Advertisement

A Four-Point Function for the Planar QCD Massive Corrections to Top-Antitop Production in the Gluon-Fusion Channel

  • Roberto BoncianiEmail author
  • Matteo Capozi
  • Paul Caucal
Chapter
Part of the Texts & Monographs in Symbolic Computation book series (TEXTSMONOGR)

Abstract

In these proceedings we present the study of a four-point function that is involved in the evaluation of the Master Integrals necessary to compute the two-loop massive QCD planar corrections to \(t\bar{t}\) production in the gluon fusuin channel, at hadron colliders. The solution involves complete elliptic integrals of the first and second kind and one- or two-fold integrations of such elliptic integrals multiplied by ratios of polynomials, inverse square roots and logarithms or dilogarithms.

References

  1. 1.
    P. Baernreuther, M. Czakon, A. Mitov, Percent level precision physics at the Tevatron: first genuine NNLO QCD corrections to \(q \bar{q} \rightarrow t \bar{t} + X\). Phys. Rev. Lett. 109, 132001 (2012)CrossRefGoogle Scholar
  2. 2.
    M. Czakon, A. Mitov, NNLO corrections to top-pair production at hadron colliders: the all-fermionic scattering channels. JHEP 12, 054 (2012)CrossRefGoogle Scholar
  3. 3.
    M. Czakon, A. Mitov, NNLO corrections to top pair production at hadron colliders: the quark-gluon reaction. JHEP 01, 080 (2013)CrossRefGoogle Scholar
  4. 4.
    M. Czakon, P. Fiedler, A. Mitov, Total top-quark pair-production cross section at Hadron Colliders through \(O(\alpha ^4_S)\). Phys. Rev. Lett. 110, 252004 (2013)CrossRefGoogle Scholar
  5. 5.
    M. Czakon, P. Fiedler, A. Mitov, Resolving the Tevatron top quark forward-backward asymmetry puzzle: fully differential next-to-next-to-leading-order calculation. Phys. Rev. Lett. 115(5), 052001 (2015)CrossRefGoogle Scholar
  6. 6.
    M. Czakon, D. Heymes, A. Mitov, High-precision differential predictions for top-quark pairs at the LHC. Phys. Rev. Lett. 116(8), 082003 (2016)CrossRefGoogle Scholar
  7. 7.
    M. Czakon, D. Heymes, A. Mitov, Dynamical scales for multi-TeV top-pair production at the LHC. JHEP 04, 071 (2017)CrossRefGoogle Scholar
  8. 8.
    M. Czakon, P. Fiedler, D. Heymes, A. Mitov, NNLO QCD predictions for fully-differential top-quark pair production at the Tevatron. JHEP 05, 034 (2016)CrossRefGoogle Scholar
  9. 9.
    S. Dittmaier, P. Uwer, S. Weinzierl, NLO QCD corrections to t anti-t \(+\) jet production at hadron colliders. Phys. Rev. Lett. 98, 262002 (2007)CrossRefGoogle Scholar
  10. 10.
    G. Bevilacqua, M. Czakon, C.G. Papadopoulos, M. Worek, Dominant QCD backgrounds in Higgs Boson analyses at the LHC: a study of pp \(->\) t anti-t \(+\) 2 jets at next-to-leading order. Phys. Rev. Lett. 104, 162002 (2010)CrossRefGoogle Scholar
  11. 11.
    G. Bevilacqua, M. Czakon, C.G. Papadopoulos, M. Worek, Hadronic top-quark pair production in association with two jets at next-to-leading order QCD. Phys. Rev. D 84, 114017 (2011)CrossRefGoogle Scholar
  12. 12.
    K. Melnikov, M. Schulze, NLO QCD corrections to top quark pair production in association with one hard jet at hadron colliders. Nucl. Phys. B 840, 129–159 (2010)CrossRefGoogle Scholar
  13. 13.
    G. Abelof, A. Gehrmann-De Ridder, P. Maierhofer, S. Pozzorini, NNLO QCD subtraction for top-antitop production in the \(q\overline{q} \) channel. JHEP 08, 035 (2014)Google Scholar
  14. 14.
    G. Abelof, A. Gehrmann-De Ridder, Light fermionic NNLO QCD corrections to top-antitop production in the quark-antiquark channel. JHEP 12, 076 (2014)CrossRefGoogle Scholar
  15. 15.
