and QCD jets at two loops, Ph.D. thesis, University of Durham, Durham, U.S.A. (2003).

  61. T. Gehrmann, M. Jaquier, E. Glover and A. Koukoutsakis, Two-loop QCD corrections to the helicity amplitudes for H→3 partons, JHEP 02 (2012) 056 [arXiv:1112.3554] [INSPIRE].

    Article  ADS  Google Scholar 

  62. R. Ree, Lie elements and an algebra associated with shuffles, Annals Math. 68 (1958) 210.

    Article  MathSciNet  MATH  Google Scholar 

  63. A.B. Goncharov, Multiple polylogarithms and mixed Tate motives, math/0103059.

  64. H. Gangl, A.B. Goncharov and A. Levin, Multiple polylogarithms, polygons, trees and algebraic cycles, Proc. Symp. Pure Math. 80 (2009) 547 [math/0508066].

    Article  MathSciNet  Google Scholar 

  65. H. Gangl and J. R. Rhodes, unpublished.

  66. A. B. Goncharov, Polylogarithms in arithmetic and geometry, in the proceedings of the International Congress of Mathematicians. Volume 1, Birkhauser, Basel, Switzerland (1995).

Download references

Author information

Authors and Affiliations

  1. Institut für theoretische Physik, ETH Zürich, Wolfgang-Paulistr. 27, CH-8093, Zürich, Switzerland

    Claude Duhr

Authors
  1. Claude Duhr
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Claude Duhr.

Additional information

ArXiv ePrint: 1203.0454

Rights and permissions

Reprints and permissions

About this article

Cite this article

Duhr, C. Hopf algebras, coproducts and symbols: an application to Higgs boson amplitudes. J. High Energ. Phys. 2012, 43 (2012). https://doi.org/10.1007/JHEP08(2012)043

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/JHEP08(2012)043

Keywords

Search

Navigation