Analytic next-to-leading order calculation of energy-energy correlation in gluon-initiated Higgs decays

  • Ming-xing Luo
  • Vladyslav Shtabovenko
  • Tong-Zhi Yang
  • Hua Xing ZhuEmail author
Open Access
Regular Article - Theoretical Physics


The energy-energy correlation (EEC) function in e+e annihilation is currently the only QCD event shape observable for which we know the full analytic result at the next-to-leading order (NLO). In this work we calculate the EEC observable for gluon initiated Higgs decay analytically at NLO in the Higgs Effective Field Theory (HEFT) framework and provide the full results expressed in terms of classical polylogarithms, including the asymptotic behavior in the collinear and back-to-back limits. This observable can be, in principle, measured at the future e+e colliders such as CEPC, ILC, FCC-ee or CLIC. It provides an interesting opportunity to simultaneously probe our understanding of the strong and Higgs sectors and can be used for the determinations of the strong coupling.


NLO Computations Jets 


Open Access

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  1. [1]
    CEPC Study Group collaboration, CEPC Conceptual Design Report: Volume 1Accelerator, arXiv:1809.00285 [INSPIRE].
  2. [2]
    CEPC Study Group collaboration, CEPC Conceptual Design Report: Volume 2Physics & Detector, arXiv:1811.10545 [INSPIRE].
  3. [3]
    T. Behnke et al., The International Linear Collider Technical Design ReportVolume 1: Executive Summary, arXiv:1306.6327 [INSPIRE].
  4. [4]
    H. Baer et al., The International Linear Collider Technical Design ReportVolume 2: Physics, arXiv:1306.6352 [INSPIRE].
  5. [5]
    TLEP Design Study Working Group collaboration, First Look at the Physics Case of TLEP, JHEP 01 (2014) 164 [arXiv:1308.6176] [INSPIRE].
  6. [6]
    M. Aicheler et al., A Multi-TeV Linear Collider Based on CLIC Technology, CERN-2012-007 [INSPIRE].
  7. [7]
    J. de Blas et al., The CLIC Potential for New Physics, arXiv:1812.02093 [INSPIRE].
  8. [8]
    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].
  9. [9]
    DELPHI collaboration, The measurement of α s from event shapes with the DELPHI detector at the highest LEP energies, Eur. Phys. J. C 37 (2004) 1 [hep-ex/0406011] [INSPIRE].
  10. [10]
    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].
  11. [11]
    OPAL collaboration, Measurement of event shape distributions and moments in e + e hadrons at 91-GeV - 209-GeV and a determination of α s, Eur. Phys. J. C 40 (2005) 287 [hep-ex/0503051] [INSPIRE].
  12. [12]
    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].
  13. [13]
    E. Farhi, A QCD Test for Jets, Phys. Rev. Lett. 39 (1977) 1587 [INSPIRE].CrossRefGoogle Scholar
  14. [14]
    L. Clavelli and D. Wyler, Kinematical Bounds on Jet Variables and the Heavy Jet Mass Distribution, Phys. Lett. 103B (1981) 383 [INSPIRE].
  15. [15]
    P.E.L. Rakow and B.R. Webber, Transverse Momentum Moments of Hadron Distributions in QCD Jets, Nucl. Phys. B 191 (1981) 63 [INSPIRE].
  16. [16]
    R.K. Ellis and B.R. Webber, QCD Jet Broadening in Hadron Hadron Collisions, Conf. Proc. C 860623 (1986) 74 [INSPIRE].
  17. [17]
    S. Catani, G. Turnock and B.R. Webber, Jet broadening measures in e + e annihilation, Phys. Lett. B 295 (1992) 269 [INSPIRE].
  18. [18]
    G. Parisi, Super Inclusive Cross-Sections, Phys. Lett. 74B (1978) 65 [INSPIRE].
  19. [19]
    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].
  20. [20]
    S. Catani, Y.L. Dokshitzer, M. Olsson, G. Turnock and B.R. Webber, New clustering algorithm for multijet cross sections in e + e annihilation, Phys. Lett. B 269 (1991) 432 [INSPIRE].
  21. [21]
    H1 collaboration, Measurement of event shape variables in deep-inelastic scattering at HERA, Eur. Phys. J. C 46 (2006) 343 [hep-ex/0512014] [INSPIRE].
