Skip to main content
SpringerLink
Log in

Colour decompositions of multi-quark one-loop QCD amplitudes

  • Published:
Journal of High Energy Physics Aims and scope Submit manuscript

Abstract

We describe the decomposition of one-loop QCD amplitudes in terms of colour-ordered building blocks. We give new expressions for the coefficients of QCD colour structures in terms of ordered objects called primitive amplitudes, for processes with up to seven partons. These results are needed in computations of high-multiplicity scattering cross sections in next-to-leading-order (NLO) QCD. We explain the origin of new relations between multi-quark primitive amplitudes which can be used to optimise efficiency of NLO computations. As a first application we compute the full-colour virtual contribution to the cross section for W + 4-jet production at the Large Hadron Collider, and verify that it is very well approximated by keeping only the leading terms in an expansion around the formal limit of a large number of colours.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

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

  2. Z. Nagy, Three jet cross-sections in hadron hadron collisions at next-to-leading order, Phys. Rev. Lett. 88 (2002) 122003 [hep-ph/0110315] [INSPIRE].

    Article  ADS  Google Scholar 

  3. Z. Nagy, Next-to-leading order calculation of three jet observables in hadron hadron collision, Phys. Rev. D 68 (2003) 094002 [hep-ph/0307268] [INSPIRE].

    ADS  Google Scholar 

  4. 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 

  5. R. Ellis, K. Melnikov and G. Zanderighi, W + 3 jet production at the Tevatron, Phys. Rev. D 80 (2009) 094002 [arXiv:0906.1445] [INSPIRE].

    ADS  Google Scholar 

  6. 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 

  7. C. Berger et al., Next-to-Leading Order QCD Predictions for Z, γ* + 3-Jet Distributions at the Tevatron, Phys. Rev. D 82 (2010) 074002 [arXiv:1004.1659] [INSPIRE].

    ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  9. A. Bredenstein, A. Denner, S. Dittmaier and S. Pozzorini, NLO QCD corrections to pp → tt bb + X at the LHC, Phys. Rev. Lett. 103 (2009) 012002 [arXiv:0905.0110] [INSPIRE].

    Article  ADS  Google Scholar 

  10. 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 

  11. T. Binoth et al., Next-to-leading order QCD corrections to pp → bb bb + X at the LHC: the quark induced case, Phys. Lett. B 685 (2010) 293 [arXiv:0910.4379] [INSPIRE].

    Article  ADS  Google Scholar 

  12. G. Bevilacqua, M. Czakon, C. Papadopoulos, R. Pittau and M. Worek, Assault on the NLO Wishlist: pp → tt bb, JHEP 09 (2009) 109 [arXiv:0907.4723] [INSPIRE].

    Article  ADS  Google Scholar 

  13. G. Bevilacqua, M. Czakon, C. Papadopoulos and M. Worek, Dominant QCD Backgrounds in Higgs Boson Analyses at the LHC: A Study of pp → tt + 2 jets at Next-To-Leading Order, Phys. Rev. Lett. 104 (2010) 162002 [arXiv:1002.4009] [INSPIRE].

    Article  ADS  Google Scholar 

  14. 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 

  15. 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 

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

    Article  ADS  Google Scholar 

  17. H. Ita et al., Precise Predictions for Z + 4 Jets at Hadron Colliders, arXiv:1108.2229 [INSPIRE].

  18. 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 

  19. F.A. Berends and W. Giele, The six gluon process as an example of Weyl-Van Der Waerden spinor calculus, Nucl. Phys. B 294 (1987) 700 [INSPIRE].

    Article  ADS  Google Scholar 

  20. M.L. Mangano, The color structure of gluon emission, Nucl. Phys. B 309 (1988) 461 [INSPIRE].

    Article  ADS  Google Scholar 

  21. Z. Bern and D.A. Kosower, Color decomposition of one loop amplitudes in gauge theories, Nucl. Phys. B 362 (1991) 389 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  22. 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 

  23. 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 

  24. 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].

    Article  ADS  Google Scholar 

  25. C. Duhr, S. Hoeche and F. Maltoni, Color-dressed recursive relations for multi-parton amplitudes, JHEP 08 (2006) 062 [hep-ph/0607057] [INSPIRE].

    Article  ADS  Google Scholar 

  26. F.A. Berends and W. Giele, Multiple soft gluon radiation in parton processes, Nucl. Phys. B 313 (1989) 595 [INSPIRE].

    Article  ADS  Google Scholar 

  27. F. Caravaglios and M. Moretti, An algorithm to compute Born scattering amplitudes without Feynman graphs, Phys. Lett. B 358 (1995) 332 [hep-ph/9507237] [INSPIRE].

    Article  ADS  Google Scholar 

  28. F. Caravaglios, M.L. Mangano, M. Moretti and R. Pittau, A new approach to multijet calculations in hadron collisions, Nucl. Phys. B 539 (1999) 215 [hep-ph/9807570] [INSPIRE].

    Article  ADS  Google Scholar 

  29. A. Kanaki and C.G. Papadopoulos, HELAC: a package to compute electroweak helicity amplitudes, Comput. Phys. Commun. 132 (2000) 306 [hep-ph/0002082] [INSPIRE].

