XXVI Physics in Collision, Buzios, Rio de Janeiro, 6-9 July 2006 1

Precision Electroweak Physics at the TevatronEric B. JamesFNAL, Batavia, IL 60510, USAon behalf of the CDF and DØ Collaborations

An overview of Tevatron electroweak measurements performed by the CDF and DØ experiments is presented.

The current status and future prospects for high precision measurements of electroweak parameters and detailed

studies of boson production are highlighted.

1. Introduction

The substantial samples of W and Z bosons cur-rently being collected by the CDF and DØ experi-ments accommodate a wide variety of precision elec-troweak measurements. The two general purpose ex-periments observe pp collisions at a center-of-mass en-ergy of 1.96 TeV generated by the Fermilab TevatronCollider. In its current operating mode, the Teva-tron operates as a W and Z boson factory. In a nor-mal week of operation the Tevatron produces roughly50,000 W boson and 5,000 Z boson events in eachlepton decay channel for each experiment. Currently,each of the experiments has recorded approximately1.5 pb−1 of data, which corresponds to about a quar-ter of the total expected Run II luminosity.

Z boson parameters have been measured to veryhigh precision at the large electron-positron collider(LEP) at CERN and the linear collider at SLAC. Forexample, the Z boson mass has been measured withan accuracy of 2 parts in 105 [1]. However, currentmeasurements of theW boson parameters are less pre-cise (the present uncertainty on the W boson mass isabout 4 parts in 104 [1]). Based on expected RunII integrated luminosities, the two Tevatron experi-ments will collect a sample of W bosons events on thesame order as the 17 million Z boson events collectedby the four LEP experiments. Using these event sam-ples, CDF and DØ will significantly reduce the currentexperimental uncertainties on the electroweak param-eters associated with the W boson.

In addition, the largeW and Z boson samples allowfor precision tests of the QCD production mechanismsfor bosons. In particular, the cross section for bosonproduction depends on both the calculable hard scat-tering parton cross sections and the Parton Distribu-tion Functions (PDFs), which describe the momen-tum fractions carried by the quarks and gluons withinthe proton. The PDFs are determined experimentally,and studies of boson production at the Tevatron canbe used to place constraints on these distributions.These constraints are important because PDF uncer-tainties significantly impact the level of precision ofTevatron measurements of electroweak parameters.

2. Detectors

The CDF and DØ detectors are designed to trig-ger on and accurately reconstruct charged particles,electrons, photons, muons, hadronic jets, and thetransverse energy imbalance associated with neutri-nos. The z-axes of the CDF and DØ coordinate sys-tems are defined to be along the direction of the in-coming protons. Particle trajectories are describedby θ, the polar angle relative to the incoming protonbeam, and φ, the azimuthal angle about the beamaxis. Pseudorapidity, η = −ln(tan(θ/2), is also usedto describe locations within the detectors.One particular strength of the CDF detector is its

beam-constrained central tracking resolution,

δ(pT )/pT ∼ 0.0005× pT (GeV/c) [|η| < 1], (1)

based on hit information from the outer open-cell driftchamber. The calorimeters of both detectors allow forhigh-resolution reconstruction of the energies of elec-trons, photons, and jets. For example, the energyresolution for clusters in the CDF central electromag-netic calorimeter is

δ(ET )/ET ∼ 13.5%⊕ 1.5% (GeV) [|η| < 1.1], (2)

which allows for high precision electron energy mea-surements. A main strength of the DØ detector isthe forward coverage provided by its calorimeters andmuon detector systems. The DØ calorimeter pro-vides hermetic coverage up to |η| < 4.2 (compared to|η| < 3.6 for CDF) and muon coverage up to |η| < 2.0(compared to |η| < 1.5 for CDF). This additional for-ward coverage results in a significantly better accep-tance for leptons from boson decays, particularly formuons.

3. Measurements of ElectroweakParameters

3.1. W Boson Mass Measurement

A precision measurement of the W boson mass isamong the highest priorities for the Tevatron experi-ments. Self-energy corrections to theW boson depend

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2 XXVI Physics in Collision, Buzios, Rio de Janeiro, 6-9 July 2006

Table I Preliminary uncertainty estimates for CDF Wboson mass measurement using 200 pb−1 of data.

Uncertainty [MeV] Electrons Muons Common

Lepton Energy Scale 27 17 17

and Resolution

Recoil Scale and 14 12 12

Resolution

Backgrounds 7 9 -

Production and 16 17 16

Decay Model

Statistics 48 53 -

Total 60 60 26

on the masses of the top quark (∝M 2top) and the Higgs

boson (∝ lnMH), as well as potential contributionsfrom non-Standard Model (SM) physics. Because ofthese dependencies, the W boson mass is a criticalinput to SM fits that constrain the mass of an unob-served Higgs boson or, subsequent to a potential Higgsdiscovery, test the consistency of the SM.

