REVIEW OF PROPERTIES OF THE TOP QUARK FROM MEASUREMENTS AT THE TEVATRON

Review of Properties of the Top Quark From Measurements at the Tevatron

MARC-ANDRÉ PLEIER
Abstract

This review summarizes the program in the physics of the top quark being pursued at Fermilab’s Tevatron proton-antiproton collider at a center of mass energy of 1.96 TeV. More than a decade after the discovery of the top quark at the two collider detectors CDF and D0, the Tevatron has been the only accelerator to produce top quarks and to study them directly.

The Tevatron’s increased luminosity and center of mass energy offer the possibility to scrutinize the properties of this heaviest fundamental particle through new measurements that were not feasible before, such as the first evidence for electroweak production of top quarks and the resulting direct constraints on the involved couplings. Better measurements of top quark properties provide more stringent tests of predictions from the standard model of elementary particle physics. In particular, the improvement in measurements of the mass of the top quark, with the latest uncertainty of 0.7% marking the most precisely measured quark mass to date, further constrains the prediction of the mass of the still to be discovered Higgs boson.

Keywords: Top Quark; Experimental Tests of the Standard Model; Hadron-induced High-energy Interactions

PACS numbers: 14.65.Ha, 12.38.Qk, 13.85.-t, 13.85.Rm, 13.38.Be, 12.60.-i

1 Introduction

The existence of a third and most massive generation of fundamental fermions was unveiled in 1975 with the discovery of the lepton at SLAC-LBL . In 1977, the discovery of the bottom quark at Fermilab extended the knowledge of a third generation into the quark sector and immediately raised the question of the existence of the top quark as the weak isospin partner of the bottom quark.

To remain self consistent, the standard model (SM) of elementary particle physics required the existence of the top quark, and electroweak precision measurements offered increasingly precise predictions of properties such as its mass. The top quark’s large mass prevented its discovery for almost two decades, but by 1994 it was indirectly constrained to be 178 11 GeV/c . After mounting experimental evidence , the top quark () was finally discovered in 1995 at Fermilab by the CDF and D0 collaborations in the mass range predicted by the standard model. The completion of the quark sector once again demonstrated the enormous predictive power of the SM.

By now, the mass of the top quark is measured to be 172.4 1.2 GeV/c , marking the most precisely measured quark mass and the most massive fundamental particle known to date. The consequent lifetime of the top quark in the SM of  s is extremely short, suggesting that it decays before hadronizing. This makes it the only quark that does not form bound states, allowing the study of an essentially bare quark with properties such as spin undisturbed by hadronization .

The measurement of top quark pair ( ) production probes our understanding of the strong interaction and predictions from perturbative QCD, while the decay of top quarks and the production of single top quarks reflect the electroweak interaction. Measuring other properties of the top quark, such as its electric charge, the helicity of the boson in decay, the branching fraction , etc., and comparing these with predictions of the SM is a very powerful tool in searching for new physics beyond the standard model.

The top quark can also be used to constrain the mass range of the last yet to be observed particle of the standard model, the Higgs boson, because their masses and the mass of the boson are linked through radiative corrections . The Higgs boson is a manifestation of the Higgs mechanism , implemented in the standard model to provide the needed breaking of the electroweak symmetry to which the top quark may be intimately connected because of its large mass.

Because of its fairly recent discovery, the top quark’s properties have not yet been explored with the same scrutiny as those of the lighter quarks. However, in the ongoing data taking at Fermilab’s Tevatron proton-antiproton collider, an integrated luminosity of more than 4 fb has already been recorded by each of the collider experiments CDF and D0, corresponding to an increase of about a factor seventy relative to the data that was available for the discovery of the top quark. The new data can be used to refine previous measurements to higher precision that starts to become limited by systematic rather than statistical uncertainties. In addition, measurements that have never been performed become feasible, such as the first evidence for electroweak production of single top quarks and the consequent first direct measurement of the CKM matrix element , recently published by D0 and CDF .

This article is intended to provide an overview of the current status of the top quark physics program pursued at the Tevatron. Results available until the LHC startup in September 2008 have been included, utilizing samples of data of up to 2.8 fb in integrated luminosity. Previous reviews of the top quark are available in Refs. . The outline of this article is as follows: The second chapter provides a brief introduction to the standard model, with emphasis on the special role played by the top quark. Chapter 3 describes production and decay modes for top quarks in the framework of the standard model. Chapter 4 outlines the experimental setup used for the measurements described in the following sections. Chapter 5 presents studies of the production of top quarks, including measurements of cross section that form the basis for other measurements of top quark characteristics. Chapter 6 elaborates on the different results for top quark decay properties, followed in Chapter 7 by a discussion of measurements of fundamental attributes of the top quark, such as its charge and mass. The final chapter (8) contains a brief summary of the achievements to date.

2 The Standard Model and the Top Quark

2.1 A brief overview of the standard model

The standard model of elementary particle physics describes very successfully the interactions of the known fundamental spin fermion constituents of matter through the exchange of spin gauge bosons.


fermion
electric generation
type charge [] 1. 2. 3.

quarks
up () charm () top ()
1.5 - 3.3 MeV/c 1.27 GeV/c 172.4 1.2 GeV/c

down () strange () bottom ()
3.5 - 6.0 MeV/c 104 MeV/c 4.20 GeV/c

leptons
0
2 eV/c 0.19 MeV/c 18.2 MeV/c
(95% C.L.) (90% C.L.) (95% C.L.)

-1
0.511 MeV/c 105.658 MeV/c 1777 MeV/c
Table 1: The known fundamental fermions and their masses .

