Journal of High Energy Physics

, 2017:198 | Cite as

Master integrals for the NNLO virtual corrections to μe scattering in QED: the planar graphs

  • Pierpaolo Mastrolia
  • Massimo Passera
  • Amedeo Primo
  • Ulrich Schubert
Open Access
Regular Article - Theoretical Physics


We evaluate the master integrals for the two-loop, planar box-diagrams contributing to the elastic scattering of muons and electrons at next-to-next-to leading-order in QED. We adopt the method of differential equations and the Magnus exponential series to determine a canonical set of integrals, finally expressed as a Taylor series around four space-time dimensions, with coefficients written as combination of generalised polylogarithms. The electron is treated as massless, while we retain full dependence on the muon mass. The considered integrals are also relevant for crossing-related processes, such as di-muon production at e+e-colliders, as well as for the QCD corrections to top-pair production at hadron colliders.


NLO Computations 


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]
    G. Backenstoss, B.D. Hyams, G. Knop, P.C. Marin and U. Stierlin, Helicity of μ mesons from π-meson decay, Phys. Rev. Lett. 6 (1961) 415 [INSPIRE].ADSCrossRefGoogle Scholar
  2. [2]
    G. Backenstoss, B.D. Hyams, G. Knop, P.C. Marin and U. Stierlin, Scattering of 8 GeV μ mesons on electrons, Phys. Rev. 129 (1963) 2759.ADSCrossRefGoogle Scholar
  3. [3]
    T. Kirk and S. Neddermeyer, Scattering of high-energy positive and negative muons on electrons, Phys. Rev. 171 (1968) 1412.ADSCrossRefGoogle Scholar
  4. [4]
    P.L. Jain and N.J. Wixon, Scattering of high-energy positive and negative muons on electrons, Phys. Rev. Lett. 23 (1969) 715 [INSPIRE].ADSCrossRefGoogle Scholar
  5. [5]
    R.F. Deery and S.H. Neddermeyer, Cloud-chamber study of hard collisions of cosmic-ray muons with electrons, Phys. Rev. 121 (1961) 1803.ADSCrossRefGoogle Scholar
  6. [6]
    I.B. McDiarmid and M.D. Wilson, The production of high-energy knock-on electrons and bremsstrahlung by μ mesons, Can. J. Phys. 40 (1962) 698.ADSCrossRefGoogle Scholar
  7. [7]
    N. Chaudhuri and M.S. Sinha, Production of knock-on electrons by cosmic-ray muons underground (148 m w.e.), Nuovo Cimento 35 (1965) 13.Google Scholar
  8. [8]
    P.D. Kearney and W.E. Hazen, Electromagnetic interactions of high-energy muons, Phys. Rev. B 138 (1965) 173.ADSCrossRefGoogle Scholar
  9. [9]
    K.P. Schuler, A muon polarimeter based on elastic muon electron scattering, AIP Conf. Proc. 187 (1989) 1401 [INSPIRE].ADSCrossRefGoogle Scholar
  10. [10]
    Spin Muon collaboration, D. Adams et al., Measurement of the SMC muon beam polarization using the asymmetry in the elastic scattering off polarized electrons, Nucl. Instrum. Meth. A 443 (2000) 1 [INSPIRE].
  11. [11]
    G. Abbiendi et al., Measuring the leading hadronic contribution to the muon g − 2 via μe scattering, Eur. Phys. J. C 77 (2017) 139 [arXiv:1609.08987] [INSPIRE].ADSCrossRefGoogle Scholar
  12. [12]
    C.M. Carloni Calame, M. Passera, L. Trentadue and G. Venanzoni, A new approach to evaluate the leading hadronic corrections to the muon g − 2, Phys. Lett. B 746 (2015) 325 [arXiv:1504.02228] [INSPIRE].ADSCrossRefGoogle Scholar
  13. [13]
