Energy flow polynomials: a complete linear basis for jet substructure

  • Patrick T. Komiske
  • Eric M. Metodiev
  • Jesse Thaler
Open Access
Regular Article - Theoretical Physics


We introduce the energy flow polynomials: a complete set of jet substructure observables which form a discrete linear basis for all infrared- and collinear-safe observables. Energy flow polynomials are multiparticle energy correlators with specific angular structures that are a direct consequence of infrared and collinear safety. We establish a powerful graph-theoretic representation of the energy flow polynomials which allows us to design efficient algorithms for their computation. Many common jet observables are exact linear combinations of energy flow polynomials, and we demonstrate the linear spanning nature of the energy flow basis by performing regression for several common jet observables. Using linear classification with energy flow polynomials, we achieve excellent performance on three representative jet tagging problems: quark/gluon discrimination, boosted W tagging, and boosted top tagging. The energy flow basis provides a systematic framework for complete investigations of jet substructure using linear methods.


Jets QCD Phenomenology 


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]
    M.H. Seymour, Tagging a heavy Higgs boson, in ECFA Large Hadron Collider Workshop, Aachen Germany, 4-9 October 1990, proceedings 2, (1991), pg. 557 [INSPIRE].
  2. [2]
    M.H. Seymour, Searches for new particles using cone and cluster jet algorithms: a comparative study, Z. Phys. C 62 (1994) 127 [INSPIRE].ADSCrossRefGoogle Scholar
  3. [3]
    J.M. Butterworth, B.E. Cox and J.R. Forshaw, W W scattering at the CERN LHC, Phys. Rev. D 65 (2002) 096014 [hep-ph/0201098] [INSPIRE].
  4. [4]
    J.M. Butterworth, J.R. Ellis and A.R. Raklev, Reconstructing sparticle mass spectra using hadronic decays, JHEP 05 (2007) 033 [hep-ph/0702150] [INSPIRE].
  5. [5]
    J.M. Butterworth, A.R. Davison, M. Rubin and G.P. Salam, Jet substructure as a new Higgs search channel at the LHC, Phys. Rev. Lett. 100 (2008) 242001 [arXiv:0802.2470] [INSPIRE].ADSCrossRefGoogle Scholar
  6. [6]
    CMS collaboration, Search for a Higgs boson in the decay channel \( H\to Z{Z}^{\left(\ast \right)}\to q\overline{q}{\ell}^{-}{\ell}^{+} \) in pp collisions at \( \sqrt{s}=7 \) TeV, JHEP 04 (2012) 036 [arXiv:1202.1416] [INSPIRE].
  7. [7]
    CMS collaboration, Search for a Standard Model-like Higgs boson decaying into \( WW\to \ell \nu q\overline{q} \) in pp collisions at \( \sqrt{s}=8 \) TeV, CMS-PAS-HIG-13-008, CERN, Geneva Switzerland, (2013).
  8. [8]
    ATLAS collaboration, Measurement of colour flow with the jet pull angle in \( t\overline{t} \) events using the ATLAS detector at \( \sqrt{s}=8 \) TeV, Phys. Lett. B 750 (2015) 475 [arXiv:1506.05629] [INSPIRE].
  9. [9]
    ATLAS collaboration, Measurement of jet charge in dijet events from \( \sqrt{s}=8 \) TeV pp collisions with the ATLAS detector, Phys. Rev. D 93 (2016) 052003 [arXiv:1509.05190] [INSPIRE].
  10. [10]
    ATLAS collaboration, Performance of jet substructure techniques in early \( \sqrt{s}=13 \) TeV pp collisions with the ATLAS detector, ATLAS-CONF-2015-035, CERN, Geneva Switzerland, (2015).
  11. [11]
    ATLAS collaboration, Measurement of the differential cross-section of highly boosted top quarks as a function of their transverse momentum in \( \sqrt{s}=8 \) TeV proton-proton collisions using the ATLAS detector, Phys. Rev. D 93 (2016) 032009 [arXiv:1510.03818] [INSPIRE].
  12. [12]
    ATLAS collaboration, Identification of boosted, hadronically decaying W bosons and comparisons with ATLAS data taken at \( \sqrt{s}=8 \) TeV, Eur. Phys. J. C 76 (2016) 154 [arXiv:1510.05821] [INSPIRE].
  13. [13]
    ATLAS collaboration, Studies of b-tagging performance and jet substructure in a high \( {p}_{\mathrm{T}}g\to b\overline{b} \) rich sample of large-R jets from pp collisions at \( \sqrt{s}=8 \) TeV with the ATLAS detector, ATLAS-CONF-2016-002, CERN, Geneva Switzerland, (2016).
