Journal of High Energy Physics

, 2018:118 | Cite as

Momentum conservation and unitarity in parton showers and NLL resummation

  • Stefan Höche
  • Daniel Reichelt
  • Frank Siegert
Open Access
Regular Article - Theoretical Physics


We present a systematic study of differences between NLL resummation and parton showers. We first construct a Markovian Monte-Carlo algorithm for resummation of additive observables in electron-positron annihilation. Approximations intrinsic to the pure NLL result are then removed, in order to obtain a traditional, momentum and probability conserving parton shower based on the coherent branching formalism. The impact of each approximation is studied, and an overall comparison is made between the parton shower and pure NLL resummation. Differences compared to modern parton-shower algorithms formulated in terms of color dipoles are analyzed.


QCD Phenomenology 


Open Access

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  1. [1]
    Y.L. Dokshitzer, D. Diakonov and S.I. Troian, Hard Processes in Quantum Chromodynamics, Phys. Rept. 58 (1980) 269 [INSPIRE].ADSCrossRefGoogle Scholar
  2. [2]
    G. Parisi and R. Petronzio, Small Transverse Momentum Distributions in Hard Processes, Nucl. Phys. B 154 (1979) 427 [INSPIRE].ADSCrossRefGoogle Scholar
  3. [3]
    J.C. Collins and D.E. Soper, Back-To-Back Jets in QCD, Nucl. Phys. B 193 (1981) 381 [Erratum ibid. B 213 (1983) 545] [INSPIRE].
  4. [4]
    G.F. Sterman, Summation of Large Corrections to Short Distance Hadronic Cross-Sections, Nucl. Phys. B 281 (1987) 310 [INSPIRE].
  5. [5]
    S. Catani and L. Trentadue, Resummation of the QCD Perturbative Series for Hard Processes, Nucl. Phys. B 327 (1989) 323 [INSPIRE].
  6. [6]
    S. Catani and L. Trentadue, Comment on QCD exponentiation at large x, Nucl. Phys. B 353 (1991) 183 [INSPIRE].
  7. [7]
    S. Catani, L. Trentadue, G. Turnock and B.R. Webber, Resummation of large logarithms in e + e event shape distributions, Nucl. Phys. B 407 (1993) 3 [INSPIRE].
  8. [8]
    S. Catani et al., QCD, in proceedings of the CERN Workshop on Standard Model Physics (and more) at the LHC (Final Plenary Meeting), Geneva, Switzerland, 14-15 October 1999, CERN-TH-2000-131 [hep-ph/0005025] [INSPIRE].
  9. [9]
    G. Dissertori et al., Determination of the strong coupling constant using matched NNLO + NLLA predictions for hadronic event shapes in e + e annihilations, JHEP 08 (2009) 036 [arXiv:0906.3436] [INSPIRE].
  10. [10]
    T. Gehrmann, M. Jaquier and G. Luisoni, Hadronization effects in event shape moments, Eur. Phys. J. C 67 (2010) 57 [arXiv:0911.2422] [INSPIRE].ADSCrossRefGoogle Scholar
  11. [11]
    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].
  12. [12]
    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].
  13. [13]
    N. Kidonakis and G.F. Sterman, Resummation for QCD hard scattering, Nucl. Phys. B 505 (1997) 321 [hep-ph/9705234] [INSPIRE].
  14. [14]
    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].
  15. [15]
    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].
  16. [16]
    R. Bonciani, S. Catani, M.L. Mangano and P. Nason, Sudakov resummation of multiparton QCD cross-sections, Phys. Lett. B 575 (2003) 268 [hep-ph/0307035] [INSPIRE].
  17. [17]
    A. Banfi, G.P. Salam and G. Zanderighi, Semi-numerical resummation of event shapes, JHEP 01 (2002) 018 [hep-ph/0112156] [INSPIRE].
  18. [18]
    A. Banfi, G.P. Salam and G. Zanderighi, Principles of general final-state resummation and automated implementation, JHEP 03 (2005) 073 [hep-ph/0407286] [INSPIRE].
  19. [19]
    A. Banfi, H. McAslan, P.F. Monni and G. Zanderighi, A general method for the resummation of event-shape distributions in e + e annihilation, JHEP 05 (2015) 102 [arXiv:1412.2126] [INSPIRE].
  20. [20]
    E. Gerwick, S. Höche, S. Marzani and S. Schumann, Soft evolution of multi-jet final states, JHEP 02 (2015) 106 [arXiv:1411.7325] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  21. [21]
    G.C. Fox and S. Wolfram, A Model for Parton Showers in QCD, Nucl. Phys. B 168 (1980) 285 [INSPIRE].
  22. [22]
    G. Marchesini and B.R. Webber, Simulation of QCD Jets Including Soft Gluon Interference, Nucl. Phys. B 238 (1984) 1 [INSPIRE].
  23. [23]
    T. Sjöstrand, A Model for Initial State Parton Showers, Phys. Lett. B 157 (1985) 321 [INSPIRE].ADSCrossRefGoogle Scholar
  24. [24]
