Fermilab-Conf-96/423UB-HET-96-06

Precision Electroweak Physics at Future Collider Experiments1

U. Baur2

Physics Department, SUNY Buffalo, Buffalo, NY 14260, USA

M. Demarteau2

Fermi National Accelerator Laboratory, P.O. Box 500, Batavia, IL 60510, USA

Working Group Members

C. Balazs(MSU), D. Errede(Urbana), S. Errede(Urbana), T. Han(Davis),

S. Keller(FNAL), Y-K. Kim (Berkeley), A.V. Kotwal (Columbia), F. Merritt (Chicago),

S. Rajagopalan(Stony Brook), R. Sobey(Davis), M. Swartz(SLAC),

D. Wackeroth(FNAL), J. Womersley(FNAL)

Abstract

We present an overview of the present status and prospects for progress in electroweak mea-surements at future collider experiments leading to precision tests of the Standard Model ofElectroweak Interactions. Special attention is paid to the measurement of theW mass, theeffective weak mixing angle, and the determination of the top quark mass. Their constraintson the Higgs boson mass are discussed.

1To appear in theProceedings of the 1996 DPF/DPB Summer Study on New Directions for High-Energy Physics(Snowmass 96), Snowmass, Colorado, June 25 – July 12, 1996Work supported in part by the U.S. Dept. of Energy under contract DE-AC02-76CHO3000 and NSF grant PHY9600770

2Co-convener

Precision Electroweak Physics at Future Collider Experimentsz

U. Baurx

Physics Department, SUNY Buffalo, Buffalo, NY 14260

M. Demarteaux

Fermi National Accelerator Laboratory, P.O. Box 500, Batavia, IL 60510

Working Group Members

C. Balazs(MSU), D. Errede(Urbana), S. Errede(Urbana), T. Han(Davis), S. Keller(FNAL),

Y-K. Kim (Berkeley), A.V. Kotwal (Columbia), F. Merritt (Chicago), S. Rajagopalan(Stony Brook),

R. Sobey(Davis), M. Swartz(SLAC), D. Wackeroth(FNAL), J. Womersley(FNAL)

ABSTRACT

We present an overview of the present status and prospectsfor progress in electroweak measurements at future collider ex-periments leading to precision tests of the Standard Model ofElectroweak Interactions. Special attention is paid to the mea-surement of theW mass, the effective weak mixing angle, andthe determination of the top quark mass. Their constraints onthe Higgs boson mass are discussed.

I. INTRODUCTION

The Standard Model (SM) of strong and electroweak interac-tions, based on the gauge groupSU(3)C � SU(2)L � U(1)Y ,has been extremely successful phenomenologically. It has pro-vided the theoretical framework for the description of a veryrich phenomenology spanning a wide range of energies, fromthe atomic scale up to theZ boson mass,MZ . It is being testedat the level of a few tenths of a percent, both at very low en-ergies and at high energies [1], and has correctly predicted therange of the top quark mass from loop corrections. However,the SM has a number of shortcomings. In particular, it does notexplain the origin of mass, the observed hierarchical pattern offermion masses, and why there are three generations of quarksand leptons. It is widely believed that at high energies, or invery high precision measurements, deviations from the SM willappear, signaling the presence of new physics.

In this report we discuss the prospects for precision tests ofthe Standard Model at future collider experiments, focussing onelectroweak measurements. The goal of these measurementsis to confront the SM predictions with experiment, and to de-rive indirect information on the mass of the Higgs boson. Theexistence of at least one Higgs boson is a direct consequenceof spontaneous symmetry breaking, the mechanism which is

zWork supported in part by the U.S. Dept. of Energy under contract DE-AC02-76CHO3000 and NSF grant PHY9600770

xCo-convener

responsible for generating mass of theW andZ bosons, andfermions in the SM. In Section II we identify some of the rel-evant parameters for precision electroweak measurements, andreview the present experimental situation. Expectations fromfuture collider experiments are discussed in Section III. We con-clude with a summary of our results.

II. CONSTRAINTS ON THE STANDARDMODEL FROM PRESENT ELECTROWEAK

MEASUREMENTS

There are three fundamental parameters measured with highprecision which play an important role as input variablesin Electroweak Physics. The fine structure constant,� =

1=137:0359895 is known with a precision of�� = 0:045 ppm.The muon decay constant,G� = 1:16639 � 10�5 GeV�2 ismeasured with�G� = 17 ppm from muon decay [2]. Finally,theZ boson mass,MZ = 91:1863GeV/c2 [1] is measured with�MZ = 22 ppm in experiments at LEP and SLC. Knowingthese three parameters, one can evaluate theW mass,MW , andthe weak mixing angle,sin2 �W , at tree level. When loop cor-rections are taken into account,MW andsin2 �W also dependon the top quark mass,Mt, and the Higgs boson mass,MH .The two parameters depend quadratically onMt, and logarith-mically onMH .

If theW mass and the top quark mass are precisely measured,information on the mass of the Higgs boson can be extracted.Constraints on the Higgs boson mass can also be obtained fromthe effective weak mixing angle andMt. The ultimate test of theSM may lie in the comparison of these indirect determinationsof MH with its direct observation at future colliders.

The mass of the top quark is presently determined by the CDFand DØ collaborations from�tt production at the Tevatron in thedi-lepton, the lepton plus jets, and the all hadronic channels [3].The combined value of the top quark mass from the lepton +

1

80

80.1

80.2

80.3

80.4

80.5

80.6

130 140 150 160 170 180 190 200

100

300

1000

Higgs Mass (GeV/c

2 )

Mt (GeV/c2)

MW

(G

eV/c

2 )

CDF+DØ (Summer 1996)

Indirect Measurements

LEP + SLD (Spring ´ 96)

CDF: MW = 80.41 ± 0.18 GeV/c2

Mt = 176.8 ± 6.5 GeV/c2

World Average

DØ: MW = 80.37 ± 0.15 GeV/c2

Mt = 169 ± 11 GeV/c2

Figure 1: Comparison of the top quark andW boson massesfrom current direct and indirect measurements with the SM pre-diction.

jets channel, which yields the most precise result, is

Mt = 175� 6 GeV=c2: (1)

TheW boson mass has been measured precisely by UA2,CDF, and DØ. Currently, the most accurate determination ofMW comes from the Tevatron CDF and DØ Run Ia analyses [4]and a preliminary DØ measurement [5] based on data taken dur-ing Run Ib. The current world average is [1]

MW = 80:356� 0:125 GeV=c2: (2)

Figure 1 compares the results of the currentMW andMt mea-surements in the(Mt;MW ) plane with those from indirect mea-surements at LEP and SLC [1], and the SM prediction for dif-ferent Higgs boson masses. The cross hatched bands show theSM prediction for the indicated Higgs boson masses. The widthof the bands is due primarily to the uncertainty on the electro-magnetic coupling constant at theZ mass scale,�(M2