    A.I. Nikishov, Radiative corrections to the scattering of μ mesons on electrons, Sov. Phys. JETP 12 (1961) 529.MathSciNetGoogle Scholar
  14. [14]
    K.E. Eriksson, Radiative corrections to muon-electron scattering, Nuovo Cim. 19 (1961) 1029.ADSCrossRefGoogle Scholar
  15. [15]
    K.E. Eriksson, B. Larsson and G.A. Rinander, Radiative corrections to muon-electron scattering, Nuovo Cim. 30 (1963) 1434.ADSCrossRefGoogle Scholar
  16. [16]
    P. Van Nieuwenhuizen, Muon-electron scattering cross-section to order alpha-to-the-third, Nucl. Phys. B 28 (1971) 429 [INSPIRE].ADSCrossRefGoogle Scholar
  17. [17]
    T.V. Kukhto, N.M. Shumeiko and S.I. Timoshin, Radiative corrections in polarized electron muon elastic scattering, J. Phys. G 13 (1987) 725 [INSPIRE].ADSCrossRefGoogle Scholar
  18. [18]
    D. Yu. Bardin and L. Kalinovskaya, QED corrections for polarized elastic muon e scattering, hep-ph/9712310 [INSPIRE].
  19. [19]
    N. Kaiser, Radiative corrections to lepton-lepton scattering revisited, J. Phys. G 37 (2010) 115005 [INSPIRE].ADSCrossRefGoogle Scholar
  20. [20]
    E. Derman and W.J. Marciano, Parity Violating Asymmetries in Polarized Electron Scattering, Annals Phys. 121 (1979) 147 [INSPIRE].ADSCrossRefGoogle Scholar
  21. [21]
    G. D’Ambrosio, Electron muon scattering in the electroweak unified theory, Lett. Nuovo Cim. 38 (1983) 593 [INSPIRE].CrossRefGoogle Scholar
  22. [22]
    J.C. Montero, V. Pleitez and M.C. Rodriguez, Left-right asymmetries in polarized e - mu scattering, Phys. Rev. D 58 (1998) 097505 [hep-ph/9803450] [INSPIRE].
  23. [23]
    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].
  24. [24]
    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].
  25. [25]
    R. Bonciani and A. Ferroglia, Two-Loop QCD Corrections to the Heavy-to-Light Quark Decay, JHEP 11 (2008) 065 [arXiv:0809.4687] [INSPIRE].ADSCrossRefGoogle Scholar
  26. [26]
    H.M. Asatrian, C. Greub and B.D. Pecjak, NNLO corrections to \( \overline{B}\to {X}_u\ell \overline{\nu} \) in the shape-function region, Phys. Rev. D 78 (2008) 114028 [arXiv:0810.0987] [INSPIRE].ADSGoogle Scholar
  27. [27]
    M. Beneke, T. Huber and X.Q. Li, Two-loop QCD correction to differential semi-leptonic bu decays in the shape-function region, Nucl. Phys. B 811 (2009) 77 [arXiv:0810.1230] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  28. [28]
    G. Bell, NNLO corrections to inclusive semileptonic B decays in the shape-function region, Nucl. Phys. B 812 (2009) 264 [arXiv:0810.5695] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  29. [29]
    T. Huber, On a two-loop crossed six-line master integral with two massive lines, JHEP 03 (2009) 024 [arXiv:0901.2133] [INSPIRE].ADSCrossRefGoogle Scholar
  30. [30]
    R. Bonciani, A. Ferroglia, T. Gehrmann, D. Maître and C. Studerus, Two-loop fermionic corrections to heavy-quark pair production: the quark-antiquark channel, JHEP 07 (2008) 129 [arXiv:0806.2301] [INSPIRE].ADSCrossRefGoogle Scholar
  31. [31]
    R. Bonciani, A. Ferroglia, T. Gehrmann and C. Studerus, Two-loop planar corrections to heavy-quark pair production in the quark-antiquark channel, JHEP 08 (2009) 067 [arXiv:0906.3671] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  32. [32]