  14. [14]
    ATLAS collaboration, Boosted Higgs \( \left(\to b\overline{b}\right) \) boson identification with the ATLAS detector at \( \sqrt{s}=13 \) TeV, ATLAS-CONF-2016-039, CERN, Geneva Switzerland, (2016).
  15. [15]
    ATLAS collaboration, Discrimination of light quark and gluon jets in pp collisions at \( \sqrt{s}=8 \) TeV with the ATLAS detector, ATLAS-CONF-2016-034, CERN, Geneva Switzerland, (2016).
  16. [16]
    CMS collaboration, Measurement of the \( t\overline{t} \) production cross section at 13 TeV in the all-jets final state, CMS-PAS-TOP-16-013, CERN, Geneva Switzerland, (2016).
  17. [17]
    ATLAS and CMS collaborations, G. Rauco, Distinguishing quark and gluon jets at the LHC, in Proceedings, Parton Radiation and Fragmentation from LHC to FCC-ee, CERN, Geneva Switzerland, 22-23 November 2016 [INSPIRE].
  18. [18]
    ATLAS collaboration, Performance of top quark and W boson tagging in run 2 with ATLAS, ATLAS-CONF-2017-064, CERN, Geneva Switzerland, (2017).
  19. [19]
    CMS collaboration, Inclusive search for a highly boosted Higgs boson decaying to a bottom quark-antiquark pair, Phys. Rev. Lett. 120 (2018) 071802 [arXiv:1709.05543] [INSPIRE].
  20. [20]
    CMS collaboration, Search for BSM \( t\overline{t} \) production in the boosted all-hadronic final state, CMS-PAS-EXO-11-006, CERN, Geneva Switzerland, (2011).
  21. [21]
    CMS collaboration, Search for anomalous \( t\overline{t} \) production in the highly-boosted all-hadronic final state, JHEP 09 (2012) 029 [Erratum ibid. 03 (2014) 132] [arXiv:1204.2488] [INSPIRE].
  22. [22]
    ATLAS, CMS collaboration, S. Fleischmann, Boosted top quark techniques and searches for \( t\overline{t} \) resonances at the LHC, J. Phys. Conf. Ser. 452 (2013) 012034 [INSPIRE].
  23. [23]
    ATLAS and CMS collaborations, J. Pilot, Boosted top quarks, top pair resonances, and top partner searches at the LHC, EPJ Web Conf. 60 (2013) 09003 [INSPIRE].
  24. [24]
    ATLAS collaboration, Performance of boosted top quark identification in 2012 ATLAS data, ATLAS-CONF-2013-084, CERN, Geneva Switzerland, (2013).
  25. [25]
    CMS collaboration, Search for pair-produced vector-like quarks of charge −1/3 decaying to bH using boosted Higgs jet-tagging in pp collisions at \( \sqrt{s}=8 \) TeV, CMS-PAS-B2G-14-001, CERN, Geneva Switzerland, (2014).
  26. [26]
    CMS collaboration, Search for top-Higgs resonances in all-hadronic final states using jet substructure methods, CMS-PAS-B2G-14-002, CERN, Geneva Switzerland, (2014).
  27. [27]
    CMS collaboration, Search for vector-like T quarks decaying to top quarks and Higgs bosons in the all-hadronic channel using jet substructure, JHEP 06 (2015) 080 [arXiv:1503.01952] [INSPIRE].
  28. [28]
    ATLAS collaboration, Search for diboson resonances in the ννqq final state in pp collisions at \( \sqrt{s}=13 \) TeV with the ATLAS detector, ATLAS-CONF-2015-068, CERN, Geneva Switzerland, (2015).
  29. [29]
    ATLAS collaboration, Search for diboson resonances in the ℓℓqq final state in pp collisions at \( \sqrt{s}=13 \) TeV with the ATLAS detector, ATLAS-CONF-2015-071, CERN, Geneva Switzerland, (2015).
  30. [30]
    CMS collaboration, Search for a massive resonance decaying into a Higgs boson and a W or Z boson in hadronic final states in proton-proton collisions at \( \sqrt{s}=8 \) TeV, JHEP 02 (2016) 145 [arXiv:1506.01443] [INSPIRE].
  31. [31]
    CMS collaboration, Search for \( t\overline{t}H \) production in the \( H\to b\overline{b} \) decay channel with \( \sqrt{s}=13 \) TeV pp collisions at the CMS experiment, CMS-PAS-HIG-16-004, CERN, Geneva Switzerland, (2016).
  32. [32]
    CMS collaboration, Search for dark matter in association with a boosted top quark in the all hadronic final state, CMS-PAS-EXO-16-017, CERN, Geneva Switzerland, (2016).
