Kirsten Tollefson Erich W. Varnes · Kirsten Tollefson Department of Physics and Astronomy,...

P1: FKZ/FOK P2: FDX/FGM QC: FDX/FFX T1: FDX

October 11, 1999 17:36 Annual Reviews AR094-11

?Annu. Rev. Nucl. Part. Sci. 1999. 49:435–79

Copyright c© 1999 by Annual Reviews. All rights reserved

DIRECT MEASUREMENT OF THE

TOP QUARK MASS

Kirsten TollefsonDepartment of Physics and Astronomy, University of Rochester, Rochester, New York14627; e-mail: [email protected]

Erich W. VarnesDepartment of Physics, Princeton University, Princeton, New Jersey 08544;e-mail: [email protected]

Key Words Tevatron, CDF, DØ, standard model, Higgs boson

■ Abstract We review the direct measurements of the top quark massMt using thesample oft t events collected by the DØ and CDF experiments at Fermilab. Measure-ments using events in the lepton plus jets, dilepton, and all-hadronict t decay modesare reviewed, as is the combination of the results to yield the current world averageMt = 174.3± 3.2 (stat.)± 4.0 (syst.) GeV/c2. We close by estimating the precisionattainable with future data sets at Fermilab and the Large Hadron Collider.

CONTENTS

1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4362. Top Production. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437

2.1 The Lepton-Plus-Jets Channel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4382.2 The Dilepton Channel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4392.3 The All-Hadronic Channel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4392.4 b-Tagging. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440

3. Detectors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4403.1 Design Features. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4403.2 Particle Identification. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441

4. Lepton Plus Jets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4454.1 Kinematic Fitting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4454.2 DØ Analyses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4454.3 CDF Analyses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454

5. Dilepton. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4595.1 DØ Analyses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4605.2 CDF Analyses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466

6. All-Hadronic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469

0163-8998/99/1201-0435$08.00 435

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?436 TOLLEFSON ■ VARNES

6.1 CDF Analyses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4697. Combined Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4718. Future Prospects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473

8.1 Fermilab Run II. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4738.2 Measurement ofMt at the Large Hadron Collider. . . . . . . . . . . . . . . . . . . 476

9. Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477

1. INTRODUCTION

The standard model (1) of particle physics unifies the strong and electroweak inter-actions into a single gauge theory based on the group SU(3)C⊗SU(2)L ⊗U(1)Y.An essential element of the model is its family or generation structure, in whichthe fundamental particles are grouped into three parallel families of leptons andquarks (shown in Table 1). Each family consists of two isospin doublets: one lep-ton doublet and one quark doublet. The eventual discovery (2, 3) of the top quarkwas not a surprise; the standard model requires there to be a weak isospin partnerto the bottom quark. What was surprising was that it took almost 20 years, afterthe discovery of the bottom quark, to find the top quark.

The prolonged search for the top quark was a result of its large mass. The topquark is the heaviest fundamental particle known; it is 35 times more massivethan its partner, theb quark. Table 2 lists the approximate masses for the threegenerations of quarks. At this scale, the top mass becomes the dominant parameterin corrections to electroweak processes and is close to the electroweak scale, whichsuggests that the top quark may play a role in the origin of fermion masses.

The mass of the top quark is a fundamental parameter of the standard model.However, like the masses of other fermions, its value is not predicted by the standardmodel. Instead, extensions to the theory relate the top quark mass to an underlyingsmaller set of fundamental parameters. With very accurate measurements of thetop mass, global fits combining it with other experimental information can be usedto test for consistency and predict unknown parameters of the standard model. One

TABLE 1 The generations of the standard model

GenerationElectric Isospin

I II III charge (Q) (I3 )

Leptonsνe νµ ντ 0 1

2

e µ τ −1 −12

Quarksu c t 2

312

d s b −13 −1

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?TOP QUARK MASS 437

TABLE 2 List of the quarkmasses in GeV/c2

Mass (GeV/c2)

u ∼0.005

d ∼0.009

s ∼0.170

c ∼1.4

b ∼4.4

t ∼175

of the most notable predictions is the mass of the Higgs boson (MH ), which is aremnant of the mechanism that gives rise to spontaneous electroweak symmetrybreaking. Direct, precision measurements of the masses of theW boson (MW) andof the top quark (Mt ) provide an indirect constraint on the Higgs boson mass via topquark and Higgs boson electroweak radiative corrections toMW. Figure 1 showsthe standard-model predictions for various Higgs boson masses as a function ofMW andMt .

CDF and DØ, the two colliding-beam detectors at Fermilab, made the firstdirect measurement of the top quark mass. They obtained a value between 170 and200 GeV/c2 (1, 2, 4) with an uncertainty of approximately 10%. Since then, eachexperiment has collected a total of∼110 pb−1 of data (this collection is referred toas Run I) and has presented several mass measurements using various techniquesin each of the top quark decay channels. We present an overview of the top quarkmass measurements from both experiments. The methods used to measure themass and its uncertainty are described for each of the decay channels. In recentmonths, CDF and DØ have worked jointly to combine their measurements, so thatthe top quark mass is known to a fractional precision better than any other quarkmass. At this writing, both experiments are preparing for the start of Run II in thespring of 2000 with the hope of collecting 2 fb−1 of data—an amount 20 timeslarger than the current sample. We conclude by estimating the precision achievableon the top mass for Run II and discuss briefly the possibilities for the Large HadronCollider (LHC).

2. TOP PRODUCTION

In pp collisions at the Tevatron, the dominant form of top quark production ist t pair production viaqq annihilation. (Heret and t refer to the top quark andantiquark;q andq refer to any of the lighter quarks and antiquarks.) In the standardmodel,t or t quarks are expected to decay through the electroweak interaction toa final state consisting of a realW boson andb quark nearly 100% of the time.

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Figure 1 Theoretically allowed values for the mass of a standard-model Higgs boson(MH ) as a function of the mass of theW boson and the top quark. The shaded bandsrepresent Higgs masses of 100, 250, 500, and 1000 GeV/c2. The current world’s bestdirect measurements ofMW andMt show the constraints placed onMH (point).

Figure 2 shows the Feynman diagram for top quark production and standard-modeltop quark decay. The subsequent decays of theW bosons lead to many final states.Top events are classified into three categories according to the decay channels ofthe two W bosons: the lepton-plus-jets channel, the dilepton channel, and theall-hadronic channel.

2.1 The Lepton-Plus-Jets Channel

In the lepton-plus-jets channel, oneW decays leptonically (W → `ν) and theother decays hadronically (W→ qq′). (Here` refers to any of the charged leptons

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?TOP QUARK MASS 439

Figure 2 Tree-level Feynman diagram for top quark production byqq annihilationand standard-model top quark decay.

e,µorτ .) Events are characterized by one lepton1 with high transverse momentum(pT ), an imbalance in energy from the undetected neutrino, and normally fourjets, each created by the hadronization of a final-state quark (q, q′, b, b). Thelepton (eorµ)-plus-jets channel represents about 30% of thet t decays but suffersfrom a large background ofW plus multijet production. In addition, a significantbackground arises from multijet events in which one jet is misidentified as a lepton,and mismeasurement of the jet energies creates a spurious missing transverseenergy,E/T .

2.2 The Dilepton Channel

In the dilepton channel, bothWs decay leptonically. Events are characterizedby two high-pT leptons of opposite charge, substantial missing energy from twoneutrinos, and two jets. This channel has significantly less background than thelepton-plus-jets channel, but it suffers from a very small branching fraction of5%. The backgrounds for this channel come from directbb, W W, Z → ττ , andDrell-Yan production, as well as lepton misidentification.

2.3 The All-Hadronic Channel

In the all-hadronic channel, bothWs decay hadronically and the final state appearsas six jets. The all-hadronic channel accounts for 44% of allt t decays. It has theadvantage that there should be no missing energy, since there are no neutrinos inthe final state. However, this channel is dominated by a very large backgroundfrom QCD multijet production.

1Because tau leptons are difficult to identify in thepp collider environment in whicht tevents are produced, only events with electrons or muons are used for the mass measurement.

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2.4 b-Tagging

The backgrounds to all of the decay channels can be greatly reduced by identifying,or “b-tagging,” the bottom quark(s) from thet or t decay. There are two methods ofb-tagging. The first method, soft-lepton tagging, searches for additional leptons inthe event originating from the semileptonic decay ofB hadrons,b→ `X or b→c→ `X, with`being a muon or electron. The secondb-tagging method, displaced-vertex tagging, uses the long lifetime ofB hadrons to locate decay vertices thatare displaced from the primary interaction point. The displaced-vertex taggingmethod has the advantage that the fraction ofB hadrons that travel a discernibledistance prior to their decay is greater than the fraction that decays semileptonically.Also, the leptons from semileptonicB decay tend to have relatively low energyand to be close to many other tracks, making them difficult to identify. Bothof these factors make displaced-vertex tagging more efficient than soft-leptontagging. Another disadvantage of soft-lepton tagging is that semileptonic decays ofB hadrons produce a neutrino in addition to the charged lepton, and this undetectedneutrino carries away energy from the jet. Fluctuations in the amount of undetectedenergy degrade the resolution for these jets.

3. DETECTORS

The Fermilab Tevatron, operating inpp collider mode, has sufficient center-of-mass energy (1.8 TeV) for the production of top quark pairs. No other facility iscurrently capable of top quark production, so the entire sample of top quark eventscomes from data recorded by the DØ and CDF experiments located at differentbeam-crossing regions of the Tevatron. This section describes the features of theselarge, multipurpose detectors. Complete descriptions of the DØ (6) and CDF (7)detectors are available elsewhere.

3.1 Design Features

The DØ and CDF detectors are of largely similar design. Both are nearly symmetricin azimuth and cover as much of the solid angle around the beam collision point as ispossible while still allowing for the passage of the beam pipe and detector services.Both consist of three major subsystems: a central detector designed primarily tomeasure the trajectories of charged particles, calorimeters to measure electron,photon, and hadron energies, and a system to detect the passage of muons. A briefsummary of the technologies used for these systems in each detector is given inTable 3, and Figures 3 and 4 illustrate their construction.