The current level of uncertainty on top quark massmeasurements from the Tevatron experiments [2] is atthe level of 2.1 GeV/c2 which corresponds to roughlya 1.2 % measurement of Mtop. To obtain equivalentconstraining power on MH , the W boson mass wouldneed to be measured to about 0.015% correspondingto a total uncertainty of about 12 MeV/c2. Due tothe needed level of precision, the W boson mass mea-surement is extremely challenging.

In order to make a measurement substantially bet-ter than 0.1 %, all aspects of W boson productionand detection need to be understood at the 10 MeVlevel. In particular, this precision must be achievedfor W boson production and decay, lepton momen-tum/energy scales and resolutions, and additional en-ergies within the event associated with hadronic recoilagainst the boson pT and underlying interactions ofthe remnant quarks and gluons. Once this detailedevent model has been constructed, the W boson masscan be determined by generating events for many dif-ferent mass values and picking the set that providesthe best match with data, in particular by fitting tothe transverse mass, MT = E`

TEνT − (E`

xEνx +E

`yE

νy ),

distribution for the W → `ν candidate events in data.

The CDF experiment is close to completing a Wmass measurement using 200 pb−1 of data collectedat the beginning of Run II. The expected uncertain-ties associated with this measurement are shown inTable I. The total uncertainty for the combined mea-surement based on events collected in both the elec-tron and muon channels is expected to be 48 MeV/c2

which would make this measurement the single mostprecise to date. More importantly, the largest compo-nent of the total uncertainty is statistical, indicating

that the result will be further improved simply by in-corporating more data. In fact, with the exceptionof the uncertainty associated with the production anddecay model, each of the uncertainty categories im-proves with additional statistics. Larger samples ofJ/Ψ, Υ, and Z boson events, for example, further im-prove the measurement of the track momentum andcalorimeter energy scales for leptons.Figure 1 shows a projection for the expected pre-

cision of the W boson mass measurement as a func-tion of integrated luminosity for a single experimentbased on Tevatron Run I measurements. The com-bined preliminary uncertainty for the 200 pb−1 CDFRun II analysis lies significantly below the expecta-tion based on the Run I results, indicating improvedunderstanding of the W boson event characteristics.With enough additional luminosity, the precision ofthe measurement will become limited by the uncer-tainties associated with the boson production and de-cay model (currently on the order of 20 MeV) which donot scale with statistics. Reducing these uncertaintiesrequires additional measurements that can constraincomponents of the production model, such as PDFsand the boson pT spectrum.A projection for the potential Tevatron constraints

on the Higgs boson mass based on 8 fb−1 of data deliv-ered to each experiment is shown in Figure 2. At theexpected level of precision, significant constraints willbe placed on non-SM particles such as those predictedby supersymmetry (SUSY).

3.2. W Boson Width Measurements

The width of the W boson is a less constraining ob-servable in global electroweak fits than the mass, butmeasuring its value confirms a basic prediction of theSM and could provide indications of new physics be-yond the SM. The Tevatron experiments make bothdirect and indirect W boson width measurements.The direct measurements have no built-in SM assump-tions and are therefore sensitive to potential contri-butions from new physics such as a heavy W ′. Indi-rect measurements are based on SM assumptions andprovide high precision results that can also be usedto place constraints on other SM parameters such asCKM matrix elements.Tevatron direct measurements of the W boson

width are extracted from the shape of the high MT

region in W → `ν events. The procedure is similar tothat used for measuring the W mass. The W bosonproduction and detector resolution effects that dis-tort the observed lineshape must be carefully modeledwithin a fast event simulation. Using the tuned simu-lation, event samples are generated based on a rangeof input values for the W boson width. The changein the shape of the high MT tail as a function of theW width is illustrated in Figure 3. DØ has made a

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XXVI Physics in Collision, Buzios, Rio de Janeiro, 6-9 July 2006 3

Figure 1: Projection for the expected precision of a single experiment W mass measurement as a function of integratedluminosity based on Run I Tevatron measurements.

Figure 2: Projection for Tevatron constraints on MH

based on the expected precision of combined top quarkand W boson mass measurements assuming 8 fb−1 ofdata collected by each experiment.

preliminary direct measurement of theW width basedon 177 pb−1 of Run II data. The measurement usesthe peak region in the MT distribution for W → eνcandidate events to normalize signal and backgroundcontributions to the sample, and then fits the shapein the tail region to determine the most likely valuefor the W width. The final result for the W widthobtained from the fit shown in Figure 4 is

ΓW = 2.011± 0.093(stat)± 0.107(syst) GeV. (3)

Indirect determinations of the W boson width areobtained from a measured ratio of production cross

(GeV)TM60 80 100 120 140 160 180 200

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Figure 3: DØ simulation of the MT distribution forW → eν events as a function of W boson width.

sections times branching fractions,

R =σ × Br(W → `ν)

σ × Br(Z → ``). (4)

