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One-loop triple collinear splitting amplitudes in QCD

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  • Published: 28 September 2015
  • Volume 2015, article number 188, (2015)
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One-loop triple collinear splitting amplitudes in QCD
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  • Simon Badger1,
  • Francesco Buciuni1 &
  • Tiziano Peraro1 

  • 473 Accesses

  • 21 Citations

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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.

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References

  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. 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].

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  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].

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  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. 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. 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].

    Article  ADS  Google Scholar 

  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].

    Article  ADS  Google Scholar 

  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].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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].

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  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].

    Article  ADS  Google Scholar 

  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. 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].

    Article  ADS  Google Scholar 

  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].

    Article  ADS  Google Scholar 

  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. 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. 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].

    Article  ADS  Google Scholar 

  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].

    Article  ADS  MATH  Google Scholar 

  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].

    Article  ADS  MATH  Google Scholar 

  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].

    Article  ADS  Google Scholar 

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

    ADS  Google Scholar 

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

    Article  ADS  MATH  Google Scholar 

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

    Article  ADS  Google Scholar 

  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].

    Article  ADS  Google Scholar 

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

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

  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].

    Article  ADS  Google Scholar 

  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].

    Article  ADS  Google Scholar 

  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].

    Article  ADS  Google Scholar 

  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].

    Article  ADS  Google Scholar 

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

    Article  ADS  MATH  Google Scholar 

  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].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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].

    Article  ADS  MATH  Google Scholar 

  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].

    Article  ADS  MathSciNet  Google Scholar 

  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].

    Article  ADS  MATH  Google Scholar 

  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].

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

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

  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. 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. 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. 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. 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. 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].

    Article  ADS  Google Scholar 

  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].

    Article  ADS  Google Scholar 

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

  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].

    Article  ADS  Google Scholar 

  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].

    Article  ADS  Google Scholar 

  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].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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].

    Article  ADS  MathSciNet  Google Scholar 

  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].

    Article  ADS  Google Scholar 

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

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

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

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

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  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].

    Article  ADS  Google Scholar 

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

    Article  ADS  MATH  Google Scholar 

  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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  1. Higgs Centre for Theoretical Physics, School of Physics and Astronomy, University of Edinburgh, Edinburgh, EH9 3JZ, Scotland, United Kingdom

    Simon Badger, Francesco Buciuni & Tiziano Peraro

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  1. Simon Badger
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  2. Francesco Buciuni
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Correspondence to Simon Badger.

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ArXiv ePrint: 1507.05070

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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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  • Received: 30 July 2015

  • Accepted: 03 September 2015

  • Published: 28 September 2015

  • DOI: https://doi.org/10.1007/JHEP09(2015)188

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Keywords

  • Scattering Amplitudes
  • Strong Coupling Expansion
  • QCD
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