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How reliably can the Higgs-Boson Mass be predicted from Electroweak Precision Data


BI-TP 96/06 KA-TP-03-96 hep-ph/9602436

How reliably can the Higgs-Boson Mass be predicted from Electroweak Precision Data??

arXiv:hep-ph/9602436v2 27 Jun 1996

S. Dittmaier, D. Schildknecht
Fakult¨t f¨r Physik, Universit¨t Bielefeld, D-33615 Bielefeld, Germany a u a

G. Weiglein
Institut f¨r Theoretische Physik, Universit¨t Karlsruhe, D-76128 Karlsruhe, Germany u a

Revised version June 1996

Abstract From the LEP precision data and the measurement of the W-boson mass, upon excluding the observables Rb , Rc in a combined ?t of the top-quark mass, mt , and the Higgs-boson < mass, MH , within the Standard Model, we ?nd the weak 1σ bound of MH ? 900 GeV. Stronger upper bounds on MH , sometimes presented in the literature, rely heavily on the inclusion of Rb in the data sample. Upon including Rb , the quality of the ?t drastically decreases, and by carefully analyzing the dependence of the ?t results on the set of experimental input data we conclude that these stronger bounds are not reliable. Moreover, the stronger bounds on MH are lost if the deviation between theory and experiment in Rb is ascribed to contributions of new physics. Replacing s2 (LEP) by the combined value ?W 2 < sW (LEP + SLD) in the data sample leads to a bound of MH ? 430 GeV at the 1σ level. ? 2 The value of sW (SLD) taken alone, however, gives rise to ?t results for MH which are in ? con?ict with MH > 65.2 GeV from direct searches. ?

? Partially supported by the EC-network contract CHRX-CT94-0579 and the Bundesministerium f¨ r u Bildung und Forschung, Bonn, Germany.

The discovery [1] of the top quark and the direct determination of its mass of mexp = t 180 ± 12 GeV open the possibility of improving the constraints on the mass of the Higgs boson, MH , from the body of the precision electroweak data at the Z-boson resonance [2, 3] and the experimental value of the W-boson mass, MW [4]. In this note we present our results for mt and MH , obtained by performing ?ts to the precision data and MW within the Standard Model (SM). The dependence of the ?ts on the experimental data on Rb , 2 Rc , and s2 is investigated, and the e?ects of varying the SM input parameters α(MZ ) ?W 2 and αs (MZ ) in the allowed range are discussed. We also examine how the results of these ? c ?ts are in?uenced if one allows for non-standard Z → bb,c? vertices. Even though several papers on this subject have appeared recently [5, 6, 7], additional investigations combined with comments on the interpretation of the results seem useful. We start from a global ?t to the available electroweak precision data. The large value of 2 χmin /d.o.f. , obtained in the ?t to be given below, requires a detailed analysis of the impact of di?erent parts of the experimental data. Accordingly, we will subsequently analyze the data in several distinct steps. In a ?rst step we concentrate on the leptonic observables Γl and s2 (LEP), and MW (the set of data to be referred to as “leptonic sector”), and include ?W the total Z-boson width, ΓT , and the Z-boson width into hadrons, Γh , from the set of hadronic observables (referring to this set of data as “all data \ Rb , Rc ”), thus ignoring in ? c the ?ts at this stage the partial Z-boson decays Z → bb,c?. In a second step we include the ? decay mode and determine mt and MH in a ?t again. In a third step we ?nally Z → bb discuss how the results of the ?ts change when the decay Z → c? is included, and we c also investigate the e?ect of replacing s2 (LEP) by s2 (SLD) and by the combined value of ?W ?W s2 (LEP+SLD) in the set of data. Within all steps we carry out two alternative ?ts, a ?rst ?W one in which the Tevatron result of mexp = 180±12 GeV is included in the ?t, and a second t one in which mt is treated as a free ?t parameter. The procedure adopted obviously allows us to identify the dependence of the results for mt and MH on the set of experimental data used in the ?ts. The various steps are motivated by the discrepancies [2] between SM ? c prediction and experiment observed in the Z → bb,c? decays and the di?erence between the LEP and SLD results for s2 . By including/excluding the experimental information on ?W mt we furthermore investigate how strongly the ?t results for mt and MH are correlated. In a ?nal step we discuss ?ts in which non-standard contributions are allowed. The set of experimental data is listed in Tab. 1. A subset of the data in Tab. 1 is referred to as “input parameters”. This is motivated by the high experimental accuracy of some of the quantities (namely G? and MZ ) and by the non-electroweak origin of others 2 2 (α(MZ ), αs (MZ )). The listed “input parameters” represent the commonly used input for theoretical predictions in the on-shell renormalization scheme. The experimental error of G? is entirely negligible with respect to the determination of mt and MH . This is also true for MZ . For completeness, this was explicitly veri?ed by treating MZ as additional 2 ?t parameter. If not otherwise indicated, the parameter α(MZ ) will be treated as ?t 2 parameter employing the constraint of Tab. 1. Finally, we note that the value of αs (MZ ) given in Tab. 1 is the result from the LEP event shape analysis [2]. Due to the fact that this value disagrees with results from di?erent experiments (e.g. deep-inelastic scattering) 2 and lattice calculations, αs (MZ ) will either be treated as free ?t parameter or the in?uence 2 of varying αs (MZ ) as input parameter will be studied separately. For the input parameter mb , the mass of the b quark, the value of mb from Tab. 1 will be inserted. A detailed 1

