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First subleading power resummation for event shapes

  • Ian Moult
  • Iain W. Stewart
  • Gherardo Vita
  • Hua Xing Zhu
Open Access
Regular Article - Theoretical Physics
  • 31 Downloads

Abstract

We derive and analytically solve renormalization group (RG) equations of gauge invariant non-local Wilson line operators which resum logarithms for event shape observables τ at subleading power in the τ ≪ 1 expansion. These equations involve a class of universal jet and soft functions arising through operator mixing, which we call θ-jet and θ-soft functions. An illustrative example involving these operators is introduced which captures the generic features of subleading power resummation, allowing us to derive the structure of the RG to all orders in αs, and provide field theory definitions of all ingredients. As a simple application, we use this to obtain an analytic leading logarithmic result for the subleading power resummed thrust spectrum for Hgg in pure glue QCD. This resummation determines the nature of the double logarithmic series at subleading power, which we find is still governed by the cusp anomalous dimension. We check our result by performing an analytic calculation up to \( \mathcal{O}\left({\alpha}_s^3\right) \). Consistency of the subleading power RG relates subleading power anomalous dimensions, constrains the form of the θ-soft and θ-jet functions, and implies an exponentiation of higher order loop corrections in the subleading power collinear limit. Our results provide a path for carrying out systematic resummation at subleading power for collider observables.