The value of R can be measured very precisely sincemany of the uncertainties associated with the individ-ual cross section measurements, in particular the sig-nificant uncertainty on the measured luminosity, can-cel in the ratio. Within the context of the SM, thisratio can also be expressed as

R =σ(W )

σ(Z)× Γ(W → `ν)

Γ(W )× Γ(Z)

Γ(Z → ``). (5)

Using this equation, a precise value for Γ(W ) canbe extracted from R using a next-to-next-to-leading

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Figure 4: DØ fit to the MT distribution of W → eνevents used to measure the W width.

order (NNLO) theoretical prediction for σ(W )/σ(Z),precision LEP measurements of Γ(Z → ``) and Γ(Z),and a SM calculation for Γ(W → `ν).CDF has made an indirect measurement of the W

boson width based on the first 72 pb−1 of data col-lected in Run II. The ratio R was measured indepen-dently in the electron and muon channels, resulting ina combined value of

R = 10.84± 0.15(stat)± 0.14(syst), (6)

which has an overall relative precision of 1.9%. Sincethe most significant contribution to the systematic un-certainty on this measurement originates from the lep-ton selection efficiency measurement made from theZ → `` data samples, it is expected that a measure-ment with a precision of better than 1% will be pos-sible using additional data statistics.The indirect value for theW boson width extracted

from the measured value of R is

Γ(W ) = 2092± 42 MeV, (7)

which is in good agreement with the SM predictionand the previously described direct measurement ofthe W boson width. A comparison of the measuredindirect width with previous results and the SM ex-pectation is shown in Figure 5. Since in the SM thetotal W boson width is a sum over partial widths forleptons and quarks, which in the case of the quarks de-pend on certain CKM matrix elements, the measuredvalue of Γ(W ) can also be used to indirectly measurethe value of the least constrained element, Vcs. Basedon world-averaged measurements of the other CKMmatrix elements that contribute to the partial widths,CDF obtains a value of

|Vcs| = 0.976± 0.030. (8)

3.3. Quark Couplings

The Tevatron experiments can extract the axial andvector neutral current light quark couplings from mea-

1.8 1.9 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8(GeV)

Γ(W)

TeVEWWG

(from R)

(from R)

Standard Model

UA1(e+µ)UA2(e)

CDF Ia(e)D0 Ia+b(e)

CDF II (e+µ)

World Average (RPP 2002)(includes Run I results)

Figure 5: Comparison of the CDF indirect widthmeasurement with previous results and the SMprediction.

Figure 6: Illustration of γ∗/Z decay in the parton-partoncenter of mass frame. Forward (backward) events aredefined as those with positive (negative) cos(θ∗).

surements of the Drell-Yan forward-backward asym-metry. This asymmetry is defined as

AFB =σF − σBσF + σB

(9)

where σF (B) is defined as the cross section for Drell-Yan events in which the positively charged lepton isproduced along (opposite) the proton’s direction ofmotion in the parton-parton center of mass frame.The decay of the γ∗/Z in this frame is illustratedin Figure 6. The sign of cos(θ∗) determines whethera given event is forward or backward (forward ifcos(θ∗) > 0).CDF and DØ have both made preliminary mea-

surements of the forward-backward asymmetry forγ∗/Z → ee events as a function of dielectron invariantmass. The CDF result based on a 364 pb−1 data sam-ple is shown in Figure 7 and the DØ result based on a177 pb−1 data sample is shown in Figure 8. As illus-trated in Figure 9, the quark couplings to the Z boson

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XXVI Physics in Collision, Buzios, Rio de Janeiro, 6-9 July 2006 5

Figure 7: CDF Measurement of the forward-backwardasymmetry in γ∗/Z → ee events as a function of thedi-electron invariant mass.

Figure 8: DØ Measurement of the forward-backwardasymmetry in γ∗/Z → ee events as a function of thedi-electron invariant mass.

can be extracted from these measurements. Althoughthe coupling measurement is less precise than that ofthe LEP experiments, it breaks a two-fold degeneracyin the LEP results, providing an important confirma-tion of the SM. The coupling values have also beendetermined from analysis of deep inelastic scatteringdata at HERA [3].

More importantly, the Tevatron experiments mea-sure AFB over a wide range of invariant masses (bothbelow and above the Z-pole). The high mass regionis of particular interest since the effects of new bosonsinterfering with the SM bosons could result in mea-sured values of AFB inconsistent with SM expecta-tions. The potential effect of a Z ′ on the predictedAFB in the high mass region is shown in Figure 8,along with the measured DØ values. With additionaldata it should be possible to distinguish between the

Figure 9: Comparison of the limits on the allowed rangeof values for the up quark axial and vector neutralcurrent couplings obtained from Tevatron (72 pb−1),HERA, and LEP measurements.

Table II Diboson final states available at the Tevatronand the trilinear couplings involved in their production.The couplings shown in parentheses are absent in the SM.