Despite the general similarity, there are significant differences in design. Mostobviously, CDF’s tracking system is immersed in a 1.4-T magnetic field whereasDØ has no field in the tracking region. Also, CDF possesses a silicon microstripvertex detector whereas DØ’s tracking uses wire chambers exclusively. These

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?TOP QUARK MASS 441

TABLE 3 Summary of parameters and performance for the DØ and CDF detectors

DØ CDF

Weight (tons) 5500 5000

Dimensions(x × y× z)a 13 m× 11 m× 17 m 8 m× 9 m× 26 m

Tracking detectors Jet cell drift chambers Silicon microstrip detectorJet cell drift chambers

Tracking coverage |η| < 3.2 Silicon: |η| < 2, |z| < 50 cmDrift chambers:|η| < 3.3

Magnetic field None 1.4 Tesla solenoid

Track resolutionb σ(φ) ≈ 2.5 mrad σ(pT )/pT = 0.0009pT ⊕ 0.0066

σ(θ) ≈ 28 mrad σ(xy imp. par.)= 17µm

Calorimeters Liquid argon sampler/ Scintillator sampler/Lead/ironUranium/copper absorber absorber

Calorimeter coverage |η| < 4 |η| < 3.6

σ(E)/E for e, γ ≈15%/√

E ≈17%/√

E

σ(E)/E for hadrons ≈41%/√

E ≈50%/√

E

Muon detectors Proportional drift tubes Proportional wire chambers

Muon coverage |η| < 1.7 |η| < 1

Acceptance ≈80% ≈70%

Particle ID Transition radiation Nonedetector (electrons)

aDØ and CDF use right-handed coordinate systems withzaxis parallel to the beam,x axis horizontal, andy axis vertical. Thepolar angleθ , azimuthal angleφ, and radial distance from the beamr are also commonly used. The polar coordinate is oftenexpressed in terms of the pseudorapidityη defined as−ln tan θ

2 .bFor all parameterizations of resolution, energies are in GeV and momenta in GeV/c.

differences, and others, affect how particles are identified (and therefore how topquark physics analyses are performed) at the two detectors.

3.2 Particle Identification

The t t final state consists in general of hadronic jets, charged leptons, and neutri-nos. The ability to efficiently identify all of these particles is crucial fort t analysis.In both detectors, identification of electrons begins with an isolated cluster of en-ergy in the electromagnetic (EM) portion of the calorimeter, with a matchingtrack from the central detector. There are two major backgrounds to this signa-ture: photon conversions in the central detector and charged hadrons for which theenergy deposition fluctuates to appear electron-like. (In DØ, the background in-cludes an additional component: low-energy hadrons with a flight path that nearly

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Figure 3 Cutaway view of the DØ detector, showing the tracking chambers, calorimetry,and muon system.

coincides with that of an energetic photon. CDF’s magnetic field allows the useof energy/momentum matching to reject such candidates; in its dilepton eventselection, DØ uses information from a transition radiation detector to suppresspions.)

CDF and DØ have several techniques in common for reducing these back-grounds. The detailed shape of the EM shower, spatial match between the showerand extrapolated track direction, and requirements that the EM cluster be isolatedfrom any hadronic energy deposition reduce the background due to hadrons. InCDF, photon conversions are suppressed by the requirement that there be a trackin the vertexing chamber and that the track cannot be combined with an oppositely

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?TOP QUARK MASS 443

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charged track to form a low-mass vertex. DØ uses dE/dx information to discrim-inate against doubly ionizing tracks. Overall, the presence of the magnetic fieldin CDF provides advantages for electron identification. On the other hand, DØhas superior acceptance (out to|η| = 2.5, compared with|η| = 1.35 for CDF),slightly better energy resolution, and simpler handling of bremsstrahlung (photonstend to travel along the same direction as their parent electron, and their energysimply contributes to the EM cluster).

The techniques for muon identification are similar for both detectors. A track inthe muon chambers is matched to the extrapolation of a central track, with energydeposition in the calorimeter consistent with the passage of a minimum ionizingparticle (CDF’s dilepton analyses permit muons that are outside the acceptanceof the muon chambers, based on tracking and calorimeter information). DØ hassufficient material in its calorimeter and muon toroid that hadronic punch-throughis negligible, whereas CDF imposes explicit isolation criteria to reduce this back-ground. CDF’s larger tracking volume also produces a larger background fromdecays of long-lived particles; requiring that the track have a small impact param-eter with respect to the interaction vertex reduces this background. The situationfor muons is the opposite of that for electrons—DØ has superior ability to identifymuons, whereas CDF, thanks to its ability to measure the momentum of the centraltrack, has far superior muon momentum resolution.

Any calorimeter energy clusters that are not identified as electrons or photonsare assumed to arise from hadronic jets. The jets are reconstructed by summingthe energy deposited in a cone of radiusR ≡

√1η2+1φ2 about the jet direc-

tion. (The direction is first estimated crudely and is reevaluated after the initialcone clustering. The process is iterated until the jet direction is stable.) The onlysignificant procedural difference between CDF and DØ is in the size of the conechosen. DØ usesR = 0.5, whereas CDF usesR = 0.4.

Neutrinos, of course, are not observed directly but are inferred from the imbal-ance of energy in the transverse plane. The technique is similar in both experiments.Superior hermeticity and coverage of the DØ calorimeter (to|η| = 4.0, comparedwith |η| = 3.6 for CDF) result in somewhat betterE/T resolution, but the differenceis not significant fort t physics.

Although the two experiments have somewhat different strengths in the identifi-cation of final-state particles, it is the ability to identify thebquarks from the decaysthat most clearly separates them. As discussed above, there are two signatures forb quarks: leptons arising from semileptonic decay and displaced vertices causedby their finite lifetime. Both experiments are about equally adept at identifyingleptons fromb decay (DØ does better at tagging muons but is unable to isolatetagging electrons, whereas CDF uses both electrons and muons), but only CDF hasa precision vertex detector allowing detection of displaced vertices. This differ-ence in detection ability has a tremendous impact on the mass measurement in thelepton-plus-jets channel, where CDF can employ a much simpler event selection,and in the challenging all-hadronic channel, where only CDF has published a topquark mass measurement.

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?TOP QUARK MASS 445

4. LEPTON PLUS JETS

4.1 Kinematic Fitting

Measurement of the top quark mass in the lepton-plus-jets sample begins by fittingeach event in the sample to the hypothesis oft t production:

pp→ t + t + X,

followed by the decays

t → W+ + b,

t → W− + b,

W± → l± + ν,and

W∓ → q + q′.

The three-momenta of the lepton and theb, b, q, andq′ quarks are measured fromthe observed lepton and four leading jets in the event. The mass of theb quark is≈5 GeV/c2, and the masses ofq andq′ are less than 1 GeV/c2. The neutrino massis assumed to be zero and its momentum is not measured, thereby yielding threeunknowns. The two tranverse momentum components ofX are measured fromthe extra jets in the event and the energy that is detected but not associated withjet or electron clusters. Five constraints are applied: the transverse momentumcomponents of the entiret t + X system must be zero, the invariant masses of thelepton-neutrino andq − q′ pairs must each equal theW boson mass, and the massof the top quark must equal that of the anti-top quark. The problem, therefore, hastwo extra constraints and is solved by a standardχ2 minimization technique. Theoutput of each event fit is an estimated top massMfit and aχ2 value quantifyinghow well the event is described by thet t hypothesis.

There are twelve distinct ways of assigning the four leading jets to the four par-tonsb, b, q, andq′. In addition, there is a quadratic ambiguity in the determinationof the longitudinal component of the neutrino momentum. This yields up to 24different configurations for reconstructing an event according to thet t hypothesis.The combinatorics are reduced in events having one or two jets tagged asb jets.In this case, only the combinations with the tagged jets assigned asb jets are con-sidered. The combination with the lowestχ2 providesMfit for the event. Eventsfor which no combination yields a fitχ2 < 10 are rejected by both DØ and CDF.

4.2 DØ Analyses

The first measurement of the top mass by DØ was based on≈50 pb−1 of data andhad a statistical precision of about 20 GeV/c2 (1). DØ’s most recent measurement,using about twice as much data, has a statistical precision of 5.6 GeV/c2 (8),

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even though the kinematic fitting technique used in the two analyses was nearlyidentical. The improvement in precision, a factor of 2/pb−1, is almost entirely dueto improvements in the way events were selected for the mass analysis.

The original analysis was based on the same events that led to the first obser-vation of the top quark. The selection criteria for these events were designed to behighly sensitive to the presence of a heavy (>160 GeV/c2) quark, and they placeda stringent requirement on the sum of the transverse energies of the jets,HT .2 Thisrequirement was effective in eliminating background events, in which the jets ariseprimarily from gluons radiated from the incoming partons. Such jets tend to havesmall transverse energies.

The drawback to such a selection for a mass analysis is thatHT is highly cor-related withMfit. Thus, the background events that do pass the selection haveMfit spectra that are difficult to distinguish from the spectrum of heavy top quarkevents.3 This difficulty leads to a higher uncertainty in the top quark massmeasurement.

The solution is to apply a new event selection, which begins with very looserequirements on the quantities necessary to perform a kinematic fit: at least fourjets with ET > 15 GeV and|η| > 2.0, an electron withET > 20 GeV and|η| <2.0 or a muon withpT > 20 GeV/c and|η|< 1.7, andE/T > 20 GeV. To suppressbackground in samples without a tagging muon, DØ requires the sum (|E`

T | + |E/T |)to be>60 GeV and theη of the reconstructed leptonically decayingW to be<2.The latter cut also eliminates a region of phase space in which theVECBOSMonteCarlo generator does not correctly reproduce the observed distribution ofW-plus-jets events (8).

A total of 91 events satisfy the above criteria, of which 77 have a kinematic fitto the t t hypothesis withχ2 < 10 and are retained for further analysis. Becausethe above cuts are so loose, about two thirds of these events are expected to arisefrom background sources.

A sensitive measurement of the top quark mass calls for additional informationabout the relative likelihood of each event to arise fromt t decay, with the require-ment that any variables used to provide this information be nearly uncorrelatedwith Mfit . The following four variables have proven useful:

1. x1 ≡ E/T . This distinguishes QCD multijet events with one jet misidentifiedas a lepton fromt t andW-plus-jet events.

