The hidden geometry of particle collisions


We establish that many fundamental concepts and techniques in quantum field theory and collider physics can be naturally understood and unified through a simple new geometric language. The idea is to equip the space of collider events with a metric, from which other geometric objects can be rigorously defined. Our analysis is based on the energy mover’s distance, which quantifies the “work” required to rearrange one event into another. This metric, which operates purely at the level of observable energy flow information, allows for a clarified definition of infrared and collinear safety and related concepts. A number of well-known collider observables can be exactly cast as the minimum distance between an event and various manifolds in this space. Jet definitions, such as exclusive cone and sequential recombination algorithms, can be directly derived by finding the closest few-particle approximation to the event. Several area- and constituent-based pileup mitigation strategies are naturally expressed in this formalism as well. Finally, we lift our reasoning to develop a precise distance between theories, which are treated as collections of events weighted by cross sections. In all of these various cases, a better understanding of existing methods in our geometric language suggests interesting new ideas and generalizations.

A preprint version of the article is available at ArXiv.


  1. [1]

    P.T. Komiske, E.M. Metodiev and J. Thaler, Metric space of collider events, Phys. Rev. Lett. 123 (2019) 041801 [arXiv:1902.02346] [INSPIRE].

    ADS  Google Scholar 

  2. [2]

    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].

  3. [3]

    N.A. Sveshnikov and F.V. Tkachov, Jets and quantum field theory, Phys. Lett. B 382 (1996) 403 [hep-ph/9512370] [INSPIRE].

  4. [4]

    G.P. Korchemsky, G. Oderda and G.F. Sterman, Power corrections and nonlocal operators, AIP Conf. Proc. 407 (1997) 988 [hep-ph/9708346] [INSPIRE].

  5. [5]

    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].

    ADS  Google Scholar 

  6. [6]

    P.S. Cherzor and N.A. Sveshnikov, Jet observables and energy momentum tensor, in the proceedings of the 12th International Workshop on High-Energy Physics and Quantum Field Theory (QFTHEP 97), September 4–10, Samara, Russia (1997). hep-ph/9710349 [INSPIRE].

  7. [7]

    F.V. Tkachov, A theory of jet definition, Int. J. Mod. Phys. A 17 (2002) 2783 [hep-ph/9901444] [INSPIRE].

  8. [8]

    G.P. Korchemsky and G.F. Sterman, Power corrections to event shapes and factorization, Nucl. Phys. B 555 (1999) 335 [hep-ph/9902341] [INSPIRE].

  9. [9]

    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].

  10. [10]

    C.F. Berger et al., Snowmass 2001: jet energy flow project, eConf C010630 (2001) P512 [hep-ph/0202207] [INSPIRE].

  11. [11]

    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].

    ADS  Google Scholar 

  12. [12]

    D.M. Hofman and J. Maldacena, Conformal collider physics: Energy and charge correlations, JHEP 05 (2008) 012 [arXiv:0803.1467] [INSPIRE].

    ADS  Google Scholar 

  13. [13]

    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].

    ADS  Google Scholar 

  14. [14]

    A.V. Belitsky et al., From correlation functions to event shapes, Nucl. Phys. B 884 (2014) 305 [arXiv:1309.0769] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  15. [15]

    P.T. Komiske, E.M. Metodiev and J. Thaler, Energy flow polynomials: a complete linear basis for jet substructure, JHEP 04 (2018) 013 [arXiv:1712.07124] [INSPIRE].

    ADS  Google Scholar 

  16. [16]

    P.T. Komiske, E.M. Metodiev and J. Thaler, Energy flow networks: deep sets for particle jets, JHEP 01 (2019) 121 [arXiv:1810.05165] [INSPIRE].

    ADS  Google Scholar 

  17. [17]

    P.T. Komiske, E.M. Metodiev and J. Thaler, Cutting multiparticle correlators down to size, Phys. Rev. D 101 (2020) 036019 [arXiv:1911.04491] [INSPIRE].

    ADS  Google Scholar 

  18. [18]

    S. Peleg, M. Werman and H. Rom, A unified approach to the change of resolution: space and gray-level, IEEE Trans. Pattern Anal. Mach. Intell. 11 (1989) 739.

    Google Scholar 

  19. [19]

    Y. Rubner, C. Tomasi and L.J. Guibas, A metric for distributions with applications to image databases, in the proceedings of the 6th International Conference on Computer Vision (ICCV’98), January 8, Washington, U.S.A. (1998).

  20. [20]

    Y. Rubner, C. Tomasi and L. J. Guibas, The Earth mover’s distance as a metric for image retrieval, Int. J. Comput. Vision 40 (2000) 99.

    MATH  Google Scholar 

  21. [21]

    O. Pele and M. Werman, A linear time histogram metric for improved SIFT matching, in the proceedings of the 10th European Conference on Computer Vision (ECCVE 2008), Ocotber 12–18, Marseille, France (2008).

