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Eur. Phys. J. C (2009) 60: 375–386 DOI 10.1140/epjc/s10052-009-0892-7 Regular Article - Theoretical Physics Top-quark pair production near threshold at LHC Y. Kiyo 1 , J.H. Kühn 1 , S. Moch 2 , M. Steinhauser 1,a , P. Uwer 3 1 Institut für Theoretische Teilchenphysik, Universität Karlsruhe (TH), Karlsruhe Institute of Technology (KIT), 76128 Karlsruhe, Germany 2 Deutsches Elektronen-Synchrotron DESY, 15738 Zeuthen, Germany 3 Institut für Physik, Humboldt-Universität zu Berlin, 10099 Berlin, Germany Received: 4 December 2008 / Revised: 7 January 2009 / Published online: 19 February 2009 © Springer-Verlag / Società Italiana di Fisica 2009 Abstract The next-to-leading order analysis for the cross section for hadroproduction of top-quark pairs close to threshold is presented. Within the framework of non- relativistic QCD a significant enhancement compared to fixed-order perturbation theory is observed which originates from the characteristic remnant of the 1S peak below pro- duction threshold of top-quark pairs. The analysis includes all color-singlet and color-octet configurations of top-quark pairs in S -wave states and, for the dominant configurations, it employs all-order soft-gluon resummation for the hard parton cross section. Numerical results for the Large Hadron Collider at s = 14 TeV and s = 10 TeV and also for the Tevatron are presented. The possibility of a top-quark mass measurement from the invariant-mass distribution of top- quark pairs is discussed. PACS 12.38.Bx · 12.38.Cy · 14.65.Ha 1 Introduction At the CERN Large Hadron Collider (LHC) the major part of top quarks are produced in pairs. Due to the experience gained at the Fermilab Tevatron [1] and the huge amount of top quarks to be produced at LHC the reconstruction of top quarks with good accuracy will be possible [2, 3]. A signif- icant fraction of top-quark pairs will be produced close to threshold. Thus a dedicated analysis of the production cross section in this region is required which is best performed within the framework of non-relativistic QCD (NRQCD) [4, 5]. The production of top–antitop-quark pairs close to the kinematical threshold has received much attention in the context of precision measurement of top-quark properties at the future International Linear Collider (ILC). Theoret- ical calculations and dedicated experimental analyses have a e-mail: [email protected] demonstrated that a precise extraction of the top-quark mass, its width and the strong coupling constant is possi- ble [6, 7] at the ILC. The complete next-to-next-to-leading order (NNLO) predictions have been available since many years [8]. (For earlier work see e.g [912].) Partial next- to-next-to-leading logarithmic (NNLL) [13, 14] and next-to- next-to-next-to-leading order (NNNLO) [1517] predictions were evaluated more recently. In contrast to the linear collider, where the physical ob- servable is the total cross section as a function of energy, at the hadron collider one considers the invariant-mass dis- tribution of the top-quark pairs. Since the expected uncer- tainty is significant larger than the one anticipated at a lin- ear collider a next-to-leading order (NLO) analysis is prob- ably sufficient. The calculation of the cross section within the NRQCD framework contains as building blocks the hard production cross section for a top-quark pair at threshold and the non-relativistic Green’s function governing the dy- namics of the would-be bound state. Both ingredients have been available in the literature since many years. In partic- ular, the hard cross section for threshold t ¯ t production can be found in [18, 19]. In [18] the NLO formulae were de- rived for quark or gluon initial states and a quarkonium in aJ PC = 0 −+ color-singlet state, plus possibly a parton. The general case, with the heavy quark system (Q ¯ Q) in S -wave singlet/triplet spin state, and with color-singlet/octet config- uration is given in [19], together with the corresponding re- sults for P -waves. The results of [18, 19] were presented for stable bound states. For unstable wide resonances it is convenient to describe the bound-state dynamics through a Green’s function. Recently a calculation of top-quark threshold hadropro- duction near threshold has appeared [20]. (For an early dis- cussion along similar lines see [21].) The basic idea of our approach is similar to the one of [20]. We aim at a de- tailed study of the top-quark production based on NLO cross section formulae in the NRQCD framework. In our set-up all NLO sub-processes have been included, i.e., also those

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Eur. Phys. J. C (2009) 60: 375–386DOI 10.1140/epjc/s10052-009-0892-7

Regular Article - Theoretical Physics

Top-quark pair production near threshold at LHC

Y. Kiyo1, J.H. Kühn1, S. Moch2, M. Steinhauser1,a, P. Uwer3

1Institut für Theoretische Teilchenphysik, Universität Karlsruhe (TH), Karlsruhe Institute of Technology (KIT), 76128 Karlsruhe, Germany2Deutsches Elektronen-Synchrotron DESY, 15738 Zeuthen, Germany3Institut für Physik, Humboldt-Universität zu Berlin, 10099 Berlin, Germany

Received: 4 December 2008 / Revised: 7 January 2009 / Published online: 19 February 2009© Springer-Verlag / Società Italiana di Fisica 2009

Abstract The next-to-leading order analysis for the crosssection for hadroproduction of top-quark pairs close tothreshold is presented. Within the framework of non-relativistic QCD a significant enhancement compared tofixed-order perturbation theory is observed which originatesfrom the characteristic remnant of the 1S peak below pro-duction threshold of top-quark pairs. The analysis includesall color-singlet and color-octet configurations of top-quarkpairs in S-wave states and, for the dominant configurations,it employs all-order soft-gluon resummation for the hardparton cross section. Numerical results for the Large HadronCollider at

√s = 14 TeV and

√s = 10 TeV and also for the

Tevatron are presented. The possibility of a top-quark massmeasurement from the invariant-mass distribution of top-quark pairs is discussed.

PACS 12.38.Bx · 12.38.Cy · 14.65.Ha

1 Introduction

At the CERN Large Hadron Collider (LHC) the major partof top quarks are produced in pairs. Due to the experiencegained at the Fermilab Tevatron [1] and the huge amount oftop quarks to be produced at LHC the reconstruction of topquarks with good accuracy will be possible [2, 3]. A signif-icant fraction of top-quark pairs will be produced close tothreshold. Thus a dedicated analysis of the production crosssection in this region is required which is best performedwithin the framework of non-relativistic QCD (NRQCD)[4, 5].