As shown in Table 1, both quarks and leptons occur in pairs, differing by one unit of electric charge , and are replicated in three generations that have a strong hierarchy in mass. The fermion masses span at least 11 orders of magnitude, with the top quark being by far the heaviest fundamental particle, which may therefore provide further insights into the process of mass generation. The origin of this breaking of the flavor symmetry and the consequent mass hierarchy is still not understood but can be accommodated in the standard model as shown below.

The forces among the fundamental fermions are mediated by the exchange of the gauge bosons of the corresponding quantized gauge fields, as listed in Table 2. The gravitational force is not included in the framework of the standard model, and will not be considered, as its strength is small compared to that of the other interactions among the fundamental fermions at energy scales considered in this article.

interaction couples affected exchange mass charge spin
type to particles boson [GeV/c] []
strong color quarks, gluon (g) 0 0 1
charge gluons
weak weak quarks, , , 80.4 +1, -1 1
charge leptons, 91.2 0 1
electro- electric electrically photon () 0 0 1
magnetic charge charged
gravitation mass, all graviton 0 0 2
energy (unobserved)
Table 2: The known fundamental interactions and their properties . Gravitation is shown separately as it is not included in the SM of elementary particles.

The standard model is a quantum field theory based on the local gauge symmetries . The theory of the strong interaction, coupling three different color charges (“red”, “green” and “blue”) carried by the quarks and the eight massless gauge bosons (gluons), is called Quantum Chromodynamics (QCD), and is based on the gauge group . This symmetry is exact, and the gluons carry both a color and an anticolor charge. At increasingly short distances (or large relative momenta), the interaction gets arbitrarily weak (asymptotically free), thereby making a perturbative treatment viable. Via the strong interaction, quarks can form bound color-singlet states called hadrons, consisting of either a quark and an antiquark (mesons) or three quarks respectively antiquarks (baryons). The fact that only color-neutral states and no free quarks are observed is referred to as the confinement of quarks in hadrons. Due to its large mass, the top quark decays faster than the typical hadronization time of QCD (), and it is therefore the only quark that does not form bound states. Its decay hence offers the unique possibility to study the properties of essentially a bare quark.

The theory of electroweak interactions developed by Glashow , Salam and Weinberg is based on the gauge group of the weak left-handed isospin  and hypercharge . Since the weak () interaction only couples to left-handed particles, the fermion fields are split up into left-handed and right-handed fields that are arranged in weak isospin doublets and singlets:

In the doublets, neutrinos and the up-type quarks () have the weak isospin , while the charged leptons and down-type quarks () carry the weak isospin . The weak hypercharge Y is then defined via electric charge and weak isospin to be . Consequently, members within a doublet carry the same hypercharge: for leptons and for quarks, as implied by the product of the two symmetry groups.

The gauge group does not accommodate mass terms for the gauge bosons or fermions without violating gauge invariance. A minimal way to incorporate these observed masses is to implement spontaneous electroweak symmetry breaking (EWSB) at energies around the mass scale of the and boson, often referred to as the “Higgs mechanism”, by introducing an SU(2) doublet of complex scalar fields . When the neutral component obtains a non-zero vacuum expectation value , the symmetry is broken to , giving mass to the three electroweak gauge bosons while keeping the photon massless, and thereby leaving the electromagnetic symmetry unbroken. From the remaining degree of freedom of the scalar doublet, we obtain an additional scalar particle, the Higgs boson.

The Higgs mechanism also provides fermion masses through fermion Yukawa couplings to the scalar field, with masses given by , for a Yukawa coupling constant for each massive fermion in the standard model. With a Yukawa coupling close to unity, the top quark may play a special role in the process of mass generation.

The mixing of flavor eigenstates in weak charged-current interactions of quarks is described by the Cabibbo-Kobayashi-Maskawa (CKM) matrix . By convention, this is a 3 x 3 unitary matrix that operates on the negatively-charged flavor states and :

(1)

This complex matrix can have 18 independent parameters. However, to conserve the probability, this matrix has to be unitary, which means that there are only nine free parameters. An additional five out of the nine can be absorbed as phases in the quark wave functions. This results in four independent parameters in total – three real Euler angles and one complex phase, the latter implementing CP violation in the standard model. Since the CKM matrix is not diagonal, charged current weak interactions can have transitions between quark generations (“generation mixing”) with coupling strengths of the boson to the physical up and down type quarks given by the above matrix elements.

From experimental evidence , neutrinos also have mass, which has led, among other things, to the introduction of an analogue leptonic mixing matrix, the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix . It contains four independent parameters as well if one assumes that neutrinos are not Majorana particles.

In summary, the standard model of elementary particle physics is a unitary, renormalizable theory , that can be used to perturbatively calculate processes at high energies. It incorporates 25 parameters that have to be provided through measurement:

  1. 12 Yukawa couplings for the fermion masses

  2. 8 parameters for the CKM and PMNS mixing matrices

  3. 3 coupling constants of and , respectively

  4. 2 parameters from EWSB: .

At currently accessible energy scales, the standard model describes successfully the interactions of fundamental fermions and gauge bosons, with only the Higgs boson remaining to be observed. For a more detailed introduction to the standard model, the reader is referred to corresponding textbooks, Refs.  on elementary particle physics and topical reviews such as Ref. .

2.2 The need for the top quark in the standard model

The existence of the top quark was postulated well before its discovery mainly for three reasons. The first argument reflects the desire to have the standard model correspond to a renormalizable theory. When expressed via a perturbation series – usually depicted in Feynman diagrams with first order “tree” diagrams and higher order “loop” terms – certain loop diagrams cause divergences that have to cancel exactly to ensure that the theory is renormalizable. One example is the fermion triangle diagram, as shown in Fig. 1.