Z), whichhas been taken to be��1(M2

Z) = 128:89� 0:10. Recent esti-mates give��(M2

Z) � 0:0004�0:0007 [6], which correspondsto ���1(M2

Z) � 0:05� 0:09.The uncertainty on�(M2

Z) is dominated by the error on thehadronic contribution to the QED vacuum polarization whichoriginates from the experimental error on the cross section fore+e� ! hadrons. Using dispersion relations [7], the hadroniccontribution to�(M2

Z) can be related to the cross section of theprocesse+e� ! hadrons via

��had(M2Z) =

�M2Z

3�PZ 1

4m2�

Rhad(s0)

s0(s0 �M2Z)

ds0 ; (3)

whereP denotes the principal value of the integral, and

Rhad =�(e+e� ! hadrons)

�(e+e� ! �+��): (4)

> 12.GeV

7 - 12 GeV 5 - 7 GeV

2.5 - 5 GeV

1.05 - 2.5 GeV

narrow resonancesρ

> 12.GeV

7 - 12 GeV

5 - 7 GeV

2.5 - 5 GeV

1.05 - 2.5 GeV

narrow re

sonance

s

ρ

contributions at MZ

in magnitude

in uncertainty

Figure 2: Relative contributions to��had(M2Z) in magnitude

and uncertainty.

The relative contributions to��had(M2Z) and the uncertainty

are detailed in Fig. 2 [6]. About 60% of the uncertainty comesfrom the energy region between 1.05 GeV and 5 GeV. Moreprecise measurements of the total hadronic cross section in thisenergy region, for example at Novosibirsk, DAP�NE or BESmay reduce the uncertainty on�(M2

Z) by about a factor 2 in thenear future.

TheW mass can also be determined indirectly from radia-tive corrections to electroweak observables at LEP and SLD,and from�N scattering experiments. The current indirect valueof MW obtained frome+e� experiments,MW = 80:337 �0:041+0:010

�0:021 GeV/c2 [1], is in excellent agreement with the re-sult obtained from direct measurements (see Fig. 1). The de-termination ofMW from �N scattering will be discussed inSection III.C.

The effective weak mixing angle,sin2 �lepteff , has been deter-mined with high precision from measurements of the forwardbackward asymmetries at LEP, and the left-right asymmetries

2

0.2305

0.231

0.2315

0.232

0.2325

0.233

79.9 80.1 80.3 80.5 80.7

LEP/SLC/CDF/D0/νN July 1996α(Mz)=1/128.89

Mt =175 ± 6 GeV

∆α

MW (GeV)

sin2θefflept

STANDARD MODEL

Mt =

220

Mt =

120

MH

= 1000

300

60

68% C.L.

99% C.L.

1σ

Figure 3: Comparison ofsin2 �lepteff and theW boson mass fromcurrent direct and indirect measurements with the SM predic-tion. The top quark and Higgs boson masses indicated in thefigure are all in GeV/c2.

at the SLC [1]. Here,sin2 �lepteff is defined by

sin2 �lepteff =1

4

�1� gV `

gA`

�; (5)

wheregV ` andgA` are the effective vector and axial vector cou-pling constants of the leptons to theZ boson, and is related tothe weak mixing angle in theMS scheme,sin2 �W (MZ), by [8]

sin2 �lepteff � sin2 �W (MZ) + 0:00028: (6)

A fit to the combined LEP and SLD asymmetry data yields

sin2 �lepteff = 0:23165� 0:00024: (7)

The experimental constraints in the(sin2 �lepteff ;MW ) plane arecompared with the SM predictions in Fig. 3. The measuredvalue of sin2 �lepteff agrees well with the SM expectation. Thestar in the lower lefthand corner of Fig. 3 indicates theW massand effective weak mixing angle predicted by taking the run-ning of � into account only. The arrow represents the currentuncertainty onMW and the effective weak mixing angle from��had(M

2Z):

� sin2 �lepteff

����

= 0:00023; (8)

�MW j�� = 12 MeV=c2: (9)

The estimated theoretical error from higher orders introduces anadditional uncertainty of [9]

� sin2 �lepteff

��th= 0:00008; (10)

140

160

180

200

102

103

MH [GeV]

Mt

[GeV

]

Preliminary

LEP Data onlyAll Data

Figure 4: The 68% confidence level contours inMt andMH

for the fits to LEP data only (dashed curve) and to all data (solidcurve).

�MW jth = 9 MeV=c2: (11)

While direct measurements ofMt andMW presently do notimpose any constraints on the Higgs boson mass, indirect mea-surements from LEP and SLD seem to indicate a preference fora relatively light Higgs boson. The 68% confidence level con-tours in theMt andMH plane for the fits to LEP data only,and to all data sets [1] (LEP, SLD, CDF and DØ), are shown inFig. 4. Taking the theoretical error due to missing higher ordercorrections into account, one obtains

MH = 149+148�82 GeV=c

2; (12)

orMH < 550 GeV=c

2at 95% CL: (13)

The results of such a fit from current data, however, shouldbe interpreted with caution. Removing one or two quantitiesfrom the fit can drastically change the predicted Higgs bosonmass range. Excluding from the fit the hadronic width of theZboson, which depends on�s, results in [10]

MH = (560� 1:5�1) GeV=c2: (14)

Omitting in addition the SLD data onALR which yield a some-what low value for the effective weak mixing angle, leads toMH = (820� 1:7�1) GeV/c2.

3

, y g pmeasurements from LEP data are expected since LEP data tak-ing at theZ peak has ceased. However, a significant reductionof the errors onMt andMW from direct experiments at LEP2,the Tevatron (Run I, Run II and TeV33), the LHC, and perhapsthe NLC and/or a�+�� collider is expected, which should re-sult in a more stable prediction forMH . This will be discussedin more detail in the next Section.

79.8

80

80.2

80.4

80.6

80.8

81

130 140 150 160 170 180 190 200

100

300

1000

Higgs Mass (GeV/c2 )

Mt (GeV/c2)

MW

(G

eV/c

2 )

MSSM Modification M W - Mt relation

Indirect MeasurementsLEP + SLD (Spring ´ 96)

CDF

CDF: MW = 80.41 ± 0.18 GeV/c2

Mt = 176.8 ± 6.5 GeV/c2

DØ:

DØ: MW = 80.37 ± 0.15 GeV/c2

Mt = 169 ± 11 GeV/c2

Figure 5: Predictions forMW as a function ofMt in the SM(shaded bands) and in the MSSM (area between the dot-dashedlines). The results from direct CDF and DØ measurements, andfrom indirect measurements at LEP and SLD are also shown.

Precise measurements ofMW andMt, if inconsistent with therange allowed by the SM, could indicate the existence of newphenomena at or above the electroweak scale, such as super-symmetry. In the near future direct and indirect measurementsof the top quark andW boson mass are expected to begin toyield useful constraints on the parameter space of the minimalsupersymmetric standard model (MSSM). This is illustrated inFig. 5, where the predictions forMW as a function ofMt in theSM (shaded bands) and in the MSSM (area between the dashedlines) are shown, together with results from direct CDF and DØmeasurements, and indirect measurements from LEP and SLD.The MSSM band has been obtained by varying the model pa-rameters so that they are consistent with current experimentaldata. In addition, it was assumed that no supersymmetric parti-cles are found at LEP2 [11].

