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One-loop triple collinear splitting amplitudes in QCD

A preprint version of the article is available at arXiv.

Abstract

We study the factorisation properties of one-loop scattering amplitudes in the triple collinear limit and extract the universal splitting amplitudes for processes initiated by a gluon. The splitting amplitudes are derived from the analytic Higgs plus four partons amplitudes. We present compact results for primitive helicity splitting amplitudes making use of super-symmetric decompositions. The universality of the collinear factorisation is checked numerically against the full colour six parton squared matrix elements.

References

  1. [1]

    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].

  2. [2]

    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].

    ADS  Article  Google Scholar 

  3. [3]

    A. Gehrmann-De Ridder, T. Gehrmann and E.W.N. Glover, Antenna subtraction at NNLO, JHEP 09 (2005) 056 [hep-ph/0505111] [INSPIRE].

    ADS  Article  Google Scholar 

  4. [4]

    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].

    ADS  Article  Google Scholar 

  5. [5]

    M. Czakon, A novel subtraction scheme for double-real radiation at NNLO, Phys. Lett. B 693 (2010) 259 [arXiv:1005.0274] [INSPIRE].

    ADS  Article  Google Scholar 

  6. [6]

    R. Boughezal, K. Melnikov and F. Petriello, A subtraction scheme for NNLO computations, Phys. Rev. D 85 (2012) 034025 [arXiv:1111.7041] [INSPIRE].

    ADS  Google Scholar 

  7. [7]

    M. Czakon and D. Heymes, Four-dimensional formulation of the sector-improved residue subtraction scheme, Nucl. Phys. B 890 (2014) 152 [arXiv:1408.2500] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  8. [8]

    V. Del Duca, C. Duhr, G. Somogyi, F. Tramontano and Z. Trócsányi, Higgs boson decay into b-quarks at NNLO accuracy, JHEP 04 (2015) 036 [arXiv:1501.07226] [INSPIRE].

    ADS  Article  Google Scholar 

  9. [9]

    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].

    ADS  Article  Google Scholar 

  10. [10]

    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].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  11. [11]

    Z. Bern, L.J. Dixon and D.A. Kosower, One loop corrections to two quark three gluon amplitudes, Nucl. Phys. B 437 (1995) 259 [hep-ph/9409393] [INSPIRE].

    ADS  Article  Google Scholar 

  12. [12]

    D.A. Kosower and P. Uwer, One loop splitting amplitudes in gauge theory, Nucl. Phys. B 563 (1999) 477 [hep-ph/9903515] [INSPIRE].

    ADS  Article  Google Scholar 

  13. [13]

    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].

    ADS  Article  Google Scholar 

  14. [14]

    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].

    ADS  Google Scholar 

  15. [15]

    C. Anastasiou, C. Duhr, F. Dulat, F. Herzog and B. Mistlberger, Higgs Boson Gluon-Fusion Production in QCD at Three Loops, Phys. Rev .Lett. 114 (2015) 212001 [arXiv:1503.06056] [INSPIRE].

    ADS  Article  Google Scholar 

  16. [16]

    C. Anastasiou et al., Higgs boson gluon-fusion production at threshold in N 3 LO QCD, Phys. Lett. B 737 (2014) 325 [arXiv:1403.4616] [INSPIRE].

    ADS  Article  Google Scholar 

  17. [17]

    C. Anastasiou et al., Higgs boson gluon-fusion production beyond threshold in N 3 LO QCD, JHEP 03 (2015) 091 [arXiv:1411.3584] [INSPIRE].

    Article  Google Scholar 

  18. [18]

    Y. Li, A. von Manteuffel, R.M. Schabinger and H.X. Zhu, Soft-virtual corrections to Higgs production at N 3 LO, Phys. Rev. D 91 (2015) 036008 [arXiv:1412.2771] [INSPIRE].

    ADS  Google Scholar 

  19. [19]

    P.A. Baikov, K.G. Chetyrkin, A.V. Smirnov, V.A. Smirnov and M. Steinhauser, Quark and gluon form factors to three loops, Phys. Rev. Lett. 102 (2009) 212002 [arXiv:0902.3519] [INSPIRE].

    ADS  Article  Google Scholar 

  20. [20]

    T. Gehrmann, E.W.N. Glover, T. Huber, N. Ikizlerli and C. Studerus, Calculation of the quark and gluon form factors to three loops in QCD, JHEP 06 (2010) 094 [arXiv:1004.3653] [INSPIRE].

