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Report of the Working Group on Precision Measurements

Conveners: Raymond Brocka, Jens Erlerb, Young-Kee Kimc, and William Marcianod

Working Group Members: William Ashmanskase, Ulrich Baurf , John Ellisong, Mark Lancasterh, LarryNodulmani, John Rhaj, David Watersk, John Womersleyl

aMichigan State University, East Lansing, MI 48824

bUniversity of Pennsylvania, Philadelphia, PA 19104

cUniversity of California, Berkeley, CA 94720

dBrookhaven National Laboratory, Upton, NY 11973

eUniversity of Chicago, Chicago, IL 60637

fState University of New York, Buffalo, NY 14260

gUniversity of California, Riverside, CA 92521

hUniversity College, London WC1E 6BT, U.K.

iArgonne National Laboratory, Argonne, IL 60439

jUniversity of California, Riverside, CA 92521

kOxford University, Oxford, OX1 3RH, U.K.

lFermilab, Batavia, IL 60510

Overview

Precision measurements of electroweak quantitiesare carried out to test the Standard Model (SM). Inparticular, measurements of the top quark mass, mtop,when combined with precise measurements of the Wmass, MW , and the weak mixing angle, sin2 θW , makeit possible to derive indirect constraints on the Higgsboson mass, MH , via top quark and Higgs boson elec-troweak radiative corrections to MW . Comparison ofthese constraints on MH with the mass obtained fromdirect observation of the Higgs boson in future colliderexperiments will be an important test of the SM.In this report, the prospects for measuring the W

parameters (mass and width) and the weak mixingangle in Run II are discussed, and a program for ex-tracting the probability distribution function of MH isdescribed. This is done in the form of three largelyseparate contributions.The first contribution describes in detail the strate-

gies of measuring MW and the W width, ΓW , athadron colliders, and discusses the statistical, theo-retical and detector specific uncertainties expected inRun II. The understanding of electroweak radiativecorrections is crucial for precision W mass measure-

ments. Recently, improved calculations of the elec-troweak radiative corrections to W and Z boson pro-duction in hadronic collisions became available. Thesecalculations are summarized and preliminary resultsfrom converting the theoretical weighted Monte Carloprogram into an event generator are described. Thetraditional method of extracting MW from the line-shape of the transverse mass distribution has been theoptimal technique for the extraction of MW in thelow luminosity environment of Run I. Other techniquesmay cancel some of the systematic and statistical un-certainties resulting in more precise measurements forthe high luminosities expected in Run II. Measuringthe W mass from fits of the transverse momentum dis-tributions of theW decay products and the ratio of thetransverse masses of the W and Z bosons are investi-gated in some detail. Finally, the precision expectedfor the W mass in Run II is compared with that fromcurrent LEP II data, and the accuracy one might hopeto achieve at the LHC and a future linear e+e− col-lider.In the second contribution, a study of the measure-

ment of the forward-backward asymmetry, AFB , ine+e− and µ+µ− events is presented. The forward-

1

backward asymmetry of ℓ+ℓ− events in Run II canyield a measurement of the effective weak mixing anglesin2 θW and can provide a test of the standard modelγ∗/Z interference at ℓ+ℓ− invariant masses well abovethe 200 GeV center of mass energy of the LEP col-lider. The asymmetry at large partonic center of massenergies can also be used to study the properties ofpossible new neutral gauge bosons, and to search forcompositeness and large extra dimensions. Estimatesof the statistical and systematic uncertainties expectedin Run II for AFB and sin2 θW are given. The un-certainty for sin2 θW is compared with the precisionexpected from LHC experiments, and from a linearcollider operating at the Z pole.The third contribution summarizes the features of

the FORTRAN package GAPP which performs a fit tothe electroweak observables and extracts the probabil-ity distribution function of MH .

2

Measurement of the W Boson Mass and Width

William Ashmanskasa, Ulrich Baurb, Raymond Brockc, Young-Kee Kimd, Mark Lancastere, Larry Nodulmanf ,David Watersg, John Womersleyh

aUniversity of Chicago, Chicago, IL 60637

bState University of New York, Buffalo, NY 14260

cMichigan State University, East Lansing, MI 48824

dUniversity of California, Berkeley, CA 94720

eUniversity College, London WC1E 6BT, U.K.

fArgonne National Laboratory, Argonne, IL 60439

gOxford University, Oxford, OX1 3RH, U.K.

hFermilab, Batavia, IL 60510

We discuss the prospects for measuring the W mass and width in Run II. The basic techniques used to measure MW are

described and the statistical, theoretical and detector-related uncertainties are discussed in detail. Alternative methods of

measuring the W mass at the Tevatron and the prospects for MW measurements at other colliders are also described.

1. Introduction

Measuring the W mass, MW , and width, ΓW areimportant objectives for the Tevatron experiments inRun II. The goal for the W mass measurement at theTevatron in Run II is determined by three factors:the direct measurement of the LEP II experiments,the indirect determination from within the StandardModel (SM), and the ultimate precision on the mea-sured top quark mass. The expectations for LEP IIappear to be an overall uncertainty of approximately±35 MeV/c2 [ 1]. The indirect determination is atthe ±30 MeV/c2 level and is not likely to significantlyimprove given the end of the LEP and SLC programs.Finally, the top quark mass precision may reach the±2 GeV/c2 level, which corresponds to a parametricuncertainty of δMW = 12 MeV/c2 [ 2]. The constraintprovided by a successful ± 30 MeV/c2 W boson massmeasurement per experiment per channel1 would havean impact on electroweak global fitting comparableto that of the LEP Z asymmetries. If the ultimateprecision on the MW determination could reach ∼±30 MeV/c2, then the bound on the Higgs boson masswould reach approximately ±30 GeV/c2 [ 3]. Withthe best fit central value close to the current LEP IIdirect search lower limit of MH > 113.2 GeV/c2 [ 4],

1While the measurements from the different channels anddifferent experiments provide cross checks, the combined mea-surement is not expected to yield a much better precision thana single measurement because of large common uncertainties.

considerable pressure can be brought to bear on theSM.This document is structured as follows. The basic

techniques used to measure the W mass and widthare briefly reviewed in section 2. The statisticaland detector-related uncertainties affecting the Wmass and width measurements are discussed in moredetail in section 3 and section 4, respectively. Anumber of systematic uncertainties clearly do not scalestatistically and these are addressed separately insection 5. The expected errors on the measured Wmass in Run II using the conventional transverse massmethod and the W width are summarized in section 6.Alternative methods of measuring the W mass at theTevatron are described in section 7 and prospects forMW measurements at other colliders are discussed insection 8. Finally, some theoretical considerations im-portant for future W mass measurements are broughtup in section 9. Section 10 concludes this document.

2. MW and ΓW Measurements from the MT

Lineshape

The determination ofMW depends on the two bodynature of the W decay: W → ℓνℓ. The kinematicalJacobian peak and sharp edge at the value ofMW /2 iseasily observed in the measurement of the transversemomentum of either of the leptons. In practice, thesituation is difficult due to both challenging exper-imental issues and the fact that phenomenological

3

Table 1Advantages and disadvantages to an MW determina-tion via transverse mass or lepton transverse momenta.

Measurable pWT sensitivity resolution sensitivityMT small significantpeT significant smallpνT significant significant

assumptions must be made in order to perform theanalysis. Because the standard measurable cannotbe written in closed form, an unbinned maximumlikelihood calculation is required. Figure 1 shows acalculation of peT (unsmeared) with pWT = 0; the effectof finite pWT ; and the inclusion of detector smearingeffects. It is apparent that peT is very sensitive to thetransverse motion of the W boson.Historically, precise understanding of pWT has been

lacking, although it is currently modeled by measur-able parameters through the resummation formalismof Collins, Soper, and Sterman [ 5]. For this reason,the transverse mass quantity was suggested [ 6] andhas been the traditional measurable. It is defined by

MT =√

2pℓT pνT (1− cos(φℓ,ν)), (1)

where φℓ,ν is the angle between the charged leptonand the neutrino in the transverse plane. The observ-ables are the lepton transverse energy or momentum~p ℓT and the non-lepton transverse energy ~u (recoil

transverse energy against the W ), from which theneutrino momentum ~p ν

T and the transverse mass MT

are derived. Figure 2 shows that the sensitivity ofMT to pWT is nearly negligible. While considerablymore stable to the phenomenology of the productionmodel, the requirement that the neutrino direction beaccurately measured leads to a set of experimentalrequirements which are difficult in practice to control.So, there are different benefits and challenges amongthe direct measurements of the transverse quantities,pℓT , p

νT , and MT . Table 1 lists these relative pros and

cons of the transverse mass and transverse momentummeasurements.Both CDF and DØ have determined the W boson

mass using the transverse mass approach. The indi-vidual measurements of both experiments are shownin Table 2 and the overall combined result is

MW = 80.452± 0.062 GeV/c2. (2)

The W boson width is precisely predicted in termsof well-measured SM masses and coupling strengths:

ΓW =GFM

3W

6√2π

[

3 + 6

(

1 +αs(MW )

π+O(α2

s)

)]

30 35 40 45 50

pT(e) (GeV)

dN/d

p T(e

)

Figure 1. The effects of resolution and the finite pWTon peT in W boson decay. The histogram shows pWTwithout detector smearing and for pWT = 0. The dotsinclude the effects of adding finite pWT , while the shadedhistogram includes the effects of detector resolutions.The effects are calculated for the DØ Run I detectorresolutions.

×(1 +O(1%))

= 2.093± 0.002 GeV (3)

where the uncertainty is dominated by the experimen-talMW precision [ 7, 8]. The mass and width of theWboson connect both theoretically and experimentally,as ΓW has been extracted from a lineshape analysisusing techniques developed for the W mass measure-ment. Combining CDF electron and muon data from1994–95 yields a result with 140 MeV precision [ 9]:

ΓW = 2.04± 0.11 (stat)± 0.09 (syst) GeV. (4)

In this measurement, u < 20 GeV is required toimprove theMT resolution and to reduce backgrounds.Figure 3 shows the dependence of the MT spectrum

on ΓW . In the region MT > 100 GeV/c2, thelineshape is sensitive to ΓW but relatively insensitiveto uncertainties in the pνT resolution. Thus, ΓW isextracted from a fit to the region 100 GeV/c2 < MT <200 GeV/c2, after signal and background templates

are normalized to the data in the region 40 GeV/c2<

MT < 200 GeV/c2. Figure 4 shows the fits to theCDF electron and muon data. The upper limit MT <200 GeV/c2 is somewhat arbitrary.The measurement of ΓW depends on a precise

determination of the transverse mass lineshape. Thus,the same error sources contribute to both the W massand width measurement. In the following we discussthese sources, concentrating on how they impact theW mass measurement. Run II projections for theindividual uncertainties contributing to the W widthmeasurement are presented in section 6.

4

Table 2Tevatron results for MW . NW is the number of W boson events observed. Scale and non-scale systematic errorsare listed separately.

Experiment∫

Ldt NW MW ± stat ± sys ± scalepb−1 GeV/c2 GeV/c2 GeV/c2 GeV/c2

CDF Run 0 e 4.4 1130 79.91 0.35 0.24 0.19CDF Run 0 µ 4.4 592 79.90 0.53 0.32 0.08CDF Run Ia e 18.2 5718 80.490 0.145 0.130 0.120DØ Run Ia e 12.8 5982 80.350 0.140 0.165 0.160CDF Run Ia µ 19.7 3268 80.310 0.205 0.120 0.050CDF Run Ib e 84 30,100 80.473 0.065 0.054 0.075DØ Run Ib e 82 28,323 80.440 0.070 0.070 0.065

DØ Run Ib e, forward 82 11,089 80.757 0.107 0.091 0.181CDF Run Ib µ 80 14,700 80.465 0.100 0.057 0.085

55 60 65 70 75 80 85 90 95

mT (GeV)

dN/d

mT

Figure 2. The effects of resolution and the finitepWT on MT in W → eν. The histogram shows MT

without detector smearing and for pWT = 0. The dotsinclude the effects of adding finite pWT , while the shadedhistogram includes the effects of detector resolutions.The effects are calculated for the DØ Run I detectorresolutions.

3. Statistical Uncertainties in the MW Determi-

nation

In order to reach the target precision for MW , con-siderable luminosity will be required. Presuming thatRun II is to deliver an integrated luminosity of 2 fb−1,the statistical precision on MW can be estimated fromthe existing data (see Table 2). Figure 5 shows theW statistical uncertainties in these measurements asa function of 1/

√NW , demonstrating a predictable

extrapolation to NW ≃ 700, 000 which correspondsto a Run II dataset per experiment per channel.The statistical uncertainty from this extrapolation isapproximately 13 MeV/c2. For a goal of ±30 MeV/c2

overall uncertainty, this leaves 27 MeV/c2 available inthe error budget which must be accounted for by allsystematic uncertainties.

MT(e,ν) (GeV)

Transverse mass lineshape(normalized to unit area)for ΓW=1.5,1.7,...,2.5 GeV

CDF Preliminary

10-5

10-4

10-3

10-2

10-1

40 60 80 100 120 140 160 180 200

Figure 3. Dependence of the MT spectrum on ΓW .Each spectrum is normalized to unit area.

4. Detector-specific Uncertainties in the MW

Determination

After the lepton energy and momentum scales, themodeling of the W recoil provided the largest system-atic uncertainty in the CDF Run Ib W mass measure-ment. Since Z statistics dominates this number, itcan be expected to be reduced significantly in Run II.Non-Z related recoil systematics were estimated toenter at the 10 MeV/c2 level, which is probablyindicative of the limiting size of this error. The increasein the average number of overlapping minimum bias

5

MT(e,ν) (GeV)

even

ts /

2GeV

1

10

10 2

10 3

40 60 80 100 120 140 160 180 200

ev /

GeV

0

500

1000

1500

2000

50 60 70 80 90 100

MT(µ,ν) (GeV)

even

ts /

2GeV

1

10

10 2

10 3

40 60 80 100 120 140 160 180 200

ev /

GeV

0

200

400

600

800

50 60 70 80 90 100

Figure 4. CDF 1994-95 e and µ data, on a semiloga-rithmic scale, with best fits for ΓW . The backgroundestimates are also shown. The insets show the Jacobianpeak regions on a linear scale.

events in Run II may seriously impact the recoil modelsystematics, although various detector improvementsmay partly compensate for this.Much of the understanding of experimental system-

atics comes from a detailed study of the Z bosons andhence as luminosity improves, systematic uncertaintiesshould diminish in kind. Certainly, the scale andresolution of the recoil energy against theW come frommeasurements of the Z system. Likewise, backgrounddetermination, underlying event studies, and selectionbiases depend critically, but not exclusively, on Zboson data. Most importantly, the lepton energyand momentum scales depend solely on the Z bosondatasets.Figure 6 shows the CDF and DØ systematic uncer-

tainties for both electrons and muons as a functionof 1/

√NW and in particular the calorimeter scale

uncertainties for electrons. This latter importantenergy scale determination is currently tied to thedetermination of fiducial di-lepton decay resonances,notably the Z boson, but also the J/ψ, Υ and theE/p dependence on the energy E, using electronsfrom W and Z decays. As the statistical precisionimproves, the dominant feature of the scale deter-mination becomes its value in the region of MW ,

0

0.05

0.1

0.15

0.2

0.25

0 0.005 0.01 0.015 0.02

Statistical UncertaintiesRun1, CDF & D0

δW(G

eV/c

2)

1/sqrt (#W)

δW (GeV/c2)= 800 + 330 L (pb -1)

Figure 5. Statistical uncertainties in Run I MW

measurements. Each circle represents either a CDFor DØ measurement. The result of a straight linefit is shown. The shaded box is the approximateextrapolation to a 2 fb−1 Run II result.

