Advertisement

Summing threshold logs in a parton shower

  • Zoltán Nagy
  • Davison E. Soper
Open Access
Regular Article - Theoretical Physics

Abstract

When parton distributions are falling steeply as the momentum fractions of the partons increases, there are effects that occur at each order in α s that combine to affect hard scattering cross sections and need to be summed. We show how to accomplish this in a leading approximation in the context of a parton shower Monte Carlo event generator.

Keywords

Perturbative QCD Resummation 

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.

References

  1. [1]
    G.F. Sterman, Summation of Large Corrections to Short Distance Hadronic Cross-Sections, Nucl. Phys. B 281 (1987) 310 [INSPIRE].ADSCrossRefGoogle Scholar
  2. [2]
    D. Appell, G.F. Sterman and P.B. Mackenzie, Soft Gluons and the Normalization of the Drell-Yan Cross-section, Nucl. Phys. B 309 (1988) 259 [INSPIRE].ADSCrossRefGoogle Scholar
  3. [3]
    S. Catani and L. Trentadue, Resummation of the QCD Perturbative Series for Hard Processes, Nucl. Phys. B 327 (1989) 323 [INSPIRE].ADSCrossRefGoogle Scholar
  4. [4]
    S. Catani, B.R. Webber and G. Marchesini, QCD coherent branching and semiinclusive processes at large x, Nucl. Phys. B 349 (1991) 635 [INSPIRE].ADSCrossRefGoogle Scholar
  5. [5]
    L. Magnea, All Order Summation and Two Loop Results for the Drell-Yan Cross-section, Nucl. Phys. B 349 (1991) 703 [INSPIRE].ADSCrossRefGoogle Scholar
  6. [6]
    G.P. Korchemsky and G. Marchesini, Resummation of large infrared corrections using Wilson loops, Phys. Lett. B 313 (1993) 433 [INSPIRE].ADSCrossRefGoogle Scholar
  7. [7]
    S. Catani, M.L. Mangano, P. Nason and L. Trentadue, The resummation of soft gluons in hadronic collisions, Nucl. Phys. B 478 (1996) 273 [hep-ph/9604351] [INSPIRE].
  8. [8]
    H. Contopanagos, E. Laenen and G.F. Sterman, Sudakov factorization and resummation, Nucl. Phys. B 484 (1997) 303 [hep-ph/9604313] [INSPIRE].
  9. [9]
    N. Kidonakis and G.F. Sterman, Resummation for QCD hard scattering, Nucl. Phys. B 505 (1997) 321 [hep-ph/9705234] [INSPIRE].
  10. [10]
    N. Kidonakis, G. Oderda and G.F. Sterman, Threshold resummation for dijet cross-sections, Nucl. Phys. B 525 (1998) 299 [hep-ph/9801268] [INSPIRE].
  11. [11]
    N. Kidonakis, G. Oderda and G.F. Sterman, Evolution of color exchange in QCD hard scattering, Nucl. Phys. B 531 (1998) 365 [hep-ph/9803241] [INSPIRE].
  12. [12]
    E. Laenen, G. Oderda and G.F. Sterman, Resummation of threshold corrections for single particle inclusive cross-sections, Phys. Lett. B 438 (1998) 173 [hep-ph/9806467] [INSPIRE].
  13. [13]
    S. Catani, M.L. Mangano and P. Nason, Sudakov resummation for prompt photon production in hadron collisions, JHEP 07 (1998) 024 [hep-ph/9806484] [INSPIRE].
  14. [14]
    H.-n. Li, Unification of the k T and threshold resummations, Phys. Lett. B 454 (1999) 328 [hep-ph/9812363] [INSPIRE].
  15. [15]
    N. Kidonakis and J.F. Owens, Soft gluon resummation and NNLO corrections for direct photon production, Phys. Rev. D 61 (2000) 094004 [hep-ph/9912388] [INSPIRE].
  16. [16]
    G.F. Sterman and W. Vogelsang, Threshold resummation and rapidity dependence, JHEP 02 (2001) 016 [hep-ph/0011289] [INSPIRE].
