DESY 08–097 TTP 08-29 SFB/CPP-08-49 July 2008 Heavy-quark pair production at two loops in QCDPresented by S.M. at Loops and Legs in Quantum Field Theory, 20–25 April 2008, Sondershausen (Germany).

Desy 08–097
Ttp 08-29
Sfb/cpp-08-49
July 2008
Heavy-quark pair production at two loops in QCDthanks: Presented by S.M. at Loops and Legs in Quantum Field Theory, 20–25 April 2008, Sondershausen (Germany).

S. Moch\addressDESY, Platanenallee 6, D–15738, Zeuthen, Germany and P. Uwer\addressInstitut für Theoretische Teilchenphysik, Universität Karlsruhe, D–76128 Karlsruhe, Germany
Abstract

We present updated predictions for the total cross section of top-quark pair production at Tevatron and LHC. For the LHC we also provide results at = 10 TeV, in view of the anticipated run in 2008 and quote numbers for the production of new heavy-quark pairs with mass in the range 0.5 – 2 TeV. Our two-loop results incorporate all logarithmically enhanced terms near threshold including Coulomb corrections as well as the exact dependence on the renormalization and factorization scale through next-to-next-to-leading order in QCD.

\readRCS

1 Introduction

Research on top-quark physics at hadron colliders has received great interest in the past years in view of the steadily improving measurements at Tevatron and the upcoming LHC (see Ref. [1] for a recent review). In this respect, the total cross section for top-quark pair production is a quantity of great importance for experimental analyses and even allows for measurements of the top-quark mass.

Moreover, on the theory side, the total cross section has been subject to numerous studies the motivation being improved predictions beyond the long-known next-to-leading order (NLO) corrections in QCD [2, 3, 4]. Recent work in this direction has aimed at completing the next-to-next-to-leading order (NNLO) QCD predictions [5, 6, 7, 8, 9], at resumming large Sudakov logarithms to next-to-next-to-leading logarithmic accuracy [10] and, at estimating bound state effects [11]. Also our knowledge on the parton distribution functions (PDFs) and the precision of the top-quark mass determination has continuously improved over the last years.

In order to study the impact of the various improvements on Tevatron and LHC predictions we build on the recent results of Ref. [10]. These approximate NNLO results for the total cross section are based on the complete logarithmic dependence on the heavy quark velocity near threshold . Moreover, they include the complete two-loop Coulomb corrections as well as the exact dependence on the renormalization and factorization scale at NNLO [12].

Recently, similar studies have appeared in Refs. [13, 14, 15]. While Ref. [13] largely follows our approach [10] to describe the total top-quark pair cross section at NNLO, Ref. [14] has limited itself to updating older predictions based on threshold resummation to next-to-leading logarithmic accuracy only. Thus, Ref. [14] necessarily arrives at larger theoretical uncertainties. The interesting study of Ref. [15] on the other hand applied consistently predictions to NLO accuracy in QCD. In doing so, it has investigated correlations of rates for top-quark pair production with many other cross sections at LHC to quantify a potential sensitivity to the gluon luminosity.

2 Total cross section

The total hadronic cross section for top-quark pair production depends on the hadronic center-of-mass energy squared and the top-quark mass . It is given by

(1)

where are the PDFs of the proton. The partonic cross section is given by and denotes the standard convolution (see e.g. Ref. [10]).

The generally adopted procedure to estimate the theoretical uncertainty for in Eq. (1) exploits the residual dependence on the renormalization and factorization scale, and , which are identified throughout this article (i.e. ). The NLO QCD corrections for the parton cross section and the PDFs provide the first instance where a meaningful error can be determined in this way. We define the range as

(2)

where is computed from the variation of the cross section with respect to the parameters of the global fit (see e.g. Refs. [15, 16, 17]).

In this contribution we employ the approximate NNLO result [10] to predict cross sections (1) and the associated uncertainty ranges (2) at Tevatron and LHC. Let us therefore briefly comment on the anticipated accuracy. Our cross section  takes along all logarithmically enhanced terms , as well as the complete Coulomb corrections () at two loops for the dominant parton channels and and adds them on top of the exact NLO predictions. In this way, our predictions rely on exact expressions in the region of phase space , where perturbative corrections receive the largest weight from the convolution with the parton luminosities, cf. Eq. (1). The effect of new parton channels opening at NNLO ( and ) is expected to be small, cf. the and channels at NLO.

The region of large energies on the other hand is inaccessible within our approach [10]. However, it is expected to give only small contributions in a full NNLO calculation in line with the observed small corrections for top-quark pair production together with an additional jet at NLO [5] which are part of the full NNLO correction for top-quark pair production.