Fig. 1: A “problematic” fermion triangle diagram that could introduce an anomaly.

The contribution for each such diagram is proportional to , with being the weak neutral current axial coupling strength, and the electric charge for the respective fermion in the loop. Since and neutrinos do not contribute, for the total strength of the anomaly to be cancelled, an equal number of lepton flavors and quark-doublets , and quarks in three colors () are required :

(2)

Consequently, the discovery of the lepton already called for an additional quark doublet to be present to keep the standard model renormalizable.

The second argument results from the fact that transitions that change the flavor but not the charge of a fermion ( or ) are observed to be strongly suppressed. The absence of such flavor-changing neutral currents (FCNC) for two quark generations could be accommodated through the GIM mechanism by postulating the existence of the charm quark – and thereby completion of the second quark doublet – years before its discovery. This mechanism can be applied in a similar way for three quark generations, requiring a sixth quark as a partner of the quark to complete the doublet.

The third argument comes from the experimental confirmation that the quark is not a weak isospin singlet but is part of an isospin doublet carrying the weak isospin and electric charge . The electric charge of the quark was measured first at the electron-positron storage ring DORIS at DESY operating at the and resonances through a measurement of the cross section for resonant hadron production . The integral over is related to the electronic partial width , the hadronic partial width , the total width and the resonance mass via . Assuming that the total width is dominated by the hadronic partial width (), a measurement of the integrated cross section and the resonance mass provides the electronic partial width of the and of the . In the framework of non-relativistic quarkonium potential models , this partial width can then be related with the bound quark’s charge.

The weak isospin of the quark was measured via the forward-backward asymmetry in the process  hadrons with the JADE detector at PETRA . The asymmetry originates from electroweak interference effects and is defined as the difference between the number of fermions produced in the forward direction (with polar angle ) and the number of fermions produced backward (), divided by their sum. is proportional to the ratio of the weak axial to the electric charge and vanishes for a weak isospin singlet. For a -quark, the predicted asymmetry is , in good agreement with the measurement of 6.0 (stat.) 2.5 (syst.)%.

As a result of these measurements, the top quark’s weak isospin and electric charge within the standard model were assigned to be , , well before its discovery. The mass of the top quark, being a free parameter in the standard model, could not be predicted. Nevertheless, the mass of the top quark can indirectly be constrained by precision electroweak measurements.

2.3 Top quark mass from precision electroweak measurements

As discussed above, the standard model comprises a set of free parameters that are a priori unknown. However, once these are measured, all physical observables can be expressed in terms of those parameters. To make optimal use of the predictive power of the theory, it is therefore crucial to measure its input parameters with highest possible precision, and thereby probe the self-consistency of the SM and any contributions beyond its scope. Being a renormalizable theory, predictions for any observable can be calculated to any order and checked with experiment.

Electroweak processes depend mainly on three parameters: the coupling constants of , respectively, and the Higgs vacuum expectation value . Since these input parameters have to be obtained from experiment, it is best to substitute them with the most precisely measured quantities of the electromagnetic fine structure constant (using electron-positron annihilations into hadrons at low center of mass energies to measure hadronic vacuum polarization corrections ), the Fermi constant (from the muon lifetime ) and the mass of the boson (from electron-positron annihilations around the pole ).

With these input values, the theoretical framework can be used to predict other quantities such as the boson mass. Given precision measurements, the boson mass is sensitive to the mass of the top quark and the mass of the Higgs boson through higher order radiative quantum corrections .

The most precise electroweak measurements to date have been performed at the Large Electron-Positron (LEP) Collider at CERN by the four experiments ALEPH , DELPHI , L3 and OPAL , and at the Stanford Linear Collider (SLC) by the SLD experiment . The LEP experiments have analyzed 17 million decays, and a sample of 600 thousand bosons produced with longitudinally polarized electron beams was analyzed by SLD.

Fig. 2: (a) Radiative contributions to the and propagators from the top quark. (b) Radiative corrections to the and propagators from the Higgs boson.

Defining the electroweak mixing angle via the vector boson masses:

(3)

the boson mass can be expressed as :

(4)

where the radiative correction is a directly observable quantum effect of electroweak theory that depends on and . The contributions from single-loop insertions containing the top quark and the Higgs boson, as depicted in Fig. 2, are :

(5)
(6)

Thus, a precise measurement of and boson masses provides access to the mass of the top quark and the Higgs boson. The top quark contribution to radiative corrections is large, primarily because of the large mass difference relative to its weak isospin partner, the quark. While the leading top quark contribution to is quadratic, it is only logarithmic in mass for the Higgs boson. Consequently, the constraints that can be derived on the mass of the top quark are much stronger than for the Higgs boson mass.

In 1994, the most stringent constraints on were based on preliminary LEP and SLD data, combined with measurements of in proton-antiproton experiments, and neutral to charged-current ratios obtained from neutrino experiments, yielding 178 11 GeV/c .

Fig. 3: (a) distributions of the standard model fit to precision electroweak data versus top quark mass for various Higgs boson masses . (b) Comparison of the indirect top quark mass measurements via radiative corrections (shaded area) with the direct measurements from the Tevatron (points) versus time .

As illustrated in Fig. 3, the central value and the first set of uncertainties are from a fit of the SM to precision electroweak data, assuming  GeV/c. The second set of uncertainties stems from the impact of varying between 60 and 1000 GeV/c.

The good agreement between predicted and observed values of , shown in Fig. 3 as a function of time , is one of the great successes of the SM. The latest prediction from precision electroweak data yields GeV/c without imposing constraints on  , and is in excellent agreement with the current world average of 172.4 1.2 GeV/c .

