%Title: Electroweak Measurements, QCD and Theory Uncertainties
%Author: J.H. Kuehn
%Published:     Proc. <EM>29th ICHEP' </EM> Vancouver, CA, 23-29 July 1998, edited by A. Astbury, D. Axen, Jacob Robinson (World Scientific, Singapore, 1999) Vol.1, 563-568. 
%UNIX --- UPDATED ON 14/8/97 
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\begin{document}

\title{
Electroweak Measurements, QCD and Theory Uncertainties
}

\author{
J.H. K\"uhn
}
\address{
Institut f\"ur Theoretische Teilchenphysik,
       Universit\"at Karlsruhe, Germany
}

\twocolumn[ \maketitle\abstracts{ Recent results for electroweak
  radiative corrections will be presented with emphasis on those
  connecting QCD and electroweak interactions. This includes purely
  elctroweak corrections at two-loop order and beyond, hadronic effects
  specific to the $b\bar b$ final state in $Z$ decays, the impact of
  perturbative QCD on the $\rho$ parameter and related observables, and
  mixed QCD and electroweak vertex corrections. Estimates for the
  theoretical uncertainties are given.  } ]


\section{Introduction}


During the past years electroweak precision measurements have become one
of the central issues in particle physics. The indirect determination of
the mass of the top quark through its impact on quantum corrections
prior to its observation in proton-antiproton collisions at the TEVATRON
can be considered one of the triumphs of the present theoretical
framework. Similarly the indirect determination of the mass of the $W$
boson is in perfect agreement with the present measurements at LEP and
the TEVATRON, which motivates a combined fit to the parameters of the
Standard Model.

Recently the measurements have become sufficiently precise to even give
first indications for a relatively small mass of the Higgs boson with an
upper limit of order 200 GeV. Significant improvements, in particular
for $M_W$ and the left-right asymmetry with polarised beams at the SLC
are expected for the next couple of years which might lead to a fairly
precise indirect determination of the Higgs mass $M_H$, thus repeating
the success of the top quark mass determination.

 
To achieve the accuracy of the theoretical predictions needed for this
program, not only the two-loop electroweak corrections have to be
incorporated --- for the dominant terms, those originating from the
large top mass, from the running of $\alpha_{\rm QED}$ or from QCD sometimes
even three-loop amplitudes have to be investigated. In this contribution a
variety of recent results connecting QCD and electroweak physics will be
presented. In section 2 purely electroweak effects at two-loop order and
beyond will be discussed. Also purely hadronic corrections may affect
the determination of precision observables, in particular ratios of
various partial decay rates of the $Z$ boson like $\Gamma_b /
\Gamma_{\rm had}$. Examples are the so-called singlet contributions and
bottom quark mass effects which are both reviewed in section 3.
The quadratic top mass dependence of the $\rho$ parameter and related
observables is crucial for the indirect $m_t$ determination. More
refined studies require a precise definition of the top quark mass and,
correspondingly, the evaluation of two- and even three-loop QCD
corrections (Sect.4). They are all related to the self-energies of the
photon, the W- and the Z-boson or their mixing, are thus
``universal'' and involve QCD and electroweak interactions
simultaneously. Another class of ``mixed'' QCD and electroweak
interactions to be discussed in section 5 is connected to the $Z q\bar
q$ vertex. They influence decays into up and down type and in particular $b$
quarks differently and, at the same time, affect the determination of
$\alpha_s$ from the $Z$ decay rate.

\section[]{Electroweak Results at Two-Loop and Beyond~\cite{weiglein}}
%
\subsection{Higgs-mass dependence of $\Delta r$ at two loop}
%
The dependence of a large class of observables on the Higgs mass is of
particular interest for the present discussion. Since a complete
calculation in two loop approximation is not available in the moment,
subtracted quantities of the form
\begin{equation}
  \Delta r_{\mathrm{subtr}}(M_{H}) = \Delta r(M_{H}) - \Delta
  r(M^0_{H})
\end{equation}
were considered~\cite{mhdepDeR} (and similarly for $\sin^2\theta_{\rm
  eff}$, $\Gamma_l$) where $\Delta r$ is defined by $M_{W}^2 (1 -
M_{W}^2/M_{Z}^2) = (\pi \alpha)/(\sqrt{2} G_\mu) (1 + \Delta r)$.  The
subtracted quantities indicate the shift in the precision observables
caused by varying the Higgs-boson mass between $M^0_{H}$ and $M_{H}$.
Values of $M_{H}$ in the interval $65\ {\rm GeV} \leq M_{H} \leq 1\ {\rm
  TeV}$ were considered.  Allowing for such a wide range of Higgs-boson
mass values may seem somewhat over-conservative regarding the fact that
the SM fits favor a light Higgs-boson with a mass smaller than about
300~GeV.  However, the form of the fit curve and thus the value of the
upper bound is affected also by the SM predictions for larger
$M_{H}$-values (and also for $M_{H}$-values below the current
experimental lower bound of $M_{H}$).

