Physics Letters B 666 (2008) 62–65

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Physics Letters B

www.elsevier.com/locate/physletb

Next-to-leading order QCD corrections to tt̄ Z production at the LHC

Achilleas Lazopoulos a, Thomas McElmurry b, Kirill Melnikov a, Frank Petriello b,∗a Department of Physics and Astronomy, University of Hawaii, 2505 Correa Rd., Honolulu, HI 96822, USAb Department of Physics, University of Wisconsin, Madison, WI 53706, USA

a r t i c l e i n f o a b s t r a c t

Article history:Received 7 June 2008Received in revised form 24 June 2008Accepted 27 June 2008Available online 9 July 2008Editor: G.F. Giudice

PACS:12.38.Bx14.65.Ha

We present a calculation of the full next-to-leading order QCD corrections to the scattering processpp → tt̄ Z . This channel will be used to measure the tt̄ Z electroweak couplings at the Large HadronCollider. These couplings cannot be directly measured in current experiments. To obtain these results,we utilize a novel, completely numerical approach to performing higher order QCD calculations. We findthat for reasonable values of the renormalization and factorization scales the QCD corrections increasethe leading order result by 35%, while reducing the theoretical error arising from uncalculated higherorder corrections to approximately ±10%.

© 2008 Elsevier B.V. All rights reserved.

Probing the properties of the top quark is an urgent prior-ity of the particle physics experimental program. The large massof the top quark, more than an order of magnitude larger thanany other fermion, suggests that it is intimately connected to themechanism of electroweak symmetry breaking and to the genera-tion of hierarchies in the flavor sector of the Standard Model (SM).Although it was discovered more than a decade ago at the Fer-milab Tevatron [1], relatively little is known about the top quark.The production process pp̄ → tt̄ at the Tevatron proceeds almostentirely through gluon exchange, and is sensitive only to the topquark mass and SU(3)C representation. The top quark decays pri-marily through t → W b, leaving only the helicity of the final stateW to be measured.

Details regarding the electroweak properties of the top quarkare unknown. This information requires measurement of the tt̄γ ,tt̄ Z , and top quark Yukawa couplings. These couplings probe manynew physics effects: they are sensitive to mixing with additionalZ ′ gauge bosons and new heavy fermions, and can provide accessto extended scalar sectors. Although inaccessible at the Tevatron,these interactions can be studied through the radiative processespp → tt̄γ , Z , H at the Large Hadron Collider (LHC). Measuringdeviations from SM predictions can provide further information re-garding new states discovered at the LHC, or can provide evidenceof new physics too heavy to produce directly.

The cross sections for the scattering channels pp → tt̄γ , Z , Hscale as σLO ∼ α2

s at leading order in the QCD perturbative expan-sion, indicating that there is significant theoretical uncertainty inpredicting their rates. It is important to quantify the effect of this

* Corresponding author.E-mail address: frankjp@physics.wisc.edu (F. Petriello).

0370-2693/$ – see front matter © 2008 Elsevier B.V. All rights reserved.doi:10.1016/j.physletb.2008.06.073

error on the measurement of the top quark couplings and deter-mine whether a next-to-leading order QCD calculation is neededto improve the theoretical prediction. A procedure for measuringthe tt̄γ , Z couplings was discussed in [2]. This study showed thatthe uncertainties in the signal cross sections hinder the extractionof anomalous top quark couplings. The theoretical error comingfrom higher order QCD corrections is the limiting factor in mea-suring tt̄ Z couplings at the LHC.

In this Letter we present a full calculation of the next-to-leadingorder (NLO) QCD corrections to the pp → tt̄ Z scattering chan-nel to facilitate accurate measurements of top quark properties atthe LHC. Computations of NLO QCD corrections to processes withfive or more external particles are notoriously difficult, and requireovercoming many technical challenges. Significant community ef-fort has been invested in devising efficient calculational algorithmsfor one-loop calculations [3], leading to a host of new results ofphenomenological relevance for the LHC [4]. In recent work wehave pioneered a completely numerical and highly flexible ap-proach to performing NLO QCD calculations [5,6]. Our methodapplies sector decomposition [7] and contour deformation [8] toFeynman parametric loop integral representations. It avoids manypitfalls present in traditional approaches to calculations of QCDcorrections to multi-particle production processes. The calculationof the QCD corrections to pp → tt̄ Z described here is the first timeour techniques have been applied to a complete scattering channelwith multiple mass scales. This mode contains the full spectrumof difficulties present in the most complex 2 → 3 processes. Sev-eral five-point topologies with two, three and four internal massivepropagators appear. In this Letter we outline the technical detailsof our calculation and present phenomenological results relevantfor LHC analyses. Preliminary results for the gg initiated partonicchannel were presented in [6].