This success of the SM also gives greater confidence in the predictions for . Since the precision of the prediction depends crucially on the accuracy of and , it provides a strong motivation for improving the corresponding measurements. The current constraints on the mass of the Higgs boson will be discussed in Section 7.3.4.

More details on precision electroweak measurements can be found in topical reviews such as given in Refs. .

3 Production and Decay of Top Quarks

The production of top quarks is only possible at highest center of mass energies , set by the scale of . The energies needed for production of top quarks in the SM are currently (and will be at least for the next decade) only accessible at hadron colliders. The Tevatron proton-antiproton collider started operation at  TeV in 1987 for a first period of data taking (“Run 0”) that lasted until 1989, with the CDF experiment recording about 4 pb of integrated luminosity. The next data taking period from 1992 until 1996 at  TeV (the so-called Run I) was utilized by both the CDF and D0 experiments and facilitated the discovery of the top quark. For the currently ongoing data taking that started in 2001 (Run II), the center of mass energy has increased to  TeV. The Tevatron will lose its monopoly for top quark production only with the turning-on of the Large Hadron Collider (LHC) that will provide proton-proton collisions at  TeV.

In the framework of the standard model, top quarks can be produced in pairs () predominantly via the strong interaction and singly via the electroweak interaction.

3.1 Top quark pair production

While hadron colliders provide the highest center of mass energies, the collision of hadrons complicates the theoretical description and prediction for processes such as production because of the composite nature of the colliding particles. These difficulties can be handled through the QCD factorization theorem that provides a way to separate hadron collisions into universal long-distance (small momentum transfer) phenomena and perturbatively calculable short-distance phenomena. The latter processes involve therefore large square of momentum transfers , and consequently the production of particles with large transverse momenta or large mass. The two components are set apart by introducing a factorization scale in the calculation.

Using this approach, the proton can be described by a collection of partons (quarks, antiquarks, gluons) that interact at a low energy scale  GeV, whereas the elementary collisions between partons of the proton (or antiproton) occur on a “hard” energy scale characterized by large transverse momenta (100 GeV).

Consequently, the partons participating in any hard process () can be considered quasi-free, and the partonic cross section of interest can be calculated using perturbative QCD, independent of the type of hadrons containing the partons. (The hatted variables denote parton quantities.) The regularization of divergences in higher order calculations (such as ultraviolet divergences from loop insertions, where the infinite range of four-momentum in the loop causes infinities in the integration from high momentum contributions) requires the introduction of a renormalization scale , along with the corresponding running coupling constant . The leading order Feynman diagrams for  production are shown in Fig. 4.

Fig. 4: Leading order Feynman diagrams for top quark pair production: (a) quark-antiquark annihilation and (b) gluon-gluon fusion.

The partons within the incoming proton (or antiproton) cannot be described by perturbative QCD, as the soft energy scale corresponding to small inherent momentum transfers implies large couplings. The distribution of the longitudinal momentum of the hadron among the partons is described by Parton Distribution Functions (PDFs): , corresponding to the probability to find a given parton inside hadron with momentum fraction when probed at an energy scale . Collinear and soft (infrared) singularities that arise in the perturbative calculation of the partonic cross section discussed above are absorbed in these PDFs.

The factorization theorem is used to describe the production cross section via an integral over the corresponding hard scattering parton cross section, folded with the parton distribution functions of the incident hadrons as follows:

(7)

The hadrons and correspond to proton and antiproton in case of the Tevatron and to protons in case of the LHC.

The physical cross section that would emerge from the evaluation of the full perturbation series does not depend on either of the two arbitrary scales for factorization and renormalization that had to be introduced for the calculation. However, the parton distribution functions and the partonic cross section do depend on these scales, and hence the result of any finite order calculation will as well. This dependence gets weaker with the inclusion of higher order terms in the calculation. In practical application, both scales are usually set to the typical momentum scale of the hard scattering process, such as the transverse momenta of the produced particles or the mass of the produced particle, so that for production, typically . The scale dependence of the result is then usually tested by varying the central scale by a factor of two; the resulting variations are interpreted as systematic uncertainties that should not be mistaken as Gaussian in nature.

The PDFs have to be determined experimentally, for example via deeply inelastic lepton scattering on nucleons, so that they can be extracted from the measured cross sections using perturbative calculations of the (hard) partonic cross sections. Once the parton densities have been measured as a function of momentum fraction at a scale , their value at a different scale can be predicted perturbatively using the DGLAP evolution equation . Since PDFs are universal and do not depend on the process they were derived from, they can be used to predict cross sections in other hard scattering processes. For consistent application, it is important that the PDFs are derived to same perturbative order and with the same renormalization scheme as the calculation of any prediction.

Fig. 5: (a) CTEQ61  parton distribution functions with their uncertainty bands, for the mass scale () for (anti-) up quarks, (anti-) down quarks and gluons in the proton. (b) Relative uncertainties on the PDFs shown in (a).

The PDFs are extracted from global fits to the available data, as is done, for example, by the CTEQ , MRST , GRV , Alekhin , H1 and ZEUS groups. Different PDFs are based on different data, different orders of perturbation theory, renormalization schemes and fitting techniques – see, for example, the overview given in Ref. . One commonly used set of PDFs derived at NLO, using the renormalization scheme , is CTEQ61 , which incorporates the Tevatron Run I data on jet production, especially important for the gluon distribution. CTEQ61 also includes an error analysis based on different sets of PDFs that describe the behavior of a global function for the fit around its minimum. The resulting error on the PDF () can be obtained by summing over the variations along/against each PDF “eigenvector” for every free parameter in the global fit: .

