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Precision bounds on mH and mt∗

arXiv:hep-ph/9203210v1 15 Mar 1992Precision bounds on mH and mt∗F. del AguilaDepartamento de F´ısica Te´orica y del CosmosUniversidad de Granada, E-18071 Granada, SpainM.

Mart´ınezPPE Division, CERNCH-1211 Geneva 23, SwitzerlandandM. Quir´os†Theory Division, CERNCH-1211 Geneva 23, SwitzerlandAbstractWe perform a fit to precise electroweak data to determine the Higgs and topmasses.

Penalty functions taking into account their production limits are included.We find mH = 65+245−4GeV and mt = 122+25−20 GeV . However whereas the top χ2distribution behaves properly near the minimum, the Higgs χ2 distribution does not,indicating a statistical fluctuation or new physics.

In fact no significative bound onthe Higgs mass can be given at present. However, if the LEP accuracy is improvedand the top is discovered in the preferred range of top masses, a meaningful boundon the Higgs mass could be obtained within the standard model framework.CERN-TH.6389/92IEM-TH-52/92February 1992CERN-TH.6389/92IEM-TH-52/92∗Work partially supported by CICYT under contracts AEN90-0139, AEN90-0683.†Permanent address: Instituto de Estructura de la Materia, CSIC, Serrano 123, E-28006 Madrid,Spain.0

IntroductionThe standard model of electroweak interactions [1] describes impressively well all experi-mental data [2]. This description depends on two unknown parameters, the top mass, mt,and the Higgs mass, mH.

The present experimental accuracy of the electroweak measure-ments justifies to look for indirect bounds on these masses. In this paper we present theχ2 distributions resulting from the most precise and recent electroweak data as functionsof mt and mH.

Similar analyses can be found in the literature [3, 4, 5]. We pay particularattention to the Higgs penalty function describing the Higgs production limit.

It playsan important role due to the low sensitivity of the data on mH and to the fact that theminimum of the χ2, if only indirect data are used, is (at present) well below the directproduction limit. Actually, the present sensitivity of the data on mH will prove to be toolow to obtain stringent, statistically significant, limits.

However the sensitivity could begreatly improved in the near future, provided that LEP errors get reduced and the top isdiscovered.DataThe experimental values used in the fits are gathered in Table 1. The detailed way LEPexcludes Higgs production can be found in Ref.

[6]. For our discussion it is enough torealize that the final result of the search is that no event with a given signature is found,whereas some events should be present if the Higgs boson would have been produced.

Ofcourse, the number of expected events decreases with increasing mH. Therefore, assuminga Poisson distribution for the number of expected events, values below mH can be excluded(for instance) at the 95% cℓif the number of expected events for mH is larger than orequal to 3.

The number of expected events at LEP, N, as a function of mH is shown inFig.1a (this plot is obtained from the compilation in Ref. [7]).

The confidence level, cℓ,corresponding to seeing no events when N events are expected is equal to 1 −e−N; andthis confidence level is transformed into the corresponding number of standard deviations,x, for a single sided N(0, 1) distribution using the relationcℓ+ 12=1√2πZ x−∞e−t2dt.In this way x can be written as a function of mH. This function is plotted in Fig.1b, andthe best linear fit,x = 15.81 −0.24 mH,is superimposed on it.

Thus, the corresponding contribution to χ2 is given by the penaltyfunction∆χ2H = 65.0 −mH(GeV )4.1!2if mH ≤65 GeV,0if mH > 65 GeV.In the case of the top quark the detailed way TEVATRON excludes top productioncan be found in Ref. [8].

In this case, however, the minimum of the χ2, is well above thedirect production limit, and then the corresponding penalty function plays no important1

role in the fit.The strategy for the top search is also different.What is studied inthe top case is the presence of an excess of events in some distributions (basically inthe transverse mass distribution) due to top production. These distributions are fittedwith a linear superposition of the standard model predictions with no top quark plusthe top production.The result of this fit for each top mass is a probability densitydistribution L(ρ) for the variable ρ defined as the ratio of the number of fitted to thenumber of expected (σt(mt)) top events for such a mass.

