Abstract
In this paper we discuss techniques, which lead to a significant improvement of the efficiency of the Monte Carlo integration, when one-loop QCD amplitudes are calculated numerically with the help of the subtraction method and contour deformation. The techniques discussed are: holomorphic and non-holomorphic division into sub-channels, optimisation of the integration contour, improvement of the ultraviolet subtraction terms, importance sampling and antithetic variates in loop momentum space, recurrence relations.
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References
S. Becker, D. Goetz, C. Reuschle, C. Schwan and S. Weinzierl, NLO results for five, six and seven jets in electron-positron annihilation, Phys. Rev. Lett. 108 (2012) 032005 [arXiv:1111.1733] [INSPIRE].
S. Becker, C. Reuschle and S. Weinzierl, Numerical NLO QCD calculations, JHEP 12 (2010) 013 [arXiv:1010.4187] [INSPIRE].
M. Assadsolimani, S. Becker, C. Reuschle and S. Weinzierl, Infrared singularities in one-loop amplitudes, Nucl. Phys. Proc. Suppl. 205-206 (2010) 224 [arXiv:1006.4609] [INSPIRE].
M. Assadsolimani, S. Becker and S. Weinzierl, A simple formula for the infrared singular part of the integrand of one-loop QCD amplitudes, Phys. Rev. D 81 (2010) 094002 [arXiv:0912.1680] [INSPIRE].
Z. Nagy and D.E. Soper, General subtraction method for numerical calculation of one loop QCD matrix elements, JHEP 09 (2003) 055 [hep-ph/0308127] [INSPIRE].
W. Gong, Z. Nagy and D.E. Soper, Direct numerical integration of one-loop Feynman diagrams for N-photon amplitudes, Phys. Rev. D 79 (2009) 033005 [arXiv:0812.3686] [INSPIRE].
C. Anastasiou, S. Beerli and A. Daleo, Evaluating multi-loop Feynman diagrams with infrared and threshold singularities numerically, JHEP 05 (2007) 071 [hep-ph/0703282] [INSPIRE].
Z. Nagy and D.E. Soper, Numerical integration of one-loop Feynman diagrams for N-photon amplitudes, Phys. Rev. D 74 (2006) 093006 [hep-ph/0610028] [INSPIRE].
D.E. Soper, Choosing integration points for QCD calculations by numerical integration, Phys. Rev. D 64 (2001) 034018 [hep-ph/0103262] [INSPIRE].
D.E. Soper, Techniques for QCD calculations by numerical integration, Phys. Rev. D 62 (2000) 014009 [hep-ph/9910292] [INSPIRE].
D.E. Soper, QCD calculations by numerical integration, Phys. Rev. Lett. 81 (1998) 2638 [hep-ph/9804454] [INSPIRE].
C. Berger et al., Precise predictions for W + 3 jet production at hadron colliders, Phys. Rev. Lett. 102 (2009) 222001 [arXiv:0902.2760] [INSPIRE].
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].
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].
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].
H. Ita et al., Precise predictions for Z + 4 jets at hadron colliders, Phys. Rev. D 85 (2012) 031501 [arXiv:1108.2229] [INSPIRE].
Z. Bern et al., Four-jet production at the Large Hadron Collider at next-to-leading order in QCD, arXiv:1112.3940 [INSPIRE].
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] [INSPIRE].
R.K. Ellis, K. Melnikov and G. Zanderighi, W + 3 jet production at the Tevatron, Phys. Rev. D 80 (2009) 094002 [arXiv:0906.1445] [INSPIRE].
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] [INSPIRE].
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].
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].
R. Frederix, S. Frixione, K. Melnikov and G. Zanderighi, NLO QCD corrections to five-jet production at LEP and the extraction of α s (M Z ), JHEP 11 (2010) 050 [arXiv:1008.5313] [INSPIRE].
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].