2. x2 ≡ aplanarity. This is a shape variable that is large for “spherical” events.Because the jets in background events arise from gluon radiation, theseevents tend to have lower aplanarity thant t events.

3. x3 ≡ HT2/Hz. HT2 is the sum of theET of all jets, excluding the leading

2The definition ofHTvaries slightly between CDF and DØ and between analyses of differentt t decay channels. Here, we defineHT specifically for each application.3The large number of possible combinations, as well as the limited jet energy resolution,make it very likely that non-t t events have acceptable fits.

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?TOP QUARK MASS 447

jet. Hz is the sum of the magnitude of the longitudinal momenta of allparticles and jets in the event. Therefore,x3 is larger for events that deposita greater fraction of their energy in the central region.

4. x4. This variable is evaluated using the pair of jets closest to each other inη−φ space. Their separation1R is multiplied by theET of the softer jet inthe pair and is divided by the sum of the charged leptonET and the missingtransverse energyE/T . Because gluon radiation tends to produce soft jetsclose to one another, this variable tends to be lower for background events.

To make maximal use of the information contained in these variables, they arecombined into a single-valued discriminant for top and background events. Thereare numerous prescriptions for such a multivariate analysis. DØ uses two differentapproaches, a classical likelihood and a neural network algorithm, to allow for across-check.

The classical likelihood, or “low-bias” discriminant,DLB, is defined as follows.For each of the variablesxi , the ratio of the signal and background probability den-sities is parameterized in the formLi ≡ si (xi )/bi (xi ). Then theLi are combinedinto a log likelihood functionL ≡ exp(

∑i wi lnLi ). Thewi are weights that are

adjusted to cancel the small residual correlation betweenL andMfit . Finally,DL B

is defined asL/(1+ L).The neural network discriminant,DNN, is determined by a three-layer feed-

forward neural network algorithm. There are four input nodes, one for each of thexi ; five hidden nodes; and a single output node. The value of the output node forany given event isDNN. The distributions ofx1–x4, along withDL B andDNN, areshown in Figure 5.

In general, the neural network is a more powerful technique because it canaccount for correlations between the input variables. However, since thexi in thiscase were chosen to be very nearly uncorrelated, the two techniques are expectedto have similar sensitivity.

At this point, each event in the data sample is described by two quantitiesMfit

andD. It is possible to select an enriched sample oft t events by discarding eventswith low values ofD. However, it is advantageous for two reasons to retain theentire sample and perform a maximum likelihood fit based on bothMfit andD. First,retaining the entire sample maximizes the statistical power of the measurement,and second, retaining information aboutD enables one to fit for the number ofsignal and background events in the sample, independent of any expectations fromthe cross-section analysis.

Therefore, a two-dimensional binned maximum likelihood fit is used to deter-mine the value ofMt and the level of background most consistent with the datasample (see 9 for the exact form of the likelihood used). The binning in the top dis-criminant axis is different in the low-bias (LB) and neural network (NN) analyses;the former divides the events into only two bins and the latter into ten. The distri-bution of the data and the best-fit Monte Carlo–generated signal and backgroundsamples for the NN analysis are shown in Figure 6.

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?Figure 5 Summary of the variables used in DØ’s event selection. For the lepton-plus-four-jets data (histograms) and Monte Carlo expectations (circles), theη of the reconstructedW→ `ν and kinematic fitχ2 are displayed in (a) and (b). (c)–( f ) show the four variablesused as input toDN N andDL B, for the lepton-plus-three-jets control sample. Finally, (g) and(h) show theDN N andDL B distributions for the four-jet data sample, compared with thebest-fit model (circles) and compared with the pure background distribution (triangles).

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?TOP QUARK MASS 449

Figure 6 Events per bin (∝ areas of boxes) versusDNN (ordinate) andMfit (abscissa) for(a) expected 172-GeV/c2 top signal, (b) expected background, and (c) DØ data.

Figure 7 displays the results of the maximum likelihood fits for the two analyses.For the LB analysis, DØ findsMt = 174.0±5.6 GeV/c2, and for the NN analysisMt = 171.3± 6.0 GeV/c2. In both cases, the errors are statistical only.

Monte Carlo tests are critical both to verify that the analyses are self-consistentand unbiased and to assess the impact of systematic effects on the measurement.The tests consist of selecting a sample of Monte Carlo–simulated events witha composition consistent with that expected for the data sample. (The Monte

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Figure 7 Summary of DØ’s maximum likelihood fit. (a)–(b), events per bin versusMfit for events in the (a) “top-like” or (b) “background-like”DL B bin. Histograms,data; filled circles, predicted mixture of top and background; open triangles, predictedbackground only. The circles and triangles are the average of the low-bias (LB) andneural network (NN) fit predictions, which differ by<10%. (c) Log of arbitrarilynormalized likelihoodL versus true top quark massMt for the LB (filled triangles)and NN (open squares) fits, with errors owing to finite top Monte Carlo statistics. Thecurves are quadratic fits to the data points near the minimum.

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?TOP QUARK MASS 451

Carlo samples contain 77 events, with the number drawn from background sourcesselected in a binomial distribution around the expected number.) The sample ofevents is passed through the same maximum likelihood technique used for the dataanalysis. By repeating this procedure many times (typically 103–104), one is ableto assess the statistical properties of the maximum likelihood mass estimate. Suchtests are performed to validate all of the top quark mass measurements discussedin this chapter.

The results of the tests for DØ’s lepton-plus-jets analysis are given in Table 4.They show that both the LB and NN analyses meet the requirement of self-consistency and that the error returned by the likelihood fit is a reasonable estimateof the statistical uncertainty of the measurement. The statistical uncertainties foundin the data analysis are somewhat smaller than the average for the Monte Carlotests but are not extremely unlikely (6% of LB analyses and 25% of NN analyseswould be expected to give smaller errors).

The systematic uncertainties in the measurement arise both from the limitedunderstanding of the detector performance and on uncertainties in the modelingof signal and background events. The top mass analysis depends critically onthese models for training of the NN or optimizing the LB selection, as well as forthe maximum likelihood fit, in which the distribution of data is compared withexpectations. Three types of events must be modeled: thet t signal, using theHERWIG generator (10);4 the W-plus-multijet background, usingVECBOS (11)with parton fragmentation provided byHERWIG; and the multijet background,using data with jets that nearly pass the lepton identification criteria. All of theMonte Carlo–generated events pass through a detailed detector simulation [basedon GEANT (R Brun, F Carminati, unpublished data)].

A variety of models are available, and these can differ significantly (for example,in the quantity and energy spectrum of radiated gluons). In the case ofW-plus-jetsevents, there is a large sample of data at lower jet multiplicities that are used tocompare and certify the model. The fact that the data and the model agree quite wellfor 2- and 3-jet events encourages confidence that 4-jet events are also correctlymodeled. Nonetheless, there are parameters inVECBOS (namely theQ2 scale atwhich the calculation is performed and the generator used for fragmentation) thatcan be varied, and no variation disagrees with the 2- and 3-jet data so stronglythat it can be ruled out. Therefore, four sets ofW-plus-jets events are prepared.The standard set used in the analyses has theQ2 scale set to the average jetpT in the event and usesHERWIG to model the parton fragmentation. Alternatesamples have theQ2 scale set to theW boson mass and/or useISAJET (F Paige,S Protopopescu, unpublished data) for fragmentation.5 The systematic uncertaintydue to uncertainties in the background model is estimated using Monte Carlo testsin which the test samples are drawn from the alternate background models butfitted to the standard models.

4DØ usesHERWIG version 5.7, and CDF uses version 5.6.5DØ usesISAJETversion 7.22, and CDF uses version 7.06.

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For t t events, the data provide little constraint on the model. As is the casefor background models, the most critical difference among the possible modelchoices is in their predictions for the amount and nature of gluon radiation. In theDØ analyses, this systematic uncertainty is estimated by parameterizingMfit asa function of the number of initial- and final-state gluon jets in the event (usingtheHERWIG model) and computing the change in fit mass that would result froma 50% increase in initial-state radiation (ISR) or final-state radiation (FSR). Theresults are added in quadrature. A cross-check on this uncertainty is obtained byperforming Monte Carlo experiments in which the signal events are drawn fromsamples generated byISAJETbut fit to the template histograms predicted byHERWIG

and noting the shift in the measured mass.In addition to the modeling of signal and background events, the jet energy

calibration is a significant source of systematic uncertainty. The calibration isdifficult because jets are composite objects, in which the fraction of energy thatappears to be electromagnetic (typically arising fromπos) varies from jet to jet,and because the lack of a discernible jet resonance in the data sample rules out adirect calibration. Instead, the strategy is to establish a relative calibration betweenthe hadronic and EM calorimeters by using events with a single jet and one ormore electromagnetic clusters. In the calibration, one must account for the energyfrom the hadronic shower that is not contained in the cone used to define the jet,underlying noise from uranium decays in the calorimeter, and unrelated particlesin the event, as well as the overall response of the hadronic calorimeter. In addition,the amount of material in front of the hadronic calorimeter varies significantly asa function ofη, and soη-dependent calibration is necessary. The largest sampleof events available for the calibration consists of single photons recoiling againstjets. The available statistics for this sample limits the relative precision to 2.5%on the jet scale. There are additional uncertainties in translating the energy scalefrom γ -plus-jets tot t events, which add a constant 0.5-GeV uncertainty to the jetcalibration. These two uncertainties are summed in quadrature to estimate theuncertainty in the calibration of a given jet. The next step is to create controlsamples in which the jet energies of all Monte Carlo events are scaled up or downby this uncertainty prior to the kinematic fit (theE/T is also changed to compensatefor this change in jet energies). Monte Carlo tests are run, taking events from thesecontrol samples as input, and the shift in the measuredMt is taken as the systematicuncertainty due to the jet energy scale.