The production of top–antitop-quark pairs close to thekinematical threshold has received much attention in thecontext of precision measurement of top-quark propertiesat the future International Linear Collider (ILC). Theoret-ical calculations and dedicated experimental analyses have

a e-mail: [email protected]

demonstrated that a precise extraction of the top-quarkmass, its width and the strong coupling constant is possi-ble [6, 7] at the ILC. The complete next-to-next-to-leadingorder (NNLO) predictions have been available since manyyears [8]. (For earlier work see e.g [9–12].) Partial next-to-next-to-leading logarithmic (NNLL) [13, 14] and next-to-next-to-next-to-leading order (NNNLO) [15–17] predictionswere evaluated more recently.

In contrast to the linear collider, where the physical ob-servable is the total cross section as a function of energy,at the hadron collider one considers the invariant-mass dis-tribution of the top-quark pairs. Since the expected uncer-tainty is significant larger than the one anticipated at a lin-ear collider a next-to-leading order (NLO) analysis is prob-ably sufficient. The calculation of the cross section withinthe NRQCD framework contains as building blocks the hardproduction cross section for a top-quark pair at thresholdand the non-relativistic Green’s function governing the dy-namics of the would-be bound state. Both ingredients havebeen available in the literature since many years. In partic-ular, the hard cross section for threshold t t production canbe found in [18, 19]. In [18] the NLO formulae were de-rived for quark or gluon initial states and a quarkonium ina JPC = 0−+ color-singlet state, plus possibly a parton. Thegeneral case, with the heavy quark system (QQ) in S-wavesinglet/triplet spin state, and with color-singlet/octet config-uration is given in [19], together with the corresponding re-sults for P -waves. The results of [18, 19] were presentedfor stable bound states. For unstable wide resonances it isconvenient to describe the bound-state dynamics through aGreen’s function.

Recently a calculation of top-quark threshold hadropro-duction near threshold has appeared [20]. (For an early dis-cussion along similar lines see [21].) The basic idea of ourapproach is similar to the one of [20]. We aim at a de-tailed study of the top-quark production based on NLO crosssection formulae in the NRQCD framework. In our set-upall NLO sub-processes have been included, i.e., also those

376 Eur. Phys. J. C (2009) 60: 375–386

which appear for the first time in O(α3s ). Furthermore, the

matching between QCD and NRQCD as performed in [20]is slightly different from the present paper. Whereas in [20]the matching has been performed for the limit where thepartonic center-of-mass energy s approaches twice the top-quark mass, we include the complete dependence on s asgiven in [18, 19]. Thus, formally, the result of [20] is onlyvalid for top-quark production where the velocity of bothquarks is small. On the other hand, in our approach the rel-ative velocity has to be small whereas the top–antitop-quarksystem can still move with high velocity. Finally, we per-form a soft-gluon resummation, which enhances the crosssection by a few per cent.

Our paper is organized as follows. In the next section de-tails of the formalism used for the calculation of the NLOcross section are provided. The effects of initial-state ra-diation and the hard contribution are discussed in Sect. 3and the soft-gluon resummation is performed in Sect. 4. Theproperties of the Green’s function are summarized in Sect. 5.In Sect. 6 the building blocks are combined and numeri-cal results for the invariant-mass distribution are presented.Theoretical uncertainties due to scale variation and unknownhigher-order corrections are estimated. A summary and con-clusions are presented in Sect. 7.

2 The production cross section

Let us denote the (quasi-) bound state of a top and anti-top quark with spin S and angular momentum L by T ≡2S+1L

[1,8]J where the superscripts [1] and [8] denote the sin-

glet and octet color states. The production rate is obtainedfrom the production cross section of a top-quark pair withinvariant mass M2 ≡ (pt +pt )

2 and its evolution to a quasi-bound state is described by the non-relativistic QCD. Theformer is a hard QCD process at a distance ∼1/mt and thuscomputable within the conventional perturbative expansionin αs.

The long-distance effects responsible for the forma-tion of a narrow bound state are described by the squaredwave function at the origin |Ψ (0)|2 or, in the language ofNRQCD, by the matrix elements

⟨(χ†Γ ψ

) · (ψ†Γ χ)⟩ = Ns Nc

∣∣Ψ (0)∣∣2

. (2.1)

Here Ns = 2S + 1 and Nc = 1,8 denote the number of spinand color degrees of freedom, respectively. We are interestedin the differential distribution dσ/dM which, for narrow res-onances with mass Mn, is proportional to δ(M − Mn). Forwide resonances, the case under consideration, it is conve-nient to convert the factor describing the sum over individual

resonances into the non-relativistic Green’s function1

n

∣∣Ψn(0)∣∣2

πδ(M − Mn)

→∑

n

ImΨn(0)Ψ ∗

n (0)

Mn − (M + iΓt )= ImG(M + iΓt), (2.2)

with G(M + iΓt ) ≡ G[1,8](�r = 0;M + iΓt ) being theGreen’s function at zero distance for the non-relativisticSchrödinger equation discussed below. Since the typicalmomentum scale governing the non-relativistic top-quarksystem mtv (with mtv

2 ≡ M + iΓt − 2mt , and v beingthe velocity of top and antitop quarks) is in the perturba-tive regime, and the large top-quark width Γt introduces anadditional cutoff scale

√mtΓt , the Green’s function can be

evaluated perturbatively. As stated above, the present pa-per is concerned with the production of top-quark pairs nearthreshold, and thus is restricted to states with L = 0, i.e.

T = 2S+1S[1,8]J . The contributions to the invariant-mass dis-

tribution with higher angular momentum are at least sup-pressed by v2, and thus of higher order (beyond NLO).

In order to obtain experimentally measurable quantitiesat a hadron collider the partonic differential cross sectiondσij→T /dM is convoluted with the luminosity function,

[dLij

](τ,μ2

f

)

=∫ 1

0dx1

∫ 1

0dx2 fi/P1

(x1,μ

2f

)fj/P2

(x2,μ

2f

)

× δ(τ − x1x2), (2.3)

where i, j refer to partons inside the hadrons P1 and P2 withthe distribution functions fi/P1 and fj/P2 . The dependenceon the factorization scale μf cancels in combination with theone contained in dσij→T /dM . The differential cross sectioncan thus be written as

MdσP1P2→T

dM

(S,M2)

=∑

i,j

∫ 1

ρ

[dLij

](τ,μ2

f

)M

dσij→T

dM

(s,M2,μ2

f

).

(2.4)

As usual s and S denote the partonic and the hadroniccenter-of-mass energy squared, respectively, and τ = s/S.The lower limit of the τ integration is given by ρ = M2/S.The partonic differential cross section dσij→T /dM consistsof a factor F that is evaluated in perturbative QCD, and that

1In the case of color-octet states we cannot take (2.2) literally but derivea corresponding formula within the framework of NRQCD.