A. Lazopoulos et al. / Physics Letters B 666 (2008) 62–65 63

We consider the process p(P1)+ p(P2) → tt̄ Z . The factorizationtheorems for hard scattering processes in QCD allow us to write

dσ =∑

i j

1∫0

dx1 dx2 f 1i (x1) f 2

j (x2)dσi j→tt̄ Z (x1x2 S), (1)

where f (1,2)i, j (x1,2) are the parton densities that give the prob-

ability to find parton i ( j) in the proton 1 (2) with momen-tum pi( j) = x1(2) P1(2) . The center-of-mass energy squared of theproton–proton collision is introduced in Eq. (1), S = 2P1 · P2. Atleading order in the αs expansion, both gg and qq̄ initiated par-tonic processes contribute. The computation of the leading ordercross section is straightforward. We use QGRAF [9] to generate therelevant Feynman diagrams and then MAPLE and FORM [10] to ma-nipulate this output. Throughout the Letter we set mt = 170.9 GeV,mZ = 91.19 GeV, and mW = 80.45 GeV. For the coupling of the Zboson to quarks, we employ

Zqq: i

√8m2

W G F√2 cos2 θW

(gq

v + gqaγ5

), (2)

where gqv = T q

32 − Q q sin2 θW , gq

a = − T q3

2 , sin2 θW = 1 − m2W /m2

Z =0.2215 is the sine squared of the electroweak mixing angle, T q

3 isthe weak isospin of the quark q, Q q is the electric charge of thequark q in units of the proton charge and G F is the Fermi constant.Numerical results for the leading order cross section are presentedbelow.

At next-to-leading order, the qg channel additionally con-tributes. Several distinct contributions to the NLO cross sectioncan be identified:

• the one-loop virtual corrections to the leading order processesgg,qq̄ → tt̄ Z ;

• the real emission corrections gg → tt̄ Z g , qq̄ → tt̄ Z g and qg →tt̄ Zq;

• the renormalization of the leading order cross section.

Both the virtual and real emission corrections are separately di-vergent. They must be combined to remove divergences arisingfrom soft gluon emission. Ultraviolet divergences are absorbed intothe definitions of the coupling constant, the top quark mass, andthe top quark wavefunction. Singularities associated with initialstate collinear gluon emission are absorbed into the definition ofthe parton distribution functions. We renormalize the top quarkmass and wavefunction on shell. We employ the MS scheme forthe parton distribution functions, and for the QCD coupling con-stant αs for the five light quarks. The ultraviolet divergence comingfrom the top quark is removed through a zero-momentum subtrac-tion. These renormalizations are performed at certain momentumscales. If computed to all orders in perturbation theory the crosssection would be independent of this scale choice; the residual de-pendence of the NLO result on the scale provides an estimate ofthe uncalculated higher order corrections.

We utilize dimensional regularization to control the singulari-ties of the virtual and radiative corrections at intermediate stages.The radiative corrections are organized using the two-cutoff slic-ing method [11]. To compute the one-loop virtual correctionswe employ the strategy we introduced in [5,6]. For each one-loop diagram interfered with the full Born amplitude, we ana-lytically perform the integration over the loop momentum andarrive at a Feynman parametric representation. While the ultra-violet divergences factorize after the integration over the loopmomentum and therefore can be simply extracted, the infraredand collinear singularities appear on the boundaries of Feynman-parameter space. We must therefore extract the infrared and

collinear singularities before integrating numerically. To do sowe sector-decompose the Feynman-parametric integrals. However,even after these divergences are extracted, it is not possible toperform the Feynman-parametric integrations numerically becausethere are singularities inside the integration region coming frominternal loop thresholds. These singularities are avoided by de-forming the integration contour into the complex plane. Once in-frared and collinear singularities are extracted and the integrationcontour is deformed, we obtain representations of the Feynmanparametric integrals suitable for numerical integration. No reduc-tion of tensor integrals is performed. This allows us to bypassmany of the issues associated when this algebraic reduction is at-tempted.