Keywords

Effective Field Theories Perturbative QCD Renormalization Group Resummation 

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]
    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].
  2. [2]
    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].
  3. [3]
    D. Pirjol and I.W. Stewart, A complete basis for power suppressed collinear ultrasoft operators, Phys. Rev. D 67 (2003) 094005 [Erratum ibid. D 69 (2004) 019903] [hep-ph/0211251] [INSPIRE].
  4. [4]
    M. Beneke and T. Feldmann, Multipole expanded soft collinear effective theory with nonAbelian gauge symmetry, Phys. Lett. B 553 (2003) 267 [hep-ph/0211358] [INSPIRE].
  5. [5]
    C.W. Bauer, D. Pirjol and I.W. Stewart, On Power suppressed operators and gauge invariance in SCET, Phys. Rev. D 68 (2003) 034021 [hep-ph/0303156] [INSPIRE].
  6. [6]
    R.J. Hill, T. Becher, S.J. Lee and M. Neubert, Sudakov resummation for subleading SCET currents and heavy-to-light form-factors, JHEP 07 (2004) 081 [hep-ph/0404217] [INSPIRE].
  7. [7]
    K.S.M. Lee and I.W. Stewart, Factorization for power corrections to BX s γ and \( B\to {X}_ul\overline{\nu} \), Nucl. Phys. B 721 (2005) 325 [hep-ph/0409045] [INSPIRE].
  8. [8]
    Yu. L. Dokshitzer, G. Marchesini and G.P. Salam, Revisiting parton evolution and the large-x limit, Phys. Lett. B 634 (2006) 504 [hep-ph/0511302] [INSPIRE].
  9. [9]
    M. Trott and A.R. Williamson, Towards the anomalous dimension to O(Λ QCD /M b) for phase space restricted \( \overline{B}\to {X}_ul\overline{\nu}\kern0.5em and\kern0.5em \overline{B}\to {X}_s\gamma \), Phys. Rev. D 74 (2006) 034011 [hep-ph/0510203] [INSPIRE].
  10. [10]
    E. Laenen, L. Magnea and G. Stavenga, On next-to-eikonal corrections to threshold resummation for the Drell-Yan and DIS cross sections, Phys. Lett. B 669 (2008) 173 [arXiv:0807.4412] [INSPIRE].ADSCrossRefGoogle Scholar
  11. [11]
    E. Laenen, G. Stavenga and C.D. White, Path integral approach to eikonal and next-to-eikonal exponentiation, JHEP 03 (2009) 054 [arXiv:0811.2067] [INSPIRE].ADSCrossRefGoogle Scholar
  12. [12]
    G. Paz, Subleading jet functions in inclusive B decays, JHEP 06 (2009) 083 [arXiv:0903.3377] [INSPIRE].ADSCrossRefGoogle Scholar
  13. [13]
    M. Benzke, S.J. Lee, M. Neubert and G. Paz, Factorization at subleading power and irreducible uncertainties in \( \overline{B}\to {X}_s\gamma \) decay, JHEP 08 (2010) 099 [arXiv:1003.5012] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  14. [14]
    E. Laenen, L. Magnea, G. Stavenga and C.D. White, Next-to-eikonal corrections to soft gluon radiation: a diagrammatic approach, JHEP 01 (2011) 141 [arXiv:1010.1860] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  15. [15]
    S.M. Freedman, Subleading corrections to thrust using effective field theory, arXiv:1303.1558 [INSPIRE].
  16. [16]
    S.M. Freedman and R. Goerke, Renormalization of subleading dijet operators in soft-collinear effective theory, Phys. Rev. D 90 (2014) 114010 [arXiv:1408.6240] [INSPIRE].ADSGoogle Scholar
  17. [17]
    D. Bonocore et al., The method of regions and next-to-soft corrections in Drell-Yan production, Phys. Lett. B 742 (2015) 375 [arXiv:1410.6406] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  18. [18]
    A.J. Larkoski, D. Neill and I.W. Stewart, Soft theorems from effective field theory, JHEP 06 (2015) 077 [arXiv:1412.3108] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    D. Bonocore, E. Laenen, L. Magnea, S. Melville, L. Vernazza and C.D. White, A factorization approach to next-to-leading-power threshold logarithms, JHEP 06 (2015) 008 [arXiv:1503.05156] [INSPIRE].ADSCrossRefGoogle Scholar
  20. [20]
    D. Bonocore et al., Non-abelian factorisation for next-to-leading-power threshold logarithms, JHEP 12 (2016) 121 [arXiv:1610.06842] [INSPIRE].ADSCrossRefGoogle Scholar
  21. [21]
    I. Moult et al., Subleading power corrections for N-jettiness subtractions, Phys. Rev. D 95 (2017) 074023 [arXiv:1612.00450] [INSPIRE].ADSGoogle Scholar
  22. [22]
    R. Boughezal, X. Liu and F. Petriello, Power corrections in the N-jettiness subtraction scheme, JHEP 03 (2017) 160 [arXiv:1612.02911] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  23. [23]
    V. Del Duca et al., Universality of next-to-leading power threshold effects for colourless final states in hadronic collisions, JHEP 11 (2017) 057 [arXiv:1706.04018] [INSPIRE].ADSCrossRefGoogle Scholar
  24. [24]
    I. Balitsky and A. Tarasov, Higher-twist corrections to gluon TMD factorization, JHEP 07 (2017) 095 [arXiv:1706.01415] [INSPIRE].ADSCrossRefGoogle Scholar
  25. [25]
    I. Moult et al., N-jettiness subtractions for ggH at subleading power, Phys. Rev. D 97 (2018) 014013 [arXiv:1710.03227] [INSPIRE].ADSGoogle Scholar
  26. [26]
    R. Goerke and M. Inglis-Whalen, Renormalization of dijet operators at order 1/Q 2 in soft-collinear effective theory, JHEP 05 (2018) 023 [arXiv:1711.09147] [INSPIRE].ADSCrossRefGoogle Scholar
