The two-loop master integrals for \( q\overline{q} \)VV

  • Thomas Gehrmann
  • Andreas von Manteuffel
  • Lorenzo Tancredi
  • Erich Weihs
Open Access


We compute the full set of two-loop Feynman integrals appearing in massless two-loop four-point functions with two off-shell legs with the same invariant mass. These integrals allow to determine the two-loop corrections to the amplitudes for vector boson pair production at hadron colliders, \( q\overline{q} \)VV , and thus to compute this process to next-to- next-to-leading order accuracy in QCD. The master integrals are derived using the method of differential equations, employing a canonical basis for the integrals. We obtain analytical results for all integrals, expressed in terms of multiple polylogarithms. We optimize our results for numerical evaluation by employing functions which are real valued for physical scattering kinematics and allow for an immediate power series expansion.


QCD Phenomenology Hadronic Colliders 


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.


  1. [1]
    J. Ohnemus, Order α s calculations of hadronic W ±γ and Zγ production, Phys. Rev. D 47 (1993) 940 [INSPIRE].ADSGoogle Scholar
  2. [2]
    U. Baur, T. Han and J. Ohnemus, QCD corrections to hadronic Wγ production with nonstandard WWγ couplings, Phys. Rev. D 48 (1993) 5140 [hep-ph/9305314] [INSPIRE].ADSGoogle Scholar
  3. [3]
    U. Baur, T. Han and J. Ohnemus, QCD corrections and anomalous couplings in Zγ production at hadron colliders, Phys. Rev. D 57 (1998) 2823 [hep-ph/9710416] [INSPIRE].ADSGoogle Scholar
  4. [4]
    L.J. Dixon, Z. Kunszt and A. Signer, Helicity amplitudes for O(α s ) production of W + W , W ± Z, ZZ, W ±γ, or Zγ pairs at hadron colliders, Nucl. Phys. B 531 (1998) 3 [hep-ph/9803250] [INSPIRE].ADSCrossRefGoogle Scholar
  5. [5]
    E. Accomando, A. Denner and A. Kaiser, Logarithmic electroweak corrections to gauge-boson pair production at the LHC, Nucl. Phys. B 706 (2005) 325 [hep-ph/0409247] [INSPIRE].ADSCrossRefGoogle Scholar
  6. [6]
    E. Accomando and A. Kaiser, Electroweak corrections and anomalous triple gauge-boson couplings in W + W and W ± Z production at the LHC, Phys. Rev. D 73 (2006) 093006 [hep-ph/0511088] [INSPIRE].ADSGoogle Scholar
  7. [7]
    E. Accomando, A. Denner and C. Meier, Electroweak corrections to Wγ and Zγ production at the LHC, Eur. Phys. J. C 47 (2006) 125 [hep-ph/0509234] [INSPIRE].ADSCrossRefGoogle Scholar
  8. [8]
    J. Baglio, L.D. Ninh and M.M. Weber, Massive gauge boson pair production at the LHC: a next-to-leading order story, Phys. Rev. D 88 (2013) 113005 [arXiv:1307.4331] [INSPIRE].ADSGoogle Scholar
  9. [9]
    A. Bierweiler, T. Kasprzik and J.H. Kühn, Vector-boson pair production at the LHC to \( \mathcal{O}\left( {{\alpha^3}} \right) \) accuracy, JHEP 12 (2013) 071 [arXiv:1305.5402] [INSPIRE].ADSCrossRefGoogle Scholar
  10. [10]
    M. Billóni, S. Dittmaier, B. Jäger and C. Speckner, Next-to-leading order electroweak corrections to ppW + W 4 leptons at the LHC in double-pole approximation, JHEP 12 (2013) 043 [arXiv:1310.1564] [INSPIRE].ADSCrossRefGoogle Scholar
  11. [11]
    S. Catani, L. Cieri, D. de Florian, G. Ferrera and M. Grazzini, Diphoton production at hadron colliders: a fully-differential QCD calculation at NNLO, Phys. Rev. Lett. 108 (2012) 072001 [arXiv:1110.2375] [INSPIRE].ADSCrossRefGoogle Scholar