Diboson Final State Trilinear Couplings

qq′ → W → Wγ WWγ only

qq′ → W → WZ WWZ only

qq → γ∗/Z → WW WWγ , WWZ

qq → γ∗/Z → Zγ (ZZγ) , (Zγγ)

qq → γ∗/Z → ZZ (ZZγ) , (ZZZ)

new physics and SM scenarios.

3.4. Trilinear Gauge Couplings

The analysis of diboson final states at the Tevatronprovides an opportunity for studying self-interactionsof the gauge bosons. These interactions are a directresult of the electroweak SU(2) structure, and the SMmakes specific predictions on the expected produc-tion cross sections for each diboson final state. Non-SM particles that couple to the electroweak bosonscan modify the expected cross sections, particularlyat high ET , and looking for potential indications ofthese anomalous couplings provides a route to uncov-ering new physics.Table II gives a summary of the diboson final states

available at the Tevatron and the trilinear gauge cou-plings that contribute to the production of each state.The Tevatron experiments are sensitive to differentcombinations of couplings than LEP and explore ahigher

√s. The couplings in the table that are en-

closed within parentheses are absent in the SM. Due

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Figure 10: Comparison of missing ET distribution forobserved data events to the combined expectation fromsignal and background in the CDF WW analysis.

to the absence of these couplings, the associated fi-nal states are ideal channels for observing effects fromnew physics.The CDF and DØ experiments have produced a

wide variety of new Run II results based on the studyof diboson final states [4] [5]. A few of these are high-lighted in detail here. The cross section for WW pro-duction, which involves both the WWγ and WWZtrilinear gauge couplings, has recently been measuredby CDF using a 825 pb−1 data sample. The analysisfocuses on the dilepton final state produced when bothW bosons decay into a lepton and neutrino. Eventsare selected with two opposite-sign leptons (electronsor muons) that satisfy the standard CDF selectioncriteria. The missing ET in the event, expected fromthe two neutrinos, is required to be above 25 GeV,greatly reducing the main expected background con-tributions from Drell-Yan, Wγ, and W plus jet pro-duction. Before looking at the signal region, events inthe low missing ET region are utilized to cross-checkthe background estimation. In the signal region, thefinal background estimate is 38 ± 5 events on top ofan expected WW signal contribution of 52±4 events.Based on 95 observed events, CDF measures a crosssection of

σ(pp→WW ) = 13.6± 3.0(stat + syst + lum), (10)

consistent with the next-to-leading order (NLO) cal-culation [6] of 12.4±0.8 pb. The final candidate eventsplotted as a function of event missing ET , along withthe expected signal and background contributions, areshown in Figure 10.Both CDF and DØ have also recently completed

measurements of WZ production. Production of thisfinal state is of particular interest because the WWZcoupling can be studied independent of theWWγ cou-pling, which also contributes to WW production. DØhas performed a search based on a data sample cor-responding to roughly 800 pb−1 of integrated lumi-nosity. This analysis uses the trilepton final state inwhich both bosons decay leptonically. A total of three

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WZ Signal MC + Backgrounds

Sum of Backgrounds

WZ Candidate Transverse Mass

Figure 11: Comparison of MT distribution determinedfrom the W boson decay products for observed dataevents with the combined expectation from signal andbackground in the DØ WZ analysis.

leptons (electrons or muons) passing the standard DØselection criteria are required. Of the three leptons,two are required to be of the same flavor and form anopposite-sign pair with an invariant mass consistentwith the Z boson mass. The event missing ET is alsorequired to be greater than 20 GeV, consistent withthat from the neutrino produced in the decay of theW boson. Taking advantage of its wider acceptancefor leptons, DØ expects to see 7.5± 1.2 signal eventson top of a background of 3.6 ± 0.2 events, and ob-serves 12 events in the data. Based on the calculatedprobability for the background to fluctuate into theobserved number of events, DØ obtains 3.3 σ evidencefor WZ production and measures

σ(pp→WZ) = 4.0+1.9−1.5(stat + syst + lum), (11)

consistent with the NLO calculation [6] of 3.68 ±0.25 pb. Figure 11 shows the transverse mass dis-tribution for the neutrino (missing ET ) and leptoncoming from the W boson decay for the DØ candi-date events compared to the combined expectationfrom signal and background.CDF completed a similar search using roughly the

same amount of data and observed only 2 events com-pared to an expectation of 3.7±0.3 signal and 0.9±0.3background events. The observation of two events wasfound to be consistent with both the background-onlyand background plus signal hypotheses. The smallernumber of expected events as compared with the DØanalysis is directly related to the reduced acceptancefor leptons in the CDF detector. In order to improvethe CDF analysis, new lepton categories were createdto take advantage of additional tracking and calorime-ter cluster information in the events to increase lep-ton acceptance. In order to increase electron coverageout to |η| < 2.8, a category for forward electron can-didates in the calorimeter with no track match wasadded. Similarly, an increase in muon coverage out to|η| < 1.6 was obtained using forward track candidatesfiducial to the calorimeter with energy deposits con-sistent with the expectation from a minimum-ionizing