III. HIGH PRECISION ELECTROWEAKPHYSICS AT FUTURE COLLIDERS

A. Measurement of the Top Quark Mass

The prospects of measuring the top quark mass in future col-lider experiments are discussed in detail in Ref. [12]. We there-fore only briefly summarize the results here.

, p y t

(RLdt = 2 fb�1) and for TeV33 (

RLdt = 10 � 30 fb�1) canbe extrapolated using current and anticipated CDF and DØ ac-ceptances and efficiencies, together with theoretical predictions.Using various different methods and techniques [13], one ex-pects thatMt can be determined to� 4GeV/c2 (� 2GeV/c2) inRun II (TeV33). The uncertainty on the top quark mass will bedominated by systematic errors. Soft and hard gluon radiation,and the jet transverse energy scale constitute the most importantsources of systematic errors in the top quark mass measurementat hadron colliders. At the LHC, one also expects a precision ofabout 2 GeV/c2 for Mt [12].

At an e+e� Linear Collider (NLC) or a�+�� collider, thetop quark mass can be determined with very high precisionfrom a threshold scan. For an integrated luminosity of 10 fb�1

(50 fb�1), the expected uncertainty onMt at the NLC is�Mt �500 MeV/c2 (200 MeV/c2) [14]. At a �+�� collider, the re-duced beamstrahlung and initial state radiation result in a betterbeam energy resolution which should make it possible to mea-sure the top quark mass with a somewhat higher precision thanat the NLC, for equal integrated luminosities. Simulations sug-gest�Mt � 300 MeV/c2 for 10 fb�1 [15].

The precision which can be achieved forMt at different col-liders is summarized in Table I. In our subsequent calculations

Table I: Expected top quark mass precision at future colliders.

Collider �Mt

Tevatron (2 fb�1) 4 GeV/c2

TeV33 (10 fb�1) 2 GeV/c2

LHC (10 fb�1) 2 GeV/c2

NLC (10 fb�1) 0.5 GeV/c2

�+�� (10 fb�1) 0.3 GeV/c2

we shall always assume that the top quark mass can be deter-mined with a precision of

�Mt = 2 GeV=c2: (15)

B. Measurement ofsin2�lepteff

1. SLD

Presently, the single most precise determination of the effec-tive weak mixing angle originates from the measurement of theleft-right asymmetry,

ALR =�L � �R

�tot(16)

at SLD. Here,�L(R) is the total production cross section for left-handed (righthanded) electrons. In the SM, the left-right asym-metry at theZ pole, ignoring photon exchange contributions, isrelated to the effective weak mixing angle by

ALR =2 (1� 4 sin2 �lepteff )

1 + (1� 4 sin2 �lepteff )2: (17)

4

p y pg [ ] ( )alized, it will be possible to collect3 � 106 Z decays over aperiod of three to four years at SLD. This should result in anuncertainty of

� sin2 �lepteff = 0:00012; (18)

which is approximately a factor 2 better than the current uncer-tainty from the fit to the combined LEP and SLD asymmetrydata (see Eq. (7)).

Further improvements could come from measurements of theleft-right forward-backward asymmetry ine+e� ! �ff ,

~AfFB(z) =

[�fL(z)� �fL(�z)]� [�fR(z)� �fR(�z)][�fL(z) + �fL(�z)] + [�fR(z) + �fR(�z)]

=2gV fgAf

g2V f + g2Af

2z

1 + z2; (19)

wherez = cos �, and� is the scattering angle.~AfFB directly

measures the coupling of the final state fermionf to theZ bo-son from which it is straightforward to determinesin2 �lepteff . Inparticular, with the self-calibrating jet-charge technique [17], aprecise measurement of theZ�bb coupling should be possible.

2. Hadron Colliders

At hadron colliders, the forward backward asymmetry,AFB ,in di-lepton production,p p

(�) ! `+`�X , (` = e; �), makes itpossible to measure the effective weak mixing angle.AFB isdefined by

AFB =F �B

F +B; (20)

where

F =

Z 1

0

d�

d cos ��d cos ��; (21)

B =

Z 0

�1

d�

d cos ��d cos ��; (22)

and cos �� is the angle between the lepton and the incomingquark in the`+`� rest frame. Inp�p collisions at Tevatron en-ergies, the flight direction of the incoming quark to a good ap-proximation coincides with the proton beam direction.cos ��

can then be related to the components of the lepton and anti-lepton four-momenta via [18]

cos �� = 2p+(`�)p�(`+)� p�(`�)p+(`+)

m(`+`�)pm2(`+`�) + p2T (`

+`�)(23)

with

p� =1p2(E � pz) : (24)

Here,m(`+`�) is the invariant mass of the lepton pair,E is theenergy, andpz is the longitudinal component of the momentumvector. In this definition ofcos ��, the polar axis is defined tobe the bisector of the proton beam momentum and the nega-tive of the anti-proton beam momentum when they are boostedinto the`+`� rest frame. The four-momenta of the quark andanti-quark cannot be determined individually. The definition of

q ( ) g gof the momentum ambiguity induced by the parton transversemomentum.

First measurements of the effective weak mixing angle usingthe forward backward asymmetry at hadron colliders have beenperformed by the UA1 and CDF collaborations [19, 20]. Fig-ure 6a shows the variation ofAFB with thee+e� invariant massin p�p! e+e� for

ps = 1:8 TeV, assumingsin2 �lepteff = 0:232.

The error bars indicate the statistical errors for 100,000 events,corresponding to an integrated luminosity of about 2 fb�1.The largest asymmetries occur at di-lepton invariant masses of

Figure 6: The forward backward asymmetry,AFB , as a func-tion of thee+e� invariant mass inp�p ! e+e� events. (a) sta-tistical error for 100,000 events, corresponding to an integratedluminosity of 2 fb�1 in an ideal detector; (b) including the ef-fects of the DØ di-electron mass resolution.

around 70 GeV/c2 and above 110 GeV/c2. A preliminary studyof the systematic errors, indicates that most sources of error aresmall compared with the statistical error. The main contributionto the systematic error originates from the uncertainty in theparton distribution functions. Since the vector and axial vec-tor couplings ofu andd quarks to theZ boson are different, themeasured asymmetry depends on the ratio ofu tod quarks in theproton. Most of the systematic errors are expected to scale with1=pN , whereN is the number of events. The effect of the elec-

tromagnetic calorimeter resolution is rather moderate, as shownin Fig. 6b. It is found that most of the sensitivity of this mea-surement tosin2 �lepteff is atm(e+e�) � MZ due to the strong

variation ofAFB with sin2 �lepteff and the high statistics in thisregion. Including QED radiative corrections, thep�p ! e+e�

forward backward asymmetry in theZ boson resonance region(75 GeV=c2 < m(e+e�) < 105 GeV=c