  12. [12]
    M. Grazzini, S. Kallweit, D. Rathlev and A. Torre, Zγ production at hadron colliders in NNLO QCD, Phys. Lett. B 731 (2014) 204 [arXiv:1309.7000] [INSPIRE].ADSCrossRefGoogle Scholar
  13. [13]
    Z. Bern, A. De Freitas and L.J. Dixon, Two loop amplitudes for gluon fusion into two photons, JHEP 09 (2001) 037 [hep-ph/0109078] [INSPIRE].ADSCrossRefGoogle Scholar
  14. [14]
    C. Anastasiou, E.W.N. Glover and M.E. Tejeda-Yeomans, Two loop QED and QCD corrections to massless fermion boson scattering, Nucl. Phys. B 629 (2002) 255 [hep-ph/0201274] [INSPIRE].ADSCrossRefGoogle Scholar
  15. [15]
    T. Gehrmann and L. Tancredi, Two-loop QCD helicity amplitudes for q \( \overline{q} \)W ±γ and q \( \overline{q} \)Z 0γ, JHEP 02 (2012) 004 [arXiv:1112.1531][INSPIRE].ADSCrossRefGoogle Scholar
  16. [16]
    T. Gehrmann, L. Tancredi and E. Weihs, Two-loop QCD helicity amplitudes for g gZ g and g gZ γ, JHEP 04 (2013) 101 [arXiv:1302.2630] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  17. [17]
    G. Chachamis, M. Czakon and D. Eiras, W pair production at the LHC. I. Two-loop corrections in the high energy limit, JHEP 12 (2008) 003 [arXiv:0802.4028] [INSPIRE].ADSCrossRefGoogle Scholar
  18. [18]
    T. Gehrmann, L. Tancredi and E. Weihs, Two-loop master integrals for q \( \overline{q} \)VV : the planar topologies, JHEP 08 (2013) 070 [arXiv:1306.6344] [INSPIRE].ADSCrossRefGoogle Scholar
  19. [19]
    J.M. Henn, K. Melnikov and V.A. Smirnov, Two-loop planar master integrals for the production of off-shell vector bosons in hadron collisions, arXiv:1402.7078 [INSPIRE].
  20. [20]
    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].ADSCrossRefGoogle Scholar
  21. [21]
    F. Chavez and C. Duhr, Three-mass triangle integrals and single-valued polylogarithms, JHEP 11 (2012) 114 [arXiv:1209.2722] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  22. [22]
    F.V. Tkachov, A Theorem on Analytical Calculability of Four Loop Renormalization Group Functions, Phys. Lett. B 100 (1981) 65 [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  23. [23]
    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
  24. [24]
    T. Gehrmann and E. Remiddi, Differential equations for two loop four point functions, Nucl. Phys. B 580 (2000) 485 [hep-ph/9912329] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  25. [25]
    S. Laporta, High precision calculation of multiloop Feynman integrals by difference equations, Int. J. Mod. Phys. A 15 (2000) 5087 [hep-ph/0102033] [INSPIRE].ADSMathSciNetGoogle Scholar
  26. [26]
    C. Studerus, Reduze-Feynman Integral Reduction in C++, Comput. Phys. Commun. 181 (2010) 1293 [arXiv:0912.2546] [INSPIRE].ADSCrossRefzbMATHMathSciNetGoogle Scholar
  27. [27]
    A. von Manteuffel and C. Studerus, Reduze 2Distributed Feynman Integral Reduction, arXiv:1201.4330 [INSPIRE].
  28. [28]
    A.V. Kotikov, Differential equations method: New technique for massive Feynman diagrams calculation, Phys. Lett. B 254 (1991) 158 [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  29. [29]
    E. Remiddi, Differential equations for Feynman graph amplitudes, Nuovo Cim. A 110 (1997) 1435 [hep-th/9711188] [INSPIRE].ADSGoogle Scholar