Potentially large $M_{H}$-dependent contributions are those associated
with the top quark due to the large Yukawa coupling $t \bar{t} H$,
and the corrections proportional to $\Delta\alpha$. Further contributions are
those from the light fermions (except the corrections already contained
in $\Delta\alpha$), and purely bosonic contributions. The latter are expected
to be rather small~\cite{mhdepDeR}. The subsequent discussion is
therefore focussed on the Higgs-dependent fermionic contributions
to the precision observables, for which exact two-loop
results are available~\cite{mhdepDeR,bsw}.

\begin{figure}
\begin{center}
\leavevmode
\epsfxsize=7.5cm
\epsffile{cmpdrhoaatbh.eps}
\caption[]{\label{fig:drmh2l}
  Higgs-mass dependent fermionic contributions to $\Delta r$ at two-loop
  order. The different curves show the contribution from the diagrams
  involving the top/bottom doublet ($\Delta r_{\mathrm{tbh}}$), the
  contribution proportional to $\Delta\alpha$ ($\Delta
  r_{\Delta\alpha}$), the contribution of the light fermions ($\Delta
  r_{\mathrm{lfh}}$), and the approximation of the top/bottom correction
  by the leading term proportional to $m_{t}^4$ ($\Delta r(m_{\rm
    t}^4)$). (From~\cite{mhdepDeR}.)  }
\end{center}
\end{figure}

Fig.~\ref{fig:drmh2l} shows the Higgs-mass dependence of the two-loop
corrections to $\Delta r$ associated with the top/bottom doublet, with
$\Delta\alpha$, and with the light fermions. The dotted line furthermore
indicates the Higgs-mass dependence of the leading $m_{t}^4$-term in
the top/bottom contribution. The top/bottom contribution gives rise to a
shift in the W-boson mass of $\Delta M^{\mathrm{top}}_{\rm W, subtr,
  (2)}(M_{H} = 1000\ {\rm GeV}) \sim 16$~MeV, which is about $10\%$
of the one-loop contribution. Its Higgs-mass dependence turns out to be
very poorly approximated by the leading $m_{t}^4$-term; the
contribution of the latter even enters with a different sign. It can be
seen from Fig.~\ref{fig:drmh2l} that the two-loop contributions to a
large extent cancel each other. The contribution of the light fermions
yields a shift in $M_{W}$ of up to $\Delta M^{\mathrm{lf}}_{\rm W,
  subtr, (2)}(M_{H} = 1000\ {\rm GeV}) \lsim 4$~MeV. In total the
two-loop contributions lead to a slight increase in the sensitivity of
$\Delta r$ to the Higgs-boson mass compared to the one-loop case.

The Higgs-mass dependence of $M_{W}$ based on the
exact result for the top/bottom doublet has been compared with the results of
the expansion in the top-quark mass up to ${\cal O}(G_\mu^2 m_{t}^2 M_{Z}^2)$
given in Ref.~\cite{DGS}.
Besides the difference in the
two-loop results also higher-order effects due to differences in the
renormalization procedure and in the treatment of non-leading
higher-order corrections enter the comparison.

Over the range of Higgs-mass values from 65~GeV to 1~TeV the difference
between the results amounts to about 8~MeV, which corresponds to $50\%$
of the two-loop top/bottom contribution, is slightly
larger than the theoretical uncertainty quoted in Ref.~\cite{degrassi}
of 2~MeV from electroweak and 5~MeV from QCD corrections.