2) can be parameter-ized in terms of the effective weak mixing angle by [21]

AFB = 3:6 (0:2464� sin2 �lepteff ): (25)

The expected precision ofsin2 �lepteff in the electron channel (perexperiment) versus the integrated luminosity at the Tevatronis shown in Fig. 7, together with the combined current uncer-tainty from LEP and SLD experiments. A similar precision isexpected in the muon channel. Combining the results of the

5

Figure 7: Projected uncertainty (per experiment) insin2 �lepteff

from the measurement ofAFB in theZ pole region at the Teva-tron versus the integrated luminosity.

electron and the muon channel, an overall uncertainty per ex-periment of

� sin2 �lepteff = 0:00013 (26)

is expected for an integrated luminosity of 30 fb�1.At the LHC, the lowest orderZ ! `+`� cross section is

approximately 1.6 nb for each lepton flavor. For the projectedyearly integrated luminosity of 100 fb�1, this results in a verylarge number ofZ ! `+`� events which, in principle, couldbe utilized to measure the forward backward asymmetry andthus sin2 �lepteff with extremely high precision [22]. Since theoriginal quark direction is unknown inpp collisions, one has toextract the angle between the lepton and the quark in the`+`�

rest frame from the boost direction of the di-lepton system withrespect to the beam axis:

cos �� = 2jpz(`+`�)jpz(`+`�)

p+(`�)p�(`+)� p�(`�)p+(`+)

m(`+`�)pm2(`+`�) + p2T (`

+`�):

(27)in order to arrive at a non-zero forward-backward asymmetry.

In contrast to Tevatron energies, sea quark effects dominateat the LHC. As a result, the probability,fq, that the quark di-rection and the boost direction of the di-lepton system coincideis significantly smaller than one. This considerably reduces theforward backward asymmetry. Events with a large rapidity ofthe di-lepton system,y(`+`�), originate from collisions whereat least one of the partons carries a large fractionx of the pro-ton momentum. Since valence quarks dominate at high valuesof x, a cut on the di-lepton rapidity increasesfq , and thus the

y y [ ] y gangle.

Imposing ajy(�+��)j > 1 cut and including QED correc-tions, the forward backward asymmetry at the LHC in the�+��

channel in theZ peak region (75 GeV=c2< m(�+��) <

105 GeV=c2) can be parameterized by

AFB = 2:10 (0:2466� sin2 �lepteff ) (28)

for an ideal detector. For an integrated luminosity of 100 fb�1,this then leads to an expected error of

� sin2 �lepteff = 4:5� 10�5: (29)

A similar precision should be achievable in the electron channel.However, electrons and muons can only be detected for pseu-

dorapiditiesj�(`)j < 2:4� 3:0 in the currently planned config-urations of the ATLAS [24] and CMS [25] experiments at theLHC. The finite pseudorapidity range available dramatically re-duces the asymmetry. In the region around theZ pole, the asym-metry is again approximately a linear function ofsin2 �lepteff with(for �+�� final states)

AFB = 0:65 (0:2488� sin2 �lepteff ) for j�(�)j < 2:4: (30)

The finite rapidity coverage also results in a reduction of thetotalZ boson cross section by roughly a factor 5. As a result, theuncertainty expected forsin2 �lepteff increases by almost a factor 7to

� sin2 �lepteff = 3:0� 10�4 for j�(�)j < 2:4: (31)

In order to improve the precision beyond that expected fromfuture SLC and Tevatron experiments, it will be necessary todetect electrons and muons in the very forward pseudorapidityrange,j�j = 3:0� 5:0, at the LHC.

3. NLC and�+�� Collider

The effective weak mixing angle can also be measured at theNLC in fixed target Møller and Bhabha scattering. In fixedtarget Møller scattering one hopes to achieve a precision of� sin2 �lepteff = 6 � 10�5 [26]. In Bhabha scattering, it shouldbe possible to measure the effective weak mixing angle with aprecision of a few�10�4 [27], depending on the energy andpolarization available. Possibilities to determine the effectiveweak mixing angle at a�+�� collider have not been investi-gated so far.

4. Constraints onMH from sin2 �lepteff andMt

The potential of extracting useful information on the Higgsboson mass from a fit to the SM radiative corrections and a pre-cise measurement ofsin2 �lepteff andMt is illustrated in Fig. 8.

Here we have assumedMt = 176 � 2 GeV/c2, sin2 �lepteff =

0:23143 � 0:00015, and��1(M2Z) = 128:89 � 0:05. From

such a measurement, one would findMH = 415+145�105 GeV/c2.

The corresponding log-likelihood function is shown in Fig. 9.From Fig. 8 it is obvious that the extracted Higgs boson mass

6

0.229

0.2295

0.23

0.2305

0.231

0.2315

0.232

0 100 200 300 400 500 600 700 800 900 1000

sin2Θefflept vs. MH

sin2Θefflept = 0.23143 ± 0.00015

MZ = 91.1884 ± 0.0022 GeV/c2

Mt = 176.0 ± 2.0 GeV/c2

MH (GeV/c2)

sin2 Θ

eff

lep

t

Figure 8: Predictedsin2 �lepteff versus the Higgs boson mass.

depends very sensitively on the central value of the effectiveweak mixing angle. The relative error on the Higgs boson mass,�MH=MH � 30%, however, depends only on the uncertaintyof higher order corrections,sin2 �lepteff ,Mt, and�(M2

Z). For the

precision ofsin2 �lepteff andMt assumed here, the theoretical er-ror from higher orders, and the uncertainty in�(M2

Z) begin tolimit the accuracy which can be achieved for the Higgs bosonmass.

C. Precision Measurement ofMW at FutureExperiments

1. Deep Inelastic Scattering and HERA

Future experiments provide a variety of opportunities to mea-sure the mass of theW boson with high precision. In�N scat-tering,MW can be determined indirectly through a measure-ment of the neutral to charged current cross section ratio

R� =�(�N ! �X)

�(�N ! ��X): (32)

In the SM,R� can be used to directly determine the weak mix-ing angle via the lowest order expression

R� =1

2� sin2 �W +

5

9(1 + r) sin4 �W + C� ; (33)

where

r =�(��N ! �+X)

�(�N ! ��X); (34)

andC� is a correction factor which incorporates, among oth-ers, effects due to charm production and longitudinal structure

0

1

2

3

4

5

6

7

8

9

10

0 100 200 300 400 500 600 700 800 900 1000

-Log(L) vs. MH

sin2Θefflept = 0.23143 ± 0.00015

MZ = 91.1884 ± 0.0022 GeV/c2

Mt = 176.0 ± 2.0 GeV/c2

MH = 415+145-105 GeV/c2

MH (GeV/c2)

-Lo

g(L

)

68% CL

90% CL95% CL

Figure 9: The negative log-likelihood function assumingsin2 �lepteff = 0:23143� 0:00015 andMt = 176� 2 GeV/c2.

functions. Electroweak radiative corrections modify the lead-ing order prediction. In the on-shell scheme, wheresin2 �W =

1 � M2W =M2

Z to all orders in perturbation theory, the (lead-ing) radiative corrections tosin2 �W andR� almost perfectlycancel [28]. This implies that, in the SM,�N scattering di-rectly measures theW mass, given the very precisely deter-minedZ boson mass. A new CCFR measurement [29] givesMW = 80:46� 0:25 GeV/c2. With the data which one hopesto collect in the NuTeV experiment during the current Fermilabfixed target run, one expects [29]

�MW � 100 MeV=c2: (35)

Figure 10 compares the current results forMW from direct mea-surements at CDF, DØ and LEP2 (see below) with indirect de-terminations from LEP and SLD via electroweak radiative cor-rections, and theW mass obtained from CCFR, other�N ex-periments [30], and the expectation for NuTeV.