  33. [33]
    CMS collaboration, Search for top quark-antiquark resonances in the all-hadronic final state at \( \sqrt{s}=13 \) TeV, CMS-PAS-B2G-15-003, CERN, Geneva Switzerland, (2015).
  34. [34]
    CMS collaboration, Searches for invisible Higgs boson decays with the CMS detector, CMS-PAS-HIG-16-016, CERN, Geneva Switzerland, (2016).
  35. [35]
    ATLAS collaboration, Searches for heavy diboson resonances in pp collisions at \( \sqrt{s}=13 \) TeV with the ATLAS detector, JHEP 09 (2016) 173 [arXiv:1606.04833] [INSPIRE].
  36. [36]
    CMS collaboration, Search for dark matter in proton-proton collisions at 8 TeV with missing transverse momentum and vector boson tagged jets, JHEP 12 (2016) 083 [Erratum ibid. 08 (2017) 035] [arXiv:1607.05764] [INSPIRE].
  37. [37]
    CMS collaboration, Search for new physics in a boosted hadronic monotop final state using 12.9 fb −1 of \( \sqrt{s}=13 \) TeV data, CMS-PAS-EXO-16-040, CERN, Geneva Switzerland, (2016).
  38. [38]
    CMS collaboration, Search for dark matter in final states with an energetic jet, or a hadronically decaying W or Z boson using 12.9 fb −1 of data at \( \sqrt{s}=13 \) TeV, CMS-PAS-EXO-16-037, CERN, Geneva Switzerland, (2016).
  39. [39]
    ATLAS collaboration, Search for dark matter produced in association with a hadronically decaying vector boson in pp collisions at \( \sqrt{s}=13 \) TeV with the ATLAS detector, Phys. Lett. B 763 (2016) 251 [arXiv:1608.02372] [INSPIRE].
  40. [40]
    CMS collaboration, Search for high-mass Zγ resonances in proton-proton collisions at \( \sqrt{s}=8 \) and 13TeV using jet substructure techniques, Phys. Lett. B 772 (2017) 363 [arXiv:1612.09516] [INSPIRE].
  41. [41]
    CMS collaboration, Search for pair production of vector-like T and B quarks in single-lepton final states using boosted jet substructure in proton-proton collisions at \( \sqrt{s}=13 \) TeV, JHEP 11 (2017) 085 [arXiv:1706.03408] [INSPIRE].
  42. [42]
    CMS collaboration, Search for massive resonances decaying into W W, W Z, ZZ, qW and qZ with dijet final states at \( \sqrt{s}=13 \) TeV, arXiv:1708.05379 [INSPIRE].
  43. [43]
    CMS collaboration, Search for low mass vector resonances decaying into quark-antiquark pairs in proton-proton collisions at \( \sqrt{s}=13 \) TeV, JHEP 01 (2018) 097 [arXiv:1710.00159] [INSPIRE].
  44. [44]
    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].
  45. [45]
    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].
  46. [46]
    A. Altheimer et al., Boosted objects and jet substructure at the LHC. Report of BOOST2012, held at IFIC Valencia, 23rd -27th of July 2012, Eur. Phys. J. C 74 (2014) 2792 [arXiv:1311.2708] [INSPIRE].
  47. [47]
    D. Adams et al., Towards an understanding of the correlations in jet substructure, Eur. Phys. J. C 75 (2015) 409 [arXiv:1504.00679] [INSPIRE].
  48. [48]
    A.J. Larkoski, I. Moult and B. Nachman, Jet substructure at the Large Hadron Collider: a review of recent advances in theory and machine learning, arXiv:1709.04464 [INSPIRE].
  49. [49]
    G. Gur-Ari, M. Papucci and G. Perez, Classification of energy flow observables in narrow jets, arXiv:1101.2905 [INSPIRE].
  50. [50]
    J. Cogan, M. Kagan, E. Strauss and A. Schwarztman, Jet-images: computer vision inspired techniques for jet tagging, JHEP 02 (2015) 118 [arXiv:1407.5675] [INSPIRE].ADSCrossRefGoogle Scholar
  51. [51]
    A.J. Larkoski, G.P. Salam and J. Thaler, Energy correlation functions for jet substructure, JHEP 06 (2013) 108 [arXiv:1305.0007] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  52. [52]
    I. Moult, L. Necib and J. Thaler, New angles on energy correlation functions, JHEP 12 (2016) 153 [arXiv:1609.07483] [INSPIRE].ADSCrossRefGoogle Scholar
  53. [53]
    A.J. Larkoski, I. Moult and D. Neill, Power counting to better jet observables, JHEP 12 (2014) 009 [arXiv:1409.6298] [INSPIRE].ADSCrossRefGoogle Scholar
  54. [54]
    J. Thaler and K. Van Tilburg, Identifying boosted objects with N -subjettiness, JHEP 03 (2011) 015 [arXiv:1011.2268] [INSPIRE].ADSCrossRefGoogle Scholar
  55. [55]
    J. Thaler and K. Van Tilburg, Maximizing boosted top identification by minimizing N -subjettiness, JHEP 02 (2012) 093 [arXiv:1108.2701] [INSPIRE].