\subsection{Fermion-loop corrections}

The dominant contributions to the electroweak precision observables at
one-loop order are the fermion-loop corrections $\Delta \alpha$ and $\Delta\rho =
N_C \frac{G_\mu m_{t}^2}{8 \pi^2 \sqrt{2}}$, which arise from the
renormalization of the electric charge and the weak mixing angle,
respectively.  One of the main issues in improving the one-loop
predictions has therefore been to find a proper resummation of these
leading one-loop contributions. In the case of $\Delta r$, for instance,
the resummation~\cite{resum}
\begin{equation}
1 + \Delta r \to \frac{1}{(1-\Delta\alpha)(1+\frac{c_W^2}{s_W^2} \Delta\rho) -
\Dri{rem}}
\label{eq:resCHJSir}
\end{equation}
correctly takes into account the terms of the form
$(\Delta\alpha)^2$, $(\Delta\rho)^2$, $(\Delta\alpha \Delta\rho)$, 
$\Delta\alpha \Delta r_{\mathrm{rem}}$ 
at two-loop order. However, even with the knowledge of
the recently evaluated terms~\cite{mt2,DGS} of 
${\cal O}(G_\mu^2 m_{t}^2 M_{Z}^2)$ different treatments of 
contributions whose 
correct resummation is not known (i.e.\ non-leading two-loop terms and
the higher-order terms except the leading term in $\Delta\alpha$) may result
in predictions for $M_{W}$ that differ from each other by several MeV.

Results~\cite{weiglein} for the pure fermion-loop contributions to
the precision observables, i.e.\ contributions containing $n$ fermion
loops at $n$-loop order (which in the case of quark loops are therefore
proportional to the $n$th power of the color factor $N_C$) have been
presented \footnote{These contributions obviously constitute an
  UV-finite and gauge-invariant subset.}.  The relevant diagrams in a
given loop order are reducible diagrams of vacuum-polarization type,
which can easily be taken into account by Dyson summation, and, in
addition, graphs with counter-term insertions into fermion-loop diagrams.

Using the on-shell renormalization scheme explicit results for the
fermion-loop contributions to the precision observables up to four-loop
order have been obtained~\cite{weiglein}. The two-loop contribution to
$\Delta r$ when compared to the $\overline{\mbox{MS}}$
result~\cite{floopmsb} is found to be in very good agreement (within
less than 1~MeV) for the $M_{W}$ prediction. Also the validity of the
resummation Eq.~\ref{eq:resCHJSir} for the non-leading two-loop and the
higher-order contributions has been investigated. Already
the first non-leading two-loop term of ${\cal O}(\Delta\rho
\log(m_{t}^2/M_{Z}^2))$
is not correctly produced by the resummation.
The terms of the form $\Delta\rho \Delta r_{\mathrm{top,rem}}$ 
and $\left(\Delta r_{\mathrm{top,rem}}\right)^2$
generated by Eq.~\ref{eq:resCHJSir} give rise to a deviation in $M_{W}$ of
about 7~MeV compared to the exact result for the top/bottom doublet.
It turns out, however, that large numerical cancellations occur between
the non-leading two-loop terms that are not correctly resummed in
Eq.~\ref{eq:resCHJSir}. While separately these terms
differ from the corresponding terms in the exact result by several MeV,
their sum approximates the full result remarkably well, up
to less than 1~MeV.

In total, due to accidental numerical cancellations, the resummation of
the fermion-loop contributions according to Eq.~\ref{eq:resCHJSir} yields
a very good numerical approximation of the complete fermion-loop result,
valid within about 2~MeV.




\section[]{QCD Corrections, Quark Mass Effects and Uncertainties in the Z Decay
  Rate~\cite{CKKPhysRep}}
%
The dominant terms of QCD corrections to the Z decay rate are
accumulated in the non-singlet corrections which result from diagrams
with the external current coupled to the same quark line. In the limit
of massless quarks one obtains the familiar correction factor
\begin{equation}
  {\Gamma(Z\to hadrons)\over \Gamma_{\rm Born}} = 1 +
  \frac{\alpha_s}{\pi} + 1.409\,
  \left(\frac{\alpha_s}{\pi}\right)^2 - 12.767\,
    \left(\frac{\alpha_s}{\pi}\right)^3
\end{equation}
which is universal for all quark species. It allows for a theoretically
clean and extremely precise determination of $\alpha_s$ from the $Z$
line shape fits. However, at the current level of precision the
theoretical and experimental analysis has to include a number of small
but nevertheless important effects which are of relevance not only for the
total rate, they even modify the partial rates into different quark
species differently. 
This is of particular importance for the partial rate into $b \bar b$
and the ratio $\Gamma_b/ \Gamma_{\rm had}$, where one hand most of the QCD
corrections cancel and which, on the other hand, is particularly sensitive
to electroweak effects specific for the $t b$ doublet.  