The W mass can also be determined from measurementsof the charged and neutral current cross sections at HERA.Moving the low � quadrupoles closer to the interaction re-gion, one hopes to achieve integrated luminosities of the or-der of 150 pb�1 per year with a 70% longitudinally polarizedelectron beam. The expected constraints onMW andMt, to-gether with the SM predictions forMH = 100 GeV/c2 andMH = 800 GeV/c2 are shown in Fig. 11 [31]. When combinedwith a measurement of the top quark mass with a precision of�Mt = 5 GeV/c2, the projected HERA results yield a precision

7

Measurements of the W Mass

79.879.9

8080.180.280.380.480.580.680.7

CDF D0 LEP2 World SLD LEP1 CCFR Other NuTeV

Direct M W Indirect e +e- νN Scattering

MW

(G

eV/c

2 )

Figure 10: A comparison of direct and indirect measurementsof theW boson mass.

of�MW � 60 MeV=c

2: (36)

Taking �Mt = 2 GeV/c2 instead only marginally improvesthe accuracy on theW mass. In deriving the result shown inEq. (36), a 1% relative systematic uncertainty of the chargedand neutral current cross sections at HERA was assumed. For asystematic error of 2%, one finds�MW � 80 MeV=c

2.

2. LEP2 and NLC

Precise measurements of theW mass at LEP2 [32] can be ob-tained using the enhanced statistical power of the rapidly vary-ing totalW+W� cross section at threshold [33], and the sharp(Breit-Wigner) peaking behaviour of the invariant mass distri-bution of theW decay products. During the recent LEP2 runatps = 161 GeV, the four LEP experiments have each ac-

cumulated approximately 10 pb�1 of data. The totalW+W�

cross section as a function of theW mass is shown in Fig. 12,together with the preliminary experimental result [34]. Com-bining the results obtained from theW+W� ! jjjj, theW+W� ! `��jj and theW+W� ! `+�`�� (` = e; �; � )channel, theW pair production cross section at

ps = 161 GeV

is measured to be�(WW ) = 3:57 � 0:46 pb. This translatesinto aW mass of [34]

MW = 80:4� 0:2� 0:1 GeV=c2: (37)

A much more accurate measurement ofMW will be possi-ble in the future through direct reconstruction methods whenLEP2 will be running at energies well above theW pair thresh-old. Here, the Breit-Wigner resonance shape is directly re-constructed from theW� final states using kinematic fittingtechniques. The potentially most important limitation in usingthis method originates from color reconnection [35] and Bose-Einstein correlations [36] in theW+W� ! jjjj channel. Tak-ing common errors into account, the expected overall precisionfrom this method at LEP2 for a total integrated luminosity of500 pb�1 per experiment is anticipated to be [32]

�MW = 35� 45 MeV=c2: (38)

79.0

79.5

80.0

80.5

81.0

81.5

0 50 100 150 200 250 300

MW

Mt

CC250 pb-1

CC + NC1000 pb-1

CC + NC1000 pb-1 andδ Mt = 5 GeV

1% systematics

e- p

GeV

GeV

Gµ(MH=100 GeV)

Gµ(MH=800 GeV)

P=-0.7

Figure 11:1� confidence contours in the(MW ;Mt) plane frompolarized electron scattering at HERA (P = �0:7), utilizingcharged current scattering alone for

RLdt = 250 pb�1 (outerellipse), and neutral and charged current scattering for 1 fb�1

(shaded ellipse). Shown is also the combination of the 1 fb�1 re-sult with a direct top mass measurement with�Mt = 5 GeV/c2

(full ellipse). The SM predictions are also shown for two valuesof MH (from Ref. [31]).

The same method can in principle also be used at the NLC.However, the beam energy spread limits the precision which onecan hope to achieve at ane+e� Linear Collider. Preliminarystudies indicate that one can hope for a precision of�MW =

20 MeV/c2 at best. No studies for a�+�� collider have beenperformed so far.

3. Tevatron

In W events produced in a hadron collider in essence onlytwo quantities are measured: the lepton momentum and thetransverse momentum of the recoil system. The latter consistsof the “hard”W -recoil and the underlying event contribution.For W -events these two are inseparable. The transverse mo-mentum of the neutrino is then inferred from these two observ-ables. Since the longitudinal momentum of the neutrino cannotbe determined unambiguously, theW -boson mass is usually ex-tracted from the distribution in transverse:

MT =p2 pT (e) pT (�) (1� cos'e�); (39)

where'e� is the angle between the electron and neutrino in thetransverse plane. TheMT distribution sharply peaks atMT �

8

MW (GeV)

W+W

− cro

ss s

ectio

n (p

b)LEP Average

σWW = 3.57 ± 0.46 pbMW = 80.4 ± 0.2 ± 0.1 (LEP) GeV

PRELIMINARY√s = 161.3 ± 0.2 GeV −

0

1

2

3

4

5

6

7

79 79.5 80 80.5 81 81.5 82

Figure 12: The totalW+W� cross section as a function of theW boson mass. The shaded band represents the cross sectionmeasured at LEP2.

MW .Both the transverse mass and lepton transverse momentum

are, by definition, invariant under longitudinal Lorentz boosts.In determining theW mass, the transverse mass is preferredover the lepton transverse momentum spectra because it is tofirst order independent of the transverse momentum of theW .Under transverse Lorentz boosts along a direction��, MT andpT (e) transform as

M2T

�= M�

T2 � �2 cos2 ��M�

L2;

pT (e) �= p�T (e) +1

2pT (W ) cos�� ;

with M�T = MW sin ��, M�

L = MW cos �� and � =

pT (W )=MW . The asterisk indicates quantities in theW restframe. The disadvantage of using the transverse mass is thatit uses the neutrino transverse momentum which is a derivedquantity. The neutrino transverse momentum is identified withthe missing transverse energy in the event, which is given by

~E/T = �Xi

~pTi = �~pT (e) � ~p recT � ~uT (L);

where~p recT is the transverse momentum of theW -recoil systemand~uT (L) the transverse energy flow of the underlying event,which depends on the luminosity. It then follows that the magni-tude of the missingET vector and the true neutrino momentumare related as

E/T = pT (�) +1

4

u2TpT (�)

: (40)

0

500

1000

1500

2000

2500

3000

3500

4000

40 50 60 70 80 90 100 110 120

IC = 1IC = 3IC = 9

Transverse Mass (GeV/c 2)

Eve

nts/

GeV

Figure 13: The effect of multiple interactions on theW trans-verse mass distribution at the Tevatron. Standard kinematic cutsof pT (e) > 25 GeV/c, j�(e)j < 1:2, E/T > 25 GeV andpT (W ) < 30 GeV/c are imposed. The effect of multiple inter-actions is simulated by adding additional minimum bias eventsto the event containing theW boson.

