NLO QCD corrections to pp->WW+jet+X
aa r X i v : . [ h e p - ph ] D ec NLO QCD corrections to pp → WW + jet + X Stefan Dittmaier ∗ a , b , Stefan Kallweit a and Peter Uwer c † a Max-Planck-Institut für Physik (Werner-Heisenberg-Institut), D-80805 München, Germany b Faculty of Physics, University of Vienna, A-1090 Vienna, Austria c Institut für Theoretische Teilchenphysik, Universität Karlsruhe, D-76128 Karlsruhe, GermanyE-mail: [email protected] , [email protected] , [email protected] We report on the calculation of the next-to-leading order QCD corrections to the production ofW-boson pairs in association with a hard jet at hadron colliders, which is an important sourceof background for Higgs and new-physics searches. If a veto against the emission of a secondhard jet is applied, the corrections stabilize the leading-order prediction for the cross sectionconsiderably. ∗ Speaker. † We (S.D. and P.U.) thank the Galileo Galilei Institute for Theoretical Physics in Florence for the hospitalityand the INFN for partial support during the completion of this work. P.U. is supported as Heisenberg Fellow of theDeutsche Forschungsgemeinschaft DFG. This work is supported in part by the European Community’s Marie-CurieResearch Training Network HEPTOOLS under contract MRTN-CT-2006-035505 and by the DFG Sonderforschungs-bereich/Transregio 9 “Computergestützte Theoretische Teilchenphysik” SFB/TR9. c (cid:13) Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
LO QCD corrections to pp → WW + jet + X Stefan Dittmaier
1. Introduction
The search for new-physics particles—including the Standard Model Higgs boson—will bethe primary task in high-energy physics after the start of the LHC that is planned for 2008. Theextremely complicated hadron collider environment does not only require sufficiently precise pre-dictions for new-physics signals, but also for many complicated background reactions that can-not entirely be measured from data. Among such background processes, several involve three,four, or even more particles in the final state, rendering the necessary next-to-leading-order (NLO)calculations in QCD very complicated. This problem lead to the creation of an “experimenters’wishlist for NLO calculations” [1] that are still missing for successful LHC analyses. The processpp → W + W − + jet + X made it to the top of this list.The process of WW+jet production is an important source for background to the productionof a Higgs boson that subsequently decays into a W-boson pair, where additional jet activity mightarise from the production or a hadronically decaying W boson [2]. WW+jet production deliversalso potential background to new-physics searches, such as supersymmetric particles, because ofleptons and missing transverse momentum from the W decays. Last but not least the process isinteresting in its own right, since W-pair production processes enable a direct precise analysis ofthe non-abelian gauge-boson self-interactions, and a large fraction of W pairs will show up withadditional jet activity at the LHC.In these proceedings we briefly report on our recent calculation [3] of NLO QCD correctionsto WW+jet production at the Tevatron and the LHC, but here we discuss results for the LHC only.Parallel to our work, another NLO study [4] of pp → W + W − + jet + X at the LHC appeared. Acomparison of results of the two groups is in progress.