  22. [22]

    O. Pele and B. Taskar, The tangent Earth mover’s distance, in the proceedings of the 1st International Conference — Geometric Science of Information (GSI 2013), August 28–30, Paris, France (2013).

  23. [23]

    L.N. Wasserstein, Markov processes over denumerable products of spaces describing large systems of automata, Probl. Inf. Trans. 5 (1969) 47.

    MathSciNet  Google Scholar 

  24. [24]

    R. L. Dobrushin, Prescribing a system of random variables by conditional distributions, Theor. Prob. Appl. 15 (1970) 458.

    MATH  Google Scholar 

  25. [25]

    P.T. Komiske et al., Exploring the space of jets with CMS open data, Phys. Rev. D 101 (2020) 034009 [arXiv:1908.08542] [INSPIRE].

    ADS  Google Scholar 

  26. [26]

    A. Mullin, H. Pacey, M. Parker, M. White and S. Williams, Does SUSY have friends? A new approach for LHC event analysis, arXiv:1912.10625 [INSPIRE].

  27. [27]

    T. Kinoshita, Mass singularities of Feynman amplitudes, J. Math. Phys. 3 (1962) 650 [INSPIRE].

    ADS  MATH  Google Scholar 

  28. [28]

    T.D. Lee and M. Nauenberg, Degenerate systems and mass singularities, Phys. Rev. 133 (1964) B1549 [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  29. [29]

    G.F. Sterman and S. Weinberg, Jets from quantum chromodynamics, Phys. Rev. Lett. 39 (1977) 1436 [INSPIRE].

    ADS  Google Scholar 

  30. [30]

    G.F. Sterman, Mass divergences in annihilation processes. 1. Origin and nature of divergences in cut vacuum polarization diagrams, Phys. Rev. D 17 (1978) 2773 [INSPIRE].

  31. [31]

    G.F. Sterman, Mass divergences in annihilation processes. 2. Cancellation of divergences in cut vacuum polarization diagrams, Phys. Rev. D 17 (1978) 2789 [INSPIRE].

  32. [32]

    G.F. Sterman, Zero mass limit for a class of jet related cross-sections, Phys. Rev. D 19 (1979) 3135 [INSPIRE].

    ADS  Google Scholar 

  33. [33]

    G. Sterman et al., Handbook of perturbative QCD, Rev. Mod. Phys. 67 (1995) 157.

    ADS  Google Scholar 

  34. [34]

    S. Weinberg, The quantum theory of fields. Volume 1: foundations, Cambridge University Press, Cambridge U.K. (2005).

  35. [35]

    R. Ellis, W. Stirling and B.R. Webber, QCD and collider physics, Cambridge University Press, Cambridge U.K. (2011) [INSPIRE].

    Google Scholar 

  36. [36]

    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].

  37. [37]

    A.J. Larkoski and J. Thaler, Unsafe but calculable: ratios of angularities in perturbative QCD, JHEP 09 (2013) 137 [arXiv:1307.1699] [INSPIRE].

    ADS  Google Scholar 

  38. [38]

    A.J. Larkoski, S. Marzani, G. Soyez and J. Thaler, Soft drop, JHEP 05 (2014) 146 [arXiv:1402.2657] [INSPIRE].

    ADS  Google Scholar 

  39. [39]

    A.J. Larkoski, S. Marzani and J. Thaler, Sudakov safety in perturbative QCD, Phys. Rev. D 91 (2015) 111501 [arXiv:1502.01719] [INSPIRE].

    ADS  Google Scholar 

  40. [40]

    S. Brandt, C. Peyrou, R. Sosnowski and A. Wroblewski, The principal axis of jets. An attempt to analyze high-energy collisions as two-body processes, Phys. Lett. 12 (1964) 57 [INSPIRE].

  41. [41]

    E. Farhi, A QCD test for jets, Phys. Rev. Lett. 39 (1977) 1587 [INSPIRE].

    ADS  Google Scholar 

  42. [42]

    H. Georgi and M. Machacek, A simple QCD prediction of jet structure in e+ e annihilation, Phys. Rev. Lett. 39 (1977) 1237 [INSPIRE].

    ADS  Google Scholar 

  43. [43]

    A.J. Larkoski, D. Neill and J. Thaler, Jet shapes with the broadening axis, JHEP 04 (2014) 017 [arXiv:1401.2158] [INSPIRE].

    ADS  Google Scholar 

  44. [44]

    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].

    ADS  Google Scholar 

  45. [45]

    C. Cesarotti and J. Thaler, A robust measure of event isotropy at colliders, arXiv:2004.06125 [INSPIRE].

  46. [46]

    S.D. Ellis et al., Jet shapes and jet algorithms in SCET, JHEP 11 (2010) 101 [arXiv:1001.0014] [INSPIRE].

