Jet shapes with the broadening axis

  • Andrew J. LarkoskiEmail author
  • Duff Neill
  • Jesse Thaler
Open Access


Broadening is a classic jet observable that probes the transverse momentum structure of jets. Traditionally, broadening has been measured with respect to the thrust axis, which is aligned along the (hemisphere) jet momentum to minimize the vector sum of transverse momentum within a jet. In this paper, we advocate measuring broadening with respect to the “broadening axis”, which is the direction that minimizes the scalar sum of transverse momentum within a jet. This approach eliminates many of the calculational complexities arising from recoil of the leading parton, and observables like the jet angularities become recoil-free when measured using the broadening axis. We derive a simple factorization theorem for broadening-axis observables which smoothly interpolates between the thrust-like and broadening-like regimes. We argue that the same factorization theorem holds for two-point energy correlation functions as well as for jet shapes based on a “winner-take-all axis”. Using kinked broadening axes, we calculate event-wide angularities in e + e collisions with next-to-leading logarithmic resummation. Defining jet regions using the broadening axis, we also calculate the global logarithms for angularities within a single jet. We find good agreement comparing our calculations both to showering Monte Carlo programs and to automated resummation tools. We give a brief historical perspective on the broadening axis and suggest ways that broadening-axis observables could be used in future jet substructure studies at the Large Hadron Collider.


Jets NLO Computations 


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.


