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

We present a calculation of the full next-to-leading order QCD corrections to the scattering process pp \to t tbar Z. This channel will be used to measure the t tbar Z electroweak couplings at the Large Hadron Collider. 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 find that for reasonable values of the renormalization and factorization scales the QCD corrections increase the leading order result by 35 percent, while reducing the theoretical error arising from uncalculated higher order corrections to approximately 10 percent.

Probing the properties of the top quark is an urgent priority of the particle physics experimental program. The large mass of the top quark, more than an order of magnitude larger than any other fermion, suggests that it is intimately connected to the mechanism of electroweak symmetry breaking and to the generation of hierarchies in the flavor sector of the Standard Model (SM). Although it was discovered more than a decade ago at the Fermilab Tevatron [1], relatively little is known about the top quark. The production process pp → tt at the Tevatron proceeds almost entirely through gluon exchange, and is sensitive only to the top quark mass and SU (3) C representation. The top quark decays primarily through t → W b, leaving only the helicity of the final state W to be measured.
Details regarding the electroweak properties of the top quark are unknown. This information requires measurement of the ttγ, ttZ, and top quark Yukawa couplings. These couplings probe many new physics effects; they are sensitive to mixing with additional Z ′ gauge bosons and new heavy fermions, and can provide access to extended scalar sectors. Although inaccessible at the Tevatron, these interactions can be studied through the radiative processes pp → ttγ, Z, H at the Large Hadron Collider (LHC). Measuring deviations from SM predictions can provide further information regarding new states discovered at the LHC, or can provide evidence of new physics too heavy to produce directly.
The cross sections for the scattering channels pp → ttγ, Z, H scale as σ LO ∼ α 2 s at leading order in the QCD perturbative expansion, indicating that there is significant theoretical uncertainty in predicting their rates. It is important to quantify the effect of this error on the measurement of the top quark couplings and determine whether a next-to-leading order QCD calculation is needed to improve the theoretical prediction. A procedure for measuring the ttγ, Z couplings was discussed in [2]. This study showed that the uncertainties in the signal cross sections hinder the extraction of anomalous top quark couplings. The theoretical error coming from higher order QCD corrections is the limiting factor in measuring ttZ couplings at the LHC.
In this Letter we present a full calculation of the nextto-leading order (NLO) QCD corrections to the pp → ttZ scattering channel to facilitate accurate measurements of top quark properties at the LHC. Computations of NLO QCD corrections to processes with five or more external particles are notoriously difficult, and require overcoming many technical challenges. Significant community effort has been invested in devising efficient calculational algorithms for one-loop calculations [3], leading to a host of new results of phenomenological relevance for the LHC [4].
Recently, a completely numerical and highly flexible approach to NLO and NNLO calculations was suggested [5,6,7,8]. It applies sector decomposition [9] and contour deformation [10] to Feynman parametric loop integral representations. It avoids many pitfalls present in traditional approaches to calculations of QCD corrections to multi-particle production processes. The calculation of the QCD corrections to pp → ttZ described here is the first time these techniques have been applied to a complete scattering channel with multiple mass scales. This mode contains the full spectrum of difficulties present in the most complex 2 → 3 processes. Several five-point topologies with two, three and four internal massive propagators appear. In this Letter we outline the technical details of our calculation and present phenomenological results relevant for LHC analyses. Preliminary results for the gg initiated partonic channel were presented in [6].
We consider the process p(P 1 ) + p(P 2 ) → ttZ. The factorization theorems for hard scattering processes in QCD allow us to write where f (1,2) i,j (x 1,2 ) are the parton densities that give the probability to find parton i(j) in the proton 1(2) with momentum p i(j) = x 1(2) P 1 (2) . The center-of-mass energy squared of the proton-proton collision is introduced in Eq. (1), S = 2P 1 · P 2 .
At leading order in the α s expansion, both gg and qq initiated partonic processes contribute. The computation of the leading order cross section is straightforward. We use QGRAF [11] to generate the relevant Feynman diagrams and then MAPLE and FORM [12] to manipulate this output. Throughout the paper we set m t = 170.9 GeV, m Z = 91.19 GeV, and m W = 80.45 GeV. For the coupling of the Z boson to quarks, we employ where 2215 is the sine squared of the electroweak mixing angle, T q 3 is the weak isospin of the quark q, Q q is the electric charge of the quark q in units of the proton charge and G F is the Fermi constant. Numerical results for the leading order cross section are presented below.