  17. [17]
    N. Kidonakis and J.F. Owens, Effects of higher order threshold corrections in high E T jet production, Phys. Rev. D 63 (2001) 054019 [hep-ph/0007268] [INSPIRE].
  18. [18]
    E. Laenen, G.F. Sterman and W. Vogelsang, Recoil and threshold corrections in short distance cross-sections, Phys. Rev. D 63 (2001) 114018 [hep-ph/0010080] [INSPIRE].
  19. [19]
    A. Kulesza, G.F. Sterman and W. Vogelsang, Joint resummation in electroweak boson production, Phys. Rev. D 66 (2002) 014011 [hep-ph/0202251] [INSPIRE].
  20. [20]
    N. Kidonakis, Next-to-next-to-next-to-leading-order soft-gluon corrections in hard-scattering processes near threshold, Phys. Rev. D 73 (2006) 034001 [hep-ph/0509079] [INSPIRE].
  21. [21]
    A. Mukherjee and W. Vogelsang, Threshold resummation for W-boson production at RHIC, Phys. Rev. D 73 (2006) 074005 [hep-ph/0601162] [INSPIRE].
  22. [22]
    P. Bolzoni, Threshold resummation of Drell-Yan rapidity distributions, Phys. Lett. B 643 (2006) 325 [hep-ph/0609073] [INSPIRE].
  23. [23]
    D. de Florian and W. Vogelsang, Resummed cross-section for jet production at hadron colliders, Phys. Rev. D 76 (2007) 074031 [arXiv:0704.1677] [INSPIRE].ADSGoogle Scholar
  24. [24]
    V. Ravindran, J. Smith and W.L. van Neerven, QCD threshold corrections to di-lepton and Higgs rapidity distributions beyond N 2 LO, Nucl. Phys. B 767 (2007) 100 [hep-ph/0608308] [INSPIRE].
  25. [25]
    V. Ravindran and J. Smith, Threshold corrections to rapidity distributions of Z and W ± bosons beyond N 2 LO at hadron colliders, Phys. Rev. D 76 (2007) 114004 [arXiv:0708.1689] [INSPIRE].ADSGoogle Scholar
  26. [26]
    M. Bonvini, S. Forte and G. Ridolfi, Soft gluon resummation of Drell-Yan rapidity distributions: Theory and phenomenology, Nucl. Phys. B 847 (2011) 93 [arXiv:1009.5691] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  27. [27]
    M. Bonvini, S. Forte and G. Ridolfi, The threshold region for Higgs production in gluon fusion, Phys. Rev. Lett. 109 (2012) 102002 [arXiv:1204.5473] [INSPIRE].ADSCrossRefGoogle Scholar
  28. [28]
    D. de Florian, P. Hinderer, A. Mukherjee, F. Ringer and W. Vogelsang, Approximate next-to-next-to-leading order corrections to hadronic jet production, Phys. Rev. Lett. 112 (2014) 082001 [arXiv:1310.7192] [INSPIRE].ADSCrossRefGoogle Scholar
  29. [29]
    S. Catani, L. Cieri, D. de Florian, G. Ferrera and M. Grazzini, Threshold resummation at N 3 LL accuracy and soft-virtual cross sections at N 3 LO, Nucl. Phys. B 888 (2014) 75 [arXiv:1405.4827] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  30. [30]
    T. Ahmed, M.K. Mandal, N. Rana and V. Ravindran, Rapidity Distributions in Drell-Yan and Higgs Productions at Threshold to Third Order in QCD, Phys. Rev. Lett. 113 (2014) 212003 [arXiv:1404.6504] [INSPIRE].ADSCrossRefGoogle Scholar
  31. [31]
    M. Bonvini and S. Marzani, Resummed Higgs cross section at N 3 LL, JHEP 09 (2014) 007 [arXiv:1405.3654] [INSPIRE].ADSCrossRefGoogle Scholar
  32. [32]
    M. Bonvini et al., Parton distributions with threshold resummation, JHEP 09 (2015) 191 [arXiv:1507.01006] [INSPIRE].ADSCrossRefGoogle Scholar
  33. [33]