    Article  ADS  MATH  Google Scholar 

  30. A. Cafarella, C.G. Papadopoulos and M. Worek, Helac-Phegas: a generator for all parton level processes, Comput. Phys. Commun. 180 (2009) 1941 [arXiv:0710.2427] [INSPIRE].

    Article  ADS  Google Scholar 

  31. M. Moretti, T. Ohl and J. Reuter, O’Mega: an optimizing matrix element generator, hep-ph/0102195 [INSPIRE].

  32. W. Kilian, T. Ohl and J. Reuter, WHIZARD: simulating multi-particle processes at LHC and ILC, Eur. Phys. J. C 71 (2011) 1742 [arXiv:0708.4233] [INSPIRE].

    Article  ADS  Google Scholar 

  33. T. Gleisberg and S. Hoeche, Comix, a new matrix element generator, JHEP 12 (2008) 039 [arXiv:0808.3674] [INSPIRE].

    Article  ADS  Google Scholar 

  34. L.J. Dixon and A. Signer, Complete O(αS ) results for e + e − → (γ, Z) → four jets, Phys. Rev. D 56 (1997) 4031 [hep-ph/9706285] [INSPIRE].

    ADS  Google Scholar 

  35. 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 

  36. Z. Bern and A. Morgan, Massive loop amplitudes from unitarity, Nucl. Phys. B 467 (1996) 479 [hep-ph/9511336] [INSPIRE].

    Article  ADS  Google Scholar 

  37. Z. Bern, L.J. Dixon, D.C. Dunbar and D.A. Kosower, One loop selfdual and N = 4 super Yang-Mills, Phys. Lett. B 394 (1997) 105 [hep-th/9611127] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  38. Z. Bern, L.J. Dixon and D.A. Kosower, On-shell recurrence relations for one-loop QCD amplitudes, Phys. Rev. D 71 (2005) 105013 [hep-th/0501240] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  39. Z. Bern, L.J. Dixon and D.A. Kosower, The last of the finite loop amplitudes in QCD, Phys. Rev. D 72 (2005) 125003 [hep-ph/0505055] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  40. 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 

  41. Z. Bern, N. Bjerrum-Bohr, D.C. Dunbar and H. Ita, Recursive calculation of one-loop QCD integral coefficients, JHEP 11 (2005) 027 [hep-ph/0507019] [INSPIRE].

    Article  ADS  Google Scholar 

  42. 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 

  43. 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 

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

    MathSciNet  ADS  Google Scholar 

  45. 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 

  46. F. del Aguila and R. Pittau, Recursive numerical calculus of one-loop tensor integrals, JHEP 07 (2004) 017 [hep-ph/0404120] [INSPIRE].

    Article  Google Scholar 

  47. R. Ellis, W. Giele and Z. Kunszt, A numerical unitarity formalism for evaluating one-loop amplitudes, JHEP 03 (2008) 003 [arXiv:0708.2398] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  48. 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 

  49. 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 

  50. A. Lazopoulos, Multi-gluon one-loop amplitudes numerically, arXiv:0812.2998 [INSPIRE].

  51. J.-C. Winter and W.T. Giele, Calculating gluon one-loop amplitudes numerically, arXiv:0902.0094 [INSPIRE].

  52. W. Giele, Z. Kunszt and J. Winter, Efficient color-dressed calculation of virtual corrections, Nucl. Phys. B 840 (2010) 214 [arXiv:0911.1962] [INSPIRE].

    Article  ADS  Google Scholar 

  53. G. Cullen et al., Automated one-loop calculations with GoSam, arXiv:1111.2034 [INSPIRE].

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

    Article  ADS  Google Scholar 

  55. G. Bevilacqua et al., Helac-nlo, arXiv:1110.1499 [INSPIRE].

  56. 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  MATH  Google Scholar 

  57. 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 

  58. Z. Kunszt, A. Signer and Z. Trócsányi, One loop helicity amplitudes for all 2 → 2 processes in QCD and N = 1 supersymmetric Yang-Mills theory, Nucl. Phys. B 411 (1994) 397 [hep-ph/9305239] [INSPIRE].

    Article  ADS  Google Scholar 

  59. Z. Kunszt, A. Signer and Z. Trócsányi, One loop radiative corrections to the helicity amplitudes of QCD processes involving four quarks and one gluon, Phys. Lett. B 336 (1994) 529 [hep-ph/9405386] [INSPIRE].

    Article  ADS  Google Scholar 

  60. Z. Bern, L.J. Dixon, D.A. Kosower and S. Weinzierl, One loop amplitudes for e + e − → qq QQ, Nucl. Phys. B 489 (1997) 3 [hep-ph/9610370] [INSPIRE].

    Article  ADS  Google Scholar 

  61. 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 

  62. anc/partials *.dat, in the source of the arXiv version of this manuscript or viewed directly following the instructions at http://arxiv.org/help/ancillary files.