    R. Bonciani, A. Ferroglia, T. Gehrmann, A. von Manteuffel and C. Studerus, Two-loop leading color corrections to heavy-quark pair production in the gluon fusion channel, JHEP 01 (2011) 102 [arXiv:1011.6661] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  33. [33]
    R. Bonciani et al., 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
  34. [34]
    F.V. Tkachov, A theorem on analytical calculability of four loop renormalization group functions, Phys. Lett. B 100 (1981) 65.ADSMathSciNetCrossRefGoogle Scholar
  35. [35]
    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
  36. [36]
    S. Laporta, High precision calculation of multiloop Feynman integrals by difference equations, Int. J. Mod. Phys. A 15 (2000) 5087 [hep-ph/0102033] [INSPIRE].
  37. [37]
    A.V. Kotikov, Differential equations method: new technique for massive Feynman diagrams calculation, Phys. Lett. B 254 (1991) 158 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  38. [38]
    E. Remiddi, Differential equations for Feynman graph amplitudes, Nuovo Cim. A 110 (1997) 1435 [hep-th/9711188] [INSPIRE].ADSGoogle Scholar
  39. [39]
    T. Gehrmann and E. Remiddi, Differential equations for two loop four point functions, Nucl. Phys. B 580 (2000) 485 [hep-ph/9912329] [INSPIRE].
  40. [40]
    J.M. Henn, Multiloop integrals in dimensional regularization made simple, Phys. Rev. Lett. 110 (2013) 251601 [arXiv:1304.1806] [INSPIRE].ADSCrossRefGoogle Scholar
  41. [41]
    M. Argeri et al., Magnus and Dyson series for master integrals, JHEP 03 (2014) 082 [arXiv:1401.2979] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  42. [42]
    S. Di Vita, P. Mastrolia, U. Schubert and V. Yundin, Three-loop master integrals for ladder-box diagrams with one massive leg, JHEP 09 (2014) 148 [arXiv:1408.3107] [INSPIRE].ADSCrossRefGoogle Scholar
  43. [43]
    R. Bonciani, S. Di Vita, P. Mastrolia and U. Schubert, Two-loop master integrals for the mixed EW-QCD virtual corrections to Drell-Yan scattering, JHEP 09 (2016) 091 [arXiv:1604.08581] [INSPIRE].ADSCrossRefGoogle Scholar
  44. [44]
    S. Di Vita, P. Mastrolia, A. Primo and U. Schubert, Two-loop master integrals for the leading QCD corrections to the Higgs coupling to a W pair and to the triple gauge couplings ZWW and γ * WW, JHEP 04 (2017) 008 [arXiv:1702.07331] [INSPIRE].CrossRefGoogle Scholar
  45. [45]
    A. Goncharov, Polylogarithms in arithmetic and geometry, Proc. Int. Congree Math. 1,2 (1995) 374.Google Scholar
  46. [46]
    E. Remiddi and J.A.M. Vermaseren, Harmonic polylogarithms, Int. J. Mod. Phys. A 15 (2000) 725 [hep-ph/9905237] [INSPIRE].
  47. [47]
    T. Gehrmann and E. Remiddi, Numerical evaluation of harmonic polylogarithms, Comput. Phys. Commun. 141 (2001) 296 [hep-ph/0107173] [INSPIRE].
  48. [48]
    J. Vollinga and S. Weinzierl, Numerical evaluation of multiple polylogarithms, Comput. Phys. Commun. 167 (2005) 177 [hep-ph/0410259] [INSPIRE].
  49. [49]
    C.W. Bauer, A. Frink and R. Kreckel, Introduction to the GiNaC framework for symbolic computation within the C++ programming language, cs/0004015.
  50. [50]
    S. Borowka et al., SecDec-3.0: numerical evaluation of multi-scale integrals beyond one loop, Comput. Phys. Commun. 196 (2015) 470 [arXiv:1502.06595] [INSPIRE].
  51. [51]
    A. von Manteuffel and C. Studerus, Reduze 2 — Distributed Feynman Integral Reduction, arXiv:1201.4330 [INSPIRE].