    ADS  Google Scholar 

  47. [47]

    J. Thaler and K. Van Tilburg, Identifying boosted objects with N -subjettiness, JHEP 03 (2011) 015 [arXiv:1011.2268] [INSPIRE].

    ADS  Google Scholar 

  48. [48]

    J. Thaler and K. Van Tilburg, Maximizing Boosted Top Identification by Minimizing N-subjettiness, JHEP 02 (2012) 093 [arXiv:1108.2701] [INSPIRE].

    ADS  Google Scholar 

  49. [49]

    I.W. Stewart et al., XCone: N -jettiness as an exclusive cone jet algorithm, JHEP 11 (2015) 072 [arXiv:1508.01516] [INSPIRE].

    ADS  Google Scholar 

  50. [50]

    J. Thaler and T.F. Wilkason, Resolving boosted jets with XCone, JHEP 12 (2015) 051 [arXiv:1508.01518] [INSPIRE].

    ADS  Google Scholar 

  51. [51]

    S. Catani, Y.L. Dokshitzer, M.H. Seymour and B.R. Webber, Longitudinally invariant Kt clustering algorithms for hadron hadron collisions, Nucl. Phys. B 406 (1993) 187 [INSPIRE].

    ADS  Google Scholar 

  52. [52]

    S.D. Ellis and D.E. Soper, Successive combination jet algorithm for hadron collisions, Phys. Rev. D 48 (1993) 3160 [hep-ph/9305266] [INSPIRE].

  53. [53]

    D. Bertolini, T. Chan and J. Thaler, Jet observables without jet algorithms, JHEP 04 (2014) 013 [arXiv:1310.7584] [INSPIRE].

    ADS  Google Scholar 

  54. [54]

    G. Salam, \( {E}_t^{\infty } \) scheme, unpublished.

  55. [55]

    M. Cacciari and G.P. Salam, Pileup subtraction using jet areas, Phys. Lett. B 659 (2008) 119 [arXiv:0707.1378] [INSPIRE].

    ADS  Google Scholar 

  56. [56]

    M. Cacciari, G.P. Salam and G. Soyez, The catchment area of jets, JHEP 04 (2008) 005 [arXiv:0802.1188] [INSPIRE].

    ADS  Google Scholar 

  57. [57]

    G. Soyez, G.P. Salam, J. Kim, S. Dutta and M. Cacciari, Pileup subtraction for jet shapes, Phys. Rev. Lett. 110 (2013) 162001 [arXiv:1211.2811] [INSPIRE].

    ADS  Google Scholar 

  58. [58]

    P. Berta, M. Spousta, D.W. Miller and R. Leitner, Particle-level pileup subtraction for jets and jet shapes, JHEP 06 (2014) 092 [arXiv:1403.3108] [INSPIRE].

    ADS  Google Scholar 

  59. [59]

    D. Bertolini, P. Harris, M. Low and N. Tran, Pileup per particle identification, JHEP 10 (2014) 059 [arXiv:1407.6013] [INSPIRE].

    ADS  Google Scholar 

  60. [60]

    G. Soyez, Pileup mitigation at the LHC: a theorist’s view, Phys. Rept. 803 (2019) 1 [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  61. [61]

    P. Berta, L. Masetti, D.W. Miller and M. Spousta, Pileup and underlying event mitigation with iterative constituent subtraction, JHEP 08 (2019) 175 [arXiv:1905.03470] [INSPIRE].

    ADS  Google Scholar 

  62. [62]

    J.M. Ortega and W.C. Rheinboldt, Iterative solution of nonlinear equations in several variables, Society for Industrial and Applied Mathematics , U.S.A. (2000).

  63. [63]

    D. Gilbarg and N. S. Trudinger, Elliptic partial differential equation of second order, Springer, Berlin Germany (2001).

  64. [64]

    J. Brewer, J.G. Milhano and J. Thaler, Sorting out quenched jets, Phys. Rev. Lett. 122 (2019) 222301 [arXiv:1812.05111] [INSPIRE].

    ADS  Google Scholar 

  65. [65]

    CMS collaboration, Performance of quark/gluon discrimination in 8 TeV pp data, CMS-PAS-JME-13-002 (2013).

  66. [66]

    J.D. Bjorken and S.J. Brodsky, Statistical model for electron-positron annihilation into hadrons, Phys. Rev. D 1 (1970) 1416 [INSPIRE].

    ADS  Google Scholar 

  67. [67]

    L. Clavelli and D. Wyler, Kinematical bounds on jet variables and the heavy jet mass distribution, Phys. Lett. B 103 (1981) 383 [INSPIRE].

    ADS  Google Scholar 

  68. [68]

    M. Dasgupta, A. Fregoso, S. Marzani and G.P. Salam, Towards an understanding of jet substructure, JHEP 09 (2013) 029 [arXiv:1307.0007] [INSPIRE].

