NLO QCD corrections to pp -> top anti-top + jet + X
aa r X i v : . [ h e p - ph ] A p r NLO QCD corrections to pp → t¯t + jet + X Stefan Dittmaier a , Peter Uwer ∗ b , Stefan Weinzierl c † a Max-Planck-Institut für Physik (Werner-Heisenberg-Institut), D-80805 München, Germany b Institut fur Theoretische Teilchenphysik, Universitat Karlsruhe, 76128 Karlsruhe, Germany c Universität Mainz, D-55099 Mainz, GermanyE-mail: [email protected] , [email protected] , [email protected] We discuss the production of top–anti-top quark pairs in association with a hard jet at the Tevatronand at the LHC and we report on the calculation of the next-to-leading order QCD corrections tothis process. Numerical results for the t¯t+jet cross section and the forward–backward chargeasymmetry are presented. The corrections stabilize the leading-order prediction for the crosssection. In contrast, the charge asymmetry receives large corrections. The dependence of thecross section as well as the asymmetry on the minimum transverse momenta used to define theadditional jet is studied in detail for the Tevatron. ∗ Speaker. † P.U. is supported as Heisenberg Fellow of the Deutsche Forschungsgemeinschaft DFG. This work is supportedin part by the European Community’s Marie-Curie Research Training Network HEPTOOLS under contract MRTN-CT-2006-035505 and by the DFG Sonderforschungsbereich/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 → t ¯ t + jet + X Peter Uwer
1. Introduction
The top quark is the heaviest of the known elementary particles. More than ten years afterits discovery, the dynamics and many properties of the top quark, such as its electroweak quantumnumbers, are not yet precisely measured. It is widely believed that the top quark plays a key rolein extensions of the Standard Model. This renders experimental investigations of the top quarkparticularly important. Up to now the main (direct) source of information on top quarks are top-quark pairs produced at the Tevatron. Only recently first evidence for single-top production hasbeen found [1]. It is important to note that in the inclusive t¯t sample a significant fraction comprisest¯t+jet events. An investigation of the process of t¯t production in association with a hard jet can thusimprove our knowledge about the top quark.In this context, the forward–backward charge asymmetry of the top (or anti-top) quark [2,3, 4, 5] is of particular interest. In inclusive t¯t production it appears first at one loop, because itresults from interferences of C-odd with C-even parts of double-gluon exchange between initialand final states. This means that the available prediction for t¯t production—although of one-looporder—describes this asymmetry only at leading-order (LO) accuracy. In t¯t + jet production theasymmetry appears already in LO. Thus, the next-to-leading order (NLO) calculation described inthe following provides a true NLO prediction for the asymmetry. Our calculation will, therefore,be an important tool in the experimental analysis of this observable at the Tevatron where theasymmetry is measureable as discussed in Ref. [5].Measuring the cross section of the related process of t¯t + g production provides direct accessto the electric charge of the top quark. Obviously NLO QCD predictions to this process are im-portant for a reliable analysis. They can be obtained from t¯t + jet production presented here viasimple substitutions. Finally, a signature of t¯t in association with a hard jet represents an impor-tant background process for searches at the LHC, such as the search for the Higgs boson in theweak-vector-boson fusion or t¯tH channels.The above-mentioned issues clearly underline the case for an NLO calculation for t¯t + jet pro-duction at hadron colliders. We report here on a first calculation of this kind as presented in Ref. [6].