    G. Abelof, A. Gehrmann-De Ridder, I. Majer, Top quark pair production at NNLO in the quark-antiquark channel. JHEP 12, 074 (2015)Google Scholar
  16. 16.
    R. Bonciani, S. Catani, M. Grazzini, H. Sargsyan, A. Torre, The \(q_T\) subtraction method for top quark production at hadron colliders. Eur. Phys. J. C 75(12), 581 (2015)CrossRefGoogle Scholar
  17. 17.
    J.G. Korner, Z. Merebashvili, M. Rogal, NNLO \(O(\alpha _s^{4})\) results for heavy quark pair production in quark-antiquark collisions: the one-loop squared contributions. Phys. Rev. D 77, 094011 (2008). [Erratum: Phys. Rev. D 85, 119904 (2012)]Google Scholar
  18. 18.
    B. Kniehl, Z. Merebashvili, J.G. Korner, M. Rogal, Heavy quark pair production in gluon fusion at next-to-next-to-leading \(O(\alpha _s^{4)}\) order: one-loop squared contributions. Phys. Rev. D 78, 094013 (2008)CrossRefGoogle Scholar
  19. 19.
    C. Anastasiou, S.M. Aybat, The one-loop gluon amplitude for heavy-quark production at NNLO. Phys. Rev. D 78, 114006 (2008)CrossRefGoogle Scholar
  20. 20.
    M. Czakon, Tops from light quarks: full mass dependence at two-loops in QCD. Phys. Lett. B 664, 307–314 (2008)CrossRefGoogle Scholar
  21. 21.
    A. Ferroglia, M. Neubert, B.D. Pecjak, L.L. Yang, Two-loop divergences of scattering amplitudes with massive partons. Phys. Rev. Lett. 103, 201601 (2009)CrossRefGoogle Scholar
  22. 22.
    A. Ferroglia, M. Neubert, B.D. Pecjak, L.L. Yang, Two-loop divergences of massive scattering amplitudes in non-abelian gauge theories. JHEP 11, 062 (2009)CrossRefGoogle Scholar
  23. 23.
    A.B. Goncharov, Polylogarithms in arithmetic and geometry, in Proceedings of the International Congress of Mathematicians, vol. 1, 2 (1995), pp. 374–387CrossRefGoogle Scholar
  24. 24.
    A.B. Goncharov, Multiple polylogarithms and mixed Tate motives (2007)Google Scholar
  25. 25.
    E. Remiddi, J.A.M. Vermaseren, Harmonic polylogarithms. Int. J. Mod. Phys. A15, 725–754 (2000)MathSciNetCrossRefGoogle Scholar
  26. 26.
    J. Vollinga, S. Weinzierl, Numerical evaluation of multiple polylogarithms. Comput. Phys. Commun. 167, 177 (2005)MathSciNetCrossRefGoogle Scholar
  27. 27.
    R. Bonciani, A. Ferroglia, T. Gehrmann, D. Maitre, C. Studerus, Two-loop Fermionic corrections to heavy-quark pair production: the quark-antiquark channel. JHEP 07, 129 (2008)CrossRefGoogle Scholar
  28. 28.
    R. Bonciani, A. Ferroglia, T. Gehrmann, C. Studerus, Two-loop planar corrections to heavy-quark pair production in the quark-antiquark channel. JHEP 08, 067 (2009)CrossRefGoogle Scholar
  29. 29.
    P. Baernreuther, M. Czakon, P. Fiedler, Virtual amplitudes and threshold behaviour of hadronic top-quark pair-production cross sections. JHEP 02, 078 (2014)CrossRefGoogle Scholar
  30. 30.
    R. Bonciani, A. Ferroglia, T. Gehrmann, A. von Manteuffel, C. Studerus, Two-loop leading color corrections to heavy-quark pair production in the Gluon fusion channel. JHEP 01, 102 (2011)CrossRefGoogle Scholar
  31. 31.
    A. von Manteuffel, C. Studerus, Massive planar and non-planar double box integrals for light Nf contributions to gg->tt. JHEP 10, 037 (2013)CrossRefGoogle Scholar
  32. 32.