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γZ

CDF Run II Preliminary -1L dt = 1.1 fb∫

Figure 12: Comparison of missing ET distribution forobserved data events with the combined expectationfrom signal and background in the CDF WZ analysis.The arrow on the figure indicates the signal region forthis search (missing ET > 25 GeV).

particle. In addition, the tracks pointing at calorime-ter cracks were placed into a flavor-neutral categoryof leptons which could be assigned as either electronsor muons. With the additional lepton categories inplace, CDF performed a new search for WZ produc-tion using 1.1 fb−1 of data. Including the improvedlepton acceptance, CDF observes 16 events with sig-nal and background expectations of 12.5± 0.9 eventsand 2.7±0.4 events, respectively. Based on the proba-bility of the background fluctuating into the observedsignal, CDF obtains a 5.9 σ observation of WZ pro-duction and measures

σ(pp→WZ) = 5.0+1.8−1.6(stat + syst + lum), (12)

which is also consistent with the NLO calculation.The final candidate events plotted as a function ofevent missing ET , along with the expected signal andbackground contributions, are shown in Figure 12.As mentioned previously, diboson production is sen-

sitive to new physics appearing in the trilinear gaugecouplings. Potential new physics contributions canbe incorporated in the Lagrangian using a standardmethodology that introduces two parameters, λ and∆κ, which are zero in the SM and non-zero in thecase of additional new physics contributions. Gen-erally, the effect of anomalous couplings on dibosonproduction is a net increase in the cross section athigh ET . Figure 13 illustrates how the shape ofthe diboson cross section as a function of ET variesfor different values of λ and ∆κ. The added termsin the Lagrangian violate unitarity unless an upperlimit (Λ) on the scale for the new physics is imposed.A common approach is to use the parameterizationα(s) = α0/(1 + s/Λ2)2 which causes the effect of the

Figure 13: The predicted shape of a generic dibosoncross section as a function of ET for different values of λand ∆κ.

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= -0.4λ = -0.4, κ∆ = 1.0λ = 1.0, κ∆

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Figure 14: Comparison of leading lepton pT distributionfor DØ WW candidate events observed in the dileptonfinal state with SM and non-SM expectations.

anomalous couplings to “turn-off” as the upper limiton the energy scale is approached.

DØ has performed a preliminary analysis to setanomalous couplings limits based on a measurement oftheWW cross section using dilepton final states. Theanalysis sets limits on anomalous WWγ and WWZtrilinear gauge couplings under the assumption thatthe two couplings are equal and Λ = 2 TeV. Figure 14shows the DØ data and both SM and non-SM expec-tations plotted as a function of the pT of the highestpT lepton. Based on the observed agreement betweendata and the SM prediction, DØ obtains the followinglimits:

−0.32 < ∆κ < 0.45,−0.29 < λ < 0.45. (13)

These preliminary limits can be improved significantlywith larger data samples and incorporating informa-tion from other final states.

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Figure 15: Illustration of W boson production at the Tevatron. A u quark in the proton annihilates with a d quark inthe anti-proton at a squared center of mass s = Q2 to produce a W+. The energies of the u and d are xpEp and xpEp,respectively.

4. Studies of Boson Production

4.1. Boson Production at the Tevatron

A typical example of boson production at the Teva-tron is shown in Figure 15. At leading order (LO), aquark and anti-quark pair annihilate to create a W orZ boson, which subsequently decays into a quark orlepton pair. The production cross section is calculatedas a sum of partial cross sections (dσqq), convolutedwith the PDFs that describe the distributions of theproton momentum fraction (xp) carried by each of theconstituent quarks and gluons. The cross section canbe written as

σ =

∫ ∑i,j

[fqi (xp)fqj (xp)+f

qi (xp)f

qj (xp)]×dσqqdxpdxp

(14)where i and j denote the different possible quark fla-vor combinations. The longitudinal momentum of theproduced boson is directly related to the PDFs. Inparticular, if one of the two annihilating quarks car-ries a significantly larger fraction of proton momen-tum, the boson will be produced with momentum inthe same direction as the incident proton.The effects of QCD and QED NLO corrections are

also important. QCD corrections give rise to finalstates that contain multiple partons, sometimes withhigh pT , and modify the overall boson productionkinematics, including the boson pT spectrum. Themost important effect originating from NLO QED cor-rections is photon radiation from final state chargedleptons, which have a significant effect on lepton iden-tification and kinematics. QED radiation from theinitial state quarks and from the boson itself (in thecase ofW bosons) also contributes to the overall eventkinematics.