No top events corresponds toρ = 0. To exclude a top mass (for instance) with a 95% cℓwe associate to each massmt the value of ρ0(mt) which defines for the probability density corresponding to thismass an area,R ρ0(mt)0L(ρ)dρ, equal to the 95% of the total area or probability,R ∞0 L(ρ)dρ.Then ρ0(mt) = 1 gives the mt limit below which the top mass is excluded.

To makethis computation for different confidence levels we have assumed that ρ0(mt)σt(mt) isindependent of mt in the region of interest. (Note that this is equivalent to assumethat the shape of the top distribution is independent of the top mass in this region.

)To translate the cℓto which a given top mass is excluded into a number of standarddeviations we proceed as in the case of the Higgs mass. The corresponding fit gives theapproximate penalty function∆χ2t = 108 −mt(GeV )9!2if mt ≤108 GeV,0if mt > 108 GeV.The χ2 also includes neutral current data: neutrino-quark data, νq [9], the latestneutrino-electron results, νµe [10], and the parity violation in atoms data, eH [11, 12]; theW mass limits: MW [13] and MWMZ [14]; the LEP results presented at the Joint InternationalLepton–Photon Symposium and Europhysics Conference on High–Energy Physics in Au-gust 1991 [15] (with the correlation matrix given in Ref.

[16]); and the strong couplingconstant, αs [17]. We take for αs the latest (most precise) ALEPH measurement usinghadronic Z decays.

The definite αs value is important to fix the χ2 value at the minimumbut modifies little the relative χ2 distribution and does not affect the conclusions.TheoryWe use the on-shell scheme of Ref. [18] with the electromagnetic coupling constant, α, andthe Fermi constant, Gµ, [9] as input parameters.

For the calculation of the observables inthe Z physics sector two independent electroweak libraries have been used: the one fromG. Burgers and W. Hollik [19] and the one from the Dubna-Zeuthen group [20].

In spiteof the fact that they use a completely different calculational scheme, their results are invery good agreement [21]. We have upgraded the first one to incorporate:• The missing relevant pieces of two-loop corrections [22];• The dominant QCD corrections to mt dependent terms [23, 24];• The updated calculation of the QCD corrections to the decay Z →b¯b [24]; and,2

• O(α3s) corrections to the decay widths [25].Similar upgrades have been implemented in the Dubna-Zeuthen library by their authors.A comparison of the present versions of both programs shows an even better agreementthan the one previously quoted. While the numbers and plots shown in this study havebeen obtained with the first library, we have explicitly checked that the changes arenegligible if the Dubna-Zeuthen library is used instead.

In our fits we use the explicitexpressions for all the observables, in particular for the neutral current ones [26]. We havechecked that reducing the full experimental information for the latter to the equivalentelectroweak mixing angle value, our fits and conclusions change little.InterpretationOur results are summarized in Figs.2-8.

Fig.2a shows the contribution to the χ2 distribu-tion of neutral current and p¯p collider data; Fig.3a that of LEP data and αs; Fig.4a theirsum; and Fig.5a that of all the data including the penalty functions. Thus Figs.2a-4agive partial contributions to Fig.5a.

At each point in the mt −mH plane we minimizewith respect to MZ and αs (Gµ and α being fixed). Thus, the total number of degrees offreedom is 20 −2 = 18.

(The two degrees of freedom corresponding to the penalty func-tions are taken care by the two parameters, mt and mH, with respect to which we do notminimize.) Figs.2b-5b show the same fits but replacing the central experimental valuesof the different observables by their predicted (standard model) values near the minimumof the χ2 distribution.

In particular we have taken mH = 70 GeV , mt = 130 GeV . Thecorresponding standard model predictions for the observables used in the fits are given inTable 2.

Comparing Figs.2a-4a and Fig.5a we see that the Higgs penalty function movesthe minimum from mH = 10 to 70 GeV . However, the main observation to be elaboratedbelow when comparing Fig.3a and Fig.3b is the apparent change in the LEP distribution,and then in the global one.

This tells us that some experimental central values manifesta statistical fluctuation or indicate new physics. This comparison also proves the actuallack of sensitivity of the data on mH.

