search for the standard model higgs boson at lep · 3 the authors are listed in phys. lett. b 517...

e

. The searchotheses:omputed asonor various

Physics Letters B 565 (2003) 61–75

www.elsevier.com/locate/np

Search for the Standard Model Higgs boson at LEP

ALEPH Collaboration1

DELPHI Collaboration2

L3 Collaboration3

OPAL Collaboration4

The LEP Working Group for Higgs Boson Searches5

Received 7 March 2003; received in revised form 25 April 2003; accepted 28 April 2003

Editor: L. Rolandi

Abstract

The four LEP Collaborations, ALEPH, DELPHI, L3 and OPAL, have collected a total of 2461 pb−1 of e+e− collision data atcentre-of-mass energies between 189 and 209 GeV. The data are used to search for the Standard Model Higgs bosonresults of the four Collaborations are combined and examined in a likelihood test for their consistency with two hypthe background hypothesis and the signal plus background hypothesis. The corresponding confidences have been cfunctions of the hypothetical Higgs boson mass. A lower bound of 114.4 GeV/c2 is established, at the 95% confidence level,the mass of the Standard Model Higgs boson. The LEP data are also used to set upper bounds on the HZZ coupling fassumptions concerning the decay of the Higgs boson. 2003 Elsevier B.V. Open access under CC BY license.

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1. Introduction

The Higgs mechanism [1] plays a central rolethe unification of the electromagnetic and weak intactions by providing mass to the W and Z intermeate vector bosons without violating local gauge invaance. Within the Standard Model [2], the Higgs meanism is invoked to break the electroweak symme

1 The authors are listed in Phys. Lett. B 526 (2002) 191.2 The authors are listed in hep-ex/0303013.3 The authors are listed in Phys. Lett. B 517 (2001) 319.4 The authors are listed in hep-ex/0209078.5 The LEP Working Group for Higgs Boson Searches cons

of members of the four LEP Collaborations and of theorists amwhom S. Heinemeyer and G. Weiglein are authors of this Letter

0370-2693 2003 Elsevier B.V.doi:10.1016/S0370-2693(03)00614-2

Open access under CC BY license.

it implies the existence of a single neutral scalar pticle, the Higgs boson. The mass of this particle isspecified, but indirect experimental limits are obtainfrom precision measurements of the electroweakrameters which depend logarithmically on the Higboson mass through radiative corrections. Currethese measurements predict that the Standard MHiggs boson mass ismH = 81+52

−33 GeV/c2 and con-strain its value to less than 193 GeV/c2 at the 95%confidence level [3].

The data collected by the four LEP Collabortions prior to the year 2000 gave no direct indicatof the production of the Standard Model Higgs bson [4] and allowed a lower bound of 107.9 GeV/c2

to be set, at the 95% confidence level, on the mDuring the last year of the LEP programme (t

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62 The LEP Working Group for Higgs Boson Searches / Physics Letters B 565 (2003) 61–75

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Table 1Integrated luminosities of the data samples of the four experimand their sum (LEP). The subsets taken at energies excee206 GeV and 208 GeV are listed separately

Integrated luminosities in pb−1

ALEPH DELPHI L3 OPAL LEP√

s � 189 GeV 629 608 627 596 2461√s � 206 GeV 130 138 139 129 536√s � 208 GeV 7.5 8.8 8.3 7.9 32.5

year 2000), substantial data samples were collectecentre-of-mass energies(

√s ) exceeding 206 GeV, ex

tending the search sensitivity to Higgs boson masof about 115 GeV/c2 through the Higgsstrahlunprocess e+e− → HZ. In their initial analyses of thefull data sets, ALEPH [5] observed an excess of eveconsistent with the production of a Standard MoHiggs boson with a mass of 115 GeV/c2; L3 [6] andOPAL [7], while being consistent with the backgrouhypothesis, slightly favoured the signal plus baground hypothesis in this mass region; DELPHIreported a slight deficit with respect to the backgrouexpectation. The final results from the four Collabotions have now been published [9–12]. These are baon final calibrations of the detectors and LEP beenergies and, in some cases, on revised analysiscedures. In this Letter we present the results fromLEP-wide combination based on these new publtions. The data span the range of centre-of-mass egies from 189 GeV to 209 GeV. The integrated lumnosities of the data samples are given in Table 1the full range of energies used and for the subsetsenergies larger than 206 GeV and 208 GeV.