    ADS  Article  MATH  Google Scholar 

  21. [21]

    T. Gehrmann, E.W.N. Glover, T. Huber, N. Ikizlerli and C. Studerus, The quark and gluon form factors to three loops in QCD through to O(eps 2), JHEP 11 (2010) 102 [arXiv:1010.4478] [INSPIRE].

    ADS  Article  MATH  Google Scholar 

  22. [22]

    C. Anastasiou, C. Duhr, F. Dulat, F. Herzog and B. Mistlberger, Real-virtual contributions to the inclusive Higgs cross-section at N 3 LO, JHEP 12 (2013) 088 [arXiv:1311.1425] [INSPIRE].

    ADS  Article  Google Scholar 

  23. [23]

    W.B. Kilgore, One-loop single-real-emission contributions to ppH + X at next-to-next-to-next-to-leading order, Phys. Rev. D 89 (2014) 073008 [arXiv:1312.1296] [INSPIRE].

    ADS  Google Scholar 

  24. [24]

    C. Duhr and T. Gehrmann, The two-loop soft current in dimensional regularization, Phys. Lett. B 727 (2013) 452 [arXiv:1309.4393] [INSPIRE].

    ADS  Article  MATH  Google Scholar 

  25. [25]

    Y. Li and H.X. Zhu, Single soft gluon emission at two loops, JHEP 11 (2013) 080 [arXiv:1309.4391] [INSPIRE].

    ADS  Article  Google Scholar 

  26. [26]

    C. Duhr, T. Gehrmann and M. Jaquier, Two-loop splitting amplitudes and the single-real contribution to inclusive Higgs production at N 3 LO, JHEP 02 (2015) 077 [arXiv:1411.3587] [INSPIRE].

    ADS  Article  Google Scholar 

  27. [27]

    F. Dulat and B. Mistlberger, Real-Virtual-Virtual contributions to the inclusive Higgs cross section at N3LO, arXiv:1411.3586 [INSPIRE].

  28. [28]

    Y. Li, A. von Manteuffel, R.M. Schabinger and H.X. Zhu, N 3 LO Higgs boson and Drell-Yan production at threshold: The one-loop two-emission contribution, Phys. Rev. D 90 (2014) 053006 [arXiv:1404.5839] [INSPIRE].

    ADS  Google Scholar 

  29. [29]

    C. Anastasiou et al., Soft Expansion of Double-Real-Virtual Corrections to Higgs Production at N 3 LO, arXiv:1505.04110 [INSPIRE].

  30. [30]

    C. Anastasiou, C. Duhr, F. Dulat and B. Mistlberger, Soft triple-real radiation for Higgs production at N3LO, JHEP 07 (2013) 003 [arXiv:1302.4379] [INSPIRE].

    ADS  Article  Google Scholar 

  31. [31]

    O.V. Tarasov, A.A. Vladimirov and A. Yu. Zharkov, The Gell-Mann-Low Function of QCD in the Three Loop Approximation, Phys. Lett. B 93 (1980) 429 [INSPIRE].

    ADS  Article  Google Scholar 

  32. [32]

    S.A. Larin and J.A.M. Vermaseren, The Three loop QCD β-function and anomalous dimensions, Phys. Lett. B 303 (1993) 334 [hep-ph/9302208] [INSPIRE].

    ADS  Article  Google Scholar 

  33. [33]

    T. van Ritbergen, J.A.M. Vermaseren and S.A. Larin, The Four loop β-function in quantum chromodynamics, Phys. Lett. B 400 (1997) 379 [hep-ph/9701390] [INSPIRE].

    ADS  Article  Google Scholar 

  34. [34]

    M. Czakon, The Four-loop QCD β-function and anomalous dimensions, Nucl. Phys. B 710 (2005) 485 [hep-ph/0411261] [INSPIRE].

    ADS  Article  MATH  Google Scholar 

  35. [35]

    A. Vogt, S. Moch and J.A.M. Vermaseren, The Three-loop splitting functions in QCD: The Singlet case, Nucl. Phys. B 691 (2004) 129 [hep-ph/0404111] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  36. [36]

    S. Moch, J.A.M. Vermaseren and A. Vogt, The Three loop splitting functions in QCD: The Nonsinglet case, Nucl. Phys. B 688 (2004) 101 [hep-ph/0403192] [INSPIRE].