    A.V. Manohar, Deep inelastic scattering as x → 1 using soft collinear effective theory, Phys. Rev. D 68 (2003) 114019 [hep-ph/0309176] [INSPIRE].
  34. [34]
    A. Idilbi and X.-d. Ji, Threshold resummation for Drell-Yan process in soft-collinear effective theory, Phys. Rev. D 72 (2005) 054016 [hep-ph/0501006] [INSPIRE].
  35. [35]
    T. Becher and M. Neubert, Threshold resummation in momentum space from effective field theory, Phys. Rev. Lett. 97 (2006) 082001 [hep-ph/0605050] [INSPIRE].
  36. [36]
    T. Becher, M. Neubert and B.D. Pecjak, Factorization and Momentum-Space Resummation in Deep-Inelastic Scattering, JHEP 01 (2007) 076 [hep-ph/0607228] [INSPIRE].
  37. [37]
    T. Becher, M. Neubert and G. Xu, Dynamical Threshold Enhancement and Resummation in Drell-Yan Production, JHEP 07 (2008) 030 [arXiv:0710.0680] [INSPIRE].ADSCrossRefGoogle Scholar
  38. [38]
    T. Becher and M.D. Schwartz, Direct photon production with effective field theory, JHEP 02 (2010) 040 [arXiv:0911.0681] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  39. [39]
    I.W. Stewart, F.J. Tackmann and W.J. Waalewijn, Factorization at the LHC: From PDFs to Initial State Jets, Phys. Rev. D 81 (2010) 094035 [arXiv:0910.0467] [INSPIRE].ADSGoogle Scholar
  40. [40]
    M. Beneke, P. Falgari and C. Schwinn, Threshold resummation for pair production of coloured heavy (s)particles at hadron colliders, Nucl. Phys. B 842 (2011) 414 [arXiv:1007.5414] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  41. [41]
    C.W. Bauer, F.J. Tackmann, J.R. Walsh and S. Zuberi, Factorization and Resummation for Dijet Invariant Mass Spectra, Phys. Rev. D 85 (2012) 074006 [arXiv:1106.6047] [INSPIRE].ADSGoogle Scholar
  42. [42]
    Y. Wang, C.S. Li, Z.L. Liu and D.Y. Shao, Threshold resummation for W ± Z and ZZ pair production at the LHC, Phys. Rev. D 90 (2014) 034008 [arXiv:1406.1417] [INSPIRE].ADSGoogle Scholar
  43. [43]
    Y. Li, D. Neill and H.X. Zhu, An Exponential Regulator for Rapidity Divergences, submitted to Phys. Rev. D (2016) [arXiv:1604.00392] [INSPIRE].
  44. [44]
    G. Lustermans, W.J. Waalewijn and L. Zeune, Joint transverse momentum and threshold resummation beyond NLL, arXiv:1605.02740 [INSPIRE].
  45. [45]
    M. Bonvini, S. Forte, M. Ghezzi and G. Ridolfi, Threshold Resummation in SCET vs. Perturbative QCD: An Analytic Comparison, Nucl. Phys. B 861 (2012) 337 [arXiv:1201.6364] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  46. [46]
    G. Sterman and M. Zeng, Quantifying Comparisons of Threshold Resummations, JHEP 05 (2014) 132 [arXiv:1312.5397] [INSPIRE].ADSCrossRefGoogle Scholar
  47. [47]
    M. Bonvini, S. Forte, G. Ridolfi and L. Rottoli, Resummation prescriptions and ambiguities in SCET vs. direct QCD: Higgs production as a case study, JHEP 01 (2015) 046 [arXiv:1409.0864] [INSPIRE].
  48. [48]
    T. Sjöstrand, A Model for Initial State Parton Showers, Phys. Lett. B 157 (1985) 321 [INSPIRE].ADSCrossRefGoogle Scholar
  49. [49]
    G. Marchesini and B.R. Webber, Simulation of QCD Jets Including Soft Gluon Interference, Nucl. Phys. B 238 (1984) 1 [INSPIRE].ADSCrossRefGoogle Scholar
  50. [50]
    G. Gustafson, Dual Description of a Confined Color Field, Phys. Lett. B 175 (1986) 453 [INSPIRE].ADSCrossRefGoogle Scholar
  51. [51]