We also present upper bounds on the HZZ coupfor non-standard models with various assumpticoncerning the decay of the Higgs boson. In ordecover the low-mass domain, the data collected duthe LEP1 phase at the Z resonance are combinedLEP2 data.

2. Analysis and combination procedure

At LEP, the Standard Model Higgs boson is epected to be produced mainly in association withZ boson through the Higgsstrahlung process e+e− →HZ [13]. Small additional contributions are expectat the end of and beyond the kinematic range of

t

-

-

Higgsstrahlung process from W and Z boson fusiwhich produce a Higgs boson and a pair of neutrior electrons, respectively, in the final state [14]. Tsignal processes are simulated using the HZHA gerator [15], which includes the fusion processestheir interference with the HZ final states. For Higboson masses which are relevant at LEP, the StanModel Higgs boson is expected to decay mainly ibb̄ quark pairs (the branching ratio is 74% for a mof 115 GeV/c2) while decays toτ+τ−, WW∗, gg(≈ 7% each), and c̄c (≈ 4%) constitute the rest of thdecay width. The final-state topologies are determiby the decay properties of the Higgs boson andthose of the associated Z boson. The searches atencompass the four-jet final state(H → bb̄)(Z → qq̄),the missing energy final state(H → bb̄)(Z → νν̄), theleptonic final state(H → bb̄)(Z → +−) where de-notes an electron or a muon, and the tau leptonnal states(H → bb̄)(Z → τ+τ−) and(H → τ+τ−) ×(Z → qq̄).

A preselection is applied by each experimentreduce some of the main backgrounds, in particufrom two-photon processes and from the radiareturn to the Z boson, e+e− → Zγ (γ ). The remainingbackground, mainly from fermion pairs and Wor ZZ production, possibly with photon or gluoradiation, is further reduced either with the heof more selective cuts or by applying multivariatechniques such as likelihood analyses and nenetworks. The identification of b-quarks in the decof the Higgs boson plays an important role in tdiscrimination between signal and background,does the reconstructed Higgs boson candidate mThe detailed implementation of these analyses bydifferent experiments is described in Refs. [9–12] ain earlier references quoted therein.

The input from the four experiments which is usin the combination procedure is provided channelchannel. The word “channel” designates any subsethe data where a Higgs boson search has beenried out. These subsets may correspond to spefinal-state topologies, to data sets collected at difent centre-of-mass energies or to the subsets ofcollected by different experiments. In most channthe input is binned in two variables: the reconstrucHiggs boson massmrec

H , and a variableG which com-bines many event features such as b-tagging varialikelihood functions or neural network outputs, whi

The LEP Working Group for Higgs Boson Searches / Physics Letters B 565 (2003) 61–75 63

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allow discrimination on a statistical basis betweenHiggs boson signal and the background processes

For each given channel and bin in the(mrecH ,G)

plane, the experiments provide the number oflected data events, the number of expected backgroevents, and the number of expected signal eventsset of hypothetical Higgs boson masses. The expesignal and background estimates make use of detaMonte Carlo simulations carried out by each of tfour experiments. These take into account all knoexperimental features such as the centre-of-massergies and integrated luminosities of the data sples, cross-sections and decay branching ratios fosignal and background processes, selection efficcies and experimental resolutions with possible nGaussian contributions. Systematic errors with thcorrelations are also included. Since the simulatiare done at fixed centre-of-mass energies and Hboson masses, interpolations are applied (see, foample, [16]) to obtain the rates and distributionsarbitrary energies and masses. In order to avoid plems which arise in some bins due to low Monte Castatistics, smoothing procedures are applied (seeexample, [17]) which combine the available informtion with the information in the neighbouring bins.

3. Hypothesis testing

The observed data configuration in the (mrecH ,G)

plane is subjected to a likelihood ratio test of two hpothetical scenarios. In the background scenarioassumed that the data receive contributions frombackground processes only, while in the signal pbackground scenario the contributions from a Stdard Model Higgs boson of test massmH are assumedin addition. The expressions for the correspondbinned likelihoodsLb andLs+b are given in Appen-dix A.

