Advertisement

Soft functions for generic jet algorithms and observables at hadron colliders

  • Daniele Bertolini
  • Daniel Kolodrubetz
  • Duff Neill
  • Piotr PietrulewiczEmail author
  • Iain W. Stewart
  • Frank J. Tackmann
  • Wouter J. Waalewijn
Open Access
Regular Article - Theoretical Physics

Abstract

We introduce a method to compute one-loop soft functions for exclusive N - jet processes at hadron colliders, allowing for different definitions of the algorithm that determines the jet regions and of the measurements in those regions. In particular, we generalize the N -jettiness hemisphere decomposition of ref. [1] in a manner that separates the dependence on the jet boundary from the observables measured inside the jet and beam regions. Results are given for several factorizable jet definitions, including anti-k T , XCone, and other geometric partitionings. We calculate explicitly the soft functions for angularity measurements, including jet mass and jet broadening, in ppL + 1 jet and explore the differences for various jet vetoes and algorithms. This includes a consistent treatment of rapidity divergences when applicable. We also compute analytic results for these soft functions in an expansion for a small jet radius R. We find that the small-R results, including corrections up to \( \mathcal{O}\left({R}^2\right) \), accurately capture the full behavior over a large range of R.

Keywords

Jets NLO Computations 

Notes

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.

