Advertisement

Factorization and resummation for generic hierarchies between Jets

  • Piotr PietrulewiczEmail author
  • Frank J. Tackmann
  • Wouter J. Waalewijn
Open Access
Regular Article - Theoretical Physics

Abstract

Jets are an important probe to identify the hard interaction of interest at the LHC. They are routinely used in Standard Model precision measurements as well as in searches for new heavy particles, including jet substructure methods. In processes with several jets, one typically encounters hierarchies in the jet transverse momenta and/or dijet invariant masses. Large logarithms of the ratios of these kinematic jet scales in the cross section are at present primarily described by parton showers. We present a general factorization framework called SCET+, which is an extension of Soft-Collinear Effective Theory (SCET) and allows for a systematic higher-order resummation of such kinematic logarithms for generic jet hierarchies. In SCET+ additional intermediate soft/collinear modes are used to resolve jets arising from additional soft and/or collinear QCD emissions. The resulting factorized cross sections utilize collinear splitting amplitudes and soft gluon currents and fully capture spin and color correlations. We discuss how to systematically combine the different kinematic regimes to obtain a complete description of the jet phase space. To present its application in a simple context, we use the case of e + e → 3 jets. We then discuss in detail the application to N -jet processes at hadron colliders, considering representative classes of hierarchies from which the general case can be built. This includes in particular multiple hierarchies that are either strongly ordered in angle or energy or not.

