Journal of High Energy Physics

, 2013:88 | Cite as

Real-virtual contributions to the inclusive Higgs cross-section at N3LO

  • Charalampos Anastasiou
  • Claude Duhr
  • Falko Dulat
  • Franz Herzog
  • Bernhard Mistlberger
Open Access


We compute the contributions to the N3LO inclusive Higgs boson cross-section from the square of one-loop amplitudes with a Higgs boson and three QCD partons as external states. Our result is a Taylor expansion in the dimensional regulator ϵ, where the coefficients of the expansion are analytic functions of the ratio of the Higgs boson mass and the partonic center of mass energy and they are valid for arbitrary values of this ratio. We also perform a threshold expansion around the limit of soft-parton radiation in the final state. The expressions for the coefficients of the threshold expansion are valid for arbitrary values of the dimension. As a by-product of the threshold expansion calculation, we have developed a soft expansion method at the integrand level by identifying the relevant soft and collinear regions for the loop-momentum.


QCD Phenomenology NLO Computations 


  1. [1]
    C. Anastasiou, S. Buehler, F. Herzog and A. Lazopoulos, Inclusive Higgs boson cross-section for the LHC at 8 TeV, JHEP 04 (2012) 004 [arXiv:1202.3638] [INSPIRE].ADSCrossRefGoogle Scholar
  2. [2]
    S. Buehler and A. Lazopoulos, Scale dependence and collinear subtraction terms for Higgs production in gluon fusion at N 3 LO, JHEP 10 (2013) 096 [arXiv:1306.2223] [INSPIRE].ADSCrossRefGoogle Scholar
  3. [3]
    P. Baikov, K. Chetyrkin, A. Smirnov, V. Smirnov and M. Steinhauser, Quark and gluon form factors to three loops, Phys. Rev. Lett. 102 (2009) 212002 [arXiv:0902.3519] [INSPIRE].ADSCrossRefGoogle Scholar
  4. [4]
    T. Gehrmann, E. Glover, T. Huber, N. Ikizlerli and C. Studerus, Calculation of the quark and gluon form factors to three loops in QCD, JHEP 06 (2010) 094 [arXiv:1004.3653] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  5. [5]
    T. Gehrmann, E. Glover, T. Huber, N. Ikizlerli and C. Studerus, The quark and gluon form factors to three loops in QCD through to O(𝜖 2 ), JHEP 11 (2010) 102 [arXiv:1010.4478] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  6. [6]
    A. Pak, M. Rogal and M. Steinhauser, Production of scalar and pseudo-scalar Higgs bosons to next-to-next-to-leading order at hadron colliders, JHEP 09 (2011) 088 [arXiv:1107.3391] [INSPIRE].ADSCrossRefGoogle Scholar
  7. [7]
    C. Anastasiou, S. Buehler, C. Duhr and F. Herzog, NNLO phase space master integrals for two-to-one inclusive cross sections in dimensional regularization, JHEP 11 (2012) 062 [arXiv:1208.3130] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  8. [8]
    M. Höschele, J. Hoff, A. Pak, M. Steinhauser and T. Ueda, Higgs boson production at the LHC: NNLO partonic cross sections through order ǫ and convolutions with splitting functions to N 3 LO, Phys. Lett. B 721 (2013) 244 [arXiv:1211.6559] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  9. [9]
    C. Anastasiou, C. Duhr, F. Dulat and B. Mistlberger, Soft triple-real radiation for Higgs production at N 3 LO, JHEP 07 (2013) 003 [arXiv:1302.4379] [INSPIRE].ADSCrossRefGoogle Scholar
  10. [10]
    C. Duhr and T. Gehrmann, The two-loop soft current in dimensional regularization, arXiv:1309.4393 [INSPIRE].
  11. [11]
    Y. Li and H.X. Zhu, Single soft gluon emission at two loops, JHEP 11 (2013) 080 [arXiv:1309.4391] [INSPIRE].ADSCrossRefGoogle Scholar
  12. [12]
    C. Anastasiou and K. Melnikov, Higgs boson production at hadron colliders in NNLO QCD, Nucl. Phys. B 646 (2002) 220 [hep-ph/0207004] [INSPIRE].ADSCrossRefGoogle Scholar
  13. [13]
    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].ADSCrossRefGoogle Scholar
  14. [14]
    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].ADSCrossRefGoogle Scholar
  15. [15]
    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].ADSGoogle Scholar
  16. [16]