Figure 5 shows the most important parton distributions within protons for  production at the Tevatron or LHC, and their corresponding uncertainties. (For antiprotons, quarks and antiquarks have to be interchanged in Fig. 5.) All PDFs vanish at large momentum fractions , and the gluon density starts to dominate over the valence-quark densities near . There is no flavor symmetry between the up and down quark distributions, neither on the valence nor the sea quark level (the latter is best seen at low ). At -values below 0.1, typical relative uncertainties on the PDFs of valence quarks and gluons are 5%. At larger -values, these uncertainties increase drastically, especially for gluons.

To produce a top quark pair, the squared center of mass energy at the parton level must at least equal . Assuming yields as threshold for  production:

(8)

Since large momentum fractions are required for  production at the Tevatron, the process is dominated by quark-antiquark annihilation (Fig. 4 (a)) of the valence quarks. For Run I energies, quark-antiquark annihilation contributes roughly 90% of the total  production rate, and for Run II energies this fraction is 85% .

At the LHC, gluon-gluon fusion dominates (Fig. 4 (b)) with a contribution of about 90% , because a small momentum fraction suffices for  production. This means that proton-proton collisions at the LHC have production cross sections comparable to rates, thereby obviating the need for the major technical challenge of producing an intense antiproton beam.

The increase in the center of mass energy by 10% between Run I and Run II at the Tevatron and the correspondingly smaller minimum momentum fraction provide an increase in the  production rate of 30%. At the LHC, the rate increases by roughly a factor of 100 compared to that of the Tevatron.

The highest-order complete perturbative calculations for heavy quark pair production have been available at next-to-leading order (NLO) – to order – since the late 1980s from Nason et al.  and Beenakker et al. . These calculations can be refined by the inclusion of large logarithmic corrections from soft-gluon emission that are particularly important for the production of heavy quarks close to the kinematic threshold (). The contributions of these logarithms are positive at all orders when evaluated at the heavy quark mass scale and their inclusion therefore increases the production cross section above the NLO level.

The impact of soft-gluon resummation on the  production cross section has been studied by Berger and Contopanagos , Laenen, Smith and van Neerven and Catani, Mangano, Nason and Trentadue at the leading logarithmic (LL) level. Studies including even higher level corrections as carried out by Cacciari et al. , based on work by Bonciani et al. (BCMN) , and Kidonakis and Vogt are summarized in Table 3.

Hadron Collider Processes [pb] Group
Tevatron Run I 90% 5.19 Cacciari et al. 
( TeV) 10% Kidonakis et al. 

Tevatron Run II
85% 6.70 Cacciari et al. 
( TeV) 15% Kidonakis et al. 
LHC 10% 833 Bonciani et al. 
( TeV) 90% 873 Kidonakis et al. 
Table 3: NLO cross section predictions including soft-gluon resummations beyond LL for  production at the Tevatron and the LHC for a top quark mass of 175 GeV/c. For the different sources of the quoted uncertainties please refer to the text.

In the case of  production at the Tevatron, the inclusion of leading and next-to-leading logarithmic (NLL) soft-gluon resummation affects the cross sections only mildly by (indicating production occurs not too close to the threshold), while significantly reducing the scale dependence of the predictions by roughly a factor of two to a level of 5% . At the LHC,  production takes place even further away from the kinematic threshold, but since gluon fusion dominates there, the enhancement of the total production rate from soft-gluon resummation and the reduction of scale dependence stay at the same level as at the Tevatron.

The results of Cacciari et al.  for the Tevatron use the NLO calculation with LL and NLL resummation at all orders of perturbation theory as carried out by Bonciani et al. (BCMN) , but are based on the more recent PDF sets with error analysis CTEQ6 and MRST2001E and also MRST2001 , which includes varied values in the PDF fit. The updated PDFs cause an increase in the central values of about 3% relative to Ref. . While the central values are very similar for the MRST2001E and CTEQ6 PDFs, the uncertainties for CTEQ6 are almost twice as large as for MRST2001E, unless the variations in in MRST2001 are also included. For the determination of the uncertainty on the cross section, Cacciari et al. combine linearly the uncertainty due to scale variation by a factor of two with the PDF uncertainty evaluated at that scale. As central values, the CTEQ6M results are chosen, and the maximum lower (upper) uncertainties given stem from the CTEQ6 PDF variation (the variation in MRST2001). The PDF uncertainties and variation contribute about 45% and 80% respectively to the total quoted uncertainty, including the scale variations, which emphasizes the importance of considering uncertainties in PDF fits. The PDF uncertainties are in turn dominated by the uncertainty of the gluon PDF at large values, causing, for example, the gluon fusion contribution to the total production rate to fluctuate between 11% and 21% for  TeV. Despite the large uncertainties on the  production rate, the ratio of production cross sections for the two center of mass energies at the Tevatron is very stable and predicted with high precision: for top quark masses between 170 and 180 GeV/c .

A prediction for the  production rate at the LHC applying the same level of soft-gluon resummation is given by Bonciani et al. using the MRS(R) PDF . Since no PDF uncertainties were available for Ref. , the quoted uncertainty in Table 3 comes from changing the scale by factors of two alone. Since gluon fusion is the dominant contribution to the total rate, uncertainties on the gluon PDFs alone lead to an uncertainty of on the total production cross section .