Keywords

Jets QCD Phenomenology 

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]
    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].
  2. [2]
    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].
  3. [3]
    C.W. Bauer and I.W. Stewart, Invariant operators in collinear effective theory, Phys. Lett. B 516 (2001) 134 [hep-ph/0107001] [INSPIRE].
  4. [4]
    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].
  5. [5]
    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].
  6. [6]
    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].
  7. [7]
    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
  8. [8]
    M. Procura, W.J. Waalewijn and L. Zeune, Resummation of Double-Differential Cross sections and Fully-Unintegrated Parton Distribution Functions, JHEP 02 (2015) 117 [arXiv:1410.6483] [INSPIRE].ADSCrossRefGoogle Scholar
  9. [9]
    A.J. Larkoski, I. Moult and D. Neill, Non-Global Logarithms, Factorization and the Soft Substructure of Jets, JHEP 09 (2015) 143 [arXiv:1501.04596] [INSPIRE].ADSCrossRefGoogle Scholar
  10. [10]
    D. Neill, The Edge of Jets and Subleading Non-Global Logs, arXiv:1508.07568 [INSPIRE].
  11. [11]
    A.J. Larkoski, I. Moult and D. Neill, Analytic Boosted Boson Discrimination, JHEP 05 (2016) 117 [arXiv:1507.03018] [INSPIRE].ADSCrossRefGoogle Scholar
  12. [12]
    A.J. Larkoski, I. Moult and D. Neill, Power Counting to Better Jet Observables, JHEP 12 (2014) 009 [arXiv:1409.6298] [INSPIRE].ADSCrossRefGoogle Scholar
  13. [13]
    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
  14. [14]
    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
  15. [15]
    M.H. Seymour, Jet shapes in hadron collisions: Higher orders, resummation and hadronization, Nucl. Phys. B 513 (1998) 269 [hep-ph/9707338] [INSPIRE].
  16. [16]
    W. M.-Y. Cheung, M. Luke and S. Zuberi, Phase Space and Jet Definitions in SCET, Phys. Rev. D 80 (2009) 114021 [arXiv:0910.2479] [INSPIRE].ADSGoogle Scholar
  17. [17]
    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
  18. [18]
    A. Banfi, M. Dasgupta, K. Khelifa-Kerfa and S. Marzani, Non-global logarithms and jet algorithms in high-pT jet shapes, JHEP 08 (2010) 064 [arXiv:1004.3483] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  19. [19]
    R. Kelley, J.R. Walsh and S. Zuberi, Abelian Non-Global Logarithms from Soft Gluon Clustering, JHEP 09 (2012) 117 [arXiv:1202.2361] [INSPIRE].ADSCrossRefGoogle Scholar
  20. [20]
    A. von Manteuffel, R.M. Schabinger and H.X. Zhu, The Complete Two-Loop Integrated Jet Thrust Distribution In Soft-Collinear Effective Theory, JHEP 03 (2014) 139 [arXiv:1309.3560] [INSPIRE].ADSCrossRefGoogle Scholar
  21. [21]
    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
  22. [22]
    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
  23. [23]
    J. Thaler and T.F. Wilkason, Resolving Boosted Jets with XCone, JHEP 12 (2015) 051 [arXiv:1508.01518] [INSPIRE].ADSCrossRefGoogle Scholar
  24. [24]
    S. Catani, B.R. Webber, Y.L. Dokshitzer and F. Fiorani, Average multiplicities in two and three jet e + e annihilation events, Nucl. Phys. B 383 (1992) 419 [INSPIRE].ADSCrossRefGoogle Scholar
  25. [25]
    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].
  26. [26]
    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].ADSGoogle Scholar
  27. [27]
    C.W. Bauer and M.D. Schwartz, Event Generation from Effective Field Theory, Phys. Rev. D 76 (2007) 074004 [hep-ph/0607296] [INSPIRE].
  28. [28]
    M. Baumgart, C. Marcantonini and I.W. Stewart, Systematic Improvement of Parton Showers with Effective Theory, Phys. Rev. D 83 (2011) 034011 [arXiv:1007.0758] [INSPIRE].ADSGoogle Scholar
  29. [29]