2. Details of the NLO calculation
At leading order (LO), hadronic WW + jet production receives contributions from the partonicprocesses q ¯ q → W + W − g, q g → W + W − q , and ¯ q g → W + W − ¯ q , where q stands for up- or down-type quarks. Note that the amplitudes for q = u , d are not the same, even for vanishing light quarkmasses. All three channels are related by crossing symmetry. The LO diagrams for a specificpartonic process are shown in Figure 1.In order to prove the correctness of our results we have evaluated each ingredient twice us-ing independent calculations based—as far as possible—on different methods, yielding results inmutual agreement. The virtual corrections modify the partonic processes that are already present at LO. At NLOthese corrections are induced by self-energy, vertex, box (4-point), and pentagon (5-point) correc- u¯u du W + W − g u¯u dd W + gW − u¯u ud gW + W − u¯u u γ/ Z W + W − g u¯u u γ/ Z gW + W − Figure 1:
LO diagrams for the partonic process u¯u → W + W − g. LO QCD corrections to pp → WW + jet + X Stefan Dittmaier u¯u uu du W + W − gg u¯u uu dd W + gW − g u¯u uu ud gW + W − g g¯u gu ud uW + W − g Figure 2:
Pentagon diagrams for the partonic process u¯u → W + W − g. tions. For illustration the pentagon graphs, which are the most complicated diagrams, are shown inFigure 2 for a specific partonic channel. At one loop WW+jet production also serves as an off-shellcontinuation of the loop-induced process of Higgs+jet production with the Higgs boson decayinginto a W-boson pair. In this subprocess the off-shell Higgs boson is coupled via a heavy-quark loopto two gluons. Version 1 of the virtual corrections is essentially obtained as for the related processes of t¯tH [5]and t¯t + jet [6] production. Feynman diagrams and amplitudes are generated with FeynArts 1.0[7] and further processed with in-house Mathematica routines, which automatically create an out-put in Fortran. The IR (soft and collinear) singularities are treated in dimensional regularizationand analytically separated from the finite remainder as described in Refs. [5, 8]. The pentagontensor integrals are directly reduced to box integrals following Ref. [9]. This method does not in-troduce inverse Gram determinants in this step, thereby avoiding numerical instabilities in regionswhere these determinants become small. Box and lower-point integrals are reduced à la Passarino–Veltman [10] to scalar integrals, which are either calculated analytically or using the results ofRefs. [11]. Sufficient numerical stability is already achieved in this way, but further improvementswith the methods of Ref. [12] are in progress. Version 2 of the evaluation of loop diagrams starts with the generation of diagrams and ampli-tudes via FeynArts 3.2 [13], which is independent of version 1.0 [7]. The amplitudes are furthermanipulated with FormCalc5.2 [14] and eventually automatically translated into Fortran code. Thewhole reduction of tensor to scalar integrals is done with the help of the LoopTools library [14],which also employs the method of Ref. [9] for the 5-point tensor integrals, Passarino–Veltman [10]reduction for the lower-point tensors, and the FF package [15] for the evaluation of regular scalarintegrals. The dimensionally regularized soft or collinear singular 3- and 4-point integrals had tobe added to this library. To this end, the explicit results of Ref. [8] for the vertex and of Ref. [16]for the box integrals (with appropriate analytical continuations) are taken.
The matrix elements for the real corrections are given by 0 → W + W − q ¯ q gg and 0 → W + W − q ¯ qq ′ ¯ q ′ with a large variety of flavour insertions for the light quarks q and q ′ . The par-tonic processes are obtained from these matrix elements by all possible crossings of quarks andgluons into the initial state. The evaluation of the real-emission amplitudes is performed in twoindependent ways. Both evaluations employ (independent implementations of) the dipole subtrac-tion formalism [17] for the extraction of IR singularities and for their combination with the virtualcorrections. Version 1 employs the Weyl–van-der-Waerden formalism (as described in Ref. [18]) for the cal-culation of the helicity amplitudes. The phase-space integration is performed by a multi-channel3
LO QCD corrections to pp → WW + jet + X Stefan Dittmaier process s [ pb ] s Sherpa [ pb ] D s / stat. errorpp → WW + . ( ) . ( ) + . → WW + . ( ) . ( ) − . Table 1:
Comparison of LO cross sections with Sherpa (taken from Ref. [24]).
Monte Carlo integrator [19] with weight optimization [20] written in C++, which is constructedsimilar to RacoonWW [21]. The results for cross sections with two resolved hard jets have beenchecked against results obtained with Whizard 1.50 [22] and Sherpa 1.0.8 [23]. Details on this partof the calculation can be found in Ref. [24], the comparison to Sherpa results is briefly illustratedin Table 1. In order to improve the integration, additional channels are included for the integrationof the difference of the real-emission matrix elements and the subtraction terms.
Version 2 is based on scattering amplitudes calculated with Madgraph [25] generated code. Thecode has been modified to allow for a non-diagonal quark mixing matrix and the extraction of therequired colour and spin structure. The latter enter the evaluation of the dipoles in the Catani–Seymour subtraction method. The evaluation of the individual dipoles was performed using a C++library developed during the calculation of the NLO corrections for t¯t + jet [6]. For the phase-spaceintegration a simple mapping has been used where the phase space is generated from a sequentialsplitting.