Eur. Phys. J. C (2009) 60: 375–386 377

can be deduced from [18, 19], and a second factor, the imag-inary part of the Green’s function G[1,8],

Mdσij→T

dM

(s,M2,μ2

f

)

= Fij→T

(s,M2,μ2

f

) 1

m2t

ImG[1,8](M + iΓt), (2.5)

where the superscript of the Green’s function refers to thecolor state of T . Equations (2.4) and (2.5) constitute ourmaster formulae, which contain several scales and vari-ous physics contributions of different origin in factorizedform. In particular, the soft dynamics of the parton distri-bution and real radiation is contained in the convolution ofFij→T with the parton luminosity; the bound-state effectsare described by G. Note that at NLO the Green’s func-tion G[1,8](M + iΓt ) and the convolution of F with the par-ton luminosity (L ⊗ F) are individually independent of therenormalization scale μr. Thus we can discuss the two partsseparately in the following two sections. Furthermore, it issimpler to assess the uncertainties for the individual contri-butions.

Let us at this point make a comment concerning the va-lidity of (2.5), which makes use of the NRQCD expansionassuming v 1, thus being limited to the threshold region.For larger invariant masses, conventional perturbation the-ory is applicable (see [22–24] and [25–28] for recent com-pilations of the total cross section and [29] for a proposalto measure the top-quark mass from the shape of dσ/dM).In the transition region the predictions from both methodsare expected to coincide, as will be discussed below (cf.Fig. 6.3).

3 Hard cross section

In this section the ingredients for the NLO corrections to thehard cross section will be collected, which are taken from[18, 19]. We parameterize the function Fij→T , represent-ing the hard cross section for ij → T X (X stands for addi-tional partons in the inclusive cross sections), in the follow-ing form:

Fij→T

(s,M2,μ2

f

)

= Nij→T

π2α2s (μr)

3s

(1 + αs(μr)

πCh

)

×[δij→T δ(1 − z) + αs(μr)

π

(Ac(z) + Anc(z)

)]. (3.1)

Here δgg→1S

[1,8]0

= δqq→3S

[8]1

= 1 and zero for all other

2 → 1 processes, and z = M2/s. The quantities Ac, Anc,and Ch all depend on i, j , and T ; the functions A in addi-tion depend on z.

The coefficients Ch originate from the hard corrections tothe production process. The functions Ac contain the realcorrections with collinear parton splitting from one of theinitial partons i, j , and are governed by the Altarelli–Parisisplitting functions, Anc originates from non-collinear realemission. These individual contributions are manifest al-ready in [18] and the Appendix of [19] and will be listed inthe following. Note that in (3.1) we have split off the factor(1 + (αs/π)Ch), which we attribute to hard corrections andthus treat as a multiplicative factor to the terms in squarebrackets.

In Table 3.1 we collect all processes of the type ij → T X

at NLO which contribute in our analysis and list the corre-sponding normalization factors Nij→T . Note that the pro-

duction of a spin-triplet color-singlet state 3S[1]1 via gq or qq

scattering is zero up to and including NLO. This is becausein these channels the heavy quarks are produced throughgluon splitting g∗ → t t , which is only possible if the t t isin an octet state.

The coefficients Ch are non-vanishing only for the proc-esses which are present also in lowest order [18, 19]:

Ch[gg → 1S

[1]0

]

= β0

2ln

(μ2

r

M2

)+ CF

(π2

4− 5

)+ CA

(1 + π2

12

),

Ch[gg → 1S

[8]0

]

= β0

2ln

(μ2

r

M2

)+ CF

(π2

4− 5

)+ CA

(3 − π2

24

),

Ch[qq → 3S

[8]1

]

= β0

2ln

(μ2

r

M2

)+ CF

(π2

3− 8

)

+ CA

(59

9+ 2 ln 2

3− π2

4

)− 10

9nfTF − 16

9TF ,

(3.2)

where β0 = (11/3)CA − (4/3)nfTF and CF = 4/3, CA = 3,TF = 1/2, nf = 5. The last term in Ch[qq → 3S

[8]1 ], arising

from non-decoupling of the top quark in the gluon propaga-tor, has been observed and discussed in [20]; see also [20,footnote 3, p. 73]. For the other processes the hard correc-tions that are of higher order; thus Ch is zero at NLO:

Ch[gq → 1S

[1,8]0

] = Ch[qq → 1S

[1,8]0

] = Ch[gg → 3S

[1,8]1

]

= Ch[gq → 3S

[8]1

] = 0. (3.3)

The function Ac is conveniently expressed using theAltarelli–Parisi splitting functions Pij (z) introduced below[18, 19]:

378 Eur. Phys. J. C (2009) 60: 375–386

Table 3.1 Normalizationfactors Nij→T for each processfor [singlet, octet] color states.(Nc = 3 is used.)

gg → 1S[1,8]0 gq → 1S

[1,8]0 qq → 1S

[1,8]0 gg → 3S

[1,8]1 gq → 3S

[1,8]1 qq → 3S

[1,8]1

[1,5/2] [1,5/2] [3/4,6] [9/4,18] [0,32/3] [0,32/3]

Ac[gg → 1S

[1,8]0

]

= (1 − z)Pgg(z)

{2

[ln(1 − z)

1 − z

]

++

[1

1 − z

]

+ln

(M2

zμ2f

)}

− β0

2δ(1 − z) ln

(μ2

f

M2

),

Ac[gq → 1S

[1,8]0

] = 1

2Pgq(z) ln

(M2(1 − z)2

zμ2f

)+ CF

2z,

Ac[qq → 1S

[1,8]0

] = 0,

Ac[gg → 3S

[1,8]1

] = 0,

Ac[gq → 3S

[8]1

]

= 1

2Pqg(z) ln

(M2(1 − z)2

zμ2f

)+ TF z(1 − z),

Ac[qq → 3S

[8]1

]

= (1 − z)Pqq(z)

{2

[ln(1 − z)

1 − z

]

++

[1

1 − z

]

+ln

(M2

zμ2f

)}

+ CF (1 − z) − 3CF

2δ(1 − z) ln

(μ2

f

M2

), (3.4)

where the conventional plus-distribution2 was employed to

regularize the singularity at z = 1. The splitting functions

Pij (z) are given by

Pgg(z) = 2CA

[1

1 − z+ 1

z+ z(1 − z) − 2

],

Pgq(z) = CF

[1 + (1 − z)2

z

],

Pqg(z) = TF

[z2 + (1 − z)2],

Pqq(z) = 2CF

[1

1 − z− 1 + z

2

].