  56. [56]
    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].
  57. [57]
    K. Datta and A. Larkoski, How much information is in a jet?, JHEP 06 (2017) 073 [arXiv:1704.08249] [INSPIRE].ADSCrossRefGoogle Scholar
  58. [58]
    J. Gallicchio, J. Huth, M. Kagan, M.D. Schwartz, K. Black and B. Tweedie, Multivariate discrimination and the Higgs + W/Z search, JHEP 04 (2011) 069 [arXiv:1010.3698] [INSPIRE].ADSCrossRefGoogle Scholar
  59. [59]
    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
  60. [60]
    J. Gallicchio and M.D. Schwartz, Quark and gluon jet substructure, JHEP 04 (2013) 090 [arXiv:1211.7038] [INSPIRE].ADSCrossRefGoogle Scholar
  61. [61]
    P. Baldi, P. Sadowski and D. Whiteson, Searching for exotic particles in high-energy physics with deep learning, Nature Commun. 5 (2014) 4308 [arXiv:1402.4735] [INSPIRE].ADSCrossRefGoogle Scholar
  62. [62]
    P. Baldi, P. Sadowski and D. Whiteson, Enhanced Higgs boson to τ + τ search with deep learning, Phys. Rev. Lett. 114 (2015) 111801 [arXiv:1410.3469] [INSPIRE].
  63. [63]
    L. de Oliveira, M. Kagan, L. Mackey, B. Nachman and A. Schwartzman, Jet-images — deep learning edition, JHEP 07 (2016) 069 [arXiv:1511.05190] [INSPIRE].CrossRefGoogle Scholar
  64. [64]
    J. Barnard, E.N. Dawe, M.J. Dolan and N. Rajcic, Parton shower uncertainties in jet substructure analyses with deep neural networks, Phys. Rev. D 95 (2017) 014018 [arXiv:1609.00607] [INSPIRE].
  65. [65]
    P.T. Komiske, E.M. Metodiev and M.D. Schwartz, Deep learning in color: towards automated quark/gluon jet discrimination, JHEP 01 (2017) 110 [arXiv:1612.01551] [INSPIRE].ADSCrossRefGoogle Scholar
  66. [66]
    G. Kasieczka, T. Plehn, M. Russell and T. Schell, Deep-learning top taggers or the end of QCD?, JHEP 05 (2017) 006 [arXiv:1701.08784] [INSPIRE].ADSCrossRefGoogle Scholar
  67. [67]
    L.G. Almeida, M. Backović, M. Cliche, S.J. Lee and M. Perelstein, Playing tag with ANN: boosted top identification with pattern recognition, JHEP 07 (2015) 086 [arXiv:1501.05968] [INSPIRE].ADSCrossRefGoogle Scholar
  68. [68]
    P. Baldi, K. Bauer, C. Eng, P. Sadowski and D. Whiteson, Jet substructure classification in high-energy physics with deep neural networks, Phys. Rev. D 93 (2016) 094034 [arXiv:1603.09349] [INSPIRE].
  69. [69]
    D. Guest, J. Collado, P. Baldi, S.-C. Hsu, G. Urban and D. Whiteson, Jet flavor classification in high-energy physics with deep neural networks, Phys. Rev. D 94 (2016) 112002 [arXiv:1607.08633] [INSPIRE].
  70. [70]
    G. Louppe, K. Cho, C. Becot and K. Cranmer, QCD-aware recursive neural networks for jet physics, arXiv:1702.00748 [INSPIRE].
  71. [71]
    J. Pearkes, W. Fedorko, A. Lister and C. Gay, Jet constituents for deep neural network based top quark tagging, arXiv:1704.02124 [INSPIRE].
  72. [72]
    A. Butter, G. Kasieczka, T. Plehn and M. Russell, Deep-learned top tagging with a Lorentz layer, arXiv:1707.08966 [INSPIRE].
  73. [73]
    J.A. Aguilar-Saavedra, J.H. Collins and R.K. Mishra, A generic anti-QCD jet tagger, JHEP 11 (2017) 163 [arXiv:1709.01087] [INSPIRE].ADSCrossRefGoogle Scholar
  74. [74]
    F.V. Tkachov, Measuring multi-jet structure of hadronic energy flow or what is a jet?, Int. J. Mod. Phys. A 12 (1997) 5411 [hep-ph/9601308] [INSPIRE].