This relation can be interpreted as the definition of the neutrinomomentum scale. Note that the underlying event gives rise toa bias in themeasuredneutrino momentum with respect to thetrue neutrino momentum. In case there are more interactionsper crossing,j~uT j behaves as a two-dimensional random walkand is proportional to

pIC , whereIC is the average number of

interactions per crossing. The shift in measured neutrino mo-mentum is thus directly proportional to the number of interac-tions per crossing. The resolution increases as

pIC .

The above equation for the missing transverse energy de-serves some more attention. The two components directly re-lated to theW decay,~pT (e) and~p recT , are only indirectly af-fected by multiple interactions through the underlying event.It is the measurement of~uT (L) which governs the luminos-ity dependence. Because of multiple interactions,~uT (L) willshow a dependence on luminosity following Poisson statistics,with the two effects indicated above:i) a degradation of theE/T resolution andii) a shift in the measured neutrino momen-tum. This is demonstrated in Fig. 13 where we show theMT

distribution for various values ofIC at the Tevatron. For Run IIone expectsIC � 3, and at TeV33,IC � 6 � 9 [37]. Botheffects, of course, propagate into the measurement of the trans-verse mass and the uncertainty onMW will not follow the sim-ple 1=

pN rule anymore [38]. In addition, however, the detec-

tor response to high luminosities needs to be folded in. In the

9

Figure 14: Comparison of the CDFW asymmetry measurementwith recent NLO parton distribution function predictions.

above discussion it was assumed that the detector response islinear to the number of multiple interactions which in generalis not the case. The effects of pile-up in the calorimeter andoccupancy in the tracking detectors produce a� 7% shift inpT for an electron with transverse momentum of 40 GeV/c atL = 1033 cm�2 s�1, which will further affect the uncertaintyon theW mass adversely [39].

Another uncertainty that will not, and has not in the past,scaled with luminosity is the theoretical uncertainty comingfrom thepT (W ) model and the uncertainty on the proton struc-ture. Parton distributions and the spectrum inpT (W ) are corre-lated. The DØ experiment has addressed this correlation in thedetermination of its uncertainty on theW mass [4, 5]. The par-ton distribution functions are constrained by varying the CDFmeasuredW charge asymmetry within the measurement errors,while at the same time utilizing all the available data. Newparametrizations of the CTEQ 3M parton distribution functionwere obtained that included in the fit the CDFW asymmetrydata from Run Ia [40], where all data points had been movedcoherently up or down by one standard deviation. In additionone of the parameters, which describes theQ2-dependence ofthe parameterization of the non-perturbative functions describ-ing thepT (W ) spectrum [41], was varied. The constraint onthis parameter was provided by the measurement of thepT (Z)spectrum. The uncertainty due to parton distribution functionsand thepT (W ) input spectrum was then assessed by varying si-multaneously the parton distribution function, as determined byvarying the measuredW charge asymmetry, and the parameterdescribing the non-perturbative part of thepT (W ) spectrum.

The CDF experiment uses their measurement of theW chargeasymmetry as the sole constraint on the uncertainty due to theparton distribution functions. Figure 14 compares the prelimi-nary CDFW charge asymmetry measurement [42] with severalrecent fits to parton distribution functions. Figure 15 shows the

-150

-125

-100

-75

-50

-25

0

25

50

75

100

-10 -7.5 -5 -2.5 0 2.5 5 7.5 10

Figure 15: The correlation between the uncertainty in theWmass and the deviation between the average measured asymme-try for Run Ia and Ib CDF data for several recent parton distri-bution functions .

correlation between the uncertainty on theW mass,�MW , and

��(A(�)) =hAPDF (�)i � hAdata(�)i

�Adata(�); (41)

the deviation between the average measured asymmetry forRun Ia and Ib data and various recent NLO parton distributionfunction fits [42]. The fittedW mass is seen to be stronglycorrelated with theW charge asymmetry. TheW charge asym-metry, however, is mainly sensitive to the slope of the ratio oftheu andd quark parton distribution functions

A(yW ) / d(x2) = u(x2) � d(x1) = u(x1)

d(x2) = u(x2) + d(x1) = u(x1)(42)

and does not probe the full parameter range describing the par-ton distribution functions .

Future measurements of thepT (Z) distribution will providea constraint on thepT distribution of theW boson. Moreover,the measurements of theW charge asymmetry, together withmeasurements from deep inelastic scattering experiments, willprovide further constraints on the parton distribution functions.An effort needs to be made, though, to provide the experimentswith parton distributions with associated uncertainties.

At high luminosities alternate methods to determine theW -mass may be advantageous. Because of the similarity ofW and

10

p , q ,such as the charged lepton transverse momenta are particularlyinteresting [43, 44]. The ratio of the leptonpT distributions isthought to be very promising for fitting theW mass in the highluminosity regime since the procedure is independent of manyresolution effects. However, the shapes of the lepton transversemomentum distributions are sensitive to the differences in theW andZ production mechanisms, which need to be better un-derstood.

Here we concentrate on a similar method which utilizes thetransverse mass ratio ofW andZ bosons [44]. Preliminary re-sults from an analysis of the transverse mass ratio have recentlybeen presented by the DØ Collaboration [45]. Only the electronchannel will be discussed in the following, although the methodis expected to work for muon final states as well.

The transverse mass ratio method treats theZ ! e+e� sam-ple similar to theW ! e� sample, thus cancelling many ofthe common systematic uncertainties. A transverse mass for theZ boson is constructed with one of the decay electrons, whiletheE/T is derived by adding the transverse energy of the otherelectron to the residualE/T in the event. Hence, two such com-binations can be formed for eachZ event.

TheZ transverse mass distribution is scaled down in finitesteps and compared with theMT distribution of theW bo-son. TheW mass is then determined from the scale factor(MW =MZ) which gives the best agreement between theMT

distributions using a Kolmogorov test. Since differences in theproduction mechanism, acceptances and resolution effects be-tween theW and theZ sample lead to differences in the shapesof the transverse mass distributions, one has to correct for theseeffects.

The dominant systematic uncertainty arises from the uncer-tainty on the underlying event. Electromagnetic and hadronicresolution effects mostly cancel in the transverse mass ratio, asexpected. The systematic uncertainty due to the parton distribu-tion functions and the transverse momentum of theW boson isreduced by more than a factor 3 compared with that found us-ing the conventionalW transverse mass method [4]. The totalsystematic error from the DØ Run Ia data sample is estimatedto be 75 MeV/c2. For comparison, the total systematic errorobtained using the transverse mass distribution of theW usingDØ Run Ia data is 165 MeV/c2 [4].