Another shortcoming of the Monte Carlo models is that they simulate the resultsof a singlepp interaction. However, at the Tevatron, it is quite common for two ormore interactions to occur during a single bunch crossing, and the particles fromall these interactions will be recorded as one event. Particles from the additionalinteractions can contribute energy into the jet cones, biasing the measurement. Theadditional tracks also complicate the process of determining thezvertex of the maininteraction, and mismeasurement of this quantity can impact the measurement ofall the momenta in the event. To estimate the effect of the additional interactions,small samples of Monte Carlo events are generated with one or two soft interactions

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?TOP QUARK MASS 453

TABLE 4 Results of Monte Carlo tests of the DØ analyses based on an ensemble of 10,000Monte Carlo samples, each containing 77 events∗

Input Mass Mean Width(GeV/c2) 〈ns〉〈nbg〉 (GeV/c2) (GeV/c2)

Low-bias likelihood 175 23.8 53.2 175.0 8.7

Neural network 172 28.8 48.2 171.6 8.0

∗The Monte Carlo top quark mass used and the signal and background composition of the samples assumed reflect the fitvalues obtained from the data.

overlaid on the main interaction. A composite sample is then created that matchesthe distribution of events with zero, one, and two additional interactions expectedin the data sample. The systematic uncertainty is based on the shift inMfit for thissample relative to the standard sample with no additional interactions.

Small systematic uncertainties also arise from the finite number of Monte Carloevents available (estimated by Monte Carlo experiments in which the Monte Carlohistograms are fluctuated according to Poisson statistics) and possible biases in thelikelihood fit method (estimated by comparing the results obtained when differentfunctional forms are used to fit the likelihood points). In addition, there may bebiases in either the LB or NN analyses; half the difference between the results isincluded as a systematic uncertainty.

The systematic uncertainties for the LB and NN analyses are very similar andsum in quadrature to 5.5 GeV/c2 (see Table 5). To reach a final result, DØ com-bines the LB and NN analyses—assuming that the statistical uncertainties are77% correlated, as measured by Monte Carlo experiments, and that the system-atic uncertainties are 100% correlated—to arrive atMt = 173.3± 5.6(stat.)±5.5(syst.) GeV/c2.

DØ has also performed an analysis in whichMt is set to an assumed value andevents are reconstructed using a three-constraint kinematic fit. The smallestχ2

TABLE 5 Systematic uncertainties (in GeV/c2) for DØ’s measurement ofMt in the lepton-plus-jets channel for two different analyses

Low-bias Neural network

Jet energy scale 4.2 3.8

t t generator 1.9 1.9

W+ jets generator 2.5 2.5

Noise/multiple interactions 1.3 1.3

Monte Carlo statistics 0.6 1.1

LB/NN result difference 0.8 0.8

Likelihood fit 1.0 1.0

Total 5.6 5.4

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obtained for any jet permutation is recorded, and repeating this process for a rangeof assumed masses produces a distribution ofχ2 versusMt for each event. (Thespirit of this approach is similar to that used in the reconstruction of dilepton events,as discussed in the following section.) The top quark mass is then measured bysubtracting the expected background contribution from the summedχ2 distributionfor events in the top-enhanced bin of the LB distribution and determining the masscorresponding to the minimum of the background-subtracted distribution.6 Thisanalysis yieldsMt = 176.0± 7.9(stat)± 4.8(syst)GeV/c2, where the sources ofsystematic uncertainty and the methods of calculation are similar to those in DØ’sother analyses.

4.3 CDF Analyses

CDF selects top candidate events in the lepton-plus-jets channel by requiring thepresence of a single isolated electron (muon) withET (pT ) ≥ 20 GeV(GeV/c)in the central region of the detector (|η| < 1) and missing transverse energy,E/T ≥ 20 GeV, indicating the presence of a neutrino. At least four jets are requiredin each event, three of which must have an observedET ≥ 15 GeV and|η| ≤ 2.In order to increase the acceptance, the requirements are relaxed on the fourth jetto ET ≥ 8 GeV and|η| ≤ 2.4, provided one of the four leading jets is tagged asa b jet. Jets formed by the fragmentation ofb quarks can be identified (tagged)either by reconstructing secondary vertices from B hadron decays with the siliconvertex detector (SVX tagging) or by finding additional leptons from semileptonicb decays (SLT tagging). (See 2 and 4 for the SVX and SLT tagging algorithmsused by CDF.) SVX tags are allowed only on jets with observedET ≥ 15 GeV,whereas SLT tags are allowed on jets with observedET ≥ 8 GeV. If no such tagis present, the fourth jet must satisfy the sameET andη requirements as the otherthree. After applying these cuts, CDF has 76 events in the lepton-plus-jets sample.

After selection, the events are fit to thet t hypothesis according to the formuladescribed in Section 3.1. Electron energies and muon momenta entering the fit aremeasured with the calorimeter and tracking chambers, respectively. Jet energiesare corrected for losses in cracks between detector components, absolute energyscale, contributions from the underlying event and multiple interactions, and lossesoutside the clustering cone. These corrections are determined from a combinationof Monte Carlo simulations and data (13). The four leading jets in at t candidateevent undergo an additional energy correction that depends on the type of partonthey are assigned to in the fit—a light quark, a hadronically decayingb quark, ora b quark that decayed semileptonically (4). This parton-specific correction wasderived from a study oft t events generated with theHERWIGMonte Carlo program(10, 14).7 When all parton-jet assignments are correctly made, the resolution ofthe reconstructed mass is 17 GeV/c2 for a top mass of 175 GeV/c2.

6Monte Carlo experiments indicate that this method of measuring the mass has an intrinsicbias, which is corrected for in the calculation ofMt and its uncertainty.7The default set of structure functions used in all Monte Carlo calculations for CDF isMRSD0′.

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?TOP QUARK MASS 455

A maximum likelihood method is used to extract a top-mass measurement fromthis sample. An essential ingredient of the likelihood function is the probabilitydensity fs(Mfit | Mt ) to reconstruct a massMfit from a t t event if the true topmass isMt . In past publications (2, 4), CDF estimatedfs for a discrete set ofMt

values by smoothing histograms ofMfit for events from aHERWIG Monte Carlosimulation. In its latest analysis, CDF parameterizesfs as a smooth function ofboth Mfit and Mt (15). Figure 8 shows an example of the parameterization ofHERWIG Monte Carlo samples that have an SVXb tag for input top masses in therange of 155–195 GeV/c2. This new approach yields a consistent,Mt -dependentway of dealing with low statistics in the tails of theMfit histograms and produces

Figure 8 Reconstructed mass distributions for events with a SVXb tag fromHERWIGMonteCarlo for various input top masses ranging from 155 GeV/c2 to 195 GeV/c2. The lines area parameterized fit of the points, which represent the reconstructed mass value for a given5-GeV/c2 bin.

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a continuous likelihood shape from which the top mass and its uncertainty can beextracted. The probability densityfbg(Mfit) for reconstructing a massMfit from abackground event is obtained by fitting a smooth function to a mass distributiongenerated withVECBOS.

The likelihood function is the product of three factors:

L = Lshape× Lbackgr× Lparam, 1.

whereLshaperepresents the joint probability density for a sample ofN reconstructedmasses,Mi , to be drawn from a population with a background fractionxbg:

Lshape=N∏

i=1

[(1− xbg) fs(Mi | Mt )+ xbg fbg(Mi )]. 2.

The fractionxbg is constrained by an independent measurement that is summarizedby the background likelihoodLbackgr. The functionLparamallows the parameter-izations of fs and fbg to vary within the uncertainties returned by the fits to theHERWIG andVECBOS histograms forMfit . By includingLparam in the likelihooddefinition, the uncertainty owing to the finite statistics of these histograms is in-corporated into the statistical uncertainty on the measured top mass. The likelihoodL is maximized with respect toMt , xbg, and the parameters that define the shapesof fs and fbg.

The precision of the top quark mass measurement is expected to increase withthe number of observed events, the signal-over-background ratio, and the de-creasing width of the reconstructed mass distribution. These characteristics varysignificantly between samples with differentb-tagging requirements. Therefore,to make optimal use of all the available information, CDF partitions the mass sam-ple into nonoverlapping subsamples, defines subsample likelihoods according toEquation 1, and maximizes the product of these likelihoods to determine the topmass and its uncertainty (15). The use of nonoverlapping subsamples ensures thatthe corresponding likelihoods are statistically uncorrelated. Monte Carlo experi-ments show that an optimum partition is made up of four subsamples: events witha single SVX tag, events with two SVX tags, events with an SLT tag but no SVXtag, and events with no tag but with the tighter kinematic requirement of four jetswith ET ≥ 15 GeV and|η| ≤ 2.

The calculation of the expected background content of each subsample startsfrom the background calculation performed on theW-plus-≥ 3-jet sample for theCDF t t cross-section measurement (16). The extrapolation to the mass subsamplestakes into account the additional requirement of a fourth jet, theχ2 cut on eventreconstruction, and the fact that SVX and SLT tags are only counted if they areassociated with one of the four leading jets. The efficiencies of these requirementsare determined from Monte Carlo studies. They are used along with backgroundrates and tagging efficiencies from the cross-section analysis to predict the totalnumber of events in each mass subsample as a function of the unknown num-bers of t t andW-plus-jet events in the combined sample. These unknowns are

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?TOP QUARK MASS 457

TABLE 6 Number of observed events Nobs, expected background fraction xobg,

and measured top mass Mt for each CDF lepton-plus-jets subsample

Subsample Nobs xbog% Measured Mt (GeV/c2 )

Two vertex tags 5 5 ± 3 170.1 ± 9.3

One vertex tag 15 13 ± 5 178.0 ± 7.9

Soft lepton tag 14 40 ± 9 142+33−14

No tag (ET ( j4) ≥ 15 GeV) 42 56 ± 15 181.0 ± 9.0

Uncertainties on the measured top mass are statistical only.

*

*

estimated by maximizing a multinomial likelihood that constrains the predictedsubsample sizes to the observed ones. This procedure generates the expected back-ground fractions shown in Table 6 and the background likelihoodLbackgr used inEquation 1. Approximately 67% of the background in the entire sample comesfrom W-plus-jet events. Another 20% consists of multijet events in which a jet ismisidentified as a lepton andbb events with ab hadron decaying semileptonically.The remaining 13% is made up ofZ-plus-jet events in which theZ boson decaysleptonically, events withW W,W Z, or Z Z diboson production, and single-top pro-duction. CDF has compared the reconstructed mass distributions inVECBOSanddata for a number of event selections that are expected to be depleted int t events(17). All selections show good agreement betweenVECBOSand data, so CDF usesa VECBOScalculation to determine the shape offbg for the likelihood function.