The peculiarities of $\Gamma_b$ are corrections due to the relatively
large bottom mass, $4m_b^2/M_Z^2\approx 10^{-2}$, and the singlet
contributions of the axial correlator, with gluons as intermediate
states only. Quark mass effects are conveniently evaluated through an
expansion in $m_b^2 / M_Z^2$. For the quadratic terms which are
sufficient at the scale of $M_Z$ the logarithms can be absorbed in
the running quark mass and the perturbative series has been
evaluated~\cite{CK1,CK2} to
order $\alpha_s^3$ (apart from the tiny $m_b$ dependent singlet terms
which are available~\cite{CKw} in order $\alpha_s^2$ only). These quark mass
effects lead to a reduction of $\Gamma_b$ by 1.53 MeV with the dominant
uncertainty from the error in the quark mass, $m_b=4.7 \pm 0.2$ MeV.

Singlet diagrams are mainly relevant for the axial part of $\Gamma_b$.
They result from the incomplete cancellation between top and bottom
quark loops in ``anomalous'' triangle diagrams and have been
calculated~\cite{KnKu90,CK93,CheTar94} in order $\alpha_s^2$ and
$\alpha_s^3$. The former is still plagued
by a considerable scale uncertainty ($M_Z$ vs.\ $m_t$!) which is largely
removed by the latter. The singlet piece thus leads to a further
reduction $\Delta \Gamma_{\rm singlet} = -1.82$~MeV and the remaining
uncertainty of $\Delta \Gamma_{\rm singlet} = 0.1$~MeV is entirely due
to the parametric uncertainty in $\alpha_s$.

In total one finds
\begin{equation}
\Delta R_b \equiv \frac{\Delta \Gamma_{\rm mass} + \Delta
\Gamma_{\rm singlet}}{\Gamma_{\rm had}}= - 0.2\%,
\end{equation}
a shift comparable to the current experimental precision of
0.34\%. The remaining uncertainty $\delta R_b = 1-2\cdot 10^{-4}$ can
safely be neglected. It is also worth mentioning that the reduction of the
total width by 3.3 MeV also affects the determination of $\alpha_s$,
leading to a shift in $\alpha_s$ of 0.006, i.e. twice the current error.

\section{Gauge boson self-energies, the mass of the top quark and QCD}

The indirect determination of the top quark through quantum corrections
prior to its observation in hadronic collisions can be considered one of
the triumphs of the Standard Model. The experimental precision of the
key observables, the masses of the top quark and the $W$ boson together
with the weak
mixing angle as determined by asymmetry measurements has increased
during the past years and this process will continue in the foreseeable
future. In order to control the influence of the top quark at an
adequate level the inclusion of QCD corrections in the top and bottom
induced self energies is mandatory. The dominant terms are characterised
by the $\rho$ parameter which, in lowest order, is given by~\cite{veltman}
\begin{equation}
\Delta\rho = 3 \sqrt{2} \frac{G_F m_t^2}{16 \pi^2}.
\end{equation}
In view of the large difference between pole and running mass at scale
$m_t$
\begin{equation} 
m_t(pole)=\bar m_t(m_t) \left(1+\frac{4}{3}\frac{\alpha_s}{\pi}+\cdots\right)
\end{equation} 
inclusion of two and even three loop QCD corrections to $\Delta\rho$ is
mandatory. 