leptonic sector Γl = 83.93 ± 0.14 MeV

hadronic sector R = 20.788 ± 0.032 σh = 41.488 ± 0.078 Rb = 0.2219 ± 0.0017 Rc = 0.1543 ± 0.0074 ΓT = 2496.3 ± 3.2 MeV Γh = 1744.8 ± 3.0 MeV Γb = 387.2 ± 3.0 MeV Γc = 269 ± 13 MeV

s2 (LEP) = 0.23186 ± 0.00034 ?W s2 (LEP + SLD) = 0.23143 ± 0.00028 ?W MW = 80.26 ± 0.16 GeV input parameters MZ = 91.1884 ± 0.0022 GeV
2 α(MZ )?1 = 128.89 ± 0.09 2 αs (MZ ) = 0.123 ± 0.006

s2 (SLD) = 0.23049 ± 0.00050 ?W

correlation matrices

G? = 1.16639(2) · 10?5 GeV?2 σh R

σh 1.00 0.15

R

ΓT Rb Rb Rc

0.15 ?0.12 1.00 ?0.01 1.00

mb = 4.5 GeV mt = 180 ± 12 GeV

1.00 ?0.34 1.00

ΓT ?0.12 ?0.01

Rc ?0.34

Table 1: The precision data used in the ?ts, consisting of the LEP data [2], the SLD value [3] for s2 , and the world average [4] for MW . The partial widths Γl , Γh , Γb , and ?W 2 Γc are obtained from the observables R = Γh /Γl , σh = (12πΓl Γh )/(MZ Γ2 ), Rb = Γb /Γh , T Rc = Γc /Γh , and ΓT using the given correlation matrices. The data in the upper left-hand column (using s2 (LEP) if not otherwise speci?ed) will be referred to as “leptonic sector” ?W in the ?ts. Inclusion of the data in the upper right-hand column will be referred to as ?tting “all data”. The theoretical predictions are based on the input parameters [1, 2, 8] given in the lower left-hand column of the table. analysis reveals that the results for mt and MH are independent of the precise value of mb for any reasonable changes of mb . Otherwise, the notation in Tab. 1 is standard. The partial Z-boson width into a lepton and an anti-lepton, assuming universality, is ? denoted by Γl . The partial widths for Z → bb and Z → c? are given by Γb and Γc . c 2 Finally, the e?ective electroweak angle, sW , in Tab. 1 is de?ned by the e?ective vector ? and axial vector couplings (gV,l and gA,l, respectively) of the Z boson to leptons at the lept Z resonance, s2 ≡ sin2 θe? ≡ (1 ? gV,l /gA,l )/4. It is accordingly extracted from the ?W asymmetry measurements at LEP [2] and SLD [3]. The theoretical SM results at the one-loop level, taking into account leading two-loop contributions, are taken from Refs. [9, 10]1 . Therefore, we provide an analysis which is completely independent of results presented by other authors [5, 6, 7].
We have supplemented the analytical results given in Refs. [9, 10] by the O(G? m2 α2 ) corrections [11] t s to the ρ-parameter.
1

2

We obtain for the global ?t to the complete set of data listed in Tab. 1 mt = 167+11 GeV, ?9
2 α(MZ )?1 = 128.90 ± 0.09,

MH = 81+144 GeV, ?52
2 αs (MZ ) = 0.121 ± 0.004,

χ2 /d.o.f. = 17/9, min

(1)

2 where the combined value s2 (LEP + SLD) has been used. In this ?t αs (MZ ) has been ?W 2 used as a free ?t parameter, while the experimental constraints on α(MZ ) and mt have been included. The result (1) is in good agreement with the corresponding results given in Refs. [6, 7].2 While the low central value and the rather tight 1σ bounds obtained for MH in this ?t seem to indicate evidence for a light Higgs-boson mass, the high value of χ2 /d.o.f. = 17/9 gives rise to the question of how reliable this bound actually is. min In order to investigate the dependence of the ?t results on inclusion/exclusion of di?erent parts of the experimental data, we now turn to an analysis in distinct steps as outlined 2 2 above. The results for the corresponding ?ts of the parameters (mt , MH , α(MZ ), αs (MZ )) 2 within the SM are presented in Tab. 2 and in the (MH , ?χ )-plots of Fig. 1. In these ?ts 2 2 αs (MZ ) is treated as a free ?t parameter while α(MZ ) is ?tted including the experimental 2 constraint from Tab. 1. Note that the values obtained for αs (MZ ) in these ?ts practically coincide with the value of Tab. 1 which is deduced by the entirely di?erent method of an event-shape (jet production) analysis. As mentioned above, mt is treated in two di?erent ways in the ?ts. Treating mt as a free ?t parameter allows to compare its ?t result with its actual experimental value, while using this information in the ?t from the start leads to a certain “compromise” result which might be more di?cult to interpret. In Fig. 2 we furthermore investigate the dependence of the ?t results on variations in 2 2 α(MZ ) and αs (MZ ). To this end these parameters are not ?tted but kept as ?xed values which are varied within the 1σ bounds of their experimental values given in Tab. 1. The top-quark mass is treated as a free ?t parameter in this ?gure. In the last row of Fig. 2 the e?ect of replacing s2 (LEP) by s2 (LEP + SLD) and by s2 (SLD) is studied. ?W ?W ?W We ?rst of all concentrate on the results of the ?rst step of our analysis, namely the ?ts in Tab. 2a and Fig. 2 based on the data sets s2 (LEP), Γl and MW (“leptonic ?W 2 sector”) and sW (LEP), Γl , MW , ΓT , Γh (“all data \ Rb , Rc ”). Both ?ts yield an excellent ? 2 2 χ2 /d.o.f. < 1, independently of whether α(MZ ) and αs (MZ ) are ?tted or whether they are min taken as ?xed input parameters that are varied within one standard deviation according to Tab. 1. Figure 2 shows that the results of the ?ts are strongly a?ected by variations 2 2 of α(MZ )?1 . For instance, lowering α(MZ )?1 = 128.89 by one standard deviation to 2 α(MZ )?1 = 128.80 also lowers the central value of MH by approximately one standard 2 2 2 deviation. Varying αs (MZ ) from αs (MZ ) = 0.123 to αs (MZ ) = 0.117 shifts the upper 1σ limit of MH from ? 1 TeV to 265 GeV in the ?t in which Γh and ΓT are included (second column of Fig. 2). We also note the somewhat low values of the top-quark mass 2 of mt = 157 GeV and mt = 162 GeV obtained for the lower values of α(MZ )?1 and 2 αs (MZ ), respectively, which are below the 1σ lower limit of mt = 168 GeV from the direct measurement of mt . The ?t results in the leptonic sector are stable under variation in 2 2 the strong coupling constant, αs (MZ ), since αs (MZ ) only enters at the two-loop level.

