New results for t bar t production at hadron colliders
aa r X i v : . [ h e p - ph ] J u l DESY-09-104SFB/CPP-09-61HU-EP-09/31
New results for t ¯ t production at hadron colliders U. Langenfeld , S. Moch , and P. Uwer ∗
1- DESY ZeuthenPlatanenallee 6, 15738 Zeuthen - Germany2- Humboldt-Universit¨at zu Berlin, Institut f¨ur PhysikNewtonstr. 15, 12489 Berlin - GermanyWe present new theoretical predictions for the t ¯ t production cross section at NNLOat the Tevatron and the LHC. We discuss the scale uncertainty and the errors due tothe parton distribution functions (PDFs). For the LHC, we present a fit formula forthe pair production cross section as a function of the center of mass energy and weprovide predictions for the pair production cross section of a hypothetical heavy fourthgeneration quark t ′ . t ¯ t - Production at Tevatron and LHC The experimental measurements of the top mass m t and the t ¯ t production cross section havereached a relative accuracy of 0 .
75% [2] and 9% [3], respectively. Therefore it is necessaryto provide improved theoretical predictions for the total cross section at the Tevatron andthe LHC in perturbative QCD.The total hadronic cross section for t ¯ t production depends on the top mass m t , the centerof mass energy s = E , the factorisation scale µ f , the renormalization scale µ r , and the PDFset and it is given by σ ( s, m t , µ r , µ f ) = X i,j = g,q, ¯ q f i/p ( µ f ) ⊗ f j/p ( µ f ) ⊗ ˆ σ ( m t , µ r , µ f ) , (1)where f i/p are the proton PDFs. In the following, we discuss these dependencies.We have updated the results from Refs. [4, 5] as follows. To obtain more reliable estimatesof the scale uncertainty we have used the full dependence on µ r and µ f . We have performeda consistent singlet - octet - decomposition when matching our NNLO threshold expansionat NLO. Further corrections (electroweak contributions [6, 7, 8], QCD bound state effectsnear threshold [9], and new parton channels qq , ¯ q ¯ q , and q i ¯ q j for unlike quarks opening atNNLO [10, 11] ) are generally small and have been estimated.As a new result we have studied the dependence of the t ¯ t production cross section on thedefinition of the mass parameter. We have used the MS mass scheme as an alternative massdescription exploiting the conversion relation between the pole mass m t and the MS mass m ( µ r ) [12]. We find that the convergence of the perturbation expansion through NNLO isimproved using the MS mass. This expansion has a considerably reduced scale dependenceeven at NLO. The NLO and NNLO corrections in the MS scheme are much smaller thanthe corresponding corrections in the pole mass scheme. Therefore, we find good propertiesof convergence of the perturbation series. From the measured t ¯ t cross section at the Teva-tron [13] we derive a MS mass m = 160 . +3 . − . GeV, which corresponds to a pole mass of ∗ Talk given by U. Langenfeld at the XVII International Workshop on Deep-Inelastic Scattering andRelated Subjects, DIS 2009, 26-30 April 2009, Madrid, see Ref. [1]
DIS 2009 . +3 . − . GeV. More details of this analysis are presented in [14]. Throughout this article,we have chosen the PDF set CTEQ6.6 [15]. In Ref. [14], results for the PDF set MSTWNNLO 2008 [16] can be found. The top mass is the pole mass and is set to the most recentvalue m t = 173 GeV [2] if not otherwise stated.We have analysed the dependence of the cross section on the renormalisation and fac-torisation scale. In Fig. 1, we display the result for the Tevatron and the LHC. At theTevatron, the gradient is nearly parallel to the diagonal, and we find errors of −
