One-loop angularity distributions with recoil using Soft-Collinear Effective Theory

  • Ankita Budhraja
  • Ambar Jain
  • Massimiliano ProcuraEmail author
Open Access
Regular Article - Theoretical Physics


Angularities are event shapes whose sensitivity to the splitting angle of a collinear emission is controlled by a continuous parameter b, with −1 < b < ∞. When measured with respect to the thrust axis, this class of QCD observables includes thrust (b = 1) and jet broadening (b = 0), the former being insensitive to the recoil of soft against collinear radiation, while the latter being maximally sensitive to it. Presently available analytic results for angularity distributions with b ≠ 0 can be applied only close to the thrust limit since recoil effects have so far been neglected. As a first step to establish a comprehensive theoretical framework based on Soft-Collinear Effective Theory valid for all recoil-sensitive angularities, we compute for the first time angularity distributions at one-loop order in αs for all values of b taking into account recoil effects. In the differential cross section, these amount to novel sub-leading singular contributions and/or power corrections, where the former are characterized by fractional powers of the angularity and contribute appreciably close to the peak region, also for b ≳ 0.5. Our calculations are checked against various limits known in the literature and agree with the numerical output of the Event2 generator.


Jets NLO Computations 


Open Access

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  1. [1]
    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].
  2. [2]
    T. Becher and M.D. Schwartz, A precise determination of α sfrom LEP thrust data using effective field theory, JHEP07 (2008) 034 [arXiv:0803.0342] [INSPIRE].Google Scholar
  3. [3]
    R.A. Davison and B.R. Webber, Non-Perturbative Contribution to the Thrust Distribution in e +e Annihilation, Eur. Phys. J.C 59 (2009) 13 [arXiv:0809.3326] [INSPIRE].Google Scholar
  4. [4]
    R. Abbate, M. Fickinger, A.H. Hoang, V. Mateu and I.W. Stewart, Thrust at N 3LL with Power Corrections and a Precision Global Fit for α s(m Z), Phys. Rev.D 83 (2011) 074021 [arXiv:1006.3080] [INSPIRE].
  5. [5]
    T. Gehrmann, G. Luisoni and P.F. Monni, Power corrections in the dispersive model for a determination of the strong coupling constant from the thrust distribution, Eur. Phys. J.C 73 (2013) 2265 [arXiv:1210.6945] [INSPIRE].
  6. [6]
    A.H. Hoang, D.W. Kolodrubetz, V. Mateu and I.W. Stewart, Precise determination of α sfrom the C-parameter distribution, Phys. Rev. D 91 (2015) 094018 [arXiv:1501.04111] [INSPIRE].
  7. [7]
    A. Banfi, G.P. Salam and G. Zanderighi, Resummed event shapes at hadron-hadron colliders, JHEP08 (2004) 062 [hep-ph/0407287] [INSPIRE].
  8. [8]
    A. Banfi, G.P. Salam and G. Zanderighi, Phenomenology of event shapes at hadron colliders, JHEP06 (2010) 038 [arXiv:1001.4082] [INSPIRE].ADSCrossRefGoogle Scholar
  9. [9]
    A. Banfi, G. Marchesini and G. Smye, Away from jet energy flow, JHEP08 (2002) 006 [hep-ph/0206076] [INSPIRE].
  10. [10]
    M. Dasgupta and G.P. Salam, Resummation of nonglobal QCD observables, Phys. Lett. B 512 (2001) 323 [hep-ph/0104277] [INSPIRE].
  11. [11]