In Ref. [6] also the available low-energy data were included in the analysis, which shows that the e?ect of these data on the results of the SM ?ts is rather small.

2

3

Table 2a: using s2 (LEP) ?W leptonic sector + mexp t leptonic sector all data + mexp \ Rb , Rc t all data all data + mexp \ Rb t all data all data + mexp \ Rc t all data all data + mexp t all data Table 2b: using s2 (LEP + SLD) ?W leptonic sector + mexp t leptonic sector all data + mexp \ Rb , Rc t all data all data + mexp \ Rb t all data all data + mexp \ Rc t all data all data + mexp t all data \ Rc \ Rb \ Rb , Rc mt / GeV 175+12 ?11 165+18 ?10 176+12 ?12 161+21 ?12 175+12 ?11 160+19 ?12 167+11 ?9 152+11 ?11 167+11 ?9 153+11 ?11 MH / GeV 152+282 ?106 64+223 ?37 154+273 ?108 51+214 ?31 148+263 ?103 49+174 ?29 76+136 ?49 34+46 ?17 81+144 ?52 35+50 ?18
2 αs (MZ )

mt / GeV 179+12 ?11 174+37 ?19 179+12 ?12 167+45 ?20 178+12 ?12 164+40 ?18 169+11 ?11 148+14 ?12 170+11 ?11 149+15 ?12

MH / GeV 353+540 ?224 248?194 356+543 ?227 163?126 343+523 ?219 133?97 54+93 ?30 197+291 ?126 57+104 ?32
+
> ?

2 αs (MZ )

χ2 /d.o.f. min 0.2/5 0.2/4 0.7/7 0.6/6 6.6/8 6.4/7 15/8 12/7 16/9 14/8

0.123 (?xed) 0.123 (?xed) 0.124+0.004 ?0.004 0.123+0.007 ?0.004 0.124+0.004 ?0.004 0.123+0.006 ?0.004 0.123+0.004 ?0.004 0.122+0.004 ?0.004 0.123+0.004 ?0.004 0.122+0.004 ?0.004

+

> ?

1000

\ Rb , Rc \ Rb \ Rc

+

> ?

1000

1000

186+277 ?119

χ2 /d.o.f. min 1.0/5 0.3/4 1.5/7 0.7/6 7.3/8 6.5/7 15/8 12/7 17/9 14/8

0.123 (?xed) 0.123 (?xed) 0.122+0.004 ?0.004 0.121+0.007 ?0.004 0.122+0.004 ?0.004 0.121+0.004 ?0.004 0.121+0.004 ?0.004 0.122+0.004 ?0.004 0.121+0.004 ?0.004 0.121+0.004 ?0.004

2 2 Table 2: The results obtained in (mt , MH , α(MZ ), αs (MZ )) ?ts to di?erent sets of experimental data, as indicated (see text). The results in Tab. 2a are based on s2 (LEP), ?W 2 while the results in Tab. 2b are based on sW (LEP + SLD). For each set of experimental ? data, the ?t results given in the lower row are obtained by treating mt as a free ?t parameter, while the results in the upper row include the constraint mexp = 180 ± 12 GeV. t 2 Note that the ?t results on α(MZ ) are not explicitly stated, because they range between 2 2 α(MZ )?1 = 128.89 ± 0.09 and α(MZ )?1 = 128.91 ± 0.09 for all cases, thus reproducing the input value from Tab. 1.