To understand this change it is illuminating torewrite (in obvious notation)χ2(mH, mt, ...)=Xi(Xexpi−Xthi (mH, mt, ...))2σ2i=Xi(Xexpi−X0i + ǫi)2σ2i=Xi(Xexpi−X0i )2 −2(Xexpi−X0i )ǫi + ǫ2iσ2i,where X0i are the standard model predictions at the χ2 minimum and ǫi are the differencesbetween the predicted values at a given point and the values at the minimum.Thelast expression shows that a statistical fluctuation or a relatively large change in thecentral value of any observable contributes to the χ2 with a relatively large constant term(Xexpi−X0i )2σ2iand with a linear term of pronounced slope −2(Xexpi−X0i )ǫiσ2i. However,what gives the sensitivity of the observable is ǫ2iσ2i.

Comparing Figs.2a-5a and Figs.2b-5b3

we see that indirect LEP data are very insensitive to mH but some observables have alarge discrepancy between their measured central values and the predicted ones. Theglobal fit to all data (Fig.5a) givesmH =65+245−4GeV,mt =122+25−20 GeV.Although the fits in Figs.2a-5a give the actual experimental information, those in Figs.2b-5b give the actual sensitivity of the data on mH and mt.

Hence present indirect elec-troweak data are not sensitive to mH. Actually, the present LEP discrepancy can betraced back to the value of the ratio Rℓand the bottom forward–backward asymmetryAbbF B.

This is to say that the data show a higher sensitivity than expected on mH due tothe pronounced slope terms (introduced above) resulting from Rℓand AbbF B. If this dis-crepancy is a statistical fluctuation and it disappears when more data are available, thesubsequent fit will approach Figs.2b-5b.

However, it can also happen that the central val-ues of other observables fluctuate, in which case our discussion would apply to them. Atany rate this will be always the case (if new physics is discarded) when the preferred mHvalue from indirect LEP data lies below the production limit.

In Figs.6a,b we show the χ2distribution as a function of mH (minimization with respect to mt is understood) for theglobal fits in Figs.5a,b. Plotting the same distribution for the different observables it canbe explicitly proven that only Rℓand AbbF B show the pronounced slope behaviour for thereal data, as can be seen in Fig.6a, whereas the other observables show a small variation,which we do not plot.

Fig.6b confirms that the sensitivity of Rℓand AbbF B on mH is verylow, as it is the sensitivity of the other observables and of the full set. In fact we plot ∆χ2in Fig.6a, subtracting from each χ2 contribution its value at the global minimum: 10.79,0.88, 2.73 for the global, Rℓ, AbbF B contributions, respectively.

Using higher values for αswould arrange the Rℓdeviation from the standard model prediction. This would translateinto a modification of the mt value at the minimum in order to conserve the agreementbetween the measured and the predicted ΓZ values.

However AbbF B would still constrainthe fit and keep unchanged the value of mH at the minimum and the conclusions. Lowervalues of αs make worse the disagreement between the measured and the predicted Rℓvalues.

It is worth to note that although the top mass limit is significative, it is stronglycorrelated to the Higgs mass as can be seen in Fig.5. For instance if the Higgs mass wereknown to be 300 ± 30 GeV , the present data would imply mt = 139+21−25 GeV (see Fig.5a).ConclusionsAs stressed above, today’s data show a fictitious mH sensitivity due to a statistical fluctu-ation or to new physics manifested by the experimental values of Rℓand AbbF B (Figs.2a-6a).As a matter of fact, the actual sensitivity of the considered observables on mH is verylow (Figs.2b-6b).

The Higgs penalty function plays an important role in the fits, push-ing up the Higgs mass value at the minimum. What is more interesting is the expectedimprovement of the mH sensitivity in the near future [27].

Figs.2-5 show the correlationbetween mH and mt which should translate, once the top quark is discovered and its mass4

determined, into a definite bound on mH. In fact, if present LEP accuracy is improved asassumed in Table 2, and the top mass is in the preferred range given above, a meaningfulbound on the Higgs mass (within the standard model framework) could be obtained.

Thisfact is illustrated in Figs.7 and 8. In Fig.7 we plot the χ2-distribution using the standardmodel predictions as central values and the set of LEP improved errors quoted in Table2.