References

  1. [1]
    T.T. Jouttenus, I.W. Stewart, F.J. Tackmann and W.J. Waalewijn, The Soft Function for Exclusive N-Jet Production at Hadron Colliders, Phys. Rev. D 83 (2011) 114030 [arXiv:1102.4344] [INSPIRE].ADSGoogle Scholar
  2. [2]
    C.W. Bauer, S. Fleming and M.E. Luke, Summing Sudakov logarithms in BX s γ in effective field theory, Phys. Rev. D 63 (2000) 014006 [hep-ph/0005275] [INSPIRE].
  3. [3]
    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].
  4. [4]
    C.W. Bauer and I.W. Stewart, Invariant operators in collinear effective theory, Phys. Lett. B 516 (2001) 134 [hep-ph/0107001] [INSPIRE].
  5. [5]
    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].
  6. [6]
    C.W. Bauer and A.V. Manohar, Shape function effects in BX s γ and \( B\to {X}_u\ell \overline{\nu} \) decays, Phys. Rev. D 70 (2004) 034024 [hep-ph/0312109] [INSPIRE].
  7. [7]
    S. Fleming, A.K. Leibovich and T. Mehen, Resumming the color octet contribution to e + e J/ψ + X, Phys. Rev. D 68 (2003) 094011 [hep-ph/0306139] [INSPIRE].
  8. [8]
    T. Becher and M. Neubert, Toward a NNLO calculation of the \( \overline{B}\to {X}_s\gamma \) decay rate with a cut on photon energy. II. Two-loop result for the jet function, Phys. Lett. B 637 (2006) 251 [hep-ph/0603140] [INSPIRE].
  9. [9]
    T. Becher and M.D. Schwartz, Direct photon production with effective field theory, JHEP 02 (2010) 040 [arXiv:0911.0681] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  10. [10]
    T. Becher and G. Bell, The gluon jet function at two-loop order, Phys. Lett. B 695 (2011) 252 [arXiv:1008.1936] [INSPIRE].ADSCrossRefGoogle Scholar
  11. [11]
    A.J. Larkoski, D. Neill and J. Thaler, Jet Shapes with the Broadening Axis, JHEP 04 (2014) 017 [arXiv:1401.2158] [INSPIRE].ADSCrossRefGoogle Scholar
  12. [12]
    I.W. Stewart, F.J. Tackmann and W.J. Waalewijn, The Quark Beam Function at NNLL, JHEP 09 (2010) 005 [arXiv:1002.2213] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  13. [13]
    C.F. Berger, C. Marcantonini, I.W. Stewart, F.J. Tackmann and W.J. Waalewijn, Higgs Production with a Central Jet Veto at NNLL+NNLO, JHEP 04 (2011) 092 [arXiv:1012.4480] [INSPIRE].ADSCrossRefGoogle Scholar
  14. [14]
    T. Becher and M. Neubert, Drell-Yan Production at Small q T , Transverse Parton Distributions and the Collinear Anomaly, Eur. Phys. J. C 71 (2011) 1665 [arXiv:1007.4005] [INSPIRE].ADSCrossRefGoogle Scholar
  15. [15]
    T. Becher and M. Neubert, Factorization and NNLL Resummation for Higgs Production with a Jet Veto, JHEP 07 (2012) 108 [arXiv:1205.3806] [INSPIRE].ADSCrossRefGoogle Scholar
  16. [16]
    J.-Y. Chiu, A. Jain, D. Neill and I.Z. Rothstein, A Formalism for the Systematic Treatment of Rapidity Logarithms in Quantum Field Theory, JHEP 05 (2012) 084 [arXiv:1202.0814] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    T. Gehrmann, T. Luebbert and L.L. Yang, Calculation of the transverse parton distribution functions at next-to-next-to-leading order, JHEP 06 (2014) 155 [arXiv:1403.6451] [INSPIRE].ADSCrossRefGoogle Scholar
  18. [18]
    J.R. Gaunt, M. Stahlhofen and F.J. Tackmann, The Quark Beam Function at Two Loops, JHEP 04 (2014) 113 [arXiv:1401.5478] [INSPIRE].ADSCrossRefGoogle Scholar
  19. [19]
    J. Gaunt, M. Stahlhofen and F.J. Tackmann, The Gluon Beam Function at Two Loops, JHEP 08 (2014) 020 [arXiv:1405.1044] [INSPIRE].ADSCrossRefGoogle Scholar
  20. [20]
    T. Lübbert, J. Oredsson and M. Stahlhofen, Rapidity renormalized TMD soft and beam functions at two loops, JHEP 03 (2016) 168 [arXiv:1602.01829] [INSPIRE].ADSCrossRefGoogle Scholar
  21. [21]
    I. Moult, I.W. Stewart, F.J. Tackmann and W.J. Waalewijn, Employing Helicity Amplitudes for Resummation, Phys. Rev. D 93 (2016) 094003 [arXiv:1508.02397] [INSPIRE].ADSGoogle Scholar
  22. [22]
    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].ADSCrossRefGoogle Scholar
  23. [23]
    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].ADSGoogle Scholar
  24. [24]
    J. Thaler and K. Van Tilburg, Maximizing Boosted Top Identification by Minimizing N-subjettiness, JHEP 02 (2012) 093 [arXiv:1108.2701] [INSPIRE].ADSCrossRefGoogle Scholar
  25. [25]
    T.T. Jouttenus, I.W. Stewart, F.J. Tackmann and W.J. Waalewijn, 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].ADSGoogle Scholar
  26. [26]
    J. Thaler and K. Van Tilburg, Identifying Boosted Objects with N-subjettiness, JHEP 03 (2011) 015 [arXiv:1011.2268] [INSPIRE].ADSCrossRefGoogle Scholar
  27. [27]
    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
  28. [28]
