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Primary Feynman rules to calculate the -dimensional integrand of any 1-loop amplitude

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Abstract

When using dimensional regularization/reduction the -dimensional numerator of the 1-loop Feynman diagrams gives rise to rational contributions. I list the set of fundamental rules that allow the extraction of such terms at the integrand level in any theory containing scalars, vectors and fermions, such as the electroweak standard model, QCD and SUSY.

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References

  1. 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  MathSciNet  ADS  Google Scholar 

  2. Z. Bern, L.J. Dixon, D.C. Dunbar and D.A. Kosower, Fusing gauge theory tree amplitudes into loop amplitudes, Nucl. Phys. B 435 (1995) 59 [hep-ph/9409265] [INSPIRE].

    Article  ADS  Google Scholar 

  3. R. Britto, F. Cachazo and B. Feng, Generalized unitarity and one-loop amplitudes in N = 4 super-Yang-Mills, Nucl. Phys. B 725 (2005) 275 [hep-th/0412103] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  4. G. Ossola, C.G. Papadopoulos and R. Pittau, Reducing full one-loop amplitudes to scalar integrals at the integrand level, Nucl. Phys. B 763 (2007) 147 [hep-ph/0609007] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

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

    MathSciNet  ADS  Google Scholar 

  6. G. Ossola, C.G. Papadopoulos and R. Pittau, CutTools: a program implementing the OPP reduction method to compute one-loop amplitudes, JHEP 03 (2008) 042 [arXiv:0711.3596] [INSPIRE].

    Article  ADS  Google Scholar 

  7. C. Berger et al., An automated implementation of on-shell methods for one-loop amplitudes, Phys. Rev. D 78 (2008) 036003 [arXiv:0803.4180] [INSPIRE].

    ADS  Google Scholar 

  8. W. Giele and G. Zanderighi, On the numerical evaluation of one-loop amplitudes: the gluonic case, JHEP 06 (2008) 038 [arXiv:0805.2152] [INSPIRE].

    Article  ADS  Google Scholar 

  9. R. Ellis, K. Melnikov and G. Zanderighi, Generalized unitarity at work: first NLO QCD results for hadronic W + 3 jet production, JHEP 04 (2009) 077 [arXiv:0901.4101] [INSPIRE].

    Article  ADS  Google Scholar 

  10. A. van Hameren, C. Papadopoulos and R. Pittau, Automated one-loop calculations: a proof of concept, JHEP 09 (2009) 106 [arXiv:0903.4665] [INSPIRE].

    Article  ADS  Google Scholar 

  11. C. Berger et al., Next-to-leading order QCD predictions for W +3 jet distributions at hadron colliders, Phys. Rev. D 80 (2009) 074036 [arXiv:0907.1984] [INSPIRE].

    ADS  Google Scholar 

  12. V. Hirschi et al., Automation of one-loop QCD corrections, JHEP 05 (2011) 044 [arXiv:1103.0621] [INSPIRE].

    Article  ADS  Google Scholar 

  13. G. Bevilacqua et al., HELAC-NLO, arXiv:1110.1499 [INSPIRE].

  14. V. Hirschi, New developments in MadLoop, arXiv:1111.2708 [INSPIRE].

  15. G.P. Salam, Perturbative QCD for the LHC, PoS(ICHEP 2010)556 [arXiv:1103.1318] [INSPIRE].

  16. R. Ellis, Z. Kunszt, K. Melnikov and G. Zanderighi, One-loop calculations in quantum field theory: from Feynman diagrams to unitarity cuts, arXiv:1105.4319 [INSPIRE].

  17. T. Binoth, G. Ossola, C. Papadopoulos and R. Pittau, NLO QCD corrections to tri-boson production, JHEP 06 (2008) 082 [arXiv:0804.0350] [INSPIRE].

    Article  ADS  Google Scholar 

  18. R. Ellis, W. Giele, Z. Kunszt, K. Melnikov and G. Zanderighi, One-loop amplitudes for W + 3 jet production in hadron collisions, JHEP 01 (2009) 012 [arXiv:0810.2762] [INSPIRE].