    G. Marchesini and B.R. Webber, Monte Carlo Simulation of General Hard Processes with Coherent QCD Radiation, Nucl. Phys. B 310 (1988) 461 [INSPIRE].
  25. [25]
    S. Catani, B.R. Webber and G. Marchesini, QCD coherent branching and semiinclusive processes at large x, Nucl. Phys. B 349 (1991) 635 [INSPIRE].
  26. [26]
    V.N. Gribov and L.N. Lipatov, Deep inelastic ep scattering in perturbation theory, Sov. J. Nucl. Phys. 15 (1972) 438 [INSPIRE].Google Scholar
  27. [27]
    L.N. Lipatov, The parton model and perturbation theory, Sov. J. Nucl. Phys. 20 (1975) 94 [INSPIRE].Google Scholar
  28. [28]
    Y.L. Dokshitzer, Calculation of the Structure Functions for Deep Inelastic Scattering and e + e Annihilation by Perturbation Theory in Quantum Chromodynamics, Sov. Phys. JETP 46 (1977) 641 [INSPIRE].
  29. [29]
    G. Altarelli and G. Parisi, Asymptotic Freedom in Parton Language, Nucl. Phys. B 126 (1977) 298 [INSPIRE].ADSCrossRefGoogle Scholar
  30. [30]
    D. Amati, A. Bassetto, M. Ciafaloni, G. Marchesini and G. Veneziano, A Treatment of Hard Processes Sensitive to the Infrared Structure of QCD, Nucl. Phys. B 173 (1980) 429 [INSPIRE].
  31. [31]
    J. Botts and G.F. Sterman, Hard Elastic Scattering in QCD: Leading Behavior, Nucl. Phys. B 325 (1989) 62 [INSPIRE].
  32. [32]
    N. Kidonakis and G.F. Sterman, Subleading logarithms in QCD hard scattering, Phys. Lett. B 387 (1996) 867 [INSPIRE].ADSCrossRefGoogle Scholar
  33. [33]
    N. Kidonakis, G. Oderda and G.F. Sterman, Threshold resummation for dijet cross-sections, Nucl. Phys. B 525 (1998) 299 [hep-ph/9801268] [INSPIRE].
  34. [34]
    G. Oderda, Dijet rapidity gaps in photoproduction from perturbative QCD, Phys. Rev. D 61 (2000) 014004 [hep-ph/9903240] [INSPIRE].
  35. [35]
    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].
  36. [36]
    M. Dasgupta and G.P. Salam, Resummation of nonglobal QCD observables, Phys. Lett. B 512 (2001) 323 [hep-ph/0104277] [INSPIRE].
  37. [37]
    S. Höche, S. Schumann and F. Siegert, Hard photon production and matrix-element parton-shower merging, Phys. Rev. D 81 (2010) 034026 [arXiv:0912.3501] [INSPIRE].
  38. [38]
    L. Lönnblad, Fooling Around with the Sudakov Veto Algorithm, Eur. Phys. J. C 73 (2013) 2350 [arXiv:1211.7204] [INSPIRE].
  39. [39]
    E. Farhi, A QCD Test for Jets, Phys. Rev. Lett. 39 (1977) 1587 [INSPIRE].ADSCrossRefGoogle Scholar
  40. [40]
    C.F. Berger, T. Kucs and G.F. Sterman, Interjet energy flow/event shape correlations, Int. J. Mod. Phys. A 18 (2003) 4159 [hep-ph/0212343] [INSPIRE].
  41. [41]
    C.F. Berger, T. Kucs and G.F. Sterman, Event shape/energy flow correlations, Phys. Rev. D 68 (2003) 014012 [hep-ph/0303051] [INSPIRE].
  42. [42]
    S. Höche, Introduction to parton-shower event generators, in proceedings of the Theoretical Advanced Study Institute in Elementary Particle Physics: Journeys Through the Precision Frontier: Amplitudes for Colliders. (TASI 2014), Boulder, Colorado, 2-27 June 2014, World Scientific (2015), p. 235 [arXiv:1411.4085] [INSPIRE].
  43. [43]
    S. Catani and M.H. Seymour, A General algorithm for calculating jet cross-sections in NLO QCD, Nucl. Phys. B 485 (1997) 291 [Erratum ibid. B 510 (1998) 503] [hep-ph/9605323] [INSPIRE].
  44. [44]
    S. Höche and S. Prestel, The midpoint between dipole and parton showers, Eur. Phys. J. C 75 (2015) 461 [arXiv:1506.05057] [INSPIRE].
  45. [45]
    R.K. Ellis, D.A. Ross and A.E. Terrano, The Perturbative Calculation of Jet Structure in e + e Annihilation, Nucl. Phys. B 178 (1981) 421 [INSPIRE].
  46. [46]
    S. Höche and S. Prestel, Triple collinear emissions in parton showers, Phys. Rev. D 96 (2017) 074017 [arXiv:1705.00742] [INSPIRE].
  47. [47]
    S. Jadach and M. Skrzypek, Exact solutions of the QCD evolution equations using Monte Carlo method, Acta Phys. Polon. B 35 (2004) 745 [hep-ph/0312355] [INSPIRE].
  48. [48]
    M. Procura and I.W. Stewart, Quark Fragmentation within an Identified Jet, Phys. Rev. D 81 (2010) 074009 [Erratum ibid. D 83 (2011) 039902] [arXiv:0911.4980] [INSPIRE].
  49. [49]
    A. Jain, M. Procura and W.J. Waalewijn, Parton Fragmentation within an Identified Jet at NNLL, JHEP 05 (2011) 035 [arXiv:1101.4953] [INSPIRE].ADSCrossRefMATHGoogle Scholar

Copyright information

© The Author(s) 2018

Authors and Affiliations

  1. 1.SLAC National Accelerator LaboratoryMenlo ParkU.S.A.
  2. 2.Institut für Kern- und TeilchenphysikTechnische Universität DresdenDresdenGermany

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