(3.5)

2The plus-distribution follows the prescription∫ 1

0 dz[ lnn(1−z)1−z

]+f (z) ≡∫ 1

0 dzlnn(1−z)

1−z[f (z) − f (1)], where f (z) is an arbitrary test function

which is regular at z = 1. It is related to the ρ-prescription used in [19]

by [ lnn(1−z)1−z

]+ = [ lnn(1−z)1−z

]ρ + lnn+1(1−ρ)n+1 δ(1 − z).

The functions Anc are obtained from the non-collinear con-tributions. For spin-singlet states, we have

Anc[gg → 1S

[1]0

]

= −CA

6z(1 − z)2(1 + z)3

[12 + 11z2 + 24z3 − 21z4

− 24z5 + 9z6 − 11z8 + 12(−1 + 5z2 + 2z3 + z4

+ 3z6 + 2z7)

ln z],

Anc[gg → 1S

[8]0

]

= −CA

6z(1 − z)(1 + z)3

[12 + 23z2 + 30z3 − 21z4

− 24z5 + 9z6 − 6z7 − 23z8 + (−12 + 60z2

+ 24z3 + 36z4 + 60z6 + 24z7) ln z]( 1

1 − z

)

+,

Anc[gq → 1S

[1,8]0

] = −CF

1

z(1 − z)(1 − ln z),

Anc[qq → 1S

[1]0

] = 32CF

3N2c

z(1 − z),

Anc[qq → 1S

[8]0

] = 32BF

3N2c

z(1 − z),

(3.6)

where BF = (N2c − 4)/(4Nc) with Nc = 3. Note that

Anc[gg → 1S[8]0 ] is singular at z = 1, and regularized by

the plus-prescription. For spin-triplet states, one obtains

Anc[gg → 3S

[1]1

]

= 256BF

6CF N2c

z

(1 − z)2(1 + z)3

× [2 + z + 2z2 − 4z4 − z5 + 2z2(5 + 2z + z2) ln z

],

Anc[gg → 3S

[8]1

]

= 1

36z(1 − z)2(1 + z)3

[108 + 153z + 400z2 + 65z3

− 356z4 − 189z5 − 152z6 − 29z7 + (108z

+ 756z2 + 432z3 + 704z4 + 260z5 + 76z6) ln z],

Anc[gq → 3S

[8]1

]

= TF

4(1 − z)(1 + 3z) + CA

4CF

1

z

[(1 − z)

(2 + z + 2z2)

+ 2z(1 + z) ln z],

Anc[qq → 3S

[8]1

]

= −[CF (1 − z)2 + CA

3

(1 + z + z2)

](1

1 − z

)

+.

(3.7)

Eur. Phys. J. C (2009) 60: 375–386 379

Table 3.2 The convolutionL ⊗ F for LHC at the referencepoint M = 2mt , for theproduction of color singlet andoctet states. The three columnscorrespond to the scale choicesμr = μf = (mt ,2mt ,4mt)

L ⊗ F [ij → T [1]] × 106 [GeV−2] L ⊗ F [ij → T [8]] × 106 [GeV−2]

gg → 1S[1,8]0 20.7 21.2 20.9 63.2 62.7 60.2

gq → 1S[1,8]0 −0.795 −1.74 −2.19 −1.99 −4.36 −5.47

qq → 1S[1,8]0 0.00664 0.00509 0.00398 0.0166 0.0127 0.00995

gg → 3S[1,8]1 0.175 0.127 0.0936 6.06 4.26 3.07

gq → 3S[8]1 – – – 3.99 1.68 0.279

qq → 3S[8]1 – – – 23.1 23.8 23.6

Total: (1S0 + 3S1)[1,8] 20.0 19.6 18.8 94.3 88.1 81.8

The function Anc[qq → 3S[8]1 ] is also defined with the plus-

prescription. The leading singular behavior of Anc is given

by Anc(z)z→1∼ −CA/(1 − z)+ both for gg → 1S

[8]0 and

qq → 3S[8]1 . In the soft limit its behavior is insensitive to

the details of the bound state and only depends on its colorconfiguration.

It is instructive to discuss the relation between the nor-malizations of different processes leading to the same boundstate. For instance, the normalization Nij→T (see Table 3.1)

for the process gq → 1S[1,8]0 X is fixed by gg → 1S

[1,8]0 , be-

cause in the collinear limit this cross section factorizes intothe corresponding LO process and the Pgq splitting func-tion. As a consequence, the cancellation of the factoriza-tion scale dependence happens among the gg and gq initi-ated reactions. Similarly, the normalization of gq → 3S

[8]1 is

fixed by qq → 3S[8]1 . In contrast, the processes qq → 1S

[1,8]0

and gg → 3S[1,8]1 are forbidden at LO; hence the corrections

have to be collinearly finite. In comparison to [20] the com-binations Ac + Anc include terms that vanish in the limitz → 1. Furthermore, subprocesses that appear for the firsttime in O(α3

s ) were neglected in [20]. The relative size ofthese terms will be addressed below.

Let us now start the numerical analysis. The partoniccross sections have to be convoluted with the parton-distribution functions (PDFs) in order to arrive at thehadronic cross section. We use the CTEQ6.5 [30] set forthe PDFs and take α

(5)s (MZ) = 0.118, mt = 172.4 GeV and√

S = 14 TeV as input values. The running of α(5)s (μr),

which is the input for the partonic cross sections, is eval-uated with the help of RunDec [31], using the four-loopapproximation of the β function. This leads to α

(5)s (μr) =

(0.1077,0.09832,0.09050) for μr = (mt ,2mt,4mt). Fur-thermore, we identify renormalization and factorizationscales (μf = μr).

As stated above, the cross section factors into the con-volution L ⊗ F and the Green’s function. To discuss therelative importance of the various contributions individ-ually, the results for the subprocesses without the factorImG(M + iΓt )/m2

t are given in Table 3.2. Note that color-singlet t t production is dominated by gg → 1S

[1]0 . Color-

octet production is dominated by gg → 1S[8]0 plus a 25%

contribution from qq → 3S[8]1 . The size of the remaining

subprocesses (neglected in [20]) amounts to five to ten per-cent and is strongly scale dependent. The variation of μ (re-call μ = μf = μr) between mt and 4mt leads to changes ofL ⊗ F by ±3% and ±7% for the total singlet and octet pro-duction, respectively. In these channels the real radiation ofpartons contains large logarithmic contributions in the NLOcorrections. In combination with the rapidly varying partonluminosity these logarithms make up for a major part of thenumbers quoted in Table 3.2. The origin of these large log-arithms can be traced to the singular behavior of the crosssection near z ≈ 1, regularized by plus-distributions. Thereexists well established technology for the resummation ofthese large logarithms to all orders in perturbation theory.We will address this issue next.