Analytic results are available in two loop approximation not only for
the leading term~\cite{djouadi1} in $\Delta\rho$ but also for all self
energies, with arbitrary top and bottom masses~\cite{Kniehl}. The
resulting shift in the prediction for $M_W$ for fixed $G_F$,
$\alpha_{\rm QED}$, $M_Z$ and $m_t=175$~GeV amounts to 68 MeV, well
comparable to the present precision and significantly larger than the
anticipated accuracy of roughly 30 MeV. To arrive at a precise
prediction for the central value and to control the theoretical
uncertainty, three-loop QCD contributions to $\Delta\rho$ as well as to
$\Delta r$ are required. The $\rho$ parameter~\cite{rhoparam} can again
be expressed through diagrams with vanishing external momentum (vacuum
or tadpole diagrams), the remaining quantities involve two point
functions at non-vanishing external momentum and can be
obtained~\cite{deltar} through an expansion in the small mass ratio
$M_Z^2/m_t^2$. The leading three-loop term corresponds to a shift of
$-10.9$~MeV.  The first three terms, amounting to $-13.7$~MeV, are adequate
for a prediction with an accuracy better than 1 MeV (Table~\ref{tab2}).
Conversely, this combined shift is equivalent to a reduction of the
effective top mass by about 1.6 GeV.
\begin{table}[th]
\renewcommand{\arraystretch}{1.3}
\begin{center}

\begin{tabular}{|l||r|r|r|}
\hline
$\delta M_W$ in MeV  & $\alpha_s^0$ & $\alpha_s^1$ & $\alpha_s^2$ \\
\hline
\hline
$M_t^2$             & 611.9 & -61.3 & -10.9 \\
const.              & 136.6 & -6.0  & -2.6  \\
$1/M_t^2$           & -9.0  & -1.0  & -0.2  \\
\hline
\end{tabular}

\end{center}
\caption[]{\label{tab2} The change in $M_W$ separated according
                      to powers in $\alpha_s$ and $M_t$ in the on-shell
                      scheme (From~\cite{deltar}.)}
\end{table}

To exploit the precision expected from a future linear collider which
will pin down $m_t$ to better than 200 or perhaps even 100 MeV, the
inclusion of $\alpha_s^2$ terms is evidently mandatory.

Similar considerations~\cite{deltar} are valid for the effective
weak mixing angle which can be deduced from the left right asymmetry or
the forward backward asymmetry in a straightforward way.

An important issue in this connection is the size of uncertainties,
arising either from uncalculated higher orders, or from the
``parametric'' uncertainties in $\alpha_s$ and $m_t$. Shifts of
$\delta\alpha_s = 0.003$ and $\delta m_t = 5$~GeV lead to changes 
in $M_W$ of $-2.4$~MeV and 35~MeV respectively. The uncertainty from 
uncalculated higher orders is
completely negligible. A $W$-mass determination with a
precision in the 10~MeV range should therefore be accompanied by a top
mass measurement with a precision around or better than 1~GeV.

\section{Mixed QCD and electroweak vertex corrections}

As stated above, gauge boson self energies, in particular those induced
by fermion loops, give rise to the dominant radiative corrections for
electroweak precision observables. Nevertheless, for a complete
treatment of ${\cal O}(\alpha\alpha_s)$ the inclusion of irreducible
vertex corrections is necessary. These have to be distinguished from the
reducible ones which are easily incorporated, if the electroweak result,
including one-loop weak terms, is simply multiplied by the QCD
correction factor $(1+\alpha_s/\pi + \cdots)$. 

The physics and the techniques of calculation are markedly different for
vertices leading to light ($u$, $d$, $s$ and $c$) quark pairs on one
hand~\cite{CzarK} and for decays into $b\bar b$ on the other
hand~\cite{list}, a consequence of the presence of top quarks with their
enhanced contribution to the vertex proportional $m_t^2/M_W^2$. The
irreducible one-loop vertex diagrams are shown in Fig.~\ref{fig2l}.

\begin{figure}[t]
\leavevmode
\centering
\epsfxsize=7.5cm
\epsffile[105 558 503 717]{zbbdias.ps}
\label{fig2l}
\caption{Diagrams contributing to $\delta\Gamma^W_b$
  in ${\cal O}(\alpha)$. Thin lines correspond to bottom quarks, thick
  lines to top quarks, dotted lines to Goldstone bosons and inner wavy
  lines represent $W$ bosons.}
\end{figure}