At next-to-leading order, the qg channel additionally contributes. Several distinct contributions to the NLO cross section can be identified: • the one-loop virtual corrections to the leading order processes gg, qq → ttZ; • the real emission corrections gg → ttZg, qq → ttZg and qg → ttZq; • the renormalization of the leading order cross section.
Both the virtual and real emission corrections are separately divergent. They must be combined to remove divergences arising from soft gluon emission. Ultraviolet divergences are absorbed into the definitions of the coupling constant, the top quark mass, and the top quark wavefunction. Singularities associated with initial state collinear gluon emission are absorbed into the definition of the parton distribution functions. We renormalize the top quark mass and wavefunction on shell. We employ the MS scheme for the parton distribution functions, and for the QCD coupling constant α s for the five light quarks. The ultraviolet divergence coming from the top quark is removed through a zero-momentum subtraction. These renormalizations are performed at certain momentum scales. If computed to all orders in perturbation theory the cross section would be independent of this scale choice; the residual dependence of the NLO result on the scale provides an estimate of the uncalculated higher order corrections. We utilize dimensional regularization to control the singularities of the virtual and radiative corrections at intermediate stages. The radiative corrections are organized using the two-cutoff slicing method [13]. To compute the one-loop virtual corrections we employ the strategy we introduced in [5,6]. For each one-loop diagram interfered with the full Born amplitude, we analytically perform the integration over the loop momentum and arrive at a Feynman parametric representation. While the ultraviolet divergences factorize after the integration over the loop momentum and therefore can be simply extracted, the infrared and collinear singularities appear on the boundaries of Feynman-parameter space. We must therefore extract the infrared and collinear singularities before integrating numerically. To do so we sectordecompose the Feynman-parametric integrals. However, even after these divergences are extracted, it is not possible to perform the Feynman-parametric integrations numerically because there are singularities inside the integration region coming from internal loop thresholds. These singularities are avoided by deforming the integration contour into the complex plane. Once infrared and collinear singularities are extracted and the integration contour is deformed, we obtain representations of the Feynman parametric integrals suitable for numerical integration. No reduction of tensor integrals is performed. This allows us to bypass many of the issues associated when this algebraic reduction is attempted.
We present below phenomenological results of our NLO QCD calculation of pp → ttZ. We employ the Martin-Roberts-Stirling-Thorne parton distribution functions [14] at the appropriate order in the perturbative expansion. All numerical results are obtained using the adaptive Monte Carlo integration algorithm VE-GAS [15] as implemented in the CUBA library [16]. We have checked that our results satisfy several consistency checks. The leading order result obtained with our code matches that obtained using the program MadEvent [17]. Using an eikonal approximation, the divergent parts of the one-loop virtual corrections arising from soft gluon exchange can be calculated in a simple analytic form. We have checked that our full results agree with this form. All divergences cancel once we assemble the separate components of the computation discussed above. The virtual corrections are independent of the size of our contour deformation of the Feynman parametric integrals. Finally, we have implemented all parts of our calculation in several independent codes which agree for all observables studied.
We present in Fig. (1) the inclusive cross section at both leading order and next-to-leading order in the perturbative expansion. We have equated the renormalization and factorization scales to a common value µ = µ R = µ F , and have varied them from µ 0 /8 to 2µ 0 with µ 0 = 2m t +m Z . The dependence of the pp → ttZ rate on this unphysical scale parameter is significantly lessened when the NLO corrections are included. Choosing a scale too different from the typical momenta and energies that give the dominant contributions to the process leads to large logarithms that spoil the convergence of the perturbative expansion. Therefore, to estimate a central value and theoretical error for the cross section, we should a pick a scale roughly equal to the typical transverse momenta 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 [µ 0 /4, µ 0 ] a reasonable range of scale variation with µ = µ 0 /2 a good central value. This yields a cross section of 1.09 pb with a theoretical error of ±11% at NLO. The result at LO is 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 at NLO to that at LO, is K inc = 1.35 for µ = µ 0 /2. The variation of K inc with scale is also shown in Fig. (1); it changes from 1.1 to 1.6 as µ varies from µ 0 /4 to µ 0 . Also included in Fig. (1) are the separate contributions of the gg, qg, and qq partonic processes at NLO. The significant scale dependence of the qg component, which first appears at this order in the perturbative expansion, is noteworthy. In addition to the inclusive cross section, the impact of higher order corrections on differential distributions must be studied. An interesting question to consider is whether their effect is completely described by the inclusive K inc -factor. If so, the NLO corrections can be accurately and simply included in leading order simulation codes by an overall reweighting of event rates. To investigate this question we present in Fig. (2) the binintegrated transverse momentum spectrum of the Z boson at both LO and NLO for the scale choice µ = µ 0 /2. Most of the cross section comes from events with p Z T less than 200 GeV. Included in this plot is the ratio of the NLO p Z T distribution over the LO spectrum, K pT . It is flat to within a few percent over the entire range, and is equal to the inclusive value K inc = 1.35. The small impact of higher order corrections on the p Z T distribution can be roughly understood by noting that at tree level, pp → ttZ is already a three-body process. Including additional partonic radiation does not open up new regions of phase space as the Z boson can already recoil against the tt pair. This intuitive argument leads us to expect that the shape of many other kinematic distributions will also be approximately unchanged by NLO corrections.