    Article  ADS  Google Scholar 

  19. C. Berger et al., Precise predictions for W +3 jet production at hadron colliders, Phys. Rev. Lett. 102 (2009) 222001 [arXiv:0902.2760] [INSPIRE].

    Article  ADS  Google Scholar 

  20. A. Bredenstein, A. Denner, S. Dittmaier and S. Pozzorini, NLO QCD corrections to \( pp \to t\overline t b\overline b + X \) at the LHC, Phys. Rev. Lett. 103 (2009) 012002 [arXiv:0905.0110] [INSPIRE].

    Article  ADS  Google Scholar 

  21. G. Bevilacqua, M. Czakon, C. Papadopoulos, R. Pittau and M. Worek, Assault on the NLO wishlist: \( pp \to t\overline t b\overline b \), JHEP 09 (2009) 109 [arXiv:0907.4723] [INSPIRE].

    Article  ADS  Google Scholar 

  22. A. Bredenstein, A. Denner, S. Dittmaier and S. Pozzorini, NLO QCD corrections to top anti-top bottom anti-bottom production at the LHC: 2. Full hadronic results, JHEP 03 (2010) 021 [arXiv:1001.4006] [INSPIRE].

    Article  ADS  Google Scholar 

  23. G. Bevilacqua, M. Czakon, C. Papadopoulos and M. Worek, Dominant QCD backgrounds in Higgs boson analyses at the LHC: a study of \( pp \to t\overline t + 2 \) jets at next-to-leading order, Phys. Rev. Lett. 104 (2010) 162002 [arXiv:1002.4009] [INSPIRE].

    Article  ADS  Google Scholar 

  24. SM and NLO Multileg Working Group collaboration, J. Andersen et al., The SM and NLO Multileg Working Group: summary report, arXiv:1003.1241 [INSPIRE].

  25. K. Melnikov and M. Schulze, NLO QCD corrections to top quark pair production in association with one hard jet at hadron colliders, Nucl. Phys. B 840 (2010) 129 [arXiv:1004.3284] [INSPIRE].

    Article  ADS  Google Scholar 

  26. C. Berger et al., Precise predictions for W + 4 jet production at the Large Hadron Collider, Phys. Rev. Lett. 106 (2011) 092001 [arXiv:1009.2338] [INSPIRE].

    Article  ADS  Google Scholar 

  27. A. Denner, S. Dittmaier, S. Kallweit and S. Pozzorini, NLO QCD corrections to WWbb production at hadron colliders, Phys. Rev. Lett. 106 (2011) 052001 [arXiv:1012.3975] [INSPIRE].

    Article  ADS  Google Scholar 

  28. G. Bevilacqua, M. Czakon, A. van Hameren, C.G. Papadopoulos and M. Worek, Complete off-shell effects in top quark pair hadroproduction with leptonic decay at next-to-leading order, JHEP 02 (2011) 083 [arXiv:1012.4230] [INSPIRE].

    Article  ADS  Google Scholar 

  29. T. Melia, K. Melnikov, R. Rontsch and G. Zanderighi, NLO QCD corrections for W + W pair production in association with two jets at hadron colliders, Phys. Rev. D 83 (2011) 114043 [arXiv:1104.2327] [INSPIRE].

    ADS  Google Scholar 

  30. R. Frederix et al., Scalar and pseudoscalar Higgs production in association with a top-antitop pair, Phys. Lett. B 701 (2011) 427 [arXiv:1104.5613] [INSPIRE].

    ADS  Google Scholar 

  31. F. Campanario, C. Englert, M. Rauch and D. Zeppenfeld, Precise predictions for Wγγ + jet production at hadron colliders, Phys. Lett. B 704 (2011) 515 [arXiv:1106.4009] [INSPIRE].

    ADS  Google Scholar 

  32. R. Frederix et al., W and Z/γ∗ boson production in association with a bottom-antibottom pair, JHEP 09 (2011) 061 [arXiv:1106.6019] [INSPIRE].

    Article  ADS  Google Scholar 

  33. K. Arnold et al., VBFNLO: a parton level Monte Carlo for processes with electroweak bosonsmanual for version 2.5.0, arXiv:1107.4038 [INSPIRE].