4

mt / GeV ?xed

144

168

180

192

MH / GeV (χ2 /d.o.f. ) min leptonic sector all data \ Rb , Rc all data \ Rb all data \ Rc all data 47+30 (4.2/3) 160+104 (0.2/3) 362+206 (0.2/3) 792+444 (0.4/3) ?18 ?69 ?136 ?280 44+34 (2.1/5) 172+110 (0.6/5) 349+196 (0.8/5) 682+368 (1.3/5) ?20 ?73 ?131 ?239 44+34 (7.7/6) 174+111 (6.4/6) 353+199 (6.7/6) 689+375 (7.3/6) ?20 ?73 ?133 ?242 45+35 (12/6) ?21 45+35 (14/7) ?21 176+112 (14/6) ?74 176+112 (15/7) ?74 355+199 (16/6) ?133 355+199 (17/7) ?133 685+368 (19/6) ?239 686+369 (20/7) ?240

Table 3: Fits of MH to various sets of experimental data in the SM for ?xed values of mt . 2 2 In all ?ts α(MZ )?1 = 128.89 and αs (MZ ) = 0.123 are kept ?xed, and the LEP value of s2 ?W is used in the input data.
2 2 Altogether, we thus conclude that varying α(MZ )?1 and αs (MZ ) within the 1σ bounds given in Tab. 1 leads to a considerable e?ect concerning the boundaries in the (mt , MH ) plane. Consequently, concerning the range of MH allowed by the results from the leptonic sector (with s2 (LEP)) and ΓT , Γh , i.e. by ?tting the SM only to those data that agree with ?W the theoretical predictions, according to the foregoing discussion of Tab. 2a and Fig. 2, it < seems hardly possible to deduce stronger limits than MH ? 900 GeV at the 1σ level, even exp upon taking into account the constraint of mt = 180±12 GeV from the direct observation of the top quark. We turn to the second step of our analysis and include Rb in the ?t, which is thus based on the leptonic sector in conjunction with ΓT , Γh and Rb . According to Tab. 2a ? and Fig. 2, taking into account the data for the Z → bb partial width leads to an increase 2 of χmin /d.o.f. by about an order of magnitude. Comparing the third column in Fig. 2 with the ?rst and second columns, one observes a considerable shrinkage of the 1σ regions in the (mt , MH ) plane and a drastic shift towards lower values of mt and MH . The sensitivity 2 2 against variations of α(MZ )?1 and αs (MZ ) is considerably weaker in this sample of data. The central ?t-value for the top-quark mass of mt = 148+14 GeV (where the experimental ?12 constraint on mt has not been taken into account in the ?t) is signi?cantly below the central value of the direct measurement of mexp = 180 ± 12 GeV, and the central value obtained t for the Higgs-boson mass, MH = 54+93 GeV, lies in the vicinity of the experimental lower ?30 bound MH > 65.2 GeV. The large increase of χ2 /d.o.f. when including Rb in the ?t signals the large discrepancy min between theory and experiment in this ?t. In particular, when evaluating Rb for the best-?t values of (mt , MH ) = (148+14 GeV, 54+93 GeV), the resulting (MH -insensitive) theoretical ?12 ?30 SM prediction, Rb = 0.2164?0.0005 (with the errors indicating the changes by varying mt +0.0004 within the 1σ limits), still lies more than 3σ below the experimental value of Rb = 0.2219± 0.0017. In connection with the low central ?t value of mt = 148 GeV, it is illuminating to consider the results of single-parameter MH ?ts, where mt is kept ?xed at certain (assumed)

5

values. In Tab. 3, again for the previously selected sets of data, results of single-parameter MH ?ts are shown. The known strong (mt , MH ) correlation in SM ?ts leads to a remarkable stability of the resulting ?t values for MH . Once mt is ?xed, there is almost no dependence of the ?t value for MH on which set of input data is actually used in the ?t. In particular, whenever a low value of mt is chosen, one obtains a low value for MH , independently of whether Rb is included in the ?t or not.3 Since the (MH -insensitive) SM prediction for Rb increases with decreasing mt , in the combined ?t of (mt , MH ) the inclusion of Rb lowers the ?t value of mt , and via the (mt , MH ) correlation also the value of MH . As discussed above, the result of mt = 148 GeV is nothing but a kind of compromise, as it still leads to a 3σ discrepancy between theory and experiment in Rb . Moreover, this result for mt is disfavored by the Tevatron result of mexp = 180 ± 12 GeV. t While the problematic features of the ?ts where Rb is included are easy to see in the case where mt is used as a free ?t parameter, they are somewhat hidden in the ?ts where the experimental information on mt is used. It partially compensates the tendency of the ?ts towards low values of mt and leads to the more moderate looking result of mt = 169 ± 11 GeV and MH = 186+277 GeV. In view of the foregoing discussion, however, ?119 the result for MH obtained in this way appears to be rather questionable. In summary, the large value of χ2 /d.o.f. and the low ?t value for mt (when mt is min treated as free ?t parameter) that is at variance with the Tevatron result, lead to the conclusion that the low value and tight bound obtained for MH when including the data for Rb does not seem reliable. It is an artifact of the procedure of describing the “nonstandard” value of Rb by the unmodi?ed SM in conjunction with the (mt , MH ) correlation. This conclusion is strengthened by the fact that a simple phenomenological modi?cation ? of the Z → bb vertex, to be discussed below, leads to values of mt compatible with the Tevatron result and removes the stringent upper bounds on MH . We turn to the third step of our analysis and consider the impact of the observable Rc . As can be seen in Tab. 2, the results for mt and MH are hardly a?ected by including Rc . This is a consequence of the fact that the contribution of Rc to χ2 depends only very weakly on mt and MH , because the experimental error for Rc is much larger than the change in the SM prediction for Rc induced by varying mt and MH . Similarly to the case of Rb , including Rc in the set of data (and omitting Rb ) leads to an enhanced value of χ2 /d.o.f. and to a tendency towards lower values of MH . min So far the analysis has been based on the LEP experimental value of s2 (LEP). Table 2b ?W 2 2 and the last row of Fig. 2 show the e?ect of replacing sW (LEP) by sW (LEP + SLD). In ? ? Fig. 2 also contours are shown that are based on taking s2 (SLD) alone. The change ?W in the allowed (mt , MH ) plane occurring as a consequence of these replacements is very strong. For the “leptonic sector” and “all data \ Rb , Rc ” the ?t value of mt ? 170 GeV (where mexp has not been included in the ?t) is consistent with the value from the direct t measurements, mexp = 180±12 GeV, while the values for MH resulting from using s2 (SLD) ?W t now have decreased to MH = 18+28 GeV and MH = 16+27 GeV, respectively. The ?t to “all ?9 ?9 data” using s2 (SLD) yields mt = 161+10 GeV and a similarly low value for MH , namely ?W ?11
Conversely, if MH is ?xed, the values of mt obtained in the ?t are fairly stable, independently of whether Rb or Rc are included in the ?t or not. This is consistent with the results of Ref. [2], where MH = 300 GeV is kept ?xed when ?tting mt .
3