FIG. 2: Transverse momentum spectrum for pp → ttZ for the scale choice µ = µ0/2 = mt + mZ/2. Included are the LO and NLO results. The pT dependent K-factor for each bin, Kp T , is also shown; the value for this should be read from the axis on the right of the plot.
We can use these results to estimate the improvement in the measurement of ttZ couplings at the LHC after NLO corrections are included. Assuming approximate CP conservation, four relevant ttZ couplings exist: dimension-four vector and axial couplings, and two dimension-five dipole couplings. Although the measurements of these parameters are correlated, the analysis of Ref. [2] indicates that the dipole couplings are expected to be measured with a precision of ±50% at the LHC and ±25% with the super-LHC luminosity upgrade, while the axial coupling should be measured with ±15% precision at the LHC and with ±5% at the super-LHC. This study also found that the uncertainty arises almost entirely from the signal normalization and statistics; the backgrounds are negligible. This analysis utilized a scale choice µ = m t , for which we find K inc ≈ 1.3. This yields a small improvement of the relative statistical error. The theoretical uncertainty assumed in this analysis of [2] was ±30%. The authors also studied the expected improvement possible if higher order corrections reduced this error to ±10%, and concluded that improvements in the precisions quoted above could reach a factor of two at the LHC. A conservative estimate of the remaining theory uncertainty from our prediction for pp → ttZ is ±15%. This accounts for imprecise knowledge of parton distribution functions. While the full factor-of-two improvement in the precision of the ttZ couplings seems slightly out of reach with our current knowledge of the higher order corrections, we still expect a significant improvment once this reduced theoretical error is propagated through the full analysis. Presumably the improvement is more significant with the super-LHC luminosity upgrade, since the relative importance of the statistical errors should be decreased.
In summary, we have presented the complete calculation of the next-to-leading order QCD corrections to the pp → ttZ scattering process at the LHC. Study of this channel allows measurements of the ttZ electroweak couplings, which cannot be directly accessed with current experiments. These couplings probe many forms of physics beyond the Standard Model, such as small mixings of extra heavy gauge bosons or vector-like fermions with the Standard Model top quark and Z boson. To compute these corrections we present a novel approach to perturbative calculations in QCD powerful enough to handle the significant complexities that occur when 2 → 3 and more complicated scattering processes are studied. This method is highly automated and flexible, and is based on a completely numerical algorithm for computing loop integrals. This is the first time it has been applied to a scattering process that contains the full set of complexities that occur in multi-leg one-loop calculations. We find that the NLO QCD corrections increase the leading order pp → ttZ cross section by a factor of 1.35 for canonical choices of the factorization and renormalization scales. We estimate the remaining theoretical uncertainty from uncalculated higher order corrections to be ±11%. Previous studies based on leading order cross sections have found that the normalization uncertainty of the pp → ttZ rate is the dominant error in extracting ttZ couplings at the LHC. We estimate that the relative precision of the ttZ couplings will be improved by a factor of 1.5-2 once our results are incorporated into these analyses. We find that the NLO corrections do not significantly change the shape of the kinematic distributions we have studied, indicating that they can be accounted for by an overall scaling of the LO result.
We are excited by the potential of our approach to NLO QCD calculations presented here. We anticipate and look forward to using it to understand other processes of interest at future collider experiments.
The work of A. L. and K. M. was supported by the US Department of Energy under contract DE-FG03-94ER-40833. The work of T. M. and F. P. was supported by the DOE grant DE-FG02-95ER40896, Outstanding Junior Investigator Award, by the University of Wisconsin Research Committee with funds provided by the Wisconsin Alumni Research Foundation, and by the Alfred P. Sloan Foundation.