  22. [22]
    CDF collaboration, Measurement of Event Shapes in Proton-Antiproton Collisions at Center-of-Mass Energy 1.96 TeV, Phys. Rev. D 83 (2011) 112007 [arXiv:1103.5143] [INSPIRE].
  23. [23]
    A. Banfi, G.P. Salam and G. Zanderighi, Phenomenology of event shapes at hadron colliders, JHEP 06 (2010) 038 [arXiv:1001.4082] [INSPIRE].CrossRefzbMATHGoogle Scholar
  24. [24]
    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].CrossRefGoogle Scholar
  25. [25]
    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].
  26. [26]
    S. Weinzierl, Event shapes and jet rates in electron-positron annihilation at NNLO, JHEP 06 (2009) 041 [arXiv:0904.1077] [INSPIRE].CrossRefGoogle Scholar
  27. [27]
    A. Gehrmann-De Ridder, T. Gehrmann, E.W.N. Glover and G. Heinrich, EERAD3: Event shapes and jet rates in electron-positron annihilation at order α s3, Comput. Phys. Commun. 185 (2014) 3331 [arXiv:1402.4140] [INSPIRE].
  28. [28]
    V. Del Duca et al., Jet production in the CoLoRFulNNLO method: event shapes in electron-positron collisions, Phys. Rev. D 94 (2016) 074019 [arXiv:1606.03453] [INSPIRE].
  29. [29]
    D. de Florian and M. Grazzini, The Back-to-back region in e + e energy-energy correlation, Nucl. Phys. B 704 (2005) 387 [hep-ph/0407241] [INSPIRE].
  30. [30]
    T. Becher and M.D. Schwartz, A precise determination of α s from LEP thrust data using effective field theory, JHEP 07 (2008) 034 [arXiv:0803.0342] [INSPIRE].CrossRefGoogle Scholar
  31. [31]
    Y.-T. Chien and M.D. Schwartz, Resummation of heavy jet mass and comparison to LEP data, JHEP 08 (2010) 058 [arXiv:1005.1644] [INSPIRE].CrossRefzbMATHGoogle Scholar
  32. [32]
    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 alphas(mZ), Phys. Rev. D 83 (2011) 074021 [arXiv:1006.3080] [INSPIRE].
  33. [33]
    P.F. Monni, T. Gehrmann and G. Luisoni, Two-Loop Soft Corrections and Resummation of the Thrust Distribution in the Dijet Region, JHEP 08 (2011) 010 [arXiv:1105.4560] [INSPIRE].CrossRefzbMATHGoogle Scholar
  34. [34]
    T. Becher and G. Bell, NNLL Resummation for Jet Broadening, JHEP 11 (2012) 126 [arXiv:1210.0580] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  35. [35]
    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].
  36. [36]
    A.H. Hoang, D.W. Kolodrubetz, V. Mateu and I.W. Stewart, C-parameter distribution at N 3 LLincluding power corrections, Phys. Rev. D 91 (2015) 094017 [arXiv:1411.6633] [INSPIRE].
  37. [37]
    A. Banfi, H. McAslan, P.F. Monni and G. Zanderighi, The two-jet rate in e + e at next-to-next-to-leading-logarithmic order, Phys. Rev. Lett. 117 (2016) 172001 [arXiv:1607.03111] [INSPIRE].
  38. [38]
    Z. Tulipánt, A. Kardos and G. Somogyi, Energy-energy correlation in electron-positron annihilation at NNLL + NNLO accuracy, Eur. Phys. J. C 77 (2017) 749 [arXiv:1708.04093] [INSPIRE].
  39. [39]
    I. Moult and H.X. Zhu, Simplicity from Recoil: The Three-Loop Soft Function and Factorization for the Energy-Energy Correlation, JHEP 08 (2018) 160 [arXiv:1801.02627] [INSPIRE].CrossRefGoogle Scholar
  40. [40]
    A. Kardos, S. Kluth, G. Somogyi, Z. Tulipánt and A. Verbytskyi, Precise determination of α S (M Z) from a global fit of energy-energy correlation to NNLO+NNLL predictions, Eur. Phys. J. C 78 (2018) 498 [arXiv:1804.09146] [INSPIRE].