  34. H. Ita et al., Precise predictions for Z + 4 jets at hadron colliders, arXiv:1108.2229 [INSPIRE].

  35. G. Bevilacqua, M. Czakon, C. Papadopoulos and M. Worek, Hadronic top-quark pair production in association with two jets at next-to-leading order QCD, Phys. Rev. D 84 (2011) 114017 [arXiv:1108.2851] [INSPIRE].

    ADS  Google Scholar 

  36. R. Frederix et al., Four-lepton production at hadron colliders: aMC@NLO predictions with theoretical uncertainties, arXiv:1110.4738 [INSPIRE].

  37. R. Frederix et al., aMC@NLO predictions for Wjj production at the Tevatron, arXiv:1110.5502 [INSPIRE].

  38. T. Binoth, J. Guillet and G. Heinrich, Algebraic evaluation of rational polynomials in one-loop amplitudes, JHEP 02 (2007) 013 [hep-ph/0609054] [INSPIRE].

    Article  ADS  Google Scholar 

  39. G. Ossola, C.G. Papadopoulos and R. Pittau, On the rational terms of the one-loop amplitudes, JHEP 05 (2008) 004 [arXiv:0802.1876] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  40. A. Bredenstein, A. Denner, S. Dittmaier and S. Pozzorini, NLO QCD corrections to ttbb production at the LHC: 1. Quark-antiquark annihilation, JHEP 08 (2008) 108 [arXiv:0807.1248] [INSPIRE].

    Article  ADS  Google Scholar 

  41. F. Campanario, Towards pp → V V jj at NLO QCD: bosonic contributions to triple vector boson production plus jet, JHEP 10 (2011) 070 [arXiv:1105.0920] [INSPIRE].

    Article  ADS  Google Scholar 

  42. W.T. Giele, Z. Kunszt and K. Melnikov, Full one-loop amplitudes from tree amplitudes, JHEP 04 (2008) 049 [arXiv:0801.2237] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  43. Z. Bern, L.J. Dixon and D.A. Kosower, Bootstrapping multi-parton loop amplitudes in QCD, Phys. Rev. D 73 (2006) 065013 [hep-ph/0507005] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

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

    Article  MathSciNet  ADS  Google Scholar 

  45. S. Badger, Direct extraction of one loop rational terms, JHEP 01 (2009) 049 [arXiv:0806.4600] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  46. P. Draggiotis, M. Garzelli, C. Papadopoulos and R. Pittau, Feynman rules for the rational part of the QCD 1-loop amplitudes, JHEP 04 (2009) 072 [arXiv:0903.0356] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  47. M. Garzelli, I. Malamos and R. Pittau, Feynman rules for the rational part of the electroweak 1-loop amplitudes, JHEP 01 (2010) 040 [Erratum ibid. 10 (2010) 097] [arXiv:0910.3130] [INSPIRE].

    Article  ADS  Google Scholar 

  48. M. Garzelli, I. Malamos and R. Pittau, Feynman rules for the rational part of the electroweak 1-loop amplitudes in the R x i gauge and in the unitary gauge, JHEP 01 (2011) 029 [arXiv:1009.4302] [INSPIRE].

    Article  ADS  Google Scholar 

  49. M. Garzelli and I. Malamos, R2SM: a package for the analytic computation of the R 2 rational terms in the standard model of the electroweak interactions, Eur. Phys. J. C 71 (2011) 1605 [arXiv:1010.1248] [INSPIRE].

    Article  ADS  Google Scholar 

  50. H.-S. Shao, Y.-J. Zhang and K.-T. Chao, Feynman rules for the rational part of the standard model one-loop amplitudes in thet Hooft-Veltman γ5 scheme, JHEP 09 (2011) 048 [arXiv:1106.5030] [INSPIRE].

    Article  ADS  Google Scholar 

  51. G. Ossola, C.G. Papadopoulos and R. Pittau, Numerical evaluation of six-photon amplitudes, JHEP 07 (2007) 085 [arXiv:0704.1271] [INSPIRE].