so offsets and any low energy nonlinearities becomerelatively less important and hence reliance on the lowmass resonances is reduced. On the other hand, forthe muon momentum scale determination, where theobservable is the curvature, low mass resonances arealso important. Figure 6 suggests that this uncer-tainty is truly statistical in nature and extrapolatesto approximately the 15 MeV/c2 level. The ability tobound non-linearities using collider data may become alimiting source of error in Run II. Hence, the remainingsystematic uncertainties must be controlled to a level ofapproximately 22 MeV/c2 in order to reach the overallgoal of ±30 MeV/c2.Figure 6 also shows the non-scale systematic uncer-

tainties from both the CDF and DØ electron mea-surements of MW and the CDF muon measurement.Here the extrapolation is not as straightforward, butthere is clearly a distinct statistical nature to theseerrors. That they appear to extrapolate to negativevalues suggests that the systematic uncertainties maycontain a statistically independent component for boththe muon and the electron analyses.For both MW and ΓW analyses, the Z → ℓℓ data

constrain both the lepton scales and resolutions and

6

0

0.05

0.1

0.15

0.2

0.25

0 0.005 0.01 0.015 0.02

Systematic and Scale UncertaintiesRun 1a,b CDF & D0

muon systematics

electron systematics

electron scale

δW(G

eV/c

2)

1/sqrt (#W)

Figure 6. Systematic uncertainties for each Run IMW measurement. The open squares are the fourelectron measurements from CDF and DØ, the circlesare the scale uncertainties from two DØ electronmeasurements and the Run Ib CDF measurement,and the diamonds are the systematic uncertainties(excluding scale) for the CDF muon measurements.The large box is the position of the extrapolatedstatistical uncertainties to the Run II luminosity. Thelines are linear fits to each set of points.

an empirical model of the hadronic recoil measurement.QED corrections are an issue in measuring the Z mass,and the discussion of these corrections should be interms of the W/Z mass ratio. In a high-precisionwidth measurement, more effort will also be neededto place bounds on possible tails in the lepton andrecoil resolution functions. Uncertainty in the recoilmeasurement is predominantly statistical in how wellmodel parameters are determined. Several cross checkswhich improve with statistics independently ensure theefficacy of the model.Selection biases can be studied with various control

samples, notably the second lepton originating fromZ decays. The QCD background can be also studiedby varying cuts and studying control samples. Thebackground from W → τν is well understood, and thebackground from Z → ℓℓ will be reduced for Run IIsince the tracking and muon coverages are improvedfor both experiments.

5. Theoretical Uncertainties in the MW Deter-

mination

The MT lineshape simulation requires a theoretical

model, as a function of ΓW and MW , of d3σds dy dpT

,including correlations between pT and s. For pro-ducing high-statistics fitting templates, a weightedMonte Carlo generator is useful, so thatMW , ΓW , andthe pT spectrum can be varied simply by reweightingevents. Because the measurement of the recoil energyagainst theW , ~u, is modeled empirically, the generatordoes not have to describe the recoil energy at theparticle level. A detailed description of final-stateQED radiation is important, because bremsstrahlungaffects the isolation variables needed to select a cleanW sample.The W and Z pT spectra are not calculable using

perturbation theory at low pT . In this region, theperturbative calculation must be augmented by anon-perturbative contribution which depends on threeparameters (see section 5.2.1) which are tuned to fitthe Z → ℓℓ data. Theoretical guidance is useful forchoosing an appropriate set of parameters to vary. Astrategy such as has been used in the CDF Run Ibanalysis to use theory to extrapolate from the Z pTdistribution to the W pT distribution seems to limitthe effect of theoretical assumptions to ±5 MeV/c2.The parameters of parton distribution functions are

also empirical, and seldom have quoted uncertainties.PDF uncertainties seem under control for Run I databut will need improvement to avoid becoming dom-inant in Run II. More work is needed to determinehow both to minimize the impact of PDF uncertainties(e.g. by extending the lepton rapidity coverage of themeasurements as done in the DØ analysis [ 10]) and toevaluate the effects of PDF uncertainties in precisionmeasurements.To date, ad hoc event generators have been used in

theW mass and width measurements. In Run II, thesemeasurements will reach a precision of tens of MeV/c2,requiring much more attention to detail in Monte Carlocalculations. Precision electroweak measurements inRun II should strive to use (possibly to develop) pub-lished, well documented Monte Carlo programs thatare common to both collider experiments. In particu-lar, theMW and ΓW measurements would benefit froma unified generator that incorporates state-of-the-artQED and electroweak calculations, uses a boson pTmodel tunable to Run II data, and correctly handlesW bosons that are produced far off-shell.The W width uncertainty in the MW measurement

could become significant but assuming the SM MW -ΓW relation, it won’t be a dominant uncertainty.

7

5.1. Parton Distribution Functions

The transverse mass distribution is invariant underthe longitudinal boost of the W boson. However, theincomplete η coverage of the detectors introduces adependence of the measured MT distribution on thelongitudinal momentum distribution of the producedW ’s, determined by the PDF’s. Therefore, quantifyingthe uncertainties in PDFs and the resulting uncertain-ties in the W mass measurement is crucial.

5.1.1. Constraining PDFs from the Tevatron

data

The measurement of the W charge asymmetry atthe Tevatron, which is sensitive to the ratio of d tou quark densities in the proton, is of direct benefit inconstraining PDF effects in the W mass measurement.This has been demonstrated by the CDF experiment.Following Ref. [ 11], they made parametric modifica-tions to the MRS family of PDFs. These modificationswith retuned parameters are listed in Table 3 andtheir predictions are compared to the W lepton chargeasymmetry measurement and the NMC d/u data [ 12]in Fig. 7. From the variation among the six referencePDFs, an uncertainty of 15 MeV/c2 was taken whichis common to the electron and muon analyses.Since the Run Ib charge asymmetry data is domi-

nated by statistical uncertainties, we expect a smalleruncertainty for the Run II measurement. Measure-ments of Drell-Yan production at the Tevatron can beused to get further constraints on PDFs.

Table 3Reference PDFs and modificationsPDF set ModificationMRS-T d/u→ d/u× (1.07− 0.07e−8x)MRS-R2 d/u→ d/u+ 0.11x× (1 + x)

MRS-R1 d/u→ d/u× (1.00− 0.04e−12 (

(x−0.07)0.015 )2)

5.1.2. Reducing the PDF uncertainty with a

larger η coverage

Since the PDF uncertainty comes from the finite ηcoverage of the detectors, it is expected to decreasewith the more complete rapidity coverage of the Run IIdetectors. The advantage of a larger rapidity coveragehas been demonstrated by the DØ experiment: theuncertainty on the W mass measurement using theircentral calorimeter was 11 MeV/c2, while that usingboth the central and end calorimeter was 7 MeV/c2.With the upgraded calorimeters and trackers for therange |η| > 1, the CDF experiment can measure theW mass over a larger rapidity range in Run II.

Figure 7. (a) The CDF measurement of the W leptoncharge asymmetry. (b) The NMC d/u data evolvedto Q2 = M2

W . The gray bands represent the rangespanned by the six reference PDFs considered.

5.1.3. A Global Approach

There has been a systematic effort to map outthe uncertainties allowed by available experimentalconstraints, both on the PDFs themselves and onphysical observables derived from them. This approachwill provide a more reliable estimate and may be thebest course of action for precision measurements suchas theW mass or theW production cross section. Thishas been emphasized at this workshop by the PartonDistributions Working Group [ 13].

5.2. W Boson Transverse Momentum

The neutrino transverse momentum is estimated bycombining the measured lepton transverse momentumand the W recoil: ~p ν

T = −(~p lT + ~u). It is clear

therefore that an understanding of both the underlyingW boson transverse momentum distribution and thecorresponding detector response, usually called therecoil model, is crucial for a precisionW mass measure-ment. For the CDF Run Ib W mass measurement, thesystematic uncertainties from these two sources wereestimated to be 15− 20 MeV/c2 and 35− 40 MeV/c2,respectively, in each channel [ 14].

8

5.2.1. Extracting the pWT Distribution

The strategy employed in Run I, which is expectedto be used also in Run II, is to extract the underlyingpWT distribution from the measured pZT distribution (Yis the weak boson rapidity):

d2σ

dpWT dY=

d2σ

dpZT dY× d2σ/dpWT dY

d2σ/dpZT dY, (5)

where the ratio of the W and Z differential distri-butions is obtained from theory. This method relieson the fact that the observed pZT distribution suffersrelatively little from detector smearing effects, allowingfits to be performed for the true distribution. TheCDF Run Ib data and the results of a Monte Carlosimulation using the best fit parameters are comparedin Fig. 8.

0

20

40

60

80

100

120

140

160

0 5 10 15 20 25 30 35 40 45

(a)

Z→ ee

χ2 / dof = 0.82

pTZ(ee) (GeV)

Eve

nts

/ 1 G

eV

0

50

100

150

200

250

300

350

0 5 10 15 20 25 30 35 40 45 50

pTZ(µµ) (GeV)

Eve

nts

/ 2 G

eV (b)

Z→ µµ

χ2 / dof = 1.28

Figure 8. The observed pZT distributions in theelectron and muon channels using the CDF Run Ibdata. Also shown are the curves for the Monte Carlosimulation using the best fit parameters for the inputpZT distribution.

The experimental uncertainties, as in many as-pects of the W mass measurement, are dominatedby the available Z → ℓ+ℓ− statistics and shouldscale correspondingly with the delivered luminosity inRun II. Theoretical uncertainties in the ratio of W

to Z transverse momentum distributions contributea further O(5) MeV/c2 to the overall error. Thetwo sources of uncertainty are compared for the CDFRun Ib W → µν analysis in Fig. 9.

-50-40-30-20-10

01020304050

0 5 10 15 20 25 30 35 40 45

pTW (GeV)

Unc

erta

inty

in p

TW

(%

)

(a)Uncertainty fromZ→ µν statistics

Theory Uncertainty(PDFs, αS,Input pT,..)

0

0.02

0.04

0 5 10 15 20 25 30

pTW (GeV)

Eve

nt F

ract

ion

in M

C

(b)∆ pT

W in W→ µν generation

Figure 9. (a) A comparison of the two sources ofuncertainty on the derived pWT distribution. (b) ThepWT distribution extracted for the CDF Run Ib W →µν mass measurement.

The ratio of W to Z transverse momentum distri-butions used in Eq. (5) is taken from resummationcalculations, which attempt to resum terms corre-sponding to multiple soft and collinear gluon emissionto all orders. They thereby include the dominantcontribution to the cross section at small boson pTthat is missing in fixed order calculations. Theseperturbative calculations need to be augmented by anon-perturbative contribution which, in the case ofimpact-parameter (b) space resummations, is typicallyparameterized as a Sudakov form-factor with the fol-lowing form:

FNP = exp [−g1b2 − g2b2 ln(Q/2Q0)

−g1g3b ln(100x1x2) ] , (6)

where Q0 is a low scale of O(few) GeV and theparameters g1, g2 and g3 must be obtained from fitsto the data [ 15]. DØ has shown that the Run I pZT

9

data is as sensitive to g1 and g2 as the low-energyDrell-Yan data that has largely been used to constrainthese parameters in the past [ 16]. The Run II datawill therefore provide significant new constraints on theform of the non-perturbative contribution to the pWTdistribution.Moreover, recent theoretical developments in com-

bining the advantages of b−space and pT−space resum-mation formalisms may provide a better theoreticalframework for extracting the underlying pWT distribu-tion in Run II [ 17].In short, the precision Z data available in Run II

together with further theoretical advances will reducein a number of ways the systematic uncertainties dueto the knowledge of the pWT distribution, perhaps downto the level of ∼ 5 MeV/c2.

5.3. QCD Higher Order Effects

The W bosons are treated as spin-one particles anddecay via the weak interaction into a charged lepton(e, µ or τ) and a neutrino. The charged leptonsare produced from the W decay with an angulardistribution determined by the O(α2

s) calculation of [18] which, for W+ bosons with a helicity of –1 withrespect to the proton direction, has the form :

dσ

d cos θCS∝ 1 + a1(pT ) cos θCS + a2(pT ) cos

2 θCS (7)

where pT is the transverse momentum of the W andθCS is the polar direction of the charged lepton withrespect to the proton direction in the Collins-Soperframe [ 19]. a1 and a2 are pT dependent parameters.For pT = 0, a1 = 2 and a2 = 1, providing theangular distribution of aW boson fully polarized alongthe proton direction. For the pWT values relevant tothe W mass analysis (pWT < 30 GeV/c), the changein W polarization as pWT increases only causes amodest change in the angular distribution of the decayleptons [ 18].While the uncertainty associated with the change in

the angular distribution of the W decay lepton dueto higher order QCD corrections (a few MeV/c2) hasbeen negligible for the Run I measurements, it cannot be ignored for the Run II measurements (see thePhoton and Weak Boson Physics working group reportfor more details).

5.4. QED Radiative Effects

5.4.1. Introduction

The understanding of QED radiative corrections iscrucial for precision W mass measurements at theTevatron. The dominant process is final state radiation(FSR) from the charged lepton, the effect of whichstrongly depends on the lepton identification criteriaand the energy or momentum measurement methods

employed. Calorimetric energy measurements, suchas those employed in the electron channel, are moreinclusive than track based momentum measurementsused in the muon channel and the effect of FSRis consequently reduced. In the CDF Run Ib Wmass measurement the mass shifts due to radiativeeffects were estimated to be −65 ± 20 MeV/c2 and−168±10 MeV/c2 for the electron and muon channels,respectively [ 14]. These effects will be larger inRun II due to increase in tracker material in CDF andmagnetic tracking in DØ.The Monte Carlo program used for the Run I W

mass measurement incorporated a calculation of QEDcorrections by Berends and Kleiss [ 20]. This treat-ment, however, does not include initial state radiation(ISR) and has a maximum of one final state photon.The effect of multiple photon emission was estimatedby comparing the calculation of Berends and Kleissto PHOTOS [ 21], a universal Monte Carlo programfor QED radiative corrections that can generate amaximum of two final state photons. Likewise, theeffect of ISR and other missing diagrams was estimatedby comparing the calculation of Berends and Kleissto a full O(α) calculation by Baur et al. [ 22]. Theresulting systematic uncertainties on the W mass areestimated to be 20 MeV/c2 and 10 MeV/c2 in theelectron and muon channels, respectively [ 14]. Clearlythese systematic uncertainties become much moresignificant in the context of statistical uncertainties ofO(10) MeV/c2 expected for 2 fb−1 in Run II.The next section describes in more detail the calcu-

lation by Baur et al., which forms the basis for a newevent generator. Some studies of the effects of QEDradiation on the W mass measurement are presentedin section 5.4.4. Section 5.4.5 briefly outlines somework in progress that should further reduce systematicuncertainties due to radiative corrections in Run II.

5.4.2. WGRAD

WGRAD is a program for calculating O(α) electroweakradiative corrections to the process qq′ →W± → ℓ±ν,including the real photon contribution qq′ → ℓ±νγ.Both ISR from the incoming quarks, FSR from thefinal state charged lepton, and interference terms areincluded. Many more details can be found in [ 22].The most important generator level cuts are on

the final state photon energy and collinearity forradiative events. The photon energy cut, controlledby the parameter δs, is made on the fraction ofthe parton’s energy carried by the emitted photonin the parton-parton center of mass system: E∗

γ >

δs√s/2. The photon collinearity cut, controlled by

the parameter δc, is made on the angle between thecharged fermion and the emitted photon in the same

10

Table 4The fraction of qq′ → µν(γ) events containing a finalstate photon for different final state photon soft andcollinear cuts. Events are generated with ISR only,FSR only, and with a full treatment of QED radiation.