6

MH = 18+24 GeV. Accordingly, using the SLD value for s2 leads to very low ?t results ?W ?9 for MH , independently of whether Rb and Rc are included in the data set or not, and of whether use is made of the experimental information on mt . Comparing these results to the lower bound from the direct Higgs-boson search, MH > 65.2 GeV [12], one arrives at a serious con?ict between the unmodi?ed SM and experiment. The discrepancy is weakened if the combined value of s2 (LEP + SLD) from Tab. 2b is used. In this case one obtains ?W mt = 153 ± 11 GeV and MH = 35+50 GeV for “all data” (where again mexp has not been t ?18 included). A resolution of the LEP–SLD discrepancy on s2 is obviously one of the most ?W important tasks with respect to the issue of MH bounds via radiative corrections. As a summary of the present situation concerning MH , in Fig. 1 we present the result 2 2 of selected (mt , MH , α(MZ ), αs (MZ )) ?ts according to Tab. 2 in a (MH , ?χ2 ) plot. The quantitative in?uence on the ?t value of MH resulting from inclusion of mexp = 180 ± t 12 GeV can be seen to agree with the qualitative expectations from Fig. 2. Other features of the results for MH previously read o? from Fig. 2, such as the correlation between MH and the input for s2 , or the e?ect of ignoring the experimental results for Rb , Rc can ?W obviously also be seen in Fig. 1. The plots in Fig. 1 clearly illustrate the di?culty of establishing a unique bound on MH . The most reliable bound, from “all data \ Rb , Rc ”, < < but including mexp yields MH ? 430 GeV based on s2 (LEP + SLD), and MH ? 900 GeV ?W t 2 based on sW (LEP). ? In order to accommodate the experimental result for Rb , we now allow for a modi?cation ? of the Z → bb vertex by a parameter ?yb , as introduced in Ref. [10].4 The possible origin of this modi?cation of the SM predictions is left open for the time being, but in particular ? it includes the impact of new particles in conjunction with loop corrections at the Z → bb 2 2 vertex. We allow for values of αs (MZ ) di?erent [14] from the LEP value of αs (MZ ) = 0.123, in order to compensate for the enhanced theoretical value of the total hadronic Z-boson width, Γh , resulting from the enlarged theoretical value of Γb which is adjusted to be in agreement with experiment. Deviations of ?yb from its (mt -dependent) SM value SM ?yb [10] lead to an extra contribution Xb [2] in the prediction for Γb , Xb = Γb ? ΓSM = b
2 α(MZ )MZ 2 SM (2s0 ? 3) RQED RQCD ?yb ? ?yb 24s2 c2 0 0

√ 2 2 where s2 c2 = s2 (1 ? s2 ) = πα(MZ )/ 2G? MZ , RQED = 1 + α/12π, and RQCD = 1 + 0 0 0 0 2 2 2 αs (MZ )/π + 1.41(αs (MZ )/π)2 ? 12.8(αs (MZ )/π)3 according to Ref. [15]. 2 In Tab. 4 we present our results for four-parameter (mt , MH , ?yb , α(MZ )) ?ts with ?xed 2 2 2 values of αs (MZ ), as well as the results of ?ve-parameter (mt , MH , ?yb , α(MZ ), αs (MZ )) ?ts. Table 4a is based on the data set “all data + mexp \ Rc ” (experimental information t on mt included), while in Tab. 4b we have used “all data \ Rc ” (mt treated as a free ?t parameter). The conclusion from Tab. 4 is simple: once one allows for a modi?cation of Rb by the parameter ?yb , the bounds on MH obtained by ?tting within the unmodi?ed SM are lost. The quality of the ?t is improved considerably, if one allows for a value
The parameter ?yb is related to the parameter εb introduced in Ref. [13] via ?yb = ?2εb ? 0.2 × 10?3 (see Ref. [10]).
4

SM = ?0.421 GeV × RQCD ?yb ? ?yb ,

(2)

7

Table 4a (“all data + mexp \ Rc ”): t
2 αs (MZ )

mt / GeV 179+11 ?11 179+11 ?11 179+12 ?11 179+12 ?12

MH / GeV
? 582?324 ? 523?302

?yb /10?3 3.9+4.6 ?4.6 ?18.6+4.6 ?4.6 ?20.9+8.9 ?8.9 ?yb /10?3 3.8+4.6 ?4.6 ?18.6+4.7 ?4.6 ?21.0+8.9 ?9.0 ?8.8+4.6 ?4.6 ?8.8+4.6 ?4.6

χ2 /d.o.f. min 11/8 3.4/8 0.9/8 0.9/8

0.123 ?xed 0.110 ?xed 0.100 ?xed 0.098 ± 0.008 ?tted
2 αs (MZ )

+ > 1000 + > 1000

+ > 1000 ? 472?284

459?281

+

> ?