    A.J. Larkoski and I. Moult, The Singular Behavior of Jet Substructure Observables, Phys. Rev. D 93 (2016) 014017 [arXiv:1510.08459] [INSPIRE].ADSGoogle Scholar
  30. [30]
    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].
  31. [31]
    Z. Ligeti, I.W. Stewart and F.J. Tackmann, Treating the b quark distribution function with reliable uncertainties, Phys. Rev. D 78 (2008) 114014 [arXiv:0807.1926] [INSPIRE].ADSGoogle 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 α s(m Z ), Phys. Rev. D 83 (2011) 074021 [arXiv:1006.3080] [INSPIRE].ADSGoogle Scholar
  33. [33]
    J. Thaler and K. Van Tilburg, Identifying Boosted Objects with N-subjettiness, JHEP 03 (2011) 015 [arXiv:1011.2268] [INSPIRE].ADSCrossRefGoogle Scholar
  34. [34]
    J. Thaler and K. Van Tilburg, Maximizing Boosted Top Identification by Minimizing N-subjettiness, JHEP 02 (2012) 093 [arXiv:1108.2701] [INSPIRE].ADSCrossRefGoogle Scholar
  35. [35]
    A.J. Larkoski, D. Neill and J. Thaler, Jet Shapes with the Broadening Axis, JHEP 04 (2014) 017 [arXiv:1401.2158] [INSPIRE].ADSCrossRefGoogle Scholar
  36. [36]
    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
  37. [37]
    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
  38. [38]
    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
  39. [39]
    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
  40. [40]
    I.W. Stewart, F.J. Tackmann and W.J. Waalewijn, Combining Fixed-Order Helicity Amplitudes With Resummation Using SCET, arXiv:1211.2305 [INSPIRE].
  41. [41]
    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
  42. [42]
    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
  43. [43]
    R.K. Ellis, D.A. Ross and A.E. Terrano, The Perturbative Calculation of Jet Structure in e + e Annihilation, Nucl. Phys. B 178 (1981) 421 [INSPIRE].ADSCrossRefGoogle Scholar
  44. [44]
    C.W. Bauer and A.V. Manohar, Shape function effects in BX s gamma and \( B\to {X}_u\ell \overline{\nu} \) decays, Phys. Rev. D 70 (2004) 034024 [hep-ph/0312109] [INSPIRE].
  45. [45]
    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].
  46. [46]
    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].
  47. [47]
    T. Becher and M.D. Schwartz, Direct photon production with effective field theory, JHEP 02 (2010) 040 [arXiv:0911.0681] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  48. [48]
    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
  49. [49]
    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
  50. [50]
    R. Boughezal, X. Liu and F. Petriello, N -jettiness soft function at next-to-next-to-leading order, Phys. Rev. D 91 (2015) 094035 [arXiv:1504.02540] [INSPIRE].ADSGoogle Scholar
  51. [51]
    O. Almelid, C. Duhr and E. Gardi, Three-loop corrections to the soft anomalous dimension in multi-leg scattering, arXiv:1507.00047 [INSPIRE].
  52. [52]
    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].
  53. [53]
    A.V. Manohar, Deep inelastic scattering as x → 1 using soft collinear effective theory, Phys. Rev. D 68 (2003) 114019 [hep-ph/0309176] [INSPIRE].
  54. [54]
    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].
  55. [55]
    F.A. Berends and W.T. Giele, Recursive Calculations for Processes with n Gluons, Nucl. Phys. B 306 (1988) 759 [INSPIRE].ADSCrossRefGoogle Scholar
  56. [56]
    M.L. Mangano and S.J. Parke, Multiparton amplitudes in gauge theories, Phys. Rept. 200 (1991) 301 [hep-th/0509223] [INSPIRE].ADSCrossRefGoogle Scholar
  57. [57]
    Z. Bern, L.J. Dixon, D.C. Dunbar and D.A. Kosower, One loop n point gauge theory amplitudes, unitarity and collinear limits, Nucl. Phys. B 425 (1994) 217 [hep-ph/9403226] [INSPIRE].
  58. [58]
