There are now sufficiently many observables that one can
determine 
, 
, and 
from the Z-pole and other indirect precision data
simultaneously. For example, 
can be determined
from the asymmetries, 
from the W and Z masses, and
from the hadronic Z-widths. In practice all
of these quantities are determined from a simultaneous fit. The
results of fits to various sets of data are shown in
Table 3.
Table 3: Results for the electroweak parameters in the standard model from
various sets of data. The central values assume 
GeV, while the
second errors are for 
and 
. The last column is
the increase in the overall 
of the fit as 
increases from 60
to 1000.
The first row of the table shows the global fit to all data,
including the direct production constraint 
GeV
from CDF and D0. The second row uses the indirect data only.
The predicted value of 
GeV
is in remarkable agreement with the CDF/D0 value.
The other fits show the sensitivity to the various data sets.
The LEP data allows a determination of the strong
coupling constant 
at the Z-pole with a small experimental
error,
where the second uncertainty is from 
. 
is almost
uncorrelated with the other parameters. It is determined mainly from the
ratio 
, which is
insensitive to 
(except in the 
vertex), and also from
. This determination is very clean theoretically, at least
within the standard model. It is the Z-pole version of the long held
view that the ratio of hadronic to leptonic rates in 
would be a
``gold plated'' extraction of 
and test of QCD. Using a recent
estimate [20] of the 
corrections to 
,
i.e. 
, one can estimate that higher-order terms
lead to an additional uncertainty 
in the 
value in (4). It should be cautioned, however, that the
lineshape value is rather sensitive to the presence of some types of new
physics which affect the Z-hadron width, as is discussed below.
The lineshape value of 
is in excellent agreement with the
independent value 
extracted from jet
event shapes at LEP using resummed QCD [24]. It is also in
agreement with the prediction
of supersymmetric grand unification
[21,22].
As can be seen in Table 4,
however, it is somewhat larger than some of the low energy determinations
of 
(which are then extrapolated theoretically to the Z-pole),
in particular those from deep inelastic scattering and the lattice
calculations of the charmonium and bottomonium spectra.
This slight discrepancy has led
some authors to suggest that there might be a light gluino which would
modify the running of 
,
or that there is a problem in the high energy determinations [23].
I think, however, that it is premature
to draw such strong conclusions, especially since most of the
determinations are dominated by theoretical uncertainties.
There is, however, one significant uncertainty in the
lineshape value: if the high
experimental value of 
is due to a new physics contribution
to the 
vertex, and not just a fluctuation,
then the formulae for R and 
are affected, and the value
of 
extracted from the lineshape is reduced [3].
Allowing for that possibility, one finds the lower value
, in better agreement with some
of the low energy determinations.
One could also consider the possibility that the low value of 
is due to new physics. However, allowing for that possibility, one obtains
or 
, where the
former (latter) value does not (does) allow for new physics
in 
as well. The first value in particular is in clear
disagreement with other determinations, so I will take the
view that 
is a statistical fluctuation.
Table 4: Values of 
at the Z-pole extracted from
various methods.