Photon Cuts ISR FSR Full

Photon Cuts ISR FSR Full

δs = 0.01, δc = 0.01 1.6% 9.4% 11.1%

δs = 0.01, δc = 0.001 2.5% 9.4% 12.0%

δs = 0.001, δc = 0.001 4.1% 15.5% 20.0%

δs = 0.001, δc = 0.0001 5.2% 15.5% 21.3%

frame: cos θ∗ < 1 − δc. However, final state collinearsingularities are regulated by the finite lepton massesand the above cut is only implemented for quarkonicradiation when ISR is included. The fraction ofradiative events corresponding to different photon cutsis given in Table 4 for the process qq′ → µν(γ) at√s = 2 TeV. Loose fiducial cuts pµT > 10 GeV/c,

|ηµ| < 2 and pνT > 10 GeV/c have been applied. Theinclusion of ISR increases the photon yield by around30%, depending on the soft and collinear photon cutsapplied. The fractions are significantly higher for theprocess qq′ → eν(γ) in the cases that FSR is included.The effect on the fitted W mass of the inclusion of ISRis examined in section 5.4.4.

5.4.3. Event Generation

WGRAD has been turned into an event generatorthrough a suitable unweighting scheme described ex-tensively in [ 23]. A significant complication is thepresence of negative qq′ → ℓ±ν event weights inthe program which, while expected to cancel withpositive qq′ → ℓ±νγ event weights in the calculationof physical observables, nevertheless appear separatelyin the unweighting procedure. The approach herehas been to unweight the negative weight events in asimilar manner to the positive weight events, such thatthe output consists of both positive and negative unitweight events. The fraction of negative weight events,plotted in Fig. 10 for the process qq′ → µν(γ), dependsstrongly on the soft and collinear photon cuts applied.It is not significantly different for qq′ → eν(γ) events.The effect of negative weights on the fitted W mass isexamined in the next section.

Muon pT (GeV)Muon pT (GeV)Muon pT (GeV)

Neg

ativ

e W

eigh

t F

ract

ion

δs = 0.001, δc = 0.0001

δs = 0.001, δc = 0.001

δs = 0.01, δc = 0.001

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0 5 10 15 20 25 30 35 40 45 50

Figure 10. The negative weight fraction versus pµT fordifferent soft and collinear photon cuts.

5.4.4. The Effect of QED Radiation on the

Measured W Mass

WGRAD has been used to generate large W → µνevent samples for the purposes of investigating theeffect of QED radiation on the measurement of theW mass. The events have been generated at

√s =

1.8 TeV in order to make use of the CDF Run IW production model and detector smearing param-eterizations. The W production model, extractedfrom the Run I Drell-Yan data, is used to smearthe true W transverse momentum. The CDF recoilmodel is then used to translate this into a measuredpT (W ), which is combined with the smeared leptonand photon momenta to form a realistic transversemass distribution. Loose fiducial cuts pµT > 20 GeV/c,|ηµ| < 2 and missing-ET > 20 GeV are applied. Tosimulate the CDF muon identification criteria, eventsare rejected if a photon with Eγ > 2.0 GeV is foundwithin an η − φ cone of radius 0.25 around the muon.Low energy photons inside the cone are not includedin the measurement of the muon pT , as is the caseexperimentally.The unweighted event samples, all generated with

MW = 80.4 GeV/c2 and ΓW = 2.1 GeV, are dividedinto “data” and “Monte Carlo” sub-samples and fittedagainst one another in pseudo-experiments. The fit isto the transverse mass distribution in the range 50 <MT < 100 GeV/c2. For a number of events in thetransverse mass fit region equal to that in the CDFRun IbW → µν analysis, the resulting statistical erroris very similar.As a cross check of this procedure, “data” and

11

Table 5The results of pseudo-experiments generated by WGRAD with different treatments of QED radiative effects. Furtherdetails are given in the text.

“Data” “Monte Carlo” Fit Result

(a) FULL; δs = 0.01, δc = 0.01 FULL; δs = 0.01, δc = 0.01 80.4030± 0.0069 GeV/c2

(b) FULL; δs = 0.01, δc = 0.01 FULL no neg.; δs = 0.01, δc = 0.01 80.4027± 0.0069 GeV/c2

(c) FSR only; δs = 0.01, δc = 0.01 FULL; δs = 0.01, δc = 0.01 80.392± 0.006 GeV/c2

(d) FSR only; δs = 0.001, δc = 0.001 FULL; δs = 0.01, δc = 0.01 80.381± 0.006 GeV/c2

(e) FULL; δs = 0.001, δc = 0.001 FULL; δs = 0.01, δc = 0.01 80.389± 0.009 GeV/c2

(f) FULL no neg.; δs = 0.001, δc = 0.001 FULL; δs = 0.01, δc = 0.001 80.399± 0.009 GeV/c2

“Monte Carlo” samples generated with identical cutsare fitted against one another, with the result shownin Table 5(a). It is interesting to note that if thenegative weight events, which occur at the 0.2% level inthe “Monte Carlo” sample, are removed, the fit resultchanges by less than 0.5 MeV/c2 (Table 5(b)).Table 5(c) shows the result of fitting “data” gener-

ated with FSR only. The shift in the fitted W massof ≈ 8 MeV/c2 is consistent with the estimate givenin [ 14] of the effect of ISR on the fitted W mass,although the uncertainties here are rather large. Ifthe soft and collinear cuts are reduced in the “data”sample, as shown in Table 5(d), the fitted W massshifts significantly downwards. This is to be expectedsince the track based muon pT measurement doesnot incorporate collinear photons. The setting ofsoft and collinear photon cuts is therefore particularlyimportant in the generation of W → µν Monte Carlosamples.The fits shown in Table 5(e) and (f) are performed

in order to examine the effect of negative weights onthe fit when, as in the case of this “data” sample,negative weights are present at the 5% level. Whenthe negative weight events are excluded from the fit,the result changes by 10 MeV/c2. The larger shiftin the fitted W mass with respect to Table 5(b) iscommensurate with the larger negative weight fractionin this sample.

5.4.5. Work in Progress

A remaining source of systematic uncertainty dueto QED radiation is the effect of multiple photonemission. As discussed above, this has previously beenestimated by comparing the Berends and Kleiss singlephoton calculation with the results of running thePHOTOS algorithm. Recently, however, complete matrixelement calculations of the processes qq′ → ℓ±νγγ and

qq → ℓ+ℓ−γγ have been performed [ 24]. It may bepossible in the future to do detailed comparisons of theresults of these calculations and the PHOTOS algorithm,in order to arrive at a better constrained systematicuncertainty due to multiple photon emission.Furthermore, a complete set of O(α) electroweak

radiative corrections to the process qq → ℓ+ℓ−,including the real photon contribution qq → ℓ+ℓ−γ,will soon be available. This will enable a consistentMonte Carlo description of theW data and the Z data,upon which the W mass analysis crucially depends forthe understanding of gauge boson production and thecalibration of the detectors.

5.4.6. Summary and Conclusions

Systematic uncertainties due to QED radiative ef-fects currently run at the level of ≈ 20 MeV/c2 inthe electron channel and ≈ 10 MeV/c2 in the muonchannel. A large contribution to this uncertainty isthe effect of ISR and interference terms, which are notpresent in the Berends and Kleiss calculation and thePHOTOS algorithm that have previously been used inWproduction Monte Carlo programs.A full O(α) calculation by Baur et al. has been

used as the basis for a new event generator. Theresults of several pseudo-experiments generated withdifferent treatments of QED radiative effects agreewith previous estimates. They show that negativeweights need to be treated carefully, especially in thecase of very small soft and collinear photon cuts.Further studies of QED radiative corrections to W

production will continue as new calculations becomeavailable. It is clear, however, that the use of newprograms such as WGRAD could significantly reducesystematic uncertainties due to QED radiative cor-rections in Run II, either through explicit correctionsbeing applied to the extracted W mass, or through

12

their use in new Monte Carlo event generators. Theremaining systematic uncertainties due to QED correc-tions might then be reduced to the level of 5 MeV/c2

and 10 MeV/c2 in the muon and electron channels,respectively.

6. Summary of Run II Expectations

As has been discussed in previous sections, manyof the systematic uncertainties in the W mass mea-surement approximately scale with statistics. Theseare listed in Table 6 for the Run Ib CDF muonanalysis and should scale to ≈ 20 MeV/c2 for anintegrated luminosity of 2 fb−1. With reasonable

Table 6Errors on the CDF Run Ib muon W mass which scalestatistically, in MeV/c2.

Source errorFit statistics 100Recoil model 35Momentum resolution 20Selection bias 18Background 25Momentum scale 85

assumptions for the size of non-scaling systematicssuch as those due to PDFs and higher order QEDeffects, a 40 MeV/c2 measurement in the muon channelby each experiment seems achievable. The systematicuncertainties in the electron channel are less easy toextrapolate given the particular sensitivity to calorime-ter scale non-uniformities in this channel and the extramaterial in the Run II tracking detectors. The detailedunderstanding of detector performance is of coursedifficult to anticipate, although it is clear that bothscalable and non-scaling systematics would be easierto understand if fast Monte Carlo generators includingall the relevant effects were available.The individual uncertainties for the Run Ib ΓW

measurement are listed in Table 7 together with theirprojections for 2 fb−1. All but the last three sourcesof error are constrained directly from collider data,and hence should scale roughly as 1/

√L. While the

last three uncertainties may decrease somewhat as newmeasurements and calculations become available, theywill not scale statistically with the Run II dataset.Assuming no improvement in these three uncertainties,while all others scale statistically, each experiment can

Table 8Dominant uncertainties for contrasting components ofthe DØ MW determination. The quantities shownare the shift in MW for a 1σ change in the releventparameter. The EM resolution term refers to thesampling term for the resolution function. Taken fromRef. [ 25].

Source δMW (MT ) δMW (peT )pWT 10 50

EM resolution 23 14hadron scale 20 16

hadron resolution 25 10backgrounds 10 20

make a ∼ 40 MeV width measurement, combining eand µ channels for a 2 fb−1 dataset.

7. Other Methods of Determining MW at the

Tevatron

While the traditional transverse mass determinationhas been the optimal technique for the extraction ofMW in the low-luminosity running at hadron colliders,other techniques have been or may be employed in thefuture. These methods may shuffle or cancel some ofthe systematic and statistical uncertainties resulting inmore precise measurements.

7.1. Transverse Momentum Fitting

As noted above, the most obvious extensions of thetraditional transverse mass approach to determiningMW are fits of the Jacobian kinematical edge fromthe transverse momentum of both leptons. DØ hasmeasured MW using all three distributions and theuncertainties are indeed ordered as one would expect:The fractional uncertainties on MW from the DØRun I measurements for the three methods of fittingare: 0.12% (MT ), 0.15% (peT ), and 0.21% (pνT ). Asexpected, the peT method is slightly less precise thanthe transverse mass. However, for a central electron(|η| < 1), the uncertainty in the peT measurementdue to the pWT model is 5 times that in the MT

measurement. As can be seen from Table 8, this isnearly balanced by effects from electron and hadronresponse and resolutions which are relatively worse forMT . Accordingly, when there are sufficient statisticsto enable cuts on the measured hadronic recoil, themeasurement uncertainty from the pWT model might bebetter controlled and enable the peT measurement tocompete favorably with the MT measurement whichrelies so heavily on modeling of the hadronic recoil.In order to optimize the advantages of all three mea-surements, the DØ final Run I determination of MW

13

Table 7Sources of error for CDF 1994-95 ΓW measurement and extrapolations to 2 fb−1. The last three uncertainties arecommon to the e and µ analyses.

CDF 1994-95 (→ 2fb−1)Source ∆Γ (e, MeV) ∆Γ (µ, MeV)Statistics 125 (→ 30) 195 (→ 45)Lepton E or pT non-linearity 60 5Recoil model 60 90W PT 55 70Backgrounds 30 50Detector modeling, lepton ID 30 40Lepton E or pT scale 20 15Lepton resolution 10 20PDFs (common) 15 15MW (common) 10 10QED (common) 10 10Uncorrelated systematic 112 (→ 25) 133 (→ 30)Correlated systematic 21 21Total systematic 115 (→ 33) 135 (→ 37)Total stat + syst 170 (→ 45) 235 (→ 60)

combined the separate results [ 25].The resolution sensitivity for muon measurements is

even less than that for electrons so that has the benefitof slightly favoring a transverse mass measurementwith muons over that for electrons.

7.2. Ratio Method

DØ has preliminarily determinedMW by considerationof ratios of W and Z boson distributions which arecorrelated with MW [ 26]. The principle is that onecan cancel common scale factors in ratios and directlydetermine the quantity rmeas ≡ MW

MZ, which can be

compared with the precise LEPMLEPZ . The quantities

that have been considered are:

1. r(MT ) and r(pT ), which has the advantage ofbeing well-studied [ 27]. There are challengeswith this approach which will be discussed below.

2. r(Ee) which has the advantage that the peak ofthe distribution is precisely correlated with MW ,but the disadvantage that statistical uncertaintywashes out the position of that peak.

3. The difference of transverse mass distributions(not as precise as ratios).

The procedure is to compare two distributions, one forW bosons and a similarly constructed one for Z bosons,for example, fW,Z(x) as a function of a given variable,such as x = MT or x = peT . Practically speaking, theZ boson decay electrons are scaled by a factor s andfZ(x, s) is compared with fW (x) as a function of x,for different trial values of s. A statistical measure

(the Kolmogorov-Smirnov test) is calculated for eachs and the value of the highest Kolmogorov-Smirnovprobability, sbest, is declared to be rmeas and thedesired mass is then extracted from MW = rmeas ×MLEP

Z . In principle, minimal Monte Carlo fitting isrequired, as the measurement is performed with data.Figure 11 shows the idea with an unsmeared Z boson

transverse mass distribution compared to a simulated(unsmeared) W boson distribution. Various values ofs lead to various mismatches between fW (MT ) andfZ(MT ) which can be characterized by a Kolmogorov-Smirnov probability as a function ofMW = s×MLEP

W .This probability distribution for an ensemble of 100Monte Carlo experiments is shown in Fig. 12 resultingin an RMS of 40 MeV/c2.However, there are challenges to be faced using this

technique.

• Many systematic effects cancel in this method,such as electromagnetic scale, hadronic scale,angular scale, luminosity effects. However, theseare first-order cancellations, some of which inthe end are not sufficient: the second ordereffects from these quantities must be considered.Likewise, most resolutions have additive termswhich do not cancel in a ratio.

• The statistical precision of the Z sample isdirectly propagated into the resultant overallδMW , in contrast to the traditional approachwhere the Z boson statistics is a component ofvarious of the measured resolutions.

• The detector modeling must take into account

14

Figure 11. For unsmeared Monte Carlo events, thetransverse mass of simulated W bosons (histogram) isoverlayed with that of Z boson (dots) events in whichthe electron has been scaled by a factor which producesthe best match.

small, but important differences between Z andW events such as underlying event, resolutions,efficiencies, acceptances, and the effects of the“extra” electron in Z boson events which compli-cates underlying event and recoil measurements.

• From the physics model, there are also differencesbetween the two samples which must be consid-ered, such as the fact that the production of ZandW bosons take place from the annihilation oflike and unlike flavored quarks, respectively andthat weak asymmetries lead to different decayangular distributions.

• Particularly difficult is the need to “extra-smear”the electrons from Z boson decays. This is dueto the fact that peT values for the heavier Zboson are harder, resulting in a different averageresolution smearing. This same effect is true forthe recoil distributions between Z andW bosons.

