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Triple-real contribution to the quark beam function in QCD at next-to-next-to-next-to-leading order

  • K. Melnikov
  • R. RietkerkEmail author
  • L. Tancredi
  • C. Wever
Open Access
Regular Article - Theoretical Physics
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Abstract

We compute the three-loop master integrals required for the calculation of the triple-real contribution to the N3LO quark beam function due to the splitting of a quark into a virtual quark and three collinear gluons, qq* + ggg. This provides an important ingredient for the calculation of the leading-color contribution to the quark beam function at N3LO.

Keywords

NLO Computations QCD Phenomenology 

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

References

  1. [1]
    ATLAS collaboration, Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC, Phys. Lett. B 716 (2012) 1 [arXiv:1207.7214] [INSPIRE].
  2. [2]
    CMS collaboration, Observation of a new boson at a mass of 125 GeV with the CMS experiment at the LHC, Phys. Lett. B 716 (2012) 30 [arXiv:1207.7235] [INSPIRE].
  3. [3]
    ATLAS collaboration, Measurement of W ± and Z-boson production cross sections in pp collisions at \( \sqrt{s} \) = 13 TeV with the ATLAS detector, Phys. Lett. B 759 (2016) 601 [arXiv:1603.09222] [INSPIRE].
  4. [4]
    CMS collaboration, Measurement of inclusive W and Z boson production cross sections in pp collisions at \( \sqrt{s} \) = 8 TeV, Phys. Rev. Lett. 112 (2014) 191802 [arXiv:1402.0923] [INSPIRE].
  5. [5]
    ATLAS collaboration, Measurements of inclusive and differential fiducial cross-sections of \( t\overline{t}\gamma \) production in leptonic final states at \( \sqrt{s} \) = 13 TeV in ATLAS, Eur. Phys. J. C 79 (2019) 382 [arXiv:1812.01697] [INSPIRE].
  6. [6]
    CMS collaboration, Measurement of the \( \mathrm{t}\overline{\mathrm{t}} \) production cross section, the top quark mass and the strong coupling constant using dilepton events in pp collisions at \( \sqrt{s} \) = 13 TeV, Eur. Phys. J. C 79 (2019) 368 [arXiv:1812.10505] [INSPIRE].
  7. [7]
    ATLAS collaboration, Measurement of W ± Z production cross sections and gauge boson polarisation in pp collisions at \( \sqrt{s} \) = 13 TeV with the ATLAS detector, arXiv:1902.05759 [INSPIRE].
  8. [8]
    CMS collaboration, Measurements of the ppWZ inclusive and differential production cross section and constraints on charged anomalous triple gauge couplings at \( \sqrt{s} \) = 13 TeV, JHEP 04 (2019) 122 [arXiv:1901.03428] [INSPIRE].
  9. [9]
    ATLAS and CMS collaborations, Combinations of single-top-quark production cross-section measurements and |f LV V tb| determinations at \( \sqrt{s} \) = 7 and 8 TeV with the ATLAS and CMS experimentsCombinations of single-top-quark production cross-section measurements and |f LV V tb| determinations at \( \sqrt{s} \) = 7 and 8 TeV with the ATLAS and CMS experiments, JHEP 05 (2019) 088 [arXiv:1902.07158] [INSPIRE].
  10. [10]
    J. Kubar, M. Le Bellac, J.L. Meunier and G. Plaut, QCD Corrections to the Drell-Yan Mechanism and the Pion Structure Function, Nucl. Phys. B 175 (1980) 251 [INSPIRE].ADSCrossRefGoogle Scholar
  11. [11]
    P. Nason, S. Dawson and R.K. Ellis, The Total Cross-Section for the Production of Heavy Quarks in Hadronic Collisions, Nucl. Phys. B 303 (1988) 607 [INSPIRE].ADSCrossRefGoogle Scholar
  12. [12]
    J. Ohnemus, An Order α s calculation of hadronic W ± Z production, Phys. Rev. D 44 (1991) 3477 [INSPIRE].ADSGoogle Scholar
  13. [13]