  30. [30]
    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].ADSGoogle Scholar
  31. [31]
    J.M. Henn, Multiloop integrals in dimensional regularization made simple, Phys. Rev. Lett. 110 (2013) 251601 [arXiv:1304.1806] [INSPIRE].ADSCrossRefGoogle Scholar
  32. [32]
    M. Argeri et al., Magnus and Dyson Series for Master Integrals, JHEP 03 (2014) 082 [arXiv:1401.2979] [INSPIRE].ADSCrossRefGoogle Scholar
  33. [33]
    S. Caron-Huot and J.M. Henn, Iterative structure of finite loop integrals, arXiv:1404.2922 [INSPIRE].
  34. [34]
    A. von Manteuffel and C. Studerus, Massive planar and non-planar double box integrals for light N f contributions to ggt \( \overline{t} \), JHEP 10 (2013) 037 [arXiv:1306.3504] [INSPIRE].CrossRefGoogle Scholar
  35. [35]
    F. Brown, The Massless higher-loop two-point function, Commun. Math. Phys. 287 (2009) 925 [arXiv:0804.1660] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  36. [36]
    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].ADSCrossRefMathSciNetGoogle Scholar
  37. [37]
    C. Duhr, H. Gangl and J.R. Rhodes, From polygons and symbols to polylogarithmic functions, JHEP 10 (2012) 075 [arXiv:1110.0458] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  38. [38]
    C. Duhr, Hopf algebras, coproducts and symbols: an application to Higgs boson amplitudes, JHEP 08 (2012) 043 [arXiv:1203.0454] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  39. [39]
    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
  40. [40]
    R. Bonciani, A. Ferroglia, T. Gehrmann, A. von Manteuffel and C. Studerus, Light-quark two-loop corrections to heavy-quark pair production in the gluon fusion channel, JHEP 12 (2013) 038 [arXiv:1309.4450] [INSPIRE].ADSCrossRefGoogle Scholar
  41. [41]
    J.A.M. Vermaseren, New features of FORM, math-ph/0010025 [INSPIRE].
  42. [42]
    W. Research, Mathematica, 8.0 edition, Wolfram Reserach, Champaign U.S.A. (2010).Google Scholar
  43. [43]
    C.W. Bauer, A. Frink and R. Kreckel, Introduction to the GiNaC framework for symbolic computation within the C++ programming language, J. Symol. Comput. 33 (2002) 1 [cs/0004015].CrossRefzbMATHMathSciNetGoogle Scholar
  44. [44]
    R. Lewis, Computer Algebra System Fermat,∼lewis.
  45. [45]
    K.-T. Chen, Iterated path integrals, Bull. Am. Math. Soc. 83 (1977) 831 [INSPIRE].CrossRefzbMATHGoogle Scholar
  46. [46]
    J.M. Henn and V.A. Smirnov, Analytic results for two-loop master integrals for Bhabha scattering I, JHEP 11 (2013) 041 [arXiv:1307.4083] [INSPIRE].ADSCrossRefGoogle Scholar
  47. [47]
    E. Remiddi and J.A.M. Vermaseren, Harmonic polylogarithms, Int. J. Mod. Phys. A 15 (2000) 725 [hep-ph/9905237] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  48. [48]
    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].ADSCrossRefGoogle Scholar
  49. [49]
    D. Zagier, Polylogarithms, Dedekind zeta functions and the algebraic K-theory of fields, in Prog. Math. Vol. 89: Arithmetic Algebraic Geometry, G.v.d. Geer, F. Oort and J. Steenbrink eds., Birkhäuser, Boston U.S.A. (1991), pg. 391.Google Scholar
  50. [50]
    A.B. Goncharov, Geometry of configurations, polylogarithms, and motivic cohomology, Adv. Math. 114 (1995) 197.CrossRefzbMATHMathSciNetGoogle Scholar