4 Soft-gluon resummation

The parton channels which exhibit enhancement due to soft-gluon emission are gg → 1S

[1]0 , gg → 1S

[8]0 , and qq →

3S[8]1 (see (3.4) and (3.7)). The relevant logarithms are con-

tained both in Ac (from initial-state radiation) and Anc (fromFSR) and read for the three leading processes

Athrlog[gg → 1S

[1]0

]

= 4CAD1 − 2CA ln

(μ2

f

M2

)D0 − β0

2δ(1 − z) ln

(μ2

f

M2

),

Athrlog[gg → 1S

[8]0

]

= Athrlog[gg → 1S

[1]0

] − CAD0,

Athrlog[qq → 3S

[8]1

]

= 4CF D1 −(

2CF ln

(μ2

f

M2

)+ CA

)D0

− 3CF

2δ(1 − z) ln

(μ2

f

M2

),

(4.1)

where Dl = [lnl (1 − z)/(1 − z)]+ denote the plus-distribu-tions and all lnμ2

f /M2 parts are included in the definition of

380 Eur. Phys. J. C (2009) 60: 375–386

the threshold logarithm. Whether the threshold logarithmsare enhanced or not depends on the behavior of the partonluminosity functions near the kinematical end point τ = ρ.To investigate the size of the threshold logarithms, we eval-uate the contribution of the factorized hard-scattering con-tribution convoluted with the PDFs, i.e. L ⊗ F separatelyfor the three contributions of tree-level, singular and reg-ular terms. (The hard corrections (1 + (αs/π)C) are com-mon to all.) The threshold-enhanced contributions are de-fined in (4.1) and correspond exactly to the terms includedin [20], while regular terms correspond to the remainder ofAc + Anc in (3.4) and (3.7) without plus-distributions. ForM = 2mt and

√S = 14 TeV we obtain the following re-

sults:

(L ⊗ F)[gg → 1S

[1]0

]

=⎧⎨

14.5 + (4.53 + 1.68)A14.0 + (5.66 + 1.58)A13.0 + (6.37 + 1.48)A

⎫⎬

⎭× 10−6 GeV−2,

(L ⊗ F)[gg → 1S

[8]0

]

=⎧⎨

39.3 + (16.6 + 7.26)A37.4 + (18.8 + 6.52)A34.4 + (20.0 + 5.83)A

⎫⎬

⎭× 10−6 GeV−2,

(L ⊗ F)[qq → 3S

[8]1

]

=⎧⎨

16.7 + (3.50 + 2.91)A16.8 + (3.41 + 3.56)A16.4 + (3.28 + 3.97)A

⎫⎬

⎭× 10−6 GeV−2.

(4.2)

The three lines correspond to μ = μf = μr =(mt ,2mt,4mt). We note that in all three cases the con-tribution of the threshold-enhanced terms from (4.1) islarge, although the regular terms in the case of qq → 3S

[8]1

are of the same order. Technically the matching appliedin [20] corresponds to neglecting all terms which vanishexactly at threshold, that is, for z = 1, i.e. (3.6) and (3.7)of Sect. 3. The regular terms in (4.2), which have notbeen accounted for in the recent analysis of [20], are ofthe same order as the NLO sub-processes as given in Ta-ble 3.2.

Threshold resummation proceeds conveniently in Mellinspace. To that end we calculate the Mellin moments withrespect to z = M2/s according to

FNij→T

(M2,μ2

f

) =∫ 1

0dz zN−1Fij→T

(s,M2,μ2

f

). (4.3)

Then, the Mellin-space expression for the threshold-enhanced terms listed in (4.1) read (see also [32, 33])

ANthrlog

[gg → 1S

[1]0

]

= 2CA ln2 N + CA lnN

(4γE − 2 ln

(M2

μ2f

))

+ CA

(2ζ2 +2γ 2

E −2γE ln

(M2

μ2f

))+ 1

2β0 ln

(M2

μ2f

),

ANthrlog

[gg → 1S

[8]0

]

= ANthrlog

[gg → 1S

[8]0

] + CA lnN + CAγE,

ANthrlog

[qq → 3S

[8]1

]

= 2CF ln2 N + CF lnN

(4γE − 2 ln

(M2

μ2f

))+ CA lnN

+ CF

(2ζ2 + 2γ 2

E + 3

2ln

(M2

μ2f

)− 2γE ln

(M2

μ2f

))

+ CAγE,

(4.4)

where we have kept all dominant terms in the large-N limitand neglected power-suppressed terms of order 1/N . γE isthe Euler–Mascheroni constant (γE = 0.577215 . . .).

The resummed expressions (defined in the MS-scheme)for the individual color structures of the hard cross sectionsF of (3.1) are given by a single exponential in Mellin space(see e.g. [33–35])

FNij→T (M2,μ2

f )

F(0),Nij→T (M2,μ2

f )= g0

ij→T

(m2

t ,μ2f ,μ

2r

)�N+1

ij→T

(m2

t ,μ2f ,μ

2r

)

+ O(N−1 lnn N

), (4.5)

where F(0),Nij→T denotes the tree-level term in (3.1) and the

exponents are commonly expressed as

ln�Nij→T = lnN · g1

ij (λ) + g2ij→T (λ) + · · · , (4.6)

where λ = β0αs lnN/(4π). To next-to-leading logarithmic(NLL) accuracy the (universal) functions g1

ij as well as the

functions g2ij→T are relevant in (4.6); see [25] for the exten-

sion to NNLL accuracy. Explicit expressions are

g1qq = A

(1)q

β0

[2 − 2 ln(1 − 2λ) + λ−1 ln(1 − 2λ)

],

g2qq→T [1]

=(

A(1)q β1

β30

− A(2)q

β20

)[2λ + ln(1 − 2λ)

]

+ A(1)q β1

2β30

ln2(1 − 2λ) − 2A

(1)q

β0γE ln(1 − 2λ)

+ ln

(M2

μ2r

)A

(1)q

β0ln(1 − 2λ) + 2 ln

(μ2

f

μ2r

)A

(1)q

β0λ,

g2qq→T [8] = g2

qq→T [1] −D

(1)