  63. 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 

  64. L.J. Dixon, Calculating scattering amplitudes efficiently, hep-ph/9601359 [INSPIRE].

  65. Z. Bern, L.J. Dixon and D.A. Kosower, Progress in one loop QCD computations, Ann. Rev. Nucl. Part. Sci. 46 (1996) 109 [hep-ph/9602280] [INSPIRE].

    Article  ADS  Google Scholar 

  66. 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].

  67. P. Cvitanovic, P. Lauwers and P. Scharbach, Gauge invariance structure of quantum chromodynamics, Nucl. Phys. B 186 (1981) 165 [INSPIRE].

    Article  ADS  Google Scholar 

  68. 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 

  69. F.A. Berends and W. Giele, Recursive calculations for processes with n gluons, Nucl. Phys. B 306 (1988) 759 [INSPIRE].

    Article  ADS  Google Scholar 

  70. D.A. Kosower, Color factorization for fermionic amplitudes, Nucl. Phys. B 315 (1989) 391 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  71. W. Giele and E. Glover, Higher order corrections to jet cross-sections in e + e − annihilation, Phys. Rev. D 46 (1992) 1980 [INSPIRE].

    ADS  Google Scholar 

  72. P. Nogueira, Automatic Feynman graph generation, J. Comput. Phys. 105 (1993) 279 [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  73. J. Vermaseren, New features of FORM, math-ph/0010025 [INSPIRE].

  74. 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 

  75. 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].

    Article  ADS  Google Scholar 

  76. Wolfram Research, Inc., Mathematica, Version 6.0, Champaign U.S.A. (2007).

  77. S. Catani, The singular behavior of QCD amplitudes at two loop order, Phys. Lett. B 427 (1998) 161 [hep-ph/9802439] [INSPIRE].

    Article  ADS  Google Scholar 

  78. T. Gleisberg et al., Event generation with SHERPA 1.1, JHEP 02 (2009) 007 [arXiv:0811.4622] [INSPIRE].

    Article  ADS  Google Scholar 

  79. F. Krauss, R. Kuhn and G. Soff, AMEGIC++ 1.0: a matrix element generator in C++, JHEP 02 (2002) 044 [hep-ph/0109036] [INSPIRE].

    Article  ADS  Google Scholar 

  80. T. Gleisberg and F. Krauss, Automating dipole subtraction for QCD NLO calculations, Eur. Phys. J. C 53 (2008) 501 [arXiv:0709.2881] [INSPIRE].

    Article  ADS  Google Scholar 

  81. 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 

  82. L.J. Dixon, J.M. Henn, J. Plefka and T. Schuster, All tree-level amplitudes in massless QCD, JHEP 01 (2011) 035 [arXiv:1010.3991] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  83. Z. Bern et al., Left-handed W bosons at the LHC, Phys. Rev. D 84 (2011) 034008 [arXiv:1103.5445] [INSPIRE].

    ADS  Google Scholar 

  84. Z. Bern et al., Driving missing data at next-to-leading order, Phys. Rev. D 84 (2011) 114002 [arXiv:1106.1423] [INSPIRE].

    ADS  Google Scholar 

  85. W. Siegel, Supersymmetric dimensional regularization via dimensional reduction, Phys. Lett. B 84 (1979) 193 [INSPIRE].

    Article  ADS  Google Scholar 

  86. D. Capper, D. Jones and P. van Nieuwenhuizen, Regularization by dimensional reduction of supersymmetric and nonsupersymmetric gauge theories, Nucl. Phys. B 167 (1980) 479 [INSPIRE].

    Article  ADS  Google Scholar 

  87. L. Avdeev and A. Vladimirov, Dimensional regularization and supersymmetry, Nucl. Phys. B 219 (1983) 262 [INSPIRE].

    Article  ADS  Google Scholar 

  88. Z. Bern and D.A. Kosower, The computation of loop amplitudes in gauge theories, Nucl. Phys. B 379 (1992) 451 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  89. G. ’t Hooft and M. Veltman, Regularization and renormalization of gauge fields, Nucl. Phys. B 44 (1972) 189 [INSPIRE].

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Physics and Astronomy, University of California (UCLA), Los Angeles, CA, 90095-1547, U.S.A.

    Harald Ita & Kemal Ozeren

  2. Niels Bohr International Academy and Discovery Center, Niels Bohr Institute (NBI), Blegdamsvej 17, DK-2100, Copenhagen, Denmark

    Harald Ita

  3. Raymond and Beverly Sackler School of Physics and Astronomy, Tel-Aviv University, Tel-Aviv, 69978, Israel

    Harald Ita

Authors
  1. Harald Ita
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Kemal Ozeren
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Harald Ita.

Additional information

ArXiv ePrint: 1111.4193

Electronic supplementary material

Below is the link to the electronic supplementary material.

ESM 1

(TAR 290 kb)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ita, H., Ozeren, K. Colour decompositions of multi-quark one-loop QCD amplitudes. J. High Energ. Phys. 2012, 118 (2012). https://doi.org/10.1007/JHEP02(2012)118

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/JHEP02(2012)118

Keywords

Search

Navigation