    ADS  Google Scholar 

  69. [69]

    J. Pumplin, How to tell quark jets from gluon jets, Phys. Rev. D 44 (1991) 2025 [INSPIRE].

    ADS  Google Scholar 

  70. [70]

    C.F. Berger, T. Kucs and G.F. Sterman, Event shape/energy flow correlations, Phys. Rev. D 68 (2003) 014012 [hep-ph/0303051] [INSPIRE].

  71. [71]

    G. Parisi, Super inclusive cross-sections, Phys. Lett. B 74 (1978) 65 [INSPIRE].

    ADS  Google Scholar 

  72. [72]

    J.F. Donoghue, F.E. Low and S.-Y. Pi, Tensor analysis of hadronic jets in quantum chromodynamics, Phys. Rev. D 20 (1979) 2759 [INSPIRE].

    ADS  Google Scholar 

  73. [73]

    R. 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].

    ADS  Google Scholar 

  74. [74]

    S. Catani and B.R. Webber, Infrared safe but infinite: soft gluon divergences inside the physical region, JHEP 10 (1997) 005 [hep-ph/9710333] [INSPIRE].

  75. [75]

    A.J. Larkoski, G.P. Salam and J. Thaler, Energy correlation functions for jet substructure, JHEP 06 (2013) 108 [arXiv:1305.0007] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  76. [76]

    A.J. Larkoski, I. Moult and D. Neill, Power counting to better jet observables, JHEP 12 (2014) 009 [arXiv:1409.6298] [INSPIRE].

    ADS  Google Scholar 

  77. [77]

    I. Moult, L. Necib and J. Thaler, New angles on energy correlation functions, JHEP 12 (2016) 153 [arXiv:1609.07483] [INSPIRE].

    ADS  Google Scholar 

  78. [78]

    C. Frye, A.J. Larkoski, M.D. Schwartz and K. Yan, Precision physics with pile-up insensitive observables, arXiv:1603.06375 [INSPIRE].

  79. [79]

    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].

    ADS  Google Scholar 

  80. [80]

    S. Marzani, L. Schunk and G. Soyez, A study of jet mass distributions with grooming, JHEP 07 (2017) 132 [arXiv:1704.02210] [INSPIRE].

    ADS  Google Scholar 

  81. [81]

    S. Marzani, L. Schunk and G. Soyez, The jet mass distribution after Soft Drop, Eur. Phys. J. C 78 (2018) 96 [arXiv:1712.05105] [INSPIRE].

    ADS  Google Scholar 

  82. [82]

    A.H. Hoang, S. Mantry, A. Pathak and I.W. Stewart, Nonperturbative corrections to soft drop jet mass, JHEP 12 (2019) 002 [arXiv:1906.11843] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  83. [83]

    ATLAS collaboration, Impact of alternative inputs and jet grooming on large-R jet performance, ATL-PHYS-PUB-2019-027 (2019).

  84. [84]

    ATLAS collaboration, Measurement of soft-drop jet observables in pp collisions with the ATLAS detector at \( \sqrt{s} \) = 13 TeV, Phys. Rev. D 101 (2020) 052007 [arXiv:1912.09837] [INSPIRE].

  85. [85]

    G.F. Sterman, Review of theoretical status: the long and short of high energy Jets, hep-ph/0606032 [INSPIRE].

  86. [86]

    B.T. Elder et al., Generalized fragmentation functions for fractal jet observables, JHEP 06 (2017) 085 [arXiv:1704.05456] [INSPIRE].

    ADS  Google Scholar 

  87. [87]

    S. Alioli et al., Combining higher-order resummation with multiple NLO calculations and parton showers in GENEVA, JHEP 09 (2013) 120 [arXiv:1211.7049] [INSPIRE].

    ADS  Google Scholar 

  88. [88]

    S. Alioli et al., Matching fully differential NNLO calculations and parton showers, JHEP 06 (2014) 089 [arXiv:1311.0286] [INSPIRE].

    ADS  Google Scholar 

  89. [89]

    S. Alioli et al., Drell-Yan production at NNLL’+NNLO matched to parton showers, Phys. Rev. D 92 (2015) 094020 [arXiv:1508.01475] [INSPIRE].

    ADS  Google Scholar 

  90. [90]

    A.J. Larkoski and I. Moult, The singular behavior of jet substructure observables, Phys. Rev. D 93 (2016) 014017 [arXiv:1510.08459] [INSPIRE].

    ADS  Google Scholar 

  91. [91]

    A. De Rujula, J.R. Ellis, E.G. Floratos and M.K. Gaillard, QCD Predictions for Hadronic Final States in e+ e Annihilation, Nucl. Phys. B 138 (1978) 387 [INSPIRE].

