Advertisement

Journal of High Energy Physics

, 2013:41 | Cite as

Analytic results for two-loop master integrals for Bhabha scattering I

  • Johannes M. HennEmail author
  • Vladimir A. Smirnov
Article

Abstract

We evaluate analytically the master integrals for one of two types of planar families contributing to massive two-loop Bhabha scattering in QED. As in our previous paper, we apply a recently suggested new strategy to solve differential equations for master integrals for families of Feynman integrals. The crucial point of this strategy is to use a new basis of the master integrals where all master integrals are pure functions of uniform weight. This allows to cast the differential equations into a simple canonical form, which can straightforwardly be integrated order by order in ϵ. The boundary conditions are also particularly transparent in this setup. We identify the class of functions relevant to this problem to all orders in ϵ. We present the results up to weight four for all except one integrals in terms of a subset of Goncharov polylogarithms, which one may call two-dimensional harmonic polylogarithms. For one integral, and more generally at higher weight, the solution is written in terms of Chen iterated integrals.

Keywords

NLO Computations 

References

  1. [1]
    R. Bonciani, A. Ferroglia, P. Mastrolia, E. Remiddi and J. van der Bij, Planar box diagram for the (N(F) = 1) two loop QED virtual corrections to Bhabha scattering, Nucl. Phys. B 681 (2004) 261 [Erratum ibid. B 702 (2004) 364-366] [hep-ph/0310333] [INSPIRE].
  2. [2]
    A. Kotikov, Differential equations method: New technique for massive Feynman diagrams calculation, Phys. Lett. B 254 (1991) 158 [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  3. [3]
    E. Remiddi, Differential equations for Feynman graph amplitudes, Nuovo Cim. A 110 (1997) 1435 [hep-th/9711188] [INSPIRE].ADSGoogle Scholar
  4. [4]
    T. Gehrmann and E. Remiddi, Differential equations for two loop four point functions, Nucl. Phys. B 580 (2000) 485 [hep-ph/9912329] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  5. [5]
    M. Argeri and P. Mastrolia, Feynman Diagrams and Differential Equations, Int. J. Mod. Phys. A 22 (2007) 4375 [arXiv:0707.4037] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  6. [6]
    V.A. Smirnov, Analytic tools for Feynman integrals, Springer Tr. Mod. Phys. 250 (2012) 1.CrossRefGoogle Scholar
  7. [7]
    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].ADSCrossRefGoogle Scholar
  8. [8]
    V.A. Smirnov, Analytical result for dimensionally regularized massive on-shell planar double box, Phys. Lett. B 524 (2002) 129 [hep-ph/0111160] [INSPIRE].ADSCrossRefGoogle Scholar
  9. [9]
    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].ADSCrossRefGoogle Scholar
  10. [10]
    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].ADSGoogle Scholar
  11. [11]
    M. Czakon, J. Gluza and T. Riemann, Harmonic polylogarithms for massive Bhabha scattering, Nucl. Instrum. Meth. A 559 (2006) 265 [hep-ph/0508212] [INSPIRE].ADSCrossRefGoogle Scholar
  12. [12]
    M. Czakon, J. Gluza and T. Riemann, On the massive two-loop corrections to Bhabha scattering, Acta Phys. Polon. B 36 (2005) 3319 [hep-ph/0511187] [INSPIRE].ADSGoogle Scholar
  13. [13]
    M. Czakon, J. Gluza and T. Riemann, The planar four-point master integrals for massive two-loop Bhabha scattering, Nucl. Phys. B 751 (2006) 1 [hep-ph/0604101] [INSPIRE].ADSCrossRefGoogle Scholar
  14. [14]