    M. Cacciari, G.P. Salam and G. Soyez, The anti-k t jet clustering algorithm, JHEP 04 (2008) 063 [arXiv:0802.1189] [INSPIRE].ADSCrossRefGoogle Scholar
  29. [29]
    J. Thaler and T.F. Wilkason, Resolving Boosted Jets with XCone, JHEP 12 (2015) 051 [arXiv:1508.01518] [INSPIRE].ADSGoogle Scholar
  30. [30]
    J.R. Walsh and S. Zuberi, Factorization Constraints on Jet Substructure, arXiv:1110.5333 [INSPIRE].
  31. [31]
    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].ADSCrossRefGoogle Scholar
  32. [32]
    J.R. Gaunt, Glauber Gluons and Multiple Parton Interactions, JHEP 07 (2014) 110 [arXiv:1405.2080] [INSPIRE].ADSCrossRefGoogle Scholar
  33. [33]
    M. Zeng, Drell-Yan process with jet vetoes: breaking of generalized factorization, JHEP 10 (2015) 189 [arXiv:1507.01652] [INSPIRE].ADSCrossRefGoogle Scholar
  34. [34]
    I.Z. Rothstein and I.W. Stewart, An Effective Field Theory for Forward Scattering and Factorization Violation, JHEP 08 (2016) 025 [arXiv:1601.04695] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  35. [35]
    D. Kang, C. Lee and I.W. Stewart, Using 1-Jettiness to Measure 2 Jets in DIS 3 Ways, Phys. Rev. D 88 (2013) 054004 [arXiv:1303.6952] [INSPIRE].ADSGoogle Scholar
  36. [36]
    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].
  37. [37]
    T. Kasemets, W.J. Waalewijn and L. Zeune, Calculating Soft Radiation at One Loop, JHEP 03 (2016) 153 [arXiv:1512.00857] [INSPIRE].ADSCrossRefGoogle Scholar
  38. [38]
    T. Becher and G. Bell, Analytic Regularization in Soft-Collinear Effective Theory, Phys. Lett. B 713 (2012) 41 [arXiv:1112.3907] [INSPIRE].ADSCrossRefGoogle Scholar
  39. [39]
    Y. Li, D. Neill and H.X. Zhu, An Exponential Regulator for Rapidity Divergences, submitted to Phys. Rev. D (2016) [arXiv:1604.00392] [INSPIRE].
  40. [40]
    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].ADSCrossRefGoogle Scholar
  41. [41]
    I.W. Stewart, F.J. Tackmann and W.J. Waalewijn, Dissecting Soft Radiation with Factorization, Phys. Rev. Lett. 114 (2015) 092001 [arXiv:1405.6722] [INSPIRE].ADSCrossRefGoogle Scholar
  42. [42]
    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
  43. [43]
    S. 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].ADSGoogle Scholar
  44. [44]
    A.H. Hoang, D.W. Kolodrubetz, V. Mateu and I.W. Stewart, C-parameter distribution at N 3 LL including power corrections, Phys. Rev. D 91 (2015) 094017 [arXiv:1411.6633] [INSPIRE].ADSGoogle Scholar
  45. [45]
    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
  46. [46]
    A. Hornig, Y. Makris and T. Mehen, Jet Shapes in Dijet Events at the LHC in SCET, JHEP 04 (2016) 097 [arXiv:1601.01319] [INSPIRE].ADSGoogle Scholar
  47. [47]
    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].ADSGoogle Scholar
  48. [48]
    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].ADSCrossRefGoogle Scholar
  49. [49]
    M. Dasgupta, K. Khelifa-Kerfa, S. Marzani and M. Spannowsky, On jet mass distributions in Z+jet and dijet processes at the LHC, JHEP 10 (2012) 126 [arXiv:1207.1640] [INSPIRE].ADSCrossRefGoogle Scholar
  50. [50]
    Z.L. Liu, C.S. Li, J. Wang and Y. Wang, Resummation prediction on the jet mass spectrum in one-jet inclusive production at the LHC, JHEP 04 (2015) 005 [arXiv:1412.1337] [INSPIRE].CrossRefGoogle Scholar
  51. [51]
    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].ADSGoogle Scholar
  52. [52]
    A.J. Larkoski, I. Moult and D. Neill, Analytic Boosted Boson Discrimination, JHEP 05 (2016) 117 [arXiv:1507.03018] [INSPIRE].ADSCrossRefGoogle Scholar
  53. [53]
    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
  54. [54]
    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

Copyright information

© The Author(s) 2017

Authors and Affiliations

  • Daniele Bertolini
    • 1
    • 2
  • Daniel Kolodrubetz
    • 3
  • Duff Neill
    • 3
    • 4
  • Piotr Pietrulewicz
    • 5
    Email author
  • Iain W. Stewart
    • 3
  • Frank J. Tackmann
    • 5
  • Wouter J. Waalewijn
    • 6
    • 7
  1. 1.Berkeley Center for Theoretical PhysicsUniversity of CaliforniaBerkeleyU.S.A.
  2. 2.Theoretical Physics GroupLawrence Berkeley National LaboratoryBerkeleyU.S.A.
  3. 3.Center for Theoretical PhysicsMassachusetts Institute of TechnologyCambridgeU.S.A.
  4. 4.Theoretical DivisionLos Alamos National LaboratoryLos AlamosU.S.A.
  5. 5.Theory Group, Deutsches Elektronen-Synchrotron (DESY)HamburgGermany
  6. 6.Institute for Theoretical Physics Amsterdam and Delta Institute for Theoretical PhysicsUniversity of AmsterdamAmsterdamThe Netherlands
  7. 7.Nikhef, Theory GroupAmsterdamThe Netherlands

Personalised recommendations