• Finally, the acceptances for the two bosons aredifferent since there are potentially two opportu-nities to select a Z boson event at the triggerand event selection stages. Similarly, there isan acceptance difference in the opposite directiondue to electrons in Z bosons being lost in cracksbetween the CC and EC calorimeters in the DØdetector.

Figure 12. The Kolmogorov-Smirnov probability dis-tribution for various scales in a comparison of Wand Z boson unsmeared events corresponding to 100experiments. The RMS is 40 MeV/c2 for 20000 events.

An analysis from Run Ia data from the DØ experimenthas been done [ 26]. Figure 13 shows data for the scaledcomparison and the unscaled original distributions.Electrons from the W boson events were selected tohave peT > 30 GeV/c, while those from Z bosonevents, must satisfy peT > 34.1 GeV/c. Electronsfrom the W sample and at least one electron fromthe Z samples were required to be in the centralcalorimeter. This results in 5244 W bosons and 535 Zboson events. Backgrounds are subtracted accordingto the traditional analysis. “Extra-smearing” is donefor each accepted Z boson event (twice, for bothelectrons) 1000 times, using a different random seedfor each smearing. Differences in the W and Z bosonproduction mechanisms and acceptances result in aneffective correction of 109 MeV/c2, while the differencein radiative corrections results in an effective correctionof −116 MeV/c2. The magnitude of these correctionsis not very different from corrections within the tra-ditional technique and the demand on knowing theuncertainties in them is similarly stringent. Figure 14shows the probability distribution for the result. Thepreliminary result from this analysis for central, Run Iaelectrons is

MW = 80.160± 0.360± 0.075 GeV/c2.

Comparison with the traditional Run Ia result fromthe same data is readily made, but most appreciated

15

Figure 13. (a) The transverse mass distributionof W (solid) and scaled Z (dots) bosons is shownalong with the hatched fit window. (b) The originaldistributions are shown, along with their difference.The Z distribution has been normalized to that of theW boson sample.

with a slightly different interpretation of the Run Iauncertainties. The Run Ia result [ 28] from Table 2 is

80.350± 0.140± 0.165± 0.160 GeV/c2

where the first error is the statistical uncertainty(from W events), the second is the systematic un-certainty and the third is the electron scale deter-mination. It is important to note that the scaleuncertainty is almost completely dominated by theZ boson statistics. Therefore, as a statistical uncer-tainty, it can be combined with the W uncertaintyof 140 MeV/c2 for the purposes of comparison withthe ratio method. This results in an overall “statis-tical” uncertainty of 212 MeV/c2. Now, the strongersystematic power of the ratio method is apparent (75versus 165 MeV/c2) and the poorer statistical power(360 versus 212 MeV/c2) is also evident.

7.2.1. Prospects for Run II

This apparent systematic power of the ratio methodcan only fully be realized in high luminosity running,such as Run II. The ratio method analysis of the

Figure 14. The Kolmogorov probability distribution(hatched) is shown as a function of the W bosonmass used as a scale factor. The dotted curve is theKolmogorov Likelihood and the dashed curve is theχ2/ndf distribution (right axis).

DØ Run Ib data was recently completed [ 29]. TheRun Ib sample has 82 pb−1 of data (1994–1995 dataset), 33,137 W and 4,588 Z events (electrons inboth Central and End Calorimeters of DØ) after thestandard electron selection cuts. TheW mass resultingfrom the ratio fit is MW = 80.115 ± 0.211 (stat.) ±0.050 (syst.) GeV/c2. The statistical uncertainty is ingood agreement with an ensemble study of 50 MonteCarlo samples of the same size (80.36± 0.25 GeV/c2).Early efforts at predicting the results for a Run II

sample of 100,000 W bosons is shown in Fig. 15 withfull detector acceptances and resolutions taken intoaccount. The statistical precision from this fit is of theorder of 20 MeV/c2 and the systematic uncertaintiesmay be nearly negligible.

8. Prospects for Measuring MW at Other Ac-

celerators

8.1. LEP II

The prospects for determination of MW at LEP IIhave become fully understood in the last year withthe accumulation of hundreds of pb−1 at four centerof mass energies. Here we review the status as ofthe Winter 2000 conferences and project the prospectsthrough to the completion of electron-positron runningat CERN. For a review, see Ref. [ 30, 31].

8.1.1. Data Accumulation

The annihilation of e+e− into W boson pairs occursvia three diagrams: a t-channel neutrino exchange ands-channel Z or γ exchange. The final states from thedecays of the twoW bosons are: both W bosons decayinto hadrons (qqqq, “4-q” mode); one W decays into

16

Figure 15. Same as Fig. 13a, but for 100,000W bosons.

quarks, and the other into leptons (e, µ , τ andtheir neutrinos, “qqℓν”); and both W bosons decayleptonically. Collectively, the latter two modes arereferred to as “non-q” modes. The efficiencies andsample purities are typically quite high, as shown inTable 9.

Table 9Efficiencies and sample purity for a representativeLEP II experiment (OPAL).

Channel ǫ (%) purity (%)qqqq 85 80qqeν 85 80qqµν 87 80-90qqτν 66 80

The results by Spring 2000 come from running atcenter of mass energies: 172, 183, 189, and severalenergies between 190 GeV and 200 GeV. There isrecent running above 200 GeV for a total of morethan 400 pb−1 accumulated per experiment. MW

results are final for all four experiments for the 172and 183 GeV sets [ 32, 33, 34, 35] and preliminaryfor the 1998 189 GeV running [ 36, 37, 38, 39]. Inaddition, ALEPH [ 40], L3 [ 41], and OPAL [ 42] havepreliminary results from the collection of runs in therange from 190 GeV to 200 GeV. Table 10 shows theapproximate accumulated running to date (July 2000).There are broadly two methods employed for de-

termining MW at LEP II. The first method is themeasurement of the threshold of theWW cross sectionand the second is the set of constrained fits possiblefor the various measured final states. The latter set of

Table 10Approximate accumulated running per experiment.The 2000 totals are current as of the first week inJuly 2000.

year beam energy (GeV)∫

Ldt (pb−1)1996 80.5-86 251997 91-92 751998 94.5 2001999 96-102 2502000 100-104 100

Table 11Typical systematic uncertainties on MW for a genericLEP II experiment from data sets corresponding tothe 189 GeV running. Entries are approximate andbroad averages meant to give a relative sense of scaleonly. The Source labels are generally self-explanatory,with CR/BE standing for “Color Reconnection” and“Bose-Einstein” respectively.

Source qqℓν (MeV/c2) qqqq (MeV/c2)ISR 15 15frag 25 30

4 fermion 20 20detector 30 35

fit 30 30bias 25 25

bckgrnd < 5 15MC stat 10 10Subtotal 61 67LEP 17 17

CR/BE 60Total 63 85

methods constitute the prominent results and employconstruction of invariant masses making use of thebeam constraints. There are a variety of methods,some of which make use of the constraintMW1 =MW2

and some of which involve sophisticated multivariateanalyses. The spirit of approach is much like thestrategies employed in the top quark mass analyses ofCDF and DØ.The results are treated separately for the qqℓν and

qqqq final states due to the significant differences insystematic uncertainties. Typical uncertainty contri-butions are listed in Table 11 [ 43]. Many of theexperimental uncertainties, such as scale, background,and Monte Carlo generation, are statistically limited.For example, there is a fixed amount of

√s = MZ

running in each running period and that contributes astatistical component to the energy scale uncertainty.The dominant uncertainty comes from the final state

17

effects in the qqqq channel. Because the outgoingquarks can have color connections among them, thefragmentation of the ensemble of quarks into hadronsare not independent. This leads to an theoreticaluncertainty called “Color Reconnection” (CR). Inaddition, since the hadronization regions of the W+

and W− overlap, coherence effects between identicallow-momentum bosons originating from different W ’sdue to Bose-Einstein (BE) correlations may be present.The combined total of these two effects is currentlyaccepted to contribute 52 MeV/c2 of uncertainty to theqqqq results. Ultimately, the non-CR/BE uncertaintywill likely be the uncertainty in modeling single-quarkfragmentation and associated QCD emission effects.

8.1.2. Results, April 2000

The preliminary results for MW from the combineddata taking through 1999 running period are shownin Table 12. The combined LEP result for the qqℓνchannels is [ 44]:

M qqℓνW = 80.398± 0.039± 0.031± 0.017 GeV/c2

where the first error is statistical, the second sys-tematic, and the third the LEP energy scale. Thecombined preliminary result for the qqqq channel is:

M4qW = 80.408± 0.037± 0.031± 0.016± 0.052 GeV/c2

where the first three errors are the same as for theqqℓν result and the fourth error is due to the combinedCR/BE theoretical uncertainty. Taking into accountthe correlations, the combined preliminary result fromconstrained fitting for all channels is:

M4fW = 80.401± 0.027± 0.031± 0.017± 0.018 GeV/c2

where the four errors are in the same order as for theqqqq result. The current overall result comes fromcombining the above with that from the thresholdmeasurement of

Mσ(E)W = 80.400± 0.220± 0.025 GeV/c2.

Here the first error is combined statistical and system-atic and the second error is the error due to LEP energyscale. This results in the preliminary overall LEP II(April 2000) value of

MLEPW = 80.401± 0.048 GeV/c2.

8.1.3. Prospects for the Future

The current results are preliminary and running isunderway at this writing with the end of LEP II sched-uled for the beginning of October, 2000. Eventually,the 1999 data will be fully analyzed and, with theaccumulation of the final 2000 running, should result

Table 12Preliminary LEP II results forMW by experiment andaccording to reconstructed channel. The results arefrom the combination of 1996-1998 running (all exper-iments) plus preliminary results from 1999 running forALEPH, DELPHI, L3, and OPAL.

MW (GeV/c2)Experiment qqℓν qqqqALEPH 80.435± 0.079 80.467± 0.086DELPHI 80.230± 0.140 80.360± 0.115

L3 80.282± 0.102 80.489± 0.132OPAL 80.483± 0.078 80.380± 0.103

in a combined statistical and systematic uncertainty(excluding the CR/BE and LEP contributions) ofapproximately 35 MeV/c2 [ 1]. With the overallcontribution of 18 MeV/c2 and 17 MeV/c2 from theCR/BE and LEP errors respectively, the ultimate limitfrom LEP II W boson pair determination of MW

should be approximately 40 MeV/c2.

8.2. LHC

It was pointed out several years ago [ 45] that theLHC has the potential to provide an even more precisemeasurement of MW . This suggestion was based onthe observations that the precision measurement ofMW at hadron colliders has been demonstrated to bepossible; that the statistical power of the LHC datasetwill be huge; and that triggering will not be a problem.These authors estimated thatMW could be determinedto better than 15 MeV/c2. More recently, the ATLAScollaboration has studied the question in more detail [46] and arrived at an uncertainty of 25 MeV/c2, perexperiment, in the electron channel alone.Achieving such precision will require substantial

further reduction of theoretical and systematic un-certainties, all of which must be reduced to the ≤10 MeV level. This includes the contributions fromthe W production model, parton distributions, andradiative decays, as well as experimental systematicssuch as the energy-momentum scale of the detector andany complications from underlying energy depositioneven in the low luminosity running. While somehave questioned whether such “heroic” progress willever be possible, we would argue that it is futile todebate the question at this time. Rather, the bestindicator of future LHC precision will be to see howwell the Fermilab experiments manage to deal with thesignificant improvements in systematics which will benecessary in order to match the anticipated precisionfor mt. The point to be made is this: should it provenecessary to determine the W mass to a precision of10–20 MeV/c2, the LHC will have the statistical power

18

to continue the hadron collider measurements into thisdomain. The success of such a program will thendepend on

• Consensus in the field that such precision isneeded. One such justification might be to distin-guish among different models of supersymmetry-breaking using global fits including MW , the topmass and the light Higgs mass. It is likely thata big parallel effort to push down the top massuncertainty to the 1 GeV/c2 level would also thenbe needed;

• A major, multi-year effort within the LHC ex-periments in order to understand their detectorsand their response to leptons, missing transverseenergy and recoil hadrons at the required level.This is a measurement which places heavy bur-dens on manpower as it requires an understand-ing of the detector which is more precise than forany other measurement;

• A comparable major effort to reduce the theo-retical uncertainties through better calculations,through control-sample measurements, and workon parton distributions.

This is not a program that will be undertaken lightly.But should it turn out to be necessary, the experienceof Run II at the Tevatron will be invaluable in carryingit out.

8.3. A Linear Collider

The W mass can be measured at a Linear Col-lider (LC) in W+W− production either in a dedi-cated threshold scan operating the machine at

√s ≈

161 GeV, or via direct reconstruction of the W bosonsin the continuum (

√s = 0.5−1.5 TeV). Both strategies

have been used with success at LEP II.In the threshold region, the W+W− cross section is

very sensitive to theW mass. The sensitivity is largestin the region around

√s = 161 GeV [ 47] at which point

the statistical uncertainty is given by

δM statW ≈ 90 MeV/c2

[

ε∫

Ldt100 pb−1

]−1/2

. (8)

Here, ε is the efficiency for detecting W bosons. Forε = 0.67 and an integrated luminosity of 100 fb−1, onefinds from Eq. (8)

δM statW ≈ 3.5 MeV/c2. (9)

Assuming that the efficiency and the integratedluminosity can be determined with a precision of ∆ε =0.25% and ∆L = 0.1%, MW can be measured with anuncertainty of [ 48]

δMW ≈ 6 MeV/c2, (10)

provided that the theoretical uncertainty on theW+W− cross section is smaller than about 0.1% inthe region of interest.Presently, the W pair cross section in the threshold

region is known with an accuracy of about 1.4% [49]. In order to reduce the theoretical uncertainty ofthe cross section to the desired level, the full O(α)electroweak corrections in the threshold region areneeded. This calculation is extremely difficult. Inparticular, currently no practicable solution of thegauge invariance problem associated with finite Wwidth effects in loop calculations exists. The existingcalculations which take into account O(α) electroweakcorrections all ignore non-resonant diagrams [ 50].If one (pessimistically) assumes that the theoretical

uncertainty of the cross section will not improve, theuncertainty of the W mass obtained from a thresholdscan is completely dominated by the theoretical error,and the precision of the W mass is limited to [ 47]

δMW ≈ δM theorW ≈ 17 MeV/c2

[

∆σ

σ× 100%

]

(11)

≈ 24 MeV/c2.