Figure 9 shows the reconstructed mass distribution of the sum of the foursubsamples. The inset shows the shape of the corresponding sum of negative log-likelihoods as a function of top mass. CDF measuresMt = 175.9± 4.8 GeV/c2

(18). Monte Carlo experiments yield an 11% probability for obtaining a statisticaluncertainty of this size or smaller. The background fractionsxbg returned by the fitagree with the expectations from the cross-section measurement, which are listedin Table 6. To judge the goodness of the fit of the combinedMfit distribution, CDFperformed a Kolmogorov-Smirnov test and obtained a confidence level of 64%.The mass measurements obtained from these fits are consistent with each other,as shown in Table 6.

The systematic uncertainties on the mass measurement, summarized in Table 7,are estimated by varying the relevant quantities in Monte Carlo experiments. Thelargest uncertainty comes from the jet energy measurement. Each of the jet energycorrections described above carries a separate, energy-dependent uncertainty (13).Recent studies of soft gluon radiation outside the jet clustering cone have reducedthe uncertainty from this source to 2.5% for a jet with observedET >40 GeV. For anobserved jetET of 40 GeV, the total uncertainty on the correctedET varies between3.4% and 5.6% depending on the proximity of the jet to cracks between detectorcomponents. CDF has checked the jet correction procedure and the evaluation ofthe jet energy scale uncertainty with events containing a leptonically decaying

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TABLE 7 Systematic uncertainties for CDF’s measurement ofMt in the lepton-plus-jets channel

Source Error (GeV/c2 )

Jet energy scale 4.4

Inital- and final-state radiation 2.6

Monte Carlo modeling (PDF)a 0.5

Monte Carlo generator 0.1

Background shape 1.3

Total 5.3

aPDF, parton distribution function.

Figure 9 Reconstructed mass distributions of the four CDF lepton-plus-jets masssubsamples combined. The data (points) are compared with the result of the combinedfit to top (dark shading) plus background components (light shading). The inset showsthe variation of the negative log-likelihoods with Mt .

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?TOP QUARK MASS 459

Z boson and one jet. A study of how the transverse momentum of the jet bal-ances that of theZ-decay products finds that the observed ratio of [pT (Z) −pT (jet)]/pT (Z) differs by 3.2± 1.5 (stat.)± 4.1 (syst.)% from the Monte Carlosimulations. The 4.1% systematic uncertainty is due to the jet energy scale only.Because the difference is consistent with zero, this study independently confirmsthe soundness of CDF’s estimate of the jet-energy–scale uncertainty. CDF obtainedfurther confirmation by reconstructing the invariant mass of the jets coming fromthe decay of one of theW bosons. A measurement of theW boson mass fromthese jets yields 77.2± 3.5 (stat.)± 2.9 (syst.) GeV/c2 (19).

The second largest systematic uncertainty arises from high–transverse-momen-tum gluons that are radiated from the initial or final state of at t event and some-times take the place of at t decay product among the four leading jets. This uncer-tainty was determined with thePYTHIA Monte Carlo calculation (20) by separatelystudying the effect of extra jets coming from initial- and final-state radiation.8

The uncertainty in the modeling of the background mass distribution was es-timated by varying theQ2 scale inVECBOS. Additional sources of uncertaintyinclude the kinematical bias introduced byb tagging and the choice of parton dis-tribution functions (CTEQ4L (21) versus MRSD0′). The sum in quadrature of allthe systematic uncertainties is 5.3 GeV/c2.

5. DILEPTON

The dilepton final state is the cleanest signature oft t production, but its smallbranching fraction limits the statistics for mass measurement. However, such ameasurement is interesting as a test of the hypothesis that the excess of eventsobserved by CDF and DØ in the lepton-plus-jets and dilepton final states are bothdue to t t production. In addition, much of the systematic uncertainty in the topquark mass measurement arises from the presence of jets in the final state. Becausedilepton events have fewer jets, one may expect the mass measurement using suchevents to have smaller systematic uncertainty and thus to be the technique of choicein future high-precision measurements.

The presence of two unmeasured neutrinos in the dilepton final state renders akinematic reconstruction underconstrained (six unknown quantities with only fiveconstraints). Nonetheless, the measured energies and angles of the jets and leptonscontain information about the mass of the top quark.

A method for extracting this information, first proposed by Kondo (22), beginswith assuming a value for the top quark mass. The neutrino momenta can then bereconstructed (up to a fourfold ambiguity arising from a quartic equation in thereconstruction and a twofold ambiguity arising from the arbitrary choice of whichlepton and jet come from the same top quark). The next step is to evaluate the like-lihood of the final-state configuration to be compatible with at t event with the

8CDF usesPYTHIA version 5.7.

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assumed mass. By scanning over a range of assumed masses, one estimates therelative likelihood of the observed final state as a function of top quark mass. DØand CDF employ variants of this dynamical likelihood technique in their dileptonmass measurements.

5.1 DØ Analyses

In contrast to the lepton-plus-jets analysis, the event selection for the dileptonchannels is nearly identical to that for the cross-section analysis (23). Events arerequired to have two leptons withET > 15(20) GeV for theeµ andµµ (ee)channels, with|ηe| < 2.5 and|ηµ| < 1.7, and two or more jets9 with ET >

20 GeV and|η| < 2.5. For theeeandeµ channels, significantE/T is required todiscriminate against background sources that have no final-state neutrinos, whereasfor theµµ analysis,Z boson background is reduced by rejecting events for whichthe χ2 probability of a fit to theZ → µµ hypothesis is>1%. Much of theremaining background is rejected using the quantitiesHT ≡

∑jets ET andHe

T ≡HT + ET (e1), where all jets withET > 15 GeV and|η| < 2.5 enter the sum, ande1 is the leadingET electron. The selection requiresHe

T (HT ) > 120(100) GeV fortheeeandeµ (µµ) channels.

When this selection is applied, there is one event that only nominally belongs inthe lepton-plus-jets sample. One of its so-called jets is an electromagnetic energycluster without an associated reconstructed track but with hits in the layers ofthe central tracking system between the interaction vertex and the cluster. Forpurposes of mass analysis, this cluster is interpreted as arising from an electron,and the event is moved to the dilepton sample. The signal-over-background ratiofor similar events is≈8/1 (“background” here includest t events in the lepton-plus-jets mode). The sample for mass analysis consists of threeeµevents, twoeeevents,and oneµµ event, with expected backgrounds of 0.21 ± 0.16, 0.51 ± 0.09, and0.73±0.25 events, respectively [background sources are detailed elsewhere (23)].

These events are processed using two dynamical likelihood techniques (as inDØ’s lepton-plus-jets analysis, this allows a cross-check of the result). The first,matrix element weighting (MWT), is an extension of the procedure given by Dalitz& Goldstein (24). This method requires the sum of the neutrinoEpT ’s to equal themeasuredEE/T . The solution likelihood is estimated as follows:

Wo(Mt ) = A(Mt ) f (x) f (x)P(EC M`1 | mt

)P(EC M`2 | Mt

),

where f (x) is the CTEQ3M (25) parton distribution function evaluated atthe proton (antiproton) momentum fractionx(x) required by the solution, andP(EC M

` | Mt ) is the probability density for the lepton energy in the top quark rest

9The calibration of jets is similar to that in the lepton-plus-jets analysis except that theaverage correction for gluon radiation is not applied. Instead, the dilepton analyses attemptto account explicitly for radiation, as described below. Because this correction is appliedsymmetrically to data and Monte Carlo events, it does not affect the jet scale uncertainty.

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?TOP QUARK MASS 461

frame (it is the latter term that generally provides the stronger constraint). Thefactor A(Mt ) normalizes the average weight of accepted events to unity, indepen-dent of the top quark mass. This term corrects for the fact that thef (x) termsare correlated with thet t production cross section and thus favor lower top quarkmass solutions.

The other weighting scheme, neutrino weighting (νWT) (26), steps the assumedη for each neutrino through 20 values at eachMt . Each step spans an equal fractionof the neutrinoη distribution expected int t production.10 At each step a weight isassigned, based on the extent to which each component of theEE/T measured in theevent (EEm

T ) agrees with the corresponding component of the sum of the neutrinoEpTs in the solution (EEp

T):

g(Mt , η1, η2) =∏

k=x,y

exp

(−(E/ p

Tk− E/mTk

)2

2σ 2

). 3.

The Gaussian resolution of each component of theEE/T is 4 GeV. If a particularset ofmt , η1, η2 yields no real solutions,g is set to 0. Summing the values ofgover the choices of neutrinoη and the solution ambiguity (which is up to eightfold)gives

Wo(Mt ) =∑η1,η2

∑Solutions

g(Mt , η1, η2) 4.

In both theMWT andνWT analyses, the assumedMt is scanned from 80–280 GeV/c2 in 4-GeV/c2 steps.

One potential drawback of solving for the neutrino momenta (rather than fittingfor them, as is possible in the lepton-plus-jets analyses) is that the detector resolu-tion is not explicitly considered in the solution. To overcome this, a Monte Carlointegration is performed for each event. All the measured energies in the eventsare varied according to their resolutions 100 (5000) times for Monte Carlo (col-lider data) events. TheWo distributions obtained for each variation are summed.The Gaussian resolutions for electrons and muons areσE/E = 15%/

√E and

σ(1/p) = 0.18(p − 2)/p2 ⊕ 0.003, respectively, withE (p) in GeV (GeV/c).Jets are smeared by double Gaussians designed to model both the inherent energyresolution of the hadronic calorimeter (narrow Gaussian) and the contribution oflarge-angle gluon radiation to the resolution (wide Gaussian). TheEE/T is then re-computed to reflect the changes in jet and lepton energies, and each component isfluctuated with a 4-GeV Gaussian.

Initial- and final-state gluon radiation (ISR and FSR) can create additional jetsthat complicate the final state. DØ sums over all combinations of the three leadingET jets. The sum is weighted by exp[−ET sinθi /(25 GeV)] if jet i is assumed to be

10This distribution is taken fromt t events generated byHERWIGand is found to be approx-imately Gaussian with width decreasing from 1.10 to 0.87 asMt increases from 80 GeV/c2

to 280 GeV/c2.

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Figure 10 W(Mt ) distributions, normalized to unity, for DØ’s dilepton candidates,using theMWT (dashed) andνWT (solid) event reconstruction methods.