V. Hirschi et al., Automation of one-loop QCD corrections, JHEP 05 (2011) 044 [arXiv:1103.0621] [INSPIRE].
G. Bevilacqua et al., HELAC-NLO, arXiv:1110.1499 [INSPIRE].
G. Cullen et al., Golem95C: a library for one-loop integrals with complex masses, Comput. Phys. Commun. 182 (2011) 2276 [arXiv:1101.5595] [INSPIRE].
G. Cullen et al., Automated one-loop calculations with GoSam, Eur. Phys. J. C 72 (2012) 1889 [arXiv:1111.2034] [INSPIRE].
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].
T. Binoth, J.-P. Guillet, G. Heinrich, E. Pilon and T. Reiter, Golem95: a numerical program to calculate one-loop tensor integrals with up to six external legs, Comput. Phys. Commun. 180 (2009)2317 [arXiv:0810.0992] [INSPIRE].
A. van Hameren, OneLOop: for the evaluation of one-loop scalar functions, Comput. Phys. Commun. 182 (2011) 2427 [arXiv:1007.4716] [INSPIRE].
A. Denner and S. Dittmaier, Scalar one-loop 4-point integrals, Nucl. Phys. B 844 (2011) 199 [arXiv:1005.2076] [INSPIRE].
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].
S. Catani and M. Seymour, A general algorithm for calculating jet cross-sections in NLO QCD, Nucl. Phys. B 485 (1997) 291 [Erratum ibid. B 510 (1998) 503-504] [hep-ph/9605323] [INSPIRE].
S. Dittmaier, A general approach to photon radiation off fermions, Nucl. Phys. B 565 (2000) 69 [hep-ph/9904440] [INSPIRE].
L. Phaf and S. Weinzierl, Dipole formalism with heavy fermions, JHEP 04 (2001) 006 [hep-ph/0102207] [INSPIRE].
S. Catani, S. Dittmaier, M.H. Seymour and Z. Trócsányi, The dipole formalism for next-to-leading order QCD calculations with massive partons, Nucl. Phys. B 627 (2002) 189 [hep-ph/0201036] [INSPIRE].
G. ’t Hooft, A planar diagram theory for strong interactions, Nucl. Phys. B 72 (1974) 461 [INSPIRE].
F. Maltoni, K. Paul, T. Stelzer and S. Willenbrock, Color flow decomposition of QCD amplitudes, Phys. Rev. D 67 (2003) 014026 [hep-ph/0209271] [INSPIRE].
S. Weinzierl, Automated computation of spin- and colour-correlated Born matrix elements, Eur. Phys. J. C 45 (2006) 745 [hep-ph/0510157] [INSPIRE].
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].
S. Weinzierl, Introduction to Monte Carlo methods, hep-ph/0006269 [INSPIRE].
G.P. Lepage, A new algorithm for adaptive multidimensional integration, J. Comput. Phys. 27 (1978) 192 [INSPIRE].
G.P. Lepage, VEGAS: an adaptive multidimensional integration program, CLNS-80/447 (1980).
F.A. Berends and W. Giele, Recursive calculations for processes with n gluons, Nucl. Phys. B 306 (1988) 759 [INSPIRE].
P. Draggiotis et al., Recursive equations for arbitrary scattering processes, Nucl. Phys. Proc. Suppl. 160 (2006) 255 [hep-ph/0607034] [INSPIRE].
A. van Hameren, Multi-gluon one-loop amplitudes using tensor integrals, JHEP 07 (2009) 088 [arXiv:0905.1005] [INSPIRE].
F. Cascioli, P. Maierhofer and S. Pozzorini, Scattering amplitudes with open loops, Phys. Rev. Lett. 108 (2012) 111601 [arXiv:1111.5206] [INSPIRE].
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ArXiv ePrint: 1205.2096
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Becker, S., Reuschle, C. & Weinzierl, S. Efficiency improvements for the numerical computation of NLO corrections. J. High Energ. Phys. 2012, 90 (2012). https://doi.org/10.1007/JHEP07(2012)090
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DOI: https://doi.org/10.1007/JHEP07(2012)090