Using direct reconstruction of W bosons and as-suming an integrated luminosity of 500 fb−1 at

√s =

500 GeV, one expects a statistical error of δM statW ≈

3.5 MeV/c2 [ 51]. Systematic errors are dominatedby jet resolution effects. Using Zγ, Z → 2 jetevents where the photon is lost in the beam pipe forcalibration, a systematic error δM syst

W < 10 MeV/c2 isexpected to be achieved. The resulting overall preci-sion of theW boson mass from directW reconstructionat a Linear Collider operating at an energy well abovethe W pair threshold is

δMW ≈ 10 MeV/c2. (12)

9. Theoretical Issues at high√s

Future hadron and lepton collider experiments areexpected to measure the W boson mass with a preci-sion of δMW ≈ 10 − 20 MeV/c2. For values of δMW

smaller than about 40 MeV/c2, the precise definition ofthe W mass and width become important when thesequantities are extracted.In a field theoretical description, finite width effects

are taken into account in a calculation by resummingthe imaginary part of the W vacuum polarization.This leads to an energy dependent width. However,the simple resumming procedure carries the risk ofbreaking gauge invariance. Gauge invariance worksorder by order in perturbation theory. By resummingthe self energy corrections one only takes into accountpart of the higher order corrections. Apart from being

19

theoretically unacceptable, breaking gauge invariancemay result in large numerical errors in cross sectioncalculations.In order to restore gauge invariance, one can adopt

the strategy of finding the minimal set of Feynman di-agrams that is necessary for compensating those termscaused by an energy dependent width which violategauge invariance [ 52]. This is relatively straightfor-ward for a simple process such as qq′ →W → ℓν [ 53],but more tricky for e+e− → W+W− → 4 fermions, inparticular when higher order corrections are included.The so-called complex mass scheme [ 54], which usesa constant, ie. an energy independent width, offersa convenient alternative. At LEP II energies,

√s ≈

200 GeV, the differences in the e+e− → 4 fermionscross section using an energy dependent and a constantwidth are small. However, at Linear Collider energies,√s = 0.5−2 TeV, the terms associated with an energy

dependent width which break gauge invariance leadto an overestimation of the cross section by up to afactor 3 [ 54].For qq′ → W → ℓν, the parameterizations of the

W resonance in terms of an energy dependent and aconstant W width are equivalent. The W resonanceparameters in the constant width scenario, MW andΓW , and the corresponding quantities,MW and ΓW , ofthe parameterization using an energy dependent widthare related by a simple transformation [ 55]

MW = MW

(

1 + γ2)−1/2

, (13)

ΓW = ΓW

(

1 + γ2)−1/2

, (14)

where γ = ΓW /MW . The W mass obtained in theconstant width scenario thus is about 27 MeV/c2

smaller than that extracted using an energy dependentwidth.In the past, an energy dependent W width has been

used in measurements of the W mass at the Tevatron [56, 57]. The Monte Carlo programs available for theW mass analysis at LEP II (see Ref. [ 50] for anoverview) in contrast use a constant W width. Sincethe difference between the W mass obtained usinga constant and an energy dependent width is of thesame size or larger than the expected experimentaluncertainty, it will be important to correct for thisdifference in future measurements.

10. Conclusions

The measurements of the W mass and width inRun I already represent great experimental achieve-ments and contribute significantly to their world av-erage determinations. Close inspection of the varioussystematic error sources leads us to believe that a Wmass measurement in Run II at the 30 MeV/c

2level

per experiment is achievable, and this compares wellto the expected uncertainty on the W mass measuredat LEP II. Each experiment is expected to measure theW width to a similar precision with 2 fb−1 of data.Alternative methods for determining MW at the

Tevatron have been discussed and may turn out to bemore appropriate in the Run II operating environmentthan the traditional transverse mass fitting approach.Determination of the W mass at the LHC will beextremely challenging, using detectors that are not op-timized for this measurement. A future linear collidershould do significantly better. Clearly, the W massand width measurements at the Tevatron in Run IIwill remain the best hadron collider determinations ofthese quantities for many years and will compete withthe best measurements made elsewhere.

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11. “Parton Distributions, d/u, and Higher Twistsat High x”, U.K. Yang, A. Bodek and Q. Fan,Proceedings of the 33th Rencontres de Moriond:QCD and High-energy Hadronic Interactions, LesArcs, France, April 1998 and UR-1518 (1998).

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17. A. Kulesza and W. J. Stirling, J. Phys. G: Nucl.Part. Phys. 26, 637 (2000) and references therein.

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23. M. Lancaster and D. Waters, CDF-Note 5240.24. U. Baur and T. Stelzer, Phys. Rev. D 61, 073007

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Lett. 84, 222 (2000) and Phys. Rev. D 62, 092006(2000).

26. S. Rajagopalan and M. Rijssenbeek, “Measure-ment of the W Mass Using the Transverse MassRatio of theW and Z”, DØ Note 3000, June, 1996;S. Rajagopalan, “Measurement of the W MassUsing the Transverse Mass Ratio of theW and Z”,Division of Particles and Fields Conference, 1996.

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28. Phys. Rev. Lett. 77, 3309 (1996); Phys. Rev. D58, 092003 (1998).

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Thesis, Stony Brook, August 2000.30. “LEP W -pair Cross Section and W Mass and

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32. R. Barate et al. [ALEPH Collaboration], Phys.Lett. B 453, 121 (1999).

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53. D. Wackeroth and W. Hollik, Phys. Rev. D 55,6788 (1997).

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57. S. Abachi et al. [D0 Collaboration], Phys. Rev.Lett. 77, 3309 (1996); B. Abbott et al. [D0 Collab-

21

oration], Phys. Rev. D 58, 012002 (1998); Phys.Rev. Lett. 80, 3008 (1998); Phys. Rev. D 58,092003 (1998); Phys. Rev. D 62, 092006 (2000)and Phys. Rev. Lett. 84, 222 (2000).

22

Measurement of the Forward-Backward Asymmetry in e+e− and µ+µ− events

with DØ in Run II

Ulich Baura, John Ellisonb and John Rhab

aState University of New York, Buffalo, NY 14260

bUniversity of California, Riverside, CA 92521

The forward-backward asymmetry of ℓ+ℓ− events in Run II can yield a measurement of the effective weak mixing angle

sin2 θW and can provide a test of the standard model γ∗/Z interference at ℓ+ℓ− invariant masses well above the 200 GeV

center of mass energy of the LEP II collider. The asymmetry at large partonic center of mass energies can also be used to

study the properties of possible new neutral gauge bosons. We describe an updated study of the forward-backward asymmetry

and give estimates of the statistical and systematic uncertainties expected in Run II. The prospects for measuring the weak

mixing angle at the LHC and a linear collider operating at√s = MZ are also briefly described.

1. Introduction

In this note we present an updated study of theprospects for measurement of the forward-backwardasymmetry in pp → γ∗/Z → ℓ+ℓ− events. This workextends our earlier study described in the TeV2000 re-port [ 1] in several respects: (i) we include the effects ofQED corrections; (ii) we include the effects of expectedRun II DØ detector resolutions and efficiencies; (iii)we consider systematic errors in more detail; and (iv)we include a simulation of the muon channel processpp→ γ∗/Z → µ+µ−.The forward-backward asymmetry (AFB) in pp →

γ∗/Z → ℓ+ℓ− events arises from the parton levelprocess qq → γ∗/Z → ℓ+ℓ−. This asymmetry dependson the vector and axial-vector couplings of the quarksand leptons to the Z boson and is therefore sensitiveto the effective weak mixing angle sin2 θW . Thecurrent world average value of sin2 θW from LEP andSLD asymmetry measurements is sin2 θW = 0.23147±0.00017 [ 2]. As will be seen from our results it willbe necessary to achieve high luminosity (> 10 fb−1)and combine the results from the electron and muonchannels and the results from DØ and CDF to achievea precision comparable to this.The SM tree level prediction [ 3] for AFB as a

function of s for qq → γ∗/Z → e+e− is shown inFig. 1 for u and d quarks. These are the same asymme-tries as encountered in the inverse e+e− annihilationreactions. The largest asymmetries occur at partoncenter-of-mass energies of around 70 GeV and above110 GeV. At the Z-pole the asymmetry is dominatedby the couplings of the Z boson and arises from theinterference of the vector and axial components ofits coupling. The asymmetry is proportional to thedeviation of sin2 θW from 1

4 . At large invariant mass,

the asymmetry is dominated by γ∗/Z interference andis almost constant (≈ 0.6), independent of invariantmass.

Figure 1. The standard model tree level predictionof the forward-backward asymmetry as a function ofe+e− invariant mass for uu→ e+e− and dd→ e+e−.

With sufficient statistics in Run II the forward-backward asymmetry can be used to measure sin2 θW ,which in turn can provide a constraint on the standardmodel complementary to the measurement of the Wboson mass. The Tevatron also allows measurementof the asymmetry at partonic center-of-mass energiesabove the center of mass energy of LEP II. Thismeasurement can be used, not only to confirm thestandard model γ∗/Z interference which dominates

23

in this region, but also to investigate possible newphenomena which may alter AFB , such as new neutralgauge bosons [ 4] or large extra dimensions [ 5].CDF have measured the forward-backward asymme-

try at the Tevatron using e+e− pairs in 110 pb−1 ofdata at

√s = 1.8 TeV [ 6]. They obtain AFB = 0.070±

0.016 in the mass region 75 GeV < me+e− < 105 GeV,and AFB = 0.43±0.10 in the regionme+e− > 105 GeV.The much larger Run II statistics will enable AFB tobe measured with an uncertainty reduced by over anorder of magnitude.

2. Simulation

The simulations presented here use the zgrad

Monte Carlo program [ 7], which includes O(α) QEDradiative corrections to the process pp → γ∗/Z →ℓ+ℓ−. We simulate this process at

√s = 2.0 TeV

using the MRST parton distributions as our defaultset. Since the radiative corrections are included inzgrad, we denote the process of interest by pp →γ∗/Z → ℓ+ℓ−(γ) in the remainder of this paper. Thezgrad program includes real and virtual correctionsin the initial and final states.In our simulations, the effects of detector resolution

are modeled by smearing the 4-momenta of the parti-cles from zgrad according to the estimated resolutionof the Run II DØ detector. We smear the 4-momentaof electrons, positrons and photons according to theenergy resolution σEM of the calorimeters, which havebeen parametrized using constant, sampling and noiseterms as

(σEM

E

)2

=

c2EM +

(

sEM√

ET

)2

+(

nEME

)2

Central Calorimeter

c2EM +

(

sEM√E

)2

+(

nEME

)2

End Calorimeters

(1)

where we use the parameters relevant for the Run Idetector, cEM = 0.0115, sEM = 0.135, and nEM =0.43 for the CC, and cEM = 0.0100, sEM = 0.157, andnEM = 0.29 for the EC. With the addition of the 2 Tsolenoidal magnetic field in Run II, only minor changesin these parameters are expected. The transversemomentum of muons in the Run II detector will bemeasured in the central tracking system, consistingof the Central Fiber Tracker (CFT) and the SiliconMicrostrip Tracker (SMT). The momentum resolutionof the tracking system has been studied using thefast Monte Carlo mcfast. From these studies the

resolution in 1/pT is parametrized as:

σ

(

1

pT

)

=

√

√

√

√

( α

L2

)2

+

(

γ

pT√

L| sin θ|

)2

(2)

where

L =

{

1 0 < θ ≤ θctan θtan θc

θc < θ < 90◦

}

. (3)

Here α = 0.0017 GeV−1, γ = 0.018, L is the fraction ofthe projection of the track length in the bending planewhich is measured in the Tracker, and θc ≈ 23◦ is thepolar angle beyond which the number of CFT layerscrossed by a track starts to decrease. The first termin Eq. (3) is due to the detector resolution while thesecond term is due to multiple scattering.Figure 2 shows the transverse momentum resolution

as a function of detector pseudorapidity |ηdet| for trackswith a pT of 1, 20 and 100 GeV, while Fig. 3 shows theresolution as a function of pT and |ηdet| in the formof a contour plot. For central tracks (η = 0) withpT = 45 GeV, the resolution is σ(pT )/pT = 8%, to becompared with the calorimeter energy resolution for45 GeV electrons of σ(E)/E = 2.5%.

Pseudorapidity

0.0 0.5 1.0 1.5 2.0 2.5

σ (p

T)/

pT

0.02

0.030.040.050.060.08

0.2

0.30.40.50.60.8

2

0.01

0.1

1

pT

=100 GeV/c

pT

=20 GeV/c

pT

=1 GeV/c

ParametrizationSMT Technical Design Report

Figure 2. Parametrized transverse momentumresolution for the DØ Run II tracking system (dottedlines). The solid lines are the results of a simple MonteCarlo simulation taken from the DØ SMT TechnicalDesign Report [ 8].

24

Pseudorapidity

0.0 0.5 1.0 1.5 2.0 2.5

Tra

nsve

rse

Mom

entu

m (G

eV/

c)

2

34568

20

3040506080

200

1

10

100

0.5

0.5

0.60.4

0.4

0.3

0.2 0.2

0.2

0.2

0.08 0.08

0.08

0.4

0.06 0.06

0.06

0.06

0.05 0.05

0.05

0.05

0.04 0.04

0.04

0.0 4

0.3 0.3

0.030.03

0.03

0.40.40.5

0.02

0.02

0.1 0.1

0.1

0.1

σ(pT)/pT

Figure 3. Transverse momentum resolution forthe central tracking system plotted as a function oftransverse momentum and pseudorapidity.

We assume an overall detection efficiency of 75%for e+e− events and 65% for µ+µ− events. Theseefficiencies are rough estimates of the effects of trig-ger and particle identification efficiencies expected inRun II. The results can be updated once more realisticnumbers for these efficiencies become available.The zgrad program generates weighted events. Due

to the occurrence of negative weights, we did notattempt to unweight the events. Thus, we work withweighted events and properly account for the weightsin our calculations of the forward-backward asymmetryerrors. The forward-backward asymmetry is defined by

AFB =σF − σBσF + σB

(4)

where σF and σB are the forward and backward crosssections, defined by

σF =

∫ 1

0

dσ

d(cosθ∗)d(cosθ∗)

σB =

∫ 0

−1

dσ

d(cosθ∗)d(cosθ∗) (5)

and θ∗ is the angle of the lepton in the Collins-Soperframe [ 9].The statistical error on AFB is given by

δAFB = 2

√

σ2B(δσF )

2 + σ2F (δσB)

2

(σF + σB)2(6)

where δσF , δσB are the uncertainties in the forwardand backward cross sections. For unweighted events,

this simplifies to

δAFB =2

NF +NB

√

NFNB

NF +NB(7)

where NF , NB are the numbers of forward andbackward events. However, zgrad generates weightedevents and, therefore, we use Eq. (6) where δσFand δσB , are calculated using the appropriate eventweights.The selection cuts used in our study are summarized

in Table 1. In the electron channel we require one of theelectrons to be in the CC (|ηdet| < 1.0), while the otherelectron may be in the CC or in the EC (|ηdet| < 1.0or 1.5 < |ηdet| < 2.5).In the muon channel we require both muons to be

within |ηdet| < 1.7. In Run II the muon coverage isexpected to extend up to |ηdet| = 2.0. We chose to limitthe muon acceptance to |ηdet| = 1.7 since Monte Carloevents were already generated with this restrictionand large CPU time would have been required tore-generate the events.We account in our simulation for the granularity

of the DØ calorimeter. If the photon is very closeto the electron its energy will be merged with thatof the electron cluster. Thus, in the simulation wecombine the photon and electron 4-momenta to forman effective electron 4-momentum if the photon is

within ∆Reγ ≡√

∆η2eγ +∆φ2eγ < 0.2. If the photon

falls within 0.2 < ∆Reγ < 0.4, we reject the event ifEγ/(Ee+Eγ) > 0.15, since the event will not pass thestandard isolation criterion imposed on electrons.If a photon is very close to a muon and it deposits

sufficient energy in the calorimeter close to the muontrack, the energy deposition in the calorimeter will notbe consistent with the passage of a minimum ionizingmuon. Therefore, in the simulation we reject events if∆Rµγ < 0.2 and Eγ > 2 GeV.

Table 1Selection criteria for e+e− and µ+µ− events.

e+e− µ+µ−

Selection cut e1 e2 µ1 µ2

pT (GeV) > 25 > 25 > 20 > 15|ηdet| < 1.0 < 1.0 < 1.7 < 1.7

or 1.5− 2.5mℓ+ℓ− (GeV) > 40 > 40 > 40 > 40

25

3. Results

The ℓ+ℓ− invariant mass distributions for pp →γ∗/Z → ℓ+ℓ−(γ) at

√s = 2.0 TeV from the

zgrad simulations, using the MRST parton distri-bution functions are shown in Fig. 4. The thin lineshows dσ/dmℓ+ℓ− without any kinematic cuts appliedand with no detector acceptance or resolution effectsincluded. In order to obtain sufficient statisticalprecision a large number of events were generated inmultiple runs covering overlapping regions of mℓ+ℓ− .The thick line shows dσ/dmℓ+ℓ− after kinematic cutsand detector effects are included. The error barsrepresent the statistical errors only, calculated fromEq. (6), assuming an integrated luminosity of 10 fb−1.

