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Master integrals for the NNLO virtual corrections to \( q\overline{q}\to t\overline{t} \) scattering in QCD: the non-planar graphs

  • Stefano Di VitaEmail author
  • Thomas Gehrmann
  • Stefano Laporta
  • Pierpaolo Mastrolia
  • Amedeo Primo
  • Ulrich Schubert
Open Access
Regular Article - Theoretical Physics
  • 49 Downloads

Abstract

We complete the analytic evaluation of the master integrals for the two-loop non-planar box diagrams contributing to the top-pair production in the quark-initiated channel, at next-to-next-to-leading order in QCD. The integrals are determined from their differential equations, which are cast into a canonical form using the Magnus exponential. The analytic expressions of the Laurent series coefficients of the integrals are expressed as combinations of generalized polylogarithms, which we validate with several numerical checks. We discuss the analytic continuation of the planar and the non-planar master integrals, which contribute to q qtt in QCD, as well as to the companion QED scattering processes ee → μμ and eμ → eμ.

Keywords

NLO Computations 

Notes

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.

Supplementary material

13130_2019_10795_MOESM1_ESM.tgz (145 kb)
ESM 1 (TGZ 145 kb)

References

  1. [1]
    ATLAS collaboration, Measurements of top-quark pair differential cross-sections in the lepton+jets channel in pp collisions at \( \sqrt{s}=8 \) using the ATLAS detector, Eur. Phys. J. C 76 (2016) 538 [arXiv:1511.04716] [INSPIRE].
  2. [2]
    ATLAS collaboration, Measurements of top-quark pair differential cross-sections in the lepton+jets channel in pp collisions at \( \sqrt{s}=13 \) TeV using the ATLAS detector, JHEP 11 (2017) 191 [arXiv:1708.00727] [INSPIRE].
  3. [3]
    CMS collaboration, Measurement of the differential cross section for top quark pair production in pp collisions at \( \sqrt{s}=8 \) TeV, Eur. Phys. J. C 75 (2015) 542 [arXiv:1505.04480] [INSPIRE].
  4. [4]
    CMS collaboration, Measurement of differential cross sections for the production of top quark pairs and of additional jets in lepton+jets events from pp collisions at \( \sqrt{s}=13 \) TeV, Phys. Rev. D 97 (2018) 112003 [arXiv:1803.08856] [INSPIRE].
  5. [5]
    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].Google Scholar
  6. [6]
    M. Czakon, P. Fiedler and A. Mitov, Resolving the Tevatron top quark forward-backward asymmetry puzzle: fully differential next-to-next-to-leading-order calculation, Phys. Rev. Lett. 115 (2015) 052001 [arXiv:1411.3007] [INSPIRE].ADSCrossRefGoogle Scholar
  7. [7]
    M. Czakon, D. Heymes and A. Mitov, High-precision differential predictions for top-quark pairs at the LHC, Phys. Rev. Lett. 116 (2016) 082003 [arXiv:1511.00549] [INSPIRE].ADSCrossRefGoogle Scholar
  8. [8]
    M. Czakon, D. Heymes and A. Mitov, Dynamical scales for multi-TeV top-pair production at the LHC, JHEP 04 (2017) 071 [arXiv:1606.03350] [INSPIRE].ADSCrossRefGoogle Scholar
  9. [9]
    S. Catani, S. Devoto, M. Grazzini, S. Kallweit, J. Mazzitelli and H. Sargsyan, Top-quark pair hadroproduction at next-to-next-to-leading order in QCD, Phys. Rev. D 99 (2019) 051501 [arXiv:1901.04005] [INSPIRE].ADSGoogle Scholar
  10. [10]
    P. Bärnreuther, M. Czakon and P. Fiedler, Virtual amplitudes and threshold behaviour of hadronic top-quark pair-production cross sections, JHEP 02 (2014) 078 [arXiv:1312.6279] [INSPIRE].ADSCrossRefGoogle Scholar
  11. [11]
    L. Chen, M. Czakon and R. Poncelet, Polarized double-virtual amplitudes for heavy-quark pair production, JHEP 03 (2018) 085 [arXiv:1712.08075] [INSPIRE].ADSCrossRefGoogle Scholar
  12. [12]
    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
  13. [13]
    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].ADSCrossRefGoogle Scholar
  14. [14]
    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].ADSCrossRefGoogle Scholar
  15. [15]
    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
  16. [16]
    G. Abelof, A. Gehrmann-De Ridder, P. Maierhofer and S. Pozzorini, NNLO QCD subtraction for top-antitop production in the \( q\overline{q} \) channel, JHEP 08 (2014) 035 [arXiv:1404.6493] [INSPIRE].Google Scholar
  17. [17]
    G. Abelof, A. Gehrmann-De Ridder and I. Majer, Top quark pair production at NNLO in the quark-antiquark channel, JHEP 12 (2015) 074 [arXiv:1506.04037] [INSPIRE].ADSGoogle Scholar
