Journal of High Energy Physics

, 2018:107 | Cite as

Local analytic sector subtraction at NNLO

  • L. Magnea
  • E. Maina
  • G. Pelliccioli
  • C. Signorile-Signorile
  • P. Torrielli
  • S. UcciratiEmail author
Open Access
Regular Article - Theoretical Physics


We present a new method for the local subtraction of infrared divergences at next-to-next-to-leading order (NNLO) in QCD, for generic infrared-safe observables. Our method attempts to conjugate the minimal local counterterm structure arising from a sector partition of the radiation phase space with the simplifications following from analytic integration of the counterterms. In this first implementation, the method applies to final-state massless particles. We show how our method compactly organises infrared subtraction at NLO, we deduce in detail the general structure of the subtraction terms at NNLO, and we provide a proof of principle with a complete application to a simple process at NNLO.


QCD Phenomenology Jets 


Open Access

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  1. [1]
    J.R. Andersen et al., Les Houches 2017: Physics at TeV Colliders Standard Model Working Group Report, in 10th Les Houches Workshop on Physics at TeV Colliders (PhysTeV 2017) Les Houches, France, June 5-23, 2017, arXiv:1803.07977 [INSPIRE].
  2. [2]
    J.C. Collins, Sudakov form-factors, Adv. Ser. Direct. High Energy Phys. 5 (1989) 573 [hep-ph/0312336] [INSPIRE].
  3. [3]
    G.F. Sterman, Partons, factorization and resummation, TASI 95, in QCD and beyond. Proceedings, Theoretical Advanced Study Institute in Elementary Particle Physics, TASI-95, Boulder, U.S.A., June 4-30, 1995, pp. 327-408, hep-ph/9606312 [INSPIRE].
  4. [4]
    S. Catani, The singular behavior of QCD amplitudes at two loop order, Phys. Lett. B 427 (1998)161 [hep-ph/9802439] [INSPIRE].
  5. [5]
    G.F. Sterman and M.E. Tejeda-Yeomans, Multiloop amplitudes and resummation, Phys. Lett. B 552 (2003) 48 [hep-ph/0210130] [INSPIRE].
  6. [6]
    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].CrossRefGoogle Scholar
  7. [7]
    E. Gardi and L. Magnea, Factorization constraints for soft anomalous dimensions in QCD scattering amplitudes, JHEP 03 (2009) 079 [arXiv:0901.1091] [INSPIRE].CrossRefGoogle Scholar
  8. [8]
    E. Gardi and L. Magnea, Infrared singularities in QCD amplitudes, Nuovo Cim. C32N5-6 (2009)137 [arXiv:0908.3273] [INSPIRE].
  9. [9]
    T. Becher and M. Neubert, Infrared singularities of scattering amplitudes in perturbative QCD, Phys. Rev. Lett. 102 (2009) 162001 [Erratum ibid. 111 (2013) 199905] [arXiv:0901.0722] [INSPIRE].
  10. [10]
    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].
  11. [11]
    I. Feige and M.D. Schwartz, Hard-Soft-Collinear Factorization to All Orders, Phys. Rev. D 90 (2014)105020 [arXiv:1403.6472] [INSPIRE].
  12. [12]
    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].CrossRefGoogle Scholar
  13. [13]
    Ø Almelid, C. Duhr, E. Gardi, A. McLeod and C.D. White, Bootstrapping the QCD soft anomalous dimension, JHEP 09 (2017) 073 [arXiv:1706.10162] [INSPIRE].
  14. [14]
    F. Bloch and A. Nordsieck, Note on the Radiation Field of the electron, Phys. Rev. 52 (1937) 54 [INSPIRE].CrossRefzbMATHGoogle Scholar
  15. [15]
    T. Kinoshita, Mass singularities of Feynman amplitudes, J. Math. Phys. 3 (1962) 650 [INSPIRE].CrossRefzbMATHGoogle Scholar
  16. [16]
    T.D. Lee and M. Nauenberg, Degenerate Systems and Mass Singularities, Phys. Rev. 133 (1964) B1549 [INSPIRE].MathSciNetCrossRefGoogle Scholar