3. Numerical results
We consistently use the CTEQ6 [26] set of parton distribution functions (PDFs), i.e. we takeCTEQ6L1 PDFs with a 1-loop running a s in LO and CTEQ6M PDFs with a 2-loop running a s in NLO. We do not include bottom quarks in the initial or final states, because the bottom PDF issuppressed w.r.t. to the others; outgoing b¯b pairs add little to the cross section and can be exper-imentally further excluded by anti-b-tagging. Quark mixing between the first two generations isintroduced via a Cabibbo angle q C = . N F =
5, and the respective QCD parameters are L LO5 =
165 MeV and L MS5 =
226 MeV.The top-quark loop in the gluon self-energy is subtracted at zero momentum. The running of a s is, thus, generated solely by the contributions of the light quark and gluon loops. The top-quarkmass is m t = . M W = .
425 GeV, M Z = . M H =
150 GeV. The weak mixing angle is set to itson-shell value, i.e. fixed by c = − s = M / M , and the electromagnetic coupling constant a is derived from Fermi’s constant G m = . × − GeV − according to a = √ G m M s / p .We apply the jet algorithm of Ref. [27] with R = p T , jet > p T , jet , cut . In contrast to the realcorrections the LO prediction and the virtual corrections are not influenced by the jet algorithm. Inour default setup, a possible second hard jet (originating from the real corrections) does not affect The input parameters of Ref. [24] are set as below, apart from a s ( M Z ) = . m ren = m fact = M W ), and a CKM matrix in Wolfenstein parametrization (to 2nd order in l ) with l = .
22. The transversemomenta of additional jets are restricted by p T , jet > M ( jet , jet ) > LO QCD corrections to pp → WW + jet + X Stefan Dittmaier
NLO (CTEQ6M) no 2nd separable jetNLO (CTEQ6M)LO (CTEQ6L1)pT;jet > 100GeVps = 14 TeVpp ! W+W(cid:0)+jet+X(cid:22)=MW(cid:27)[pb℄ 1010.12520151050
NLO (CTEQ6M) no 2nd separable jetNLO (CTEQ6M)LO (CTEQ6L1)0:5MW < (cid:22) < 2MWps = 14 TeVpp ! W+W(cid:0)+jet+XpT;jet; ut[GeV℄(cid:27)[pb℄ 200180160140120100806010
Figure 3:
LO and NLO cross sections for WW + jet production at the LHC: scale dependence with renor-malization and factorization scales set equal to m (lhs) and cut dependence (rhs) (taken from Ref. [3]). the event selection, but alternatively we also consider mere WW+jet events with “no 2 nd separablejet” where only the first hard jet is allowed to pass the p T , jet cut but not the second.The left-hand side of Figure 3 shows the scale dependence of the integrated LO and NLOcross sections at the LHC for p T , jet , cut =
100 GeV. The renormalization and factorization scalesare identified here ( m = m ren = m fact ), and the variation ranges from m = . M W to m = M W .The dependence is rather large in LO, illustrating the well-known fact that the LO predictions canonly provide a rough estimate. Varying the scales simultaneously by a factor of 4 (10) changes theLO cross section by about 35% (70%).Only a modest reduction of the scale dependence to 25% (60%) is observed in the transi-tion from LO to NLO if W pairs in association with two hard jets are taken into account. Thislarge residual scale dependence in NLO, which is mainly due to q g-scattering channels, can besignificantly suppressed upon applying the veto of having “no 2 nd separable jet”. In this case