We minimize at each point with respect to MZ and αs. Fig.7 shows that fixing mt arelatively strong upper bound on mH can be deduced.

This is explicitly shown in Fig.8,where we fix the top mass, mt = 130 ± 1 GeV . In this case mH < 315 GeV at 95% cℓ.AcknowledgementsOne of us (M.M.) would like to thank J. Steinberger for triggering the interest for thisstudy as well as for many suggestions and A. Blondel and G. Rolandi for discussions.We have benefitted from discussions with J. Ellis, with W. Hollik about the upgradeof his libraries and also with D. Schaile concerning the actual interpretation of the χ2distribution.

We acknowledge W. Hollik and the Dubna-Zeuthen group for making usavailable their computer programs.5

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Table captionsTable 1 Experimental values of the observables used in the fits.Table 2 Standard model predictions for the observables used in the fits and for mH =70 GeV and mt = 130 GeV . The improved error column gives a guess of futureLEP errors.Figure captionsFig.1a[b] Number of expected events [standard deviations] at LEP as a function of mH.Fig.2a[b] Level contours for the contribution to the χ2-distribution of neutral currentand p¯p collider data in Table 1 [standard model predictions in Table 2 with thepresent experimental errors in Table 1].

Minimization with respect to MZ and αs isunderstood.Fig.3a[b] The same as in Fig.2a[b] but for LEP observables including αs.Fig.4a[b] The same as in Fig.2a[b] but for all observables in Table 1.Fig.5a[b] The same as in Fig.2a[b] but for all observables in Table 1 plus the top andHiggs penalty functions given in the text.Fig.6a[b] χ2-distributions as functions of mH for the global fit in Fig.5a[b] (solid curve).The Rℓ(dashed curve) and the Ab¯bF B (dotted curve) contributions are also shown.Minimization with respect to MZ, αs and mt is understood.Fig.7 Level contours for the total χ2-distribution assuming the standard model predic-tions as central values and the set of improved errors given in Table 2. Minimizationwith respect to MZ and αs is understood.Fig.8 χ2-distribution as a function of mH for the global fit in Fig.7 and mt = 130±1 GeV .Minimization with respect to MZ, αs and mt is understood.8

QuantityExperimental ValueCorrelation Matrixg2L0.2977 ± 0.00421.νqg2R0.0317 ± 0.00341.θL2.50 ± 0.031.θR4.59+0.44−0.271.νµegeA−0.503 ± 0.0181.−0.05geV−0.025 ± 0.0201.C1u + C1d0.144 ± 0.0071.eHC1u −C1d−0.60 ± 0.091.C2u −C2d−0.05 ± 0.111.p¯pMW79.91 ± 0.39 GeVMW/MZ0.8813 ± 0.0041MZ91.175 ± 0.021 GeV1.0.090.010.00ΓZ2487 ± 10 MeV1.–0.25–0.07σ0h41.36 ± 0.23 nb1.0.18Rℓ20.92 ± 0.111.LEPFrom AF B: gℓVgℓA(MZ)!20.0048 ± 0.00121.From Apol(τ):gℓVgℓA(MZ)0.072 ± 0.0171.From b¯b-asymmetry:Ab¯bF B(MZ)0.132 ± 0.0221.From q¯q-asymmetry:sin2 θW(MZ)0.2303 ± 0.00351.αs0.125 ± 0.005Table 19

QuantityStandard model predictionImproved errorsg2L0.30120.0042νqg2R0.03020.0034θL2.460.03θR5.180.44νµegeA−0.5050.018geV−0.0480.020C1u + C1d0.1460.007eHC1u −C1d−0.540.09C2u −C2d−0.090.11p¯pMW80.14 GeV0.06 GeVMW/MZ0.87900.0041MZ91.175 GeV0.005 GeVΓZ2489 MeV4 MeVσ0h41.41 nb0.08 nbRℓ20.810.02LEPFrom AF B: gℓVgℓA(MZ)!20.00480.0006From Apol(τ):gℓVgℓA(MZ)0.0690.009From b¯b-asymmetry:Ab¯bF B(MZ)0.0970.010From q¯q-asymmetry:sin2 θW(MZ)0.23200.0035αs0.1250.005Table 210

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