    D.A. Kosower, All order collinear behavior in gauge theories, Nucl. Phys. B 552 (1999) 319 [hep-ph/9901201] [INSPIRE].
  59. [59]
    D.A. Kosower and P. Uwer, One loop splitting amplitudes in gauge theory, Nucl. Phys. B 563 (1999) 477 [hep-ph/9903515] [INSPIRE].
  60. [60]
    Z. Bern, V. Del Duca, W.B. Kilgore and C.R. Schmidt, The infrared behavior of one loop QCD amplitudes at next-to-next-to leading order, Phys. Rev. D 60 (1999) 116001 [hep-ph/9903516] [INSPIRE].
  61. [61]
    G.F.R. Sborlini, D. de Florian and G. Rodrigo, Double collinear splitting amplitudes at next-to-leading order, JHEP 01 (2014) 018 [arXiv:1310.6841] [INSPIRE].ADSCrossRefGoogle Scholar
  62. [62]
    Z. Bern, V. Del Duca and C.R. Schmidt, The infrared behavior of one loop gluon amplitudes at next-to-next-to-leading order, Phys. Lett. B 445 (1998) 168 [hep-ph/9810409] [INSPIRE].
  63. [63]
    S. Catani and M. Grazzini, The soft gluon current at one loop order, Nucl. Phys. B 591 (2000) 435 [hep-ph/0007142] [INSPIRE].
  64. [64]
    C. Duhr and T. Gehrmann, The two-loop soft current in dimensional regularization, Phys. Lett. B 727 (2013) 452 [arXiv:1309.4393] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  65. [65]
    Y. Li and H.X. Zhu, Single soft gluon emission at two loops, JHEP 11 (2013) 080 [arXiv:1309.4391] [INSPIRE].ADSCrossRefGoogle Scholar
  66. [66]
    J.-y. Chiu, R. Kelley and A.V. Manohar, Electroweak Corrections using Effective Field Theory: Applications to the LHC, Phys. Rev. D 78 (2008) 073006 [arXiv:0806.1240] [INSPIRE].ADSGoogle Scholar
  67. [67]
    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
  68. [68]
    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
  69. [69]
    S.M. Aybat, L.J. Dixon and G.F. Sterman, The two-loop soft anomalous dimension matrix and resummation at next-to-next-to leading pole, Phys. Rev. D 74 (2006) 074004 [hep-ph/0607309] [INSPIRE].
  70. [70]
    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
  71. [71]
    J.R. Gaunt, Glauber Gluons and Multiple Parton Interactions, JHEP 07 (2014) 110 [arXiv:1405.2080] [INSPIRE].ADSCrossRefGoogle Scholar
  72. [72]
    M. Zeng, Drell-Yan process with jet vetoes: breaking of generalized factorization, JHEP 10 (2015) 189 [arXiv:1507.01652] [INSPIRE].ADSCrossRefGoogle Scholar
  73. [73]
    I.Z. Rothstein and I.W. Stewart, An Effective Field Theory for Forward Scattering and Factorization Violation, arXiv:1601.04695 [INSPIRE].
  74. [74]
    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
  75. [75]
    J.R. Andersen and J.M. Smillie, Constructing All-Order Corrections to Multi-Jet Rates, JHEP 01 (2010) 039 [arXiv:0908.2786] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  76. [76]
    W.T. Giele, E.W.N. Glover and D.A. Kosower, Higher order corrections to jet cross-sections in hadron colliders, Nucl. Phys. B 403 (1993) 633 [hep-ph/9302225] [INSPIRE].
  77. [77]
    L.J. Dixon, Calculating scattering amplitudes efficiently, hep-ph/9601359 [INSPIRE].
  78. [78]
    L.J. Dixon, A brief introduction to modern amplitude methods, arXiv:1310.5353 [INSPIRE].
  79. [79]
    C.M. Arnesen, J. Kundu and I.W. Stewart, Constraint equations for heavy-to-light currents in SCET, Phys. Rev. D 72 (2005) 114002 [hep-ph/0508214] [INSPIRE].
  80. [80]
    D. Kang, O.Z. Labun and C. Lee, Equality of hemisphere soft functions for e + e , DIS and pp collisions at \( \mathcal{O}\left({\alpha}_s^2\right) \), Phys. Lett. B 748 (2015) 45 [arXiv:1504.04006] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  81. [81]
    T. Becher and M. Neubert, On the Structure of Infrared Singularities of Gauge-Theory Amplitudes, JHEP 06 (2009) 081 [Erratum ibid. 1311 (2013) 024] [arXiv:0903.1126] [INSPIRE].
  82. [82]
