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Feynman rules for the rational part of the standard model one-loop amplitudes in the ’tHooft-Veltman \(\gamma_{5}\) scheme. (English) Zbl 1301.81360
Summary: We study Feynman rules for the rational part \(R\) of the Standard Model amplitudes at one-loop level in the ’t Hooft-Veltman \(\gamma_{5}\) scheme. Comparing our results for quantum chromodynamics and electroweak 1-loop amplitudes with that obtained based on the Kreimer-Korner-Schilcher (KKS) \(\gamma_{5}\) scheme, we find the latter result can be recovered when our \(\gamma_{5}\) scheme becomes identical (by setting \(g5s = 1\) in our expressions) with the KKS scheme. As an independent check, we also calculate Feynman rules obtained in the KKS scheme, finding our results in complete agreement with formulae presented in the literature. Our results, which are studied in two different \(\gamma_{5}\) schemes, may be useful for clarifying the \(\gamma_{5}\) problem in dimensional regularization. They are helpful to eliminate or find ambiguities arising from different dimensional regularization schemes.

MSC:
81V22 Unified quantum theories
81T15 Perturbative methods of renormalization applied to problems in quantum field theory
81T18 Feynman diagrams
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[1] Bredenstein, A.; Denner, A.; Dittmaier, S.; Pozzorini, S., NLO QCD corrections to top anti-top bottom anti-bottom production at the LHC: 1. quark-antiquark annihilation, JHEP, 08, 108, (2008)
[2] Bredenstein, A.; Denner, A.; Dittmaier, S.; Pozzorini, S., NLO QCD corrections to top anti-top bottom anti-bottom production at the LHC: 2. full hadronic results, JHEP, 03, 021, (2010)
[3] Bevilacqua, G.; Czakon, M.; Papadopoulos, CG; Pittau, R.; Worek, M., Assault on the NLO wishlist: \( pp → t\bar{t}b\bar{b} \), JHEP, 09, 109, (2009)
[4] Ellis, RK; Melnikov, K.; Zanderighi, G., \(W\) + 3 jet production at the tevatron, Phys. Rev., D 80, 094002, (2009)
[5] Berger, CF; etal., Next-to-leading order QCD predictions for \(W\) + 3-jet distributions at hadron colliders, Phys. Rev., D 80, 074036, (2009)
[6] Berger, CF; etal., Next-to-leading order QCD predictions for \(Z\), \(γ\)\^{}{*} + 3-jet distributions at the tevatron, Phys. Rev., D 82, 074002, (2010)
[7] Berger, CF; etal., Precise predictions for \(W\) + 4 jet production at the large hadron collider, Phys. Rev. Lett., 106, 092001, (2011)
[8] Melia, T.; Melnikov, K.; Rontsch, R.; Zanderighi, G., Next-to-leading order QCD predictions for \(W\) + \(W\) + jj production at the LHC, JHEP, 12, 053, (2010)
[9] Bozzi, G.; Jager, B.; Oleari, C.; Zeppenfeld, D., Next-to-leading order QCD corrections to \(W\) + \(Z\) and \(W\) − \(Z\) production via vector-boson fusion, Phys. Rev., D 75, 073004, (2007)
[10] Jager, B.; Oleari, C.; Zeppenfeld, D., Next-to-leading order QCD corrections to Z boson pair production via vector-boson fusion, Phys. Rev., D 73, 113006, (2006)
[11] Jager, B.; Oleari, C.; Zeppenfeld, D., Next-to-leading order QCD corrections to \(W\)\^{}{+}\(W\)\^{}{−} production via vector-boson fusion, JHEP, 07, 015, (2006)
[12] F. Campanario, Towards pp → VVjj at NLO QCD: bosonic contributions to triple vector boson production plus jet, arXiv:1105.0920 [SPIRES].
[13] Bevilacqua, G.; Czakon, M.; Papadopoulos, CG; Worek, M., Dominant QCD backgrounds in Higgs boson analyses at the LHC: a study of \( pp → t\bar{t} + 2 \) jets at next-to-leading order, Phys. Rev. Lett., 104, 162002, (2010)
[14] Bevilacqua, G.; etal., NLO QCD calculations with HELAC-NLO, Nucl. Phys. Proc. Suppl., 205-206, 211, (2010)
[15] M. Worek, Recent developments in NLO QCD calculations: the particular case of\( pp → t\bar{t} \)jj, arXiv:1012.4987 [SPIRES].