    ADS  Google Scholar 

  92. [92]

    D.P. Barber et al., Tests of quantum chromodynamics and a direct measurement of the strong coupling constant αS at \( \sqrt{s} \) = 30 GeV, Phys. Lett. B 89 (1979) 139 [INSPIRE].

    ADS  Google Scholar 

  93. [93]

    JADE collaboration, Observation of planar three jet events in e+ e annihilation and evidence for gluon Bremsstrahlung, Phys. Lett. B 91 (1980) 142 [INSPIRE].

  94. [94]

    TASSO collaboration, Jet production and fragmentation in e+ e annihilation at 12 GeV to 43 GeV, Z. Phys. C 22 (1984) 307 [INSPIRE].

  95. [95]

    D. Bender et al., Study of quark fragmentation at 29 GeV: global jet parameters and single particle distributions, Phys. Rev. D 31 (1985) 1 [INSPIRE].

    ADS  Google Scholar 

  96. [96]

    MARK-II collaboration, First measurements of hadronic decays of the Z boson, Phys. Rev. Lett. 63 (1989) 1558 [INSPIRE].

  97. [97]

    AMY collaboration, Multi-hadron event properties in e+ e annihilation at \( \sqrt{s} \) = 52 GeV to 57 GeV, Phys. Rev. D 41 (1990) 2675 [INSPIRE].

  98. [98]

    ALEPH collaboration, Measurement of αs from the structure of particle clusters produced in hadronic Z decays, Phys. Lett. B 257 (1991) 479 [INSPIRE].

  99. [99]

    TASSO collaboration, Global jet properties at 14 GeV to 44 GeV Center-of-mass Energy in e+ e annihilation, Z. Phys. C 47 (1990) 187 [INSPIRE].

  100. [100]

    SLD collaboration, Measurement of αs (\( {M}_Z^2 \)) from hadronic event observables at the Z0 resonance, Phys. Rev. D 51 (1995) 962 [hep-ex/9501003] [INSPIRE].

  101. [101]

    ALEPH collaboration, Studies of QCD at e+ e centre-of-mass energies between 91 GeV and 209,GeV, Eur. Phys. J. C 35 (2004) 457 [INSPIRE].

  102. [102]

    DELPHI collaboration, 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].

  103. [103]

    L3 collaboration, 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].

  104. [104]

    OPAL collaboration, Measurement of event shape distributions and moments in e+ e → hadrons at 91 GeV–209 GeV and a determination of αs , Eur. Phys. J. C 40 (2005) 287 [hep-ex/0503051] [INSPIRE].

  105. [105]

    A. Gehrmann-De Ridder, T. Gehrmann, E.W.N. Glover and G. Heinrich, Second-order QCD corrections to the thrust distribution, Phys. Rev. Lett. 99 (2007) 132002 [arXiv:0707.1285] [INSPIRE].

    ADS  Google Scholar 

  106. [106]

    A. Gehrmann-De Ridder, T. Gehrmann, E.W.N. Glover and G. Heinrich, NNLO corrections to event shapes in e+ e annihilation, JHEP 12 (2007) 094 [arXiv:0711.4711] [INSPIRE].

    ADS  Google Scholar 

  107. [107]

    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].

    ADS  Google Scholar 

  108. [108]

    S. Weinzierl, Event shapes and jet rates in electron-positron annihilation at NNLO, JHEP 06 (2009) 041 [arXiv:0904.1077] [INSPIRE].

    ADS  Google Scholar 

  109. [109]

    R. Abbate et al., Thrust at N3LL with Power Corrections and a Precision Global Fit for αs (mZ), Phys. Rev. D 83 (2011) 074021 [arXiv:1006.3080] [INSPIRE].

    ADS  Google Scholar 

  110. [110]

    R. Abbate et al., Precision thrust cumulant moments at N3LL, Phys. Rev. D 86 (2012) 094002 [arXiv:1204.5746] [INSPIRE].

    ADS  Google Scholar 

  111. [111]

    P.E.L. Rakow and B.R. Webber, Transverse momentum moments of hadron distributions in QCD jets, Nucl. Phys. B 191 (1981) 63 [INSPIRE].

    ADS  Google Scholar 

  112. [112]

    R. Ellis and B.R. Webber, QCD jet broadening in hadron hadron collisions, Conf. Proc. C860623 (1986) 74 [INSPIRE].

    Google Scholar 

  113. [113]

    S. Catani, G. Turnock and B.R. Webber, Jet broadening measures in e+ e annihilation, Phys. Lett. B 295 (1992) 269 [INSPIRE].

    ADS  Google Scholar 

  114. [114]

    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].

  115. [115]

    S. Brandt and H.D. Dahmen, Axes and scalar measures of two-jet and three-jet events, Z. Phys. C 1 (1979) 61 [INSPIRE].

    ADS  Google Scholar 

  116. [116]

    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].