    R.K. Ellis, G. Marchesini and B.R. Webber, Soft Radiation in Parton Parton Scattering, Nucl. Phys. B 286 (1987) 643 [Erratum ibid. B 294 (1987) 1180] [INSPIRE].
  52. [52]
    G. Gustafson and U. Pettersson, Dipole Formulation of QCD Cascades, Nucl. Phys. B 306 (1988) 746 [INSPIRE].ADSCrossRefGoogle Scholar
  53. [53]
    G. Marchesini, B.R. Webber, G. Abbiendi, I.G. Knowles, M.H. Seymour and L. Stanco, HERWIG: A Monte Carlo event generator for simulating hadron emission reactions with interfering gluons. Version 5.1 — April 1991, Comput. Phys. Commun. 67 (1992) 465 [INSPIRE].
  54. [54]
    T. Sjöstrand, High-energy physics event generation with PYTHIA 5.7 and JETSET 7.4, Comput. Phys. Commun. 82 (1994) 74 [INSPIRE].ADSCrossRefGoogle Scholar
  55. [55]
    T. Gleisberg, S. Hoeche, F. Krauss, A. Schalicke, S. Schumann and J.-C. Winter, SHERPA 1. alpha: A proof of concept version, JHEP 02 (2004) 056 [hep-ph/0311263] [INSPIRE].
  56. [56]
    T. Sjöstrand and P.Z. Skands, Transverse-momentum-ordered showers and interleaved multiple interactions, Eur. Phys. J. C 39 (2005) 129 [hep-ph/0408302] [INSPIRE].
  57. [57]
    Z. Nagy and D.E. Soper, A new parton shower algorithm: Shower evolution, matching at leading and next-to-leading order level, in New trends in HERA physics: Proceedings of the Ringberg Workshop, Tegernsee, Germany, 2-7 October 2005, G. Grindhammer et al. eds., World Scientific, Hackensack, U.S.A. (2006), hep-ph/0601021 [INSPIRE].
  58. [58]
    M. Dinsdale, M. Ternick and S. Weinzierl, Parton showers from the dipole formalism, Phys. Rev. D 76 (2007) 094003 [arXiv:0709.1026] [INSPIRE].ADSGoogle Scholar
  59. [59]
    S. Schumann and F. Krauss, A parton shower algorithm based on Catani-Seymour dipole factorisation, JHEP 03 (2008) 038 [arXiv:0709.1027] [INSPIRE].ADSCrossRefGoogle Scholar
  60. [60]
    M. Bahr et al., HERWIG++ Physics and Manual, Eur. Phys. J. C 58 (2008) 639 [arXiv:0803.0883] [INSPIRE].ADSCrossRefGoogle Scholar
  61. [61]
    T. Gleisberg et al., Event generation with SHERPA 1.1, JHEP 02 (2009) 007 [arXiv:0811.4622] [INSPIRE].
  62. [62]
    S. Platzer and S. Gieseke, Coherent Parton Showers with Local Recoils, JHEP 01 (2011) 024 [arXiv:0909.5593] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  63. [63]
    S. Platzer and S. Gieseke, Dipole Showers and Automated NLO Matching in HERWIG++, Eur. Phys. J. C 72 (2012) 2187 [arXiv:1109.6256] [INSPIRE].ADSCrossRefGoogle Scholar
  64. [64]
    P.C. Stichel and W.J. Zakrzewski, General relativistic, nonstandard model for the dark sector of the Universe, Eur. Phys. J. C 75 (2015) 9 [arXiv:1409.1363] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  65. [65]
    T. Sjöstrand et al., An introduction to PYTHIA 8.2, Comput. Phys. Commun. 191 (2015) 159 [arXiv:1410.3012] [INSPIRE].
  66. [66]
    Z. Nagy and D.E. Soper, On the transverse momentum in Z-boson production in a virtuality ordered parton shower, JHEP 03 (2010) 097 [arXiv:0912.4534] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  67. [67]
    Z. Nagy and D.E. Soper, Parton showers with quantum interference, JHEP 09 (2007) 114 [arXiv:0706.0017] [INSPIRE].ADSCrossRefGoogle Scholar
  68. [68]