We present below phenomenological results of our NLO QCDcalculation of pp → tt̄ Z . We employ the Martin–Roberts–Stirling–Thorne parton distribution functions [12] at the appropriate orderin the perturbative expansion. All numerical results are obtainedusing the adaptive Monte Carlo integration algorithm VEGAS [13]as implemented in the CUBA library [14]. We have checked thatour results satisfy several consistency checks. The leading orderresult obtained with our code matches that obtained using theprogram MadEvent [15]. Using an eikonal approximation, the di-vergent parts of the one-loop virtual corrections arising from softgluon exchange can be calculated in a simple analytic form. Wehave checked that our full results agree with this form. All di-vergences cancel once we assemble the separate components ofthe computation discussed above. We note that the divergencestructure of this process is identical to that of pp → tt̄ H ; detailsregarding the ultraviolet counterterm structure and poles arisingfrom soft and collinear radiation can be found in Ref. [16]. Thevirtual corrections are independent of the size of our contour de-formation of the Feynman parametric integrals. Finally, we haveimplemented all parts of our calculation in several independentcodes which agree for all observables studied. We have generatedseveral tens of thousands of phase-space points with 2 → 3 kine-matics and several million points with 2 → 4 kinematics to obtainour numerical results. This required roughly one week of runningon a farm of approximately 100 3.0GHz processors.

We present in Fig. 1 the inclusive cross section at both lead-ing order and next-to-leading order in the perturbative expansion.We have equated the renormalization and factorization scales to acommon value μ = μR = μF , and have varied them from μ0/8 to2μ0 with μ0 = 2mt + mZ . The dependence of the pp → tt̄ Z rateon this unphysical scale parameter is significantly lessened whenthe NLO corrections are included. Choosing a scale too differentfrom the typical momenta and energies that give the dominantcontributions to the process leads to large logarithms that spoilthe convergence of the perturbative expansion. Therefore, to esti-mate a central value and theoretical error for the cross section, weshould a pick a scale roughly equal to the typical transverse mo-menta and masses in the final state and vary μ around this value.As these momenta and masses are approximately 100–200 GeV,we consider μ in [μ0/4,μ0] a reasonable range of scale variationwith μ = μ0/2 a good central value. This yields a cross section of1.09 pb with a theoretical error of ±11% at NLO. The result at LOis 0.808 pb with an uncertainty of ±25–35%. The inclusive K inc-factor for this process, defined as the ratio of the cross section atNLO to that at LO, is K inc = 1.35 for μ = μ0/2. The variation ofK inc with scale is also shown in Fig. 1; it changes from 1.1 to 1.6as μ varies from μ0/4 to μ0. Also included in Fig. 1 are the sepa-rate contributions of the gg , qg , and qq̄ partonic processes at NLO.The significant scale dependence of the qg component, which firstappears at this order in the perturbative expansion, is noteworthy.To facilitate comparison with our results, we present in Table 1 theresults separated into the various partonic channels at both LO andNLO.

64 A. Lazopoulos et al. / Physics Letters B 666 (2008) 62–65

Fig. 1. Inclusive cross section for pp → tt̄ Z as a function of the scale choice μ.Included are the LO and NLO results, as well as the contributions of the gg , qq̄,and qg partonic channels at NLO. The dotted line shows the inclusive K -factor; thevalue for this should be read from the axis on the right of the plot.

Table 1Contribution of the various components of each partonic channel to the inclusivecross section for various choices of μ using the renormalization scheme describedin the text. LO denotes the leading order results, while NLO denotes the combina-tion of the virtual, real radiation, and counterterm contributions at next-to-leadingorder. All results are in femtobarns

σx (fb) μ = μ0/4 μ = μ0/2 μ = μ0

ggLO 738 537 400ggNLO 740 739 695qq̄LO 340 271 220qq̄NLO 267 283 278qgNLO 206 70.6 0.2

In addition to the inclusive cross section, the impact of higherorder corrections on differential distributions must be studied. Aninteresting question to consider is whether their effect is com-pletely described by the inclusive K inc-factor. If so, the NLO cor-rections can be accurately and simply included in leading ordersimulation codes by an overall reweighting of event rates. To inves-tigate this question we present in Fig. 2 the bin-integrated trans-verse momentum spectrum of the Z boson at both LO and NLO forthe scale choice μ = μ0/2. Most of the cross section comes fromevents with p Z

T less than 200 GeV. Included in this plot is the ra-tio of the NLO p Z

T distribution over the LO spectrum, K pT . It isflat to within a few percent over the entire range, and is equal tothe inclusive value K inc = 1.35. The small impact of higher ordercorrections on the p Z

T distribution can be roughly understood bynoting that at tree level, pp → tt̄ Z is already a three-body process.Including additional partonic radiation does not open up new re-gions of phase space as the Z boson can already recoil against thett̄ pair. This intuitive argument leads us to expect that the shapeof many other kinematic distributions will also be approximatelyunchanged by NLO corrections.

