Abstract
We present the complete set of Feynman rules producing the rational terms of kind R2 needed to perform any 1-loop calculation in the Electroweak Standard Model. Our formulae are given both in the R ξ gauge and in the Unitary gauge, therefore completing the results in the ’t Hooft-Feynman gauge already presented in a previous publication.
As a consistency check, we verified, in the case of the process H → γγ and in a few other physical cases, the independence of the total Rational Part (R1 +R2) on the chosen gauge. In addition, we explicitly checked the equivalence of the limits ξ→∞ after or before the loop momentum integration in the definition of the Unitary gauge at 1-loop.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
P. Mastrolia, G. Ossola, C.G. Papadopoulos and R. Pittau, Optimizing the reduction of one-loop amplitudes, JHEP 06 (2008) 030 [arXiv:0803.3964] [SPIRES].
T. Binoth, G. Ossola, C.G. Papadopoulos and R. Pittau, NLO QCD corrections to tri-boson production, JHEP 06 (2008) 082 [arXiv:0804.0350] [SPIRES].
A. van Hameren, C.G. Papadopoulos and R. Pittau, Automated one-loop calculations: a proof of concept, JHEP 09 (2009) 106 [arXiv:0903.4665] [SPIRES].
R.K. 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] [SPIRES].
C.F. Berger et al., Precise predictions for W + 3 jet production at hadron colliders, Phys. Rev. Lett. 102 (2009) 222001 [arXiv:0902.2760] [SPIRES].
T. Binoth et al., A proposal for a standard interface between Monte Carlo tools and one-loop programs, Comput. Phys. Commun. 181 (2010) 1612 [arXiv:1001.1307] [SPIRES].
SM and NLO Multileg Working Group collaboration, J. R. Andersenetal., The SM and NLO multileg working group: summary report, arXiv:1003.1241 [SPIRES].
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] [SPIRES].
R. Pittau, Testing and improving the numerical accuracy of the NLO predictions, Comput. Phys. Commun. 181 (2010) 1941 [arXiv:1006.3773] [SPIRES].
G. Heinrich, G. Ossola, T. Reiter and F. Tramontano, Tensorial Reconstruction at the Integrand Level, JHEP 10 (2010) 105 [arXiv:1008.2441] [SPIRES].
F. del Aguila and R. Pittau, Recursive numerical calculus of one-loop tensor integrals, JHEP 07 (2004) 017 [hep-ph/0404120] [SPIRES].
R. Pittau, Formulae for a numerical computation of one-loop tensor integrals, hep-ph/0406105 [SPIRES].
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] [SPIRES].
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] [SPIRES].
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] [SPIRES].
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] [SPIRES].
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] [SPIRES].
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] [SPIRES].
Z. Bern and A.G. Morgan, Massive loop amplitudes from unitarity, Nucl. Phys. B 467 (1996) 479 [hep-ph/9511336] [SPIRES].
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] [SPIRES].
C. Anastasiou, R. Britto, B. Feng, Z. Kunszt and P. Mastrolia, D-dimensional unitarity cut method, Phys. Lett. B 645 (2007) 213 [hep-ph/0609191] [SPIRES].
D. Forde, Direct extraction of one-loop integral coefficients, Phys. Rev. D 75 (2007) 125019 [arXiv:0704.1835] [SPIRES].
W.T. Giele, Z. Kunszt and K. Melnikov, Full one-loop amplitudes from tree amplitudes, JHEP 04 (2008) 049 [arXiv:0801.2237] [SPIRES].
R.K. Ellis, W.T. Giele, Z. Kunszt and K. Melnikov, Masses, fermions and generalized D-dimensional unitarity, Nucl. Phys. B 822 (2009) 270 [arXiv:0806.3467] [SPIRES].
G. Ossola, C.G. Papadopoulos and R. Pittau, Numerical evaluation of six-photon amplitudes, JHEP 07 (2007) 085 [arXiv:0704.1271] [SPIRES].
G. Bevilacqua, M. Czakon, C.G. Papadopoulos, R. Pittau and M. Worek, A ssault on the NLO wishlist: pp → ttbb, JHEP 09 (2009) 109 [arXiv:0907.4723] [SPIRES].
G. Bevilacqua, M. Czakon, C.G. 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] [SPIRES].
T. Melia, K. Melnikov, R. Rontsch and G. Zanderighi, Next-to-leading order QCD predictions for W+W+jj production at the LHC, JHEP 12 (2010) 053 [arXiv:1007.5313] [SPIRES].
C.F. Berger et al., Precise predictions for W +4 jet production at the Large Hadron Collider, arXiv:1009.2338 [SPIRES].
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] [SPIRES].
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] [SPIRES].
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] [SPIRES].
C.F. Berger, Z. Bern, L.J. Dixon, D. Forde and D.A. Kosower, Bootstrapping one-loop QCD amplitudes with general helicities, Phys. Rev. D 74 (2006) 036009 [hep-ph/0604195] [SPIRES].
G. Ossola, C.G. Papadopoulos and R. Pittau, On the rational terms of the one-loop amplitudes, JHEP 05 (2008) 004 [arXiv:0802.1876] [SPIRES].
P. Draggiotis, M.V. Garzelli, C.G. Papadopoulos and R. Pittau, Feynman rules for the rational part of the QCD 1-loop amplitudes, JHEP 04 (2009) 072 [arXiv:0903.0356] [SPIRES].
M.V. Garzelli, I. Malamos and R. Pittau, Feynman rules for the rational part of the electroweak 1-loop amplitudes, JHEP 01 (2010) 040 [arXiv:0910.3130] [SPIRES].
T. Binoth, J.P. Guillet and G. Heinrich, Algebraic evaluation of rational polynomials in one-loop amplitudes, JHEP 02 (2007) 013 [hep-ph/0609054] [SPIRES].
A. Bredenstein, A. Denner, S. Dittmaier and S. Pozzorini, NLO QCD corrections to top anti-top bottom anti-bottom production at the LHC: 1. quark-antiquark annihilation, JHEP 08 (2008) 108 [arXiv:0807.1248] [SPIRES].
Z. Bern and D.A. Kosower, The computation of loop amplitudes in gauge theories, Nucl. Phys. B 379 (1992) 451 [SPIRES].
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] [SPIRES].
S. Catani, M.H. Seymour and Z. Trócsányi, Regularization scheme independence and unitarity in QCD cross sections, Phys. Rev. D 55 (1997) 6819 [hep-ph/9610553] [SPIRES].
Z. Bern, A. De Freitas, L.J. Dixon and H.L. Wong, Supersymmetric regularization, two-loop QCD amplitudes and coupling shifts, Phys. Rev. D 66 (2002) 085002 [hep-ph/0202271] [SPIRES].
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] [SPIRES].
J.A.M. Vermaseren, New features of FORM, math-ph/0010025 [SPIRES].
J.A.M. Vermaseren, The FORM project, Nucl. Phys. Proc. Suppl. 183 (2008) 19 [arXiv:0806.4080] [SPIRES].
D.Y. Bardin and G. Passarino, The standard model in the making: precision study of the electroweak interactions, Clarendon Press, Oxford U.K. (1999), p. 685 [SPIRES]
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint:1009.4302
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Garzelli, M.V., Malamos, I. & Pittau, R. Feynman rules for the rational part of the electroweak 1-loop amplitudes in the Rξ gauge and in the unitary gauge. J. High Energ. Phys. 2011, 29 (2011). https://doi.org/10.1007/JHEP01(2011)029
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP01(2011)029