In the analysis of the Run Ia data sample, electrons fromWandZ decay are identified as in the conventionalW mass anal-ysis.W candidates are selected by requiringpT (e) > 30GeV/candpT (�) > 30 GeV/c, while electrons fromZ decays are re-quired to havepT (e) > 34 GeV/c, since they are eventuallyscaled down. Electrons fromW decay and at least one electronfrom Z decay are required to be in the central pseudorapidityregion,j�(e)j < 1:1. Z events are used twice if both electronsfall in the central region. The shape comparison is performedin the fitting window65 GeV=c2 < MT < 100 GeV=c

2. TheselectedZ sample is scaled down in finite steps and, at everystep, the shape of theZ andW MT distribution is comparedusing the Kolmogorov test. Figure 16 shows theMT (Z) distri-bution superimposed on theMT (W ) distribution for one of thefits. The preliminary result forMW from Run Ia data is

Figure 16: The Run Ia DØMT (W ) distribution (histogram)with the scaledMT (Z) distribution (points) superimposed.

MW = 80:160� 0:360(stat)� 0:075(syst) GeV=c2: (43)

The limitation of the method described here comes entirely fromthe limitedZ statistics, which is expected to scale exactly as1=pN in future experiments.

The power of theMT ratio method becomes apparent whenone compares the uncertainty onMW expected for 1 fb�1 and10 fb�1 with that expected from the traditionalW transversemass analysis [38]. The results for both methods are listed inTable II. To calculate the projected statistical (systematic) errorsin the transverse mass ratio method, we have taken the errorsof Eq. (43) and scaled them with1=

pN (

pIC=N ), assuming

IC = 3 (IC = 9) for 1 fb�1 (10 fb�1). Both, electron andmuon channels are combined in Table II, assuming that the twochannels yield the same precision inMW .

Table II: Projected statistical and systematic errors (per experi-ment) on theW mass at the Tevatron, combining theW ! e�andW ! �� channel.

traditionalMT analysisRLdt = 1 fb�1RLdt = 10 fb�1

�MW IC = 3 IC = 9

statistical 29 MeV=c2

17 MeV=c2

systematic 42 MeV=c2

23 MeV=c2

total 51 MeV=c2

29 MeV=c2

W=Z transverse mass ratioRLdt = 1 fb�1RLdt = 10 fb�1

�MW IC = 3 IC = 9

statistical 29 MeV=c2

9 MeV=c2

systematic 10 MeV=c2

6 MeV=c2

total 31 MeV=c2

11 MeV=c2

TheW mass can also be determined from the transverse en-ergy (momentum) distribution of the electron (muon) inW !

11

e ( � �) , p W = p pof a precise measurement ofMW from theET (e) distributionin Run II and at TeV33 have been investigated in Ref. [39].The measurement of the lepton four-momentum vector is in-dependent of theE/T resolution, and the electronET resolu-tion is dominated by the intrinsic calorimeter resolution. Hencethe statistical uncertainty of theW mass measurement from theET (e) distribution is expected to scale approximately as1=

pN .

Simulations have shown that a sample of 30,000 events (similarto the DØ Run Ib data sample) gives a statistical error on theWmass of 100 MeV/c2 from theET (e) fit. This is in agreementwith the result of the preliminary DØ Run IbW mass analy-sis [46]. The systematic error from this method is expected tobe about 170 MeV/c2 for the same number of events. Scalingthe total uncertainty as1=

pN , the projected uncertainty ofMW

from the electronET fit is:

�MW = 55 MeV=c2

for 1 fb�1;

�MW = 18 MeV=c2

for 10 fb�1: (44)

In estimating the uncertainties given in Eq. (44) and Table II,we have assumed that the current uncertainty from parton distri-bution functions and the theoretical uncertainty originating fromhigher order electroweak corrections can be drastically reducedin the future. In order to measureMW with high precision, itis crucial to fully control higher order electroweak (EW) cor-rections. So far, only the final stateO(�) photonic correctionshave been calculated [47], using an approximation which indi-rectly estimates the soft + virtual part from the inclusiveO(�2)W ! `�( ) width and the hard photon bremsstrahlung contri-bution. Using this approximation, electroweak corrections werefound to shift theW mass by about�65MeV/c2 in the electron,and�170 MeV/c2 in the muon channel [4, 5].

Currently, a more complete calculation of theO(�) EW cor-rections, which takes into account initial and final state correc-tions, is being carried out [48]. The calculation is performedusing standard Monte Carlo phase space slicing techniques forNLO calculations. In calculating the initial state radiative cor-rections, mass (collinear) singularities are absorbed into theparton distribution functions through factorization, in completeanalogy to the QCD case. QED corrections to the evolution ofthe parton distribution function are not taken into account. Astudy of the effect of QED on the evolution indicates that thechange in the scale dependence of the PDF is small [49]. Totreat the QED radiative corrections in a consistent way, theyshould be incorporated in the global fitting of the PDF. The rel-ative size and the characteristics of the various contributions tothe EW corrections toW production is shown in Fig. 17.

Initial state (photon and weak) radiative corrections are foundto be uniform and, therefore, are expected to have little effect ontheW boson mass extracted. While initial state photon radia-tion increases the cross section by0:9%, weak one-loop correc-tions almost completely cancel the initial state photonic correc-tions. The completeO(�) initial state EW corrections reducethe leading order (LO) cross section by about 0.1%. Initial andfinal state photon radiation interfere very little. The interferenceeffects are uniform and have essentially no effect on theMT dis-tribution. Final state photon radiation changes the shape of the

Figure 17: The ratio of the NLO to LOMT (e�e) distributionfor various individual contributions: the QED-like initial or fi-nal state contributions (solid), the completeO(�) initial and fi-nal state contributions (short dashed) and the initial–final stateinterference contribution (long dashed).

transverse mass distribution and reduces the LO cross sectionby up to1:4% in theW resonance region. Weak correctionsagain have no influence on the lineshape, but reduce the crosssection by about 1%. TheW mass obtained from theMT dis-tribution including the full EW one-loop corrections is expectedto be several MeV/c2 smaller than that extracted employing theapproximate calculation of Ref. [47].

Since final state photon radiation introduces a significant shiftin theW mass, one also has to worry about multiple photonradiation. A calculation ofp�p ! �� [50] which includesall initial and final state radiation and finite muon mass effectsshows that approximately 0.8% of allW ! �� events containtwo photons withET ( ) > 0:1 GeV (the approximate towerthreshold of the electromagnetic calorimeters of CDF and DØ)and�R( ; ) > 0:14. This suggests that the additional shiftin MW from multiple photon radiation may not be negligible ifone aims at a measurement with a precision ofO(10 MeV=c

2).