  41. [41]
    A. Banfi, B.K. El-Menoufi and P.F. Monni, The Sudakov radiator for jet observables and the soft physical coupling, JHEP 01 (2019) 083 [arXiv:1807.11487] [INSPIRE].CrossRefGoogle Scholar
  42. [42]
    G. Bell, A. Hornig, C. Lee and J. Talbert, e + e angularity distributions at NNLL accuracy, JHEP 01 (2019) 147 [arXiv:1808.07867] [INSPIRE].
  43. [43]
    A. Verbytskyi et al., High precision determination of α s from a global fit of jet rates, [arXiv:1902.08158] [INSPIRE].
  44. [44]
    Z. Nagy, Three jet cross-sections in hadron hadron collisions at next-to-leading order, Phys. Rev. Lett. 88 (2002) 122003 [hep-ph/0110315] [INSPIRE].
  45. [45]
    Z. Nagy, Next-to-leading order calculation of three jet observables in hadron hadron collision, Phys. Rev. D 68 (2003) 094002 [hep-ph/0307268] [INSPIRE].
  46. [46]
    S. Catani and M.H. Seymour, The dipole formalism for the calculation of QCD jet cross-sections at next-to-leading order, Phys. Lett. B 378 (1996) 287 [hep-ph/9602277] [INSPIRE].
  47. [47]
    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].
  48. [48]
    J. Gao, Probing light-quark Yukawa couplings via hadronic event shapes at lepton colliders, JHEP 01 (2018) 038 [arXiv:1608.01746] [INSPIRE].CrossRefGoogle Scholar
  49. [49]
    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].CrossRefGoogle Scholar
  50. [50]
    F. Wilczek, Decays of Heavy Vector Mesons Into Higgs Particles, Phys. Rev. Lett. 39 (1977) 1304 [INSPIRE].CrossRefGoogle Scholar
  51. [51]
    M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, Remarks on Higgs Boson Interactions with Nucleons, Phys. Lett. 78B (1978) 443 [INSPIRE].CrossRefGoogle Scholar
  52. [52]
    T. Inami, T. Kubota and Y. Okada, Effective Gauge Theory and the Effect of Heavy Quarks in Higgs Boson Decays, Z. Phys. C 18 (1983) 69 [INSPIRE].
  53. [53]
    B.A. Kniehl and M. Spira, Low-energy theorems in Higgs physics, Z. Phys. C 69 (1995) 77 [hep-ph/9505225] [INSPIRE].
  54. [54]
    P.A. Baikov, K.G. Chetyrkin and J.H. Kühn, Five-Loop Running of the QCD coupling constant, Phys. Rev. Lett. 118 (2017) 082002 [arXiv:1606.08659] [INSPIRE].
  55. [55]
    K.G. Chetyrkin, B.A. Kniehl and M. Steinhauser, Hadronic Higgs decay to order α S4, Phys. Rev. Lett. 79 (1997) 353 [hep-ph/9705240] [INSPIRE].
  56. [56]
    L.J. Dixon, M.-X. Luo, V. Shtabovenko, T.-Z. Yang and H.X. Zhu, Analytical Computation of Energy-Energy Correlation at Next-to-Leading Order in QCD, Phys. Rev. Lett. 120 (2018) 102001 [arXiv:1801.03219] [INSPIRE].CrossRefGoogle Scholar
  57. [57]
    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].
  58. [58]
    A.V. Belitsky, S. Hohenegger, G.P. Korchemsky, E. Sokatchev and A. Zhiboedov, Event shapes in \( \mathcal{N}=4 \) super-Yang-Mills theory, Nucl. Phys. B 884 (2014) 206 [arXiv:1309.1424] [INSPIRE].
  59. [59]
    A.V. Belitsky, S. Hohenegger, G.P. Korchemsky, E. Sokatchev and A. Zhiboedov, Energy-Energy Correlations in N = 4 Supersymmetric Yang-Mills Theory, Phys. Rev. Lett. 112 (2014) 071601 [arXiv:1311.6800] [INSPIRE].
  60. [60]
    J.M. Henn, E. Sokatchev, K. Yan and A. Zhiboedov, Energy-energy correlations at next-to-next-to-leading order, arXiv:1903.05314 [INSPIRE].
  61. [61]
    J. Gao, Y. Gong, W.-L. Ju and L.L. Yang, Thrust distribution in Higgs decays at the next-to-leading order and beyond, JHEP 03 (2019) 030 [arXiv:1901.02253] [INSPIRE].CrossRefGoogle Scholar
  62. [62]
    C. Anastasiou and K. Melnikov, Higgs boson production at hadron colliders in NNLO QCD, Nucl. Phys. B 646 (2002) 220 [hep-ph/0207004] [INSPIRE].