    R. Bonciani, A. Ferroglia, T. Gehrmann, A. von Manteuffel, C. Studerus, Light-quark two-loop corrections to heavy-quark pair production in the gluon fusion channel. JHEP 12, 038 (2013)CrossRefGoogle Scholar
  33. 33.
    L. Adams, E. Chaubey, S. Weinzierl, The planar double box integral for top pair production with a closed top loop to all orders in the dimensional regularisation parameter (2018)Google Scholar
  34. 34.
    L. Adams, E. Chaubey, S. Weinzierl, Analytic results for the planar double box integral relevant to top-pair production with a closed top loop (2018)Google Scholar
  35. 35.
    A. von Manteuffel, L. Tancredi, A non-planar two-loop three-point function beyond multiple polylogarithms. JHEP 06, 127 (2017)MathSciNetCrossRefGoogle Scholar
  36. 36.
    U. Aglietti, R. Bonciani, G. Degrassi, A. Vicini, Analytic results for virtual QCD corrections to Higgs production and decay. JHEP 01, 021 (2007)CrossRefGoogle Scholar
  37. 37.
    C. Anastasiou, S. Beerli, S. Bucherer, A. Daleo, 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, 082 (2007)CrossRefGoogle Scholar
  38. 38.
    C. Anastasiou, A. Lazopoulos, Automatic integral reduction for higher order perturbative calculations. JHEP 07, 046 (2004)CrossRefGoogle Scholar
  39. 39.
    R.N. Lee, Presenting LiteRed: a tool for the Loop InTEgrals REDuction (2012)Google Scholar
  40. 40.
    R.N. Lee, LiteRed 1.4: a powerful tool for reduction of multiloop integrals. J. Phys. Conf. Ser. 523, 012059 (2014)CrossRefGoogle Scholar
  41. 41.
    P. Maierhoefer, J. Usovitsch, P. Uwer, Kira - a Feynman integral reduction program. Comput. Phys. Commun. 230, 99–112 (2018)CrossRefGoogle Scholar
  42. 42.
    A.V. Smirnov, Algorithm FIRE - Feynman integral reduction. JHEP 10, 107 (2008)CrossRefGoogle Scholar
  43. 43.
    A.V. Smirnov, V.A. Smirnov, FIRE4, LiteRed and accompanying tools to solve integration by parts relations. Comput. Phys. Commun. 184, 2820–2827 (2013)CrossRefGoogle Scholar
  44. 44.
    A.V. Smirnov, FIRE5: a C++ implementation of Feynman integral reduction. Comput. Phys. Commun. 189, 182–191 (2014)CrossRefGoogle Scholar
  45. 45.
    C. Studerus, Reduze-Feynman integral reduction in C++. Comput. Phys. Commun. 181, 1293–1300 (2010)MathSciNetCrossRefGoogle Scholar
  46. 46.
    A. von Manteuffel, C. Studerus, Reduze 2 - distributed Feynman integral reduction (2012)Google Scholar
  47. 47.
    F.V. Tkachov, A theorem on analytical calculability of four loop renormalization group functions. Phys. Lett. B 100, 65–68 (1981)MathSciNetCrossRefGoogle Scholar
  48. 48.
    K.G. Chetyrkin, F.V. Tkachov, Integration by parts: the algorithm to calculate beta functions in 4 loops. Nucl. Phys. B 192, 159–204 (1981)CrossRefGoogle Scholar
  49. 49.
    S. Laporta, High precision calculation of multiloop Feynman integrals by difference equations. Int. J. Mod. Phys. A 15, 5087–5159 (2000)MathSciNetzbMATHGoogle Scholar
  50. 50.
    T. Gehrmann, E. Remiddi, Differential equations for two loop four point functions. Nucl. Phys. B 580, 485–518 (2000)MathSciNetCrossRefGoogle Scholar
  51. 51.
    R. Bonciani, P. Mastrolia, E. Remiddi, Vertex diagrams for the QED form-factors at the two loop level. Nucl. Phys. B 661, 289–343 (2003)CrossRefGoogle Scholar
  52. 52.
    R. Bonciani, P. Mastrolia, E. Remiddi, Master integrals for the two loop QCD virtual corrections to the forward backward asymmetry. Nucl. Phys. B 690, 138–176 (2004)CrossRefGoogle Scholar
  53. 53.
    R. Bonciani, A. Ferroglia, Two-loop QCD corrections to the heavy-to-light quark decay. JHEP 11, 065 (2008)CrossRefGoogle Scholar
  54. 54.