[16] Denner, A.; Dittmaier, S.; Kallweit, S.; Pozzorini, S., NLO QCD corrections to \( WWb\bar{b} \) production at hadron colliders, Phys. Rev. Lett., 106, 052001, (2011)
[17] Bevilacqua, G.; Czakon, M.; Hameren, A.; Papadopoulos, CG; Worek, M., Complete off-shell effects in top quark pair hadroproduction with leptonic decay at next-to-leading order, JHEP, 02, 083, (2011)
[18] Binoth, T.; etal., Next-to-leading order QCD corrections to \( pp → b\bar{b}b\bar{b} + X \) at the LHC: the quark induced case, Phys. Lett., B 685, 293, (2010)
[19] N. Greiner, A. Guffanti, T. Reiter and J. Reuter, NLO QCD corrections to the production of two bottom-antibottom pairs at the LHC, arXiv:1105.3624 [SPIRES].
[20] F. Campanario, C. Englert, M. Rauch and D. Zeppenfeld, Precise predictions for W γ γ + jet production at hadron colliders, arXiv:1106.4009 [SPIRES].
[21] SM and NLO Multileg Working Group collaboration, J.R. Andersen et al., The SM and NLO multileg working group: summary report, arXiv:1003.1241 [SPIRES].
[22] Bern, Z.; Dixon, LJ; Kosower, DA, One loop corrections to five gluon amplitudes, Phys. Rev. Lett., 70, 2677, (1993)
[23] Bern, Z.; Dixon, LJ; Dunbar, DC; Kosower, DA, One-loop n-point gauge theory amplitudes, unitarity and collinear limits, Nucl. Phys., B 425, 217, (1994)
[24] Bern, Z.; Dixon, LJ; Dunbar, DC; Kosower, DA, Fusing gauge theory tree amplitudes into loop amplitudes, Nucl. Phys., B 435, 59, (1995)
[25] Bern, Z.; Dixon, LJ; Kosower, DA, One loop corrections to two quark three gluon amplitudes, Nucl. Phys., B 437, 259, (1995)
[26] Britto, R.; Cachazo, F.; Feng, B., Generalized unitarity and one-loop amplitudes in \(N\) = 4 super-Yang-Mills, Nucl. Phys., B 725, 275, (2005)
[27] Aguila, F.; Pittau, R., Recursive numerical calculus of one-loop tensor integrals, JHEP, 07, 017, (2004)
[28] R. Pittau, Formulae for a numerical computation of one-loop tensor integrals, hep-ph/0406105 [SPIRES].
[29] Ossola, G.; Papadopoulos, CG; Pittau, R., Reducing full one-loop amplitudes to scalar integrals at the integr and level, Nucl. Phys., B 763, 147, (2007)
[30] Ossola, G.; Papadopoulos, CG; Pittau, R., Numerical evaluation of six-photon amplitudes, JHEP, 07, 085, (2007)
[31] Ossola, G.; Papadopoulos, CG; Pittau, R., Cuttools: a program implementing the OPP reduction method to compute one-loop amplitudes, JHEP, 03, 042, (2008)
[32] Mastrolia, P.; Ossola, G.; Papadopoulos, CG; Pittau, R., Optimizing the reduction of one-loop amplitudes, JHEP, 06, 030, (2008)
[33] Ossola, G.; Papadopoulos, CG; Pittau, R., On the rational terms of the one-loop amplitudes, JHEP, 05, 004, (2008)
[34] Binoth, T.; Guillet, JP; Heinrich, G., Algebraic evaluation of rational polynomials in one-loop amplitudes, JHEP, 02, 013, (2007)
[35] Passarino, G.; Veltman, MJG, One loop corrections for \(e\)\^{}{+}\(e\)\^{}{−} annihilation into \(μ\)\^{}{+}\(μ\)\^{}{−} in the Weinberg model, Nucl. Phys., B 160, 151, (1979)
[36] Denner, A.; Dittmaier, S., Reduction of one-loop tensor 5-point integrals, Nucl. Phys., B 658, 175, (2003)
[37] Denner, A.; Dittmaier, S., Reduction schemes for one-loop tensor integrals, Nucl. Phys., B 734, 62, (2006)
[38] Draggiotis, P.; Garzelli, MV; Papadopoulos, CG; Pittau, R., Feynman rules for the rational part of the QCD 1-loop amplitudes, JHEP, 04, 072, (2009)
[39] Garzelli, MV; Malamos, I.; Pittau, R., Feynman rules for the rational part of the electroweak 1-loop amplitudes, JHEP, 01, 040, (2010)
[40] Garzelli, MV; Malamos, I.; Pittau, R., Feynman rules for the rational part of the electroweak 1-loop amplitudes in the \(R\)_{\(ξ\)} gauge and in the unitary gauge, JHEP, 01, 029, (2011)
[41] Garzelli, MV; Malamos, I., R2SM: a package for the analytic computation of the R2 rational terms in the standard model of the electroweak interactions, Eur. Phys. J., C 71, 1605, (2011)
[42] Siegel, W., Supersymmetric dimensional regularization via dimensional reduction, Phys. Lett., B 84, 193, (1979)
[43] Bern, Z.; Kosower, DA, The computation of loop amplitudes in gauge theories, Nucl. Phys., B 379, 451, (1992)
[44] Kunszt, Z.; Signer, A.; Trócsányi, Z., One loop helicity amplitudes for all 2 → 2 processes in QCD and \(N\) = 1 supersymmetric Yang-Mills theory, Nucl. Phys., B 411, 397, (1994)
[45] Catani, S.; Seymour, MH; Trócsányi, Z., Regularization scheme independence and unitarity in QCD cross sections, Phys. Rev., D 55, 6819, (1997)
[46] Bern, Z.; Freitas, A.; Dixon, LJ; Wong, HL, Supersymmetric regularization, two-loop QCD amplitudes and coupling shifts, Phys. Rev., D 66, 085002, (2002)
[47] ’t Hooft, G.; Veltman, MJG, Regularization and renormalization of gauge fields, Nucl. Phys., B 44, 189, (1972)
[48] D. Kreimer, The role of γ_{5}in dimensional regularization, hep-ph/9401354 [SPIRES].