theuncertainty is 10% (15%) if the scale is varied by a factor of 4 (10). The relevance of a jet veto in or-der to suppress the scale dependence at NLO was also realized [28] for genuine W-pair productionat hadron colliders.Finally, we show the integrated LO and NLO cross sections as functions of p T , jet , cut on theright-hand side of Figure 3. The widths of the bands, which correspond to scale variations within M W / < m < M W , reflect the behaviour discussed above for fixed value of p T , jet , cut . For the LHCthe reduction of the scale uncertainty is only mild unless WW + References [1] C. Buttar et al. , arXiv:hep-ph/0604120;J. M. Campbell, J. W. Huston and W. J. Stirling, Rept. Prog. Phys. (2007) 89[arXiv:hep-ph/0611148].[2] B. Mellado, W. Quayle and S. L. Wu, Phys. Rev. D (2007) 093007 [arXiv:0708.2507 [hep-ph]]. Results for p T , jet , cut = LO QCD corrections to pp → WW + jet + X Stefan Dittmaier[3] S. Dittmaier, S. Kallweit and P. Uwer, arXiv:0710.1577 [hep-ph].[4] J. M. Campbell, R. K. Ellis and G. Zanderighi, arXiv:0710.1832 [hep-ph].[5] W. Beenakker et al., Nucl. Phys. B (2003) 151 [arXiv:hep-ph/0211352].[6] S. Dittmaier, P. Uwer and S. Weinzierl, Phys. Rev. Lett. (2007) 262002 [arXiv:hep-ph/0703120].[7] J. Küblbeck, M. Böhm and A. Denner, Comput. Phys. Commun. (1990) 165;H. Eck and J. Küblbeck, Guide to FeynArts 1.0 , University of Würzburg, 1992.[8] S. Dittmaier, Nucl. Phys. B (2003) 447 [arXiv:hep-ph/0308246].[9] A. Denner and S. Dittmaier, Nucl. Phys. B (2003) 175 [hep-ph/0212259].[10] G. Passarino and M. Veltman, Nucl. Phys. B (1979) 151.[11] G. ’t Hooft and M. Veltman, Nucl. Phys. B (1979) 365;W. Beenakker and A. Denner, Nucl. Phys. B (1990) 349;A. Denner, U. Nierste and R. Scharf, Nucl. Phys. B (1991) 637.[12] A. Denner and S. Dittmaier, Nucl. Phys. B (2006) 62 [arXiv:hep-ph/0509141].[13] T. Hahn, Comput. Phys. Commun. (2001) 418 [hep-ph/0012260].[14] T. Hahn and M. Pérez-Victoria, Comput. Phys. Commun. (1999) 153 [hep-ph/9807565];T. Hahn, Nucl. Phys. Proc. Suppl. (2000) 231 [hep-ph/0005029].[15] G. J. van Oldenborgh and J. A. M. Vermaseren, Z. Phys. C (1990) 425;G. J. van Oldenborgh, Comput. Phys. Commun. (1991) 1.[16] Z. Bern, L. J. Dixon and D. A. Kosower, Nucl. Phys. B (1994) 751 [arXiv:hep-ph/9306240].[17] S. Catani and M. H. Seymour, Nucl. Phys. B (1997) 291 [Erratum-ibid. B (1998) 503][arXiv:hep-ph/9605323].[18] S. Dittmaier, Phys. Rev. D (1999) 016007 [arXiv:hep-ph/9805445].[19] F. A. Berends, R. Pittau and R. Kleiss, Nucl. Phys. B (1994) 308 [arXiv:hep-ph/9404313].[20] R. Kleiss and R. Pittau, Comput. Phys. Commun. (1994) 141 [arXiv:hep-ph/9405257].[21] A. Denner et al., Nucl. Phys. B (1999) 33 [arXiv:hep-ph/9904472].[22] W. Kilian, T. Ohl and J. Reuter, arXiv:0708.4233 [hep-ph].[23] T. Gleisberg et al., JHEP (2004) 056 [arXiv:hep-ph/0311263].[24] S. Kallweit, diploma thesis (in German), LMU Munich, 2006.[25] T. Stelzer and W. F. Long, Comput. Phys. Commun. (1994) 357 [arXiv:hep-ph/9401258].[26] J. Pumplin et al., JHEP (2002) 012 [arXiv:hep-ph/0201195];D. Stump et al., JHEP (2003) 046 [arXiv:hep-ph/0303013].[27] S. D. Ellis and D. E. Soper, Phys. Rev. D (1993) 3160 [arXiv:hep-ph/9305266].[28] L. J. Dixon, Z. Kunszt and A. Signer, Phys. Rev. D (1999) 114037 [arXiv:hep-ph/9907305].(1999) 114037 [arXiv:hep-ph/9907305].