    S. Laporta, E. Remiddi, Analytic treatment of the two loop equal mass sunrise graph. Nucl. Phys. B 704, 349–386 (2005)MathSciNetCrossRefGoogle Scholar
  55. 55.
    L. Adams, C. Bogner, S. Weinzierl, The two-loop sunrise graph with arbitrary masses. J. Math. Phys. 54, 052303 (2013)MathSciNetCrossRefGoogle Scholar
  56. 56.
    E. Remiddi, L. Tancredi, Differential equations and dispersion relations for Feynman amplitudes. The two-loop massive sunrise and the kite integral. Nucl. Phys. B 907, 400–444 (2016)MathSciNetCrossRefGoogle Scholar
  57. 57.
    S. Bloch, P. Vanhove, The elliptic dilogarithm for the sunset graph. J. Number Theor. 148, 328–364 (2015)MathSciNetCrossRefGoogle Scholar
  58. 58.
    S. Bloch, M. Kerr, P. Vanhove, A Feynman integral via higher normal functions. Compos. Math. 151(12), 2329–2375 (2015)MathSciNetCrossRefGoogle Scholar
  59. 59.
    L. Adams, C. Bogner, S. Weinzierl, The iterated structure of the all-order result for the two-loop sunrise integral. J. Math. Phys. 57(3), 032304 (2016)MathSciNetCrossRefGoogle Scholar
  60. 60.
    C. Bogner, A. Schweitzer, S. Weinzierl, Analytic continuation and numerical evaluation of the kite integral and the equal mass sunrise integral. Nucl. Phys. B 922, 528–550 (2017)MathSciNetCrossRefGoogle Scholar
  61. 61.
    A.V. Kotikov, Differential equations method: new technique for massive Feynman diagrams calculation. Phys. Lett. B 254, 158–164 (1991)MathSciNetCrossRefGoogle Scholar
  62. 62.
    E. Remiddi, Differential equations for Feynman graph amplitudes. Nuovo Cim. A 110, 1435–1452 (1997)Google Scholar
  63. 63.
    M. Argeri, P. Mastrolia, Feynman diagrams and differential equations. Int. J. Mod. Phys. A 22, 4375–4436 (2007)MathSciNetCrossRefGoogle Scholar
  64. 64.
    J.M. Henn, Lectures on differential equations for Feynman integrals. J. Phys. A 48, 153001 (2015)MathSciNetCrossRefGoogle Scholar
  65. 65.
    J.M. Henn, Multiloop integrals in dimensional regularization made simple. Phys. Rev. Lett. 110, 251601 (2013)CrossRefGoogle Scholar
  66. 66.
    M. Argeri, S. Di Vita, P. Mastrolia, E. Mirabella, J. Schlenk et al., Magnus and Dyson series for master integrals. JHEP 1403, 082 (2014)MathSciNetCrossRefGoogle Scholar
  67. 67.
    S. Di Vita, P. Mastrolia, U. Schubert, V. Yundin, Three-loop master integrals for ladder-box diagrams with one massive leg. JHEP 09, 148 (2014)CrossRefGoogle Scholar
  68. 68.
    J.M. Henn, A.V. Smirnov, V.A. Smirnov, Evaluating single-scale and/or non-planar diagrams by differential equations. JHEP 1403, 088 (2014)CrossRefGoogle Scholar
  69. 69.
    T. Gehrmann, A. von Manteuffel, L. Tancredi, E. Weihs, The two-loop master integrals for \(q\overline{q} \rightarrow VV\). JHEP 1406, 032 (2014)CrossRefGoogle Scholar
  70. 70.
    R.N. Lee, Reducing differential equations for multiloop master integrals. JHEP 04, 108 (2015)MathSciNetCrossRefGoogle Scholar
  71. 71.
    L. Adams, E. Chaubey, S. Weinzierl, Simplifying differential equations for multiscale Feynman integrals beyond multiple polylogarithms. Phys. Rev. Lett. 118(14), 141602 (2017)CrossRefGoogle Scholar
  72. 72.
    J. Ablinger, C. Schneider, A. Behring, J. Blümlein, A. de Freitas, Algorithms to solve coupled systems of differential equations in terms of power series. PoS(LL2016), 005, arXiv:1608.05376 [cs.SC]
  73. 73.