4.2. Parton Distribution Functions

The functional forms of the PDFs originate fromnon-perturbative QCD interactions and are thereforeincalculable. Instead, they are parameterized usingdata from deep inelastic scattering, fixed target, andhadron collider experiments. Two standard parame-terizations come from the CTEQ [7] and MRST [8]groups. In the case of the CTEQ group, the partonmomentum fraction distributions are parameterizedas

xfa(x,Q0) = A0xA1(1− x)A2eA3x(1 +A4x)

A5 (15)

for five categories of quark/gluon proton constituents(valence u and d quarks, sea u and d quark com-binations, and gluons). This configuration gives atotal of thirty free parameters in the fit to the ex-perimental data, although the CTEQ group choosesto leave ten of these at fixed values. The remainingfree parameters are determined for a low energy scale,Q0 = 1.3 GeV, and the Q2 dependence is obtainedfrom QCD evolution equations.A recent development is that each group also pro-

vides a set of “error” PDFs that are intended to mapout the allowable parameter space for the PDFs withinthe experimental data uncertainties. The twenty freeparameters used in the fit are found to be correlatedwith one another. To facilitate uncertainty calcula-tions, these correlations are removed by forming eigen-vectors within the Ai-space. For each of the twentyeigenvectors, two complete PDF sets are generatedcorresponding to a given increase in χ2 of the overallfit (∆χ2 = 100 for the CTEQ group). The MRSTgroup follows a similar procedure using a slightly dif-ferent parameterization that results in only fifteen freeparameters for their fit. The MRST group also uses a

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PDF CTEQ6 Eigenvector5 10 15 20 25 30 35 40

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Figure 16: An example of how CTEQ “error” PDF sets are used to determine an overall PDF uncertainty. The shift inthe measured W boson mass from its central value is obtained for Monte Carlo templates generated with each of the 40error PDF sets. The observed shifts associated with each of the twenty orthogonal eigenvectors are added inquadrature to determine the total uncertainty.

smaller ∆χ2 = 50 to construct its version of the errorPDFs.An example of how error PDFs are used to deter-

mine an overall PDF uncertainty for a specific analysisis shown in Figure 16 for the case of theW boson massmeasurement. The shift in the measured mass fromits central value is obtained using Monte Carlo tem-plates generated with each of the forty error PDF sets.Since the twenty eigenvectors are orthogonal to eachother by design, the observed shifts associated witheach can be added in quadrature to determine a totalPDF model uncertainty. Although each eigenvectortypically contains information about multiple fit pa-rameters, there is a strong correlation in some casesbetween a given fit parameter and an eigenfunction.For example, the eigenvector corresponding to errorPDFs 1 and 2 in Figure 16 has a significant correla-tion with the A1 (low-xp) parameter associated withvalence u quarks. These correlations give an indica-tion of the experimental inputs to the fits which needto be improved to reduce the overall PDF uncertaintyfor a specific analysis.

4.3. Inclusive Production Cross Sections

Because many electroweak measurements at theTevatron are sensitive to uncertainties in the PDFmodel, both CDF and DØ perform studies of boson

production to constrain the PDF model. The sim-plest of these studies are measurements of the inclu-sive boson production cross sections. The Tevatronexperiments measure inclusiveW and Z cross sectionsusing each of the lepton (e, µ, and τ) decay chan-nels. The dominant uncertainty in these results is as-sociated with the integrated luminosity measurementsmade by each experiment (∼ 6%). Within this uncer-tainty, the measured cross sections are found to be ingood agreement with the NNLO theoretical calcula-tions [9]. The agreement between CDF and DØ mea-sured values and theoretical predications are shownin Figures 17 and 18. Since the theoretical uncertain-ties are significantly smaller than the measurementuncertainties, no additional constraints on the bosonproduction model can be obtained from these mea-surements.

4.4. Forward W Boson Cross Section

Differential cross section measurements contain ad-ditional information that can be used to constrainPDFs. CDF performs a simple differential measure-ment by independently evaluating the W boson crosssection using W → eν events with electrons observedin the central and forward regions of the detector. Fig-ure 19 shows the W boson acceptance as a function

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10 XXVI Physics in Collision, Buzios, Rio de Janeiro, 6-9 July 2006

(TeV)cmE1.2 1.4 1.6 1.8 2 2.2 2.4

BR

(nb

)× σ

1

2

D0 I (e)

D0 II (e)

) µ D0 II (

CDF I (e)

) µ CDF II (e +

|< 2.8)η CDF II (e,1.2<|

NNLO theory curves:Martin, Roberts, Stirling, Thorne

Figure 17: Summary of Tevatron inclusive W boson crosssection measurements as a function of ECM compared toa NNLO theoretical calculation (solid black line).