To obtain all one-particle irreducible vertex diagrams in two-loop
approximation , these have to be dressed with gluons in all conceivable
ways. The resulting amplitudes are first studied for arbitrary $q^2$ by
considering expansions in the ratio $x_Z=q^2/M_Z^2$ and $x_W=
q^2/m_W^2$ (or for some diagrams, in $1/x_{Z,W}$). Even for the limiting
values $x_Z=1$ and $x_W= M_Z^2/M_W^2 $ the exact results are well
approximated, once sufficiently many terms are included. Using $\alpha_s
=0.12$, $\alpha=1/129$, $\sin\theta_W = 0.223$ $M_Z = 91.19$~GeV, it is
found~\cite{CzarK} that the net effect of the nonfactorizable corrections
is
\begin{eqnarray}
\lefteqn{\Gamma^{\mbox{\small (2 loop EW/QCD)}} - {\alpha_s\over \pi}
\Gamma^{\mbox{\small (1 loop EW)}}}
\nonumber \\
&& \qquad\qquad = 
\left\{
\begin{array}{l}
-1.13(4)\times 10^{-4}  \mbox{ GeV}\quad \mbox{for $Z\to \bar uu$}
\\
 -1.60(6)\times 10^{-4}  \mbox{ GeV}\quad \mbox{for $Z\to \bar dd$}
\end{array}
\right.
\end{eqnarray}
The total change in the partial width $\Gamma(Z\to hadrons)$ is
obtained by summing over 2 down-type and 2 up-type quarks:
\begin{eqnarray}
\Delta \Gamma(Z\to u,d,s,c)= -0.55(3) \mbox{ MeV}
\end{eqnarray}
which translates into the change of the strong coupling constant
determined at LEP 1 equal to
\begin{eqnarray}
\Delta\alpha_s = 
-\pi{\Delta \Gamma(Z\to hadrons)\over \Gamma(Z\to hadrons) }
=\pi{0.55\over 1741} \approx 0.001
\end{eqnarray}
This shift is somewhat smaller but still of the same order of
magnitude as the 
experimental accuracy and should
to be taken into account in the  final analysis of LEP 1 data.
Electroweak parameters extracted from $Z$ decays are not affected
by this correction.

The two-loop corrections to the $Zb\bar b$ vertex are dominated by terms
quadratic in $m_t$. The QCD corrections to these have been evaluated
already some time ago~\cite{FJRT+}.  The sub-leading terms $\propto \ln
m_t^2$ were obtained in~\cite{KwiSte95} and are of comparable magnitude,
thus indicating relatively slow convergence of the series. The complete
evaluation is thus mandatory and has been recently been
performed~\cite{HarSeiSte97}. In contrast to the previous case one is
confronted with three different scales, the masses of top, the $W$ and
the $Z$ boson.

Using the \hmp\ for $m_t^2\gg M_Z^2,M_W^2$, one may factor out the
$m_t$--dependence. However, for a part of the diagrams one still is left
with two-scale and even three-scale integrals involving $M_Z^2$ and
$M_W^2$ and $\xi_W M_W^2$, where $\xi_W$ is the electroweak gauge
parameter which we want to keep.  Although they appear to be only
one-loop integrals, their exact evaluation up to ${\cal O}(\epsilon)$
produces inconvenient results.  Instead, the results~\cite{HarSeiSte97}
were obtained by applying the \hmp\ to these kinds of diagrams once
more, this time using $\xi_W M_W^2, M_W^2\gg M_Z^2$. This seemingly
unrealistic choice of scales can be well justified: It is not possible
for an expansion to distinguish the inequality $M_W^2\gg M_Z^2$ from
$4M_W^2\gg M_Z^2$ or $(m_t+M_W)^2\gg M_Z^2$, the latter ones being
perfectly alright. The only matter is to perform the expansion on the
appropriate side of all thresholds, and here one is concerned with
thresholds at $2M_W$ and at $m_t+M_W$. Therefore, the choice $M_W^2\gg
M_Z^2$ is to be understood purely in this technical sense. Graphically
this continued expansion looks as follows:
\begin{eqnarray*}
&& \epsfxsize=5em 
\raisebox{-1.3em}{\epsffile[120 260 460 450]{T1hmp.ps}}
%
\stackrel{m_t^2\to \infty}{\longrightarrow}
\,\,\,\,\,
\epsfxsize=6em \raisebox{-1.3em}{\epsffile[120 260 560 450]{T3czbb.ps}} 
  \star
%
\epsfxsize=.7em 
\raisebox{-1.3em}{\epsffile[260 260 310 450]{T11hmp.ps}} 
\,\,+ \,\,\cdots \\[.5em]&&\hspace{6em}
%
\stackrel{\xi M_W^2\to \infty}{\longrightarrow}
%
  \bigg(\!\!\!\!
  \epsfxsize=5em
  \raisebox{-1.3em}{\epsffile[120 260 460 450]{T31cczbb.ps}}
  \star 
  \epsfxsize=4em \raisebox{-1.em}{\epsffile[120 260 460
  450]{T31czbb.ps}} \hspace{-2em}\bigg)
\star
\epsfxsize=.7em 
\raisebox{-1.3em}{\epsffile[260 260 310 450]{T11hmp.ps}} + \cdots\,,
\end{eqnarray*}
%
where only those terms are displayed which are relevant in the
discussion above and all others contributing to the \hmp\ are merged
into the ellipse.  The thick plain line is the top quark, the thick wavy one
a Goldstone boson with mass squared $\xi_W M_W^2$, for example.  The thin
plain lines are $b$--quarks, the inner thin wavy lines are $W$--bosons,
the outer ones $Z$-bosons. The spring-line is a gluon. The mass
hierarchy is assumed to be $m_t^2\gg \xi_W M_W^2 \gg M_W^2 \gg
M_Z^2$. The freedom in choosing the magnitude of $\xi_W$ provides a
welcome check of the routines and the results.

