# Predictions for energy correlators probing substructure of groomed heavy quark jets

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## Abstract

We develop an effective field theory (EFT) framework to perform an analytic calculation for energy correlator observables computed on groomed heavy-quark jets. A soft-drop grooming algorithm is applied to a jet initiated by a massive quark to minimize soft contamination effects such as pile-up and multi-parton interactions. We specifically consider the two-particle energy correlator as an initial application of this EFT framework to compute heavy quark jet substructure. We find that there are different regimes for the event shapes, depending on the size of the measured correlator observable, that require the use of different EFT formulations, in which the quark mass and grooming parameters may be relevant or not. We use the EFT to resum large logarithms in the energy correlator observable in terms of the momentum of a reconstructed heavy hadron to NLL′ accuracy and subsequently match it to a full QCD \( \mathcal{O} \)(*α*_{s}) cross section, which we also compute. We compare our predictions to simulations in Pythia for *e*^{+}*e*^{−} collisions. We find a good agreement with partonic simulations, as well as hadronic ones with an appropriate shape function used to describe nonperturbative effects and the heavy quark hadron decay turned off. We also predict the scaling behavior for the leading nonperturbative power correction due to hadronization. Consequently, we can give a prediction for the energy correlator distribution at the level of the reconstructed heavy hadron. This work provides a general framework for the analysis of heavy quark jet substructure observables.

## Keywords

Jets QCD Phenomenology## Notes

### **Open Access**

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## References

- [1]CMS collaboration,
*Search for a Higgs boson in the decay channel H*→*ZZ*^{(∗)}→*q*\( \overline{q} \)*ℓ*^{−}*ℓ*^{+}*in pp collisions at*\( \sqrt{s} \) = 7*TeV*,*JHEP***04**(2012) 036 [arXiv:1202.1416] [INSPIRE]. - [2]CMS collaboration,
*Search for a Standard Model-like Higgs boson decaying into WW*→*l*ν*q*\( \overline{q}^{\prime } \)*in pp collisions at*\( \sqrt{s} \) = 8*TeV*, CMS-PAS-HIG-13-008 (2013) [INSPIRE]. - [3]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]. - [4]CMS collaboration,
*Search for BSM t*\( \overline{t} \)*Production in the Boosted All-Hadronic Final State*, CMS-PAS-EXO-11-006 (2011) [INSPIRE]. - [5]ATLAS and CMS collaborations,
*Boosted top quark techniques and searches for t*\( \overline{t} \)*resonances at the LHC*,*J. Phys. Conf. Ser.***452**(2013) 012034 [INSPIRE]. - [6]ATLAS and CMS collaborations,
*Boosted Top Quarks, Top Pair Resonances and Top Partner Searches at the LHC*,*EPJ Web Conf.***60**(2013) 09003 [INSPIRE]. - [7]ATLAS collaboration,
*Performance of boosted top quark identification in 2012 ATLAS data*, ATLAS-CONF-2013-084 (2013) [INSPIRE]. - [8]H.A. Andrews et al.,
*Novel tools and observables for jet physics in heavy-ion collisions*, arXiv:1808.03689 [INSPIRE]. - [9]J.D. Bjorken,
*Energy Loss of Energetic Partons in Quark-Gluon Plasma: Possible Extinction of High p*_{T}*Jets in Hadron-Hadron Collisions*, FERMILAB-PUB-82-059-THY (1982) [INSPIRE]. - [10]