    ADS  Article  MATH  Google Scholar 

  37. [37]

    C. Anastasiou, S. Buehler, C. Duhr and F. Herzog, NNLO phase space master integrals for two-to-one inclusive cross sections in dimensional regularization, JHEP 11 (2012) 062 [arXiv:1208.3130] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  38. [38]

    M. Höschele, J. Hoff, A. Pak, M. Steinhauser and T. Ueda, Higgs boson production at the LHC: NNLO partonic cross sections through order ϵ and convolutions with splitting functions to N 3 LO, Phys. Lett. B 721 (2013) 244 [arXiv:1211.6559] [INSPIRE].

    ADS  Article  MATH  Google Scholar 

  39. [39]

    S. Buehler and A. Lazopoulos, Scale dependence and collinear subtraction terms for Higgs production in gluon fusion at N3LO, JHEP 10 (2013) 096 [arXiv:1306.2223] [INSPIRE].

    ADS  Article  Google Scholar 

  40. [40]

    D.A. Kosower, All order collinear behavior in gauge theories, Nucl. Phys. B 552 (1999) 319 [hep-ph/9901201] [INSPIRE].

    ADS  Article  Google Scholar 

  41. [41]

    Z. Bern, L.J. Dixon and D.A. Kosower, Two-loop ggg splitting amplitudes in QCD, JHEP 08 (2004) 012 [hep-ph/0404293] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  42. [42]

    S.D. Badger and E.W.N. Glover, Two loop splitting functions in QCD, JHEP 07 (2004) 040 [hep-ph/0405236] [INSPIRE].

  43. [43]

    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].

  44. [44]

    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].

  45. [45]

    T.G. Birthwright, E.W.N. Glover, V.V. Khoze and P. Marquard, Multi-gluon collinear limits from MHV diagrams, JHEP 05 (2005) 013 [hep-ph/0503063] [INSPIRE].

  46. [46]

    T.G. Birthwright, E.W.N. Glover, V.V. Khoze and P. Marquard, Collinear limits in QCD from MHV rules, JHEP 07 (2005) 068 [hep-ph/0505219] [INSPIRE].

  47. [47]

    S. Catani, D. de Florian and G. Rodrigo, The Triple collinear limit of one loop QCD amplitudes, Phys. Lett. B 586 (2004) 323 [hep-ph/0312067] [INSPIRE].

  48. [48]

    G.F.R. Sborlini, D. de Florian and G. Rodrigo, Triple collinear splitting functions at NLO for scattering processes with photons, JHEP 10 (2014) 161 [arXiv:1408.4821] [INSPIRE].

    ADS  Article  Google Scholar 

  49. [49]

    G.F.R. Sborlini, D. de Florian and G. Rodrigo, Polarized triple-collinear splitting functions at NLO for processes with photons, JHEP 03 (2015) 021 [arXiv:1409.6137] [INSPIRE].

    ADS  Article  Google Scholar 

  50. [50]

    C.F. Berger, V. Del Duca and L.J. Dixon, Recursive Construction of Higgs-Plus-Multiparton Loop Amplitudes: The Last of the Phi-nite Loop Amplitudes, Phys. Rev. D 74 (2006) 094021 [Erratum ibid. D 76 (2007) 099901] [hep-ph/0608180] [INSPIRE].

  51. [51]

    S.D. Badger and E.W.N. Glover, One-loop helicity amplitudes for Hgluons: The All-minus configuration, Nucl. Phys. Proc. Suppl. 160 (2006) 71 [hep-ph/0607139] [INSPIRE].

  52. [52]

    S.D. Badger, E.W.N. Glover and K. Risager, One-loop phi-MHV amplitudes using the unitarity bootstrap, JHEP 07 (2007) 066 [arXiv:0704.3914] [INSPIRE].

    ADS  Article  Google Scholar 

  53. [53]

    E.W.N. Glover, P. Mastrolia and C. Williams, One-loop phi-MHV amplitudes using the unitarity bootstrap: The General helicity case, JHEP 08 (2008) 017 [arXiv:0804.4149] [INSPIRE].