  51. [51]
    A.B. Goncharov, Multiple polylogarithms and mixed Tate motives, math/0103059 [INSPIRE].
  52. [52]
    A.B. Goncharov, Galois symmetries of fundamental groupoids and noncommutative geometry, Duke Math. J. 128 (2005) 209 [math/0208144] [INSPIRE].CrossRefzbMATHMathSciNetGoogle Scholar
  53. [53]
    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].ADSCrossRefMathSciNetGoogle Scholar
  54. [54]
    L. Adams, C. Bogner and S. Weinzierl, The two-loop sunrise graph with arbitrary masses, arXiv:1302.7004 [INSPIRE].
  55. [55]
    E. Remiddi and L. Tancredi, Schouten identities for Feynman graph amplitudes; The Master Integrals for the two-loop massive sunrise graph, Nucl. Phys. B 880 (2014) 343 [arXiv:1311.3342] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  56. [56]
    S. Bloch and P. Vanhove, The elliptic dilogarithm for the sunset graph, arXiv:1309.5865 [INSPIRE].
  57. [57]
    J.M. Henn, A.V. Smirnov and V.A. Smirnov, Analytic results for planar three-loop four-point integrals from a Knizhnik-Zamolodchikov equation, JHEP 07 (2013) 128 [arXiv:1306.2799] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  58. [58]
    J.M. Henn, A.V. Smirnov and V.A. Smirnov, Evaluating single-scale and/or non-planar diagrams by differential equations, JHEP 03 (2014) 088 [arXiv:1312.2588] [INSPIRE].ADSCrossRefGoogle Scholar
  59. [59]
    A. von Manteuffel, R.M. Schabinger and H.X. Zhu, The Complete Two-Loop Integrated Jet Thrust Distribution In Soft-Collinear Effective Theory, JHEP 03 (2014) 139 [arXiv:1309.3560] [INSPIRE].ADSCrossRefGoogle Scholar
  60. [60]
    W.L. van Neerven, Dimensional Regularization of Mass and Infrared Singularities in Two Loop On-shell Vertex Functions, Nucl. Phys. B 268 (1986) 453 [INSPIRE].ADSCrossRefGoogle Scholar
  61. [61]
    A. von Manteuffel, Mathematica package for multiple polylogarithms.Google Scholar
  62. [62]
    E. Weihs, Mathematica package for multiple polylogarithms.Google Scholar
  63. [63]
    J. Vollinga and S. Weinzierl, Numerical evaluation of multiple polylogarithms, Comput. Phys. Commun. 167 (2005) 177 [hep-ph/0410259] [INSPIRE].ADSCrossRefzbMATHMathSciNetGoogle Scholar
  64. [64]
    S. Borowka, J. Carter and G. Heinrich, Numerical Evaluation of Multi-Loop Integrals for Arbitrary Kinematics with SecDec 2.0, Comput. Phys. Commun. 184 (2013) 396 [arXiv:1204.4152] [INSPIRE].ADSCrossRefGoogle Scholar
  65. [65]
    S. Borowka and G. Heinrich, Massive non-planar two-loop four-point integrals with SecDec 2.1, Comput. Phys. Commun. 184 (2013) 2552 [arXiv:1303.1157] [INSPIRE].ADSCrossRefGoogle Scholar
  66. [66]
    F. Caola, J.M. Henn, K. Melnikov and V.A. Smirnov, Non-planar master integrals for the production of two off-shell vector bosons in collisions of massless partons, arXiv:1404.5590 [INSPIRE].

Copyright information

© The Author(s) 2014

Authors and Affiliations

  • Thomas Gehrmann
    • 1
  • Andreas von Manteuffel
    • 2
  • Lorenzo Tancredi
    • 1
  • Erich Weihs
    • 1
  1. 1.Physik-InstitutUniversität ZürichZürichSwitzerland
  2. 2.PRISMA Cluster of Excellence & Institute of PhysicsJohannes Gutenberg UniversityMainzGermany

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