    J.M. Campbell and E.W.N. Glover, Double unresolved approximations to multiparton scattering amplitudes, Nucl. Phys. B 527 (1998) 264 [hep-ph/9710255] [INSPIRE].
  83. [83]
    S. Catani and M. Grazzini, Infrared factorization of tree level QCD amplitudes at the next-to-next-to-leading order and beyond, Nucl. Phys. B 570 (2000) 287 [hep-ph/9908523] [INSPIRE].
  84. [84]
    V. Del Duca, A. Frizzo and F. Maltoni, Factorization of tree QCD amplitudes in the high-energy limit and in the collinear limit, Nucl. Phys. B 568 (2000) 211 [hep-ph/9909464] [INSPIRE].
  85. [85]
    D.A. Kosower, Antenna factorization in strongly ordered limits, Phys. Rev. D 71 (2005) 045016 [hep-ph/0311272] [INSPIRE].
  86. [86]
    G. Somogyi, Z. Trócsányi and V. Del Duca, Matching of singly- and doubly-unresolved limits of tree-level QCD squared matrix elements, JHEP 06 (2005) 024 [hep-ph/0502226] [INSPIRE].
  87. [87]
    S. Catani and M. Grazzini, Collinear factorization and splitting functions for next-to-next-to-leading order QCD calculations, Phys. Lett. B 446 (1999) 143 [hep-ph/9810389] [INSPIRE].
  88. [88]
    S. Alioli, C.W. Bauer, C.J. Berggren, A. Hornig, F.J. Tackmann, C.K. Vermilion et al., Combining Higher-Order Resummation with Multiple NLO Calculations and Parton Showers in GENEVA, JHEP 09 (2013) 120 [arXiv:1211.7049] [INSPIRE].ADSCrossRefGoogle Scholar
  89. [89]
    S. Alioli, C.W. Bauer, C. Berggren, F.J. Tackmann, J.R. Walsh and S. Zuberi, Matching Fully Differential NNLO Calculations and Parton Showers, JHEP 06 (2014) 089 [arXiv:1311.0286] [INSPIRE].ADSCrossRefGoogle Scholar
  90. [90]
    S. Alioli, C.W. Bauer, C. Berggren, F.J. Tackmann and J.R. Walsh, Drell-Yan production at NNLL +NNLO matched to parton showers, Phys. Rev. D 92 (2015) 094020 [arXiv:1508.01475] [INSPIRE].ADSGoogle Scholar
  91. [91]
    K. Hamilton, P. Nason and G. Zanderighi, MINLO: Multi-Scale Improved NLO, JHEP 10 (2012) 155 [arXiv:1206.3572] [INSPIRE].ADSCrossRefGoogle Scholar
  92. [92]
    K. Hamilton, P. Nason, C. Oleari and G. Zanderighi, Merging H/W/Z + 0 and 1 jet at NLO with no merging scale: a path to parton shower + NNLO matching, JHEP 05 (2013) 082 [arXiv:1212.4504] [INSPIRE].ADSCrossRefGoogle Scholar
  93. [93]
    R. Frederix and K. Hamilton, Extending the MINLO method, JHEP 05 (2016) 042 [arXiv:1512.02663] [INSPIRE].ADSCrossRefGoogle Scholar
  94. [94]
    W.E. Caswell, Asymptotic Behavior of Nonabelian Gauge Theories to Two Loop Order, Phys. Rev. Lett. 33 (1974) 244 [INSPIRE].ADSCrossRefGoogle Scholar
  95. [95]
    D.R.T. Jones, Two Loop Diagrams in Yang-Mills Theory, Nucl. Phys. B 75 (1974) 531 [INSPIRE].ADSCrossRefGoogle Scholar
  96. [96]
    A. Idilbi, X.-d. Ji, J.-P. Ma and F. Yuan, Threshold resummation for Higgs production in effective field theory, Phys. Rev. D 73 (2006) 077501 [hep-ph/0509294] [INSPIRE].
  97. [97]
    A. Idilbi, X.-d. Ji and F. Yuan, Resummation of threshold logarithms in effective field theory for DIS, Drell-Yan and Higgs production, Nucl. Phys. B 753 (2006) 42 [hep-ph/0605068] [INSPIRE].
  98. [98]
    G. ’t Hooft and M.J.G. Veltman, Regularization and Renormalization of Gauge Fields, Nucl. Phys. B 44 (1972) 189 [INSPIRE].
  99. [99]
    S.D. Badger and E.W.N. Glover, Two loop splitting functions in QCD, JHEP 07 (2004) 040 [hep-ph/0405236] [INSPIRE].

Copyright information

© The Author(s) 2016

Authors and Affiliations

  • Piotr Pietrulewicz
    • 1
    Email author
  • Frank J. Tackmann
    • 1
  • Wouter J. Waalewijn
    • 2
    • 3
  1. 1.Theory Group, Deutsches Elektronen-Synchrotron (DESY)HamburgGermany
  2. 2.ITFA, University of AmsterdamAmsterdamNetherlands
  3. 3.Nikhef, Theory GroupAmsterdamNetherlands

Personalised recommendations