  27. [27]
    I. Balitsky and A. Tarasov, Power corrections to TMD factorization for Z-boson production, JHEP 05 (2018) 150 [arXiv:1712.09389] [INSPIRE].ADSCrossRefGoogle Scholar
  28. [28]
    M. Beneke et al., Anomalous dimension of subleading-power N-jet operators, JHEP 03 (2018) 001 [arXiv:1712.04416] [INSPIRE].ADSCrossRefGoogle Scholar
  29. [29]
    I. Feige et al., A complete basis of helicity operators for subleading factorization, JHEP 11 (2017) 142 [arXiv:1703.03411] [INSPIRE].ADSCrossRefGoogle Scholar
  30. [30]
    I. Moult, I.W. Stewart and G. Vita, A subleading operator basis and matching for ggH, JHEP 07 (2017) 067 [arXiv:1703.03408] [INSPIRE].ADSCrossRefGoogle Scholar
  31. [31]
    C.-H. Chang, I.W. Stewart and G. Vita, A Subleading Power Operator Basis for the Scalar Quark Current, JHEP 04 (2018) 041 [arXiv:1712.04343] [INSPIRE].ADSCrossRefGoogle Scholar
  32. [32]
    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].
  33. [33]
    C.W. Bauer et al., An effective field theory for collinear and soft gluons: heavy to light decays, Phys. Rev. D 63 (2001) 114020 [hep-ph/0011336] [INSPIRE].
  34. [34]
    C.W. Bauer and I.W. Stewart, Invariant operators in collinear effective theory, Phys. Lett. B 516 (2001) 134 [hep-ph/0107001] [INSPIRE].
  35. [35]
    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].
  36. [36]
    E. Farhi, A QCD test for jets, Phys. Rev. Lett. 39 (1977) 1587 [INSPIRE].ADSCrossRefGoogle Scholar
  37. [37]
    V.V. Sudakov, Vertex parts at very high-energies in quantum electrodynamics, Sov. Phys. JETP 3 (1956) 65 [Zh. Eksp. Teor. Fiz. 30 (1956) 87] [INSPIRE].
  38. [38]
    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].
  39. [39]
    J.C. Collins, D.E. Soper and G.F. Sterman, Factorization for short distance hadron-hadron scattering, Nucl. Phys. B 261 (1985) 104 [INSPIRE].ADSCrossRefGoogle Scholar
  40. [40]
    J.C. Collins, D.E. Soper and G.F. Sterman, Soft Gluons and Factorization, Nucl. Phys. B 308 (1988) 833 [INSPIRE].ADSCrossRefGoogle Scholar
  41. [41]
    J.C. Collins, D.E. Soper and G.F. Sterman, Factorization of hard processes in QCD, Adv. Ser. Direct. High Energy Phys. 5 (1989) 1 [hep-ph/0409313] [INSPIRE].
  42. [42]
    G.F. Sterman, Summation of Large Corrections to Short Distance Hadronic Cross-Sections, Nucl. Phys. B 281 (1987) 310 [INSPIRE].ADSCrossRefGoogle Scholar
  43. [43]
    C.W. Bauer et al., Hard scattering factorization from effective field theory, Phys. Rev. D 66 (2002) 014017 [hep-ph/0202088] [INSPIRE].
  44. [44]
    S. 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].
  45. [45]
    M.D. Schwartz, Resummation and NLO matching of event shapes with effective field theory, Phys. Rev. D 77 (2008) 014026 [arXiv:0709.2709] [INSPIRE].ADSGoogle Scholar
  46. [46]
    G.P. Korchemsky and G.F. Sterman, Power corrections to event shapes and factorization, Nucl. Phys. B 555 (1999) 335 [hep-ph/9902341] [INSPIRE].
  47. [47]
    S. Catani, G. Turnock, B.R. Webber and L. Trentadue, Thrust distribution in e + e annihilation, Phys. Lett. B 263 (1991) 491 [INSPIRE].ADSCrossRefGoogle Scholar
  48. [48]
    S. Catani, L. Trentadue, G. Turnock and B.R. Webber, Resummation of large logarithms in e + e event shape distributions, Nucl. Phys. B 407 (1993) 3 [INSPIRE].ADSCrossRefGoogle Scholar
  49. [49]
    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].ADSCrossRefGoogle Scholar
  50. [50]
    R. Abbate et al., Thrust at N 3 LL with power corrections and a precision global fit for α s(m Z), Phys. Rev. D 83 (2011) 074021 [arXiv:1006.3080] [INSPIRE].ADSGoogle Scholar
  51. [51]
    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].ADSCrossRefzbMATHGoogle Scholar
  52. [52]
    A.H. Hoang et al., C-parameter distribution at N 3 LLincluding power corrections, Phys. Rev. D 91 (2015) 094017 [arXiv:1411.6633] [INSPIRE].ADSGoogle Scholar
  53. [53]
    I. Moult and H.X. Zhu, Simplicity from recoil: the three-loop soft function and factorization for the energy-energy correlation, arXiv:1801.02627 [INSPIRE].
  54. [54]
    M. Beneke and T. Feldmann, Factorization of heavy to light form-factors in soft collinear effective theory, Nucl. Phys. B 685 (2004) 249 [hep-ph/0311335] [INSPIRE].
  55. [55]
    R. Boughezal, A. Isgrò and F. Petriello, Next-to-leading-logarithmic power corrections for N -jettiness subtraction in color-singlet production, Phys. Rev. D 97 (2018) 076006 [arXiv:1802.00456] [INSPIRE].ADSGoogle Scholar
  56. [56]
    I. Moult et al., Fermionic Glauber operators and quark reggeization, JHEP 02 (2018) 134 [arXiv:1709.09174] [INSPIRE].ADSCrossRefGoogle Scholar
  57. [57]
    R. Brüser, S. Caron-Huot and J.M. Henn, Subleading Regge limit from a soft anomalous dimension, JHEP 04 (2018) 047 [arXiv:1802.02524] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  58. [58]