    S. Frixione, P. Nason and G. Ridolfi, Strong corrections to W Z production at hadron colliders, Nucl. Phys. B 383 (1992) 3 [INSPIRE].ADSCrossRefGoogle Scholar
  14. [14]
    G. Ossola, C.G. Papadopoulos and R. Pittau, Reducing full one-loop amplitudes to scalar integrals at the integrand level, Nucl. Phys. B 763 (2007) 147 [hep-ph/0609007] [INSPIRE].
  15. [15]
    C.F. Berger et al., An Automated Implementation of On-Shell Methods for One-Loop Amplitudes, Phys. Rev. D 78 (2008) 036003 [arXiv:0803.4180] [INSPIRE].ADSGoogle Scholar
  16. [16]
    F. Cascioli, P. Maierhofer and S. Pozzorini, Scattering Amplitudes with Open Loops, Phys. Rev. Lett. 108 (2012) 111601 [arXiv:1111.5206] [INSPIRE].ADSCrossRefGoogle Scholar
  17. [17]
    J. Alwall et al., The automated computation of tree-level and next-to-leading order differential cross sections and their matching to parton shower simulations, JHEP 07 (2014) 079 [arXiv:1405.0301] [INSPIRE].ADSCrossRefGoogle Scholar
  18. [18]
    S. Actis, A. Denner, L. Hofer, J.-N. Lang, A. Scharf and S. Uccirati, RECOLA: REcursive Computation of One-Loop Amplitudes, Comput. Phys. Commun. 214 (2017) 140 [arXiv:1605.01090] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  19. [19]
    C. Anastasiou, C. Duhr, F. Dulat, F. Herzog and B. Mistlberger, Higgs Boson Gluon-Fusion Production in QCD at Three Loops, Phys. Rev. Lett. 114 (2015) 212001 [arXiv:1503.06056] [INSPIRE].ADSCrossRefGoogle Scholar
  20. [20]
    C. Anastasiou et al., High precision determination of the gluon fusion Higgs boson cross-section at the LHC, JHEP 05 (2016) 058 [arXiv:1602.00695] [INSPIRE].ADSCrossRefGoogle Scholar
  21. [21]
    B. Mistlberger, Higgs boson production at hadron colliders at N 3 LO in QCD, JHEP 05 (2018) 028 [arXiv:1802.00833] [INSPIRE].ADSCrossRefGoogle Scholar
  22. [22]
    F. Dulat, B. Mistlberger and A. Pelloni, Precision predictions at N 3 LO for the Higgs boson rapidity distribution at the LHC, Phys. Rev. D 99 (2019) 034004 [arXiv:1810.09462] [INSPIRE].ADSGoogle Scholar
  23. [23]
    C. Anastasiou and K. Melnikov, Higgs boson production at hadron colliders in NNLO QCD, Nucl. Phys. B 646 (2002) 220 [hep-ph/0207004] [INSPIRE].
  24. [24]
    L. Cieri, X. Chen, T. Gehrmann, E.W.N. Glover and A. Huss, Higgs boson production at the LHC using the q T subtraction formalism at N 3 LO QCD, JHEP 02 (2019) 096 [arXiv:1807.11501] [INSPIRE].ADSCrossRefGoogle Scholar
  25. [25]
    S. Catani and M. Grazzini, An NNLO subtraction formalism in hadron collisions and its application to Higgs boson production at the LHC, Phys. Rev. Lett. 98 (2007) 222002 [hep-ph/0703012] [INSPIRE].
  26. [26]
    I.W. Stewart, F.J. Tackmann and W.J. Waalewijn, N-Jettiness: An Inclusive Event Shape to Veto Jets, Phys. Rev. Lett. 105 (2010) 092002 [arXiv:1004.2489] [INSPIRE].ADSCrossRefGoogle Scholar
  27. [27]
    J. Gaunt, M. Stahlhofen, F.J. Tackmann and J.R. Walsh, N-jettiness Subtractions for NNLO QCD Calculations, JHEP 09 (2015) 058 [arXiv:1505.04794] [INSPIRE].CrossRefGoogle Scholar
  28. [28]
    R. Boughezal, C. Focke, X. Liu and F. Petriello, W -boson production in association with a jet at next-to-next-to-leading order in perturbative QCD, Phys. Rev. Lett. 115 (2015) 062002 [arXiv:1504.02131] [INSPIRE].ADSCrossRefGoogle Scholar
  29. [29]
    R. Boughezal, F. Caola, K. Melnikov, F. Petriello and M. Schulze, Higgs boson production in association with a jet at next-to-next-to-leading order, Phys. Rev. Lett. 115 (2015) 082003 [arXiv:1504.07922] [INSPIRE].ADSCrossRefGoogle Scholar