In a search experiment, the likelihood ratio

(1)Q = Ls+b/Lb

makes efficient use of the information containedthe event configuration. For convenience, the logarmic form−2 lnQ is used as the test statistic since tquantity is approximately equal to the difference inχ2

when the data configuration is compared to the baground hypothesis and to the signal plus backgro

-

Fig. 1. Observed and expected behaviour of the test statistic−2 lnQ

as a function of the test massmH, obtained by combining the data othe four LEP experiments. The full curve represents the observathe dashed curve shows the median background expectationdark and light shaded bands represent the 68% and 95% probabands about the median background expectation. The dash-dcurve indicates the expectation for−2 lnQ, given the signal plusbackground hypothesis, when the signal mass on the abscistested. For the expected behaviours, the medians of the simudistributions are shown.

hypothesis (it becomes exactly equal in the limithigh statistics). Furthermore,−2 lnQ can be writtenas a sum of contributions from the individual observevents (see Eq. (A.3) in Appendix A).

Fig. 1 shows the test statistic−2 lnQ as a func-tion of the test mass for the LEP-wide combinatioThe expected curves are obtained by replacing theserved data configuration by a large number of simlated event configurations for the two hypotheses.the background hypothesis the 68% and 95% probility bands are also shown. There is a broad minimin the observed−2 lnQ starting at about 115 GeV/c2.The negative values in this mass range indicate thahypothesis including a Standard Model Higgs bosof such a mass is more favoured than the backgrohypothesis, albeit at low significance. Note also tthe median expectation for the signal plus backgrohypothesis crosses the observed curve in this mrange. The fact that the observed curve slightly deates from the background expectation over the wh


isxcept the

Fig. 2. Observed and expected behaviour of the test statistic−2 lnQ as a function of the test massmH when the combination procedureapplied to subsets of the LEP data. Plots (a) to (d): data sets from individual experiments; (e): the four-jet final state and (f): all efour-jet final state, with the data of the four experiments combined. The same notation as in Fig. 1 is used.


-of-n,events

Table 2Properties of the candidates with the largest contribution to−2 lnQ at mH = 115 GeV/c2. For each candidate, the experiment, the centremass energy, the final-state topology, the reconstructed Higgs boson mass and the weight atmH = 115 GeV/c2 are listed. The applied selectioln(1+ s/b) � 0.18 (i.e.,s/b � 0.2) atmH = 115 GeV/c2, retains 17 candidates while the expected numbers of signal and backgroundare 8.4 and 15.8, respectively

Experiment√

s (GeV) Final state topology mrecH (GeV/c2) ln(1+ s/b)

at 115 GeV/c2

1 ALEPH 206.6 Four-jet 114.1 1.762 ALEPH 206.6 Four-jet 114.4 1.443 ALEPH 206.4 Four-jet 109.9 0.594 L3 206.4 Missing energy 115.0 0.535 ALEPH 205.1 Leptonic 117.3 0.496 ALEPH 208.0 Tau 115.2 0.457 OPAL 206.4 Four-jet 111.2 0.438 ALEPH 206.4 Four-jet 114.4 0.419 L3 206.4 Four-jet 108.3 0.30

10 DELPHI 206.6 Four-jet 110.7 0.2811 ALEPH 207.4 Four-jet 102.8 0.2712 DELPHI 206.6 Four-jet 97.4 0.2313 OPAL 201.5 Missing energy 108.2 0.2214 L3 206.4 Missing energy 110.1 0.2115 ALEPH 206.5 Four-jet 114.2 0.1916 DELPHI 206.6 Four-jet 108.2 0.1917 L3 206.6 Four-jet 109.6 0.18

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mass range of the figure can also be explained by lupward fluctuations of the background and by lonrange effects due to the experimental mass resoluThe mass resolution is channel dependent and isically a Gaussian of widths 2–3 GeV/c2 with size-able asymmetric tails, accentuated by the proximof the kinematic limit of the HZ signal process athe Z boson mass constraint applied during the recstruction.

In Fig. 2 the likelihood test is applied to susets of the LEP data from individual experimenand final-state topologies. A signal-like deviation byond the 95% confidence level is only observedthe ALEPH data. For a given test mass, the dtance between the background expectation and thenal plus background expectation, compared to thspreads, is a measure of the discriminating powethe corresponding data set. These figures thus itrate the relative power of the subsets and the radecrease in discriminating power as the test massproaches the kinematic limit of the HZ signal proceOne should note that no individual LEP experime

has the statistical power to distinguish betweentwo hypotheses for a test mass larger than aboutGeV/c2 at the level of two standard deviations (sthe intersections of the signal plus background cuwith the lower edge of the light-shaded 95% condence level bands). Regarding the final-state topgies, the combined LEP data in the four-jet chanhave about the same discriminating power as allother final states together. The comparison of Figand 2 illustrates the gain in sensitivity when the dof the four experiments are combined in all chanels.