    A. Hornig, Y. Makris and T. Mehen, Jet Shapes in Dijet Events at the LHC in SCET, JHEP04 (2016) 097 [arXiv:1601.01319] [INSPIRE].ADSGoogle Scholar
  12. [12]
    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].
  13. [13]
    C.F. Berger, T. Kucs and G.F. Sterman, Event shape/energy flow correlations, Phys. Rev. D 68 (2003) 014012 [hep-ph/0303051] [INSPIRE].
  14. [14]
    C.F. Berger and L. Magnea, Scaling of power corrections for angularities from dressed gluon exponentiation, Phys. Rev. D 70 (2004) 094010 [hep-ph/0407024] [INSPIRE].
  15. [15]
    A. Hornig, C. Lee and G. Ovanesyan, Effective Predictions of Event Shapes: Factorized, Resummed and Gapped Angularity Distributions, JHEP05 (2009) 122 [arXiv:0901.3780] [INSPIRE].ADSCrossRefGoogle Scholar
  16. [16]
    S.D. Ellis, C.K. Vermilion, J.R. Walsh, A. Hornig and C. Lee, Jet Shapes and Jet Algorithms in SCET, JHEP11 (2010) 101 [arXiv:1001.0014] [INSPIRE].ADSCrossRefGoogle Scholar
  17. [17]
    P. Achard et al., Generalized event shape and energy flow studies in e +e annihilation at \( \sqrt{s} \) = 91.2 GeV-208.0-GeV, JHEP10 (2011) 143 [INSPIRE].
  18. [18]
    E. Farhi, A QCD Test for Jets, Phys. Rev. Lett.39 (1977) 1587 [INSPIRE].ADSCrossRefGoogle Scholar
  19. [19]
    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].ADSCrossRefGoogle Scholar
  20. [20]
    P.E.L. Rakow and B.R. Webber, Transverse Momentum Moments of Hadron Distributions in QCD Jets, Nucl. Phys.B 191 (1981) 63 [INSPIRE].ADSCrossRefGoogle Scholar
  21. [21]
    S. Catani, G. Turnock and B.R. Webber, Jet broadening measures in e +e annihilation, Phys. Lett.B 295 (1992) 269 [INSPIRE].Google Scholar
  22. [22]
    G. Bell, A. Hornig, C. Lee and J. Talbert, e +e angularity distributions at NNLLaccuracy, JHEP01 (2019) 147 [arXiv:1808.07867] [INSPIRE].
  23. [23]
    Y.L. Dokshitzer, A. Lucenti, G. Marchesini and G.P. Salam, On the QCD analysis of jet broadening, JHEP01 (1998) 011 [hep-ph/9801324] [INSPIRE].
  24. [24]
    A.J. Larkoski, D. Neill and J. Thaler, Jet Shapes with the Broadening Axis, JHEP04 (2014) 017 [arXiv:1401.2158] [INSPIRE].
  25. [25]
    M. Procura, W.J. Waalewijn and L. Zeune, Joint resummation of two angularities at next-to-next-to-leading logarithmic order, JHEP10 (2018) 098 [arXiv:1806.10622] [INSPIRE].ADSCrossRefGoogle Scholar
  26. [26]
    A. Banfi, B.K. El-Menoufi and P.F. Monni, The Sudakov radiator for jet observables and the soft physical coupling, [arXiv:1807.11487] [INSPIRE].
  27. [27]
    C.W. Bauer, S. Fleming and M.E. Luke, Summing Sudakov logarithms in BX() in effective field theory, Phys. Rev. D 63 (2000) 014006 [hep-ph/0005275] [INSPIRE].
  28. [28]
    C.W. Bauer, S. Fleming, D. Pirjol and I.W. Stewart, An Effective field theory for collinear and soft gluons: Heavy to light decays, Phys. Rev. D 63 (2001) 114020 [hep-ph/0011336] [INSPIRE].
  29. [29]
    C.W. Bauer and I.W. Stewart, Invariant operators in collinear effective theory, Phys. Lett. B 516 (2001) 134 [hep-ph/0107001] [INSPIRE].
  30. [30]
    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].
  31. [31]
    C.W. Bauer, S. Fleming, D. Pirjol, I.Z. Rothstein and I.W. Stewart, Hard scattering factorization from effective field theory, Phys. Rev. D 66 (2002) 014017 [hep-ph/0202088] [INSPIRE].
  32. [32]
    A. Budhraja, A. Jain and M. Procura, NLL resummation of recoil-sensitive angularities using SCET, in preparation.Google Scholar