The studies performed by Kidonakis and Vogt consider soft-gluon corrections up to next-to-next-to-next-to leading logarithmic (NNNLL) terms at NNLO in a truncated resummation, resulting in a reduced sensitivity of 3% to scale variations. For the Tevatron, the  production cross section is evaluated using MRST2002 NNLO and CTEQ6M NLO parton densities. Two different parton-level conditions are considered for the scattering process: (i) one-particle inclusive (1PI) and (ii) pair-invariant mass (PIM) kinematics  . While both sets of PDFs give very similar results, the variations from the difference in kinematics are significant. Consequently, the average of 1PI and PIM kinematics for both PDFs is used as the central value in Table 3, while the separate averages over the PDFs for 1PI and PIM are quoted as uncertainties. For the predicted LHC rate, which is dominated by gluon fusion, the 1PI kinematics is considered more appropriate, and the value in Table 3 gives the corresponding result based on MRST2002 NNLO PDFs, using scale changes by factors of two for estimating the uncertainties.

All results in Table 3 are evaluated for a top quark mass of 175 GeV/c, and the Run II values serve as the main predictions for CDF and D0. To improve comparability of the uncertainties on the different predictions, the calculation by Kidonakis and Vogt has an additional uncertainty obtained from the maximum simultaneous changes in scale and PDFsaaaThe PDF uncertainties in this case stem from CTEQ6 sets “129” and “130” alone. added in quadrature with the uncertainty due to the dependence on kinematics .

In spring 2008, Cacciari et al.  and Kidonakis et al.  updated their predictions using more recent PDFs such as CTEQ6.6M , which had only little impact on the results. In addition, Moch and Uwer  have now performed a complete NNLL soft-gluon resummation and provide an approximation of the NNLO cross section also based on CTEQ6.6M. To illustrate the dependence of the predictions on the top quark mass, Fig. 6 shows the central values and uncertainties from References  for Tevatron Run II versus . An exponential form, as suggested in Ref. , is applied in a fit to the central values and uncertainties for Kidonakis et al., while third-order polynomials, as provided by the authors, are used for the other references. The total uncertainties are obtained by linearly combining the provided uncertainties.

Fig. 6: Dependence of the  production cross section on in Run II of the Tevatron. The current world-averaged top quark mass and the resulting expected  production cross section are indicated by the vertical and horizontal lines, respectively. The predictions are based on the CTEQ6.6M PDFs .

For the current world-averaged top quark mass of 172.4 1.2 GeV/c , the predicted  production cross section is  pb for Cacciari et al.,  pb for Kidonakis et al., and  pb for Moch et al.. An additional uncertainty of  pb arises from the uncertainty on the top quark mass for all three predictions. It should be noted that these predictions based on MRST 2006 NNLO PDFs  yield about 6% higher central values and exhibit smaller uncertainties from PDFs.

Fig. 7: Cross sections for various processes at hadron colliders as a function of center of mass energy . denotes the  production cross section.

A precise measurement of the  production cross section provides a test of the predictions for physics beyond the SM. Together with a precise mass measurement, the self-consistency of the predictions can also be examined. Because  production is a major source of background for single top production (to be discussed in the next section), standard model Higgs boson production and many other phenomena beyond the SM, its accurate understanding is crucial for such studies.

Figure 7 illustrates the production rates of various processes versus center of mass energy for proton-antiproton collisions below  TeV and for proton-proton collisions above  TeV. As can be appreciated from the plot,  production is suppressed by ten orders of magnitude relative to the total interaction rate at the Tevatron and eight orders of magnitude at the LHC. While the LHC is often referred to as a “top-factory” because of the increased production cross section by two orders of magnitude, extraction of the signal from the large background is a challenge at both hadron colliders, requiring efficient triggers and selection methods. The  cross section measurements performed in Run II of the Tevatron will be described in Section 5.1.

3.2 Single top quark production

Fig. 8: Representative Feynman diagrams for electroweak single top quark production: (a) -channel, (b,c) -channel and (d,e) associated production.

In addition to the strong pair production discussed in the previous section, top quarks can also be produced singly via the electroweak interaction through a vertex (see Fig. 8). and vertices are strongly CKM suppressed (see Section 3.3.1). There are three different production modes, classified via the virtuality (negative of the square of the four-momentum ) of the participating boson :

  1. The Drell-Yan-like -channel production proceeds via quark-antiquark annihilation into a time-like virtual boson (), as illustrated in Fig. 8: .

  2. In the -channel “flavor excitation” process, a space-like virtual boson () couples to a quark from the nucleon’s sea to produce a top quark, as shown in Fig. 8 for . A higher order contribution of comes from gluon splitting, as depicted in Fig. 8, which is also referred to as W-gluon fusion for .

  3. In associated production, an on-shell boson () is produced together with a top quark from a quark and a gluon, as illustrated in Figs. 8 and 8 for .

In the above discussion, charge conjugate processes are implied for each production mode, and represents a light-flavor quark. All three modes differ in both their initial and final states, and the processes are simply denoted as -channel (), -channel () and associated () production. The corresponding signatures can be used to discriminate between the production modes: The -channel is characterized by an additional quark accompanying the top quark, the -channel by a forward light quark, and associated production by the decay products of the boson in addition to those of the top quark. Due to the incoming quark and gluon, the -channel and -channel rates are especially sensitive to the corresponding PDFs, which are known with less precision than the PDFs for the valence quarks of the proton. The measured cross sections will therefore provide further constraints on the quark and gluon PDFs.

The cross sections for all three modes have been evaluated at NLO, including radiative corrections of : s-channel , t-channel , and -channel . (The most recent references provide differential distributions.) Subsequent calculations also include top quark decay at NLO for s-channel , -channel , and -channel , and latest NLO calculations include higher-order soft-gluon corrections up to NNNLO at NLL accuracy .