    M. Czakon, J. Gluza, K. Kajda and T. Riemann, Differential equations and massive two-loop Bhabha scattering: The B5l2m3 case, Nucl. Phys. Proc. Suppl. 157 (2006) 16 [hep-ph/0602102] [INSPIRE].ADSCrossRefGoogle Scholar
  15. [15]
    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].ADSCrossRefGoogle Scholar
  16. [16]
    M. Czakon and A. Mitov, Inclusive Heavy Flavor Hadroproduction in NLO QCD: The Exact Analytic Result, Nucl. Phys. B 824 (2010) 111 [arXiv:0811.4119] [INSPIRE].ADSCrossRefGoogle Scholar
  17. [17]
    A. von Manteuffel and C. Studerus, Massive planar and non-planar double box integrals for light N f contributions to gg\( t\overline{t} \), JHEP 10 (2013) 037 [arXiv:1306.3504] [INSPIRE].CrossRefGoogle Scholar
  18. [18]
    A.A. Penin, Two-loop corrections to Bhabha scattering, Phys. Rev. Lett. 95 (2005) 010408 [hep-ph/0501120] [INSPIRE].ADSCrossRefGoogle Scholar
  19. [19]
    Z. Bern, L.J. Dixon and A. Ghinculov, Two loop correction to Bhabha scattering, Phys. Rev. D 63 (2001) 053007 [hep-ph/0010075] [INSPIRE].ADSGoogle Scholar
  20. [20]
    S. Actis, M. Czakon, J. Gluza and T. Riemann, Planar two-loop master integrals for massive Bhabha scattering: N(f)=1 and N(f)=2, Nucl. Phys. Proc. Suppl. 160 (2006) 91 [hep-ph/0609051] [INSPIRE].ADSCrossRefGoogle Scholar
  21. [21]
    S. Actis, M. Czakon, J. Gluza and T. Riemann, Two-loop fermionic corrections to massive Bhabha scattering, Nucl. Phys. B 786 (2007) 26 [arXiv:0704.2400] [INSPIRE].ADSCrossRefGoogle Scholar
  22. [22]
    S. Actis, M. Czakon, J. Gluza and T. Riemann, Fermionic NNLO contributions to Bhabha scattering, Acta Phys. Polon. B 38 (2007) 3517 [arXiv:0710.5111] [INSPIRE].ADSGoogle Scholar
  23. [23]
    S. Actis, M. Czakon, J. Gluza and T. Riemann, Virtual hadronic and leptonic contributions to Bhabha scattering, Phys. Rev. Lett. 100 (2008) 131602 [arXiv:0711.3847] [INSPIRE].ADSCrossRefGoogle Scholar
  24. [24]
    S. Actis, T. Riemann, M. Czakon and J. Gluza, Two-loop heavy fermion corrections to Bhabha scattering, eConf C0705302 (2007) TEV02.Google Scholar
  25. [25]
    S. Actis, M. Czakon, J. Gluza and T. Riemann, Virtual Hadronic and Heavy-Fermion O(α 2) Corrections to Bhabha Scattering, Phys. Rev. D 78 (2008) 085019 [arXiv:0807.4691] [INSPIRE].ADSGoogle Scholar
  26. [26]
    R. Bonciani, A. Ferroglia and A. Penin, Heavy-flavor contribution to Bhabha scattering, Phys. Rev. Lett. 100 (2008) 131601 [arXiv:0710.4775] [INSPIRE].ADSCrossRefGoogle Scholar
  27. [27]
    R. Bonciani, A. Ferroglia and A. Penin, Calculation of the Two-Loop Heavy-Flavor Contribution to Bhabha Scattering, JHEP 02 (2008) 080 [arXiv:0802.2215] [INSPIRE].ADSCrossRefGoogle Scholar
  28. [28]
    R. Bonciani and A. Ferroglia, Two-loop Bhabha scattering in QED, Phys. Rev. D 72 (2005) 056004 [hep-ph/0507047] [INSPIRE].ADSGoogle Scholar
  29. [29]
    J.M. Henn, Multiloop integrals in dimensional regularization made simple, Phys. Rev. Lett. 110 (2013) 251601 [arXiv:1304.1806] [INSPIRE].ADSCrossRefGoogle Scholar
  30. [30]
    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
  31. [31]
    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
  32. [32]
    K.-T. Chen, Iterated path integrals, Bull. Am. Math. Soc. 83 (1977) 831 [INSPIRE].CrossRefzbMATHGoogle Scholar
  33. [33]
    F. Brown, Iterated integrals in quantum field theory, http://www.math.jussieu.fr/∼brown/.
  34. [34]
  35. [35]
    J. Ablinger, J. Blümlein and C. Schneider, Analytic and Algorithmic Aspects of Generalized Harmonic Sums and Polylogarithms, J. Math. Phys. 54 (2013) 082301 [arXiv:1302.0378] [INSPIRE].ADSCrossRefGoogle Scholar
  36. [36]