Fig. 5 shows the forward-backward asymmetry as afunction of mℓ+ℓ− . The solid line shows AFB withoutany kinematic cuts applied and with no detector accep-tance or resolution effects included. The solid pointsshow AFB after kinematic cuts and detector effects areincluded. The error bars represent the statistical errorsonly, calculated from Eq. (6), assuming an integratedluminosity of 10 fb−1. As can be seen, detectorresolution and acceptance effects significantly alter theshape of the AFB vs. mℓ+ℓ− curve, especially at lowdi-lepton invariant masses. In this region, the effectof CC/EC acceptance increases AFB, while restrictingµ+µ− events to be in the central region decreasesthe asymmetry. This is also true of e+e− events ifonly CC/CC events are considered. In the vicinityof the Z-pole the energy resolution is better than thepT resolution, and hence the Ae+e−

FB is altered less

than Aµ+µ−

FB . In these plots the AFB shown is thereconstructed AFB without corrections for acceptanceor resolution effects.In order to obtain a measurement of the weak mixing

angle we assume the relationship

AFB = a+ b sin2 θW (8)

so that the statistical error on sin2 θW is given by

δ sin2 θW =δAFB

b. (9)

The quantity b is determined by varying sin2 θW in theMonte Carlo simulations. Since AFB is determinedover a finite range of di-lepton invariant mass, wehave investigated the effect of the lower and uppermℓ+ℓ− cuts on δ sin2 θW . The optimal precision isobtained for 75 GeV < mℓ+ℓ− < 105 GeV, i.e. amass window encompassing the Z-pole. This is to beexpected because the sensitivity b is maximal at theZ-pole and this region is where the cross section peaksand hence the statistical error is smallest. Thus, the

0.001

0.01

0.1

1

10

100

50 100 150 200

e+ e– Invariant Mass (GeV)

G�GP

(pb

/GeV

)

CC/CC+CC/EC

No cuts or smearing

(a)

�� � �

����7H9�����IESS = H HV

γ γ+ −

−→ →=

0.001

0.01

0.1

1

10

100

50 100 150 200

µ + µ – Invariant Mass (GeV)

G�GP

(pb

/GeV

)

Central Tracking

Unsmeared, Uncut

�� � �

����7H9�����IESS =V

γ µ µ γ+ −

−→ →=

(b)

Figure 4. Invariant mass mℓ+ℓ− distributions for (a)e+e− events and (b) µ+µ− events. The thin line is thedistribution obtained with no cuts or detector effectsapplied and the thick line is the resulting distributionafter selection cuts and detector effects are included.

AFB values and errors presented in the remainder ofthis paper are all obtained with a di-lepton invariantmass cut of 75 GeV < mℓ+ℓ− < 105 GeV. Table 2shows the resulting statistical uncertainties obtainedfrom the electron and muon channels. The e+e− andµ+µ− channels yield similar uncertainties on AFB andsin2 θW . In both channels the effect of the selectioncuts is to reduce the sensitivity b from about 3.5 toabout 2.8.The effects of NLO QCD corrections to the process

pp → γ∗/Z → ℓ+ℓ− are not included in zgrad, sowe estimate these using the O(αs) event generatordescribed in [ 7]. Using the same method as describedin Section 2, we calculate the change in the forward-backward asymmetry and the shift in sensitivity due

26

Table 2Uncertainties on AFB and sin2 θW in the invariant mass range 75 GeV < mℓ+ℓ− < 105 GeV for an integratedluminosity of 10 fb−1. Also shown are the assumed event detection efficiency, the number of events passing all thecuts Nobs, the forward-backward asymmetry AFB , and the sensitivity b.

Process Selection cuts Efficiency Nobs AFB δAstatFB b δ sin2 θW

Z → e+e− no 100% 1.78× 106 0.0551 0.0008 3.43 0.0002Z → e+e− yes 75% 3.82× 105 0.0515 0.0014 2.78 0.0005Z → µ+µ− no 100% 1.87× 106 0.0534 0.0007 3.51 0.0002Z → µ+µ− yes 65% 5.67× 105 0.0420 0.0011 2.62 0.0004

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����

�� ��� ��� ��� ��� ��� ��� ��� ��� ���

H � H ± �,QYDULDQW�0DVV��*H9�

$)%

&&�&&�RU�&&�(&1R�FXWV�RU�VPHDULQJ

�D�

�� � �

����7H9�����IESS = H HV

g g+ -

-� �=

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����

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����

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±�,QYDULDQW�0DVV��*H9�

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&HQWUDO�0XRQV��_ _�������

1R�FXWV�RU�VPHDULQJ

�

� � �

����7H9�����IE

SS =

V

g m m g+ -

-

� �

=

�E�

Figure 5. Forward-backward asymmetry AFB vs.di-lepton invariant mass for (a) e+e− events and(b) µ+µ− events. The solid line is the distributionobtained with no cuts or detector effects applied andthe points are the resulting distribution after selectioncuts and detector effects are included. The error barsrepresent the statistical errors for a data sample of10 fb−1.

to NLO QCD corrections. Thus, we write

∆AFB = AO(αs)FB −ALO

FB (10)

∆b = bO(αs) − bLO (11)

where LO denotes the leading-order quantities. Forevents generated including detector effects we find theshift in AFB to be negligible for e+e− events and−13% for µ+µ− events. The shift in sensitivity is∆b/bLO ≈ −3.4% for e+e− events and ≈ −25% for(µ+µ−) events. Thus, NLO QCD effects decrease thesensitivity to sin2 θW by 3.4% in the e+e− channel and25% in the µ+µ− channel.

4. Systematic Uncertainties

4.1. Parton Distribution Functions

Since the vector and axial couplings of the u andd quarks to the Z boson are different, the leptonforward-backward asymmetry is expected to dependon the ratio of the u to d quark parton distributionfunctions. Thus, the choice of the parton distributionfunctions (PDF’s) will affect the measured leptonforward-backward asymmetry.We have run simulations with six PDF’s from the

MRS [ 10] and CTEQ [ 11] sets to study the effect ofthe PDF’s on the asymmetry. Fig. 6 shows the e+e−

and µ+µ− asymmetries and their statistical errors foreach PDF.The largest deviation from the MRST value for AFB

is 0.0018 for the e+e− channel and 0.0015 for the µ+µ−

channel. While these numbers are of the same orderas the statistical error expected on AFB for 10 fb−1,we expect that in Run II our knowledge of the PDF’swill improve considerably, e.g. from the constraintsimposed by the Run II W asymmetry measurements.Thus, we expect a significantly decreased systematicerror due to the uncertainty in the PDF’s whichwill likely render it insignificant compared with thestatistical error in the measurement. For example, ifthe PDF uncertainty scales as 1/

√N , the uncertainty

in AFB would be δAFB ≈ 0.00018 (0.00015) for an

27

Variation of Forward-Backward Asymmetry with PDF

0.040

0.042

0.044

0.046

0.048

0.050

0.052

0.054

MR

ST

MR

S(A

’)

MR

SR

2

CT

EQ

3M

CT

EQ

4M

CT

EQ

5M

MR

ST

MR

S(A

’)

MR

SR

2

CT

EQ

3M

CT

EQ

4M

CT

EQ

5M

AF

B

� � �SS = H Hγ γ+ −→ → � � �SS =γ µ µ γ+ −→ →

Figure 6. Variation of the forward-backward asym-metry for e+e− and µ+µ− events, including selectioncuts and detector effects.

integrated luminosity of 10 fb−1 in the e+e− (µ+µ−)channel.

4.2. Energy Scale Calibration Uncertainties

The energy scale uncertainty, or the uncertainty inmapping the calorimeter response to the true electronenergy, affects the forward-backward asymmetry bycausing a shift in the ℓ+ℓ− invariant mass range overwhich we integrate AFB . The effect is significant inthe Z-pole region, but at large invariant masses AFB

is essentially constant and the energy scale uncertaintydoes not play a role. For electrons, the measuredenergy can be related to the true energy by

Emeas = αEtrue + δ (12)

where the scale factor α and offset δ are determinedby calibration of the calorimeters. In Run I DØ deter-mined α = 0.9533± 0.0008 and δ = −0.16+0.03

−0.21 GeV.Assuming these uncertainties in α and δ, we find a

systematic error of δAFB = 0.0002 due to the overallenergy scale uncertainty.

4.3. Uncertainty due to Backgrounds

Backgrounds are not included in the simulationsabove, but we can estimate the uncertainty due tobackgrounds as follows. If we assume that the fractionof observed events which are due to backgrounds isα± δα, then the uncertainty in the forward-backwardasymmetry will be

δAFB = 2√2

NFNB

(NF +NB)2δα (13)

where NF , NB are the numbers of background-subtracted forward and backward events, and we haveassumed that the background events are symmetric incosθ∗. If we assume that the error in the backgroundfraction δα scales as the inverse of the integratedluminosity, we can extrapolate from the uncertaintiesin the Run I CDF and DØ data samples to estimate theerror. The Run I uncertainties were δα ≈ 0.1− 1.0%.Thus, for an integrated luminosity of 10 fb−1, weobtain an uncertainty on AFB of δAFB = 0.00014.

4.4. Summary of Uncertainties in sin2 θWTable 3 summarizes the statistical and individ-

ual systematic uncertainty estimates expected with10 fb−1 of data. We estimated the uncertainties dueto electron energy resolution and muon transversemomentum resolution to be negligible.

4.5. Conclusions

The measurement of the forward-backward asymme-try in e+e− and µ+µ− events in Run II provides ameans to test the standard model γ∗/Z interference atℓ+ℓ− invariant masses well above the center of massenergy of the LEP II collider. The estimated DØprecision on AFB achievable with 10 fb−1 integratedluminosity is shown in Fig. 5.In the vicinity of the Z-pole this measurement can

also be used to determine the effective weak mixingangle sin2 θW . The optimal precision on sin2 θW isobtained for 75 GeV < mℓ+ℓ− < 105 GeV, i.e. amass window encompassing the Z-pole. This is tobe expected because the sensitivity b is maximal atthe Z-pole and this region is where the cross sectionpeaks and hence the statistical error is smallest. For10 fb−1 we estimate that the total error on sin2 θWwill be 0.0005 in the electron channel and 0.0006 inthe muon channel, assuming that systematic errorsscale as the inverse of the square root of the integratedluminosity. One would expect similar precision fromCDF, and combining the results of the two experimentsin both channels the overall uncertainty would beδ sin2 θW ≈ 0.00028. Therefore, if integrated luminosi-ties in excess of 10 fb−1 can be achieved in Run II,it appears that the determination of sin2 θW will havecomparable precision to the current world average ofthe measurements from LEP and SLD.

5. Measuring AFB at the LHC

At the LHC, the Z → ℓ+ℓ− cross section is approxi-mately a factor 7 larger than at the Tevatron. However,the measurement of the forward backward asymmetryis complicated by several factors. In pp collisions, thequark direction in the initial state has to be extracted

28

Table 3Summary of uncertainties on AFB and sin2 θW in the invariant mass range 75 GeV < mℓ+ℓ− < 105 GeV for anintegrated luminosity of 10 fb−1. The effects of O(α) QED corrections and NLO QCD corrections have been takeninto account.

e+e− µ+µ−

Error source δAFB δ sin2 θW δAFB δ sin2 θW

Statistical 0.0014 0.0005 0.0011 0.0006

Systematics 0.0003 0.0001 0.0002 0.00010

PDF 0.00018 0.00007 0.00015 0.00008EM scale 0.0002 0.00007 − −Backgrounds 0.00014 0.00005 0.00014 0.00007

Total uncertainty 0.0014 0.0005 0.0011 0.0006

from the boost direction of the ℓ+ℓ− system withrespect to the beam axis. At LHC energies, the sea-seaquark flux is much larger than at the Tevatron. As aresult, the probability, fq, that the quark direction andthe boost direction of the di-lepton system coincideis rather small. The forward backward asymmetryis therefore smaller than at the Tevatron, and thesensitivity to sin2 θW at the LHC with 100 fb−1 perlepton channel and experiment [ 7, 12] is similar to thatestimated for the Tevatron with 10 fb−1 (see Sec. 4.5).Restricting the AFB measurement to events which

satisfy |y(ℓ+ℓ−)| > 1 in addition to the |y(ℓ)| < 2.5cut improves the significance of the measurement byabout a factor 1.5. Events with a large di-leptonrapidity originate from collisions where at least oneof the partons carries a large fraction x of the protonmomentum. Since valence quarks dominate at highvalues of x, a cut on y(ℓ+ℓ−) increases fq and thus theasymmetry. However, the gain due to the larger asym-metry is partially cancelled by the loss of statistics,leaving a modest improvement only.In order to achieve a precision better than the

current value of δ sin2 θW = 1.7× 10−4 [ 13], it will benecessary to detect one of the leptons in the rapidityrange up to |y(ℓ)| < 5 at the LHC. If this can bedone, one expects that the weak mixing angle can bedetermined with a precision of

δ sin2 θW = 1.4× 10−4, (14)

per lepton channel and experiment for an integratedluminosity of 100 fb−1. In order to reach the precisiongiven in Eq. (14), a jet rejection factor of 10 − 100has to be achieved in the forward rapidity region2.5 < |y(ℓ)| < 5, and the lepton acceptance times thereconstruction efficiency as a function of y(ℓ) has to beknown to 0.1% or better [ 12].For comparison, at a Linear Collider operating at

√s =MZ with a luminosity of a few ×1033 cm−2 s−1,

it is expected that the weak mixing angle be deter-mined with a precision of about δ sin2 θW = 1× 10−5 [14].

REFERENCES

1. “Future Electroweak Physics at the Fermilab Teva-tron: Report of the TeV 2000 Study Group”,edited by D. Amidei and R. Brock, FERMILAB-PUB-96/082, pp.75-79 (1996).

2. J. Mnich, “Tests of the Standard Model”, CERN-EP/99-143 (1999), to appear in Proceedings ofthe international Europhysics Conference on HighEnergy Physics, Tampere, Finland, 15-21 July,1999.

3. J.L. Rosner, Phys. Lett. B221, 85 (1989)4. J.L. Rosner, Phys. Rev. D35, 2244 (1987);

J.L. Rosner, Phys. Rev. D54, 1078 (1996).5. I. Antoniadis, et al., Phys. Lett. B436, 257 (1998);

J.L. Hewett, Phys. Rev. Lett. 82, 4765 (1999).6. F. Abe, et al., (CDF Collaboration), Phy. Rev.

Lett. 77, 2616 (1996).7. U. Baur, S. Keller, and W. K. Sakumoto, Phy. Rev.

D57, 199 (1998).8. DØ Silicon Tracker Technical Design Report, (DØ

Collaboration), DØ Note 2169 (1994).9. J.C. Collins and D.E. Soper, Phys. Rev. D16, 2219

(1977).10. A.D. Martin, R.G. Roberts, W.J. Stirling, Phys.

Rev. D50, 6734 (1994); ibid. D51, 4756(1995); A.D. Martin, R.G. Roberts, W.J. Stir-ling, Phys. Lett. B387, 419 (1996); A.D. Martin,R.G. Roberts, W.J. Stirling, R.S. Thorne, Phys.Lett. B443, 301 (1998).