  30. [30]
    A. Gehrmann-De Ridder, T. Gehrmann, E.W.N. Glover, A. Huss and T.A. Morgan, Precise QCD predictions for the production of a Z boson in association with a hadronic jet, Phys. Rev. Lett. 117 (2016) 022001 [arXiv:1507.02850] [INSPIRE].ADSCrossRefGoogle Scholar
  31. [31]
    R. Boughezal, C. Focke, W. Giele, X. Liu and F. Petriello, Higgs boson production in association with a jet at NNLO using jettiness subtraction, Phys. Lett. B 748 (2015) 5 [arXiv:1505.03893] [INSPIRE].ADSCrossRefGoogle Scholar
  32. [32]
    X. Chen, J. Cruz-Martinez, T. Gehrmann, E.W.N. Glover and M. Jaquier, NNLO QCD corrections to Higgs boson production at large transverse momentum, JHEP 10 (2016) 066 [arXiv:1607.08817] [INSPIRE].ADSCrossRefGoogle Scholar
  33. [33]
    A. Gehrmann-De Ridder, T. Gehrmann, E.W.N. Glover, A. Huss and D.M. Walker, Next-to-Next-to-Leading-Order QCD Corrections to the Transverse Momentum Distribution of Weak Gauge Bosons, Phys. Rev. Lett. 120 (2018) 122001 [arXiv:1712.07543] [INSPIRE].ADSCrossRefGoogle Scholar
  34. [34]
    C.F. Berger, C. Marcantonini, I.W. Stewart, F.J. Tackmann and W.J. Waalewijn, Higgs Production with a Central Jet Veto at NNLL+NNLO, JHEP 04 (2011) 092 [arXiv:1012.4480] [INSPIRE].ADSCrossRefGoogle Scholar
  35. [35]
    J.R. Gaunt, M. Stahlhofen and F.J. Tackmann, The Quark Beam Function at Two Loops, JHEP 04 (2014) 113 [arXiv:1401.5478] [INSPIRE].ADSCrossRefGoogle Scholar
  36. [36]
    J. Gaunt, M. Stahlhofen and F.J. Tackmann, The Gluon Beam Function at Two Loops, JHEP 08 (2014) 020 [arXiv:1405.1044] [INSPIRE].ADSCrossRefGoogle Scholar
  37. [37]
    C.W. Bauer, S. Fleming and M.E. Luke, Summing Sudakov logarithms in BX s γ in effective field theory, Phys. Rev. D 63 (2000) 014006 [hep-ph/0005275] [INSPIRE].
  38. [38]
    C.W. Bauer, S. Fleming, D. Pirjol and I.W. Stewart, An Effective field theory for collinear and soft gluons: Heavy to light decays, Phys. Rev. D 63 (2001) 114020 [hep-ph/0011336] [INSPIRE].
  39. [39]
    C.W. Bauer and I.W. Stewart, Invariant operators in collinear effective theory, Phys. Lett. B 516 (2001) 134 [hep-ph/0107001] [INSPIRE].
  40. [40]
    C.W. Bauer, D. Pirjol and I.W. Stewart, Soft collinear factorization in effective field theory, Phys. Rev. D 65 (2002) 054022 [hep-ph/0109045] [INSPIRE].
  41. [41]
    C.W. Bauer, S. Fleming, D. Pirjol, I.Z. Rothstein and I.W. Stewart, Hard scattering factorization from effective field theory, Phys. Rev. D 66 (2002) 014017 [hep-ph/0202088] [INSPIRE].
  42. [42]
    K. Melnikov, R. Rietkerk, L. Tancredi and C. Wever, Double-real contribution to the quark beam function at N 3 LO QCD, JHEP 02 (2019) 159 [arXiv:1809.06300] [INSPIRE].ADSCrossRefGoogle Scholar
  43. [43]
    M. Ritzmann and W.J. Waalewijn, Fragmentation in Jets at NNLO, Phys. Rev. D 90 (2014) 054029 [arXiv:1407.3272] [INSPIRE].ADSGoogle Scholar
  44. [44]
    S. Catani and M. Grazzini, Infrared factorization of tree level QCD amplitudes at the next-to-next-to-leading order and beyond, Nucl. Phys. B 570 (2000) 287 [hep-ph/9908523] [INSPIRE].
  45. [45]
    P. Nogueira, Automatic Feynman graph generation, J. Comput. Phys. 105 (1993) 279.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  46. [46]
    J.A.M. Vermaseren, New features of FORM, math-ph/0010025 [INSPIRE].
  47. [47]
    F.V. Tkachov, A Theorem on Analytical Calculability of Four Loop Renormalization Group Functions, Phys. Lett. 100B (1981) 65 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  48. [48]
    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