1000

Table 4b (“all data \ Rc ”): mt / GeV 173+28 ?22 174+32 ?23 172+37 ?24 171+39 ?24 MH / GeV
? 414?294

χ2 /d.o.f. min 11/7 3.3/7 0.9/7 0.8/7

0.123 ?xed 0.110 ?xed 0.100 ?xed 0.098 ± 0.008 ?tted

+ > 1000 +
> ?

375?284 269?216
+
> ?

1000

+ > 1000 ? 300?236 1000

Table 4: The results of four-parameter and ?ve-parameter ?ts to “all data \ Rc ” with 2 2 mexp in-/excluded. In the four-parameter (mt , MH , ?yb , α(MZ )) ?ts αs (MZ ) is kept ?xed t 2 2 as indicated (?rst three rows), while in the ?ve-parameter (mt , MH , ?yb , α(MZ ), αs (MZ )) 2 2 ?ts (last row) αs (MZ ) is treated as a free ?t parameter. The ?t results for α(MZ ) are 2 ?1 2 ?1 again omitted, since they merely vary between α(MZ ) = 128.90 ± 0.09 and α(MZ ) = 128.92 ± 0.09.
2 of αs (MZ ) substantially below the LEP result from the event shape measurement [2] of 2 2 αs (MZ ) = 0.123 ± 0.006. Fitting also αs (MZ ) leads to the extremely low best-?t value 2 2 of αs (MZ ) = 0.098 ± 0.008, which is even lower than the extrapolated value of αs (MZ ) 2 from low-energy deep inelastic scattering data, αs (MZ ) = 0.112 ± 0.004 [15], and the value 2 obtained from lattice QCD calculations, αs (MZ ) = 0.115 ± 0.003 [15]. The values of mt in Tab. 4b roughly coincide with the ones obtained in the SM ?ts to the “leptonic sector” 2 given in Tab. 2a. For low αs (MZ ) also the MH bounds in Tab. 4b are similar to the results of the SM ?t obtained for the “leptonic sector” (Tab. 2a). As in the previous case of the pure SM ?ts, the results do not change qualitatively 2 2 when Rc is included in the data set. In the (mt , MH , ?yb , α(MZ ), αs (MZ )) ?t to “all data + mexp ” we obtain t

mt = 179 ± 12 GeV,
2 αs (MZ ) = 0.102 ± 0.008,

? MH = 440?270

+ > 1000

GeV,

α(M2 )?1 = 128.90 ± 0.09, Z χ2 /d.o.f. = 5.8/9. min (3)

?yb = ( ?15.2+8.5 ) × 10?3, ?8.6

This is in good agreement with the results presented in Ref. [2] for a ?t of (mt , αs , Xb ) for MH ?xed at MH = 300 GeV. The increased value of χ2 /d.o.f. in (3) relative to min the corresponding value in Tab. 4 is of course a consequence of the 2.5σ discrepancy [2] in Rc . However, it does not seem to be meaningful to introduce an additional non8

2 αs (MZ )

0.123 0

0.099

0.123 9.3/6 9.9 ± 4.2 7.0 ± 4.3 ?4.2 ± 1.5

0.099 0.5/6 10.1 ± 4.2 5.7 ± 4.3 ?5.0 ± 1.5

χ2 /d.o.f. min ?xexp /10?3 ?y exp /10?3 εexp /10?3
exp ?yb /10?3 exp ?yh /10?3 exp ?yν /10?3

10.1 ± 4.2 5.4 ± 4.3 ?5.3 ± 1.6 ?14.6 ± 6.9 ?20.9 ± 7.0 4.9 ± 2.3 ?1.7 ± 2.3 0.6 ± 5.2

0.7 ± 4.7 ?20.5 ± 4.7 ?1.4 (from theory) ?3.0 (from theory)

2 Table 5: Experimental results for the e?ective parameters for αs (MZ ) = 0.123 and the 2 low value αs (MZ ) = 0.099. The entries on the left-hand side are obtained by determining the six e?ective parameters from the six observables Γl , s2 (LEP), MW , ΓT , Γh , and Γb . ?W On the right-hand side ?yh and ?yν have been taken from theory, and the remaining 2 2 parameters have been ?tted twice, namely for ?xed αs (MZ ) = 0.123 and with αs (MZ ) as 2 additional ?t parameter, resulting in αs (MZ ) = 0.099.