    Z. Nagy and D.E. Soper, Parton showers with quantum interference: Leading color, spin averaged, JHEP 03 (2008) 030 [arXiv:0801.1917] [INSPIRE].ADSCrossRefGoogle Scholar
  69. [69]
    Z. Nagy and D.E. Soper, Parton showers with quantum interference: Leading color, with spin, JHEP 07 (2008) 025 [arXiv:0805.0216] [INSPIRE].ADSCrossRefGoogle Scholar
  70. [70]
    Z. Nagy and D.E. Soper, Parton shower evolution with subleading color, JHEP 06 (2012) 044 [arXiv:1202.4496] [INSPIRE].ADSCrossRefGoogle Scholar
  71. [71]
    Z. Nagy and D.E. Soper, A parton shower based on factorization of the quantum density matrix, JHEP 06 (2014) 097 [arXiv:1401.6364] [INSPIRE].ADSCrossRefGoogle Scholar
  72. [72]
    Z. Nagy and D.E. Soper, Ordering variable for parton showers, JHEP 06 (2014) 178 [arXiv:1401.6366] [INSPIRE].ADSCrossRefGoogle Scholar
  73. [73]
    Z. Nagy and D.E. Soper, Parton distribution functions in the context of parton showers, JHEP 06 (2014) 179 [arXiv:1401.6368] [INSPIRE].ADSCrossRefGoogle Scholar
  74. [74]
    Z. Nagy and D.E. Soper, Effects of subleading color in a parton shower, JHEP 07 (2015) 119 [arXiv:1501.00778] [INSPIRE].ADSCrossRefGoogle Scholar
  75. [75]
    J.C. Collins and D.E. Soper, Parton Distribution and Decay Functions, Nucl. Phys. B 194 (1982) 445 [INSPIRE].ADSGoogle Scholar
  76. [76]
    S. Dulat et al., New parton distribution functions from a global analysis of quantum chromodynamics, Phys. Rev. D 93 (2016) 033006 [arXiv:1506.07443] [INSPIRE].ADSGoogle Scholar
  77. [77]
    J.M. Campbell and R.K. Ellis, Radiative corrections to \( Zb\overline{b} \) production, Phys. Rev. D 62 (2000) 114012 [hep-ph/0006304] [INSPIRE].
  78. [78]
    J.C. Collins, D.E. Soper and G.F. Sterman, Transverse Momentum Distribution in Drell-Yan Pair and W and Z Boson Production, Nucl. Phys. B 250 (1985) 199 [INSPIRE].ADSCrossRefGoogle Scholar
  79. [79]
    G.A. Ladinsky and C.P. Yuan, The Nonperturbative regime in QCD resummation for gauge boson production at hadron colliders, Phys. Rev. D 50 (1994) R4239 [hep-ph/9311341] [INSPIRE].
  80. [80]
    F. Landry, R. Brock, P.M. Nadolsky and C.P. Yuan, Tevatron Run-1 Z boson data and Collins-Soper-Sterman resummation formalism, Phys. Rev. D 67 (2003) 073016 [hep-ph/0212159] [INSPIRE].
  81. [81]
    M. Cacciari, G.P. Salam and G. Soyez, The anti-k t jet clustering algorithm, JHEP 04 (2008) 063 [arXiv:0802.1189] [INSPIRE].ADSCrossRefGoogle Scholar
  82. [82]
    M. Cacciari, G.P. Salam and G. Soyez, FastJet User Manual, Eur. Phys. J. C 72 (2012) 1896 [arXiv:1111.6097] [INSPIRE].ADSCrossRefGoogle Scholar
  83. [83]
    S.D. Ellis, Z. Kunszt and D.E. Soper, The One Jet Inclusive Cross-section at Order α s3 Quarks and Gluons, Phys. Rev. Lett. 64 (1990) 2121 [INSPIRE].ADSCrossRefGoogle Scholar
  84. [84]
    J.R. Forshaw, A. Kyrieleis and M.H. Seymour, Gaps between jets in the high energy limit, JHEP 06 (2005) 034 [hep-ph/0502086] [INSPIRE].
  85. [85]
    M. Krämer and D.E. Soper, Next-to-leading order numerical calculations in Coulomb gauge, Phys. Rev. D 66 (2002) 054017 [hep-ph/0204113] [INSPIRE].

Copyright information

© The Author(s) 2016

Authors and Affiliations

  1. 1.DESYHamburgGermany
  2. 2.Institute of Theoretical ScienceUniversity of OregonEugeneU.S.A.

Personalised recommendations