  63. [63]
    C. Anastasiou, L.J. Dixon, K. Melnikov and F. Petriello, Dilepton rapidity distribution in the Drell-Yan process at NNLO in QCD, Phys. Rev. Lett. 91 (2003) 182002 [hep-ph/0306192] [INSPIRE].
  64. [64]
    K.G. Chetyrkin and F.V. Tkachov, Integration by Parts: The Algorithm to Calculate β-functions in 4 Loops, Nucl. Phys. B 192 (1981) 159 [INSPIRE].
  65. [65]
    F.V. Tkachov, A Theorem on Analytical Calculability of Four Loop Renormalization Group Functions, Phys. Lett. 100B (1981) 65 [INSPIRE].MathSciNetCrossRefGoogle Scholar
  66. [66]
    A.V. Kotikov, Differential equation method: The calculation of N point Feynman diagrams, Phys. Lett. B 267 (1991) 123 [Erratum ibid. B 295 (1992) 409] [INSPIRE].
  67. [67]
    A.V. Kotikov, Differential equations method: New technique for massive Feynman diagrams calculation, Phys. Lett. B 254 (1991) 158 [INSPIRE].
  68. [68]
    A.V. Kotikov, Differential equations method: The calculation of vertex type Feynman diagrams, Phys. Lett. B 259 (1991) 314 [INSPIRE].
  69. [69]
    Z. Bern, L.J. Dixon and D.A. Kosower, Dimensionally regulated pentagon integrals, Nucl. Phys. B 412 (1994) 751 [hep-ph/9306240] [INSPIRE].
  70. [70]
    E. Remiddi, Differential equations for Feynman graph amplitudes, Nuovo Cim. A 110 (1997) 1435 [hep-th/9711188] [INSPIRE].
  71. [71]
    T. Gehrmann and E. Remiddi, Differential equations for two loop four point functions, Nucl. Phys. B 580 (2000) 485 [hep-ph/9912329] [INSPIRE].
  72. [72]
    J.M. Henn, Multiloop integrals in dimensional regularization made simple, Phys. Rev. Lett. 110 (2013) 251601 [arXiv:1304.1806] [INSPIRE].CrossRefGoogle Scholar
  73. [73]
    O. Gituliar and S. Moch, Fuchsia and Master Integrals for Energy-Energy Correlations at NLO in QCD, Acta Phys. Polon. B 48 (2017) 2355 [arXiv:1711.05549] [INSPIRE].
  74. [74]
    P. Nogueira, Automatic Feynman graph generation, J. Comput. Phys. 105 (1993) 279 [INSPIRE].
  75. [75]
    T. Hahn, Generating Feynman diagrams and amplitudes with FeynArts 3, Comput. Phys. Commun. 140 (2001) 418 [hep-ph/0012260] [INSPIRE].
  76. [76]
    A. Alloul, N.D. Christensen, C. Degrande, C. Duhr and B. Fuks, FeynRules 2.0A complete toolbox for tree-level phenomenology, Comput. Phys. Commun. 185 (2014) 2250 [arXiv:1310.1921] [INSPIRE].
  77. [77]
    R. Mertig, M. Böhm and A. Denner, FEYN CALC: Computer algebraic calculation of Feynman amplitudes, Comput. Phys. Commun. 64 (1991) 345 [INSPIRE].MathSciNetCrossRefGoogle Scholar
  78. [78]
    V. Shtabovenko, R. Mertig and F. Orellana, New Developments in FeynCalc 9.0, Comput. Phys. Commun. 207 (2016) 432 [arXiv:1601.01167] [INSPIRE].
  79. [79]
    J.A.M. Vermaseren, New features of FORM, math-ph/0010025 [INSPIRE].
  80. [80]
    T. van Ritbergen, A.N. Schellekens and J.A.M. Vermaseren, Group theory factors for Feynman diagrams, Int. J. Mod. Phys. A 14 (1999) 41 [hep-ph/9802376] [INSPIRE].
  81. [81]
    F. Feng, $Apart: A Generalized Mathematica Apart Function, Comput. Phys. Commun. 183 (2012) 2158 [arXiv:1204.2314] [INSPIRE].