2. Details of the NLO calculation
At LO, hadronic t¯t + jet production receives contributions from the partonic processes q ¯ q → t¯t g , qg → t¯t q , ¯ qg → t¯t ¯ q , and gg → t¯t g . The first three channels are related by crossing symmetry to theamplitude 0 → t¯t q ¯ qg . Evaluating 2 → 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-tions. The most complicated diagrams are the pentagon diagrams.2
LO QCD corrections to pp → t ¯ t + jet + X Peter Uwer
Version 1 of the virtual corrections is essentially obtained following the method described inRef. [7], where t¯tH production at hadron colliders was considered. Feynman diagrams and ampli-tudes have been generated with the FeynArts package [8, 9] and further processed with in-houseMathematica routines, which automatically create an output in Fortran. The IR (soft and collinear)singularities are analytically separated from the finite remainder as described in Refs. [7, 10]. Thetensor integrals appearing in the pentagon diagrams are directly reduced to box integrals followingRef. [11]. This method does not introduce inverse Gram determinants in this step, thereby avoid-ing notorious numerical instabilities in regions where these determinants become small. Box andlower-point integrals are reduced à la Passarino–Veltman [12] to scalar integrals, which are eithercalculated analytically or using the results of Refs. [13, 14, 15]. Sufficient numerical stability isalready achieved in this way. Nevertheless the integral evaluation is currently further refined byemploying the more sophisticated methods described in Ref. [16] in order to numerically stabilizethe tensor integrals in exceptional phase-space regions.
Version 2 of the evaluation of loop diagrams starts with the generation of diagrams and ampli-tudes via QGRAF[17], which are then further manipulated with Form[18] and eventually automat-ically translated into C++ code. The reduction of the the 5-point tensor integrals to scalar integralsis performed with an extension of the method described in Ref. [19]. In this procedure also in-verse Gram determinents of four four-momenta are avoided. The lower-point tensor integrals arereduced using an independent implementation of the Passarino–Veltman procedure. The IR-finitescalar integrals are evaluated using the FF package [20, 21].
The matrix elements for the real corrections are given by 0 → t¯t gggg , 0 → t¯t q ¯ qgg , 0 → t¯t q ¯ qq ′ ¯ q ′ and 0 → t¯t q ¯ qq ¯ q . The various partonic processes are obtained from these matrix elements by allpossible crossings of light particles into the initial state.The evaluation of the real-emission amplitudes is performed in two independent ways. Bothevaluations employ the dipole subtraction formalism [22, 23, 24] for the extraction of IR singular-ities and for their combination with the virtual corrections. Version 1 results from a fully automated calculation based on helicity amplitudes, as described inRef. [25]. Individual helicity amplitudes are computed with the help of Berends–Giele recurrencerelations [26]. The evaluation of color factors and the generation of subtraction terms is automated.For the channel gg → t¯t gg a dedicated soft-insertion routine [27] is used for the generation of thephase space. Version 2 uses for the LO 2 → gg → t¯t gg process optimized code obtainedfrom a Feynman diagramatic approach. As in version 1 standard techniques like color decompo-sition and the use of helicity amplitudes are employed. For the 2 →
3. Numerical results
In the following we consistently use the CTEQ6 [29, 30] set of parton distribution functions(PDFs). In detail, we take CTEQ6L1 PDFs with a 1-loop running a s in LO and CTEQ6M PDFs3 LO QCD corrections to pp → t ¯ t + jet + X Peter Uwer
LO (CTEQ6L1)NLO (CTEQ6M) p T , jet > √ s = 1 .
96 TeVp¯p → t¯t+jet+X µ/m t σ [pb] . LO (CTEQ6L1)NLO (CTEQ6M) p T , jet > √ s = 14 TeVpp → t¯t+jet+X µ/m t σ [pb] . Figure 1:
Scale dependence of the LO and NLO cross sections for t¯t + jet production at the Tevatron (left)and at the LHC (right) as taken from Ref. [6], where the renormalization scale ( m r ) and the factorizationscale ( m f ) are set equal to m . LO (CTEQ6L1)NLO (CTEQ6M) p T , jet > √ s = 1 .
