model dependence, one can measure the four*i*from the basic observables of LEP
physics .*Z*! ^{C} ^{}/,*A*_{FB}, and*R**b*on the*Z*peak plus*m**W*. With increasing model
dependence, one can include other measurements in the fit for the*i*. For example,
one can use lepton universality to average the with the*e*and£ final states, or
include all lepton asymmetries and so on. The present experimental values of the*i*,
obtained from a fit of all LEP1-SLD measurements plus*m**W*, are [142]

_{1}10^{3}D5:6˙1:0 ; _{2}10^{3}D 7:8˙0:9 ;

310^{3}D5:6˙0:9 ; *b*10^{3}D 5:8˙1:3 : (3.109)
Note that theparameters are of order a few10^{}^{3}and 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]:

^{SM}_{1} 10^{3}D5:21˙0:08 ; _{2}^{SM}10^{3}D 7:37˙0:03 ;

^{SM}_{3} 10^{3}D5:279˙0:004 ; *b*^{SM}10^{3}D 6:94˙0:15 : (3.110)
All models of new physics must be compared with these findings and pass this
difficult test.

*g*_{af} and *g*_{vf} 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
parameter*A**f*, defined in terms of the real parts of the effective coupling constants
*g*_{af} and*g*_{vf} by

*A**f* D2 *g*v*f**g*a*f*

*g*^{2}_{vf}C*g*^{2}_{af} D2 *g*v*f*=*g*a*f*

1C.*g*v*f*=*g*a*f*/^{2} ; *A*^{0;}_{FB}^{f} D 3

4*A**e**A**f* : (3.113)
The measurements are the forward–backward asymmetry (*A*^{0;}_{FB}^{f}), the tau polarization
(*A*_{£}) and its forward–backward asymmetry (*A**e*) measured at LEP, as well as the
left–right and left–right forward–backward asymmetry measured at SLC (*A**e* and
*A**f*, 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

sin^{2}eff 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 %. The^{2} is pushed up by the two most
precise measurements of sin^{2}eff, namely those derived from the measurements of
*A**l*by SLD, dominated by the left–right asymmetry*A*^{0}_{LR}, and measurements of the
forward–backward asymmetry*A*^{0;}_{FB}^{b}measured in*bb*Nproduction 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 all*Z*
pole data plus*m**W* and*W* (this is interesting as it shows the value of*m**t* obtained

**Fig. 3.16** Summary of
sin^{2}effprecision
measurements at high
*Q*^{2} [350]

**Table 3.2** Standard Model fits of electroweak data [350]

Fit 1 2 3

Measurements *m*_{W},*W* *m*_{t} *m*_{t},*m*_{W},*W*

*m*_{t}.GeV/ 178:1^{C10:9}_{7:8} 173:2˙0:9 173:26˙0:89

*m*_{H}.GeV/ 148^{C237}_{81} 122^{C59}_{41} 94^{C29}_{24}

log Œ*m*_{H}.GeV/ 2:17˙ C0:38 2:09˙0:17 1:97˙0:12

˛s.*m*_{Z}/ 0:1190˙0:0028 0:1191˙0:0027 0:1185˙0:0026

*m*_{W}.MeV/ 80381˙13 80363˙20 80377˙12

All fits use the*Z*pole results and˛^{.5/}had.*m*^{2}_{Z}/, as listed in Fig.3.15. In addition, the measurements
listed at the top of each column are included in that case. The fitted*W*mass is also shown [350]

(the directly measured value is*m*_{W}D80 385˙15MeV)

indirectly from radiative corrections, to be compared with the value of*m**t*measured
in production experiments), in column 2, a fit of all*Z* pole data plus*m**t*(here it is
*m**W* 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 constraining*m*_{H}).

From the fit in column 1 we see that the extracted value of *m**t* is in good
agreement with the direct measurement (see Fig 3.15). Similarly, we see that
the experimental measurement of *m**W* 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 on*m*_{H}. 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

log_{10}*m*_{H}.GeV/D1:97˙0:12 ; or *m*_{H}D94^{C}24^{29}GeV:

This result on the Higgs mass is truly remarkable. The value of log_{10}*m*H.GeV/

is compatible with the small window between 2and 3which is allowed, on
the one side, by the direct search limit*m*H > 114GeV from LEP2 [350], and on
the other side by the theoretical upper limit on the Higgs mass in the minimal SM,
*m*H.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 level*m*H . 152GeV
(which becomes*m*H . 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, log*m*His 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 sin^{2}W D 1*m*^{2}_{W}=*m*^{2}_{Z}from 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 sin^{2}W. 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 sin^{2}eff from*A*_{LR} and*A*^{b}_{FB}differ by about3. In general, there
appears to be a discrepancy between sin^{2}effmeasured from leptonic asymmetries,
denoted.sin^{2}eff/l, and from hadronic asymmetries, denoted.sin^{2}eff/h. In fact,
the result from*A*_{LR} 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 *A*^{b}_{FB}. These two results for sin^{2}eff are shown in
Fig.3.17 [210]. Each of them is plotted at the*m*_{H} value that would correspond
to it given the central value of*m**t*. Of course, the value for*m*_{H}indicated by each
sin^{2}eff has a horizontal ambiguity determined by the measurement error and the
width of the˙1band for*m**t*.

Even taking this spread into account, it is clear that the implications for*m*_{H}are
significantly different. One might imagine that some new physics effect could be
hidden in the*Zbb*N vertex. For instance, for the top quark mass there could be other
non-decoupling effects from new heavy states or a mixing of the*b*quark 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 the*Zbb*Nvertex. 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 the*b*quark with a new heavy vector-
like quark [140], or some mixing of the*Z*with ad hoc heavy states [170]. But then
this effect should normally also appear in the direct measurement of*A**b*performed at
SLD using the left–right polarized*b*asymmetry, even within the moderate accuracy
of this result. The measurements of neither*A**b*at SLD nor*R**b*confirm 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

M_{H} [GeV]

sin^{2}θ^{lept}_{eff} world av.

sin2θ lept eff

M_{t} +1σ
M_{t} -1σ

M_{t}=172.6 GeV

hadr. asymm

**Fig. 3.17** The data for sin^{2}eff^{lept} are plotted vs*m*_{H}. The theoretical prediction for the measured
value of*m*_{t}is also shown. For presentation purposes the measured points are each shown at the*m*_{H}
value that would ideally correspond to it, given the central value of*m*_{t}. Adapted from [210]. New
version courtesy of P. Gambino

such a large effect (recently a numerical calculation of NLO corrections to*R**b* [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 sin^{2}effis 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 *m**W* (with good
agreement between LEP and the Tevatron) is a bit high compared to the SM
prediction (see Fig.3.18). The value of*m*H indicated by*m**W* is on the low side,
just in the same interval as for sin^{2}_{eff}^{lept}measured 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 *M*_{W}^{2} D *M*^{2}_{Z}cos^{2}W 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 limit*m*H & 114:5GeV (at 95% confidence level) was

**Fig. 3.18** The data for*m**W*

are plotted vs*m*_{t} [350]

80.3 80.4 80.5

155 175 195

LHC excluded

m_{H}[GeV]

114 300 600 1000
m_{t} [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 for*m*Hin the
SM from the EW tests depends on the value of the top quark mass*m**t*. The CDF
and D0 combined value after Run II is at present*m**t* D 173:2˙0:9GeV [350].

As a consequence, the limit on*m*H from the LEP and Tevatron measurements is
rather stringent [350]:*m*H< 171GeV (at 95% confidence level, after including the
information from the 114.5 GeV direct bound).