  18. [18]
    A. von Manteuffel and L. Tancredi, A non-planar two-loop three-point function beyond multiple polylogarithms, JHEP 06 (2017) 127 [arXiv:1701.05905] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  19. [19]
    L. Adams, E. Chaubey and S. Weinzierl, Planar double box integral for top pair production with a closed top loop to all orders in the dimensional regularization parameter, Phys. Rev. Lett. 121 (2018) 142001 [arXiv:1804.11144] [INSPIRE].ADSCrossRefGoogle Scholar
  20. [20]
    L. Adams, E. Chaubey and S. Weinzierl, Analytic results for the planar double box integral relevant to top-pair production with a closed top loop, JHEP 10 (2018) 206 [arXiv:1806.04981] [INSPIRE].ADSCrossRefGoogle Scholar
  21. [21]
    L.-B. Chen and J. Wang, Master integrals of a planar double-box family for top-quark pair production, Phys. Lett. B 792 (2019) 50 [arXiv:1903.04320] [INSPIRE].ADSCrossRefGoogle Scholar
  22. [22]
    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
  23. [23]
    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
  24. [24]
    P. Mastrolia, M. Passera, A. Primo and U. Schubert, Master integrals for the NNLO virtual corrections to μe scattering in QED: the planar graphs, JHEP 11 (2017) 198 [arXiv:1709.07435] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  25. [25]
    S. Di Vita, S. Laporta, P. Mastrolia, A. Primo and U. Schubert, Master integrals for the NNLO virtual corrections to μe scattering in QED: the non-planar graphs, JHEP 09 (2018) 016 [arXiv:1806.08241] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  26. [26]
    J.M. Henn, Multiloop integrals in dimensional regularization made simple, Phys. Rev. Lett. 110 (2013) 251601 [arXiv:1304.1806] [INSPIRE].ADSCrossRefGoogle Scholar
  27. [27]
    M. Argeri et al., Magnus and Dyson series for master integrals, JHEP 03 (2014) 082 [arXiv:1401.2979] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  28. [28]
    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
  29. [29]
    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
  30. [30]
    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
  31. [31]
    A. Primo, G. Sasso, G. Somogyi and F. Tramontano, Exact top Yukawa corrections to Higgs boson decay into bottom quarks, Phys. Rev. D 99 (2019) 054013 [arXiv:1812.07811] [INSPIRE].ADSGoogle Scholar
  32. [32]
    F.V. Tkachov, A theorem on analytical calculability of four loop renormalization group functions, Phys. Lett. B 100 (1981) 65 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  33. [33]
    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
  34. [34]
    S. Laporta, High precision calculation of multiloop Feynman integrals by difference equations, Int. J. Mod. Phys. A 15 (2000) 5087 [hep-ph/0102033] [INSPIRE].
  35. [35]
    G. Barucchi and G. Ponzano, Differential equations for one-loop generalized Feynman integrals, J. Math. Phys. 14 (1973) 396 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  36. [36]
    A.V. Kotikov, Differential equations method: new technique for massive Feynman diagrams calculation, Phys. Lett. B 254 (1991) 158 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  37. [37]
    E. Remiddi, Differential equations for Feynman graph amplitudes, Nuovo Cim. A 110 (1997) 1435 [hep-th/9711188] [INSPIRE].ADSGoogle Scholar
  38. [38]
    T. Gehrmann and E. Remiddi, Differential equations for two loop four point functions, Nucl. Phys. B 580 (2000) 485 [hep-ph/9912329] [INSPIRE].
  39. [39]
    A. Goncharov, Polylogarithms in arithmetic and geometry, Proc. Internat. Congress Math. 1,2 (1995) 374.MathSciNetzbMATHGoogle Scholar
  40. [40]
    E. Remiddi and J.A.M. Vermaseren, Harmonic polylogarithms, Int. J. Mod. Phys. A 15 (2000) 725 [hep-ph/9905237] [INSPIRE].
  41. [41]
    T. Gehrmann and E. Remiddi, Numerical evaluation of harmonic polylogarithms, Comput. Phys. Commun. 141 (2001) 296 [hep-ph/0107173] [INSPIRE].
  42. [42]
    J. Vollinga and S. Weinzierl, Numerical evaluation of multiple polylogarithms, Comput. Phys. Commun. 167 (2005) 177 [hep-ph/0410259] [INSPIRE].
  43. [43]
    R.N. Lee and K.T. Mingulov, Master integrals for two-loop C-odd contribution to e + e → ℓ+ process, arXiv:1901.04441 [INSPIRE].
  44. [44]