  33. [33]
    C.F. Berger and G.F. Sterman, Scaling rule for nonperturbative radiation in a class of event shapes, JHEP09 (2003) 058 [hep-ph/0307394] [INSPIRE].
  34. [34]
    J.-Y. Chiu, A. Jain, D. Neill and I.Z. Rothstein, A Formalism for the Systematic Treatment of Rapidity Logarithms in Quantum Field Theory, JHEP05 (2012) 084 [arXiv:1202.0814] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  35. [35]
    L.G. Almeida, S.D. Ellis, C. Lee, G. Sterman, I. Sung and J.R. Walsh, Comparing and counting logs in direct and effective methods of QCD resummation, JHEP04 (2014) 174 [arXiv:1401.4460] [INSPIRE].ADSCrossRefGoogle Scholar
  36. [36]
    A.H. Hoang, D.W. Kolodrubetz, V. Mateu and I.W. Stewart, C-parameter distribution at N 3LLincluding power corrections, Phys. Rev.D 91 (2015) 094017 [arXiv:1411.6633] [INSPIRE].
  37. [37]
    A. Banfi, H. McAslan, P.F. Monni and G. Zanderighi, A general method for the resummation of event-shape distributions in e +e annihilation, JHEP05 (2015) 102 [arXiv:1412.2126] [INSPIRE].Google Scholar
  38. [38]
    T. Becher and G. Bell, NNLL Resummation for Jet Broadening, JHEP11 (2012) 126 [arXiv:1210.0580] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  39. [39]
    G.P. Salam and D. Wicke, Hadron masses and power corrections to event shapes, JHEP05 (2001) 061 [hep-ph/0102343] [INSPIRE].
  40. [40]
    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].
  41. [41]
    J.-y. Chiu, A. Jain, D. Neill and I.Z. Rothstein, The Rapidity Renormalization Group, Phys. Rev. Lett.108 (2012) 151601 [arXiv:1104.0881] [INSPIRE].
  42. [42]
    T. Becher, G. Bell and M. Neubert, Factorization and Resummation for Jet Broadening, Phys. Lett.B 704 (2011) 276 [arXiv:1104.4108] [INSPIRE].ADSCrossRefGoogle Scholar
  43. [43]
    A. Idilbi and I. Scimemi, Singular and Regular Gauges in Soft Collinear Effective Theory: The Introduction of the New Wilson Line T, Phys. Lett.B 695 (2011) 463 [arXiv:1009.2776] [INSPIRE].ADSCrossRefGoogle Scholar
  44. [44]
    A.V. Manohar, Deep inelastic scattering as x → 1 using soft collinear effective theory, Phys. Rev. D 68 (2003) 114019 [hep-ph/0309176] [INSPIRE].
  45. [45]
    C.W. Bauer, C. Lee, A.V. Manohar and M.B. Wise, Enhanced nonperturbative effects in Z decays to hadrons, Phys. Rev. D 70 (2004) 034014 [hep-ph/0309278] [INSPIRE].
  46. [46]
    M.D. Schwartz, Resummation and NLO matching of event shapes with effective field theory, Phys. Rev.D 77 (2008) 014026 [arXiv:0709.2709] [INSPIRE].
  47. [47]
    A.V. Manohar and I.W. Stewart, The Zero-Bin and Mode Factorization in Quantum Field Theory, Phys. Rev. D 76 (2007) 074002 [hep-ph/0605001] [INSPIRE].
  48. [48]
    M.D. Schwartz, Quantum Field Theory and the Standard Model, Cambridge University Press, Cambridge U.K. (2014).Google Scholar
  49. [49]
    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].

Copyright information

© The Author(s) 2019

Authors and Affiliations

  • Ankita Budhraja
    • 1
  • Ambar Jain
    • 1
  • Massimiliano Procura
    • 2
    • 3
    Email author
  1. 1.Indian Institute of Science Education and ResearchBhopalIndia
  2. 2.Fakultät für PhysikUniversität WienWienAustria
  3. 3.Theoretical Physics DepartmentCERNGeneva 23Switzerland

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