The outcome of this procedure is a nested series: The coefficients of
the $M_W/m_t$--expansion are in turn series in $M_Z/M_W$. Note that in
contrast to the decay into $u,d,s,c$ there is no threshold at $M_W$
which makes an additional expansion in $M_W/M_Z$ unnecessary.

In view of this calculation the procedure of successive application of
the \hmp\ resp.\ the \lmp\ has been implemented in a Fortran 90 program
named {\tt EXP}~\cite{SeiDiplom}. Therefore, given an arbitrary
hierarchy of mass scales, the computation of a three-loop two-point
function can now be done fully automatically. Even more, the link to the
Feynman diagram generator {\tt QGRAF}~\cite{qgraf} in a common
environment called {\tt GEFICOM}~\cite{geficom} allows to obtain the
result of a whole physical process without any human interference except
for specification of the process and final renormalization.

Finally, let us present the result for the $W$--induced corrections to
the $Z$--decay rate $\delta\Gamma^W(Z\to b\bar b)$ in the form of the
renormalization scheme independent difference to the decay rate into
$d\bar d$. Inserting the on-shell top mass $m_t = 175$~GeV, the
$Z$--mass $M_Z=91.91$~GeV and $\sin^2\theta_W = 0.223$
gives~\cite{HarSeiSte97}
\begin{eqnarray}
&&\delta\Gamma^W(Z\to b\bar b) - \delta\Gamma^W(Z\to d\bar d) =
\nonumber\\&&\mbox{}
=    \Gamma^0 {1\over \sin^2\theta_W} {\alpha\over \pi}
  \bigg\{ - 0.50  + (0.71 -0.48)
  \nonumber\\[0em]&&\mbox{}
  + (0.08 - 0.29) + (-0.01 - 0.07) + (-0.007 - 0.006) 
%
  \nonumber\\[0em]&&\mbox{}
%
  + {\alpha_s\over \pi} \bigg[ 1.16 + (1.21 - 0.49) + (0.30 - 0.65)
  \nonumber\\[0em]&&\mbox{}
%
  +  (0.02 - 0.21 + 0.01) + (-0.01 - 0.04 + 0.004) \bigg] \bigg\} = 
%
\nonumber\\&&\mbox{\hspace{0em}}
%
=\Gamma^0 {1\over \sin^2\theta_W} {\alpha\over \pi} \bigg\{- 0.50 - 0.07 +
{\alpha_s\over \pi} \bigg[ 1.16 + 0.13 \bigg]\bigg\}\,,\nonumber\\
\label{eqgamnum}
\end{eqnarray}
where the factor $\Gamma^0\alpha/(\pi\sin^2\theta_W)$ with $\Gamma^0 =
M_Z \alpha/(4 \sin^2\theta_W \cos^2\theta_W)$ has been pulled out for
convenience. The numbers after the first equality sign correspond to
successively increasing orders in $1/m_t^2$, where the brackets collect
the corresponding constant, $\log m_t$ and, if present, $\log^2
m_t$--terms. The numbers after the second equality sign represent the
leading $m_t^2$--term and the sum of the sub-leading ones.  The ${\cal
  O}(\alpha)$ and ${\cal O}(\alpha\alpha_s)$--results are displayed
separately. Comparison of this expansion of the one-loop terms to the
exact result~\cite{AkhBarRie86} shows agreement up to $0.01\%$
which gives quite some confidence in the $\alpha\alpha_s$--contribution.
One can see that although the $m_t^2$--, $m_t^0$-- and $m_t^0\log
m_t$--terms are of the same order of magnitude, the final result is
surprisingly well represented by the $m_t^2$--term, since the sub-leading
terms largely cancel among each other.