    P. Maierhöfer, J. Usovitsch and P. Uwer, Kiraa Feynman integral reduction program, Comput. Phys. Commun. 230 (2018) 99 [arXiv:1705.05610] [INSPIRE].ADSCrossRefGoogle Scholar
  45. [45]
    R.N. Lee, Presenting LiteRed: a tool for the Loop InTEgrals REDuction, arXiv:1212.2685 [INSPIRE].
  46. [46]
    R.N. Lee, LiteRed 1.4: a powerful tool for reduction of multiloop integrals, J. Phys. Conf. Ser. 523 (2014) 012059 [arXiv:1310.1145] [INSPIRE].
  47. [47]
    A. von Manteuffel and C. Studerus, Reduze 2 — distributed Feynman integral reduction, arXiv:1201.4330 [INSPIRE].
  48. [48]
    A. von Manteuffel, E. Panzer and R.M. Schabinger, A quasi-finite basis for multi-loop Feynman integrals, JHEP 02 (2015) 120 [arXiv:1411.7392] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  49. [49]
    C. 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].
  50. [50]
    S. Borowka, G. Heinrich, S.P. Jones, M. Kerner, J. Schlenk and T. Zirke, SecDec-3.0: numerical evaluation of multi-scale integrals beyond one loop, Comput. Phys. Commun. 196 (2015) 470 [arXiv:1502.06595] [INSPIRE].
  51. [51]
    M. Becchetti, R. Bonciani, V. Casconi, A. Ferroglia, S. Lavacca and A. von Manteuffel, Master integrals for the two-loop, non-planar QCD corrections to top-quark pair production in the quark-annihilation channel, arXiv:1904.10834 [INSPIRE].
  52. [52]
    J.C. Collins and J.A.M. Vermaseren, Axodraw version 2, arXiv:1606.01177 [INSPIRE].
  53. [53]
    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].
  54. [54]
    R. Bonciani, P. Mastrolia and E. Remiddi, Vertex diagrams for the QED form-factors at the two loop level, Nucl. Phys. B 661 (2003) 289 [Erratum ibid. B 702 (2004) 359] [hep-ph/0301170] [INSPIRE].
  55. [55]
    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].
  56. [56]
    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
  57. [57]
    H.M. Asatrian, C. Greub and B.D. Pecjak, NNLO corrections to \( \overline{B}\to {X}_ul\overline{v} \) in the shape-function region, Phys. Rev. D 78 (2008) 114028 [arXiv:0810.0987] [INSPIRE].Google Scholar
  58. [58]
    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].ADSCrossRefGoogle Scholar
  59. [59]
    G. Bell, NNLO corrections to inclusive semileptonic B decays in the shape-function region, Nucl. Phys. B 812 (2009) 264 [arXiv:0810.5695] [INSPIRE].ADSCrossRefGoogle Scholar
  60. [60]
    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
  61. [61]
    A. von Manteuffel and C. Studerus, Massive planar and non-planar double box integrals for light N f contributions to ggtt, JHEP 10 (2013) 037 [arXiv:1306.3504] [INSPIRE].CrossRefGoogle Scholar
  62. [62]
    J.M. Henn, Lectures on differential equations for Feynman integrals, J. Phys. A 48 (2015) 153001 [arXiv:1412.2296] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  63. [63]
    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].
  64. [64]
    T. Gehrmann and E. Remiddi, Analytic continuation of massless two loop four point functions, Nucl. Phys. B 640 (2002) 379 [hep-ph/0207020] [INSPIRE].
  65. [65]
    J.B. Tausk, Nonplanar massless two loop Feynman diagrams with four on-shell legs, Phys. Lett. B 469 (1999) 225 [hep-ph/9909506] [INSPIRE].
  66. [66]
    D. Gaiotto, J. Maldacena, A. Sever and P. Vieira, Pulling the straps of polygons, JHEP 12 (2011) 011 [arXiv:1102.0062] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  67. [67]
    S. Abreu, R. Britto and H. Grönqvist, Cuts and coproducts of massive triangle diagrams, JHEP 07 (2015) 111 [arXiv:1504.00206] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  68. [68]
    O.V. Tarasov, Connection between Feynman integrals having different values of the space-time dimension, Phys. Rev. D 54 (1996) 6479 [hep-th/9606018] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  69. [69]
    R.N. Lee, Space-time dimensionality D as complex variable: calculating loop integrals using dimensional recurrence relation and analytical properties with respect to D, Nucl. Phys. B 830 (2010) 474 [arXiv:0911.0252] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.INFNSezione di MilanoMilanoItaly
  2. 2.Department of PhysicsUniversity of ZürichZürichSwitzerland
  3. 3.Dipartimento di Fisica ed AstronomiaUniversità di PadovaPadovaItaly
  4. 4.INFNSezione di PadovaPadovaItaly
  5. 5.Department of Physics, University at BuffaloThe State University of New YorkBuffaloU.S.A.

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