3.1. Contributions from single candidates

The contribution to the test statistic−2 lnQ froman individual candidate event can be evaluateding the binned likelihood functions that appear in Apendix A. We refer to this contribution as the eveweight which, in simplified notation, can be writteas ln(1 + s/b), wheres andb refer to the signal andbackground estimates in the bins of(mrec

H ,G) where


eight ofiontest mass

Fig. 3. Evolution of the event weight ln(1 + s/b) with test massmH for the events which have the largest contributions to−2 lnQ atmH = 115 GeV/c2. The labels correspond to the candidate numbers in the first column of Table 2. The sudden increase in the wthe OPAL missing energy candidate labelled “13” atmH = 107 GeV/c2 is due to switching from the low-mass to high-mass optimizatof the search at that mass. A similar increase is observed in the case of the L3 four-jet candidate labelled “17” which is due to thedependent attribution of the jets to the Z and Higgs bosons.

ates15sets,

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the candidate event is reconstructed. The candidwhich have the highest weight for a test mass of 1GeV/c2, chosen throughout this Letter for the purpoof illustration, are listed in Table 2. For these eventhe evolution of ln(1 + s/b) with test mass is showin Fig. 3. Typically, the weight is largest formH closeto the reconstructed mass but there is also a sizeweight over a large domain of test masses due toexperimental resolution, as mentioned before. Fotest mass of 115 GeV/c2, the events listed in Tableand shown in Fig. 3 contribute with about 40% to t

total weight in the likelihood ratio. The distributionsevent weights, shown in Fig. 4 for two test masses,in agreement with the expectation for the backgrohypothesis.

3.2. The reconstructed Higgs boson mass

The reconstructed Higgs boson massmrecH is one of

the crucial variables which contribute to the discriination between the signal and the backgroundthus to the test statistic−2 lnQ. In Fig. 5 the distri-


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Fig. 4. Expected and observed distributions of the event weln(1 + s/b) for test massesmH of (a) 110 and (b) 115 GeV/c2.Dashed line histograms: expected distributions for the backgroshaded histograms: expected distributions for the signal; pointserror bars: selected data.

butions for this discriminating variable are showntwo levels of signal purity.6 There is a clear peak ithe background prediction in the vicinity ofmZ due to

6 These distributions do not enter directly into the hypothetesting but have been produced to illustrate the level of agreembetween the data and the Monte Carlo simulation.

Fig. 5. Distributions of the reconstructed Higgs boson mass,mrecH ,

obtained from two selections with different expected signal puritThe histograms show the Monte Carlo predictions, lightly shafor the background, heavily shaded for an assumed Standard MHiggs boson of mass 115 GeV/c2. The points with error bars showthe data. In the loose and tight selections the cuts are adjustsuch a way as to obtain, for a Higgs boson of mass 115 GeV/c2,approximately 0.5 or 2 times more expected signal than backgroevents when integrated over the regionmrec

H > 109 GeV/c2. In thesearches where the event selection depends on the test masAppendix A), its value is set at 115 GeV/c2.

the e+e− → ZZ background process which is reprduced by the data.


Thermass

Fig. 6. Probability density functions corresponding to fixed test massesmH, for the background and signal plus background hypotheses.observed values of the test statistic−2 lnQ are indicated by the vertical lines. The light shaded areas, 1− CLb, measure the confidence fothe background hypothesis and the dark shaded areas, CLs+b, the confidence for the signal plus background hypothesis. Plot (a): testmH = 115 GeV/c2; (b): mH = 110 GeV/c2; (c): mH = 120 GeV/c2.

inichr-nc-

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4. Results of the hypothesis testing

The expected distributions of the test statistic−2×lnQ from the combined LEP analysis are shownFig. 6 for three test masses. These distributions, whcan be thought of as “slices” of Fig. 1 at the coresponding test masses, are probability density fu

tions (PDF) for the background and the signal pbackground hypotheses and include both the effof random statistical variations in the numbersevents and the systematic uncertainties affectingexpected rates. Systematic uncertainties are incorated by randomly varying the signal and backgrouestimates in each channel. For a given source of


esean-rre-ticsrorsriseorstheon-

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certainty, correlations are addressed by applying thrandom variations simultaneously to all those chnels where the uncertainty is relevant. The uncolated errors are dominated by the limited statisof the simulated background event samples. Erwhich are correlated between the experiments amainly from using the same Monte Carlo generatand cross-section calculations, for example, forsignal processes. The three parts of Fig. 6 demstrate a significant discriminating power of the cobined LEP data formH = 110 GeV/c2, a moderateone formH = 115 GeV/c2 and the rapid decreasethe discriminating power towards the end of thevestigated range,mH = 120 GeV/c2. The inclusion ofthe systematic errors led to some widening of thetributions in Fig. 6 thus slightly reducing the searsensitivity.