  17. [17]
    G. Grammer Jr. and D.R. Yennie, Improved treatment for the infrared divergence problem in quantum electrodynamics, Phys. Rev. D 8 (1973) 4332 [INSPIRE].
  18. [18]
    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].
  19. [19]
    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].
  20. [20]
    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].
  21. [21]
    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].
  22. [22]
    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].
  23. [23]
    D.A. Kosower, All order collinear behavior in gauge theories, Nucl. Phys. B 552 (1999) 319 [hep-ph/9901201] [INSPIRE].
  24. [24]
    S. Catani and M. Grazzini, The soft gluon current at one loop order, Nucl. Phys. B 591 (2000)435 [hep-ph/0007142] [INSPIRE].
  25. [25]
    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].
  26. [26]
    C. Duhr and T. Gehrmann, The two-loop soft current in dimensional regularization, Phys. Lett. B 727 (2013) 452 [arXiv:1309.4393] [INSPIRE].
  27. [27]
    Y. Li and H.X. Zhu, Single soft gluon emission at two loops, JHEP 11 (2013) 080 [arXiv:1309.4391] [INSPIRE].CrossRefGoogle Scholar
  28. [28]
    P. Banerjee, P.K. Dhani and V. Ravindran, Gluon jet function at three loops in QCD, Phys. Rev. D 98 (2018) 094016 [arXiv:1805.02637] [INSPIRE].
  29. [29]
    R. Brüser, Z.L. Liu and M. Stahlhofen, Three-Loop Quark Jet Function, Phys. Rev. Lett. 121 (2018)072003 [arXiv:1804.09722] [INSPIRE].
  30. [30]
    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].
  31. [31]
    W.T. Giele, E.W.N. Glover and D.A. Kosower, The inclusive two jet triply differential cross-section, Phys. Rev. D 52 (1995) 1486 [hep-ph/9412338] [INSPIRE].
  32. [32]
    S. Frixione, Z. Kunszt and A. Signer, Three jet cross-sections to next-to-leading order, Nucl. Phys. B 467 (1996) 399 [hep-ph/9512328] [INSPIRE].
  33. [33]
    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].
  34. [34]
    J.M. Campbell and R.K. Ellis, An update on vector boson pair production at hadron colliders, Phys. Rev. D 60 (1999) 113006 [hep-ph/9905386] [INSPIRE].
  35. [35]
    T. Gleisberg and F. Krauss, Automating dipole subtraction for QCD NLO calculations, Eur. Phys. J. C 53 (2008) 501 [arXiv:0709.2881] [INSPIRE].
  36. [36]
    R. Frederix, T. Gehrmann and N. Greiner, Automation of the Dipole Subtraction Method in MadGraph/MadEvent, JHEP 09 (2008) 122 [arXiv:0808.2128] [INSPIRE].CrossRefGoogle Scholar
  37. [37]
    M. Czakon, C.G. Papadopoulos and M. Worek, Polarizing the Dipoles, JHEP 08 (2009) 085 [arXiv:0905.0883] [INSPIRE].CrossRefGoogle Scholar
  38. [38]
    K. Hasegawa, S. Moch and P. Uwer, AutoDipole: Automated generation of dipole subtraction terms, Comput. Phys. Commun. 181 (2010) 1802 [arXiv:0911.4371] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  39. [39]
    R. Frederix, S. Frixione, F. Maltoni and T. Stelzer, Automation of next-to-leading order computations in QCD: The FKS subtraction, JHEP 10 (2009) 003 [arXiv:0908.4272] [INSPIRE].CrossRefGoogle Scholar
  40. [40]
    S. Alioli, P. Nason, C. Oleari and E. Re, A general framework for implementing NLO calculations in shower Monte Carlo programs: the POWHEG BOX, JHEP 06 (2010) 043 [arXiv:1002.2581] [INSPIRE].CrossRefzbMATHGoogle Scholar
  41. [41]
    S. Platzer and S. Gieseke, Dipole Showers and Automated NLO Matching in HERWIG++, Eur. Phys. J. C 72 (2012) 2187 [arXiv:1109.6256] [INSPIRE].