  75. [75]
    A.J. Larkoski, J. Thaler and W.J. Waalewijn, Gaining (mutual) information about quark/gluon discrimination, JHEP 11 (2014) 129 [arXiv:1408.3122] [INSPIRE].ADSCrossRefGoogle Scholar
  76. [76]
    M.H. Stone, The generalized Weierstrass approximation theorem, Math. Magazine 21 (1948) 237.MathSciNetCrossRefGoogle Scholar
  77. [77]
    N.L. Zhang and D. Poole, Exploiting causal independence in Bayesian network inference, J. Artificial Intel. Res. 5 (1996) 301.MathSciNetCrossRefGoogle Scholar
  78. [78]
    T. Kinoshita, Mass singularities of Feynman amplitudes, J. Math. Phys. 3 (1962) 650 [INSPIRE].ADSCrossRefGoogle Scholar
  79. [79]
    T.D. Lee and M. Nauenberg, Degenerate systems and mass singularities, Phys. Rev. 133 (1964) B1549 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  80. [80]
    S. Weinberg, The quantum theory of fields. Volume 1: foundations, Cambridge University Press, Cambridge U.K., (2005) [INSPIRE].
  81. [81]
    CTEQ collaboration, R. Brock et al., Handbook of perturbative QCD: version 1.0, Rev. Mod. Phys. 67 (1995) 157 [INSPIRE].ADSCrossRefGoogle Scholar
  82. [82]
    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
  83. [83]
    A.J. Larkoski, S. Marzani and J. Thaler, Sudakov safety in perturbative QCD, Phys. Rev. D 91 (2015) 111501 [arXiv:1502.01719] [INSPIRE].
  84. [84]
    W.J. Waalewijn, Calculating the charge of a jet, Phys. Rev. D 86 (2012) 094030 [arXiv:1209.3019] [INSPIRE].
  85. [85]
    H.-M. Chang, M. Procura, J. Thaler and W.J. Waalewijn, Calculating track-based observables for the LHC, Phys. Rev. Lett. 111 (2013) 102002 [arXiv:1303.6637] [INSPIRE].ADSCrossRefGoogle Scholar
  86. [86]
    B.T. Elder, M. Procura, J. Thaler, W.J. Waalewijn and K. Zhou, Generalized fragmentation functions for fractal jet observables, JHEP 06 (2017) 085 [arXiv:1704.05456] [INSPIRE].ADSCrossRefGoogle Scholar
  87. [87]
    N.A. Sveshnikov and F.V. Tkachov, Jets and quantum field theory, Phys. Lett. B 382 (1996) 403 [hep-ph/9512370] [INSPIRE].ADSCrossRefGoogle Scholar
  88. [88]
    P.S. Cherzor and N.A. Sveshnikov, Jet observables and energy momentum tensor, in Quantum field theory and high-energy physics. Proceedings, Workshop, QFTHEP’97, Samara Russia, 4-10 September 1997, pg. 402 [hep-ph/9710349] [INSPIRE].
  89. [89]
    F.V. Tkachov, A theory of jet definition, Int. J. Mod. Phys. A 17 (2002) 2783 [hep-ph/9901444] [INSPIRE].ADSCrossRefGoogle Scholar
  90. [90]
    P. Dallot, P.D. Bristowe and M. Demazure, Reduced coordinates on the configuration space of three and four atoms, Phys. Rev. B 46 (1992) 2133.ADSCrossRefGoogle Scholar
  91. [91]
    S.S. Mandal, S. Mukherjee and K. Ray, Determination of many-electron basis functions for a quantum Hall ground state using Schur polynomials, Annals Phys. 390 (2018) 236 [arXiv:1708.07658] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  92. [92]
    M. Hogervorst, S. Rychkov and B.C. van Rees, Truncated conformal space approach in d dimensions: a cheap alternative to lattice field theory?, Phys. Rev. D 91 (2015) 025005 [arXiv:1409.1581] [INSPIRE].
  93. [93]
    A.J. Larkoski, I. Moult and D. Neill, Analytic boosted boson discrimination at the Large Hadron Collider, arXiv:1708.06760 [INSPIRE].
  94. [94]
    N.J. Sloane, The on-line encyclopedia of integer sequences, in Towards mechanized mathematical assistants, Springer, Berlin Heidelberg Germany, (2007), pg. 130.Google Scholar
  95. [95]
    F. Harary and E.M. Palmer, Graphical enumeration, Elsevier, The Netherlands, (2014).zbMATHGoogle Scholar
  96. [96]
    J. Gallicchio and M.D. Schwartz, Seeing in color: jet superstructure, Phys. Rev. Lett. 105 (2010) 022001 [arXiv:1001.5027] [INSPIRE].