96 TeVp¯p → t¯t+jet+X µ/m t A tFB . . . − . − . − . − . − . − . Figure 2:
Scale dependence of the LO and NLO forward–backward charge asymmetry of the top quark inp¯p → t¯t + jet + X at the Tevatron as taken from Ref. [6] with m = m f = m r . with a 2-loop running a s in NLO. The number of active flavours is N F =
5, and the respective QCDparameters are L LO5 =
165 MeV and L MS5 =
226 MeV. Note that the top-quark loop in the gluonself-energy is subtracted at zero momentum. In this scheme the running of a s is generated solelyby the contributions of the light quark and gluon loops. The top-quark mass is renormalized in theon-shell scheme, as numerical value we take m t =
174 GeV.We apply the jet algorithm of Ref. [31] with R = p T , jet > p cutT =
20 GeV for the hardestjet. The outgoing (anti-)top quarks are neither affected by the jet algorithm nor by the phase-spacecut. Note that the LO prediction and the virtual corrections are not influenced by the jet algorithm,but the real corrections are.In Figure 1 the scale dependence of the NLO cross sections is shown. For comparison, theLO results are included as well. As expected, the NLO corrections significantly reduce the scaledependence compared to LO. We observe that arround m ≈ m t the NLO corrections are of moderatesize for the chosen setup.We have also studied the forward–backward charge asymmetry of the top quark at the Teva-tron. In LO the asymmetry is defined by 4 LO QCD corrections to pp → t ¯ t + jet + X Peter Uwer A tFB , LO = s − LO s + LO , s ± LO = s LO ( y t > ) ± s LO ( y t < ) , (3.1)where y t denotes the rapidity of the top quark. Cross-section contributions s ( y t >< ) correspond totop quarks in the forward or backward hemispheres, respectively, where incoming protons fly intothe forward direction by definition. Denoting the corresponding NLO contributions to the crosssections by ds ± NLO , we define the asymmetry at NLO by A tFB , NLO = s − LO s + LO (cid:18) + ds − NLO s − LO − ds + NLO s + LO (cid:19) , (3.2)i.e. via a consistent expansion in a s . Note, however, that the LO cross sections in Eq. (3.2) areevaluated in the NLO setup (PDFs, a s ).Figure 2 shows the scale dependence of the asymmetry at LO and NLO. The LO result for theasymmetry is of order a s and does therefore not depend on the renormalization scale. The plotfor the LO result shows a mild residual dependence on the factorization scale, but the size of thisvariation does not reflect the theoretical uncertainty, which is much larger. The NLO correctionsto the asymmetry are of order a s and depend on the renormalization scale. It is therefore naturalto expect a stronger scale dependence of the asymmetry at NLO than at LO, as seen in the plot.The size of the asymmetry, which is about −
7% at LO, is drastically reduced by the NLO correc-tions. To investigate the origin of the large NLO corrections to the asymmetry we have studied thedependence on the cut value p cutT used to define the minimal p T of the additional jet. The resultsare shown in Table 1. We observe that both the NLO cross section as well as the NLO asymmetrydependent strongly on p cutT . This is related to the fact that the cross section becomes ill-defined inthe limit p cutT → p cutT .cross section [pb] charge asymmetry [%] p cutT [GeV] LO NLO LO NLO20 1.583(2) + . − . + . − . − . ( ) + . − . − . ( ) + . − .
30 0.984(1) + . − . + . − . − . ( ) + . − . − . ( ) + . − .
40 0.6632(8) + . − . + . − . − . ( ) + . − . − . ( ) + . − .
50 0.4670(6) + . − . + . − . − . ( ) + . − . − . ( ) + . − . Table 1:
Cross section and forward-backward charge asymmetry at the Tevatron for different values of p cutT used to define the minimal transverse momentum p T of the additional jet ( m = m f = m r = m t ). The upperand lower indices are the shifts towards m = m t / m = m t .