QQ

2β0ln(1 − 2λ),

(4.7)

Eur. Phys. J. C (2009) 60: 375–386 381

where the full dependence on μr and μf has been kept. The

gluonic expressions g1gg and g2

gg→T are obtained with the

obvious replacement A(i)q → A

(i)g . The perturbative expan-

sions of the anomalous dimensions are universal and wellknown. We have [36]

A(1)q = 4CF ,

A(2)q = 8CF

[(67

18− ζ2

)CA − 5

9nf

], (4.8)

D(1)

QQ= 4CA,

and all gluonic quantities are given by multiplying A(i)q by

CA/CF . We also give explicit results for the matching func-tions g0

ij→T in (4.5),

g0gg→T [1] = 1 + αs

π

{

CA

[2ζ2 + 2γ 2

E − 2γE ln

(M2

μ2f

)]

+ 1

2β0 ln

(M2

μ2f

)}

,

g0gg→T [8] = g0

gg→T [1] + αs

πCAγE,

g0qq→T [8] = 1 + αs

π

{

CF

[2ζ2 + 2γ 2

E + 3

2ln

(M2

μ2f

)

− 2γE ln

(M2

μ2f

)]+ CAγE

}

.

(4.9)

For phenomenological applications [37, 38] of soft-gluonresummation at the parton level one introduces an improved(resummed) hard cross section F res, which is obtained by aninverse Mellin transformation, as follows:

F resij→T

(s,M2,μ2

f

)

=∫ c+i∞

c−i∞dN

2π ix−N

(FN

ij→T

(M2,μ2

f

)

− FNij→T

(M2,μ2

f

)∣∣NLO

) + F NLOij→T

(s,M2,μ2

f

). (4.10)

Here F NLOij→T is the standard fixed-order cross section at NLO

in QCD, while FNij→T |NLO is the perturbative truncation at

the same order in αs obtained by employing (4.4). This is tosay that for the matching we have fully expanded all formu-lae consistently to O(αs). This adds the hard coefficients Ch

of (3.2) to the results (4.4) and (4.9). In this way, the right-hand side of (4.10) reproduces the fixed-order results andresums soft-gluon effects beyond NLO to NLL accuracy.

In Sect. 6 we employ (4.10) for phenomenological pre-dictions by performing the inverse Mellin transform numer-ically. To that end, one should note that the treatment of the

Table 4.1 Comparison of the NLO and resummed result of the con-volution L ⊗ F (in 10−6 GeV−2) for LHC at the reference pointM = 2mt . The three columns correspond to the scale choices μr =μf = (mt ,2mt ,4mt). The NLO results can also be found in Ta-ble 3.2

NLO Resummed

gg → 1S[1]0 20.7 21.2 20.9 22.0 23.2 24.0

gg → 1S[8]0 63.2 62.7 60.2 67.8 69.7 70.6

qq → 3S[8]1 23.1 23.8 23.6 23.8 24.0 23.6

precise numerical matching to the exact NLO hard cross sec-tion is a matter of choice since different schemes lead onlyto differences which are formally of higher order. We havefound that the application of the resummed result is welljustified when the kinetic energy of the top-quark pair is afew GeV or less, see e.g. [25], where the precise numericalvalue is not important. Another issue concerns the constantterms in (4.9), which are sometimes modified to include for-mally sub-leading (but numerically not insignificant) terms;see for instance [37, 38]. As just explained, in the presentanalysis we adopt the minimal approach, i.e. we apply (4.9)(including the hard coefficients Ch of (3.2)) and account forall regular terms in (4.2) through matching to NLO.

In Table 4.1 we compare the fixed-order NLOand re-summed result of the convolution L ⊗ F . One observes anenhancement up to about 10% depending on the process.

5 Bound-state corrections

Let us next discuss the bound-state corrections. As men-tioned above, the convolution of Fij→T with the partonluminosities provides the normalization of the differentialcross section, while its shape is mainly determined by thenon-relativistic Green’s function. The latter describes thelong-distance evolution of the top-quark pair produced nearthreshold. The kinematics of the produced top-quark pairis non-relativistic, and the dynamics is governed by ex-change of potential gluons leading to the formation of quasi-bound states. The corresponding potential is given at NLOby

V[1,8]C (�q) = −4παs(μr)C

[1,8]

�q2

×[

1 + αs(μr)

(β0 ln

μ2r

�q2+ a1

)], (5.1)

with C[1] = CF = 4/3 and C[8] = CF − CA/2 = −1/6, anda1 = (31/9)CA − (20/9)TF nf.

382 Eur. Phys. J. C (2009) 60: 375–386

The color-singlet Green’s function feels an attractiveforce, the color-octet Green’s function is governed by re-pulsion and thus does not develop a bound state. Theyare both defined as the solutions of the Schrödinger equa-tions

{2mt +

[(−i∇)2

mt

+ V[1,8]C (�r)

]− (M + iΓt )

}

× G[1,8](�r;M + iΓt) = δ(3)(�r). (5.2)

For the Green’s function at zero distance, the NLO result isknown in a compact form [39] (see also [14])

G[1,8](M + iΓt)

≡ G[1,8](�r = 0;M + iΓt )

= C[1,8]αs(μr)m2t

[gLO + αs(μr)

4πgNLO + · · ·

],

gLO = − 1

2κ+ L − ψ(0),

gNLO = β0[L2 − 2L

(ψ(0) − κψ(1)

) + κψ(2)

+ (ψ(0)

)2 − 3ψ(1) − 2κψ(0)ψ(1)

+ 44F3(1,1,1,1;2,2,1 − κ;1)]

+ a1[L − ψ(0) + κψ(1)

],

(5.3)

with

κ ≡ iC[1,8]αs(μr)

2v, v =

√M + iΓt − 2mt

mt

. (5.4)

Here L = ln(iμr/(2mtv)) and ψ(n) = ψ(n)(1 − κ) is thenth derivative of ψ(z) ≡ γE + (d/dz) lnΓ (z) with argu-ment (1 − κ). The Green’s function in (5.3) correctly re-produces all the NLO terms in NRQCD; however, it is notsufficient to describe the behavior of the Green’s functionin the vicinity of bound-state poles. It is because the ex-act solution to the Schrödinger equation has only singlepoles in the bound-state energy G[1] ∼ |Ψ (0)|2/(Mn −M −iΓt), while (5.3) is an expansion around the LO bound-state poles and thus has multiple poles of the form G ∼|Ψ (0)

n (0)|2/(M(0)n − M)k (k = 1,2 at the NLO). However,

resummation of these multiple poles into single poles isstraightforward and well known. We refer to [39] for furtherdetails.