  97. [97]
    G.P. Salam and D. Wicke, Hadron masses and power corrections to event shapes, JHEP 05 (2001) 061 [hep-ph/0102343] [INSPIRE].CrossRefGoogle Scholar
  98. [98]
    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].
  99. [99]
    I.W. Stewart, F.J. Tackmann, J. Thaler, C.K. Vermilion and T.F. Wilkason, XCone: N -jettiness as an exclusive cone jet algorithm, JHEP 11 (2015) 072 [arXiv:1508.01516] [INSPIRE].ADSCrossRefGoogle Scholar
  100. [100]
    C.F. Berger, T. Kucs and G.F. Sterman, Event shape/energy flow correlations, Phys. Rev. D 68 (2003) 014012 [hep-ph/0303051] [INSPIRE].
  101. [101]
    L.G. Almeida, S.J. Lee, G. Perez, G.F. Sterman, I. Sung and J. Virzi, Substructure of high-p T jets at the LHC, Phys. Rev. D 79 (2009) 074017 [arXiv:0807.0234] [INSPIRE].
  102. [102]
    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
  103. [103]
    A.J. Larkoski, D. Neill and J. Thaler, Jet shapes with the broadening axis, JHEP 04 (2014) 017 [arXiv:1401.2158] [INSPIRE].ADSCrossRefGoogle Scholar
  104. [104]
    J. Thaler and L.-T. Wang, Strategies to identify boosted tops, JHEP 07 (2008) 092 [arXiv:0806.0023] [INSPIRE].ADSCrossRefGoogle Scholar
  105. [105]
    S.D. Ellis, C.K. Vermilion and J.R. Walsh, Techniques for improved heavy particle searches with jet substructure, Phys. Rev. D 80 (2009) 051501 [arXiv:0903.5081] [INSPIRE].
  106. [106]
    S.D. Ellis, C.K. Vermilion and J.R. Walsh, Recombination algorithms and jet substructure: pruning as a tool for heavy particle searches, Phys. Rev. D 81 (2010) 094023 [arXiv:0912.0033] [INSPIRE].
  107. [107]
    D. Krohn, J. Thaler and L.-T. Wang, Jet trimming, JHEP 02 (2010) 084 [arXiv:0912.1342] [INSPIRE].
  108. [108]
    M. Dasgupta, A. Fregoso, S. Marzani and G.P. Salam, Towards an understanding of jet substructure, JHEP 09 (2013) 029 [arXiv:1307.0007] [INSPIRE].ADSCrossRefGoogle Scholar
  109. [109]
    A.J. Larkoski, S. Marzani, G. Soyez and J. Thaler, Soft drop, JHEP 05 (2014) 146 [arXiv:1402.2657] [INSPIRE].
  110. [110]
    B. Henning, X. Lu, T. Melia and H. Murayama, Operator bases, S-matrices and their partition functions, JHEP 10 (2017) 199 [arXiv:1706.08520] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  111. [111]
    M. Cacciari and G.P. Salam, Dispelling the N 3 myth for the k t jet-finder, Phys. Lett. B 641 (2006) 57 [hep-ph/0512210] [INSPIRE].
  112. [112]
    K.P. Murphy, Machine learning: a probabilistic perspective, MIT press, U.S.A., (2012).Google Scholar
  113. [113]
    P.T. Komiske, E.M. Metodiev, B. Nachman and M.D. Schwartz, Pileup mitigation with machine learning (PUMML), JHEP 12 (2017) 051 [arXiv:1707.08600] [INSPIRE].ADSCrossRefGoogle Scholar
  114. [114]
    L. de Oliveira, M. Paganini and B. Nachman, Learning particle physics by example: location-aware generative adversarial networks for physics synthesis, Comput. Softw. Big Sci. 1 (2017) 4 [arXiv:1701.05927] [INSPIRE].CrossRefGoogle Scholar
  115. [115]
    M. Paganini, L. de Oliveira and B. Nachman, Accelerating science with generative adversarial networks: an application to 3D particle showers in multilayer calorimeters, Phys. Rev. Lett. 120 (2018) 042003 [arXiv:1705.02355] [INSPIRE].
