model dependence, one can measure the fourifrom the basic observables of LEP physics .Z! C /,AFB, andRbon theZpeak plusmW. With increasing model dependence, one can include other measurements in the fit for thei. For example, one can use lepton universality to average the with theeand£ final states, or include all lepton asymmetries and so on. The present experimental values of thei, obtained from a fit of all LEP1-SLD measurements plusmW, are [142]
1103D5:6˙1:0 ; 2103D 7:8˙0:9 ;
3103D5:6˙0:9 ; b103D 5:8˙1:3 : (3.109) Note that theparameters are of order a few103and are known with an accuracy in the range 15–30%. These values are in agreement with the predictions of the SM with a 126 GeV Higgs [142]:
SM1 103D5:21˙0:08 ; 2SM103D 7:37˙0:03 ;
SM3 103D5:279˙0:004 ; bSM103D 6:94˙0:15 : (3.110) All models of new physics must be compared with these findings and pass this difficult test.
gaf and gvf are the real parts of the effective couplings, and contains non- factorisable mixed corrections.
Besides total cross-sections, various types of asymmetries have been measured.
The results of all asymmetry measurements are quoted in terms of the asymmetry parameterAf, defined in terms of the real parts of the effective coupling constants gaf andgvf by
Af D2 gvfgaf
g2vfCg2af D2 gvf=gaf
1C.gvf=gaf/2 ; A0;FBf D 3
4AeAf : (3.113) The measurements are the forward–backward asymmetry (A0;FBf), the tau polarization (A£) and its forward–backward asymmetry (Ae) measured at LEP, as well as the left–right and left–right forward–backward asymmetry measured at SLC (Ae and Af, respectively). Hence the set of partial width and asymmetry results allows the extraction of the effective coupling constants.
The various asymmetries determine the effective electroweak mixing angle for leptons with highest sensitivity (see Fig.3.16). The weighted average of these results, including small correlations, is
sin2eff D0:23153˙0:00016 ; (3.114) Note, however, that this average has a 2 of 11.8 for 5 degrees of freedom, corresponding to a probability of a few %. The2 is pushed up by the two most precise measurements of sin2eff, namely those derived from the measurements of Alby SLD, dominated by the left–right asymmetryA0LR, and measurements of the forward–backward asymmetryA0;FBbmeasured inbbNproduction at LEP, which differ by about3.
We now extend the discussion of the SM fit of the data. One can think of different types of fit, depending on which experimental results are included or which answers one wants to obtain. For example, in Table3.2we present in column 1 a fit of allZ pole data plusmW andW (this is interesting as it shows the value ofmt obtained
Fig. 3.16 Summary of sin2effprecision measurements at high Q2 [350]
Table 3.2 Standard Model fits of electroweak data [350]
Fit 1 2 3
Measurements mW,W mt mt,mW,W
mt.GeV/ 178:1C10:97:8 173:2˙0:9 173:26˙0:89
mH.GeV/ 148C23781 122C5941 94C2924
log ŒmH.GeV/ 2:17˙ C0:38 2:09˙0:17 1:97˙0:12
˛s.mZ/ 0:1190˙0:0028 0:1191˙0:0027 0:1185˙0:0026
mW.MeV/ 80381˙13 80363˙20 80377˙12
All fits use theZpole results and˛.5/had.m2Z/, as listed in Fig.3.15. In addition, the measurements listed at the top of each column are included in that case. The fittedWmass is also shown [350]
(the directly measured value ismWD80 385˙15MeV)
indirectly from radiative corrections, to be compared with the value ofmtmeasured in production experiments), in column 2, a fit of allZ pole data plusmt(here it is mW which is indirectly determined), and finally, in column 3, a fit of all the data listed in Fig.3.15(which is the most relevant fit for constrainingmH).
From the fit in column 1 we see that the extracted value of mt is in good agreement with the direct measurement (see Fig 3.15). Similarly, we see that the experimental measurement of mW is larger by about one standard deviation with respect to the value from the fit in column 2. We have seen that quantum corrections depend only logarithmically onmH. In spite of this small sensitivity, the measurements are still precise enough to obtain a quantitative indication of the mass range. From the fit in column 3 we obtain
log10mH.GeV/D1:97˙0:12 ; or mHD94C2429GeV:
This result on the Higgs mass is truly remarkable. The value of log10mH.GeV/
is compatible with the small window between 2and 3which is allowed, on the one side, by the direct search limitmH > 114GeV from LEP2 [350], and on the other side by the theoretical upper limit on the Higgs mass in the minimal SM, mH.600–800 GeV [320], to be discussed in Sect.3.13.
Thus the whole picture of a perturbative theory with a fundamental Higgs is well supported by the data on radiative corrections. It is important that there is a clear indication for a particularly light Higgs: at 95% confidence levelmH . 152GeV (which becomesmH . 171GeV, including the input from the LEP2 direct search result). This was quite encouraging for the LHC search for the Higgs particle.
More generally, if the Higgs couplings are removed from the Lagrangian, the resulting theory is non-renormalizable. A cutoff must be introduced. In the quantum corrections, logmHis then replaced by logplus a constant. The precise determination of the associated finite terms would be lost (that is, the value of the mass in the denominator in the argument of the logarithm). A heavy Higgs would need some unfortunate accident: the finite terms, different in the new theory from those of the SM, should by chance compensate for the heavy Higgs in a few
key parameters of the radiative corrections (mainly1 and 3, see, for example, [48]). Alternatively, additional new physics, for example in the form of effective contact terms added to the minimal SM Lagrangian, should accidentally do the compensation, which again needs some sort of conspiracy.