    C. Meyer, Transforming differential equations of multi-loop Feynman integrals into canonical form. JHEP 04, 006 (2017)MathSciNetCrossRefGoogle Scholar
  74. 74.
    A. Georgoudis, K.J. Larsen, Y. Zhang, Azurite: an algebraic geometry based package for finding bases of loop integrals (2016)Google Scholar
  75. 75.
    O. Gituliar, V. Magerya, Fuchsia: a tool for reducing differential equations for Feynman master integrals to epsilon form. Comput. Phys. Commun. 219, 329–338 (2017)MathSciNetCrossRefGoogle Scholar
  76. 76.
    A.V. Smirnov, M.N. Tentyukov, Feynman integral evaluation by a sector decomposition approach (FIESTA). Comput. Phys. Commun. 180, 735–746 (2009)CrossRefGoogle Scholar
  77. 77.
    A.V. Smirnov, FIESTA 3: cluster-parallelizable multiloop numerical calculations in physical regions. Comput. Phys. Commun. 185, 2090–2100 (2014)CrossRefGoogle Scholar
  78. 78.
    A.V. Smirnov, FIESTA4: optimized feynman integral calculations with GPU support. Comput. Phys. Commun. 204, 189–199 (2016)CrossRefGoogle Scholar
  79. 79.
    R. Bonciani, V. Del Duca, H. Frellesvig, J.M. Henn, F. Moriello, V.A. Smirnov, Two-loop planar master integrals for Higgs \(\rightarrow 3\) partons with full heavy-quark mass dependence. JHEP 12, 096 (2016)CrossRefGoogle Scholar
  80. 80.
    R.N. Lee, V.A. Smirnov, The dimensional recurrence and analyticity method for multicomponent master integrals: using unitarity cuts to construct homogeneous solutions. JHEP 12, 104 (2012)MathSciNetCrossRefGoogle Scholar
  81. 81.
    A. Primo, L. Tancredi, On the maximal cut of Feynman integrals and the solution of their differential equations. Nucl. Phys. B 916, 94–116 (2017)MathSciNetCrossRefGoogle Scholar
  82. 82.
    H. Frellesvig, C.G. Papadopoulos, Cuts of Feynman integrals in Baikov representation. JHEP 04, 083 (2017)MathSciNetCrossRefGoogle Scholar
  83. 83.
    M. Harley, F. Moriello, R.M. Schabinger, Baikov–Lee representations of cut Feynman integrals. JHEP 06, 049 (2017)MathSciNetCrossRefGoogle Scholar
  84. 84.
    J. Ablinger, J. Blümlein, A. De Freitas, M. van Hoeij, E. Imamoglu, C.G. Raab, C.S. Radu, C. Schneider, Iterated elliptic and hypergeometric integrals for Feynman diagrams. J. Math. Phys. 59(6), 062305 (2018), arXiv:1706.01299 [hep-th]MathSciNetCrossRefGoogle Scholar
  85. 85.
    J. Broedel, C. Duhr, F. Dulat, L. Tancredi, Elliptic polylogarithms and iterated integrals on elliptic curves. Part I: general formalism. JHEP 05, 093 (2018)MathSciNetCrossRefGoogle Scholar
  86. 86.
    J. Broedel, C. Duhr, F. Dulat, L. Tancredi, Elliptic polylogarithms and iterated integrals on elliptic curves II: an application to the sunrise integral. Phys. Rev. D 97(11), 116009 (2018)CrossRefGoogle Scholar
  87. 87.
    E. Remiddi, L. Tancredi, An elliptic generalization of multiple polylogarithms. Nucl. Phys. B 925, 212–251 (2017)MathSciNetCrossRefGoogle Scholar
  88. 88.
    J. Broedel, C. Duhr, F. Dulat, B. Penante, L. Tancredi, Elliptic symbol calculus: from elliptic polylogarithms to iterated integrals of Eisenstein series (2018)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Dipartimento di FisicaUniversità di Roma “La Sapienza”RomaItaly
  2. 2.INFN Sezione di RomaRomaItaly
  3. 3.Max-Planck-Institute for PhysicsMünchenGermany
  4. 4.IPhT, Université Paris-Saclay, CNRS, CEAGif-sur-YvetteFrance

Personalised recommendations