    A.B. Goncharov, Multiple polylogarithms, cyclotomy and modular complexes, Math. Res. Lett. 5 (1998) 497 [arXiv:1105.2076] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    K.T. Chen, Iterated path integrals, Bull. Amer. Math. Soc. 83 (1977) 831.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    F.C. Brown, Multiple zeta values and periods of moduli spaces \( {{\mathfrak{M}}_{0,n }} \), Annales Sci. Ecole Norm. Sup. 42 (2009) 371 [math/0606419] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    A. Goncharov, A simple construction of Grassmannian polylogarithms, arXiv:0908.2238 [INSPIRE].
  20. [20]
    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
  21. [21]
    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
  22. [22]
    A. Goncharov, Galois symmetries of fundamental groupoids and noncommutative geometry, Duke Math. J. 128 (2005) 209 [math/0208144] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  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]
    P. Baikov and K. Chetyrkin, Top quark mediated Higgs boson decay into hadrons to order \( \alpha_s^5 \), Phys. Rev. Lett. 97 (2006) 061803 [hep-ph/0604194] [INSPIRE].ADSCrossRefGoogle Scholar
  26. [26]
    E. Furlan, Gluon-fusion Higgs production at NNLO for a non-standard Higgs sector, JHEP 10 (2011) 115 [arXiv:1106.4024] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  27. [27]
    C. Anastasiou, R. Boughezal and E. Furlan, The NNLO gluon fusion Higgs production cross-section with many heavy quarks, JHEP 06 (2010) 101 [arXiv:1003.4677] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  28. [28]
    K. Chetyrkin, B.A. Kniehl and M. Steinhauser, Hadronic Higgs decay to order \( \alpha_S^4 \), Phys. Rev. Lett. 79 (1997) 353 [hep-ph/9705240] [INSPIRE].ADSCrossRefGoogle Scholar
  29. [29]
    M. Krämer, E. Laenen and M. Spira, Soft gluon radiation in Higgs boson production at the LHC, Nucl. Phys. B 511 (1998) 523 [hep-ph/9611272] [INSPIRE].ADSCrossRefGoogle Scholar
  30. [30]
    D. Graudenz, M. Spira and P. Zerwas, QCD corrections to Higgs boson production at proton proton colliders, Phys. Rev. Lett. 70 (1993) 1372 [INSPIRE].ADSCrossRefGoogle Scholar
  31. [31]
    S. Dawson, Radiative corrections to Higgs boson production, Nucl. Phys. B 359 (1991) 283 [INSPIRE].ADSCrossRefGoogle Scholar
  32. [32]
    A. Djouadi, M. Spira and P. Zerwas, Production of Higgs bosons in proton colliders: QCD corrections, Phys. Lett. B 264 (1991) 440 [INSPIRE].ADSCrossRefGoogle Scholar
  33. [33]
    M. Spira, A. Djouadi, D. Graudenz and P. Zerwas, Higgs boson production at the LHC, Nucl. Phys. B 453 (1995) 17 [hep-ph/9504378] [INSPIRE].ADSCrossRefGoogle Scholar
  34. [34]
    R.V. Harlander and W.B. Kilgore, Next-to-next-to-leading order Higgs production at hadron colliders, Phys. Rev. Lett. 88 (2002) 201801 [hep-ph/0201206] [INSPIRE].ADSCrossRefGoogle Scholar
  35. [35]
    V. Ravindran, J. Smith and W.L. van Neerven, NNLO corrections to the total cross-section for Higgs boson production in hadron hadron collisions, Nucl. Phys. B 665 (2003) 325 [hep-ph/0302135] [INSPIRE].ADSCrossRefGoogle Scholar
  36. [36]
    E. Remiddi and J. Vermaseren, Harmonic polylogarithms, Int. J. Mod. Phys. A 15 (2000) 725 [hep-ph/9905237] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  37. [37]
    T. Gehrmann and E. Remiddi, Numerical evaluation of harmonic polylogarithms, Comput. Phys. Commun. 141 (2001) 296 [hep-ph/0107173] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  38. [38]
    J. Vollinga and S. Weinzierl, Numerical evaluation of multiple polylogarithms, Comput. Phys. Commun. 167 (2005) 177 [hep-ph/0410259] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  39. [39]
    D. Maître, Extension of HPL to complex arguments, Comput. Phys. Commun. 183 (2012) 846 [hep-ph/0703052] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  40. [40]
    D. Maître, HPL, a mathematica implementation of the harmonic polylogarithms, Comput. Phys. Commun. 174 (2006) 222 [hep-ph/0507152] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  41. [41]
    S. Buehler and C. Duhr, CHAPLINcomplex harmonic polylogarithms in fortran, arXiv:1106.5739 [INSPIRE].
  42. [42]
    (RV) 2 Higgs boson cross-section webpage,
  43. [43]