We can use these results to estimate the improvement in themeasurement of tt̄ Z couplings at the LHC after NLO corrections areincluded. Assuming approximate C P conservation, four relevanttt̄ Z couplings exist: dimension-four vector and axial couplings, andtwo dimension-five dipole couplings. Although the measurementsof these parameters are correlated, the analysis of Ref. [2] indi-cates that the dipole couplings are expected to be measured with aprecision of ±50% at the LHC and ±25% with the super-LHC lumi-nosity upgrade, while the axial coupling should be measured with±15% precision at the LHC and with ±5% at the super-LHC. Thisstudy also found that the uncertainty arises almost entirely fromthe signal normalization and statistics; the backgrounds are neg-

Fig. 2. Transverse momentum spectrum for pp → tt̄ Z for the scale choiceμ = μ0/2 = mt + mZ /2. Included are the LO and NLO results. The pT dependentK -factor for each bin, K pT , is also shown; the value for this should be read fromthe axis on the right of the plot.

ligible. This analysis utilized a scale choice μ = mt , for which wefind K inc ≈ 1.3. This yields a small improvement of the relative sta-tistical error. The theoretical uncertainty assumed in this analysisof [2] was ±30%. The authors also studied the expected improve-ment possible if higher order corrections reduced this error to±10%, and concluded that improvements in the precisions quotedabove could reach a factor of two at the LHC. A conservative es-timate of the remaining theory uncertainty from our predictionfor pp → tt̄ Z is ±15%. This accounts for imprecise knowledge ofparton distribution functions. While the full factor-of-two improve-ment in the precision of the tt̄ Z couplings seems slightly out ofreach with our current knowledge of the higher order corrections,we still expect a significant improvement once this reduced the-oretical error is propagated through the full analysis. Presumablythe improvement is more significant with the super-LHC luminos-ity upgrade, since the relative importance of the statistical errorsshould be decreased.

In summary, we have presented the complete calculation ofthe next-to-leading order QCD corrections to the pp → tt̄ Z scat-tering process at the LHC. Study of this channel allows measure-ments of the tt̄ Z electroweak couplings, which cannot be directlyaccessed with current experiments. These couplings probe manyforms of physics beyond the Standard Model, such as small mix-ings of extra heavy gauge bosons or vector-like fermions with theStandard Model top quark and Z boson. To compute these correc-tions we present a novel approach to perturbative calculations inQCD powerful enough to handle the significant complexities thatoccur when 2 → 3 and more complicated scattering processes arestudied. This method is highly automated and flexible, and is basedon a completely numerical algorithm for computing loop integrals.This is the first time it has been applied to a scattering processthat contains the full set of complexities that occur in multi-legone-loop calculations. We find that the NLO QCD corrections in-crease the leading order pp → tt̄ Z cross section by a factor of1.35 for canonical choices of the factorization and renormalizationscales. We estimate the remaining theoretical uncertainty from un-calculated higher order corrections to be ±11%. Previous studiesbased on leading order cross sections have found that the normal-ization uncertainty of the pp → tt̄ Z rate is the dominant error inextracting tt̄ Z couplings at the LHC. We estimate that the relativeprecision of the tt̄ Z couplings will be improved by a factor of 1.5–2once our results are incorporated into these analyses.

We are excited by the potential of our approach to NLO QCDcalculations presented here. We anticipate and look forward to us-

A. Lazopoulos et al. / Physics Letters B 666 (2008) 62–65 65

ing it to understand other processes of interest at future colliderexperiments.

Acknowledgements

The work of A.L. and K.M. was supported by the US Depart-ment of Energy under contract DE-FG03-94ER-40833. The work ofT.M. and F.P. was supported by the DOE grant DE-FG02-95ER40896,Outstanding Junior Investigator Award, by the University of Wis-consin Research Committee with funds provided by the WisconsinAlumni Research Foundation, and by the Alfred P. Sloan Founda-tion.

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