To the list of precision tests of the SM, one should add the results on low energy tests obtained from neutrino and antineutrino deep inelastic scattering (NuTeV [353]), parity violation in Cs atoms (APV [274]), and the recent measurement of the parity-violating asymmetry in Moller scattering [354]. When these experimental results are compared with the SM predictions, the agreement is good except for the NuTeV result, which differs by three standard deviations. The NuTeV measurement is quoted as a measurement of sin2W D 1m2W=m2Zfrom the ratio of neutral to charged current deep inelastic cross-sections fromandNusing the Fermilab beams. But it has been argued, and it is now generally accepted, that the NuTeV anomaly probably simply arises from an underestimation of the theoretical uncertainty in the QCD analysis needed to extract sin2W. In fact, the lowest order QCD parton formalism upon which the analysis has been based is too crude to match the experimental accuracy.
When confronted with these results, the SM performs rather well on the whole, so that it is fair to say that no clear indication for new physics emerges from the data. However, as already mentioned, one problem is that the two most precise measurements of sin2eff fromALR andAbFBdiffer by about3. In general, there appears to be a discrepancy between sin2effmeasured from leptonic asymmetries, denoted.sin2eff/l, and from hadronic asymmetries, denoted.sin2eff/h. In fact, the result fromALR is in good agreement with the leptonic asymmetries measured at LEP, while all hadronic asymmetries, though their errors are large, are better compatible with the result of AbFB. These two results for sin2eff are shown in Fig.3.17 [210]. Each of them is plotted at themH value that would correspond to it given the central value ofmt. Of course, the value formHindicated by each sin2eff has a horizontal ambiguity determined by the measurement error and the width of the˙1band formt.
Even taking this spread into account, it is clear that the implications formHare significantly different. One might imagine that some new physics effect could be hidden in theZbbN vertex. For instance, for the top quark mass there could be other non-decoupling effects from new heavy states or a mixing of thebquark with some other heavy quark. However, it is well known that this discrepancy is not easily explained in terms of any new physics effect in theZbbNvertex. A rather large change with respect to the SM of the b quark right-handed coupling to the Z is needed in order to reproduce the measured discrepancy (in fact, a 30% change in the right-handed coupling), an effect too large to be a loop effect, but which could be produced at the tree level, e.g., by mixing of thebquark with a new heavy vector- like quark [140], or some mixing of theZwith ad hoc heavy states [170]. But then this effect should normally also appear in the direct measurement ofAbperformed at SLD using the left–right polarizedbasymmetry, even within the moderate accuracy of this result. The measurements of neitherAbat SLD norRbconfirm the need for
50 100 lept. asymm 0.231
0.2314 0.2316 0.2318 0.232 0.2322 0.2324
0.2312
200 300 400 500
MH [GeV]
sin2θlepteff world av.
sin2θ lept eff
Mt +1σ Mt -1σ
Mt=172.6 GeV
hadr. asymm
Fig. 3.17 The data for sin2efflept are plotted vsmH. The theoretical prediction for the measured value ofmtis also shown. For presentation purposes the measured points are each shown at themH value that would ideally correspond to it, given the central value ofmt. Adapted from [210]. New version courtesy of P. Gambino
such a large effect (recently a numerical calculation of NLO corrections toRb [204]
appeared at first to indicate a rather large result, but in the end the full correction turned out to be rather small). Alternatively, the observed discrepancy could simply be due to a large statistical fluctuation or an unknown experimental problem. As a consequence of this problem, the ambiguity in the measured value of sin2effis in practice greater than the nominal error, reported in (3.114), obtained from averaging all the existing determinations, and the interpretation of precision tests is less sharp than it would otherwise be.
We have already observed that the experimental value of mW (with good agreement between LEP and the Tevatron) is a bit high compared to the SM prediction (see Fig.3.18). The value ofmH indicated bymW is on the low side, just in the same interval as for sin2effleptmeasured from leptonic asymmetries.
In conclusion, the experimental information on the Higgs sector, obtained from EW precision tests at LEP1 and 2 and the Tevatron can be summarized as follows.
First, the relation MW2 D M2Zcos2W in (3.52), modified by small, computable radiative corrections, has been demonstrated experimentally. This relation means that the effective Higgs (be it fundamental or composite) is indeed a weak isospin doublet. The direct lower limitmH & 114:5GeV (at 95% confidence level) was
Fig. 3.18 The data formW
are plotted vsmt [350]
80.3 80.4 80.5
155 175 195
LHC excluded
mH[GeV]
114 300 600 1000 mt [GeV]
mW[GeV] 68% CL
Δα LEP1 and SLD LEP2 and Tevatron
March 2012
obtained from searches at LEP2. When compared to the data on precision EW tests, the radiative corrections computed in the SM lead to a clear indication of a light Higgs, not too far from the direct LEP2 lower bound. The upper limit formHin the SM from the EW tests depends on the value of the top quark massmt. The CDF and D0 combined value after Run II is at presentmt D 173:2˙0:9GeV [350].
As a consequence, the limit onmH from the LEP and Tevatron measurements is rather stringent [350]:mH< 171GeV (at 95% confidence level, after including the information from the 114.5 GeV direct bound).