    T. Liu and A.A. Penin, High-energy limit of QCD beyond the Sudakov approximation, Phys. Rev. Lett. 119 (2017) 262001 [arXiv:1709.01092] [INSPIRE].ADSCrossRefGoogle Scholar
  59. [59]
    I. Moult, I. W. Stewart and G. Vita, Subleading power factorization with radiative functions, MIT-CTP 4948, to appear.Google Scholar
  60. [60]
    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].
  61. [61]
    N.A. Sveshnikov and F.V. Tkachov, Jets and quantum field theory, Phys. Lett. B 382 (1996) 403 [hep-ph/9512370] [INSPIRE].
  62. [62]
    G.P. Korchemsky, G. Oderda and G.F. Sterman, Power corrections and nonlocal operators, AIP Conf. Proc. 407 (1997) 988 [hep-ph/9708346] [INSPIRE].
  63. [63]
    C.W. Bauer et al., 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].ADSGoogle Scholar
  64. [64]
    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].
  65. [65]
    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].ADSGoogle Scholar
  66. [66]
    G.P. Korchemsky and A.V. Radyushkin, Renormalization of the Wilson loops beyond the leading order, Nucl. Phys. B 283 (1987) 342 [INSPIRE].ADSCrossRefGoogle Scholar
  67. [67]
    I.A. Korchemskaya and G.P. Korchemsky, On lightlike Wilson loops, Phys. Lett. B 287 (1992) 169 [INSPIRE].ADSCrossRefGoogle Scholar
  68. [68]
    D.W. Kolodrubetz, I. Moult and I.W. Stewart, Building blocks for subleading helicity operators, JHEP 05 (2016) 139 [arXiv:1601.02607] [INSPIRE].ADSCrossRefGoogle Scholar
  69. [69]
    J. Chay and C. Kim, Collinear effective theory at subleading order and its application to heavy-light currents, Phys. Rev. D 65 (2002) 114016 [hep-ph/0201197] [INSPIRE].
  70. [70]
    F.E. Low, Bremsstrahlung of very low-energy quanta in elementary particle collisions, Phys. Rev. 110 (1958) 974 [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  71. [71]
    T.H. Burnett and N.M. Kroll, Extension of the low soft photon theorem, Phys. Rev. Lett. 20 (1968) 86 [INSPIRE].ADSCrossRefGoogle Scholar
  72. [72]
    A.V. Manohar, Deep inelastic scattering as x → 1 using soft collinear effective theory, Phys. Rev. D 68 (2003) 114019 [hep-ph/0309176] [INSPIRE].
  73. [73]
    M. Beneke, F. Campanario, T. Mannel and B.D. Pecjak, Power corrections to \( \overline{B}\to {X}_ul\overline{\nu}\left({X}_s\gamma \right) \) decay spectra in theshape-functionregion, JHEP 06 (2005) 071 [hep-ph/0411395] [INSPIRE].
  74. [74]
    L.W. Garland et al., The two loop QCD matrix element for e + e → 3 jets, Nucl. Phys. B 627 (2002) 107 [hep-ph/0112081] [INSPIRE].
  75. [75]
    L.W. Garland et al., Two loop QCD helicity amplitudes for e + e three jets, Nucl. Phys. B 642 (2002) 227 [hep-ph/0206067] [INSPIRE].
  76. [76]
    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].ADSCrossRefzbMATHGoogle Scholar
  77. [77]
    S. Catani, The singular behavior of QCD amplitudes at two loop order, Phys. Lett. B 427 (1998) 161 [hep-ph/9802439] [INSPIRE].
  78. [78]
    L.J. Dixon, L. Magnea and G.F. Sterman, Universal structure of subleading infrared poles in gauge theory amplitudes, JHEP 08 (2008) 022 [arXiv:0805.3515] [INSPIRE].ADSCrossRefGoogle Scholar
  79. [79]
    T. Becher and M. Neubert, On the structure of infrared singularities of gauge-theory amplitudes, JHEP 06 (2009) 081 [Erratum ibid. 11 (2013) 024] [arXiv:0903.1126] [INSPIRE].
  80. [80]
    E. Gardi and L. Magnea, Factorization constraints for soft anomalous dimensions in QCD scattering amplitudes, JHEP 03 (2009) 079 [arXiv:0901.1091] [INSPIRE].ADSCrossRefGoogle Scholar
  81. [81]
    Ø. Almelid, C. Duhr and E. Gardi, Three-loop corrections to the soft anomalous dimension in multileg scattering, Phys. Rev. Lett. 117 (2016) 172002 [arXiv:1507.00047] [INSPIRE].ADSCrossRefGoogle Scholar
  82. [82]
    B.R. Webber, A QCD model for jet fragmentation including soft gluon interference, Nucl. Phys. B 238 (1984) 492 [INSPIRE].ADSCrossRefGoogle Scholar
  83. [83]
    G. Marchesini and B.R. Webber, Simulation of QCD jets including soft gluon interference, Nucl. Phys. B 238 (1984) 1 [INSPIRE].ADSCrossRefGoogle Scholar
  84. [84]
    G. Marchesini and B.R. Webber, Monte Carlo simulation of general hard processes with coherent QCD radiation, Nucl. Phys. B 310 (1988) 461 [INSPIRE].ADSCrossRefGoogle Scholar
  85. [85]
    D. Nandan, J. Plefka and W. Wormsbecher, Collinear limits beyond the leading order from the scattering equations, JHEP 02 (2017) 038 [arXiv:1608.04730] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  86. [86]
    S. Caron-Huot et al., Bootstrapping a five-loop amplitude using Steinmann relations, Phys. Rev. Lett. 117 (2016) 241601 [arXiv:1609.00669] [INSPIRE].ADSCrossRefGoogle Scholar
  87. [87]
    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

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© The Author(s) 2018

Authors and Affiliations

  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.Department of PhysicsZhejiang UniversityHangzhouChina

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