    ADS  Google Scholar 

  117. [117]

    C.F. Berger et al., Higgs production with a central jet veto at NNLL+NNLO, JHEP 04 (2011) 092 [arXiv:1012.4480] [INSPIRE].

    ADS  Google Scholar 

  118. [118]

    T.T. Jouttenus et al., Jet mass spectra in Higgs boson plus one jet at next-to-next-to-leading logarithmic order, Phys. Rev. D 88 (2013) 054031 [arXiv:1302.0846] [INSPIRE].

    ADS  Google Scholar 

  119. [119]

    J. Rabin, J. Delon and Y. Gousseau, Transportation distances on the circle, J. Math. Imag. Vision 41 (2011) 147.

    MathSciNet  MATH  Google Scholar 

  120. [120]

    L.G. Almeida et al., Substructure of high-pT jets at the LHC, Phys. Rev. D 79 (2009) 074017 [arXiv:0807.0234] [INSPIRE].

    ADS  Google Scholar 

  121. [121]

    A.J. Larkoski, J. Thaler and W.J. Waalewijn, Gaining (mutual) information about quark/gluon discrimination, JHEP 11 (2014) 129 [arXiv:1408.3122] [INSPIRE].

    ADS  Google Scholar 

  122. [122]

    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, Phys. Rept. 841 (2020) 1 [arXiv:1709.04464] [INSPIRE].

    ADS  Google Scholar 

  123. [123]

    R. Kogler et al., Jet substructure at the Large Hadron Collider: experimental review, Rev. Mod. Phys. 91 (2019) 045003 [arXiv:1803.06991] [INSPIRE].

    ADS  Google Scholar 

  124. [124]

    S. Marzani, G. Soyez and M. Spannowsky, Looking inside jets: an introduction to jet substructure and boosted-object phenomenology, Springer, Germany (2019).

    Google Scholar 

  125. [125]

    K. Datta and A. Larkoski, How much information is in a jet?, JHEP 06 (2017) 073 [arXiv:1704.08249] [INSPIRE].

    ADS  Google Scholar 

  126. [126]

    K. Datta and A.J. Larkoski, Novel jet observables from machine learning, JHEP 03 (2018) 086 [arXiv:1710.01305] [INSPIRE].

    ADS  Google Scholar 

  127. [127]

    A.J. Larkoski and E.M. Metodiev, A theory of quark vs. gluon discrimination, JHEP 10 (2019) 014 [arXiv:1906.01639] [INSPIRE].

  128. [128]

    J. Thaler, Jet maximization, axis minimization and stable cone finding, Phys. Rev. D 92 (2015) 074001 [arXiv:1506.07876] [INSPIRE].

    ADS  Google Scholar 

  129. [129]

    G.C. Blazey et al., Run II jet physics, in the proceedings of QCD and weak boson physics in Run II, March 4–6 and June 3–4, Batavia, U.S.A. (1999).

  130. [130]

    S.D. Ellis, J. Huston and M. Tonnesmann, On building better cone jet algorithms, eConf C010630 (2001) 513 [hep-ph/0111434] [INSPIRE].

  131. [131]

    G.P. Salam and G. Soyez, A practical seedless infrared-safe cone jet algorithm, JHEP 05 (2007) 086 [arXiv:0704.0292] [INSPIRE].

    ADS  Google Scholar 

  132. [132]

    H. Georgi, A simple alternative to jet-clustering algorithms, arXiv:1408.1161 [INSPIRE].

  133. [133]

    S.-F. Ge, The Georgi algorithms of jet clustering, JHEP 05 (2015) 066 [arXiv:1408.3823] [INSPIRE].

    ADS  Google Scholar 

  134. [134]

    Y. Bai, Z. Han and R. Lu, \( {J}_{E_T} \) : a global jet finding algorithm, JHEP 03 (2015) 102 [arXiv:1411.3705] [INSPIRE].

  135. [135]

    Y. Bai, Z. Han and R. Lu, \( {J}_{E_T}^{\mathrm{II}} \) : a two-prong jet finding algorithm, arXiv:1509.07522 [INSPIRE].

  136. [136]

    A.Y. Wei, P. Naik, A.W. Harrow and J. Thaler, Quantum algorithms for jet clustering, Phys. Rev. D 101 (2020) 094015 [arXiv:1908.08949] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  137. [137]

    M. Cacciari, G.P. Salam and G. Soyez, The anti-kt jet clustering algorithm, JHEP 04 (2008) 063 [arXiv:0802.1189] [INSPIRE].

    ADS  MATH  Google Scholar 

  138. [138]

    M. Cacciari, G.P. Salam and G. Soyez, FastJet user manual, Eur. Phys. J. C 72 (2012) 1896 [arXiv:1111.6097] [INSPIRE].

    ADS  MATH  Google Scholar 

  139. [139]

    Y.L. Dokshitzer, G.D. Leder, S. Moretti and B.R. Webber, Better jet clustering algorithms, JHEP 08 (1997) 001 [hep-ph/9707323] [INSPIRE].