Hadron Collider [pb] [pb] [pb] Group
Tevatron Run I    — Sullivan 
( TeV)
Tevatron Run II    — Sullivan 
( TeV) Kidonakis 
   — Sullivan 
LHC    —
( TeV) Kidonakis 
Table 4: Cross sections for -channel (), -channel () and associated () single-top production at NLO (Sullivan), and NLO, including soft-gluon resummations (Kidonakis), expected at the Tevatron and at the LHC, for a top quark mass of 175 GeV/c. (For sources of the quoted uncertainties see the text.)

Table 4 summarizes the expected single-top production cross sections at the Tevatron and the LHC for the NLO calculations by Sullivan  (based on the work of Harris et al. in ) and NLO results including soft-gluon resummations by Kidonakis  (based on his work in Refs. and matching to the exact NLO results of Harris et al.  and Zhu ). Both results use current PDFs and include corresponding uncertainties.

While top and antitop production are identical at the Tevatron for all production modes, at the LHC this is only the case for associated production. Consequently, the results given for the Tevatron include both top and antitop production but are given separately for the LHC.

The NLO results of Sullivan are based on CTEQ5M1 PDFs  for their central values. The uncertainties for PDFs are derived from CTEQ6M , and added in quadrature with uncertainties from changes in scale by the usual factors of two, changes in top quark mass by 4.3 GeV/c (using an older world-averaged  GeV/c ), and uncertainties in quark mass and , the latter two being negligible. The rate dependence on the top quark mass is approximated as linear and is especially important for the -channel, since a change from 175 GeV/c to the current world-averaged  GeV/c raises the rates at the Tevatron by 7% for the -channel and 5% for the -channel. The observed uncertainties in scale are reduced relative to LO results, and amount to 4-6% at the Tevatron and 2-3% at the LHC.

The NLO calculations of Kidonakis that include higher order soft-gluon corrections provide single-top production cross sections based on MRST2004 NNLO PDFs . The quoted values are obtained by matching the NLO cross section to the results of Harris et al.  and Zhu , and including the additional soft-gluon corrections up to NNNLO. Exceptions are the rate at the Tevatron, where no corresponding NLO result is available, and the given value is therefore not matched, and the -channel rate at the LHC, where no soft-gluon corrections are considered and an updated NLO result with the quoted PDFs is given instead. The uncertainties given are derived from varying the scale by a factor of two, and adding in quadrature PDF uncertainties derived using the MRST2001E NLO PDFs . No uncertainty in is included. At the Tevatron, the -channel uncertainty is dominated by the uncertainty in PDFs, and corrections from soft-gluon resummations relative to LO are small (5%). In contrast, the soft-gluon corrections have a large effect (60%) for the -channel at the Tevatron and scale uncertainties dominate over those from PDFs.

At the Tevatron, -channel production dominates the total rate of single top quark production with a contribution of 65%, followed by -channel production at 30%. Associated () production at the Tevatron contributes only 5% to the total rate, and is usually neglected. At the LHC, -channel production again dominates at 74%, followed now by associated production at 23%, while -channel production contributes only 3% because of the missing contribution from valence antiquarks in the collisions, which will make it difficult to discriminate this channel from background. Despite being an electroweak process, single top production has a cross section of the same order of magnitude as  production (of of the  rate at both the Tevatron and the LHC). With only one heavy top quark to be produced, single top production is accessible at smaller and therefore better-populated momentum fractions of the partons. Furthermore, no color matching is required for the production. The fact that the observed yields of single-top and -pairs are consistent with theory is a major triumph of the SM.

The measurement of single top production offers a check of the top quark’s weak interaction, and direct access to the CKM matrix element , as the cross sections in all three production modes are proportional to . The polarization of the top quark at production is preserved due to its short lifetime and provides a test of the structure of the weak interaction via angular correlations among the decay products . All three production modes provide different sensitivity to various aspects of physics beyond the standard model (BSM) , which makes their independent reconstruction a desirable goal. The -channel is sensitive to the existence of new charged bosons (such as or charged Higgs) that couple to the top-bottom weak-isospin doublet, an effect that could be detectable through an enhancement of the observed cross section. Such effects would not be observed in the mode, where the boson is on-shell, or in the t-channel, where the virtual boson is space-like and cannot go on-shell as in the -channel. The -channel production rate could be enhanced via FCNC processes involving new couplings between the up-type quarks and a boson (Higgs, gluon, photon, ). This would be hard to observe in the -channel, since there is no quark in the final state, which is essential for discrimination of the signal in that production mode. Finally, the channel is the only mode that provides a more direct test of the vertex since the boson appears in the final state.

A thorough understanding of single top quark production will also facilitate the study of processes exhibiting a similar signature such as SM -Higgs production or BSM signals to which single top production is a background process. Despite a production rate similar to that of , the signature for single top quark production is much harder to separate from background, which has delayed first measurements until very recently. The current analyses at the Tevatron provide first evidence for production of single top quarks, and this will be described in Section 5.8.

3.3 Top quark decay

3.3.1 Top quark CKM matrix elements

Since the mass of the top quark is larger than that of the boson, decays , with being one of the down-type quarks , are dominant. The contribution of each quark flavor to the total decay width is proportional to the square of the respective CKM matrix element . Utilizing the unitarity of the CKM matrix and assuming three quark generations, the corresponding matrix elements can be constrained indirectly at 90% confidence level to :

(9)
(10)
(11)

Consequently, the decay is absolutely dominant and will be considered exclusively throughout this article, unless noted to the contrary. Potential deviations from the SM decay will be discussed in Section 6.