  1. [1]
    E. Farhi, A QCD Test for Jets, Phys. Rev. Lett. 39 (1977) 1587 [INSPIRE].ADSCrossRefGoogle Scholar
  2. [2]
    M. Dasgupta and G.P. Salam, Event shapes in e + e annihilation and deep inelastic scattering, J. Phys. G 30 (2004) R143 [hep-ph/0312283] [INSPIRE].ADSCrossRefGoogle Scholar
  3. [3]
    ALEPH collaboration, A. Heister et al., Studies of QCD at e + e centre-of-mass energies between 91-GeV and 209-GeV, Eur. Phys. J. C 35 (2004) 457 [INSPIRE].ADSGoogle Scholar
  4. [4]
    DELPHI collaboration, J. Abdallah et al., A study of the energy evolution of event shape distributions and their means with the DELPHI detector at LEP, Eur. Phys. J. C 29 (2003) 285 [hep-ex/0307048] [INSPIRE].ADSGoogle Scholar
  5. [5]
    L3 collaboration, P. Achard et al., Studies of hadronic event structure in e + e annihilation from 30-GeV to 209-GeV with the L3 detector, Phys. Rept. 399 (2004) 71 [hep-ex/0406049] [INSPIRE].ADSCrossRefGoogle Scholar
  6. [6]
    OPAL collaboration, G. Abbiendi et al., Measurement of event shape distributions and moments in e + e hadrons at 91-209 GeV and a determination of αs, Eur. Phys. J. C 40 (2005) 287 [hep-ex/0503051] [INSPIRE].ADSGoogle Scholar
  7. [7]
    T. Becher and M.D. Schwartz, A precise determination of αs from LEP thrust data using effective field theory, JHEP 07 (2008) 034 [arXiv:0803.0342] [INSPIRE].ADSCrossRefGoogle Scholar
  8. [8]
    R.A. Davison and B.R. Webber, Non-Perturbative Contribution to the Thrust Distribution in e + e Annihilation, Eur. Phys. J. C 59 (2009) 13 [arXiv:0809.3326] [INSPIRE].ADSCrossRefGoogle Scholar
  9. [9]
    R. Abbate, M. Fickinger, A.H. Hoang, V. Mateu and I.W. Stewart, Thrust at N 3 LL with Power Corrections and a Precision Global Fit for αs(m Z), Phys. Rev. D 83 (2011) 074021 [arXiv:1006.3080] [INSPIRE].ADSGoogle Scholar
  10. [10]
    J. Gallicchio and M.D. Schwartz, Quark and Gluon Tagging at the LHC, Phys. Rev. Lett. 107 (2011) 172001 [arXiv:1106.3076] [INSPIRE].ADSCrossRefGoogle Scholar
  11. [11]
    J. Gallicchio and M.D. Schwartz, Quark and Gluon Jet Substructure, JHEP 04 (2013) 090 [arXiv:1211.7038] [INSPIRE].ADSCrossRefGoogle Scholar
  12. [12]
    A.J. Larkoski, G.P. Salam and J. Thaler, Energy Correlation Functions for Jet Substructure, JHEP 06 (2013) 108 [arXiv:1305.0007] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  13. [13]
    L.G. Almeida et al., Substructure of high-p T Jets at the LHC, Phys. Rev. D 79 (2009) 074017 [arXiv:0807.0234] [INSPIRE].ADSMathSciNetGoogle Scholar
  14. [14]
    S.D. Ellis, C.K. Vermilion, J.R. Walsh, A. Hornig and C. Lee, Jet Shapes and Jet Algorithms in SCET, JHEP 11 (2010) 101 [arXiv:1001.0014] [INSPIRE].ADSCrossRefGoogle Scholar
  15. [15]
    A. Abdesselam et al., Boosted objects: A probe of beyond the Standard Model physics, Eur. Phys. J. C 71 (2011) 1661 [arXiv:1012.5412] [INSPIRE].ADSCrossRefGoogle Scholar
  16. [16]
    A. Altheimer et al., Jet Substructure at the Tevatron and LHC: New results, new tools, new benchmarks, J. Phys. G 39 (2012) 063001 [arXiv:1201.0008] [INSPIRE].ADSCrossRefGoogle Scholar
  17. [17]
    A. Altheimer et al., Boosted objects and jet substructure at the LHC, arXiv:1311.2708 [INSPIRE].
  18. [18]
    P.E.L. Rakow and B.R. Webber, Transverse Momentum Moments of Hadron Distributions in QCD Jets, Nucl. Phys. B 191 (1981) 63 [INSPIRE].ADSCrossRefGoogle Scholar
  19. [19]
    R.K. Ellis and B. Webber, QCD jet broadening in hadron-hadron collisions, Conf. Proc. C860623 (1986) 74 [FERMILAB-CONF-86-151] [INSPIRE].
  20. [20]
    S. Catani, G. Turnock and B.R. Webber, Jet broadening measures in e + e annihilation, Phys. Lett. B 295 (1992) 269 [INSPIRE].ADSCrossRefGoogle Scholar