  82. [82]
    A. Pak, The toolbox of modern multi-loop calculations: novel analytic and semi-analytic techniques, J. Phys. Conf. Ser. 368 (2012) 012049 [arXiv:1111.0868] [INSPIRE].
  83. [83]
    S. Laporta, High precision calculation of multiloop Feynman integrals by difference equations, Int. J. Mod. Phys. A 15 (2000) 5087 [hep-ph/0102033] [INSPIRE].
  84. [84]
    R.N. Lee, Presenting LiteRed: a tool for the Loop InTEgrals REDuction, arXiv:1212.2685 [INSPIRE].
  85. [85]
    A.V. Smirnov, FIRE5: a C++ implementation of Feynman Integral REduction, Comput. Phys. Commun. 189 (2015) 182 [arXiv:1408.2372] [INSPIRE].
  86. [86]
    P. Maierhöfer, J. Usovitsch and P. Uwer, KiraA Feynman integral reduction program, Comput. Phys. Commun. 230 (2018) 99 [arXiv:1705.05610] [INSPIRE].
  87. [87]
    R.N. Lee, Reducing differential equations for multiloop master integrals, JHEP 04 (2015) 108 [arXiv:1411.0911] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  88. [88]
    C. Meyer, Transforming differential equations of multi-loop Feynman integrals into canonical form, JHEP 04 (2017) 006 [arXiv:1611.01087] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  89. [89]
    C. Meyer, Algorithmic transformation of multi-loop Feynman integrals to a canonical basis, Ph.D. thesis, Humboldt U., Berlin, 2018-01-22. arXiv:1802.02419. 10.18452/18763 [INSPIRE].
  90. [90]
    O. Gituliar and V. Magerya, Fuchsia: a tool for reducing differential equations for Feynman master integrals to epsilon form, Comput. Phys. Commun. 219 (2017) 329 [arXiv:1701.04269] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  91. [91]
    M. Prausa, epsilon: A tool to find a canonical basis of master integrals, Comput. Phys. Commun. 219 (2017) 361 [arXiv:1701.00725] [INSPIRE].
  92. [92]
    C. Meyer, Algorithmic transformation of multi-loop master integrals to a canonical basis with CANONICA, Comput. Phys. Commun. 222 (2018) 295 [arXiv:1705.06252] [INSPIRE].CrossRefGoogle Scholar
  93. [93]
    R.N. Lee and A.A. Pomeransky, Normalized Fuchsian form on Riemann sphere and differential equations for multiloop integrals, arXiv:1707.07856 [INSPIRE].
  94. [94]
    E. Remiddi and J.A.M. Vermaseren, Harmonic polylogarithms, Int. J. Mod. Phys. A 15 (2000) 725 [hep-ph/9905237] [INSPIRE].
  95. [95]
    D. Maître, HPL, a mathematica implementation of the harmonic polylogarithms, Comput. Phys. Commun. 174 (2006) 222 [hep-ph/0507152] [INSPIRE].
  96. [96]
    A. Gehrmann-De Ridder, T. Gehrmann and G. Heinrich, Four particle phase space integrals in massless QCD, Nucl. Phys. B 682 (2004) 265 [hep-ph/0311276] [INSPIRE].
  97. [97]
    E. Panzer, Algorithms for the symbolic integration of hyperlogarithms with applications to Feynman integrals, Comput. Phys. Commun. 188 (2015) 148 [arXiv:1403.3385] [INSPIRE].CrossRefzbMATHGoogle Scholar
  98. [98]
    K. Konishi, A. Ukawa and G. Veneziano, A Simple Algorithm for QCD Jets, Phys. Lett. 78B (1978) 243 [INSPIRE].MathSciNetCrossRefGoogle Scholar
  99. [99]
    D.G. Richards, W.J. Stirling and S.D. Ellis, Second Order Corrections to the Energy-energy Correlation Function in Quantum Chromodynamics, Phys. Lett. 119B (1982) 193 [INSPIRE].CrossRefGoogle Scholar
  100. [100]
    T. Gehrmann, M. Jaquier, E.W.N. Glover and A. Koukoutsakis, Two-Loop QCD Corrections to the Helicity Amplitudes for H → 3 partons, JHEP 02 (2012) 056 [arXiv:1112.3554] [INSPIRE].