Moreover, we have also included the exact and scale dependence at NNLO [12] which can be constructed using renormalization group methods. For the time being, we have chosen a common value for the scales, and we will address the independent variation of and in a future publication. However, based on preliminary studies we do not expect large modifications here. In summary, we have accounted for all numerically dominant contributions and are confident that this provides a very good approximation to the unknown full NNLO result as experience from other reactions, e.g. Higgs-production in gluon fusion [18] shows.

Let us next present our results for Tevatron and LHC. In Figs. 1 and 2 we plot the uncertainty range (2) comparing NLO and NNLO accuracy.

Figure 1: The NLO and NNLO (approx) QCD prediction for the total cross section at Tevatron for  TeV. The bands denote the total uncertainty from PDF and scale variations for the MRST06nnlo set [16] according to Eq. (2).
Figure 2: Same as Fig. 1 for LHC with  TeV.

At Tevatron (Fig. 1) the central value at NNLO increases typically by 8% with respect to NLO. The residual scale dependence of  is 3%, which corresponds to a reduction by a factor of two compared to NLO. The overall uncertainty according to Eq. (2) is at NNLO (approx) about 8% for the CTEQ6.6 and 6% for the MRST06nnlo PDF set. At LHC (Fig. 2) our  leads only to a small shift of a few percent in the central value and the NNLO (approx) band is about 6% for CTEQ6.6 and about 4% for MRST06nnlo, which exhibits again a drastic reduction of the scale uncertainty as compared to the prediction based on NLO QCD.

For phenomenological applications, the results of Eqs. (1), (2) are best presented by means of simple formulae for the mass dependence of the total cross section. To that end we make the ansatz following Ref. [14]

(3)

where . The parameters are fitted to reproduce in the mass range with a typical accuracy of better than per mille. For Tevatron and LHC the respective results for various PDF sets are given in Tabs. 13. Note, that Eq.(3) uses a polynomial of degree four and also determines the parameter for the central value from the fit.

Tevatron a[pb] b[pb] c[pb] d[pb] e[pb]
CTEQ6.5 7.93923 - 0.247233 0.43143 - 6.09398 7.37166
7.49359 - 0.231436 0.39979 - 5.59529 6.73356
8.38488 - 0.263031 0.46307 - 6.59268 8.00976
7.6572 - 0.239371 0.418793 - 5.92369 7.16791
8.00918 - 0.247709 0.428962 - 6.00414 7.2013
CTEQ6.6 7.80984 - 0.243547 0.425424 - 6.01643 7.29114
7.36172 - 0.22749 0.39309 - 5.5075 6.6412
8.25797 - 0.259604 0.457757 - 6.52536 7.94108
7.53093 - 0.235746 0.412856 - 5.84596 7.08203
7.87933 - 0.244019 0.422956 - 5.92653 7.11881
MRST06nnlo 8.24268 - 0.262604 0.46727 - 6.70602 8.19512
8.02331 - 0.254847 0.451872 - 6.46408 7.88561
8.46205 - 0.27036 0.482669 - 6.94796 8.50462
7.90318 - 0.252688 0.450793 - 6.47936 7.91184
8.32601 - 0.263279 0.464701 - 6.60993 8.01315
Table 1: The coefficients of the parameterization (3) for the cross section of Ref. [10] in pb at Tevatron ( TeV) using the PDF sets CTEQ6.5 [17], CTEQ6.6 [15] and MRST06nnlo set [16].
LHC a[pb] b[pb] c[pb] d[pb] e[pb]
CTEQ6.5 418.588 - 11.8988 0.19815 - 0.267839 3.07947
399.508 - 11.4849 0.192657 - 0.261487 3.01046
437.668 - 12.3126 0.203644 - 0.274191 3.14848
413.69 - 11.7539 0.195659 - 0.264393 3.03946
400.724 - 11.3609 0.188716 - 0.254368 2.91524
CTEQ6.6 419.062 - 11.9351 0.199035 - 0.269327 3.09784
400.4 - 11.5275 0.193785 - 0.263554 3.03731
438.121 - 12.3427 0.204284 - 0.275101 3.15837
413.944 - 11.7839 0.196439 - 0.265736 3.05688
401.139 - 11.3946 0.18954 - 0.255779 2.93453
MRST06nnlo 449.93 - 12.6551 0.208761 - 0.279652 3.1917
441.788 - 12.4928 0.206866 - 0.277817 3.17472
458.073 - 12.8175 0.210656 - 0.281487 3.20868
443.176 - 12.4618 0.205521 - 0.275247 3.14118
431.685 - 12.111 0.199286 - 0.266217 3.03056
Table 2: Same as in Tab. 1 for the cross section at LHC with start-up energy  TeV.
LHC a[pb] b[pb] c[pb] d[pb] e[pb]
CTEQ6.5 917.522 - 24.8588 0.398405 - 0.520463 5.82374
887.489 - 24.2054 0.388932 - 0.507626 5.66159
947.556 - 25.5122 0.407878 - 0.5333 5.98589
908.276 - 24.5978 0.394104 - 0.514815 5.76339
878.259 - 23.7272 0.379206 - 0.493949 5.51362
CTEQ6.6 920.475 - 24.9757 0.400681 - 0.523783 5.85946
891.302 - 24.3671 0.392303 - 0.512828 5.72046
949.648 - 25.5842 0.409059 - 0.534738 5.99846
910.798 - 24.7036 0.396235 - 0.517914 5.79166
881.04 - 23.8377 0.381364 - 0.497136 5.54881
MRST06nnlo 969.128 - 25.9797 0.412622 - 0.534539 5.93782
957.674 - 25.7648 0.409906 - 0.530883 5.88547
980.582 - 26.1945 0.415337 - 0.538195 5.99016
956.58 - 25.636 0.407062 - 0.527262 5.86177
929.84 - 24.8578 0.393723 - 0.508485 5.63364
Table 3: Same as in Tab. 1 for the cross section at LHC with  TeV.