  21. [21]
    J. Gallicchio et al., Multivariate discrimination and the Higgs + W/Z search, JHEP 04 (2011) 069 [arXiv:1010.3698] [INSPIRE].ADSCrossRefGoogle Scholar
  22. [22]
    C.F. Berger, T. Kucs and G.F. Sterman, Event shape/energy flow correlations, Phys. Rev. D 68 (2003) 014012 [hep-ph/0303051] [INSPIRE].ADSGoogle Scholar
  23. [23]
    T. Becher and G. Bell, NNLL Resummation for Jet Broadening, JHEP 11 (2012) 126 [arXiv:1210.0580] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  24. [24]
    Y.L. Dokshitzer, G. Marchesini and G.P. Salam, Revisiting nonperturbative effects in the jet broadenings, Eur. Phys. J. direct C 1 (1999) 3 [hep-ph/9812487] [INSPIRE].Google Scholar
  25. [25]
    G.P. Salam and D. Wicke, Hadron masses and power corrections to event shapes, JHEP 05 (2001) 061 [hep-ph/0102343] [INSPIRE].ADSCrossRefGoogle Scholar
  26. [26]
    T. Becher and G. Bell, Enhanced non-perturbative effects through the collinear anomaly, arXiv:1312.5327 [INSPIRE].
  27. [27]
    T. Becher, G. Bell and M. Neubert, Factorization and Resummation for Jet Broadening, Phys. Lett. B 704 (2011) 276 [arXiv:1104.4108] [INSPIRE].ADSCrossRefGoogle Scholar
  28. [28]
    J.-Y. Chiu, A. Jain, D. Neill and I.Z. Rothstein, A Formalism for the Systematic Treatment of Rapidity Logarithms in Quantum Field Theory, JHEP 05 (2012) 084 [arXiv:1202.0814] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  29. [29]
    H. Georgi and M. Machacek, A Simple QCD Prediction of Jet Structure in e + e Annihilation, Phys. Rev. Lett. 39 (1977) 1237 [INSPIRE].ADSCrossRefGoogle Scholar
  30. [30]
    S. Brandt and H. Dahmen, Axes and Scalar Measures of Two-Jet and Three-Jet Events, Z. Physik C1 (1979) 61.ADSGoogle Scholar
  31. [31]
    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
  32. [32]
    J. Thaler and K. Van Tilburg, Identifying Boosted Objects with N-subjettiness, JHEP 03 (2011) 015 [arXiv:1011.2268] [INSPIRE].ADSCrossRefGoogle Scholar
  33. [33]
    J. Thaler and K. Van Tilburg, Maximizing Boosted Top Identification by Minimizing N-subjettiness, JHEP 02 (2012) 093 [arXiv:1108.2701] [INSPIRE].ADSCrossRefGoogle Scholar
  34. [34]
    A.J. Larkoski and J. Thaler, Unsafe but Calculable: Ratios of Angularities in Perturbative QCD, JHEP 09 (2013) 137 [arXiv:1307.1699] [INSPIRE].ADSCrossRefGoogle Scholar
  35. [35]
    V. Mateu, I.W. Stewart and J. Thaler, Power Corrections to Event Shapes with Mass-Dependent Operators, Phys. Rev. D 87 (2013) 014025 [arXiv:1209.3781] [INSPIRE].ADSGoogle Scholar
  36. [36]
    C.W. Bauer, S. Fleming and M.E. Luke, Summing Sudakov logarithms in \( \overrightarrow{B} \) Xsγ in effective field theory, Phys. Rev. D 63 (2000) 014006 [hep-ph/0005275] [INSPIRE].ADSGoogle Scholar
  37. [37]
    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].ADSGoogle Scholar
  38. [38]
    C.W. Bauer and I.W. Stewart, Invariant operators in collinear effective theory, Phys. Lett. B 516 (2001) 134 [hep-ph/0107001] [INSPIRE].ADSCrossRefGoogle Scholar
  39. [39]
    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].ADSGoogle Scholar
  40. [40]
    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].ADSGoogle Scholar
  41. [41]
    A. Hornig, C. Lee and G. Ovanesyan, Effective Predictions of Event Shapes: Factorized, Resummed and Gapped Angularity Distributions, JHEP 05 (2009) 122 [arXiv:0901.3780] [INSPIRE].ADSCrossRefGoogle Scholar
  42. [42]
    M. Dasgupta and G.P. Salam, Resummation of nonglobal QCD observables, Phys. Lett. B 512 (2001) 323 [hep-ph/0104277] [INSPIRE].ADSGoogle Scholar
  43. [43]