    Article  ADS  Google Scholar 

  52. A. Signer and D. Stöckinger, Using dimensional reduction for hadronic collisions, Nucl. Phys. B 808 (2009) 88 [arXiv:0807.4424] [INSPIRE].

    Article  ADS  Google Scholar 

  53. D. Stöckinger, Regularization by dimensional reduction: consistency, quantum action principle and supersymmetry, JHEP 03 (2005) 076 [hep-ph/0503129] [INSPIRE].

    Article  ADS  Google Scholar 

  54. P. Mastrolia, G. Ossola, T. Reiter and F. Tramontano, Scattering amplitudes from unitarity-based reduction algorithm at the integrand-level, JHEP 08 (2010) 080 [arXiv:1006.0710] [INSPIRE].

    Article  ADS  Google Scholar 

  55. G. Cullen et al., Automation of one-loop calculations with GoSam: present status and future outlook, arXiv:1111.3339 [INSPIRE].

  56. G. Passarino and M. Veltman, One loop corrections for e + e annihilation into μ + μ in the Weinberg model, Nucl. Phys. B 160 (1979) 151 [INSPIRE].

    Article  ADS  Google Scholar 

  57. Z. Xiao, G. Yang and C.-J. Zhu, The rational part of QCD amplitudes. I. The general formalism, Nucl. Phys. B 758 (2006) 1 [hep-ph/0607015] [INSPIRE].

    Article  ADS  Google Scholar 

  58. S. Badger, R. Sattler and V. Yundin, One-loop helicity amplitudes for \( t\overline t \) production at hadron colliders, Phys. Rev. D 83 (2011) 074020 [arXiv:1101.5947] [INSPIRE].

    ADS  Google Scholar 

  59. S. Badger, B. Biedermann and P. Uwer, NGluon: a package to calculate one-loop multi-gluon amplitudes, Comput. Phys. Commun. 182 (2011) 1674 [arXiv:1011.2900] [INSPIRE].

    Article  ADS  Google Scholar 

  60. R. Boels and C. Schwinn, CSW rules for a massive scalar, Phys. Lett. B 662 (2008) 80 [arXiv:0712.3409] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  61. E. Nigel Glover and C. Williams, One-loop gluonic amplitudes from single unitarity cuts, JHEP 12 (2008) 067 [arXiv:0810.2964] [INSPIRE].

    Article  Google Scholar 

  62. H. Elvang, D.Z. Freedman and M. Kiermaier, Integrands for QCD rational terms and N = 4 SYM from massive CSW rules, arXiv:1111.0635 [INSPIRE].

  63. N.D. Christensen and C. Duhr, FeynRulesFeynman rules made easy, Comput. Phys. Commun. 180 (2009) 1614 [arXiv:0806.4194] [INSPIRE].

    Article  ADS  Google Scholar 

  64. F. Cascioli, P. Maierhofer and S. Pozzorini, Scattering amplitudes with open loops, arXiv:1111.5206 [INSPIRE].

  65. A. Denner, Techniques for calculation of electroweak radiative corrections at the one loop level and results for W physics at LEP-200, Fortschr. Phys. 41 (1993) 307 [arXiv:0709.1075] [INSPIRE].

    Google Scholar 

  66. J. Rosiek, Complete set of Feynman rules for the MSSM: erratum, hep-ph/9511250 [INSPIRE].

  67. A. Denner, H. Eck, O. Hahn and J. Kublbeck, Feynman rules for fermion number violating interactions, Nucl. Phys. B 387 (1992) 467 [INSPIRE].

    Article  ADS  Google Scholar 

  68. A. Denner, H. Eck, O. Hahn and J. Kublbeck, Compact Feynman rules for Majorana fermions, Phys. Lett. B 291 (1992) 278 [INSPIRE].

    ADS  Google Scholar 

  69. A. Signer, Helicity method for next-to-leading order corrections in QCD, Ph.D. Thesis, ETH, Zürich Switzerland (1995).

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Pittau, R. Primary Feynman rules to calculate the -dimensional integrand of any 1-loop amplitude. J. High Energ. Phys. 2012, 29 (2012). https://doi.org/10.1007/JHEP02(2012)029

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