- [11]X.-N. Wang and M. Gyulassy,
*Gluon shadowing and jet quenching in A*+*A collisions at*\( \sqrt{s} \) = 200*GeV*,*Phys. Rev. Lett.***68**(1992) 1480 [INSPIRE].ADSCrossRefGoogle Scholar - [12]ATLAS collaboration,
*Measurements of the Nuclear Modification Factor for Jets in Pb*+*Pb Collisions at*\( \sqrt{{}^s\mathrm{NN}} \) = 2*.*76*TeV with the ATLAS Detector*,*Phys. Rev. Lett.***114**(2015) 072302 [arXiv:1411.2357] [INSPIRE]. - [13]CMS collaboration,
*Studies of jet quenching using isolated-photon*+*jet correlations in PbPb and pp collisions at*\( \sqrt{{}^s\mathrm{NN}} \) = 2*.*76*TeV*,*Phys. Lett.***B 718**(2013) 773 [arXiv:1205.0206] [INSPIRE]. - [14]CMS collaboration,
*Observation and studies of jet quenching in PbPb collisions at*\( \sqrt{{}^s\mathrm{NN}} \) = 2*.*76*TeV*,*Phys. Rev.***C 84**(2011) 024906 [arXiv:1102.1957] [INSPIRE]. - [15]ATLAS collaboration,
*Observation of a Centrality-Dependent Dijet Asymmetry in Lead-Lead Collisions at*\( \sqrt{{}^s\mathrm{NN}} \) = 2*.*77*TeV with the ATLAS Detector at the LHC*,*Phys. Rev. Lett.***105**(2010) 252303 [arXiv:1011.6182] [INSPIRE]. - [16]CMS collaboration,
*Measurement of jet fragmentation into charged particles in pp and PbPb collisions at*\( \sqrt{{}^s\mathrm{NN}} \) = 2*.*76*TeV*,*JHEP***10**(2012) 087 [arXiv:1205.5872] [INSPIRE]. - [17]CMS collaboration,
*Modification of jet shapes in PbPb collisions at*\( \sqrt{{}^s\mathrm{NN}} \) = 2*.*76*TeV*,*Phys. Lett.***B 730**(2014) 243 [arXiv:1310.0878] [INSPIRE]. - [18]ATLAS collaboration,
*Measurement of inclusive jet charged-particle fragmentation functions in P b*+*P b collisions at*\( \sqrt{{}^s\mathrm{NN}} \) = 2*.*76*TeV with the ATLAS detector*,*Phys. Lett.***B 739**(2014) 320 [arXiv:1406.2979] [INSPIRE]. - [19]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]. - [20]M. Field, G. Gur-Ari, D.A. Kosower, L. Mannelli and G. Perez,
*Three-Prong Distribution of Massive Narrow QCD Jets*,*Phys. Rev.***D 87**(2013) 094013 [arXiv:1212.2106] [INSPIRE]. - [21]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 - [22]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]. - [23]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 - [24]M. Dasgupta, A. Powling and A. Siodmok,
*On jet substructure methods for signal jets*,*JHEP***08**(2015) 079 [arXiv:1503.01088] [INSPIRE]. - [25]H.T. Li and I. Vitev,
*Inverting the mass hierarchy of jet quenching effects with prompt b-jet substructure*,*Phys. Lett.***B 793**(2019) 259 [arXiv:1801.00008] [INSPIRE].ADSCrossRefGoogle Scholar - [26]
- [27]A.J. Larkoski, I. Moult and D. Neill,
*Power Counting to Better Jet Observables*,*JHEP***12**(2014) 009 [arXiv:1409.6298] [INSPIRE]. - [28]A.J. Larkoski, I. Moult and D. Neill,
*Building a Better Boosted Top Tagger*,*Phys. Rev.***D 91**(2015) 034035 [arXiv:1411.0665] [INSPIRE]. - [29]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 - [30]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]. - [31]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 - [32]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]. - [33]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]. - [34]J.S. Conway, R. Bhaskar, R.D. Erbacher and J. Pilot,