In Fig. 5.1 we show the imaginary parts of the color-singlet and color-octet Green’s functions in the thresh-old region. As input we use mPS

t = 170.1 GeV, which toNLO accuracy corresponds to mt = 172.4 GeV [1], andΓt = 1.36 GeV [40–42]. At NLO the Green’s function isseparately renormalization-scale invariant and we are free

to chose μr independent from the hard process. A well-motivated physical scale is μs = mtCF αs(μs) = 32.21 GeV,which corresponds to twice the inverse Bohr radius. Thecorresponding αs value used in Fig. 5.1 is α

(nf=5)s (μs) =

0.1401. It has been observed that the color-singlet CoulombGreen’s function has a well-convergent perturbative seriesfor this choice of scale [43].

In order to see the effect of Coulomb resummation, weplot for both color states three lines: the full Green’s func-tion (solid line) and the expansion of G in fixed order upto O(αs) with and without top-quark width (dashed/dotted).The upper three lines in Fig. 5.1 correspond to the color-singlet case and the lower three to the color-octet one. Thecolor-singlet Green’s function shows a pronounced peakwhich corresponds to the t t resonance below 2mt , whilefor the color octet there is no enhancement. Note that thecurve for the full octet Green’s function is very close to theone-loop expansion (taking into account the finite top-quarkwidth). Thus for the color-octet state the Coulomb resum-mation effect is negligible. In addition, one more line (dash-dotted) for the color-singlet Green’s function is plotted in-cluding the NNLO Coulomb potential, which is useful toestimate yet unknown bound-state corrections to the NLOcolor-singlet Green’s function. As input value we againadopt the PS top-quark mass [44] given above. Note that inthe absence of a full NNLO result for the Green’s functionand hard correction, this improved Green’s function wouldnot be sufficient for a full NNLO prediction. Nevertheless,the difference between solid and dash-dotted curves gives an

Fig. 5.1 Imaginary part of the Green’s functions for the color-singlet(upper solid line) and color-octet (lower solid line) cases as functionsof the top-quark invariant mass. For comparison, also the expansionsof G in fixed order up to O(αs) with (dashed) and without (dotted line)Γt are plotted. The imaginary part of the NNLO Green’s function forthe color-singlet case is shown as dash-dotted line

Eur. Phys. J. C (2009) 60: 375–386 383

indication of the intrinsic uncertainties of the Green’s func-tion, which is roughly 10%.

The expansion of G up to O(αs) is obtained from the gLO

in (5.3) as

1

m2t

ImGc = Im

[v

(i + αsC

[1,8]

v

[iπ

2− lnv

])]

+ O(α2

s

). (5.5)

In the zero-width limit (iΓt → +i0), the color-singlet curvefor the expansion exhibits a step of height αsCF /8 (forM → 2mt ), and the color-octet curve formally becomesnegative for v ≤ −αsC

[8]π/2 which corresponds to M −2mt < 0.23 GeV. Both for the singlet and octet case, thefixed-order result without Γt , the imaginary part of theGreen’s function vanishes below 2mt . The qualitative dif-ference between the solid and the short-dashed curves willbe reflected in the comparison of our final results for theinvariant-mass distribution with the prediction based on afixed-order calculation: for the color-singlet curve we ob-serve a sizable excess in the region below the nominalthreshold up to roughly 5 GeV above. In the color-octet case,as a consequence of the relative smallness of C[8], the pre-diction follows roughly the Born approximation. Althoughthe color-octet Green’s function is significantly smaller thanthe singlet one, the relatively large hard scattering factorL ⊗ F for 1S

[8]0 plus 3S

[8]1 , which exceeds the one for the

singlet case by roughly a factor 4, quickly over-compensatesthe effect of the Green’s functions.

In the present paper we use the analytical result of theGreen’s function, which includes the αs correction (i.e. thesecond term in the square brackets of (5.1)) by means ofthe Rayleigh–Schrödinger perturbation approach. In [20] anumerical solution to (5.2) has been employed, which re-sums the αs corrections to all order. The numerical solutionis more stable against scale variation and is applicable over awide range of μr. However, the difference between the twoapproaches is below 2% and formally of higher order. Ex-tensive studies on higher-order effects to the color-singletGreen’s function exist in the literature (see, e.g., [8, 43]), in-cluding different implementations of the Green’s function.From the experience collected in the linear collider studieson t t production, we expect rather large corrections fromthe variation of μr for the color-singlet Green’s function ofabout 20% which is significantly bigger than the estimatefrom the NNLO Green’s function mentioned above. In con-trast to the color-singlet case the higher-order corrections tothe color-octet Green’s function are expected to be unimpor-tant, since there is no resonance enhancement and the colorcoefficient C[8] is small.

6 Invariant-mass distribution

We are now in the position to combine the results of thepreceding sections and discuss the cross section for the in-variant top-quark distribution.

In Fig. 6.1 the invariant-mass distributions for LHC(√

S = 14 TeV) is shown for the three dominant processes.The bands reflect the scale variation of the convolutionL ⊗ F, which for the color-singlet case amounts to roughly±1%. The reduction as compared to Table 3.2 and Fig. 6.1is due to a compensation of the μ dependence after includ-ing the sub-leading NLO processes. Note, however, that thecorresponding Green’s function shows an uncertainty dueto the renormalization-scale variation of about 20% whichis well known from top-quark production studies in e+e−collisions, consistent with the difference between solid anddash-dotted curves in Fig. 5.1 and thus not discussed inFig. 6.1. This pattern is also evident from Fig. 6.2, where allproduction channels as listed in Table 3.2 are included. Thewidth of the bands is obtained from varying renormaliza-tion and factorization scales in the hard cross section as de-scribed above. The additional uncertainty from the Green’sfunction, which we estimate 20% for the singlet and below5% for the octet case, is not included.