5% at( µ f , µ r ) = ( m t / , m t /
2) and +3% at (2 m t , m t ). Likewise for the LHC, the scale uncer-tainty is about 1% at about (2 m t , m t /
2) and −
4% at (2 m t , m t ). Note that in the case ofthe LHC, the cross section is not a monotonically decreasing function if µ r = µ f as it is inthe case of the Tevatron, see Ref. [14] for details.In Figs. 2 and 3, we show the mass dependence of the total hadronic cross section forboth colliders including the scale uncertainty for µ r = µ f ≡ µ = m t / µ = 2 m t .The pure PDF error ∆ O is given by∆ O = s X k =1 ,n PDF (cid:0) σ k + − σ k − (cid:1) , (2)where ∆ O is determined from the variation of the cross section with respect to the param-eters of the PDF fit. The PDF errors are added linearly. The result is presented in Fig. 4for the Tevatron and in Fig. 5 for the LHC. We show for both colliders the NLO and NNLOcross sections together with their error bands. This demonstrates the shrinking of the totalerror for the NNLO cross section.Having discussed scale uncertainty and PDF error, we present our prediction for thecross section at the Tevatron and the LHC. To obtain a more conservative error bound, wecalculate the contribution of the scale uncertainty asmin µ r ,µ f ∈ [ m t / , m t ] σ ( µ r , µ f ) ≤ σ ( µ r , µ f ) ≤ max µ r ,µ f ∈ [ m t / , m t ] σ ( µ r , µ f ) . (3)For t ¯ t production at the Tevatron, this definition changes nothing, but for t ¯ t production atthe LHC, the upper bound is shifted to larger values by a few per cent. See also Fig. 1 andthe corresponding discussion. For the CTEQ6.6 PDF set and m t = 173 GeV (pole mass),we arrive at σ ( p ¯ p → t ¯ t ) = 7 . +0 . − . pb @ Tevatron, σ ( pp → t ¯ t ) = 874 +14 − pb @ LHC 14 TeV.For the LHC, we have calculated the total hadronic cross section as a function of thecenter of mass energy E for a value of m t = 172 . σ ( E, µ ) = a + bx + cx + dx log (cid:18) E √ s (cid:19) + ex log (cid:18) E √ s (cid:19) + f x log (cid:18) E √ s (cid:19) + gx log (cid:18) E √ s (cid:19) (4)with x = E/ GeV and √ s = 14 TeV. The numerical values for the coefficients a, . . . , g canbe found in Tab. 1 for µ = m t , m t /
2, and 2 m t . The fit is valid for 3 TeV ≤ E ≤
14 TeV andhas an accuracy of better than 0 .
05% within this range. This ansatz is justified by generallimits for cross sections at high energies and is consistent with unitarity. Parametrisationsof the total cross section as a function of m t can be found for various scenarios in Ref. [14]. DIS 2009 [ × b [ × − ] c [ × − ] d [ × − ] e [ × − ] f [ × − ] g [ × − ] σ ( µ = m t ) 3 . − . . − . − . − . . σ ( µ = m t /
2) 3 . − . . − . − . − . . σ ( µ = 2 m t ) 3 . − . . − . − . − . . Table 1: Numerical values of the coefficients (in pb) of Eq. 4 for m t = 172 . log ( µ f / m t ) l og ( µ r / m t ) . ˆ =+ % . ˆ =+ % . ˆ =+ % .
34 7 . ˆ = − % . ˆ = − % . ˆ = − % . ˆ = − % log ( µ f / m t ) l og ( µ r / m t ) ˆ = + % ˆ =+ . % ˆ = − . % ˆ = − % ˆ = − . % ˆ = − % ˆ = − . % ˆ = − % Figure 1: Contour lines of the total hadronic cross section from the independent variationof renormalization and factorization scale µ r and µ f for the Tevatron with √ s = 1 .