ISR or by exp[−Mi j /(20 GeV/c2)] if jets i and j , having invariant massMi j , areassumed to arise from the samebquark by FSR. In each case, the form of the weightis based on the characteristics of gluon radiation, and the coefficient is determinedempirically such that the averageW(Mt ) from events containing ISR or FSR jetsis the same as that for events without extra jets. Figure 10 displays the distributionsof W(Mt ) [which corresponds toWo(Mt ) after accounting for resolution and jetcombinations] for the six candidate events, using the two weighting methods.

The remaining step of the analysis is to translate the set of weight distributionsinto a measurement of the top quark mass. Because the weights are only approxi-mate, and background has not yet been considered, one may not simply treat theweight distributions as probability densities for the top quark mass. The simplestway to obtain an exact solution is to select a single characteristic of each distri-bution (for example, the peak) and enter this value into a maximum likelihood fit,as is done in the lepton-plus-jets analysis. This technique clearly fails to take fulladvantage of the available information. On the other hand, taking account of all50 estimated weights for each event is numerically cumbersome.

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?TOP QUARK MASS 463

A compromise solution is to divide the 80–280-GeV/c2 range of assumedmasses into five regions and sum the weights in each interval. If the weight distri-butions are normalized to one, there are four independent quantities (the summedweights in the first four intervals) measured for each event. These quantities may betreated as elements of a four-dimensional vectorEwi , wherei is the event number.Then the likelihoodL(Mt , ns, nbg) to be maximized is

L = G(nbg)P(ns + nbg)×N∏i

ns fs( Ewi | Mt )+ nbg fbg( Ewi )

ns + nbg, 5.

wherens andnbg are the fitted signal and background levels,G(nbg) is a Gaussianconstraint thatnbg be consistent with expectations from the cross-section an-alysis,11 P(ns + nbg) is a Poisson constraint thatns + nbg be consistent withthe sample sizeN, and fs and fbg are the four-dimensional probability densitiesfor signal and background. These probability densities are estimated by summingthe contributions from four-dimensional Gaussian kernels placed at the locationof Ew for each event in the signal Monte Carlo or background samples (27). (Thecomponents ofEw are transformed to eliminate the correlations between them priorto this step.) Using the estimatedfs and fbg, L is maximized with respect tons

andnbg asMt is varied. The maximum likelihood estimate ofMt and its error aredetermined by a quadratic fit to−ln L for the nine points about the minimum. Theadditional complexity introduced to the maximum likelihood fit by consideringfour variables for each event is rewarded by a statistical uncertainty that is roughly25% smaller than the simpler procedure could offer.

Applying the maximum likelihood fit to the data, the top quark mass is deter-mined to beMt = 168.2±12.4 GeV/c2 (MWT), andMt = 170.0±14.8 GeV/c2

(νWT), where the uncertainties are statistical only (see Figure 11).As in the lepton-plus-jets analysis, Monte Carlo experiments are a vital tool to

ensure that the complex analysis methods are self-consistent and that the estimatedmass and its error are unbiased. Table 8 lists medians, widths, and means and widthsof the pulls for different Monte Carlo top quark masses. The pull widths are nearlyunity, and the median estimated top quark mass is near the input value. However,because of the small sample size, the estimated top quark mass is not Gaussian.It is important to remember that although the uncertainties quoted above define aregion in which the true top quark mass is 68% likely to lie, the 95% confidenceinterval is significantly more than twice as wide. The properties of the estimatedtop quark mass are very similar for both methods. It is found that obtaining astatistical uncertainty smaller than 12.4 GeV/c2 for theMWT analysis is 21%probable, whereas obtaining an uncertainty smaller than 14.8 GeV/c2 for theνWTanalysis is 47% probable.

11Inclusion of this term introduces a dependence on the cross-section analysis. Althoughsuch dependence is not desirable (and not introduced in DØ’s lepton-plus-jets analysis), it isneeded here because the signal and backgroundW(Mt ) distributions are not significantlydifferent.

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Figure 11 Summary of DØ’s dilepton analyses. The histograms represent the sum ofthe normalized candidate weights grouped into the five bins considered in the maximumlikelihood fit (circles) for the (a) MWT and (b) νWT analyses. The uncertainty onthese points is taken from the RMS spread of the weights in Monte Carlo studies. Alsoshown are the average weights from the best-fit background (dashed) and signal plusbackground (solid). The −ln L distributions and quadratic fits are inset.

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?TOP QUARK MASS 465

TABLE 8 Results of DØ’s Monte Carlo experiments, which explore the propertiesof the maximum likelihood estimate of the top quark mass

M MCt Median Width Pull Pull

GeV/c2 GeV/c2 GeV/c2 mean width

160 161.6 15.8 0.12 ± 0.03 1.03

MWT 170 172.2 16.7 0.11 ± 0.03 0.99

180 180.5 17.3 0.00 ± 0.03 0.98

160 161.5 14.4 0.17 ± 0.03 0.96

νWT 170 172.2 16.2 0.08 ± 0.03 0.98

180 180.5 18.1 0.03 ± 0.03 1.03

MWT, matrix element weighting; νWT, neutrino weighting.

*

*

*

The sources of systematic uncertainty, and the methods used to estimate theirmagnitudes, are analogous to those encountered in the lepton-plus-jets analysis.(An exception is the signal model uncertainty, which is estimated in the dileptonanalyses by using Monte Carlo experiments in which signal events are generatedby ISAJET rather than the standardHERWIG.) As might be expected with fewerjets in the final state, the impact of the jet scale uncertainty is less than in thelepton-plus-jets case.

The contributions from all sources are summed in quadrature to give the sys-tematic uncertainty on the measurement (see Table 9).

To arrive at a single number, DØ combines the results of theMWT andνWTanalyses. Based on Monte Carlo experiments, the results are 77% correlated, andthe result of the combination is

Mt = 168.4± 12.3 (stat.)± 3.6 (syst.) GeV/c2.

TABLE 9 Systematic errors in DØ’s measurementof Mt in the dilepton channels

Source Error (GeV/c2)


Signal model 1.8

Multiple interactions 1.3

Background model 1.1

Likelihood fit 1.1

Total 3.6

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5.2 CDF Analyses

CDF’s first published measurement ofMt using dilepton events was obtained froma comparison of data with Monte Carlo simulation oft t events for two kinematicvariables: theb-jet energies and the invariant mass of the lepton andb-jet systems(28). Such an analysis is straightforward but suffers from poor statistical precisionrelative to techniques that take into account all the kinematic information in eachevent. Therefore, CDF recently published a new analysis using theνWT methodto reconstruct events (29).

For the dilepton channel, CDF requires two high–transverse-momentum(pT> 20 GeV/c) oppositely charged leptons (eorµ) in the central detector region(|η| < 1), at least one of which is well isolated from nearby tracks and calorimeteractivity. To rejectZ → l+l−X events, CDF requires that the dilepton invariantmass,Mee or Mµµ, be outside the interval 75–105 GeV/c2, and removes eventscontaining an isolated photon withET > 10 GeV if they are consistent with radia-tive Z decays. CDF also requires at least two jets in the region|η|< 2.0, each withobservedET > 10 GeV, andE/T > 25 GeV, as a signature for missing neutrinos.To reject events in whichE/T is caused by lepton or jet energy mismeasurements,CDF requires|E/T |> 50 GeV if theE/T is close to a lepton or a jet (1φ(E/T , l or j )< 20◦). Finally, CDF requiresHT > 170 GeV, whereHT is defined as the scalarsum of the|pT | of the two leptons (ET for electrons) andET of the two highestET jets and|E/T |.

CDF obtains a sample of eight candidate events. The expected background of1.3 ± 0.3 events consists of events in which a track or a jet is misidentified as alepton, Drell-Yan production,W W production,Z → ττ decays, andZ → µµ

decays in which theµ tracks are mismeasured.These events are reconstructed using theνWT method, withη1 andη2 stepped

over 100 values each andMt scanned from 90–290 GeV/c2. The resolution oneach component ofE/T is taken to be 4

√n GeV, wheren is the average number

of pp interactions per bunch crossing for the instantaneous luminosity at whichthe event was recorded. If there are more than two jets in the event, only the twoleadingET jets are considered in the reconstruction.

As in the DØ analysis, the detector resolution for jets and leptons is takeninto account by varying the measured quantities many times within their res-olutions. The Gaussian resolutions for electrons and muons areσE/E =√(0.135)2/ET + (0.02)2 andσ(1/pT ) = 0.11%, whereET , in GeV, is the elec-

tron energy measured in the calorimeter andpT is the beam-constrained muonmomentum measured in the central tracking chamber. For jets, CDF uses anET -dependent Gaussian resolution appropriate forb partons derived fromHERWIG

t t Monte Carlo, in conjunction with a detector simulation, assuming a top massof 170 GeV/c2. The E/T is recomputed for each sampling using new jet and lep-ton energies. This procedure (along with the constant resolution term explicitlyused in theνWT reconstruction) takes into account all the uncertainties in theE/T

measurement.

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?TOP QUARK MASS 467

Figure 12 Weight distribution normalized to unity as a function ofMt for the eight CDFdilepton top candidate events.

TheW(Mt ) distribution for each of the eight candidate events, normalized tounity, is shown in Figure 12.

For each eventi , CDF uses this distribution to determine a top mass estimateMi by averaging the values ofMt corresponding to values ofW(Mt ) closest toand greater thanW(Mt )max/2 on either side of the maximum. TheMi distributionfor the eight events is shown in Figure 13, along with the Monte Carlo expectationfor background alone and for top plus background normalized to the data.

The Mi are input to a maximum likelihood fit. The form of the likelihood isthe same as in Equation 5, withfs(Mi ,Mt ) and fbg(Mi ) obtained from smoothparameterizations of the distribution of model events.

CDF uses Monte Carlo experiments to assess the reliability of the maximumlikelihood fit. For each experiment a massmexpand a statistical uncertaintyσexpare

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Figure 13 Reconstructed top mass for the eight CDF dilepton events (solid line). Also shownare the background distribution (shaded, 1.3 events) and top Monte Carlo (6.7 events) addedto background (dashed line). Inset, the negative log-likelihood distribution as a function ofthe top mass.

obtained. The distributions of the pulls,(Mexp−Mt )/σexp, have medians consistentwith zero and widths that are at most 1.1. CDF takes this width into account byincreasing the uncertainty returned by the fit by 10%. This factor is included in allstatistical uncertainties on the dilepton measurements given below.