11. H.L. Lai et al., Phys. Rev. D51, 4763 (1995); ibid.

29

D55, 1280 (1997); H.L. Lai et al., Eur. Phys. J.C12, 375 (2000).

12. S. Haywood et al., Report of the ElectroweakPhysics Working Group, hep-ph/0003275 (March2000), in Proceedings of the 1999 CERNWorkshopon Standard Model Physics (and more) at theLHC CERN Yellow Report, CERN-2000-04, eds.G. Altarelli and M. Mangano.

13. D. Abbaneo et al. (The LEP Electroweak WorkingGroup), CERN-EP-2000-016 (January 2000).

14. J. Erler, S. Heinemeyer, W. Hollik, G. Weigleinand P. M. Zerwas, Phys. Lett. B486, 125 (2000).

30

Global Fits to Electroweak Data Using GAPP

J. Erlera

aDepartment of Physics and Astronomy, University of Pennsylvania,Philadelphia, PA 19104-6396

At Run II of the Tevatron it will be possible to measure the W boson mass with a relative precision of about 2 × 10−4,

which will eventually represent the best measured observable beyond the input parameters of the SM. Proper interpretation

of such an ultra-high precision measurement, either within the SM or beyond, requires the meticulous implementation and

control of higher order radiative corrections. The FORTRAN package GAPP, described here, is specifically designed to meet

this need and to ensure the highest possible degrees of accuracy, reliability, adaptability, and efficiency.

1. PRECISION TESTS

Precision analysis of electroweak interactions followsthree major objectives: high precision tests of theSM; the determination of its fundamental parameters;and studies of indications and constraints of possiblenew physics beyond the SM, such as supersymmetryor new gauge bosons. Currently, the experimentalinformation comes from the very high precision Zboson measurements at LEP and the SLC, direct massmeasurements and constraints from the Tevatron andLEP II, and low energy precision experiments, suchas in atomic parity violation, ν scattering, and raredecays. These measurements are compared with thepredictions of the SM and its extensions. The level ofprecision is generally very high. Besides the need forhigh-order loop calculations, it is important to utilizeefficient renormalization schemes and scales to ensuresufficient convergence of the perturbative expansions.The tasks involved called for the creation of a

special purpose FORTRAN package, GAPP, shortfor the Global Analysis of Particle Properties [ 1].It is mainly devoted to the calculation of pseudo-observables, i.e., observables appropriately idealizedfrom the experimental reality. The reduction ofraw data to pseudo-observables is performed by theexperimenters with available packages (e.g., ZFITTERfor Z pole physics). For cross section and asymmetrymeasurements at LEP II (not implemented in thecurrent version, GAPP 99.7), however, this reductionis not optimal and convoluted expressions should beused instead. GAPP attempts to gather all availabletheoretical and experimental information; it allows theaddition of extra parameters describing new physics; ittreats all relevant SM inputs as global fit parameters;and it can easily be updated with new calculations,data, observables, or fit parameters. For clarity andspeed it avoids numerical integrations throughout. Itis based on the modified minimal subtraction (MS)

scheme which demonstrably avoids large expansioncoefficients.GAPP is endowed with the option to constrain

nonstandard contributions to the oblique parameters

defined to affect only the gauge boson self-energies [ 2](e.g. S, T , and U); specific anomalous Z couplings; thenumber of active neutrinos (with standard couplings tothe Z boson); and the masses, mixings, and couplingstrengths of extra Z bosons appearing in modelsof new physics. With view on the importance ofsupersymmetric extensions of the SM on one hand, andupcoming experiments on the other, I also includedthe b → sγ transition amplitude, and intend toadd the muon anomalous magnetic moment. In thelatter case, there are theoretical uncertainties fromhadronic contributions which are partially correlatedwith the renormalization group (RG) evolutions of theQED coupling and the weak mixing angle. Thesecorrelations will be partially taken into account byincluding heavy quark effects in analytical form; seeRef. [ 3] for a first step in this direction. By comparingthis scheme with more conventional ones, it will also bepossible to isolate a QCD sum rule and to rigorouslydetermine the charm and bottom quark MS masses,mc and mb, with high precision.

2. GAPP

2.1. Basic structure

In the default running mode of the current version,GAPP 99.7, a fit is performed to 41 observables, outof which 26 are from Z pole measurements at LEPand the SLC. The Fermi constant, GF (from the muonlifetime), the electromagnetic fine structure constant,α (from the quantum Hall effect), and the light fermionmasses are treated as fixed inputs. The exception ismc which strongly affects the RG running1 of α(µ)for µ > mc. I therefore treat mc as a fit parameter

1Quantities defined in the MS scheme are denoted by a caret.

31

and include an external constraint with an enhancederror to absorb hadronic threshold uncertainties ofother quark flavors, as well as theoretical uncertaintiesfrom the application of perturbative QCD at relativelylow energies. Other fit parameters are the Z bosonmass, MZ , the Higgs boson mass, MH , the top quarkmass, mt, and the strong coupling constant, αs, sothat there are 37 effective degrees of freedom. Givencurrent precisions,MZ may alternatively be treated asan additional fixed input.The file fit.f basically contains a simple call to

the minimization program MINUIT [ 4] (from theCERN program library) which is currently used in datadriven mode (see smfit.dat). It in turn calls thecore subroutine fcn and the χ2-function chi2, bothcontained in chi2.f. Subroutine fcn defines constantsand flags; initializes parts of the one-loop package FF [5, 6]; and makes the final call to subroutine values

in main.f which drives the output (written to filesmfit.out). In chi2 the user actively changes andupdates the data for the central values, errors, andcorrelation coefficients of the observables, and includesor excludes individual contributions to χ2 (right afterthe initialization, chi2 = 0.d0). To each observable(as defined at the beginning of chi2) corresponds anentry in each of the fields value, error, smval, andpull, containing the central observed value, the total(experimental and theoretical) error, the calculated fitvalue, and the standard deviation, respectively. Thefunction chi2 also contains calls to various other sub-routines where the actual observable calculations takeplace. These are detailed in the following subsections.Another entry to GAPP is provided through mh.f

which computes the probability distribution functionof MH . The probability distribution function is thequantity of interest within Bayesian data analysis (asopposed to point estimates frequently used in the con-text of classical methods), and defined as the productof a prior density and the likelihood, L ∼ exp(−χ2/2).If one chooses to disregard any further information onMH (such as from triviality considerations or directsearches) one needs a non-informative prior. It isrecommended to choose a flat prior in a variabledefined on the whole real axis, which in the case ofMH is achieved by an equidistant scan over logMH .An informative prior is obtained by activating one ofthe approximate Higgs exclusion curves from LEP IInear the end of chi2.f. These curves affect valuesof MH even larger than the corresponding quoted95% CL lower limit and includes an extrapolation tothe kinematic limit; notice that this corresponds to aconservative treatment of the upper MH limit.Contour plots can be obtained using the routine

mncontours from MINUIT. For the cases this fails,

some simpler and slower but more robust contourprograms are also included in GAPP, but these haveto be adapted by the user to the case at hand.

2.2. α, sin2 θW , MW

At the core of present day electroweak analyses isthe interdependence between GF , MZ , the W bosonmass, MW , and the weak mixing angle, sin2 θW . Inthe MS scheme it can be written as [ 7, 8],

s2 =A2

M2W (1−∆rW )

, s2c2 =A2

M2Z(1 −∆rZ)

, (1)

where,

A =

[

πα√2GF

]1/2

= 37.2805(2) GeV, (2)

s2 is the MS mixing angle, c2 = 1− s2, and where,

∆rW =α

π∆γ +

ΠWW (M2W )− ΠWW (0)

M2W

+V + B, (3)

and,

∆rZ = ∆rW + (1−∆rW )ΠZZ(M

2Z)−

ΠWW (M2W

)c2

M2Z

, (4)

collect the radiative correctionscomputed in sin2th.f. The Π indicate MS subtractedself-energies, and V + B denote the vertex and boxcontributions to µ decay. Although these relationsinvolve the MS gauge couplings they employ on-shellgauge boson masses, absorbing a large class of radiativecorrections [ 9].∆rW and ∆rZ are both dominated by the contribu-

tion ∆γ(MZ) which is familiar from the RG runningof the electromagnetic coupling,

α(µ) =α

1− απ ∆γ(µ)

, (5)

and computed in alfahat.f up to four-loop O(αα3s).

Contributions from c and b quarks are calculated usingan unsubtracted dispersion relation [ 3]. If µ is equal tothe mass of a quark, three-loop matching is performedand the definition of α changes accordingly. Pure QEDeffects are included up to next-to-leading order (NLO)while higher orders are negligible. Precise results canbe obtained for µ < 2mπ and µ ≥ mc.Besides full one-loop electroweak corrections, ∆rW

and ∆rZ include enhanced two-loop contributionsof O(α2m4

t ) [ 10] (implemented using the ana-lytic expressions of Ref. [ 11]) and O(α2m2

t ) [ 12](available as expansions in small and large MH);mixed electroweak/QCD corrections of O(ααs) [ 13]and O(αα2

sm2t ) [ 14]; the analogous mixed elec-

troweak/QED corrections of O(α2); and fermion masscorrections also including the leading gluonic andphotonic corrections.

32

2.3. Z decay widths and asymmetries

The partial width for Z → f f decays is given by,

Γff =Nf

CMZα

24s2c2|ρf |

[

1− 4|Qf |Re(κf )s2 + 8Q2f s

4|κf |2]

×[

1 + δQED + δNSQCD + δSQCD − ααs

4π2Q2

f +O(m2f )

]

. (6)

NfC is the color factor, Qf is the fermion charge,

and ρf and κf are form factors which differ fromunity through one-loop electroweak corrections [ 15]and are computed in rho.f and kappa.f, respectively.For f 6= b there are no corrections of O(α2m4

t ) andcontributions of O(α2m2

t ) to κf [ 16] and ρf [ 9] arevery small and presently neglected. On the other hand,vertex corrections of O(ααs) [ 17] are important andshift the extracted αs by ∼ 0.0007.The Z → bb vertex receives extra corrections due

to heavy top quark loops. They are large and havebeen implemented in bvertex.f based on Ref. [ 18].O(α2m4

t ) corrections [ 10, 11] are included, as well,while those of O(α2m2

t ) are presently unknown. Theleading QCD effects of O(ααsm

2t ) [ 19] and all sublead-

ing O(ααs) corrections [ 20] are incorporated into ρband κb, but not the O(αα2

sm2t ) contribution which is

presently available only for nonsinglet diagrams [ 21].In Eq. (6), δQED are the O(α) and O(α2) QED

corrections. δNSQCD are the universal QCD corrections

up to O(α3s) which include quark mass dependent

contributions due to double-bubble type diagrams [22, 23]. δSQCD are the singlet contributions to theaxial-vector and vector partial widths which start,respectively, atO(α2

s) andO(α3s), and induce relatively

large family universal but flavor non-universal mt

effects [ 23, 24]. The corrections appearing in thesecond line of Eq. (6) are evaluated in lep100.f.The dominant massless contribution to δNS

QCD canbe obtained by analytical continuation of the AdlerD-function, which (in the MS scheme) has a very wellbehaved perturbative expansion ∼ 1 +

∑

i=0 diai+1s in

as = αs(MZ)/π (see the Appendix for details). Theprocess of analytical continuation from the Euclideanto the physical region induces further terms whichare proportional to β-function coefficients, enhancedby powers of π2, and start at O(α3

s). Fortunately,these terms [ 25] involve only known coefficients up toO(α5

s), and the only unknown coefficient in O(α6s) is

proportional to the four-loop Adler function coefficient,d3. In the massless approximation,

δNSQCD ≈ as + 1.4092a2s − (0.681 + 12.086) a3s +

(d3 − 89.19) a4s + (d4 + 79.7) a5s +

(d5 − 121d3 + 3316) a6s, (7)

and terms of order a7s ∼ 10−10 are clearly negligible.Notice, that the O(α6

s) term effectively reduces thesensitivity to d3 by about 18%. Eq. (7) amounts to areorganization of the perturbative series in terms of thedi times some function of αs; a similar idea is routinelyapplied to the perturbative QCD contribution to τdecays [ 26].Final state fermion mass effects [ 22, 27] of O(m2

f )

(and O(m4b) for b quarks) are best evaluated by

expanding in m2q(MZ) thus avoiding large logarithms

in the quark masses. The singlet contribution ofO(α2

sm2b) is also included.

The dominant theoretical uncertainty in the Z line-shape determination of αs originates from the masslessquark contribution, and amounts to about ±0.0004as estimated in the Appendix. There are severalfurther uncertainties, all of O(10−4): from the O(α4

s)heavy top quark contribution to the axial-vector partof δSQCD; from the missing O(αα2

sm2t ) and O(α2m2

t )

contributions to the Zbb-vertex; from further non-enhanced but cohering O(αα2

s)-vertex corrections; andfrom possible contributions of non-perturbative origin.The total theory uncertainty is therefore,

∆αs(MZ) = ±0.0005, (8)

which can be neglected compared to the current ex-perimental error. If mb is kept fixed in a fit, then itsparametric error would add an uncertainty of ±0.0002,but this would not change the total uncertainty (8).Polarization asymmetries are (in some cases up to

a trivial factor 3/4 or a sign) given by the asymmetryparameters,

Af =1− 4|Qf |Re(κf )s2

1− 4|Qf |Re(κf )s2 + 8Q2f s

4|κf |2, (9)

and the forward-backward asymmetries by,

AFB(f) =3

4AeAf . (10)

The hadronic charge asymmetry, QFB, is the linearcombination,

QFB =∑

q=d,s,b

RqAFB(q)−∑

q=u,c

RqAFB(q), (11)

and the hadronic peak cross section, σhad, is stored insigmah, and defined by,

σhad =12πΓe+e−Γhad

M2ZΓ

2Z

. (12)

Widths and asymmetries are stored in the fieldsgamma(f), alr(f), and afb(f). The fermion index,f, and the partial width ratios, R(f), are defined inTable 1.

33

Table 1Some of the variables used in lep100.f. Γinv andΓhad are the invisible and hadronic decays widths,respectively.

0 ν gamma(0) = Γinv alr(0) = 11 e R(1) = Γhad/Γe+e− alr(1) = Ae

2 µ R(2) = Γhad/Γµ+µ− alr(2) = Aµ

3 τ R(3) = Γhad/Γτ+τ− alr(3) = Aτ

4 u R(4) = Γuu/Γhad —5 c R(5) = Γcc/Γhad alr(5) = Ac

6 t R(6) = 0 —7 d R(7) = Γdd/Γhad —8 s R(8) = Γss/Γhad alr(8) = As

9 b R(9) = Γbb/Γhad alr(9) = Ab

10 had gamma(10) = Γhad afb(10) = QFB

11 all gamma(11) = ΓZ —

2.4. Fermion masses

I use MS masses as far as QCD is concerned,but retain on-shell masses for QED since renormaloneffects are unimportant in this case. This results ina hybrid definition for quarks. Accordingly, the RGrunning of the masses to scales µ 6= mq uses pureQCD anomalous dimensions. The running massescorrespond to the functions msrun(µ), mcrun(µ), etc.which are calculated in masses.f to three-loop order.Anomalous dimensions are also available at four-looporder [ 28], but can safely be neglected. Also neededis the RG evolution of αs which is implemented tofour-loop precision [ 29] in alfas.f.I avoid pole masses for the five light quarks through-

out. Due to renormalon effects, these can be deter-mined only up to O(ΛQCD) and would therefore inducean irreducible uncertainty of about 0.5 GeV. In fact,perturbative expansions involving the pole mass showunsatisfactory convergence. In contrast, the MS massis a short distance mass which can, in principle, bedetermined to arbitrary precision, and perturbativeexpansions are well behaved with coefficients of orderunity (times group theoretical factors which grow onlygeometrically). Note, however, that the coefficients ofexpansions involving large powers of the mass, mn, arerather expected to be of O(n). This applies, e.g., todecays of heavy quarks (n = 5) and to higher orders inlight quark mass expansions.The top quark pole mass enters the analysis

when the results on mt from on-shell produced topquarks at the Tevatron are included. In subroutinepolemasses(nf,mpole) mq(mq) is converted to thequark pole mass, mpole, using the two-loop pertur-bative relation from Ref. [ 30]. The exact three-loopresult [ 31] has been approximated (for mt) by em-

ploying the BLM [ 32] scale for the conversion. Sincethe pole mass is involved it is not surprising thatthe coefficients are growing rapidly. The third ordercontribution is 31%, 75%, and 145% of the secondorder for mt (nf = 6), mb (nf = 5), and mc (nf = 4),respectively. I take the three-loop contribution tothe top quark pole mass of about 0.5 GeV as thetheoretical uncertainty, but this is currently negligiblerelative to the experimental error. At a high energylepton collider it will be possible to extract the MStop quark mass directly and to abandon quark polemasses altogether.