  140. [140]

    M. Wobisch and T. Wengler, Hadronization corrections to jet cross-sections in deep inelastic scattering, in the proceedings of the Workshop on Monte Carlo Generators for HERA Physics (Plenary Starting Meeting), April 27–30, Hamburg, Germany (1998), hep-ph/9907280 [INSPIRE].

  141. [141]

    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].

  142. [142]

    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].

    ADS  Google Scholar 

  143. [143]

    D. Krohn, M.D. Schwartz, M. Low and L.-T. Wang, Jet cleansing: pileup removal at high luminosity, Phys. Rev. D 90 (2014) 065020 [arXiv:1309.4777] [INSPIRE].

    ADS  Google Scholar 

  144. [144]

    M. Cacciari, G.P. Salam and G. Soyez, Use of charged-track information to subtract neutral pileup, Phys. Rev. D 92 (2015) 014003 [arXiv:1404.7353] [INSPIRE].

    ADS  Google Scholar 

  145. [145]

    M. Cacciari, G.P. Salam and G. Soyez, SoftKiller, a particle-level pileup removal method, Eur. Phys. J. C 75 (2015) 59 [arXiv:1407.0408] [INSPIRE].

    ADS  Google Scholar 

  146. [146]

    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].

    ADS  Google Scholar 

  147. [147]

    P. Hansen, J.W. Monk and C. Wiglesworth, A wavelet based pile-up mitigation method for the LHC upgrade, arXiv:1812.07412 [INSPIRE].

  148. [148]

    J. Arjona Martínez et al., Pileup mitigation at the Large Hadron Collider with graph neural networks, Eur. Phys. J. Plus 134 (2019) 333 [arXiv:1810.07988] [INSPIRE].

    Google Scholar 

  149. [149]

    V. Hartmann and D. Schuhmacher, Semi-discrete optimal transport-the case p = 1, arXiv:1706.07650.

  150. [150]

    L. Ambrosio, N. Gigli and G. Savare, Gradient flows in metric spaces and in the space of probability measures, Birkhäuser, Basel, Switzerland (2005).

  151. [151]

    F. Aurenhammer, R. Klein and D.-T. Lee, Voronoi diagrams and Delaunay triangulations, World Scientific, Singapore (2013).

    MATH  Google Scholar 

  152. [152]

    CMS collaboration, Jet primary dataset in AOD format from RunA of 2011 (/Jet/Run2011A-12Oct2013-v1/AOD), CERN Open Data Portal (2016).

  153. [153]

    M. Liero, A. Mielke and G. Savaré, Optimal entropy-transport problems and a new hellinger–kantorovich distance between positive measures, Inv. Math. 211 (2018) 969.

    ADS  MathSciNet  MATH  Google Scholar 

  154. [154]

    D.P. Bourne, B. Schmitzer and B. Wirth, Semi-discrete unbalanced optimal transport and quantization, arXiv:1808.01962.

  155. [155]

    M. I. Karavelas and M. Yvinec, Dynamic additively weighted Voronoi diagrams in 2D, in the proceedings of Algorithms — ESA 2002, 10th Annual European Symposium, September 17–21, Rome, Italy (2002), Lecture Notes in Computer Science volume 2641, Springer (2002).

  156. [156]

    D. Geiß, R. Klein, R. Penninger and G. Rote, Optimally solving a transportation problem using voronoi diagrams, Comput. Geom. 46 (2013) 1009.

    MathSciNet  MATH  Google Scholar 

  157. [157]

    S. Xin et al., Centroidal power diagrams with capacity constraints: computation, applications, and extension, ACM Trans. Graph. 35 (2016) 244.

    Google Scholar 

  158. [158]

    J. Erdmenger, K.T. Grosvenor and R. Jefferson, Information geometry in quantum field theory: lessons from simple examples, SciPost Phys. 8 (2020) 073 [arXiv:2001.02683] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  159. [159]

    C. Cheung, TASI lectures on scattering amplitudes, in the proceedings of Theoretical Advanced Study Institute in Elementary Particle Physics: Anticipating the Next Discoveries in Particle Physics (TASI 2016), June 6–July 1, Boulder, U.S.A. (2016), arXiv:1708.03872 [INSPIRE].

  160. [160]

    C. Frye, H. Hannesdottir, N. Paul, M.D. Schwartz and K. Yan, Infrared finiteness and forward scattering, Phys. Rev. D 99 (2019) 056015 [arXiv:1810.10022] [INSPIRE].