    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].ADSCrossRefGoogle Scholar
  44. [44]
    M. Jankowiak and A.J. Larkoski, Jet Substructure Without Trees, JHEP 06 (2011) 057 [arXiv:1104.1646] [INSPIRE].ADSCrossRefGoogle Scholar
  45. [45]
    A.J. Larkoski, QCD Analysis of the Scale-Invariance of Jets, Phys. Rev. D 86 (2012) 054004 [arXiv:1207.1437] [INSPIRE].ADSGoogle Scholar
  46. [46]
    D. Bertolini, T. Chan and J. Thaler, Jet Observables Without Jet Algorithms, arXiv:1310.7584 [INSPIRE].
  47. [47]
    Y.L. Dokshitzer, A. Lucenti, G. Marchesini and G.P. Salam, On the QCD analysis of jet broadening, JHEP 01 (1998) 011 [hep-ph/9801324] [INSPIRE].ADSCrossRefGoogle Scholar
  48. [48]
    C.W. Bauer, S.P. Fleming, C. Lee and G.F. Sterman, Factorization of e + e Event Shape Distributions with Hadronic Final States in Soft Collinear Effective Theory, Phys. Rev. D 78 (2008) 034027 [arXiv:0801.4569] [INSPIRE].ADSGoogle Scholar
  49. [49]
    G.C. Blazey et al., Run II jet physics, hep-ex/0005012 [INSPIRE].
  50. [50]
    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
  51. [51]
    G.P. Salam and G. Soyez, A Practical Seedless Infrared-Safe Cone jet algorithm, JHEP 05 (2007) 086 [arXiv:0704.0292] [INSPIRE].ADSCrossRefGoogle Scholar
  52. [52]
    S. Catani, Y.L. Dokshitzer, M.H. Seymour and B.R. Webber, Longitudinally invariant K t clustering algorithms for hadron hadron collisions, Nucl. Phys. B 406 (1993) 187 [INSPIRE].ADSCrossRefGoogle Scholar
  53. [53]
    J.M. Butterworth, J.P. Couchman, B.E. Cox and B.M. Waugh, KtJet: A C++ implementation of the K-perpendicular clustering algorithm, Comput. Phys. Commun. 153 (2003) 85 [hep-ph/0210022] [INSPIRE].ADSCrossRefGoogle Scholar
  54. [54]
    J. Chay and C. Kim, Collinear effective theory at subleading order and its application to heavy - light currents, Phys. Rev. D 65 (2002) 114016 [hep-ph/0201197] [INSPIRE].ADSGoogle Scholar
  55. [55]
    A.V. Manohar, T. Mehen, D. Pirjol and I.W. Stewart, Reparameterization invariance for collinear operators, Phys. Lett. B 539 (2002) 59 [hep-ph/0204229] [INSPIRE].ADSCrossRefGoogle Scholar
  56. [56]
    C. Lee and G.F. Sterman, Momentum Flow Correlations from Event Shapes: Factorized Soft Gluons and Soft-Collinear Effective Theory, Phys. Rev. D 75 (2007) 014022 [hep-ph/0611061] [INSPIRE].ADSGoogle Scholar
  57. [57]
    B. Grinstein and I.Z. Rothstein, Effective field theory and matching in nonrelativistic gauge theories, Phys. Rev. D 57 (1998) 78 [hep-ph/9703298] [INSPIRE].ADSGoogle Scholar
  58. [58]
    M. Beneke and T. Feldmann, Multipole expanded soft collinear effective theory with nonAbelian gauge symmetry, Phys. Lett. B 553 (2003) 267 [hep-ph/0211358] [INSPIRE].ADSCrossRefGoogle Scholar
  59. [59]
    M.D. Schwartz, Resummation and NLO matching of event shapes with effective field theory, Phys. Rev. D 77 (2008) 014026 [arXiv:0709.2709] [INSPIRE].ADSGoogle Scholar
  60. [60]
    S. Fleming, A.H. Hoang, S. Mantry and I.W. Stewart, Jets from massive unstable particles: Top-mass determination, Phys. Rev. D 77 (2008) 074010 [hep-ph/0703207] [INSPIRE].ADSGoogle Scholar
  61. [61]
    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].ADSCrossRefGoogle Scholar
  62. [62]
    S.D. Ellis and D.E. Soper, Successive combination jet algorithm for hadron collisions, Phys. Rev. D 48 (1993) 3160 [hep-ph/9305266] [INSPIRE].ADSGoogle Scholar
  63. [63]
    Y.L. Dokshitzer, G.D. Leder, S. Moretti and B.R. Webber, Better jet clustering algorithms, JHEP 08 (1997) 001 [hep-ph/9707323] [INSPIRE].ADSCrossRefGoogle Scholar
  64. [64]
    M. Wobisch and T. Wengler, Hadronization corrections to jet cross-sections in deep inelastic scattering, hep-ph/9907280 [INSPIRE].