(TeV)cmE1.2 1.4 1.6 1.8 2 2.2 2.4

BR

(nb

)× σ

-110

D0 I (e)

D0 II (e)

) µ D0 II (

) τ D0 II (

CDF I (e)

) µ CDF II (e +

) µ CDF II (

) τ CDF II (

NNLO theory curves:Martin, Roberts, Stirling, Thorne

Figure 18: Summary of Tevatron inclusive Z boson crosssection measurements as a function of ECM compared toa NNLO theoretical calculation (solid black line).

of the boson rapidity, defined as

yW =1

2log

E + pzE − pz

, (16)

for the CDF W → eν cross section measurements us-ing events with electrons reconstructed in the centraland forward calorimeter modules. Since W bosonsproduced at different rapidities probe different regionsof xp, the ratio of central to forward cross sectionsmeasurements can be a useful tool for placing con-straints on PDFs.The selection of forward electron candidates is

based on electromagnetic clusters in the calorimetermatched with tracks reconstructed primarily from sil-icon detector hits [10]. Given the selection criteria,CDF observes 48,165 candidate events in a 223 pb−1

data sample. The MT spectrum of the candidateevents is shown in Figure 20, along with the combinedexpectation for signal and background. The observedagreement indicates a good understanding of the for-ward detector systems.The measured forward (1.2 < |ηdete | < 2.8) cross

section is

σfor = 2796± 13(stat)+95−90(syst) pb, (17)

W Boson Rapidity0 0.5 1 1.5 2 2.5 3

ν e

→

Acc

epta

nce:

W

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

|<1)η Central Region (|

| < 2.8)η Forward Region (1.2 < |

Figure 19: W boson acceptance as a function of rapidityfor the CDF W → eν cross section measurements usingevents with reconstructed electrons in the central andforward parts of the detector.

Figure 20: MT distribution for candidate events in theCDF cross section measurement based on W → eν eventswith electrons in the forward detector region.

neglecting the luminosity uncertainty which will can-cel in the cross section ratio. The previously measuredcentral (|ηdete | < 0.9) cross section [11] has a value of

σcen = 2771± 14(stat)+62−56(syst) pb, (18)

also neglecting the luminosity uncertainty. The re-maining systematic uncertainties on the measure-ments are dominated by those associated with elec-tron identification and the PDF model. In order toseparate these, CDF uses visible cross sections, de-fined as

σvis = σtot ×A, (19)

where Acen is for example the kinematic and geomet-ric acceptance for W → eν events in the central cross

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XXVI Physics in Collision, Buzios, Rio de Janeiro, 6-9 July 2006 11

section measurement. Using this definition the PDFmodel uncertainties are removed from the measuredratio of cross sections,

Rexp = σcenvis /σforvis = 0.925± 0.033. (20)

CDF then compares the measured ratio with theequivalent theoretical ratio of acceptances

Rth = Acen/Afor (21)

determined from simulated event samples generatedusing both the CTEQ (Rth = 0.924 ± 0.037) andMRST (Rth = 0.941± 0.012) PDF distributions. Theuncertainties on the acceptance ratios are obtainedfrom the error PDF sets using the previously describedmethod. The uncertainty on the measured ratio is ofthe same order as the PDF uncertainties on the the-oretical ratio, suggesting that a similar measurementwith additional statistics would help to constrain thePDF models.

4.5. Differential Z Boson Cross Section

Measuring the differential boson production crosssection over the full rapidity range can further improvePDF model constraints. The dilepton decay modes ofthe Z boson allow for precise measurements, since thebackgrounds in these final states are small and the fullevent kinematics can be precisely reconstructed. Therapidity of the Z boson is closely related to the pro-ton momentum fractions carried by the two collidingquarks. As shown in Figure 21, W or Z bosons areproduced at high rapidity when the proton momen-tum fraction of one quark is significantly larger thanthat of the other. Therefore, the measured differentialcross section at high rapidity is a good probe of thePDF distributions at high xp.DØ has made a preliminary measurement of the dif-

ferential Z boson cross section based on a 337 pb−1

data sample. Using Z → ee candidate events, DØ re-constructs the differential cross section shown in Fig-ure 22. The measured cross section is observed toagree well with the NNLO prediction. The measure-ment is currently statistics-limited but can be used toconstrain PDF models using additional data.

4.6. W Boson Charge Asymmetry

A final measurement useful for constraining PDFs istheW boson charge asymmetry measurement. On av-erage the u quarks inside the proton contain a higherfraction of the proton’s momentum than the d quarks.Due to this imbalance, W+ (W−) bosons produced atthe Tevatron have a net positive (negative) rapidity,as shown in Figure 23. The V-A structure of the elec-troweak couplings dictates the angular distribution ofthe leptons in the decays of the W bosons, which is

Figure 21: The interacting partons’ momentum fractionsrequired to produce a W boson (Q = 80 GeV). Thelarger the difference between xp and xp, the greater therapidity of the produced boson.