[49] Korner, JG; Kreimer, D.; Schilcher, K., A practicable \(γ\)_{5} scheme in dimensional regularization, Z. Phys., C 54, 503, (1992)
[50] Kreimer, D., The \(γ\)_{5} problem and anomalies: a Clifford algebra approach, Phys. Lett., B 237, 59, (1990)
[51] Bollini, CG; Giambiagi, JJ, Lowest order divergent graphs in \(ν\)-dimensional space, Phys. Lett., B 40, 566, (1972)
[52] Cicuta, GM; Montaldi, E., Analytic renormalization via continuous space dimension, Nuovo Cim. Lett., 4, 329, (1972)
[53] Ashmore, JF, A method of gauge invariant regularization, Lett. Nuovo Cim., 4, 289, (1972)
[54] Breitenlohner, P.; Maison, D., Dimensional renormalization and the action principle, Commun. Math. Phys., 52, 11, (1977)
[55] Breitenlohner, P.; Maison, D., Dimensionally renormalized green’s functions for theories with massless particles. 1, Commun. Math. Phys., 52, 39, (1977)
[56] Breitenlohner, P.; Maison, D., Dimensionally renormalized green’s functions for theories with massless particles. 2, Commun. Math. Phys., 52, 55, (1977)
[57] Bonneau, G., Consistency in dimensional regularization with γ5, Phys. Lett., B 96, 147, (1980)
[58] Bonneau, G., Preserving canonical Ward identities in dimensional regularization with a nonanticommuting \(γ\)_{5}, Nucl. Phys., B 177, 523, (1981)
[59] Martin, CP; Sánchez-Ruiz, D., Action principles, restoration of BRS symmetry and the renormalization group equation for chiral non-abelian gauge theories in dimensional renormalization with a non-anticommuting \(γ\)_{5}, Nucl. Phys., B 572, 387, (2000)
[60] Schubert, C., The Yukawa model as an example for dimensional renormalization with \(γ\) − 5, Nucl. Phys., B 323, 478, (1989)
[61] Pernici, M.; Raciti, M.; Riva, F., Dimensional renormalization of Yukawa theories via Wilsonian methods, Nucl. Phys., B 577, 293, (2000)
[62] Pernici, M., Semi-naive dimensional renormalization, Nucl. Phys., B 582, 733, (2000)
[63] Pernici, M.; Raciti, M., Axial current in QED and semi-naive dimensional renormalization, Phys. Lett., B 513, 421, (2001)
[64] Ferrari, R.; Yaouanc, A.; Oliver, L.; Raynal, JC, Gauge invariance and dimensional regularization with \(γ\)_{5} in flavor changing neutral processes, Phys. Rev., D 52, 3036, (1995)
[65] Jegerlehner, F., Facts of life with \(γ\)_{5}, Eur. Phys. J., C 18, 673, (2001)
[66] Kublbeck, J.; Böhm, M.; Denner, A., Feyn arts: computer algebraic generation of Feynman graphs and amplitudes, Comput. Phys. Commun., 60, 165, (1990)
[67] Hahn, T., Generating Feynman diagrams and amplitudes with feynarts 3, Comput. Phys. Commun., 140, 418, (2001)
[68] Denner, A., Techniques for calculation of electroweak radiative corrections at the one loop level and results for W physics at LEP-200, Fortschr. Phys., 41, 307, (1993)
[69] Korner, JG; Nasrallah, N.; Schilcher, K., Evaluation of the flavor changing vertex \(b\) → sh using the breitenlohner-maison-’t Hooft-veltman \(γ\)_{5} scheme, Phys. Rev., D 41, 888, (1990)
[70] H.-S. Shao, Y.-J. Zhang and K.-T. Chao, Dijet invariant mass distribution in top quark hadronic decay with QCD corrections, arXiv:1106.5483 [SPIRES].
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