*Identification of High-Momentum Top Quarks, Higgs Bosons and W and Z Bosons Using Boosted Event Shapes*,*Phys. Rev.***D 94**(2016) 094027 [arXiv:1606.06859] [INSPIRE]. - [35]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]. - [36]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]. - [37]A.J. Larkoski, S. Marzani, G. Soyez and J. Thaler,
*Soft Drop*,*JHEP***05**(2014) 146 [arXiv:1402.2657] [INSPIRE].ADSCrossRefGoogle Scholar - [38]A.J. Larkoski, I. Moult and D. Neill,
*Analytic Boosted Boson Discrimination at the Large Hadron Collider*, arXiv:1708.06760 [INSPIRE]. - [39]A.H. Hoang, S. Mantry, A. Pathak and I.W. Stewart,
*Extracting a Short Distance Top Mass with Light Grooming*, arXiv:1708.02586 [INSPIRE]. - [40]S. Marzani, L. Schunk and G. Soyez,
*A study of jet mass distributions with grooming*,*JHEP***07**(2017) 132 [arXiv:1704.02210] [INSPIRE]. - [41]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]. - [42]C. Frye, A.J. Larkoski, M.D. Schwartz and K. Yan,
*Precision physics with pile-up insensitive observables*, arXiv:1603.06375 [INSPIRE]. - [43]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 - [44]A.J. Larkoski, S. Marzani and J. Thaler,
*Sudakov Safety in Perturbative QCD*,*Phys. Rev.***D 91**(2015) 111501 [arXiv:1502.01719] [INSPIRE].ADSGoogle Scholar - [45]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]. - [46]Y.-T. Chien and I. Vitev,
*Jet Shape Resummation Using Soft-Collinear Effective Theory*,*JHEP***12**(2014) 061 [arXiv:1405.4293] [INSPIRE]. - [47]Y.L. Dokshitzer and D.E. Kharzeev,
*Heavy quark colorimetry of QCD matter*,*Phys. Lett.***B 519**(2001) 199 [hep-ph/0106202] [INSPIRE]. - [48]A.D. Frawley, T. Ullrich and R. Vogt,
*Heavy flavor in heavy-ion collisions at RHIC and RHIC II*,*Phys. Rept.***462**(2008) 125 [arXiv:0806.1013] [INSPIRE].ADSCrossRefGoogle Scholar - [49]PHENIX collaboration,
*An Upgrade Proposal from the PHENIX collaboration*, arXiv:1501.06197 [INSPIRE]. - [50]S.P. 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]. - [51]S.P. Fleming, A.H. Hoang, S. Mantry and I.W. Stewart,
*Top Jets in the Peak Region: Factorization Analysis with NLL Resummation*,*Phys. Rev.***D 77**(2008) 114003 [arXiv:0711.2079] [INSPIRE]. - [52]Y. Makris and V. Vaidya,
*Transverse Momentum Spectra at Threshold for Groomed Heavy Quark Jets*,*JHEP***10**(2018) 019 [arXiv:1807.09805] [INSPIRE].ADSCrossRefGoogle Scholar - [53]P. Pietrulewicz, D. Samitz, A. Spiering and F.J. Tackmann,
*Factorization and Resummation for Massive Quark Effects in Exclusive Drell-Yan*,*JHEP***08**(2017) 114 [arXiv:1703.09702] [INSPIRE].ADSCrossRefGoogle Scholar - [54]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]. - [55]M. Jankowiak and A.J. Larkoski,
*Jet Substructure Without Trees*,*JHEP***06**(2011) 057 [arXiv:1104.1646] [INSPIRE]. - [56]A.J. Larkoski, G.P. Salam and J. Thaler,
*Energy Correlation Functions for Jet Substructure*,*JHEP***06**(2013) 108 [arXiv:1305.0007] [INSPIRE]. - [57]C.W. Bauer, F.J. Tackmann, J.R. Walsh and S. Zuberi,
*Factorization and Resummation for Dijet Invariant Mass Spectra*,*Phys. Rev.***D 85**(2012) 074006 [arXiv:1106.6047] [INSPIRE]. - [58]B. Dehnadi,
*Heavy quark mass determinations with sum rules and jets*, Ph.D. Thesis, University of Vienna, Vienna Austria (2016).Google Scholar - [59]A.H. Hoang, C. Lepenik and M. Stahlhofen,