standard parameter ?yc in order to accommodate the Rc discrepancy. On the one hand, ? a modi?cation of the Z → c? vertex is much less motivated than in the case of the Z → bb c vertex (see e.g. the discussion in Ref. [10]); on the other hand, a ?t in which a non-standard 2 ?yc is allowed yields the absurd value αs (MZ ) = 0.19 ± 0.04, which was also obtained e.g. in Refs. [2, 7, 15]. 2 2 As a ?nal point, we compare the (mt , MH , ?yb , α(MZ ), αs (MZ ))-?t discussed above with an analysis based on phenomenological e?ective parameters. The six observables Γl , s2 (LEP), MW and ΓT , Γh , Γb can be represented as linear combinations of six phenomeno?W logical parameters ?x, ?y, ε and ?yh = (?yu + ?yd )/2 + (s2 /6c2 )(?yd ? ?yu ), ?yb , ?yν 0 0 that describe possible sources of SU(2) violation within an e?ective Lagrangian for electroweak interactions at the Z-boson resonance [10]. We assume that the QCD corrections, such as RQCD , which enter ΓT , Γh , and Γb , have standard form. These corrections are extracted from the experimental data before the determination of the e?ective parameters 2 (see Ref. [10]), which therefore quantify all electroweak corrections to the α(MZ )-Born approximation. The results of extracting the experimental values of the six parameters ?xexp etc. from the six observables by inverting the system of linear equations is shown on the left-hand side exp exp of Tab. 5. The αs -dependence only a?ects ?yb and ?yh , since the leptonic sector by itself determines ?xexp , ?y exp and εexp . The values of ?xexp , ?y exp and εexp are in excellent agreement with their SM predictions, as discussed in detail for the data of Refs. [2, 3] in exp 2 Ref. [16]. For αs (MZ ) = 0.123, in addition to the non-standard value of ?yb , also the exp SM parameter ?yh disagrees with the theoretical prediction [10] of ?yh = ?3.0 × 10?3 .

9

Agreement between SM and experiment in ?yh is achieved, however, for low values of 2 2 αs (MZ ), such as αs (MZ ) = 0.099. Noting that the process-speci?c parameters ?yh and ?yν only depend [10] on the empirically well-established5 couplings between vector-bosons and light fermions (i.e. all fermions except for top and bottom quarks), we now impose the SM values for ?yh and ?yν and determine the remaining parameters in a ?t. According to the right-hand side 2 of Tab. 5, for ?xed αs (MZ ) = 0.123, we ?nd a rather poor quality of the ?t (χ2 /d.o.f. = min 2 9.3/6). Allowing for αs (MZ ) as additional ?t parameter, we obtain an excellent quality of 2 2 the ?t (χ2 /d.o.f. = 0.5/6), and for αs (MZ ) the low value of αs (MZ ) = 0.099 deliberately min chosen before. 2 It has thus been shown that the low value for αs (MZ ), as a consequence of the experimental value of Rb , emerges independently of much of the details of electroweak radiative corrections. The two very weak assumptions made here, namely standard form of the QCD 2 corrections and of ?yν and ?yh , already imply the very low value of αs (MZ ) = 0.099±0.008 in the ?t which includes Rb . This value is very close to the one obtained above in the 2 2 (mt , MH , ?yb , α(MZ ), αs (MZ )) ?t, where non-standard contributions have only been al? vertex. Moreover, the values of ?yb = 0.7 ± 4.7 for αs (M 2 ) = 0.123 lowed in the Z → bb Z 2 and of ?yb = ?20.5 ± 4.7 for αs (MZ ) = 0.099 (as well as ?xexp , ?y exp , εexp ) obtained in the present analysis are in good agreement with the values given in Tab. 4 for the 2 (mt , MH , ?yb , α(MZ )) ?t. In both treatments a decent value of χ2 /d.o.f. therefore remin 2 quires a value of αs (MZ ) that, in the best-?t case, is four standard deviations below 2 αs (MZ ) = 0.123 ± 0.006. In summary, from our analysis of the precision data at the Z-boson resonance and MW , we ?nd that a Higgs-boson mass lying in the perturbative regime of the Standard Model, i.e. below 1 TeV, is indeed favored at the 1σ level. Having investigated in much detail the ? c impact of the data for the Z → bb,c? decay modes and the experimental value of s2 (SLD), ?W 2 as well as the in?uence of the uncertainties connected with the input parameters α(MZ ) 2 < and αs (MZ ), we conclude that a stronger upper 1σ bound on MH than MH ? 900 GeV < based on s2 (LEP) and MH ? 430 GeV based on s2 (LEP + SLD) can hardly be justi?ed ?W ?W from the data at present. The stringent bounds on MH that are obtained when the unmodi?ed Standard Model is ?tted to the complete data sample are immediately lost when Rb and s2 (SLD) are excluded from the analysis or, as demonstrated for the case of ?W Rb , if non-standard contributions are allowed in the theoretical model. The well-known fact that allowing for a non-standard contribution to Rb gives rise to an extremely low 2 value of αs (MZ ) has been shown to emerge already under the weak theoretical assumptions of standard QCD corrections and standard form of the couplings of the gauge-bosons to the leptons and to the quarks of the ?rst two generations. Note added in proof: The most recent value from the Tevatron on mt is given by = 175 ± 9 GeV. All essential conclusions of the present work, based on mexp = t 180 ± 12 GeV, remain valid if this most recent value of mexp is used. t mexp t
5

Here we ignore the Rc problem, previously commented upon.

10

Acknowledgement
G.W. thanks A. Djouadi for useful discussions.