The vertical line in each part of Fig. 6 indicatthe observed value of−2 lnQ for the correspondingtest mass. Integrating the PDF for the backgroundpothesis from−∞ to the observed value, one otains the background confidence 1− CLb which ex-presses the incompatibility of the observation with

background hypothesis (also known as ap-value, seeRef. [18]). For a large number of simulated measuments with no signal and given the backgroundpothesis, 1− CLb is the probability to obtain a configuration of events which is less background-likemore signal plus background-like) than the oneserved. Similarly, integrating the PDF for the signplus background hypothesis from the observed vaof the test statistic to+∞, one obtains the confidenc(p-value) CLs+b which quantifies the compatibility othe observation with the signal plus backgroundpothesis.

Fig. 7 shows the expected and observed baground confidence 1− CLb for test masses in thrange from 80 to 120 GeV/c2. In the regionmH ≈98 GeV/c2 the observed value of about 0.02 tranlates into 2.3 standard deviations (see Appendixfor the conversion). Note that the number of sigevents which would produce such a deviation frombackground expectation is about an order of magtude smaller than the number expected in the StanModel for a Higgs boson of this mass. In the regof mH above 115 GeV/c2 the approximate value o

tedgnalectation. The

Fig. 7. The background confidence 1− CLb as a function of the test massmH. Full curve: observation; dashed curve: median expecbackground confidence. Dash-dotted line: the median expectation for 1− CLb, given the signal plus background hypothesis, when the simass on the abscissa is tested; the dark and light shaded bands represent the 68% and 95% probability bands about the median exphorizontal solid lines indicate the levels for 2σ and 3σ deviations from the background hypothesis (see Appendix A for the conversion).


Fig. 8. The background confidence 1−CLb as a function of the test massmH for subsets of the LEP data. The same notation as in Fig. 7 is used.Plots (a) to (d): individual experiments; (e): the four-jet and (f): all but the four-jet final state, with the data of the four experiments combined.


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0.09 translates into 1.7 standard deviations frombackground hypothesis. This deviation, althoughlow significance, is compatible with a Standard ModHiggs boson in this mass range while being alsoagreement with the background hypothesis. The vaof 1 − CLb would change in this region from abo0.09 to about 0.08 if the systematic errors werenored.

The dash-dotted line in Fig. 7 shows the medexpectation for 1− CLb, given the signal plus background hypothesis, when the signal mass on thescissa is tested. For a given mass hypothesis, the c1 − CLb versus test mass tends to have a minimclose to the hypothesized mass. For example, inpresence of a Higgs boson of mass 115 GeV/c2, thecurve showing the median of the 1−CLb results wouldhave a minimum at 115 GeV/c2, with a value of 0.009which would coincide with the dash-dotted line. Thline is an indication of the range of sensitivity of thcombined LEP data for detecting a Standard MoHiggs boson signal.

One should realise that the observed values of−CLb quoted above quantify deviations from the baground hypothesis which are local in mass. A rouestimate of the probability for such a deviation to ocur anywhere in a given mass range of interest istained by multiplying the local value of 1− CLb bythe ratio of the widths of the mass range to the mresolution. In the present case, the mass range of iest can be taken as the region exceeding 114 GeV/c2

where the observation is compatible, within the 95confidence level, with the Standard Model Higgs bson hypothesis (see the light shaded area in FigTaking 3 GeV/c2 as a crude value for the mass reslution (see Section 3), one obtains about two formultiplication factor.

Fig. 8 shows 1− CLb as a function of the test masfor subsets of the LEP data. The confidences 1− CLband CLs+b, for a test mass of 115 GeV/c2, are listedin Table 3 for all LEP data combined and for variosub-samples.