Y(Z)-3 -2 -1 0 1 2 3

/dY

σ d

σ

1/

0

0.05

0.1

0.15

0.2

0.25

0.3

Rapidity *γZ/NNLO, MRST01Data

Run II Preliminary∅D

Figure 22: Differential Z boson cross section measuredby DØ as a function of boson rapidity. The measuredcross section is in good agreement with a NNLOtheoretical prediction based on MRST PDFs (solid line).

preferentially opposite to the production asymmetry.As shown in Figure 23, the net effect of the decayasymmetry is to partially reduce the observable pro-duction asymmetry extracted from the lepton rapid-ity distributions. Because the production asymmetryoriginates from the imbalance of the momentum frac-tions carried by u and d quarks within the proton,charge asymmetry measurements provide constraintson the d/u ratio in the proton as a function of xp.Measurements are typically performed using the

charged leptons from theW boson decays. The leptonasymmetry is defined as

A(η`) =dσ+/dη` − dσ−/dη`dσ+/dη` + dσ−/dη`

= A(yW )⊗ (V −A).

(22)Both CDF and DØ have performed preliminary mea-surements of the lepton charge asymmetry. The keyexperimental issues are understanding forward lep-ton identification and charge misidentification rates,which are needed to correct the observed asymmetry.

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Figure 23: Rapidity distributions of positively andnegatively charged W bosons produced at the Tevatron,and the pseudorapidity distributions of the positively andnegatively charged leptons produced in their decays.

Figure 24: DØ lepton charge asymmetry measurementbased on W → µν events. The measurement is comparedto a theoretical calculation based on the CTEQ andMRST PDF models.

A DØ measurement of the lepton charge asymmetryusing W → µν events selected from a 230 pb−1 datasample is shown in Figure 24. The measured chargemisidentification rates for this analysis are found to bebelow 10−4 out to muon pseudorapidities of 2. Themeasured asymmetry is compared to a theoretical pre-diction based on the CTEQ PDF model. The mea-surement is observed to have some sensitivity to PDFseven at the current level of statistical sensitivity. TheCDF measurement based on W → eν events selectedfrom a 170 pb−1 data sample are shown in Figure 25.Here the data is separated into two categories basedon the ET of the electron. Comparisons with theoreti-cal predictions using the CTEQ PDF model illustratethe increased sensitivity of the high ET events to PDFvariations.

A new generation of Tevatron charge asymmetryanalyses are currently under development, with thegoal of fully exploiting the kinematic information inW events to directly reconstruct the underlying Wboson production asymmetry. Applying a W massconstraint leads to two kinematic solutions that canbe weighted by taking into account information aboutthe production and decay of the W bosons. Potential

Figure 25: CDF lepton charge asymmetry measurementbased on W → eν events. The measurement is comparedto a theoretical calculation based on the CTEQ PDFmodel.

Figure 26: Comparison of the potential PDF sensitivitiesfor lepton charge asymmetry and W boson productionasymmetry measurements made with a common set ofsimulated candidate events corresponding to a luminosityof 400 pb−1.

dependencies on the input model are resolved throughan iterative procedure. Preliminary CDF studies ofthis approach indicate significantly increased sensitiv-ity to PDFs. The potential increase in sensitivity isillustrated in Figure 26, which shows a comparisonof hypothetical lepton charge asymmetry and directW boson charge asymmetry measurements based ona common set of candidate events obtained from a400 pb−1 dataset.

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5. Conclusions

The large samples of W and Z bosons being col-lected at the Tevatron accommodate a wide variety ofelectroweak measurements. In particular, the proper-ties of the W boson can be measured with very highprecision by the CDF and DØ experiments. In ad-dition, detailed studies of boson production at theTevatron can be used to constrain PDF models andprovide important information about the boson pro-duction mechanisms. The analyses reported here arebased on only a small fraction of the expected data,so there is significant room for improving the preci-sion of the current measurements. It is important tonote that obtaining similar precision results from theLarge Hadron Collider (LHC) will be challenging andcertainly require input (such as PDF constraints) fromthe Tevatron experiments.

References

[1] W.-M. Yao et al., J. Phys. G33, 1 (2006), andreferences therein.

[2] Tevatron Electroweak Working Group (for theCDF and DØ Collaborations), “Combination ofCDF and DØ results on the Mass of the TopQuark,” hep-ex/060832.

[3] ZEUS Collaboration, “QCD and Electroweakanalysis of the ZEUS NC and CC inclu-sive and jet cross sections,” http://www-zeus.desy.de/public results/publicsearch.html,ZEUS-prel-06-003.

[4] http://www-cdf.fnal.gov/physics/ewk/[5] http://www-d0.fnal.gov/Run2Physics/wz/[6] J. Campbell and K. Ellis, Phys. Rev. D60, 113006(1999).

[7] http://www.phys.psu.edu/cteq[8] http://www-spires.dur.ac.uk/hepdata/mrs.html[9] P. Sutton, A. Martin, R. Roberts, and W. Stir-ling, Phys. Rev. D45, 2349 (1992).

[10] C. Hays et al., Nucl. Instrum. Meth. Phys. Res.A 538, 249 (2005).

[11] D. Acosta et al., Phys. Rev. Lett. 94, 091803(2005).

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