It should be noted that the above constraints on the CKM matrix elements would change dramatically (especially ) if there were more than three quark generations. Assuming the unitarity of the expanded matrix, the limits become :

(12)
(13)
(14)

It is therefore important to constrain these matrix elements through direct measurements, as outlined below.

The and matrix elements cannot be extracted from lowest-order (tree level) top decays in the framework of the standard model, but can be inferred from B-meson mixing, as shown in Fig. 9. While all up-type quarks can contribute in the depicted box diagrams, the contribution from the top quark is dominant . The oscillation frequency given by the mass difference between heavy and light mass eigenstates, for and for oscillations, is proportional to the combination of CKM matrix elements and , respectively.

Fig. 9: Feynman “box” diagrams for and mixing.

The mass difference for the system is  . Using CKM unitarity and assuming three generations, yielding , translates into  , where the uncertainty arises primarily from the theoretical uncertainty on the hadronic matrix element, which is obtained from lattice QCD calculations. To reduce these theoretical uncertainties, a measurement of the ratio, in which some uncertainties cancel, is more desirable (). With the recent first measurement of in -oscillations by D0 and CDF at the Tevatron , yielding 17 ps ps at 90% C.L. and , this ratio has now been measured for the first time as . These results are in good agreement with SM expectations.

The direct measurement of the matrix element without assuming three quark generations and unitarity of the CKM matrix is only possible via single top quark production (described in Section 3.2), because the production rate in each channel is proportional to . One way to assess the relative size of compared to and is to measure the ratio of the top quark branching fractions, which can be expressed via CKM matrix elements as

(15)

Assuming three generation unitarity, the denominator in the above expression equals one, and constraints on can be inferred. The current status of these measurements is discussed in Section 6.2.

The most precise extraction of the top quark CKM matrix elements proceeds via global fits to all available measurements, imposing the SM constraints of three generation unitarity, as done by the CKMfitter  or UTfit  groups. The CKMfitter update for summer 2008 yields :

(16)
(17)
(18)

3.3.2 Decay width of the top quark

The decay width of the top quark in the SM, including first-order QCD corrections, can be expressed as follows :

(19)

where the above formula assumes , and ignores corrections of and . While the above QCD corrections lower the width by 10%, first-order electroweak corrections increase the width by 1.7% . However, the electroweak correction is almost cancelled when the finite width of the boson is taken into account, thereby decreasing the width again by 1.5% . Corrections to the top quark width of have also been evaluated  and reduce the width by 2%. Including all these effects, the decay width is predicted to a precision of 1%. The other SM decays, and , contribute negligibly to the total decay width because of proportionality to and .

Equation 19 yields the top width to better than 2% accuracy, and the width increases with . For and  , is 1.02/1.26/1.54 GeV for top quark masses of 160/170/180 GeV/c.

The resulting lifetime of the top quark is approximately  s, and significantly shorter than the hadronization time  s. As a consequence, the top quark decays before it can form hadrons, and in particular there can be no  bound states (toponium), as was already pointed out in the 1980s . Nevertheless, although the top quark can generally be considered as a free quark, residual non-perturbative effects associated with hadronization should still be present in top quark events, and the fragmentation and hadronization processes will be influenced by the color structure of the hard interaction.

In electron-positron annihilation, top quark pairs are produced in color singlet states, so that hadronization before decay depends mainly on the mass of the top quark and collision energy. In hadronic  production, and are usually produced in color octet states and form color singlets with the proton and antiproton remnants. The energy in the color field (or in the string when using the picture of string fragmentation) is proportional to the distance between top quark and the remnant. If a characteristic length of about 1 fm is reached before the top quark decays, light hadrons can materialize out of the string’s energy. The possibility for such string fragmentation depends strongly on the center of mass energy in the hadron collisions. For Tevatron energies, this can be neglected , while it may be more important at LHC energies, where top quarks are produced with sizeable Lorentz boosts. Since heavy quarks have hard fragmentation functions and the fractional energy loss of the top quarks is therefore expected to be small, it will be difficult to experimentally establish these effects directly, even at the LHC. In case no string fragmentation takes place before the top quark decays, long-distance QCD effects will still connect the decay products of the top quark.

With top quark mass measurements aiming at uncertainties of GeV/c, it becomes more and more important to assess the impact of such non-perturbative effects on the measurements. One example that may play an important role in this context is the possibility of color reconnections before hadronization, and the corresponding modeling of the underlying event (beam-remnant interactions) .

3.3.3 Helicity of the boson

Top quark decay in the framework of the standard model proceeds via the left-handed charged current weak interaction, exhibiting a vector minus axial vector () structure. This is reflected in the observed helicity states of the boson, which can be exploited to examine the couplings at the vertex .

The emitted quark can be regarded as massless compared to the top quark, and hence expected to be predominantly of negative helicity (left-handed), meaning that its spin points opposite to its line of flight. The emitted boson, being a massive spin-1 particle, can assume any of three helicities: one longitudinal () and two transverse states (, left-handed and , right-handed). To conserve angular momentum in the decay, the spin projection of the boson onto its momentum must vanish if the quark’s spin points along the spin of the top quark, while a left-handed boson is needed if the quark’s spin points opposite to the spin of the top quark. In the limit of a massless quark, a right-handed boson cannot contribute to the decay, as illustrated in Fig. 10. For the decay of an antitop quark, a left-handed boson is forbidden.

Fig. 10: Angular momentum conservation in top quark decay does not allow right-handed bosons when quarks are assumed massless.

At lowest “Born”-level, the expected fractions of decays with different boson helicities, taking the finite quark mass into account, are given by :

(20)
(21)