    A.B. Goncharov, Multiple polylogarithms, cyclotomy and modular complexes, Math. Res. Lett. 5 (1998) 497 [arXiv:1105.2076] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  37. [37]
    A. Smirnov, Algorithm FIRE - Feynman Integral REduction, JHEP 10 (2008) 107 [arXiv:0807.3243] [INSPIRE].ADSCrossRefGoogle Scholar
  38. [38]
    A. Smirnov and V. Smirnov, FIRE4, LiteRed and accompanying tools to solve integration by parts relations, arXiv:1302.5885 [INSPIRE].
  39. [39]
    C.W. Bauer, A. Frink and R. Kreckel, Introduction to the GiNaC framework for symbolic computation within the C++ programming language, J. Symb. Comput. 33 (2002) 1 [cs/0004015].MathSciNetCrossRefzbMATHGoogle Scholar
  40. [40]
    J. Vollinga and S. Weinzierl, Numerical evaluation of multiple polylogarithms, Comput. Phys. Commun. 167 (2005) 177 [hep-ph/0410259] [INSPIRE].MathSciNetADSCrossRefzbMATHGoogle Scholar
  41. [41]
    L.D. Landau, On analytic properties of vertex parts in quantum field theory, Nucl. Phys. 13 (1959) 181 .CrossRefzbMATHGoogle Scholar
  42. [42]
    T. Gehrmann and E. Remiddi, Analytic continuation of massless two loop four point functions, Nucl. Phys. B 640 (2002) 379 [hep-ph/0207020] [INSPIRE].ADSCrossRefGoogle Scholar
  43. [43]
    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].Google Scholar
  44. [44]
    A. Goncharov, A simple construction of Grassmannian polylogarithms, arXiv:0908.2238 [INSPIRE].
  45. [45]
    E. Remiddi and J. Vermaseren, Harmonic polylogarithms, Int. J. Mod. Phys. A 15 (2000) 725 [hep-ph/9905237] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  46. [46]
    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
  47. [47]
    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].ADSCrossRefGoogle Scholar
  48. [48]
    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].MathSciNetADSCrossRefGoogle Scholar
  49. [49]
    C. Duhr, H. Gangl and J.R. Rhodes, From polygons and symbols to polylogarithmic functions, JHEP 10 (2012) 075 [arXiv:1110.0458] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  50. [50]
    T. Binoth and G. Heinrich, Numerical evaluation of multiloop integrals by sector decomposition, Nucl. Phys. B 680 (2004) 375 [hep-ph/0305234] [INSPIRE].ADSCrossRefGoogle Scholar
  51. [51]
    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
  52. [52]
    A. Smirnov and M. Tentyukov, Feynman Integral Evaluation by a Sector decomposiTion Approach (FIESTA), Comput. Phys. Commun. 180 (2009) 735 [arXiv:0807.4129] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  53. [53]
    A. Smirnov, V. Smirnov and M. Tentyukov, FIESTA 2: Parallelizeable multiloop numerical calculations, Comput. Phys. Commun. 182 (2011) 790 [arXiv:0912.0158] [INSPIRE].MathSciNetADSCrossRefzbMATHGoogle Scholar
  54. [54]
    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].MathSciNetADSCrossRefGoogle Scholar
  55. [55]
    M. Czakon, Tops from Light Quarks: Full Mass Dependence at Two-Loops in QCD, Phys. Lett. B 664 (2008) 307 [arXiv:0803.1400] [INSPIRE].ADSCrossRefGoogle Scholar
  56. [56]
    S. Wolfram, The History and Future of Special Functions, http://www.stephenwolfram.com/publications/recent/specialfunctions/.
  57. [57]
    M. Kontsevich, Vassilievs knot invariants, Adv. Soviet Math. 16 (1993) 137.MathSciNetGoogle Scholar
  58. [58]
    L. Lewin ed., Structural Properties of Polylogarithms, American Mathematical Society, (1991).Google Scholar

Copyright information

© SISSA, Trieste, Italy 2013

Authors and Affiliations

  1. 1.Institute for Advanced StudyPrincetonU.S.A.
  2. 2.Skobeltsyn Institute of Nuclear Physics of Moscow State UniversityMoscowRussia

Personalised recommendations