4. LHC

At the LHC, the cross section forW production is about afactor 4 larger than at the Tevatron. During the first year ofoperation, it is likely that the LHC will run at a reduced lu-minosity of approximatelyL = 1033 cm�2 s�1, resulting inroughly 0:9 � 107 W ! e� events with a central electron(j�(e)j < 1:2) and a transverse mass in the range65 GeV=c

2<

MT < 100 GeV=c2. A similar number ofW ! �� events

12

p , [ ] [ ],will be able to trigger on electrons and muons with a transversemomentum ofpT (`) > 15 GeV/c ( = e; �), and should befully efficient for pT (`) > 20 GeV/c. They are well-optimizedfor electron, muon andE/T detection.

At L = 1033 cm�2 s�1, the average number of interactionsper crossing at the LHC is approximatelyIC = 2, which issignificantly smaller than what one expects at the Tevatron forthe same luminosity. A precision measurement of theW massat the LHC running at a reduced luminosity, using the traditionaltransverse mass analysis, thus seems feasible [51].

QCD corrections to the transverse mass distribution at theLHC enhance the cross section by 10 – 20% in theMT rangewhich is normally used to determineMW . This is illustrated inFig. 18, where the LO and NLO QCD transverse mass distribu-tion is shown, together with the NLO to LO differential crosssection ratio. Here, apT (`) > 20 GeV/c and ap/T > 20 GeV/ccut have been imposed, and the pseudorapidity of the lepton isrequired to bej�(`)j < 1:2. The slight change in the shape oftheMT distribution induced by the NLO QCD corrections isdue to the cuts imposed.

So far, no detailed study of the precision which one mighthope to achieve forMW at the LHC has been performed. Fora crude order of magnitudeestimate, one can use the statisticaland systematic errors of the current CDF and DØ analyses [4,5], and scale them by

pIC=N . For an integrated luminosity of

10 fb�1, one obtains [51]:

�MW<� 15 MeV=c

2: (45)

In order to see whether LHC experiments can perform a mea-surement ofMW which is significantly more precise than whatone expects from TeV33 or the NLC, a more detailed studywhich also considers other quantities such as the transversemass ratio ofW andZ bosons [43, 44] has to be carried out.

5. Constraints onMH fromMW andMt

The potential of extracting useful information on the Higgsboson mass from a fit to the SM radiative corrections and aprecise measurement ofMW andMt is illustrated in Fig. 19.Here we have assumedMt = 176 � 2 GeV/c2, MW =

80:330 � 0:010 GeV/c2, and ��1(M2Z) = 128:89 � 0:05.

Such a measurement would constrain the Higgs boson massto MH = 285+65

�55 GeV/c2. The corresponding log-likelihoodfunction is shown in Fig. 20. A measurement of theW masswith a precision of�MW = 10 MeV/c2 and of the top masswith an accuracy of 2 GeV/c2 thus translates into an indirectdetermination of the Higgs boson mass with a relative error ofabout

�MH=MH � 20%: (46)

From a global analysis of all electroweak precision data onemight then expect�MH=MH < 15%.

For the precision ofMt andMW assumed here, the theoret-ical error from higher orders and the uncertainty in the electro-magnetic coupling constant�(M2

Z) become limiting factors forthe accuracy which can be achieved forMH . Efforts to calculatehigher order corrections and to significantly improve the error

Figure 18: The LO and NLO QCDW transverse mass distribu-tion at the LHC. Also shown is the NLO to LO differential crosssection ratio as a function ofMT .

on�(M2Z) beyond what one can expect from measurements at

Novosibirsk, DAP�NE, or BES, need increased emphasis fromboth experimentalists and theorists in order to be able to achievean ultimate relative precision onMH better than about 15%.

13

80.2

80.25

80.3

80.35

80.4

80.45

80.5

100 200 300 400 500 600 700 800 900 1000

MW vs. MH

MW = 80.330 ± 0.010 GeV/c2

Mt = 176.0 ± 2.0 GeV/c2

MH (GeV/c2)

MW

(G

eV

/c2 )

Figure 19: PredictedW versus Higgs boson mass forMt =

176 � 2 GeV/c2. The theoretical predictions incorporate theeffects of higher order electroweak and QCD corrections.

IV. SUMMARY AND CONCLUSIONS

In this report, we have highlighted some current high pre-cision electroweak measurements, and explored prospects forfurther improvements over the next decade. The aim of preci-sion electroweak measurements is to test the SM at the quan-tum level, and to extract indirect information on the mass of theHiggs boson. The confrontation of these indirect predictions ofMH with the results of direct searches for the Higgs boson willbe perhaps the most exciting development of the next decade inthe field of particle physics.

Although a global fit to all available precision electroweakdata yieldsMH = 149+148

�82 GeV/c2, the Higgs boson mass ex-tracted strongly depends on the input quantities used in the fit.Excluding a particular observable which displays a statisticallysignificant deviation from the SM prediction,e.g.the SLD left-right asymmetry, may easily increase the central value ofMH

by a factor 4. One therefore has to conclude that present dataare not quite sufficient to obtain a stable estimate of the Higgsboson mass.

Results of future collider experiments are expected to dras-tically change this situation. In these experiments one hopesto precisely determine three observables which are key ingre-dients in obtaining reliable indirect information on the Higgsboson mass:

� The uncertainty on the top quark mass is expected to bereduced by at least a factor 3 in Tevatron and LHC exper-iments. At the NLC or a�+�� collider, a precision of a

0

1

2

3

4

5

6

7

8

9

10

100 200 300 400 500 600 700 800 900 1000

-Log(L) vs. MH

MW = 80.330 ± 0.010 GeV/c2

Mt = 176.0 ± 2.0 GeV/c2

MH = 285+65-55 GeV/c2

MH (GeV/c2)

-Lo

g(L

)

68% CL

90% CL95% CL

Figure 20: The negative log-likelihood function assumingMW = 80:330� 0:010 GeV/c2 andMt = 176� 2 GeV/c2.

few hundred MeV/c2 may be possible.

� It should be possible to reduce the error onsin2 �lepteff byat least a factor two through measurements of the left-rightasymmetry at a luminosity upgraded SLC, and the forwardbackward asymmetry in theZ peak region at the Tevatronand LHC.

� The most profound improvement is likely to occur for theW mass, where a gain of a factor 5 seems to be withinreach. New strategies developed for extractingMW athadron colliders [43, 44] will make it possible to fully ex-ploit the expected increase in integrated luminosity at theTevatron.

From a measurement ofMt with a precision of 2 GeV/c2, andMW with an uncertainty of 10 MeV/c2 alone it should be pos-sible to constrainMH within 20%.

As the electroweak measurements improve, the theoretical er-ror from higher orders and the uncertainty in�(M2

Z) will grad-ually become more and more important limitations in the pre-cision which can be achieved. The determination of�(M2

Z)

is limited by the knowledge of the photon hadron coupling atsmall momentum transfer. An increased experimental and the-oretical effort is needed to overcome the present limitations indetermining�(M2

Z), and to calculate higher order correctionsto the electroweak observables.

14

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