5. Bounds for the Higgs boson mass and coupling

The ratio CLs = CLs+b/CLb as a function ofthe test mass, shown in Fig. 9, is used to derivlower bound on the Standard Model Higgs bos

Table 3The background confidence 1−CLb and the signal plus backgrounconfidence CLs+b for a test massmH = 115 GeV/c2, for all LEPdata combined and for various subsets. The values for the fouand all but the four-jet final states are obtained with the data ofour experiments combined.

1− CLb CLs+b

LEP 0.09 0.15

ALEPH 3.3× 10−3 0.87DELPHI 0.79 0.03L3 0.33 0.30OPAL 0.50 0.14

Four-jet 0.05 0.44All but four-jet 0.37 0.10

mass (see Appendix A). The lowest test mass givCLs = 0.05 is taken as the lower bound on tmass at the 95% confidence level. The expecand observed lower bounds are listed in TableThe expected limits provide an indication of tsensitivities of the data subsets. The observed 9confidence level lower bound on the mass ofStandard Model Higgs boson obtained by combinthe four LEP experiments is 114.4 GeV/c2, whilethe median expected limit is 115.3 GeV/c2. Thedifference reflects the slight excess observed indata with respect to the background expectationhigh masses. The observed and the expected liwould shift upwards by about 50 MeV/c2 if thesystematic errors were ignored.

The combined LEP data are also used to set 9confidence level upper bounds on the HZZ couplingnon-standard models. In the ratioξ2 = (gHZZ/gSM

HZZ)2

Table 4Expected (median) and observed 95% confidence level lobounds on the Standard Model Higgs boson mass, for all LEPcombined and for various subsets of the data. The numbers fofour-jet and all but the four-jet final states are obtained with the dof the four experiments combined.

Expected limit Observed limit(GeV/c2) (GeV/c2)

LEP 115.3 114.4

ALEPH 113.5 111.5DELPHI 113.3 114.3L3 112.4 112.0OPAL 112.7 112.8

Four-jet channel 114.5 113.3All but four-jet 114.2 114.2


hypothesisto

Fig. 9. The ratio CLs = CLs+b/CLb for the signal plus background hypothesis, as a function of the test massmH. Solid line: observation;dashed line: median background expectation. The dark and light shaded bands around the median expectation for the backgroundcorrespond to the 68% and 95% probability bands. The intersection of the horizontal line for CLs = 0.05 with the observed curve is useddefine the 95% confidence level lower bound on the mass of the Standard Model Higgs boson.

Zrd

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the variablegHZZ designates the non-standard HZcoupling andgSM

HZZ the same coupling in the StandaModel. In deriving the limits onξ2, which cover abroad range of masses, the LEP1 data collected aZ resonance [19] have been combined with LEP2 dtaken at energies between 161 and 209 GeV. In parof Fig. 10 the Higgs boson is assumed to decayfermions and bosons according to the Standard Mowhile the cross-sections for the process e+e− → HZand the fusion processes WW→ H and ZZ→ H arescaled withg2

HZZ. For masses below 12 GeV/c2, notshown in the figure, the limits quoted in [20–23] appIn parts (b) and (c) it is assumed that the Higgs bodecays exclusively into b̄b orτ+τ− pairs. In theτ+τ−case and for masses below 30 GeV/c2, the limit shownis provided by the search of Ref. [24].

6. Summary

Combining the final results from the four LEexperiments, ALEPH, DELPHI, L3 and OPAL,

lower bound of 114.4 GeV/c2 is set on the mass of thStandard Model Higgs boson at the 95% confidelevel. At the beginning of the LEP programme no solimit existed for the mass of this particle [25].

At a mass of 115 GeV/c2, where ALEPH reportedan excess compatible with the production of the Stdard Model Higgs boson, the confidence 1− CLbof the combined LEP data expressing the levelconsistency with the background hypothesis is 0while the confidence CLs+b measuring the consistency with the signal plus background hypotheis 0.15.

The LEP1 and LEP2 data have been combineset upper bounds on the HZZ coupling for a wirange of Higgs boson masses and for various assutions concerning the Higgs boson decay properties

The searches for the Standard Model Higgs bocarried out by the four LEP experiments extendthe sensitive range well beyond that anticipated atbeginning of the LEP programme [26]. This is duethe higher energy achieved and to more sophisticdetectors and analysis techniques.


diang. (a): For

Fig. 10. The 95% confidence level upper bound on the ratioξ2 = (gHZZ/gSMHZZ)2 (see text). The dark and light shaded bands around the me

expected line correspond to the 68% and 95% probability bands. The horizontal lines correspond to the Standard Model couplinHiggs boson decays predicted by the Standard Model; (b): for the Higgs boson decaying exclusively into bb̄ and (c): intoτ+τ− pairs.