Finally, we briefly quote some NNLO (approx) rates for the pair-production of new heavy quarks in the fundamental representation of the color gauge group at LHC with  TeV (see also Ref. [14]). Such particles with a mass appear in certain extensions of the Standard Model and we focus on a production model which is entirely dominated by QCD effects. Thus, our cross section provides a meaningful and accurate prediction because its numerical values arises largely from the threshold region where the logarithms dominate.

In Tabs. 5, 5 we quote the corresponding numbers in the mass range (see Ref. [14] for results to NLO accuracy). We observe that the scale dependence at NNLO accuracy is rather small, showing the expected good stability of the perturbative prediction. The relative variation of with respect to the PDFs, though, is dominating by far. Note there is the usual factor of two between the PDF uncertainty quoted by MRST06nnlo [16] and the CTEQ6.5 [17] PDF sets due to the definition of the tolerance criteria in the respective fits. The reason for the large observed PDF uncertainty is the gluon PDF being poorly constrained in the relevant region of large momentum fraction . This is a fact well-known to influence many searches for high-mass particles in gluon fusion channels (see e.g. Ref. [15] for the correlation of top-quark pair production rate with the high mass Higgs cross section).

only scale uncertainty only pdf uncertainty total uncertainty
[TeV] min max min max min max
0.5 4345. 4472. 1 4287. 4656. 4 4160. 4656. 6
0.6 1561. 1601. 1 1526. 1676. 5 1486. 1676. 6
0.7 634.1 649.2 1 616.0 682.5 5 600.8 682.5 6
0.8 282.3 288.5 1 272.6 304.4 6 266.4 304.4 7
0.9 134.5 137.2 1 129.3 145.1 6 126.6 145.1 7
1.0 67.64 68.94 1 64.81 73.08 6 63.50 73.08 7
1.1 35.45 36.17 1 33.93 38.41 6 33.22 38.41 7
1.2 19.23 19.65 1 18.38 20.91 6 17.97 20.91 8
1.3 10.74 10.99 1 10.26 11.72 7 10.01 11.72 8
1.4 6.147 6.301 1 5.862 6.741 7 5.708 6.741 8
1.5 3.589 3.687 1 3.417 3.957 7 3.319 3.957 9
1.6 2.130 2.192 1 2.021 2.363 8 1.959 2.363 9
1.7 1.282 1.322 2 1.212 1.432 8 1.172 1.432 10
1.8 0.781 0.806 2 0.735 0.878 9 0.710 0.878 11
1.9 0.480 0.497 2 0.450 0.544 9 0.433 0.544 11
2.0 0.298 0.309 2 0.277 0.340 10 0.266 0.340 12
Table 4: The NNLO (approx) cross section of Ref. [10] in fb for the pair-production of a (new) heavy quark with mass at LHC ( TeV) using the MRST06nnlo PDF set [16]. is the relative uncertainty with respect to the central value: .
only scale uncertainty only pdf uncertainty total uncertainty
[TeV] min max min max min max
0.5 3921. 4037. 1 3639. 4436. 10 3522. 4436. 11
0.6 1402. 1440. 1 1275. 1604. 11 1238. 1604. 13
0.7 568.0 582.5 1 508.6 656.5 13 494.1 656.5 14
0.8 252.4 258.2 1 222.5 294.0 14 216.6 294.0 15
0.9 120.3 122.8 1 104.5 141.0 15 102.0 141.0 16
1.0 60.48 61.66 1 51.94 71.37 16 50.76 71.37 17
1.1 31.78 32.30 1 26.95 37.64 17 26.44 37.64 17
1.2 17.25 17.57 1 14.50 20.57 17 14.21 20.57 18
1.3 9.626 9.802 1 8.023 11.58 18 7.847 11.58 19
1.4 5.503 5.614 1 4.555 6.673 19 4.444 6.673 20
1.5 3.208 3.277 1 2.630 3.925 20 2.560 3.925 21
1.6 1.902 1.946 1 1.545 2.347 21 1.502 2.347 22
1.7 1.144 1.173 1 0.921 1.426 22 0.892 1.426 23
1.8 0.696 0.715 1 0.554 0.876 23 0.535 0.876 24
1.9 0.428 0.441 1 0.337 0.544 24 0.324 0.544 25
2.0 0.265 0.274 2 0.206 0.342 25 0.198 0.342 27
Table 5: Same as in Tab. 5 using the CTEQ6.5 [17] PDF set.