  49. [49]
    S. Laporta, High precision calculation of multiloop Feynman integrals by difference equations, Int. J. Mod. Phys. A 15 (2000) 5087 [hep-ph/0102033] [INSPIRE].
  50. [50]
    A. von Manteuffel and C. Studerus, Reduze 2Distributed Feynman Integral Reduction, arXiv:1201.4330 [INSPIRE].
  51. [51]
    P. Maierhöfer, J. Usovitsch and P. Uwer, KiraA Feynman integral reduction program, Comput. Phys. Commun. 230 (2018) 99 [arXiv:1705.05610] [INSPIRE].ADSCrossRefGoogle Scholar
  52. [52]
    A.V. Kotikov, Differential equations method: New technique for massive Feynman diagrams calculation, Phys. Lett. B 254 (1991) 158 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  53. [53]
    E. Remiddi, Differential equations for Feynman graph amplitudes, Nuovo Cim. A 110 (1997) 1435 [hep-th/9711188] [INSPIRE].ADSGoogle Scholar
  54. [54]
    T. Gehrmann and E. Remiddi, Differential equations for two loop four point functions, Nucl. Phys. B 580 (2000) 485 [hep-ph/9912329] [INSPIRE].
  55. [55]
    A.V. Kotikov, The property of maximal transcendentality: calculation of master integrals, Theor. Math. Phys. 176 (2013) 913 [arXiv:1212.3732] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  56. [56]
    J.M. Henn, Multiloop integrals in dimensional regularization made simple, Phys. Rev. Lett. 110 (2013) 251601 [arXiv:1304.1806] [INSPIRE].ADSCrossRefGoogle Scholar
  57. [57]
    R.N. Lee, Reducing differential equations for multiloop master integrals, JHEP 04 (2015) 108 [arXiv:1411.0911] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  58. [58]
    O. Gituliar and V. Magerya, Fuchsia: a tool for reducing differential equations for Feynman master integrals to epsilon form, Comput. Phys. Commun. 219 (2017) 329 [arXiv:1701.04269] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  59. [59]
    E.E. Kummer, Über die Transcendenten, welche aus wiederholten Integrationen rationaler Formeln entstehen, J. Reine Angew. Math. 21 (1840) 74.MathSciNetCrossRefGoogle Scholar
  60. [60]
    A.B. Goncharov, Multiple polylogarithms, cyclotomy and modular complexes, Math. Res. Lett. 5 (1998) 497 [arXiv:1105.2076] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  61. [61]
    E. Remiddi and J.A.M. Vermaseren, Harmonic polylogarithms, Int. J. Mod. Phys. A 15 (2000) 725 [hep-ph/9905237] [INSPIRE].
  62. [62]
    A.B. Goncharov, Multiple polylogarithms and mixed Tate motives, math/0103059.
  63. [63]
    J. Vollinga and S. Weinzierl, Numerical evaluation of multiple polylogarithms, Comput. Phys. Commun. 167 (2005) 177 [hep-ph/0410259] [INSPIRE].
  64. [64]
    M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover, New York (1964).zbMATHGoogle Scholar
  65. [65]
    E. Panzer, Algorithms for the symbolic integration of hyperlogarithms with applications to Feynman integrals, Comput. Phys. Commun. 188 (2015) 148 [arXiv:1403.3385] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  66. [66]
    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
  67. [67]
    M. Czakon, D. Kosower, A. Smirnov and V. Smirnov, MB Tools, https://mbtools.hepforge.org/.

Copyright information

© The Author(s) 2019

Authors and Affiliations

  • K. Melnikov
    • 1
  • R. Rietkerk
    • 1
    Email author
  • L. Tancredi
    • 2
  • C. Wever
    • 3
  1. 1.Institute for Theoretical Particle PhysicsKITKarlsruheGermany
  2. 2.Theoretical Physics Department, CERNGeneva 23Switzerland
  3. 3.Physik-Department T31Technische Universität MünchenGarchingGermany

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