    P. Nogueira, Automatic Feynman graph generation, J. Comput. Phys. 105 (1993) 279 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  44. [44]
    J. Vermaseren, New features of FORM, math-ph/0010025 [INSPIRE].
  45. [45]
    C. Anastasiou, E.N. Glover and C. Oleari, The two loop scalar and tensor pentabox graph with lightlike legs, Nucl. Phys. B 575 (2000) 416 [Erratum ibid. B 585 (2000) 763] [hep-ph/9912251] [INSPIRE].
  46. [46]
    C. Anastasiou, T. Gehrmann, C. Oleari, E. Remiddi and J. Tausk, The tensor reduction and master integrals of the two loop massless crossed box with lightlike legs, Nucl. Phys. B 580 (2000) 577 [hep-ph/0003261] [INSPIRE].ADSCrossRefGoogle Scholar
  47. [47]
    S. Laporta, High precision calculation of multiloop Feynman integrals by difference equations, Int. J. Mod. Phys. A 15 (2000) 5087 [hep-ph/0102033] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  48. [48]
    C. Anastasiou and A. Lazopoulos, Automatic integral reduction for higher order perturbative calculations, JHEP 07 (2004) 046 [hep-ph/0404258] [INSPIRE].ADSCrossRefGoogle Scholar
  49. [49]
    C. Anastasiou, E.N. Glover and C. Oleari, Application of the negative dimension approach to massless scalar box integrals, Nucl. Phys. B 565 (2000) 445 [hep-ph/9907523] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  50. [50]
    M. Beneke and V.A. Smirnov, Asymptotic expansion of Feynman integrals near threshold, Nucl. Phys. B 522 (1998) 321 [hep-ph/9711391] [INSPIRE].ADSCrossRefGoogle Scholar
  51. [51]
    A. Pak and A. Smirnov, Geometric approach to asymptotic expansion of Feynman integrals, Eur. Phys. J. C 71 (2011) 1626 [arXiv:1011.4863] [INSPIRE].ADSCrossRefGoogle Scholar
  52. [52]
    B. Jantzen, Foundation and generalization of the expansion by regions, JHEP 12 (2011) 076 [arXiv:1111.2589] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  53. [53]
    S. Coleman and R. Norton, Singularities in the physical region, Nuovo Cim. 38 (1965) 438 [INSPIRE].CrossRefGoogle Scholar
  54. [54]
    G.F. Sterman, Mass divergences in annihilation processes. 1. Origin and nature of divergences in cut vacuum polarization diagrams, Phys. Rev. D 17 (1978) 2773 [INSPIRE].ADSGoogle Scholar
  55. [55]
    K. Chetyrkin and F. Tkachov, Integration by parts: the algorithm to calculate β-functions in 4 loops, Nucl. Phys. B 192 (1981) 159 [INSPIRE].ADSCrossRefGoogle Scholar
  56. [56]
    F. Tkachov, A theorem on analytical calculability of four loop renormalization group functions, Phys. Lett. B 100 (1981) 65 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  57. [57]
    C.W. Bauer, A. Frink and R. Kreckel, Introduction to the GiNaC framework for symbolic computation within the C++ programming language, J. Symbol. Comput. 33 (2002) 1 [cs/0004015].MathSciNetCrossRefzbMATHGoogle Scholar
  58. [58]
    T. Gehrmann and E. Remiddi, Differential equations for two loop four point functions, Nucl. Phys. B 580 (2000) 485 [hep-ph/9912329] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  59. [59]
    A. Kotikov, Differential equations method: new technique for massive Feynman diagrams calculation, Phys. Lett. B 254 (1991) 158 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  60. [60]
    T. Huber and D. Maˆıtre, HypExp 2, expanding hypergeometric functions about half-integer parameters, Comput. Phys. Commun. 178 (2008) 755 [arXiv:0708.2443] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  61. [61]
    S. Moch, P. Uwer and S. Weinzierl, Nested sums, expansion of transcendental functions and multiscale multiloop integrals, J. Math. Phys. 43 (2002) 3363 [hep-ph/0110083] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© SISSA 2013

Authors and Affiliations

  • Charalampos Anastasiou
    • 1
  • Claude Duhr
    • 2
  • Falko Dulat
    • 1
  • Franz Herzog
    • 3
  • Bernhard Mistlberger
    • 1
  1. 1.Institut für Theoretische Physik, ETH ZürichZürichSwitzerland
  2. 2.Institute for Particle Physics PhenomenologyUniversity of DurhamDurhamUnited Kingdom
  3. 3.CERN, Theory DivisionGeneva 23Switzerland

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