  116. [116]
    T. Sjöstrand, S. Mrenna and P.Z. Skands, PYTHIA 6.4 physics and manual, JHEP 05 (2006) 026 [hep-ph/0603175] [INSPIRE].
  117. [117]
    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].ADSCrossRefGoogle Scholar
  118. [118]
    T. Sjöstrand et al., An introduction to PYTHIA 8.2, Comput. Phys. Commun. 191 (2015) 159 [arXiv:1410.3012] [INSPIRE].
  119. [119]
    M. Cacciari, G.P. Salam and G. Soyez, FastJet user manual, Eur. Phys. J. C 72 (2012) 1896 [arXiv:1111.6097] [INSPIRE].
  120. [120]
    M. Cacciari, G.P. Salam and G. Soyez, The anti-k t jet clustering algorithm, JHEP 04 (2008) 063 [arXiv:0802.1189] [INSPIRE].CrossRefGoogle Scholar
  121. [121]
    R. Tibshirani, Regression shrinkage and selection via the lasso: a retrospective, J. Roy. Statist. Soc. B 73 (2011) 273.MathSciNetCrossRefGoogle Scholar
  122. [122]
    C.M. Bishop, Pattern recognition and machine learning, Springer, New York U.S.A., (2006).Google Scholar
  123. [123]
    F. Pedregosa et al., Scikit-learn: machine learning in Python, J. Machine Learning Res. 12 (2011) 2825 [arXiv:1201.0490] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  124. [124]
    J.R. Andersen et al., Les Houches 2015: physics at TeV colliders Standard Model working group report, in 9th Les Houches Workshop on Physics at TeV Colliders (PhysTeV 2015), FERMILAB-CONF-16-175-PPD-T, Les Houches France, 1-19 June 2015 [arXiv:1605.04692] [INSPIRE].
  125. [125]
    P. Gras et al., Systematics of quark/gluon tagging, JHEP 07 (2017) 091 [arXiv:1704.03878] [INSPIRE].
  126. [126]
    R.A. Fisher, The use of multiple measurements in taxonomic problems, Ann. Human Genetics 7 (1936) 179.CrossRefGoogle Scholar
  127. [127]
    V. Nair and G.E. Hinton, Rectified linear units improve restricted Boltzmann machines, in Proceedings of the 27th international conference on machine learning (ICML-10), (2010), pg. 807.Google Scholar
  128. [128]
    K. He, X. Zhang, S. Ren and J. Sun, Delving deep into rectifiers: surpassing human-level performance on imagenet classification, in Proceedings of the IEEE international conference on computer vision, IEEE, (2015), pg. 1026.Google Scholar
  129. [129]
    D. Kingma and J. Ba, Adam: a method for stochastic optimization, arXiv:1412.6980.
  130. [130]
    F. Chollet, Keras,, (2017).
  131. [131]
    J. Bergstra et al., Theano: a CPU and GPU math compiler in Python, in Proc. 9th Python in Science Conf., (2010), pg. 1.Google Scholar
  132. [132]
    S. Chang, T. Cohen and B. Ostdiek, What is the machine learning?, Phys. Rev. D 97 (2018) 056009 [arXiv:1709.10106] [INSPIRE].
  133. [133]
    E.M. Metodiev and J. Thaler, On the topic of jets, arXiv:1802.00008 [INSPIRE].
  134. [134]
    P.T. Komiske, E.M. Metodiev, B. Nachman and M.D. Schwartz, Learning to classify from impure samples, arXiv:1801.10158 [INSPIRE].
  135. [135]
    L.M. Dery, B. Nachman, F. Rubbo and A. Schwartzman, Weakly supervised classification in high energy physics, JHEP 05 (2017) 145 [arXiv:1702.00414] [INSPIRE].ADSCrossRefGoogle Scholar
  136. [136]
    E.M. Metodiev, B. Nachman and J. Thaler, Classification without labels: learning from mixed samples in high energy physics, JHEP 10 (2017) 174 [arXiv:1708.02949] [INSPIRE].ADSCrossRefGoogle Scholar
  137. [137]
    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
  138. [138]
    G.P. Korchemsky and S. Tafat, On power corrections to the event shape distributions in QCD, JHEP 10 (2000) 010 [hep-ph/0007005] [INSPIRE].
  139. [139]
    C. Frye, A.J. Larkoski, M.D. Schwartz and K. Yan, Factorization for groomed jet substructure beyond the next-to-leading logarithm, JHEP 07 (2016) 064 [arXiv:1603.09338] [INSPIRE].ADSCrossRefGoogle Scholar
  140. [140]
    I. Feige, M.D. Schwartz, I.W. Stewart and J. Thaler, Precision jet substructure from boosted event shapes, Phys. Rev. Lett. 109 (2012) 092001 [arXiv:1204.3898] [INSPIRE].ADSCrossRefGoogle Scholar
  141. [141]
    M. Dasgupta, A. Fregoso, S. Marzani and A. Powling, Jet substructure with analytical methods, Eur. Phys. J. C 73 (2013) 2623 [arXiv:1307.0013] [INSPIRE].