4. Conclusions
Predictions for t¯t + jet production at hadron colliders have been reviewed at NLO QCD. For thecross section the NLO corrections drastically reduce the scale dependence of the LO predictions,which is of the order of 100%. The charge asymmtry of the top quarks, which is going to be mea-sured at the Tevatron, is significantly decreased at NLO and is almost washed out by the residual5 LO QCD corrections to pp → t ¯ t + jet + X Peter Uwer scale dependence. In addition we have also studied the p cutT -dependence of the NLO predictions.Further refinements of the precise definition of the forward-backward asymmetry are required tostabilize the asymmetry with respect to higher order corrections. References [1] D0, V.M. Abazov et al., (2006), hep-ex/0612052 ,[2] F. Halzen, P. Hoyer and C.S. Kim, Phys. Lett. B195 (1987) 74,[3] J.H. Kühn and G. Rodrigo, Phys. Rev. D59 (1999) 054017, hep-ph/9807420 ,[4] J.H. Kühn and G. Rodrigo, Phys. Rev. Lett. 81 (1998) 49, hep-ph/9802268 ,[5] M.T. Bowen, S.D. Ellis and D. Rainwater, Phys. Rev. D73 (2006) 014008, hep-ph/0509267 ,[6] S. Dittmaier, P. Uwer and S. Weinzierl, Phys. Rev. Lett. 98 (2007) 262002, hep-ph/0703120 ,[7] W. Beenakker et al., Nucl. Phys. B653 (2003) 151, hep-ph/0211352 ,[8] J. Küblbeck, M. Böhm and A. Denner, Comput. Phys. Commun. 60 (1990) 165,[9] T. Hahn, Comput. Phys. Commun. 140 (2001) 418, hep-ph/0012260 ,[10] S. Dittmaier, Nucl. Phys. B675 (2003) 447, hep-ph/0308246 ,[11] A. Denner and S. Dittmaier, Nucl. Phys. B658 (2003) 175, hep-ph/0212259 ,[12] G. Passarino and M.J.G. Veltman, Nucl. Phys. B160 (1979) 151,[13] G. ’t Hooft and M.J.G. Veltman, Nucl. Phys. B153 (1979) 365,[14] W. Beenakker and A. Denner, Nucl. Phys. B338 (1990) 349,[15] A. Denner, U. Nierste and R. Scharf, Nucl. Phys. B367 (1991) 637,[16] A. Denner and S. Dittmaier, Nucl. Phys. B734 (2006) 62, hep-ph/0509141 ,[17] P. Nogueira, J. Comput. Phys. 105 (1993) 279,[18] J.A.M. Vermaseren, (2000), math-ph/0010025 ,[19] W.T. Giele and E.W.N. Glover, JHEP 04 (2004) 029, hep-ph/0402152 ,[20] G.J. van Oldenborgh and J.A.M. Vermaseren, Z. Phys. C46 (1990) 425,[21] G.J. van Oldenborgh, Comput. Phys. Commun. 66 (1991) 1,[22] S. Catani and M.H. Seymour, Nucl. Phys. B485 (1997) 291, hep-ph/9605323 ,[23] L. Phaf and S. Weinzierl, JHEP 04 (2001) 006, hep-ph/0102207 ,[24] S. Catani et al., Nucl. Phys. B627 (2002) 189, hep-ph/0201036 ,[25] S. Weinzierl, Eur. Phys. J. C45 (2006) 745, hep-ph/0510157 ,[26] F.A. Berends and W.T. Giele, Nucl. Phys. B306 (1988) 759,[27] S. Weinzierl and D.A. Kosower, Phys. Rev. D60 (1999) 054028, hep-ph/9901277 ,[28] T. Stelzer and W.F. Long, Comput. Phys. Commun. 81 (1994) 357, hep-ph/9401258 ,[29] J. Pumplin et al., JHEP 07 (2002) 012, hep-ph/0201195 ,[30] D. Stump et al., JHEP 10 (2003) 046, hep-ph/0303013 ,[31] S.D. Ellis and D.E. Soper, Phys. Rev. D48 (1993) 3160, hep-ph/9305266 ,,