*Two-Loop Massive Quark Jet Functions in SCET*, arXiv:1904.12839 [INSPIRE]. - [60]I. Moult, L. Necib and J. Thaler,
*New Angles on Energy Correlation Functions*,*JHEP***12**(2016) 153 [arXiv:1609.07483] [INSPIRE].ADSCrossRefGoogle Scholar - [61]A.J. Larkoski, D. Neill and J. Thaler,
*Jet Shapes with the Broadening Axis*,*JHEP***04**(2014) 017 [arXiv:1401.2158] [INSPIRE]. - [62]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]. - [63]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 - [64]S.D. Ellis and D.E. Soper,
*Successive combination jet algorithm for hadron collisions*,*Phys. Rev.***D 48**(1993) 3160 [hep-ph/9305266] [INSPIRE]. - [65]Y.L. Dokshitzer, G.D. Leder, S. Moretti and B.R. Webber,
*Better jet clustering algorithms*,*JHEP***08**(1997) 001 [hep-ph/9707323] [INSPIRE]. - [66]M. Cacciari, G.P. Salam and G. Soyez,
*The anti-k*_{t}*jet clustering algorithm*,*JHEP***04**(2008) 063 [arXiv:0802.1189] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar - [67]M. Wobisch and T. Wengler,
*Hadronization corrections to jet cross-sections in deep inelastic scattering*, in proceedings of the*Workshop on Monte Carlo Generators for HERA Physics (Plenary Starting Meeting)*, Hamburg, Germany, 27-30 April 1998, pp. 270-279 [hep-ph/9907280] [INSPIRE]. - [68]M. Wobisch,
*Measurement and QCD analysis of jet cross-sections in deep inelastic positron proton collisions at*\( \sqrt{s} \) = 300*GeV*, DESY-THESIS-2000-049 (2000) [INSPIRE]. - [69]
- [70]D. Bertolini, T. Chan and J. Thaler,
*Jet Observables Without Jet Algorithms*,*JHEP***04**(2014) 013 [arXiv:1310.7584] [INSPIRE]. - [71]C.W. Bauer, S.P. 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]. - [72]C.W. Bauer and I.W. Stewart,
*Invariant operators in collinear effective theory*,*Phys. Lett.***B 516**(2001) 134 [hep-ph/0107001] [INSPIRE]. - [73]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]. - [74]C.W. Bauer, S.P. 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]. - [75]M. Beneke, A.P. Chapovsky, M. Diehl and T. Feldmann,
*Soft collinear effective theory and heavy to light currents beyond leading power*,*Nucl. Phys.***B 643**(2002) 431 [hep-ph/0206152] [INSPIRE]. - [76]A.V. Manohar and M.B. Wise,
*Heavy quark physics*,*Camb. Monogr. Part. Phys. Nucl. Phys. Cosmol.***10**(2000) 1 [INSPIRE]. - [77]N. Isgur and M.B. Wise,
*Weak Decays of Heavy Mesons in the Static Quark Approximation*,*Phys. Lett.***B 232**(1989) 113 [INSPIRE]. - [78]N. Isgur and M.B. Wise,
*Weak transition form-factors between heavy mesons*,*Phys. Lett.***B 237**(1990) 527 [INSPIRE]. - [79]P. Pietrulewicz, F.J. Tackmann and W.J. Waalewijn,
*Factorization and Resummation for Generic Hierarchies between Jets*,*JHEP***08**(2016) 002 [arXiv:1601.05088] [INSPIRE].ADSCrossRefGoogle Scholar - [80]Y.-T. Chien, A. Hornig and C. Lee,
*Soft-collinear mode for jet cross sections in soft collinear effective theory*,*Phys. Rev.***D 93**(2016) 014033 [arXiv:1509.04287] [INSPIRE]. - [81]T. Becher, M. Neubert, L. Rothen and D.Y. Shao,
*Effective Field Theory for Jet Processes*,*Phys. Rev. Lett.***116**(2016) 192001 [arXiv:1508.06645] [INSPIRE]. - [82]T. Becher, M. Neubert, L. Rothen and D.Y. Shao,