96 TeV(left) and the LHC with √ s = 14 TeV (right) with CTEQ6.6[15]. The range of µ r and µ f corresponds to µ r , µ f ∈ [ m t / , m t ]. t ′ ¯ t ′ Production at Tevatron and LHC
We briefly present theoretical predictions for the pair production cross section of a hypo-thetical heavy fourth generation quark t ′ at the Tevatron and the LHC. In this calculationwe have set the number of light flavours to n f = 6. As one can see in Figs. 7 and 8 thecross section decreases very rapidly with increasing t ′ mass. At the Tevatron, we predict fora 200 GeV t ′ quark a cross section of σ ( p ¯ p → t ′ ¯ t ′ ) = 3 . ± . m t ′ = 500 GeV,we predict σ ( p ¯ p → t ′ ¯ t ′ ) = 1 . +0 . − . fb. Scale uncertainty and PDF error contribute roughlyequal parts to the total error. At the LHC, we can test higher m t ′ masses. We predict for m t ′ = 500 GeV a cross section of σ ( pp → t ′ ¯ t ′ ) = 4 . +0 . − . pb and for m t ′ = 2000 GeV, wehave σ ( pp → t ′ ¯ t ′ ) = 0 . +0 . − . fb. At the LHC, the PDF error is much larger than the scaleuncertainty. Most t ′ ¯ t ′ pairs are produced via the gg channel, the PDF error of the gluonPDF is large in the relevant kinematic region, i.e. at high x . The hadronic cross sections for t ′ ¯ t ′ production including the total error bands are presented in Fig. 7 for the Tevatron andin Fig. 8 for the LHC. DIS 2009
60 170 180 190 m top [GeV] s ( pp - - > tt -) [ pb ] TevatronPDF set: Cteq6.6
Theoretical uncertainty
Figure 2: NNLO t ¯ t production cross sectionat the Tevatron. The blue band indicates thescale uncertainty.
160 170 180 190 m top [GeV] s ( pp - - > tt -) [ pb ] LHCPDF set: Cteq6.6
Theoretical uncertainty
Figure 3: NNLO t ¯ t production cross sec-tion at the LHC. The blue band indicatesthe scale uncertainty.
165 170 175 180 m top [GeV] s [ pb ] NLONNLO approx s (pp- -> tt-) [pb] @ Tevatron CTEQ6.6 Figure 4: Combined scale uncertainty andPDF error for t ¯ t production at NLO (blueband) and NNLO (red band) at the Teva-tron.
165 170 175 180 m top [GeV] s [ pb ] NLONNLO approx s (pp -> tt-) [pb] @ LHC, CTEQ6.6 Figure 5: Combined scale uncertainty andPDF error for t ¯ t production at NLO (blueband) and NNLO (red band) at the LHC. E cms [GeV] -2 -1 s [ pb ] m =1/2 m top m = m top m = 2 m top s (pp -> tt-) @ LHC, m top = 172.5 GeV, CTEQ 6.6 Figure 6: t ¯ t production at the LHC as afunction of the center of mass energy E for m t = 172 . µ = 1 / m t , m t , m t , see Eq. (4) andTab. 1 for the parametrisation. DIS 2009
00 300 400 500 m t’ [GeV] -3 -2 -1 s [ pb ] s (pp- -> t’t-) s (pp- -> t’t-’) @ Tevatron, 1.96 TeV, CTEQ6.6combined scale uncertainty and PDF error
500 1000 1500 2000 m t’ [GeV] -4 -3 -2 -1 s [ pb ] s (pp -> t’t-’) s (pp -> t’t-’) @ LHC 14 TeV, CTEQ6.6combined scale uncertainty and PDF error Figure 7: Pair production cross sectionfor a hypothetical heavy fourth generationquark at the Tevatron. The violet band in-dicates the combined scale uncertainty andPDF error. Figure 8: Pair production cross sectionfor a hypothetical heavy fourth generationquark at the LHC. The violet band indi-cates the combined scale uncertainty andPDF error.
Acknowledgments
This work is supported by the Helmholtz Gemeinschaft under contract VH-NG-105 and bythe Deutsche Forschungsgemeinschaft under contract SFB/TR 9. P.U. acknowledges thesupport of the Initiative and Networking Fund of the Helmholtz Gemeinschaft, contractHA-101 (”Physics at the Terascale”).
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