2.5. ν scattering

The ratios of neutral-to-charged current cross sec-tions,

Rν =σNCνN

σCCνN

, Rν =σNCνN

σCCνN

, (13)

have been measured precisely in deep inelastic ν(ν) hadron scattering (DIS) at CERN (CDHS andCHARM) and Fermilab (CCFR). The most preciseresult was obtained by the NuTeV Collaboration atFermilab who determined the Paschos-Wolfenstein ra-tio,

R− =σNCνN − σNC

νN

σCCνN − σCC

νN

∼ Rν − rRν , (14)

with r = σCCνN /σ

CCνN . Results on Rν are frequently

quoted in terms of the on-shell weak mixing angle(or MW ) as this incidentally gives a fair descriptionof the dependences on mt and MH . One can writeapproximately,

Rν = g2L + g2Rr, Rν = g2L +g2Rr, R− = g2L − g2R, (15)

where,

g2L =1

2− sin2 θW +

5

9sin4 θW , g2R =

5

9sin4 θW . (16)

However, the study of new physics requires the im-plementation of the actual linear combinations ofeffective four-Fermi operator coefficients, ǫL,R(u) andǫL,R(d), which have been measured. With the ap-propriate value for the average momentum trans-fer, q2, as input, these are computed in the sub-routines nuh(q2,epsu L,epsd L,epsu R,epsd R) (ac-cording to Ref. [ 33]), nuhnutev, nuhccfr, andnuhcdhs, all contained in file dis.f. Note, thatthe CHARM results have been adjusted to CDHSconditions [ 34]. While the experimental correlationsbetween the various DIS experiments are believed tobe negligible, large correlations are introduced by thephysics model through charm mass threshold effects,

34

quark sea effects, radiative corrections, etc. I con-structed the matrix of correlation coefficients using theanalysis in Ref. [ 34],

R− Rν Rν Rν Rν Rν Rν

1.00 0.10 0.10 0.10 0.00 0.00 0.000.10 1.00 0.40 0.40 0.10 0.10 0.100.10 0.40 1.00 0.40 0.10 0.10 0.100.10 0.40 0.40 1.00 0.10 0.10 0.100.00 0.10 0.10 0.10 1.00 0.15 0.150.00 0.10 0.10 0.10 0.15 1.00 0.150.00 0.10 0.10 0.10 0.15 0.15 1.00

. (17)

The effective vector and axial-vector couplings, gνeVand gνeA , from elastic νe scattering are calculated insubroutine nue(q2,gvnue,ganue) in file nue.f. Themomentum transfer, q2, is currently set to zero [36]. Needed is the low energy ρ parameter, rhonc,which describes radiative corrections to the neutral-to-charged current interaction strengths. Togetherwith sin2t0 (described below) it is computed in filelowenergy.f.

2.6. Low energy observables

The weak atomic charge, Qw, from atomic parityviolation and fixed target ep scattering is computedin subroutine apv(Qw,Z,AA,C1u,C1d,C2u,C2d) whereZ and AA are, respectively, the atomic number andweight. Also returned are the coefficients from lepton-quark effective four-Fermi interactions which are cal-culated according to [ 37].These observables are sensitive to the low energy

mixing angle, sin2t0, which defines the electroweakcounterpart to the fine structure constant and is similarto the one introduced in Ref. [ 7]. There is significantcorrelation between the hadronic uncertainties fromthe RG evolutions of α and the weak mixing angle.Presently, this correlation is ignored, but with therecent progress in atomic parity violation experimentsit should be accounted for in the future.An additional source of hadronic uncertainty is in-

troduced by γZ-box diagrams which are unsuppressedat low energies. At present, this uncertainty can beneglected relative to the experimental precision.Besides apv, the file pnc.f contains in addition

the subroutine moller for the anticipated polarizedfixed target Møller scattering experiment at SLAC.Radiative corrections are included following Ref. [ 38].

2.7. b→ sγSubroutine bsgamma returns the decay ratio,

R =B(b→ sγ)

B(b→ ceν). (18)

It is given by [ 39, 40],

R =6α

π

∣

∣

∣

∣

V ∗tsVtbVcb

∣

∣

∣

∣

2S

f(z)

|D|2 +A/S + δNP + δEW

(1 + δSLNP )(1 + δSL

EW ), (19)

where |V ∗tsVtb/Vcb|2 = 0.950 is a combination of

Cabbibo-Kobayashi-Maskawa matrix elements and Sis the Sudakov factor [ 41]. δNP and δEW are non-perturbative and NLO electroweak [ 42] corrections,both for the b → sγ and the semileptonic (b → ceν)decay rates.

D = C07 +

αs(mb)

4π(C1

7 + V ), (20)

is called the reduced amplitude for the process b→ sγ,and is given in terms of the Wilson coefficient C7

at NLO. C7 and the other Ci appearing below areeffective Wilson coefficients with NLO RG evolution [43] from the weak scale to µ = mb understood. TheNLO matching conditions at the weak scale have beencalculated in Ref. [ 44]. D includes the virtual gluoncorrections,

V = r2C02 + r7C

07 + r8C

08 , (21)

so that it squares to a positive definite branchingfraction. On the other hand, the amplitude for gluonBremsstrahlung (b→ sγg),

A = αs(mb)π [C0

2 (C08f28(1) + C0

7f27(1) + C02f22(1))+

C08 (C

08f88(δ) + C0

7f78(δ)) + (C07 )

2f77(δ)], (22)

is added linearly to the cross section. The Wilsoncoefficient C0

2 is defined as in Ref. [ 45]. It enters onlyat NLO, is significantly larger than C0

7 , and dominatesthe NLO contributions. The parameter 0 ≤ δ ≤ 1 inthe coefficient functions fij characterizes the minimumphoton energy and has been set to δ = 0.9 [ 41],except for the first line in Eq. (22) where δ = 1.0corresponding to the full cross section. The f2i arecomplicated integrals which can be solved in terms ofpolylogarithms up to 5th order. In the code I usean expansion in z = m2

c/m2b and δ = 1.0. Once

experiments become more precise the correction toδ = 0.9 should be included.f(z) is the phase space factor for the semileptonic

decay rate including NLO corrections [ 46]. I definedthe MS mass ratio in z = [mc(mb)/mb(mb)]

2 at thecommon scale, µ = mb, which I also assumed for thefactor m5

b multiplying the decay widths. Since I do notre-expand the denominator this effects the phase spacefunction at higher orders. Using the O(α2

s)-estimate2

2I computed the O(α2s) coefficient for comparison only, and did

not include it in the code.

35

from Ref. [ 47], I obtain for the semileptonic decaywidth,

ΓSL ∼ m5bf0(z)

[

1 + 2.7αs(mb)

π− 1.6

(

αs

π

)2]

, (23)

where f0(z) is the leading order phase space factor. Itis amusing that the coefficients in Eq. (23) are com-fortably (and perhaps somewhat fortuitously) small,with the O(a2s)-coefficient even smaller than the onein Ref. [ 47] where a low scale running mass had beenadvocated. Moreover, using the pre-factor m5

b in thenumerator of R reduces the size of r7 in Eq. (21) andtherefore the coefficient κ(δ) = f77(δ) + r7/2 which

multiplies the term as(C0,eff7 )2. I obtain −2.1 < κ(δ) <

1.4, while with the pole mass pre-factorM5b one would

have −8.7 ≤ κ(δ) < −5.3.

Acknowledgements

It is a pleasure to thank Paul Langacker and DamienPierce for collaborations on precision analyses. I amgrateful to Francesca Borzumati and Paolo Gambinofor providing me with parts of their FORTRAN codes.

A. Uncertainties from perturbative QCD

Writing the perturbative expansion of some quantityin its general form for an arbitrary gauge group, it caneasily be decomposed into separately gauge invariantparts. Table 2 shows for some (related) examples thatafter removing the group theoretical pre-factors, allcoefficients, yi, are strictly of order unity, and thattheir mean, y, is very close to zero. In particular,there is no sign of factorial growth of coefficients.These observations offer a valuable tool to estimatethe uncertainties associated with the truncation of theloop expansion, so I would like to make them moreprecise.Assume (for simplicity) that the yi are random draws

from some normal distribution with unknown mean,µ, and variance, σ2. One can show that the marginaldistribution of µ follows a Student-t distribution withn− 1 degrees of freedom, tn−1, centered about y, andwith standard deviation,

∆µ =

√

∑

i(yi − y)2

n(n− 3). (24)

As can be seen from the Table, µ is consistent with zeroin all cases, justifying the nullification of the unknowncoefficients from higher loops. I next assert that thedistribution of σ, conditional on µ = 0, follows a scaledinverse-χ2 distribution with n degrees of freedom, from

which I obtain the estimate,

σ = σ0 ±∆σ =

√

∑

i y2i

n− 2

[

1±√

1

2(n− 4)

]

. (25)

Inspection of the Table shows indeed that σ, as thetypical size of a coefficient, is estimated to be <

∼ O(1).I now focus on the partial hadronic Z decay width.

As discussed in Section 2.3, the O(α3s) term, d2, is

much smaller than the π2 term arising from analyticalcontinuation. This is specifically true for the relevantcase of nf = 5 active flavors, where large cancellationsoccur between gluonic and fermionic loops. Notice,that the D-function, in contrast to Rhad, has oppositesigns in the leading terms proportional to C2

ACF andCACFTFnF . Indeed, the Adler D-function and theβ-function have similar structures regarding the signsand sizes of the various terms (see Table 2), and wedo expect large cancellations in the β-function. Thereason is that it has to vanish identically in the case ofN = 4 supersymmetry. Ignoring scalar contributionsthis case can be mimicked by setting TFnf = 2CA

(there are 2 Dirac fermions in the N = 4 gaugemultiplet) or nf = 12 for QCD, which is of the rightorder. In fact, all known QCD β-function coefficientsbecome very small for some value of nf between 6 and16. We therefore have a reason to expect that similarcancellations will reoccur in the di at higher orders. Asa 1σ error estimate for d3, I suggest to use the largestknown coefficient (3 × 0.71) times the largest grouptheoretical pre-factor in the next order (C3

ACF ) whichresults in

d3 = 0± 77. (26)

With Eq. (7) and αs(MZ) = 0.120 one can absorb allhigher order effects into the O(α4

s)-coefficient of Rhad,reff3 = −81 ± 63. This shifts the extracted αs fromthe Z line-shape by +0.0005 and introduces the smalluncertainty of ±0.0004.The argument given above does certainly not apply

to the quenched case, nf = 0, and indeed d2(nf = 0) isabout −73% of the π2 term, i.e., large and positive. Inthe case of nf = 3, which is of interest for the precisiondetermination of αs from τ decays, d2 is about −38%of the π2 term. If one assumes that the same is trueof d3, one would obtain d3(nf = 3) = 60. Estimatesbased on the principles of minimal sensitivity, PMS,or fastest apparent convergence, FAC, yield d3(nf =3) = 27.5 [ 25] so there might be some indications for apositive d3(nf = 3). In any case, all these estimates liewithin the uncertainty in Eq. (26) and we will have toawait the proper calculation of the O(α4

s)-coefficient totest these hypotheses. Note, that the current τ decayanalysis by the ALEPH Collaboration uses d3 = 50±50 [ 50] which is more optimistic.

36

The analogous error estimate for the five-loop β-function coefficient yields,

β4 = 0± 579. (27)

To get an estimate for the uncertainty in the RGrunning of αs, I translate Eq. (27) into

β3 = β3 ±αs(µ0)

πβ4, (28)

where µ0 is taken to be the lowest scale in-volved. This overestimates the uncertainty from β4,thereby compensating for other neglected terms ofO(αn+4

s lnn µ2/µ20). For the RG evolution from µ = mτ

to µ = MZ this yields an uncertainty of ∆αs(MZ) =±0.0005. Conversely, for fixed αs(MZ) = 0.120, Iobtain αs(mb) = 0.2313± 0.0006, αs(mτ ) = 0.3355±0.0045, and αs(mc) = 0.403±0.011, where I have usedmb = 4.24 GeV and mc = 1.31 GeV. For comparison,the ALEPH Collaboration quotes an evolution errorof ∆αs(MZ) = ±0.0010 which is twice as large. Iemphasize that it is important to adhere to consistentstandards when errors are estimated. This is especiallytrue in the context of a global analysis where theprecisions of the observables enter as their relativeweights.

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Table 2Coefficients (MS ) appearing in the β-function of asimple group [ 29]; in non-Abelian corrections to theQED β-function (denoted D) [ 3, 48]; in the AdlerD-function [ 49] (rescaled by an overall factor 1/3);and in Rhad (analytical continuation of D). The firstfour segments correspond, respectively, to the first fourloop orders of non-singlet type. The fifth segment isthe singlet (double triangle) contribution in O(α3

s). InD, D, and Rhad, an overall factor αMZ and the sumsinvolving charges or Z couplings have been dropped.The completely symmetrical tensors of rank four, dAand dF , as well as TF when appearing in parenthesis,apply to the β-function only. Each TF is understood tobe multiplied by the number of flavors, nf , except forthe singlet term involving the symmetrical structureconstants, d.

group factor β D D Rhad

CA 0.92 — — —(TF ) −0.33 −0.33 0.33 0.33C2

A 0.71 — — —CATF −0.42 — — —CF (TF ) −0.25 −0.25 0.25 0.25C3

A 0.83 — — —C2

ATF −0.82 — — —CACF (TF ) −0.36 −0.23 0.18 0.18CAT

2F 0.09 — — —

C2F (TF ) 0.03 0.03 −0.03 −0.03

CFTF (TF ) 0.08 0.08 −0.06 −0.06C4

A 1.19 — — —C3

ATF −1.67 — — —C2

ACF (TF ) −0.23 −0.28 0.32 −0.38C2

AT2F 0.50 — — —

CAC2F (TF ) −0.42 −0.07 0.51 0.51

CACFTF (TF ) 0.51 0.42 −0.71 −0.21CAT

3F 0.01 — — —

C3F (TF ) 0.18 0.18 −0.18 −0.18

C2FTF (TF ) −0.17 −0.17 0.02 0.02

CFT2F (TF ) 0.02 0.02 0.09 −0.01

d2A/NA 1.07 — — —dAdF /NA −2.38 — — —d2F /NA (T 2

Fd2/4) 0.50 0.50 −0.50 −0.50

y −0.02 −0.01 0.02 −0.01∆µ 0.17 0.09 0.11 0.09σ0 0.83 0.28 0.37 0.31∆σ 0.13 0.07 0.09 0.08

38