As expected, for M < 2mt the production of t t pairs isdominated by the singlet contribution. However, for M >

2mt one observes a strong increase of the octet contribu-tions, in particular of the gluon-induced subprocess whichfor M � 2mt + 5 GeV becomes even larger than the cor-responding singlet contribution. For the color-octet case thescale dependence of the hard scattering amounts to ±7%.Considering the threshold behavior as shown in Figs. 6.1

Fig. 6.1 (Color online) Invariant-mass distributions for leading sub-processes: gg → 1S

[1,8]0 (blue and light green, respectively) and

qq → 3S[8]1 (green). For each process the bands take into account scale

variation of the hard cross sections

384 Eur. Phys. J. C (2009) 60: 375–386

and 6.2 it is clear, that the location of the threshold is en-tirely governed by the behavior of the color-singlet (S-wave)contribution. Thus, as a matter of principle, determining thelocation of this step experimentally would allow for a top-quark mass measurement, which is conceptually very differ-ent from the one based on the reconstruction of a (colored)single quark in the decay chain t → Wb. In fact, much ofthe detailed investigations of t t threshold production at alinear collider were performed for this particular relationsbetween the location of the color-singlet quasi-bound-statepole of t t and the top-quark MS-mass. The absolute normal-ization of the cross section is also sensitive to electroweakcorrections [45–49], which are of the order of 5% close tothreshold. For example, the difference between correctionsfrom a light (Mh = 120 GeV) and a heavy (Mh = 1000 GeV)Higgs boson amounts to roughly 6% [48].

In Fig. 6.3 the prediction for dσ/dM based on NRQCDis compared with the one obtained from a fixed-order NLOcalculation for stable top quarks which is obtained using theprogram HVQMNR [50]. As expected from the compari-son of solid and dotted curves in Fig. 5.1, the two predic-tions overlap for invariant masses around 355 GeV. Above355 GeV relativistic corrections start to become important.From this comparison we find an additional contribution tothe total cross section for t t production of roughly 10 pb,which could become of relevance for precision measure-ments. Note that the band of the NRQCD-based predictiononly contains the uncertainty from the scale variation ofL ⊗ F, whereas the one of the Green’s function (which canreach up to 20%; see Sect. 5) is not shown.

The analysis of this work has concentrated on the thresh-old region and is applicable for M up 360 GeV at most.

Fig. 6.2 Invariant-mass distribution including all production channelsshown in Table 3.2. The width of the bands reflect the scale dependenceof the hard scattering parts

However, it is obvious that the overall shape of dσ/dM willbe distorted and the mean 〈M〉 shifted to smaller values,which might affect the global fit of dσ/dM . In Fig. 6.4we present for comparison the NLO prediction for dσ/dM

in the wide range up to 700 GeV. The distribution reachesquickly its maximum of 3.3 pb/GeV at around 390 GeVand then falls off slowly. It is remarkable that its value at370 GeV is already not too far from the maximum of thecurve, and the threshold modifications thus affect a sizeablepart of the distribution.

Fig. 6.3 Invariant-mass distribution dσ/dM from NRQCD and for afixed NLO for LHC with

√s = 14 TeV. The bands are due to scale

variation from mt to 4mt . For the NRQCD prediction the additionaluncertainty due to the Green’s function estimated to 20% (5%) for thecolor-singlet (-octet) contribution is not included

Fig. 6.4 Invariant-mass distribution dσ/dM from NLO calculation forLHC with

√s = 14 TeV

Eur. Phys. J. C (2009) 60: 375–386 385

Although the most detailed top-quark studies will be per-formed at the LHC at an energy of 14 TeV, a sample oftop quarks has been collected at the Tevatron in proton–antiproton collisions at 1.96 TeV. Furthermore, the firstLHC data set will be taken at 10 TeV. For this reason wegive the results for these two cases; see Figs. 6.5 and 6.6. Thecross section in Fig. 6.5 has the same characteristic shape asthe one in Fig. 6.2; however, the absolute size is consider-ably smaller. As expected, the enhancement at threshold issignificantly less pronounced for Tevatron where the color-singlet contribution is very small.

Fig. 6.5 Invariant-mass distribution dσ/dM for LHC with√s = 10 TeV

Fig. 6.6 Invariant-mass distribution dσ/dM for Tevatron with√s = 1.96 TeV. At the Tevatron qq → 1S

[8]0 dominates the cross

section, and the luminosity for the gg channels is small; thus thebound-state peak is buried by color-octet production

Our analysis confirms the findings of [20]; however,the numerical results for the cross sections as presented inFig. 6.2 are slightly higher than the corresponding correc-tions of [20], which is due to the combined effect of the soft-gluon resummation, the inclusion of the NLO sub-processesand the different matching to full QCD.

7 Summary

A NLO analysis of top-quark production near threshold athadron colliders has been performed. The large width of thetop quark in combination with the large contribution fromgluon fusion into a (loosely bound) color-singlet t t systemleads to a sizable cross section for masses of the t t sys-tem significantly below the nominal threshold. A precisemeasurement of the Mtt -distribution in this region, whichis dominated by the color-singlet configuration, could leadto a top-quark mass determination which does not involvethe systematic uncertainties inherent in the determination ofthe mass of a single (color triplet) quark. Furthermore, alsothe shape of the differential distribution dσ/dM is distortedand the mean 〈M〉 shifted towards smaller values.

The effects of initial-state radiation as well as bound-statecorrections are taken into account in a consistent manner atNLO. As compared to [20] we include the complete s de-pendence in the matching condition and also implement allNLO sub-processes. We observe a partial numerical cancel-lation between these two effects leading to similar predic-tions as [20]. Furthermore, we perform a soft-gluon resum-mation and thus include the dominant logarithmically en-hanced higher-order terms. This last step stabilizes the pre-diction. However, it enhances the cross section at most by10%.

The effects are more pronounced at the LHC with topproduction being dominated by gluon fusion and less rele-vant in proton–antiproton collisions with top quarks domi-nantly in color-octet states. Considering the threshold region(say up to Mtt = 350 GeV) separately, an integrated crosssection of 15 pb is obtained, which should be compared to5 pb as derived from the NLO predictions using a stable topquark and neglecting the binding correction. Within this rel-atively narrow region, the enhancement amounts to roughlya factor 3 and a significant shift of the threshold. Comparedto the total cross section for t t production of about 840 pb(obtained using fixed-order NLO accuracy for μ = mt ; see,e.g., [25]), the increase is relatively small, about 1%. How-ever, in view of the anticipated experimental precision ofbetter than 10%, these effects should not be ignored.

Note added

While this article was finished an analytic evaluation of thetotal cross section at NLO accuracy appeared [51], which

386 Eur. Phys. J. C (2009) 60: 375–386

has been used in [20] to clarify the existence of a non-decoupling top-quark effect overlooked in [19] (see [20,footnote 3, p. 73].

Acknowledgements This work was supported by the DFG throughSFB/TR 9, by the BMBF through contract 05HT4VKAI3 and by theHelmholtz Gemeinschaft under contract VH-NG-105.

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