  142. [142]
    A.J. Larkoski, I. Moult and D. Neill, Toward multi-differential cross sections: measuring two angularities on a single jet, JHEP 09 (2014) 046 [arXiv:1401.4458] [INSPIRE].ADSCrossRefGoogle Scholar
  143. [143]
    M. Procura, W.J. Waalewijn and L. Zeune, Resummation of double-differential cross sections and fully-unintegrated parton distribution functions, JHEP 02 (2015) 117 [arXiv:1410.6483] [INSPIRE].ADSCrossRefGoogle Scholar
  144. [144]
    A.J. Larkoski, I. Moult and D. Neill, Analytic boosted boson discrimination, JHEP 05 (2016) 117 [arXiv:1507.03018] [INSPIRE].ADSCrossRefGoogle Scholar
  145. [145]
    M. Dasgupta, L. Schunk and G. Soyez, Jet shapes for boosted jet two-prong decays from first-principles, JHEP 04 (2016) 166 [arXiv:1512.00516] [INSPIRE].ADSGoogle Scholar
  146. [146]
    M. Dasgupta, A. Powling, L. Schunk and G. Soyez, Improved jet substructure methods: Y-splitter and variants with grooming, JHEP 12 (2016) 079 [arXiv:1609.07149] [INSPIRE].ADSCrossRefGoogle Scholar
  147. [147]
    S. Marzani, L. Schunk and G. Soyez, A study of jet mass distributions with grooming, JHEP 07 (2017) 132 [arXiv:1704.02210] [INSPIRE].ADSCrossRefGoogle Scholar
  148. [148]
    A.J. Larkoski, I. Moult and D. Neill, Factorization and resummation for groomed multi-prong jet shapes, JHEP 02 (2018) 144 [arXiv:1710.00014] [INSPIRE].ADSCrossRefGoogle Scholar
  149. [149]
    C.L. Basham, L.S. Brown, S.D. Ellis and S.T. Love, Energy correlations in electron-positron annihilation: testing QCD, Phys. Rev. Lett. 41 (1978) 1585 [INSPIRE].ADSCrossRefGoogle Scholar
  150. [150]
    C.L. Basham, L.S. Brown, S.D. Ellis and S.T. Love, Energy correlations in electron-positron annihilation in quantum chromodynamics: asymptotically free perturbation theory, Phys. Rev. D 19 (1979) 2018 [INSPIRE].
  151. [151]
    A.V. Belitsky, G.P. Korchemsky and G.F. Sterman, Energy flow in QCD and event shape functions, Phys. Lett. B 515 (2001) 297 [hep-ph/0106308] [INSPIRE].ADSCrossRefGoogle Scholar
  152. [152]
    D.M. Hofman and J. Maldacena, Conformal collider physics: energy and charge correlations, JHEP 05 (2008) 012 [arXiv:0803.1467] [INSPIRE].ADSCrossRefGoogle Scholar
  153. [153]
    O.T. Engelund and R. Roiban, Correlation functions of local composite operators from generalized unitarity, JHEP 03 (2013) 172 [arXiv:1209.0227] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  154. [154]
    A. Zhiboedov, On conformal field theories with extremal a/c values, JHEP 04 (2014) 038 [arXiv:1304.6075] [INSPIRE].ADSCrossRefGoogle Scholar
  155. [155]
    A.V. Belitsky, S. Hohenegger, G.P. Korchemsky, E. Sokatchev and A. Zhiboedov, From correlation functions to event shapes, Nucl. Phys. B 884 (2014) 305 [arXiv:1309.0769] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  156. [156]
    A.V. Belitsky, S. Hohenegger, G.P. Korchemsky, E. Sokatchev and A. Zhiboedov, Event shapes in N = 4 super-Yang-Mills theory, Nucl. Phys. B 884 (2014) 206 [arXiv:1309.1424] [INSPIRE].
  157. [157]
    G.P. Korchemsky, G. Oderda and G.F. Sterman, Power corrections and nonlocal operators, AIP Conf. Proc. 407 (1997) 988 [hep-ph/9708346] [INSPIRE].
  158. [158]
    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].
  159. [159]
    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].

Copyright information

© The Author(s) 2018

Authors and Affiliations

  • Patrick T. Komiske
    • 1
  • Eric M. Metodiev
    • 1
  • Jesse Thaler
    • 1
  1. 1.Center for Theoretical Physics, Massachusetts Institute of TechnologyCambridgeU.S.A.

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