The likelihood method is applied to the data shown in Figure 13. The insetshows the negative log-likelihood as a function of the top mass, from which a topquark mass value of 167.4± 10.3 (stat) GeV/c2 is obtained. Monte Carlo testsyield an 8% probability for one such experiment to have a statistical uncertainty≤10.3 GeV/c2 at a mass of 167 GeV/c2.

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?TOP QUARK MASS 469

TABLE 10 Systematic uncertainties for CDF’s measurement ofMt in the dilepton channel

Source Uncertainty (GeV/c2)


Initial- and final-state radiation 2.7

Monte Carlo modeling (PDF) 0.6



Monte Carlo statistics 0.7

Total 4.8

The mass reconstruction procedure used here is quite different from that usedfor the lepton-plus-jets sample. In that case, there is only one missing neutrino, anda kinematic fit with two constraints is performed. TheνWT method for the dileptonevents was cross-checked by applying it to the five events of the lepton-plus-four-jets sample that had two jets tagged asb jets by the SVX. The two untagged jetsare assumed to be the products of the hadronicW decay and mimic the dileptondecay by treating the highest-ET untagged jet as a lepton and the second untaggedjet as a neutrino. The present likelihood method gives a top quark mass from thefive double tagged events of 181.5± 12.6 GeV/c2 which differs by 11 GeV/c2

from that obtained with the lepton-plus-jets kinematic fit procedures, 170.1±9.3 GeV/c2. Monte Carlo experiments show that the difference in mass obtainedby the two methods is expected to be centered at zero with a resolution of14 GeV/c2.

The systematic uncertainties on the dilepton mass measurement are estimatedwith the same general procedure as in the lepton-plus-jets mass analysis. Top massdistributions are generated by varying the appropriate quantities in the Monte Carlosimulation and then performing likelihood fits to many experiments with eightevents each, using the standard templates. The mass shifts obtained determine thesystematic uncertainties, which are summarized in Table 10. The total systematicuncertainty amounts to 4.8 GeV/c2.

6. ALL-HADRONIC

6.1 CDF Analyses

CDF searches for events in which bothW bosons decay into quark-antiquarkpairs, leading to an all-hadronic final state (30). The study of this channel, witha branching ratio of about 4/9, complements the leptonic modes, and the massmeasurement takes advantage of a fully reconstructed final state. Because the

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Figure 14 Reconstructed top mass for all-hadronic data (points) with six or more jetsand at least oneb tag. Also shown are the background distribution (shaded) and thecontribution from t t Monte Carlo events withMt = 175 GeV/c2 (open). Inset, thedifference in−ln (likelihood) and the fit used to determine the mass.

expected decay signature involves only hadronic jets, a very large background fromstandard QCD multijet production is present and dominates overt t production.CDF requires an SVX tag to reduce this background.

To determine the top quark mass, CDF applies a full kinematic reconstructionto a sample of events with six or more jets withET > 15 GeV/c2 and|η| < 2 andat least one SVXb-tag. In addition, CDF requires the total transverse energy ofthe jets (6ET ) to be greater than 200 GeV/c2 and to account for more than threefourths of the total energy,

√s, available. Events are also required to be spherical

in shape by requiring the aplanarityA to be greater than−0.00256ET + 0.54,where the sum does not include the contribution from the two highest-ET jets.

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?TOP QUARK MASS 471

TABLE 11 Systematic uncertainties for CDF’s measurement ofMt in the all-hadronic channel

Source Error (GeV/c2)


Initial- and final-state radiation 1.8

Monte Carlo modeling (PDF)a 0.2



Monte Carlo statistics 0.6

Total 5.7

aPDF, parton distribution function.

Events are reconstructed to thet t → W+bW−b hypothesis, where bothWbosons decay into a quark pair, with each quark associated to one of the sixhighest-ET jets. This corresponds to 16 four-momentum conservation equationswith 13 unknown variables, the three-momenta of the two top quarks and the twoW bosons, and the unknown top quark mass. Because all events contain at least oneb tag, CDF requires the tagged jet to be assigned to ab or b quark and not theW±.A kinematic fit like the one used for the lepton-plus-jets channel is applied andthe combination with the lowestχ2 is chosen. Figure 14 shows the reconstructedmass distribution for the 136 tagged events, along with the expected backgroundand t t contributions. The background is calculated by normalizing the spectrumof the untagged sample of 1121 events to 108± 9 events, estimated from thetag probability. A maximum likelihood method is applied to extract the top quarkmass. The experimental data are compared withHERWIG Monte Carlo samplesof t t events, in a top quark mass range from 160 GeV/c2 to 210 GeV/c2, and abackground sample from the untagged events. The difference in the log-likelihoodwith respect to the minimum is shown in the inset of Figure 14. The minimumis at 186 GeV/c2, with a statistical uncertainty of±10 GeV/c2. The systematicuncertainties, calculated exactly the same way as for the lepton-plus-jets channel,are listed in Table 11. The all-hadronic result is 186.0± 10.0± 5.7 GeV/c2.

7. COMBINED RESULTS

CDF and DØ have combined their mass results to provide the single best top massmeasurement (31).

Table 12 gives the systematic uncertainties in all channels for both CDF andDØ. Table 13 shows the mass measurements, including statistical and systematicuncertainties, for both CDF and DØ. These results are combined to give a final

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TABLE 12 Systematic uncertainties (in GeV/c2) for CDF and DØ in all channels in themeasurement ofMt

∗

CDF DØ

Lepton LeptonSource plus jets Dilepton All-Hadronic plus jets Dilepton

Jet energy scale 4.4 3.8 5.0 4.0 2.4

Signal model 2.6 2.8 1.8 1.9 1.7

Monte Carlo generator 0.1 0.6 0.8 — —

Multiple interactions — — — 1.3 1.3

Background model 1.3 0.3 1.7 2.5 1.0

Mass fitting method 0.0 0.7 0.6 1.5 1.1

Total 5.3 4.8 5.7 5.5 3.6

∗The two collaborations use somewhat different definitions of systematic uncertainty sources, as reflected by the empty entries.

result from both Tevatron experiments of 174.3± 5.1 GeV/c2. The relative con-tributions from the five results are 35% for CDF lepton plus jets, 34% for DØlepton plus jets, 11% for DØ dileptons, 10% for CDF dileptons, and 10% for CDFall-hadronic.

All statistical uncertainties are taken to be uncorrelated. The systematic uncer-tainties that arise from Monte Carlo models are assumed to be 100% correlatedbetween the two experiments (since CDF and DØ used the same models). All othersystematics are treated as uncorrelated.

It is interesting that for each of the analyses used to obtain the combined re-sult, the statistical uncertainty observed in the data sample is smaller than wouldbe expected from Monte Carlo studies (see Table 13). The probability of four

TABLE 13 Summary of all mass values and their uncertainties for CDF and DØ in all channels∗

Channel Mass (GeV/c2) Probability

CDF Lepton plus jets 175.9± 4.8(stat.)± 5.3(syst.) 0.11

Dilepton 167.4± 10.3(stat.)± 4.8(syst.) 0.08

All-hadronic 186.0± 10.0(stat.)± 5.7(syst.) N/A

DØ Lepton plus jets 173.3± 5.6(stat.)± 5.5(syst.) 0.16

Dilepton 168.4± 12.3(stat.)± 3.6(syst.) 0.34

Combined Tevatron result 174.3± 3.2(stat.)± 4.0(syst.) 0.01–0.05

∗“Probability” refers to the fraction of Monte Carlo simulated experiments that yield a statistical uncertainty less than thatobtained in the analysis (for the DØ data samples, where two analyses combine to give the final number, the average of theindividual analysis probabilities is listed; probability information is not available for CDF’s all-hadronic analysis).

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?TOP QUARK MASS 473

independent likelihoods having a smaller product (sum) than this collection is0.05 (0.01). Hence there is an indication, at the 95–99% confidence level, that thetop quark mass can be measured to a better precision than Monte Carlo studiesindicate. The most likely explanation for such an effect is that the Monte Carlogenerators produce too much gluon radiation. It will be interesting to see whetherthis indication is confirmed in future experiments, in which the top quark statisticswill be sufficient to directly compare to the Monte Carlo models.

8. FUTURE PROSPECTS

In this section, we explore the prospects for improvement of the top quark massmeasurement. Our focus is mainly on the upcoming run of an enhanced Tevatron,for which our prediction can be reasonably confident. Increasingly accurate mea-surements of the top quark mass during this run also have the benefit of constrainingthe Higgs boson mass in advance of the LHC’s turn-on.

The ultimate top quark mass precision may eventually be achieved by scanningthe t t production threshold at a future lepton collider (see 32 for a discussion ofsuch measurements).

8.1 Fermilab Run II

The DØ and CDF detectors, as well as the Tevatron, are currently undergoingupgrades in preparation for a run to begin in the spring of 2000 (33, 34). Theintegrated luminosity for this run (called Run II) is expected to reach 2 fb−1 at anenergy of 2 TeV. The increase in energy from 1.8 TeV to 2 TeV increases thet tproduction cross section by 40% (35), so that Run II is expected to produce about30 times moret t events than Run I.

Both CDF and DØ will, among other improvements, greatly enhance theirability to detectb quark jets by looking for displaced vertices. DØ will add asilicon vertex detector, while CDF will significantly extend the acceptance andcapabilities of its silicon detector. These improvements will help to reduce boththe statistical and systematic uncertainties of the top quark mass measurement.

Efficient detection ofb quark jets allows the collection of a clean sample oft t events with relatively mild kinematic cuts (withoutb-tagging, such a samplewould be overwhelmed by background). So the improved silicon detectors willincrease the overall efficiency of CDF’s and DØ’s detection oft t events. Further,since the per-jet tagging efficiency will be≈45% for each detector,≈20% oft t events will have both jets tagged. In the lepton-plus-jets case, this representssignificant additional information that will reduce the combinatorial uncertainty inthe mass fit, making each double-tagged event more statistically valuable than anuntagged event.