Nheh-nks

nsoforco-ingthe

Acknowledgements

We congratulate our colleagues from the CERAccelerator Divisions for the successful running of tLEP collider, particularly in the year 2000 at the higest energies. We would also like to express our tha

to the engineers and technicians in all our institutiofor their contributions to the excellent performancethe four LEP detectors. The LEP Working Group fHiggs Boson Searches acknowledges the fruitfuloperation between the Collaborations in developthe combination procedures and applying them toLEP data.


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Appendix A. Statistical method

The test statistic adopted in the combination of Ldata [27] is−2 lnQ whereQ is the ratio of the likeli-hood function for the signal plus background hypoesis to the likelihood function for the background hpothesis,

(A.1)Q(mH) = Ls+b

Lb.

The binned likelihood functions are defined by

L(η) =N∏

k=1

exp[−(ηsk(mH) + bk)](ηsk(mH) + bk)nk

nk!

(A.2)

×nk∏

j=1

ηsk(mH)Sk(�xjk;mH) + bkBk(�xjk)

ηsk(mH) + bk

,

whereη = 1 in the case ofLs+b andη = 0 in the caseof Lb. The indexk runs over all independent contrbutions to the combined result: from different evetopology selections, data taken at different centremass energies and data collected in different expments. The symbolN stands for the number of succontributions (“channels”);nk is the number of observed candidates in channelk and�xjk designates theposition�x of candidatej of channelk in the plane de-fined by the discriminating variablesmrec

H andG (seeSection 2). The quantitiessk(mH) andbk are the inte-grated signal and background rates in channelk. ThefunctionsSk(�xjk;mH) andBk(�xjk) are the probabil-ity density functions (PDFs) of the discriminating vaables for the signal and background. These PDFsevaluated in bins ofmrec

H andG for a set of values fomH with some interpolation and smoothing proceduapplied [16,17]. The test statistic can be written

−2 lnQ(mH) = 2N∑

k=1

[sk(mH)

(A.3)

−nk∑

j=1

ln

(1+ sk(mH)Sk(�xjk;mH)

bkBk(�xjk)

)]

thus becoming a sum of contributions (weights) frothe individual observed events. The above notaassumes that the background-related quantitiesbk andBk(�xjk) do not depend onmH. If the selection criteriain any one channel are explicitlymH dependent (the

searches of L3 and OPAL in the four-jet channel hthis property),bk andBk(�xjk) have to be replaced bbk(mH) andBk(�xjk;mH).

The presence of a signal can be inferred frombehaviour of the confidence 1− CLb for the back-ground hypothesis (also calledp-value, see Ref. [18])which is obtained, for a given test mass, by integratthe corresponding PDF for−2 lnQ from −∞ to theobserved value of the test statistic. The PDFs aretained from detailed simulations of experiments, givthe background hypothesis. If the background hypoesis is correct, 1− CLb is uniformly distributed be-tween zero and one; the median of the distributwould thus be 0.5. In the presence of a significantnal 1−CLb would be very small for the correspondintest mass.

To express a given value of 1− CLb in terms ofstandard deviations (σ ), we adopt a convention (seTable 31.1 of Ref. [18]) where 1− CLb = 2.7× 10−3

(5.7× 10−7) would indicate a 3σ (5σ ) excess beyondthe background expectation. The vertical scales onright-hand side in Figs. 7 and 8 correspond to tconvention.

The frequentist exclusion limit is usually computfrom the confidence CLs+b for the signal plus background hypothesis which, for a given test mass, istained by integrating the corresponding PDF fromobserved value of the test statistic to+∞. The signalplus background hypothesis is considered excludethe 95% confidence level if an observation is masuch that CLs+b is lower than 0.05. However, this procedure may lead to the undesired possibility thalarge downward fluctuation of the background wouallow hypotheses to be excluded for which the expment has no sensitivity due to the small expectednal rate. This problem is avoided by introducing thetio CLs = CLs+b/CLb. Since CLb is a positive numbeless than one, CLs will always be greater than CLs+band the limit obtained will therefore be conservatiWe adopt this quantity for setting exclusion limits aconsider a mass hypothesis to be excluded at theconfidence level if the corresponding value of CLs isless than 0.05.

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