    ADS  Article  Google Scholar 

  54. [54]

    S. Badger, E.W. Nigel Glover, P. Mastrolia and C. Williams, One-loop Higgs plus four gluon amplitudes: Full analytic results, JHEP 01 (2010) 036 [arXiv:0909.4475] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  55. [55]

    L.J. Dixon and Y. Sofianatos, Analytic one-loop amplitudes for a Higgs boson plus four partons, JHEP 08 (2009) 058 [arXiv:0906.0008] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  56. [56]

    S. Badger, J.M. Campbell, R.K. Ellis and C. Williams, Analytic results for the one-loop NMHV Hqqgg amplitude, JHEP 12 (2009) 035 [arXiv:0910.4481] [INSPIRE].

    ADS  Article  Google Scholar 

  57. [57]

    D.A. Kosower, Antenna factorization of gauge theory amplitudes, Phys. Rev. D 57 (1998) 5410 [hep-ph/9710213] [INSPIRE].

  58. [58]

    C. Duhr, Mathematical aspects of scattering amplitudes, arXiv:1411.7538 [INSPIRE].

  59. [59]

    H.J. Lu and C.A. Perez, Massless one loop scalar three point integral and associated Clausen, Glaisher and L functions, SLAC-PUB-5809 (1992).

  60. [60]

    Z. Bern, L.J. Dixon and D.A. Kosower, Dimensionally regulated pentagon integrals, Nucl. Phys. B 412 (1994) 751 [hep-ph/9306240] [INSPIRE].

  61. [61]

    T. Binoth, J.P. Guillet, G. Heinrich and C. Schubert, Calculation of one loop hexagon amplitudes in the Yukawa model, Nucl. Phys. B 615 (2001) 385 [hep-ph/0106243] [INSPIRE].

  62. [62]

    A. van Hameren, J. Vollinga and S. Weinzierl, Automated computation of one-loop integrals in massless theories, Eur. Phys. J. C 41 (2005) 361 [hep-ph/0502165] [INSPIRE].

  63. [63]

    R.K. Ellis and G. Zanderighi, Scalar one-loop integrals for QCD, JHEP 02 (2008) 002 [arXiv:0712.1851] [INSPIRE].

    ADS  Article  Google Scholar 

  64. [64]

    D. Forde, Direct extraction of one-loop integral coefficients, Phys. Rev. D 75 (2007) 125019 [arXiv:0704.1835] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  65. [65]

    R. Kleiss and H. Kuijf, Multi-gluon cross-sections and five jet production at hadron colliders, Nucl. Phys. B 312 (1989) 616 [INSPIRE].

    ADS  Article  Google Scholar 

  66. [66]

    V. Del Duca, L.J. Dixon and F. Maltoni, New color decompositions for gauge amplitudes at tree and loop level, Nucl. Phys. B 571 (2000) 51 [hep-ph/9910563] [INSPIRE].

  67. [67]

    M.L. Mangano, S.J. Parke and Z. Xu, Duality and multi-gluon scattering, Nucl. Phys. B 298 (1988) 653 [INSPIRE].

    ADS  Article  Google Scholar 

  68. [68]

    M.L. Mangano and S.J. Parke, Multiparton amplitudes in gauge theories, Phys. Rept. 200 (1991) 301 [hep-th/0509223] [INSPIRE].

    ADS  Article  Google Scholar 

  69. [69]

    S. Badger, B. Biedermann, P. Uwer and V. Yundin, Numerical evaluation of virtual corrections to multi-jet production in massless QCD, Comput. Phys. Commun. 184 (2013) 1981 [arXiv:1209.0100] [INSPIRE].

    ADS  Article  Google Scholar 

  70. [70]

    A. van Hameren, OneLOop: For the evaluation of one-loop scalar functions, Comput. Phys. Commun. 182 (2011) 2427 [arXiv:1007.4716] [INSPIRE].

    ADS  Article  MATH  Google Scholar 

  71. [71]

    Z. Bern, L.J. Dixon and D.A. Kosower, One loop amplitudes for e + e to four partons, Nucl. Phys. B 513 (1998) 3 [hep-ph/9708239] [INSPIRE].

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Badger, S., Buciuni, F. & Peraro, T. One-loop triple collinear splitting amplitudes in QCD. J. High Energ. Phys. 2015, 188 (2015). https://doi.org/10.1007/JHEP09(2015)188

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Keywords

  • Scattering Amplitudes
  • Strong Coupling Expansion
  • QCD