    ADS  Google Scholar 

  161. [161]

    H. Hannesdottir and M.D. Schwartz, A finite S-matrix, arXiv:1906.03271 [INSPIRE].

  162. [162]

    H. Hannesdottir and M.D. Schwartz, S-Matrix for massless particles, Phys. Rev. D 101 (2020) 105001 [arXiv:1911.06821] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  163. [163]

    I. Jubran, A. Maalouf and D. Feldman, Introduction to coresets: accurate coresets, arXiv:1910.08707.

  164. [164]

    A. W. Harrow, Small quantum computers and large classical data sets, arXiv:2004.00026.

  165. [165]

    S. Claici and J. Solomon, Wasserstein coresets for Lipschitz costs, [arXiv:1805.07412].

  166. [166]

    S. Bianchini and A. Brancolini, Estimates on path functionals over wasserstein spaces, SIAM J. Math. Anal. 42 (2010) 1179.

    MathSciNet  MATH  Google Scholar 

  167. [167]

    M. Agueh and G. Carlier, Barycenters in the Wasserstein space, SIAM J. Math. Anal. 43 (2011) 904.

    MathSciNet  MATH  Google Scholar 

  168. [168]

    J. Bertrand and B. R. Kloeckner, A geometric study of wasserstein spaces: an addendum on the boundary, in the proceedings of the 1st International Conference — Geometric Science of Information (GSI 2013), August 28–30, Paris, France (2013).

  169. [169]

    T. L. Gouic and J. Loubes, Barycenter in Wasserstein spaces: existence and consistency, in the proceedings of the 2nd International Conference — Geometric Science of Information (GSI 2013), October 28–30, Palaiseau, France (2015).

  170. [170]

    C.W. Bauer, A. Hornig and F.J. Tackmann, Factorization for generic jet production, Phys. Rev. D 79 (2009) 114013 [arXiv:0808.2191] [INSPIRE].

    ADS  Google Scholar 

  171. [171]

    A.V. Belitsky et al., Energy-energy correlations in N = 4 supersymmetric Yang-Mills theory, Phys. Rev. Lett. 112 (2014) 071601 [arXiv:1311.6800] [INSPIRE].

    ADS  Google Scholar 

  172. [172]

    L.J. Dixon, I. Moult and H.X. Zhu, Collinear limit of the energy-energy correlator, Phys. Rev. D 100 (2019) 014009 [arXiv:1905.01310] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  173. [173]

    H. Chen et al., Three point energy correlators in the collinear limit: symmetries, dualities and analytic results, arXiv:1912.11050 [INSPIRE].

  174. [174]

    H. Elvang and Y.-t. Huang, Scattering amplitudes, arXiv:1308.1697 [INSPIRE].

  175. [175]

    J.J.M. Carrasco, Gauge and gravity amplitude relations, in the proceedings of the Theoretical Advanced Study Institute in Elementary Particle Physics: Journeys Through the Precision Frontier: Amplitudes for Colliders , June 2–27, Boulder, U.S.A. (2014), arXiv:1506.00974 [INSPIRE].

  176. [176]

    M.C. Romao, N.F. Castro, J.G. Milhano, R. Pedro and T. Vale, Use of a generalized energy mover’s distance in the search for rare phenomena at colliders, arXiv:2004.09360 [INSPIRE].

  177. [177]

    A. Banfi, G.P. Salam and G. Zanderighi, Infrared safe definition of jet flavor, Eur. Phys. J. C 47 (2006) 113 [hep-ph/0601139] [INSPIRE].

  178. [178]

    A. Radovic et al., Machine learning at the energy and intensity frontiers of particle physics, Nature 560 (2018) 41.

    ADS  Google Scholar 

  179. [179]

    G. Carleo et al., Machine learning and the physical sciences, Rev. Mod. Phys. 91 (2019) 045002 [arXiv:1903.10563] [INSPIRE].

    ADS  Google Scholar 

  180. [180]

    J. Hajer, Y.-Y. Li, T. Liu and H. Wang, Novelty detection meets collider physics, Phys. Rev. D 101 (2020) 076015 [arXiv:1807.10261] [INSPIRE].

    ADS  Google Scholar 

  181. [181]

    T. Heimel, G. Kasieczka, T. Plehn and J.M. Thompson, QCD or What?, SciPost Phys. 6 (2019) 030 [arXiv:1808.08979] [INSPIRE].

    ADS  Google Scholar 

  182. [182]

    M. Farina, Y. Nakai and D. Shih, Searching for new physics with deep autoencoders, Phys. Rev. D 101 (2020) 075021 [arXiv:1808.08992] [INSPIRE].

    ADS  Google Scholar 

  183. [183]

    T.S. Roy and A.H. Vijay, A robust anomaly finder based on autoencoder, arXiv:1903.02032 [INSPIRE].

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Komiske, P.T., Metodiev, E.M. & Thaler, J. The hidden geometry of particle collisions. J. High Energ. Phys. 2020, 6 (2020).

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  • Jets
  • QCD Phenomenology