3 Conclusion

We have presented updated predictions for cross sections of top-quark pair production based on the (approximate) NNLO results of Ref. [10]. These represent the best present estimates for hadro-production of top-quark pairs, both at Tevatron and LHC. We have argued that the neglected contributions (i.e. power suppressed terms away from threshold and new parton channels) are numerically small. We have found good convergence properties of the higher order corrections and greatly improved stability of the total cross section with respect to scale variations by our NNLO (approx) result. For applications, we have presented simple formulae (3) with 0.1 per mille accuracy for the mass dependence of the total cross section in the range . Finally, we have applied our results to estimate the pair-production rates of new quarks heavier than the top-quark in the range up to .

The results of Tabs. 13 for the fit of the mass dependence of have also been coded in a C-program, which is available from the authors upon request.

Acknowledgments

S.M. is supported by the Helmholtz Gemeinschaft under contract VH-NG-105 and and P.U. is a Heisenberg fellow of Deutsche Forschungsgemeinschaft (DFG). This work is also partly supported by DFG in SFB/TR 9.

References

  • [1] W. Bernreuther, J. Phys. G35 (2008) 083001, arXiv:0805.1333 [hep-ph]
  • [2] P. Nason, S. Dawson and R.K. Ellis, Nucl. Phys. B303 (1988) 607
  • [3] W. Beenakker et al., Phys. Rev. D40 (1989) 54
  • [4] W. Bernreuther et al., Nucl. Phys. B690 (2004) 81, hep-ph/0403035
  • [5] S. Dittmaier, P. Uwer and S. Weinzierl, Phys. Rev. Lett. 98 (2007) 262002, hep-ph/0703120
  • [6] M. Czakon, A. Mitov and S. Moch, Phys. Lett. B651 (2007) 147, arXiv:0705.1975 [hep-ph]
  • [7] M. Czakon, A. Mitov and S. Moch, Nucl. Phys. B798 (2008) 210, arXiv:0707.4139 [hep-ph]
  • [8] J.G. Körner, Z. Merebashvili and M. Rogal, (2008), arXiv:0802.0106 [hep-ph]
  • [9] M. Czakon, Phys. Lett. B664 (2008) 307, arXiv:0803.1400 [hep-ph]
  • [10] S. Moch and P. Uwer, Phys. Rev. D (2008) in press, arXiv:0804.1476 [hep-ph]
  • [11] K. Hagiwara, Y. Sumino and H. Yokoya, (2008), arXiv:0804.1014 [hep-ph]
  • [12] N. Kidonakis et al., Phys. Rev. D64 (2001) 114001, hep-ph/0105041
  • [13] N. Kidonakis and R. Vogt, (2008), arXiv:0805.3844 [hep-ph]
  • [14] M. Cacciari et al., (2008), arXiv:0804.2800 [hep-ph]
  • [15] P.M. Nadolsky et al., (2008), arXiv:0802.0007 [hep-ph]
  • [16] A.D. Martin et al., Phys. Lett. B652 (2007) 292, arXiv:0706.0459 [hep-ph]
  • [17] W.K. Tung et al., JHEP 02 (2007) 053, hep-ph/0611254
  • [18] S. Moch and A. Vogt, Phys. Lett. B631 (2005) 48, hep-ph/0508265
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