  65. [65]
    M. Wobisch, Measurement and QCD analysis of jet cross-sections in deep inelastic positron proton collisions at \( \sqrt{s} \) = 300 GeV, Ph.D. Thesis, Aachen TU, Aachen, Germany (2000) DESY-THESIS-2000-049.Google Scholar
  66. [66]
    A. Banfi and M. Dasgupta, Problems in resumming interjet energy flows with k t clustering, Phys. Lett. B 628 (2005) 49 [hep-ph/0508159] [INSPIRE].ADSCrossRefGoogle Scholar
  67. [67]
    T. Sjöstrand, S. Mrenna and P.Z. Skands, PYTHIA 6.4 Physics and Manual, JHEP 05 (2006) 026 [hep-ph/0603175] [INSPIRE].ADSCrossRefGoogle Scholar
  68. [68]
    T. Sjöstrand, S. Mrenna and P.Z. Skands, A Brief Introduction to PYTHIA 8.1, Comput. Phys. Commun. 178 (2008) 852 [arXiv:0710.3820] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  69. [69]
    M. Bengtsson and T. Sjöstrand, A Comparative Study of Coherent and Noncoherent Parton Shower Evolution, Nucl. Phys. B 289 (1987) 810 [INSPIRE].ADSCrossRefGoogle Scholar
  70. [70]
    J.D. Bjorken and S.J. Brodsky, Statistical Model for electron-Positron Annihilation Into Hadrons, Phys. Rev. D 1 (1970) 1416 [INSPIRE].ADSGoogle Scholar
  71. [71]
    J.R. Ellis, M.K. Gaillard and G.G. Ross, Search for Gluons in e + e Annihilation, Nucl. Phys. B 111 (1976) 253 [Erratum ibid. B 130 (1977) 516] [INSPIRE].ADSCrossRefGoogle Scholar
  72. [72]
    N. Schmitz, Recent Results on the Hadronic Final State in Charged Current Neutrino and Antineutrino Reactions, eConf C790823 (1979) 359 [INSPIRE].
  73. [73]
    W.T. Ford et al., Measurement of α s from hadron jets in e + e annihilation at \( \sqrt{s} \) of 29 GeV, Phys. Rev. D 40 (1989) 1385 [INSPIRE].ADSGoogle Scholar
  74. [74]
    L3 collaboration, P. Achard et al., Measurement of the cross section of W-boson pair production at LEP, Phys. Lett. B 600 (2004) 22 [hep-ex/0409016] [INSPIRE].ADSCrossRefGoogle Scholar
  75. [75]
    S. Catani, G. Turnock, B.R. Webber and L. Trentadue, Thrust distribution in e + e annihilation, Phys. Lett. B 263 (1991) 491 [INSPIRE].ADSCrossRefGoogle Scholar
  76. [76]
    S. Catani, G. Turnock and B.R. Webber, Heavy jet mass distribution in e + e annihilation, Phys. Lett. B 272 (1991) 368 [INSPIRE].ADSCrossRefGoogle Scholar
  77. [77]
    M. Freytsis, T. Volansky and J. Walsh, Jet Substructure with Missing Decays, unpublished.Google Scholar
  78. [78]
    D. Curtin, R. Essig and B. Shuve, Boosted Multijet Resonances and New Color-Flow Variables, Phys. Rev. D 88 (2013) 034019 [arXiv:1210.5523] [INSPIRE].ADSGoogle Scholar
  79. [79]
    CMS collaboration, Pileup Jet Identification, CMS-PAS-JME-13-005.
  80. [80]
    G.P. Korchemsky and G.F. Sterman, Power corrections to event shapes and factorization, Nucl. Phys. B 555 (1999) 335 [hep-ph/9902341] [INSPIRE].ADSCrossRefGoogle Scholar
  81. [81]
    G.P. Korchemsky and S. Tafat, On power corrections to the event shape distributions in QCD, JHEP 10 (2000) 010 [hep-ph/0007005] [INSPIRE].ADSCrossRefGoogle Scholar
  82. [82]
    Y.L. Dokshitzer, A. Lucenti, G. Marchesini and G.P. Salam, On the universality of the Milan factor for 1/Q power corrections to jet shapes, JHEP 05 (1998) 003 [hep-ph/9802381] [INSPIRE].ADSCrossRefGoogle Scholar
  83. [83]
    Z. Ligeti, I.W. Stewart and F.J. Tackmann, Treating the b quark distribution function with reliable uncertainties, Phys. Rev. D 78 (2008) 114014 [arXiv:0807.1926] [INSPIRE].ADSGoogle Scholar

Copyright information

© The Author(s) 2014

Authors and Affiliations

  1. 1.Center for Theoretical Physics, Massachusetts Institute of TechnologyCambridgeU.S.A

Personalised recommendations