  101. [101]
    Q. Jin and G. Yang, Analytic Two-Loop Higgs Amplitudes in Effective Field Theory and the Maximal Transcendentality Principle, Phys. Rev. Lett. 121 (2018) 101603 [arXiv:1804.04653] [INSPIRE].
  102. [102]
    V. Shtabovenko, FeynHelpers: Connecting FeynCalc to FIRE and Package-X, Comput. Phys. Commun. 218 (2017) 48 [arXiv:1611.06793] [INSPIRE].CrossRefzbMATHGoogle Scholar
  103. [103]
    H.H. Patel, Package-X: A Mathematica package for the analytic calculation of one-loop integrals, Comput. Phys. Commun. 197 (2015) 276 [arXiv:1503.01469] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  104. [104]
    H.H. Patel, Package-X 2.0: A Mathematica package for the analytic calculation of one-loop integrals, Comput. Phys. Commun. 218 (2017) 66 [arXiv:1612.00009] [INSPIRE].
  105. [105]
    W.B. Kilgore, One-loop single-real-emission contributions to ppH + X at next-to-next-to-next-to-leading order, Phys. Rev. D 89 (2014) 073008 [arXiv:1312.1296] [INSPIRE].
  106. [106]
    K.G. Chetyrkin, J.H. Kühn and M. Steinhauser, RunDec: A Mathematica package for running and decoupling of the strong coupling and quark masses, Comput. Phys. Commun. 133 (2000) 43 [hep-ph/0004189] [INSPIRE].
  107. [107]
    F. Herren and M. Steinhauser, Version 3 of RunDec and CRunDec, Comput. Phys. Commun. 224 (2018) 333 [arXiv:1703.03751] [INSPIRE].CrossRefGoogle Scholar
  108. [108]
    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].
  109. [109]
    Y.L. Dokshitzer, G. Marchesini and B.R. Webber, Nonperturbative effects in the energy energy correlation, JHEP 07 (1999) 012 [hep-ph/9905339] [INSPIRE].
  110. [110]
    A. Gao, H.T. Li, I. Moult and H.X. Zhu, The Transverse Energy-Energy Correlator in the Back-to-Back Limit, arXiv:1901.04497 [INSPIRE].
  111. [111]
    M.A. Ebert, I. Moult, I.W. Stewart, F.J. Tackmann, G. Vita and H.X. Zhu, Subleading power rapidity divergences and power corrections for q T , JHEP 04 (2019) 123 [arXiv:1812.08189] [INSPIRE].CrossRefGoogle Scholar
  112. [112]
    T. Sjöstrand et al., An Introduction to PYTHIA 8.2, Comput. Phys. Commun. 191 (2015) 159 [arXiv:1410.3012] [INSPIRE].
  113. [113]
    F. An et al., Precision Higgs physics at the CEPC, Chin. Phys. C 43 (2019) 043002 [arXiv:1810.09037] [INSPIRE].
  114. [114]
    DELPHI collaboration, Tuning and test of fragmentation models based on identified particles and precision event shape data, Z. Phys. C 73 (1996) 11 [INSPIRE].
  115. [115]
    R. Brun and F. Rademakers, ROOT: An object oriented data analysis framework, Nucl. Instrum. Meth. A 389 (1997) 81 [INSPIRE].
  116. [116]
    TOPAZ collaboration, Measurements of α s in e + e Annihilation at \( \sqrt{s}=53.3 \) GeV and 59.5 GeV, Phys. Lett. B 227 (1989) 495 [INSPIRE].
  117. [117]
    P. Skands, S. Carrazza and J. Rojo, Tuning PYTHIA 8.1: the Monash 2013 Tune, Eur. Phys. J. C 74 (2014) 3024 [arXiv:1404.5630] [INSPIRE].
  118. [118]
    S. Catani, B.R. Webber and G. Marchesini, QCD coherent branching and semiinclusive processes at large x, Nucl. Phys. B 349 (1991) 635 [INSPIRE].
  119. [119]
    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].

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© The Author(s) 2019

Authors and Affiliations

  • Ming-xing Luo
    • 1
  • Vladyslav Shtabovenko
    • 1
  • Tong-Zhi Yang
    • 1
  • Hua Xing Zhu
    • 1
    Email author
  1. 1.Zhejiang Institute of Modern Physics, Department of PhysicsZhejiang UniversityHangzhouChina

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