  42. [42]
    J. Reuter et al., Automation of NLO processes and decays and POWHEG matching in WHIZARD, J. Phys. Conf. Ser. 762 (2016) 012059 [arXiv:1602.06270] [INSPIRE].
  43. [43]
    A. Gehrmann-De Ridder, T. Gehrmann, E.W.N. Glover and G. Heinrich, Jet rates in electron-positron annihilation at O(α s3) in QCD, Phys. Rev. Lett. 100 (2008) 172001 [arXiv:0802.0813] [INSPIRE].
  44. [44]
    S. Weinzierl, NNLO corrections to 3-jet observables in electron-positron annihilation, Phys. Rev. Lett. 101 (2008) 162001 [arXiv:0807.3241] [INSPIRE].CrossRefGoogle Scholar
  45. [45]
    M. Czakon, D. Heymes and A. Mitov, High-precision differential predictions for top-quark pairs at the LHC, Phys. Rev. Lett. 116 (2016) 082003 [arXiv:1511.00549] [INSPIRE].
  46. [46]
    M. Czakon, P. Fiedler, D. Heymes and A. Mitov, NNLO QCD predictions for fully-differential top-quark pair production at the Tevatron, JHEP 05 (2016) 034 [arXiv:1601.05375] [INSPIRE].CrossRefGoogle Scholar
  47. [47]
    M. Czakon, A novel subtraction scheme for double-real radiation at NNLO, Phys. Lett. B 693 (2010) 259 [arXiv:1005.0274] [INSPIRE].
  48. [48]
    R. Boughezal, F. Caola, K. Melnikov, F. Petriello and M. Schulze, Higgs boson production in association with a jet at next-to-next-to-leading order, Phys. Rev. Lett. 115 (2015) 082003 [arXiv:1504.07922] [INSPIRE].
  49. [49]
    J. Gaunt, M. Stahlhofen, F.J. Tackmann and J.R. Walsh, N-jettiness Subtractions for NNLO QCD Calculations, JHEP 09 (2015) 058 [arXiv:1505.04794] [INSPIRE].CrossRefGoogle Scholar
  50. [50]
    R. Boughezal et al., Z-boson production in association with a jet at next-to-next-to-leading order in perturbative QCD, Phys. Rev. Lett. 116 (2016) 152001 [arXiv:1512.01291] [INSPIRE].CrossRefGoogle Scholar
  51. [51]
    R. Boughezal et al., Color singlet production at NNLO in MCFM, Eur. Phys. J. C 77 (2017) 7 [arXiv:1605.08011] [INSPIRE].
  52. [52]
    S. Catani and M. Grazzini, An NNLO subtraction formalism in hadron collisions and its application to Higgs boson production at the LHC, Phys. Rev. Lett. 98 (2007) 222002 [hep-ph/0703012] [INSPIRE].
  53. [53]
    M. Grazzini, S. Kallweit and M. Wiesemann, Fully differential NNLO computations with MATRIX, Eur. Phys. J. C 78 (2018) 537 [arXiv:1711.06631] [INSPIRE].
  54. [54]
    M. Grazzini, S. Kallweit, D. Rathlev and M. Wiesemann, W ± Z production at the LHC: fiducial cross sections and distributions in NNLO QCD, JHEP 05 (2017) 139 [arXiv:1703.09065] [INSPIRE].
  55. [55]
    D. de Florian et al., Differential Higgs Boson Pair Production at Next-to-Next-to-Leading Order in QCD, JHEP 09 (2016) 151 [arXiv:1606.09519] [INSPIRE].CrossRefGoogle Scholar
  56. [56]
    A. Gehrmann-De Ridder, T. Gehrmann, E.W.N. Glover, A. Huss and T.A. Morgan, Precise QCD predictions for the production of a Z boson in association with a hadronic jet, Phys. Rev. Lett. 117 (2016) 022001 [arXiv:1507.02850] [INSPIRE].
  57. [57]
    J. Currie, T. Gehrmann and J. Niehues, Precise QCD predictions for the production of dijet final states in deep inelastic scattering, Phys. Rev. Lett. 117 (2016) 042001 [arXiv:1606.03991] [INSPIRE].
  58. [58]
    J. Currie, A. Gehrmann-De Ridder, T. Gehrmann, E.W.N. Glover, A. Huss and J. Pires, Precise predictions for dijet production at the LHC, Phys. Rev. Lett. 119 (2017) 152001 [arXiv:1705.10271] [INSPIRE].CrossRefGoogle Scholar
  59. [59]