  52. [52]
    V.A. Smirnov, Analytical result for dimensionally regularized massless on shell double box, Phys. Lett. B 460 (1999) 397 [hep-ph/9905323] [INSPIRE].
  53. [53]
    J.B. Tausk, Nonplanar massless two loop Feynman diagrams with four on-shell legs, Phys. Lett. B 469 (1999) 225 [hep-ph/9909506] [INSPIRE].
  54. [54]
    M. Czakon, Tops from light quarks: full mass dependence at two-loops in QCD, Phys. Lett. B 664 (2008) 307 [arXiv:0803.1400] [INSPIRE].
  55. [55]
    M. Czakon and A. Mitov, NNLO corrections to top pair production at hadron colliders: the quark-gluon reaction, JHEP 01 (2013) 080 [arXiv:1210.6832] [INSPIRE].ADSCrossRefGoogle Scholar
  56. [56]
    M. Czakon and A. Mitov, NNLO corrections to top-pair production at hadron colliders: the all-fermionic scattering channels, JHEP 12 (2012) 054 [arXiv:1207.0236] [INSPIRE].ADSCrossRefGoogle Scholar
  57. [57]
    P. Bärnreuther, M. Czakon and A. Mitov, Percent level precision physics at the Tevatron: first genuine NNLO QCD corrections to \( q\overline{q}\to t\overline{t}+X \), Phys. Rev. Lett. 109 (2012) 132001 [arXiv:1204.5201] [INSPIRE].ADSCrossRefGoogle Scholar
  58. [58]
    M. Czakon, P. Fiedler and A. Mitov, Total top-quark pair-production cross section at hadron colliders through O(α S4), Phys. Rev. Lett. 110 (2013) 252004 [arXiv:1303.6254] [INSPIRE].ADSCrossRefGoogle Scholar
  59. [59]
    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].
  60. [60]
    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
  61. [61]
    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].
  62. [62]
    F. Brown, On the decomposition of motivic multiple zeta values, arXiv:1102.1310 [INSPIRE].
  63. [63]
    C. Duhr, Hopf algebras, coproducts and symbols: an application to Higgs boson amplitudes, JHEP 08 (2012) 043 [arXiv:1203.0454] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  64. [64]
    C. Duhr, Mathematical aspects of scattering amplitudes, in the proceedings of the Theoretical Advanced Study Institute in Elementary Particle Physics (TASI 2014), June 2–27, Boulder, Colorado, U.S.A. (2014) arXiv:1411.7538 [INSPIRE].
  65. [65]
    D.J. Broadhurst, Massive three-loop Feynman diagrams reducible to SC * primitives of algebras of the sixth root of unity, Eur. Phys. J. C 8 (1999) 311 [hep-th/9803091] [INSPIRE].ADSGoogle Scholar
  66. [66]
    J. Zhao, Standard relations of multiple polylogarithm values at roots of unity, arXiv:0707.1459.
  67. [67]
    J.M. Henn, A.V. Smirnov and V.A. Smirnov, Evaluating multiple polylogarithm values at sixth roots of unity up to weight six, Nucl. Phys. B 919 (2017) 315 [arXiv:1512.08389] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  68. [68]
    F. Dulat and B. Mistlberger, Real-Virtual-Virtual contributions to the inclusive Higgs cross section at N3LO, arXiv:1411.3586 [INSPIRE].

Copyright information

© The Author(s) 2017

Authors and Affiliations

  • Pierpaolo Mastrolia
    • 1
    • 2
  • Massimo Passera
    • 2
  • Amedeo Primo
    • 1
    • 2
  • Ulrich Schubert
    • 3
  1. 1.Dipartimento di Fisica ed AstronomiaUniversità di PadovaPadovaItaly
  2. 2.INFN — Sezione di PadovaPadovaItaly
  3. 3.High Energy Physics DivisionArgonne National LaboratoryArgonneU.S.A.

Personalised recommendations