References
[1] CDF collaboration, F. Abe et al., Phys. Rev. Lett. 74 (1995) 2626; D? collaboration, S. Abachi et al., Phys. Rev. Lett. 74 (1995) 2632. [2] LEP Collaboration ALEPH, DELPHI, L3, OPAL and the LEP Electroweak Working Group, Data presented at the 1995 Summer Conferences, LEPEWWG/95-02. [3] SLD Collaboration: K. Abe et al., Phys. Rev. Lett. 73 (1994) 25; SLD Collaboration: K. Abe et al., contributed paper to EPS-HEP-95, Brussels, eps0654. [4] UA(2) Collaboration: J. Alitti et al., Phys. Lett. B276 (1992) 354; CDF Collaboration: F. Abe et al., Phys. Rev. Lett. 65 (1990) 2243, Phys. Rev. D43 (1991) 2070; CDF Collaboration: F. Abe et al., Phys. Rev. Lett. 75 (1995) 11; D? Collaboration: C.K. Jung, Proc. of the XXVII Int. Conf. on High Energy Physics, Glasgow, July 1994, eds. P.J. Bussey and I.G. Knowles. [5] G. Montagna, O. Nicrosini, G. Passarino and F. Piccinini, Phys. Lett. B335 (1994) 484; V.A. Novikov, L.B. Okun, A.N. Rozanov and M.I. Vysotskii, Mod. Phys. Lett. A9 (1994) 2641; J. Erler and P. Langacker, Phys. Rev. D52 (1995) 441; S. Matsumoto, Mod. Phys. Lett. A10 (1995) 2553; Z. Hioki, TOKUSHIMA 95-04, hep-ph/9510269; K. Kang and S.K. Kang, BROWN-HET-979, hep-ph/9503478. [6] J. Ellis, G.L. Fogli and E. Lisi, CERN-TH/95-202, hep-ph/9507424. [7] P.H. Chankowski and S. Pokorski, Phys. Lett. B356 (1995) 307; hep-ph/9509207. [8] H. Burkhardt and B. Pietrzyk, Phys. Lett. B356 (1995) 398; S. Eidelman and F. Jegerlehner, Z. Phys. C67 (1995) 585. [9] S. Dittmaier, K. Kolodziej, M. Kuroda, and D. Schildknecht, Nucl. Phys. B426 (1994) 249, E: B446 (1995) 334. [10] S. Dittmaier, M. Kuroda, and D. Schildknecht, Nucl. Phys. B448 (1995) 3. [11] L. Avdeev, J. Fleischer, S. Mikhailov and O. Tarasov, Phys. Lett. B336 (1994) 560, E: B349 (1995) 597; K.G. Chetyrkin, J.H. K¨ hn and M. Steinhauser, Phys. Lett. B351 (1995) 331. u

11

[12] J.F. Grivaz, LAL-95-83, Proc. International EPS Conference on High-Energy Physics, Brussels, July 1995, eds. J. Lemonne, C. Vander Velde and F. Verbeure. [13] G. Altarelli, R. Barbieri, and F. Caravaglios, Phys. Lett. B349 (1995) 145. [14] G. Altarelli, R. Barbieri and F. Caravaglios, CERN-TH-6859-93; M. Shifman, Mod. Phys. Lett. A10 (1995) 605. [15] Particle Data Group, L. Montanet et al., Phys. Rev. D50 (1994) 1173 and 1995 o?year partial update for the 1996 edition available on the PDG WWW pages (URL: http://pdg.lbl.gov/). [16] S. Dittmaier, D. Schildknecht, and G. Weiglein, Nucl. Phys. B465 (1996) 3; BI-TP 95/34, hep-ph/9511281.

12

6 5 4 3 2 1 0 6 5 4 3 2 1 0 10

s2 (LEP) W

all data + m all data all data + m all data

exp t exp t n

2 =d o f min

n R b ; Rc

R b ; Rc

16/9 14/8 0.7/7 0.6/6

: : :

2

2

all data + m all data all data + m all data

s2 (LEP + SLD) W
exp t exp n t

2 =d o f min

n R b ; Rc

R b ; Rc

17/9 14/8 1.5/7 0.7/6

: : :

100

MH = GeV

1000

2 2 Figure 1: ?χ2 = χ2 ? χ2 is plotted against MH for the (mt , MH , α(MZ ), αs (MZ )) ?t to min various sets of physical observables, as speci?ed in Tab. 2.

13

Figure 2: The results of the two-parameter (mt , MH ) ?ts within the SM are displayed in the (mt , MH ) plane. The three di?erent columns refer to the three di?erent sets of experimental data used in the corresponding ?ts, (i) “leptonic sector”: Γl , s2 (LEP), MW , ?W (ii) “all data \Rc , Rb ”: ΓT , Γh are added to set (i), (iii) “all data \Rc ”: ΓT , Γh , Γb are added to the set (i). 2 2 The second and the third rows show the shift resulting from changing α(MZ )?1 and αs (MZ ), respectively, by one standard deviation in the SM prediction. The fourth row shows the e?ect of replacing s2 (LEP) by s2 (SLD) and s2 (LEP + SLD) in the ?ts. Note that the 1σ ?W ?W ?W boundaries given in the ?rst row are repeated identically in each row, in order to facilitate comparison with other boundaries. The value of χ2 /d.o.f. indicated in the plots refers to min 2 2 the central values of α(MZ )?1 and αs (MZ ). In all plots the empirical value of the top-quark mass (not included as input of the ?ts) of mexp = 180 ± 12 GeV is also indicated. t

14

leptonic sector
1000

all data n Rc; Rb

all data n Rc
2

=1 4

100

2 =d.o.f. min
= 0 2 3

:=

2 =d.o.f. min
= 0 6 5

:=

d.o.f. = 12=6
2 = min 2 (MZ ) = 1=128::80 1=128 89 1=128:98

10

1000

100

MH = GeV

10

1000

s

= 0:117 0:123 0:129

100

10

1000

s2 (LEP) W s2 (LEP + SLD) W s2 (SLD) W

100

2 =d.o.f. min
= 1 4 3 10 140 160 180 200 220 140 160 180

:=

2 =d.o.f. min
= 2 0 5

:=

d.o.f. = 15=6
2 = min
220

200

mt = GeV
Figure 2: 15

140

160

180

200

220


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