The statistical uncertainty to be achieved in Run II can be conservatively es-timated by extrapolating the power of the current analyses to the larger data set

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TABLE 14 Expected statistical precision ofMt measurement using events from variouschannels∗

Channel σ/event (GeV) Nexp Run II σ (GeV)

Lepton plus jets, untagged 43.3–49.9 180 3.1–4.1

Lepton plus jets, single vertex tagged 31.4 550 1.3

Lepton plus jets, double vertex tagged 21.3 240 1.4

Dilepton 35.4 160 2.7

All-hadronic 52.9 1120 1.6

Total 1360 0.8

∗A top mass of 175 GeV/c2 is assumed. The values in theσ/event column list the per-event precision expected from studies ofRun I analyses (improvements in measurement techniques or in signal-over-background ratio may make the Run II analysesmore precise than shown here). The expected numbers of events are per experiment for a 2-fb−1 Run II. Estimates are basedon References (8, 15, 34, 35).

(Table 14). The statistical precision will be roughly 1 GeV per experiment for2 fb−1 of luminosity.

The challenge will be to understand the systematic issues at a comparable level.The systematic uncertainties that will limit the Run II measurement arise from thet t and background production models and the jet energy scale. The uncertaintyin both cases is due to limits on our understanding of the underlying physics.Therefore, one cannot simply assume that the uncertainties will scale as 1/

√L,

whereL is the integrated luminosity. Further, the uncertainties will be highlycorrelated between measurements in differentt t decay channels and between thetwo experiments.

The major uncertainty in thet t and background models is in the treatment ofhard gluon radiation. Such effects can in principle be predicted by QCD, but thecalculations are difficult. For the Run I analyses, gluon radiation was modeledby either an approximation (HERWIG or ISAJET) or by a leading-order calculation(VECBOS).

Higher-order models of gluon radiation may be available for Run II. Even ifnot, the larger data samples will allow more stringent constraints on the amountand nature of gluon radiation (33, 36). Here again the extremely pure sample ofevents with two jets tagged asbs will prove invaluable because any model oft tproduction that is inconsistent with the events in this sample may be discarded.In addition, samples ofbb, W-plus-jets, andγ -plus-jets events can be used toconstrain models of gluon radiation by comparing the number of jets, and theirenergies, to Monte Carlo predictions (34).

The uncertainty attributable to jet energy scale consists of two components.One is understanding the response of the calorimeter to the jets composed ofhadrons. Although calibrating this response is more difficult than calibrating an EMcalorimeter, this is not the limiting uncertainty in the top quark mass measurement.

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?TOP QUARK MASS 475

Implicit in the kinematic reconstruction oft t events is the assumption that the en-ergy and direction of jets observed in the calorimeter have a well-defined relation-ship to the energy and direction of the quarks (or radiated gluons) that arise fromtop quark decay. Hence, the jet energy scale uncertainty means the uncertainty inthe relationship between the energy of clusters observed in the calorimeter and theenergy of the parent partons.

Our understanding of this relationship is limited by our knowledge of thehadronization process of partons, which is governed by nonperturbative QCD.Variation of the hadronic composition of jets (e.g. theπo fraction), the angulardispersion of the hadrons, and the final-state interactions between partons can allalter the jet energy scale, and in all cases, our models are only approximate.

As far as possible, we must gain an empirical understanding from control datasets. The control sets most important for Run II analyses will be the following(33, 34, 36, 37):

1. t t events with both jets vertex-tagged. CDF has already demonstrated theability to use such events to reconstruct the hadronicW mass (19). Theability to uset t events in calibration is extremely important because thefinal-state interaction and radiation effects can be process dependent.CDF’s studies indicate that the hadronicW mass uncertainty will be about2% for Run II (37).

2. Z plus jets. In these events aZ boson, decaying toee, recoils against oneor more jets, which provides a means for relating the jet energy scale to theelectromagnetic energy. This was one of the calibration samples used inRun I. Although statistics will be 20 times greater in Run II, there arefundamental limitations in applying this calibration tot t events. First, onecannot expect theZ and jet to balance exactly because of the presence ofinitial-state radiation and the underlying event (spectactor quarkinteractions). Second, the final-state interaction effects could be quitedifferent from those int t events.

3. γ plus jets. This was one of the primary calibration samples in Run Ianalyses because the statistics available were much larger than those forZ-plus-jet events. However, the same fundamental limitations apply andare compounded by the difficulty of isolating a pure sample of photons.This sample will probably be used as a cross-check.

4. Z→ bb. This sample allows direct calibration ofb quark jets, whosehadronization effects differ from those of light quarks. Becausesemileptonicb decays would form a biased sample, isolation of this samplerelies on a displaced vertex trigger, which is planned for the CDF upgradeand has been proposed for DØ.

In summary, the Run II analyses will benefit from larger and more powerfulcontrol samples than were available in Run I. Nonetheless, the underlying physics iscomplex, so expecting 1/

√L scaling in the systematic uncertainties is unrealistic.

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Figure 15 Top production cross-section dependence as a function of the top mass.Three different theory calculations are shown (5). Data points are the measured crosssections from CDF and DØ (16, 23).

When the CDF and DØ upgrades were designed, it was envisioned that the topquark mass resolution attainable in Run II would be 3–4 GeV/c2 (33, 34). Massmeasurement techniques have since improved significantly, so these predictionsare quite conservative (requiring improvement by less than a factor of two in thesystematic uncertainties found in Run I). Run II is now expected to yield precisionof 2.0–2.5 GeV/c2 per experiment.

8.2 Measurement of Mt at the Large Hadron Collider

Roughly 107 t t events will be produced every year at the LHC. With this wealthof statistics, one is free to select only those events that present the cleanest mass

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?TOP QUARK MASS 477

measurement (for example, events in which thet and t decay products are wellseparated and thus have minimal combinatoric confusion) and to apply mass mea-surement techniques with poorer statistical precision but smaller systematic un-certainties than the kinematic reconstruction in use at the Tevatron. One possibleanalysis would involve selecting only events in which the top quark decays tob`ν, and theb decays semileptonically. Then one can use the invariant mass ofthe lepton pair as a measure of the top quark mass and thus be free of first-orderdependence on hadronic effects (uncertainties in modeling the fragmentation ofbquarks would still come into play).

Studies of such analyses are in the early stages. The CMS and ATLAS experi-ments at the LHC both expect to perform measurements with precisions of roughly2 GeV/c2 (38). It is too early to tell whether the LHC measurements will offer asignificant improvement over those attained in Fermilab’s Run II.

9. CONCLUSIONS

The past four years have seen the advent of top quark physics. The data samplecollected at Fermilab has allowed not only the unambiguous observation of the topquark but also the measurement of its mass with a relative precision of 3%. Thismeasurement is already sufficient to allow a meaningful comparison of thet t pro-duction cross section with QCD predictions. The fact that there is no discrepancy(see Figure 15) can be used to constrain extensions of the standard model in whicht t production is enhanced (39) or the top quark can decay to non–standard-modelparticles [charged Higgs bosons being a prime example (40)].

The direct measurements that produced this number involved a combination oftime-honored and novel analysis techniques. The concept of kinematic reconstruc-tion is nearly as old as the field of high-energy physics, but the top quark massmeasurement is the first in which jets have been treated as composite particles inkinematic fits.

It is important to note that the current measurement depends minimally ontheoretical assumptions. Therefore, the result can be used to predict properties ofthe standard model or any extensions thereof. The most obvious prediction is theconstraint on the mass of the Higgs boson in the minimal (single-Higgs) standardmodel (as shown in Figure 1). Although relatively light Higgs masses are favored,the constraint is not yet strong enough to rule out any mass.

This situation could well change prior to the start of data collection at theLHC. Fermilab’s next run, which is expected to expand thet t statistics by a factorof 30–40, should allow the top quark to be measured with a precision of about2 GeV/c2. This measurement, combined with improvements in theW boson massmeasurement, should place meaningful restrictions on the Higgs mass.

The recent past has seen the development of an array of techniques for measuringthe top mass in nearly all of its decay channels. The application of these techniques

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(or improvements on them) to the large sample oft t events soon to be availablewill be interesting at the very least, and may even be of sufficient sensitivity toexpose physics beyond the standard model.

Visit the Annual Reviews home page at http://www.AnnualReviews.org

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Annual Review of Nuclear and Particle Science Volume 49, 1999

CONTENTSSNAPSHOTS OF A PHYSICIST'S LIFE, J. David Jackson 1RECENT PROGRESS IN BARYOGENESIS, Antonio Riotto, Mark Trodden 35THE COSMIC MICROWAVE BACKGROUND AND PARTICLE PHYSICS, Marc Kamionkowski, Arthur Kosowsky 77MEASUREMENT OF SMALL ELECTRON-BEAM SPOTS, Peter Tenenbaum, Tsumoru Shintake 125PARTICLE PHYSICS FROM STARS, Georg G. Raffelt 163HIGH-ENERGY HADRON-INDUCED DILEPTON PRODUCTION FROM NUCLEONS AND NUCLEI, P. L. McGaughey, J. M. Moss, J. C. Peng 217CHARMONIUM SUPPRESSION IN HEAVY-ION COLLISIONS, C. Gerschel, J. Hüfner 255SPIN STRUCTURE FUNCTIONS, E. W. Hughes, R. Voss 303MICROPATTERN GASEOUS DETECTORS, Fabio Sauli, Archana Sharma 341LEPTOQUARK SEARCHES AT HERA AND THE TEVATRON, Darin E. Acosta, Susan K. Blessing 389DIRECT MEASUREMENT OF THE TOP QUARK MASS, Kirsten Tollefson, Erich W. Varnes 435NEUTRINO MASS AND OSCILLATION, Peter Fisher, Boris Kayser, Kevin S. McFarland 481TWO-PARTICLE CORRELATIONS IN RELATIVISTIC HEAVY-ION COLLISIONS, Ulrich Heinz, Barbara V. Jacak 529COLLECTIVE FLOW IN HEAVY-ION COLLISIONS, Norbert Herrmann, Johannes P. Wessels, Thomas Wienold 581INCLUSIVE JET AND DIJET PRODUCTION AT THE TEVATRON, Gerald C. Blazey, Brenna L. Flaugher 633

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Kirsten Tollefson Erich W. Varnes · Kirsten Tollefson Department of Physics and Astronomy,...

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