*Factorization and Resummation for Jet Processes*,*JHEP***11**(2016) 019 [*Erratum JHEP***05**(2017) 154] [arXiv:1605.02737] [INSPIRE].ADSzbMATHGoogle Scholar - [83]D.W. Kolodrubetz, P. Pietrulewicz, I.W. Stewart, F.J. Tackmann and W.J. Waalewijn,
*Factorization for Jet Radius Logarithms in Jet Mass Spectra at the LHC*,*JHEP***12**(2016) 054 [arXiv:1605.08038] [INSPIRE].ADSCrossRefGoogle Scholar - [84]S.D. Ellis, A. Hornig, C. Lee, C.K. Vermilion and J.R. Walsh,
*Consistent Factorization of Jet Observables in Exclusive Multijet Cross-Sections*,*Phys. Lett.***B 689**(2010) 82 [arXiv:0912.0262] [INSPIRE]. - [85]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]. - [86]A.K. Leibovich, Z. Ligeti and M.B. Wise,
*Comment on quark masses in SCET*,*Phys. Lett.***B 564**(2003) 231 [hep-ph/0303099] [INSPIRE]. - [87]Y. Makris, D. Neill and V. Vaidya,
*Probing Transverse-Momentum Dependent Evolution With Groomed Jets*,*JHEP***07**(2018) 167 [arXiv:1712.07653] [INSPIRE].ADSCrossRefGoogle Scholar - [88]L.G. Almeida, S.D. Ellis, C. Lee, G.F. Sterman, I. Sung and J.R. Walsh,
*Comparing and counting logs in direct and effective methods of QCD resummation*,*JHEP***04**(2014) 174 [arXiv:1401.4460] [INSPIRE].ADSCrossRefGoogle Scholar - [89]G.P. Korchemsky and A.V. Radyushkin,
*Renormalization of the Wilson Loops Beyond the Leading Order*,*Nucl. Phys.***B 283**(1987) 342 [INSPIRE]. - [90]T. Becher, M. Neubert and B.D. Pecjak,
*Factorization and Momentum-Space Resummation in Deep-Inelastic Scattering*,*JHEP***01**(2007) 076 [hep-ph/0607228] [INSPIRE]. - [91]T. Becher and M. Neubert,
*Threshold resummation in momentum space from effective field theory*,*Phys. Rev. Lett.***97**(2006) 082001 [hep-ph/0605050] [INSPIRE]. - [92]G. Bell, A. Hornig, C. Lee and J. Talbert,
*e*^{+}*e*^{−}*angularity distributions at NNLL*′*accuracy*,*JHEP***01**(2019) 147 [arXiv:1808.07867] [INSPIRE]. - [93]Y.L. Dokshitzer and B.R. Webber,
*Calculation of power corrections to hadronic event shapes*,*Phys. Lett.***B 352**(1995) 451 [hep-ph/9504219] [INSPIRE]. - [94]Y.L. Dokshitzer and B.R. Webber,
*Power corrections to event shape distributions*,*Phys. Lett.***B 404**(1997) 321 [hep-ph/9704298] [INSPIRE]. - [95]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]. - [96]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]. - [97]C. Lee and G.F. Sterman,
*Universality of nonperturbative effects in event shapes*,*eConf***C 0601121**(2006) A001 [hep-ph/0603066] [INSPIRE]. - [98]F.R. Ore Jr. and G.F. Sterman,
*An operator approach to weighted cross-sections*,*Nucl. Phys.***B 165**(1980) 93 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [99]N.A. Sveshnikov and F.V. Tkachov,
*Jets and quantum field theory*,*Phys. Lett.***B 382**(1996) 403 [hep-ph/9512370] [INSPIRE]. - [100]G.P. Korchemsky, G. Oderda and G.F. Sterman,
*Power corrections and nonlocal operators*,*AIP Conf. Proc.***407**(1997) 988 [hep-ph/9708346] [INSPIRE]. - [101]G.P. Korchemsky and G.F. Sterman,
*Power corrections to event shapes and factorization*,*Nucl. Phys.***B 555**(1999) 335 [hep-ph/9902341] [INSPIRE]. - [102]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]. - [103]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]. - [104]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].