    Z. Tulipánt, A. Kardos and G. Somogyi, Energy-energy correlation in electron-positron annihilation at NNLL + NNLO accuracy, Eur. Phys. J. C 77 (2017) 749 [arXiv:1708.04093] [INSPIRE].
  60. [60]
    V. Del Duca, C. Duhr, A. Kardos, G. Somogyi and Z. Trócsányi, Three-Jet Production in Electron-Positron Collisions at Next-to-Next-to-Leading Order Accuracy, Phys. Rev. Lett. 117 (2016)152004 [arXiv:1603.08927] [INSPIRE].
  61. [61]
    V. Del Duca et al., Jet production in the CoLoRFulNNLO method: event shapes in electron-positron collisions, Phys. Rev. D 94 (2016) 074019 [arXiv:1606.03453] [INSPIRE].
  62. [62]
    M. Cacciari, F.A. Dreyer, A. Karlberg, G.P. Salam and G. Zanderighi, Fully Differential Vector-Boson-Fusion Higgs Production at Next-to-Next-to-Leading Order, Phys. Rev. Lett. 115 (2015)082002 [Erratum ibid. 120 (2018) 139901] [arXiv:1506.02660] [INSPIRE].
  63. [63]
    F. Caola, K. Melnikov and R. Röntsch, Nested soft-collinear subtractions in NNLO QCD computations, Eur. Phys. J. C 77 (2017) 248 [arXiv:1702.01352] [INSPIRE].
  64. [64]
    F. Caola, G. Luisoni, K. Melnikov and R. Röntsch, NNLO QCD corrections to associated WH production and \( H\to b\overline{b} \) decay, Phys. Rev. D 97 (2018)074022 [arXiv:1712.06954] [INSPIRE].
  65. [65]
    G.F.R. Sborlini, F. Driencourt-Mangin and G. Rodrigo, Four-dimensional unsubtraction with massive particles, JHEP 10 (2016) 162 [arXiv:1608.01584] [INSPIRE].CrossRefGoogle Scholar
  66. [66]
    F. Herzog, Geometric IR subtraction for final state real radiation, JHEP 08 (2018) 006 [arXiv:1804.07949] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  67. [67]
    F.A. Dreyer and A. Karlberg, Vector-Boson Fusion Higgs Production at Three Loops in QCD, Phys. Rev. Lett. 117 (2016) 072001 [arXiv:1606.00840] [INSPIRE].
  68. [68]
    F. Dulat, B. Mistlberger and A. Pelloni, Differential Higgs production at N 3 LO beyond threshold, JHEP 01 (2018) 145 [arXiv:1710.03016] [INSPIRE].
  69. [69]
    J. Currie, T. Gehrmann, E.W.N. Glover, A. Huss, J. Niehues and A. Vogt, N 3 LO corrections to jet production in deep inelastic scattering using the Projection-to-Born method, JHEP 05 (2018)209 [arXiv:1803.09973] [INSPIRE].
  70. [70]
    L. Magnea, E. Maina, G. Pelliccioli, C. Signorile-Signorile, P. Torrielli and S. Uccirati, Factorisation and Subtraction beyond NLO, arXiv:1809.05444 [INSPIRE].
  71. [71]
    S. Frixione and M. Grazzini, Subtraction at NNLO, JHEP 06 (2005) 010 [hep-ph/0411399] [INSPIRE].
  72. [72]
    A. Gehrmann-De Ridder, T. Gehrmann and E.W.N. Glover, Infrared structure of e + e → 2 jets at NNLO, Nucl. Phys. B 691 (2004) 195 [hep-ph/0403057] [INSPIRE].
  73. [73]
    R. Hamberg, W.L. van Neerven and T. Matsuura, A complete calculation of the order α s2 correction to the Drell-Yan K factor, Nucl. Phys. B 359 (1991) 343 [Erratum ibid. B 644 (2002)403] [INSPIRE].
  74. [74]
    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].
  75. [75]
    A. Gehrmann-De Ridder, Radiative corrections to the photon + 1 jet rate at LEP, Durham theses, Durham University, (1997),
  76. [76]
    A. Gehrmann-De Ridder, T. Gehrmann and G. Heinrich, Four particle phase space integrals in massless QCD, Nucl. Phys. B 682 (2004) 265 [hep-ph/0311276] [INSPIRE].

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© The Author(s) 2018

Authors and Affiliations

  1. 1.Dipartimento di Fisica and Arnold-Regge CenterUniversità di TorinoTorinoItaly
  2. 2.INFN, Sezione di TorinoTorinoItaly

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