The uncertainty from uncalculated higher order QCD terms is far below
the foreseeable experimental precision, and the parametric uncertainty
from $\alpha_s$ and $m_t$ dominates. An important lesson to be learned
from the $Z b\bar b$ vertex in one and two loop approximation concerns
the interplay between ``dominant'' and ``sub-leading'' pieces: whenever
leading and sub-leading terms are of comparable magnitude, inclusion of
a sizeable number of terms in the expansion is required. The estimate of
the final result or of theoretical uncertainties based on the first two
terms of the series may lead to a wrong or misleading result.

Summary:
The interplay between electroweak interactions and QCD is crucial for
the evaluation and interpretation of precision measurements. Terms of
order $\alpha\alpha_s$ are under complete control, the top contribution
to the $\rho$ parameter and related observables are even available in
order $\alpha\alpha_s^2$. These results are important for the
interpretation of the current data and measurements to be performed in
the near future. The remaining theoretical uncertainties from the usage
of expansion techniques and uncalculated higher orders are far below the
anticipated experimental precision. In fact even the parametric
uncertainties from $\alpha_s$ and $m_t$ which dominate the current error
can be ignored in the foreseeable future.

\section{Acknowledgments:}
I would like to thank K.~Chetyrkin, R.~Harlander, Th.~Seidensticker,
M.~Steinhauser, and G. Weiglein for providing important input material
and for instructive discussions. \TeX nical help by R.~Harlander is
gratefully acknowledged.

\def\app#1#2#3{{\it Act.~Phys.~Pol.~}{\bf B #1} (#2) #3}
\def\apa#1#2#3{{\it Act.~Phys.~Austr.~}{\bf#1} (#2) #3}
\def\cpc#1#2#3{{\it Comp.~Phys.~Commun.~}{\bf #1} (#2) #3}
\def\fortp#1#2#3{{\it Fortschr.~Phys.~}{\bf#1} (#2) #3}
\def\npb#1#2#3{{\it Nucl.~Phys.~}{\bf B #1} (#2) #3}
\def\plb#1#2#3{{\it Phys.~Lett.~}{\bf B #1} (#2) #3}
\def\prc#1#2#3{{\it Phys.~Reports }{\bf #1} (#2) #3}
\def\prd#1#2#3{{\it Phys.~Rev.~}{\bf D #1} (#2) #3}
\def\pR#1#2#3{{\it Phys.~Rev.~}{\bf #1} (#2) #3}
\def\prl#1#2#3{{\it Phys.~Rev.~Lett.~}{\bf #1} (#2) #3}
\def\nima#1#2#3{{\it Nucl.~Inst.~Meth.~}{\bf A #1} (#2) #3}
\def\pr#1#2#3{{\it Phys.~Reports }{\bf #1} (#2) #3}
\def\sovnp#1#2#3{{\it Sov.~J.~Nucl.~Phys.~}{\bf #1} (#2) #3}
\def\yadfiz#1#2#3{{\it Yad.~Fiz.~}{\bf #1} (#2) #3}
\def\jetp#1#2#3{{\it JETP~Lett.~}{\bf #1} (#2) #3}
\def\zpc#1#2#3{{\it Z.~Phys.~}{\bf C #1} (#2) #3}
\def\ptp#1#2#3{{\it Prog.~Theor.~Phys.~}{\bf #1} (#2) #3}
\def\nca#1#2#3{{\it Nuovo~Cim.~}{\bf #1A} (#2) #3}
\def\mpl#1#2#3{{\it Mod.~Phys.~Lett.~}{\bf A #1} (#2) #3}
\def\tmf#1#2#3{{\it Teor.~Mat.~Fiz.~}{\bf #1} (#2) #3}
\def\ibid#1#2#3{{ibid.~}{\bf #1} (#2) #3}
\def\cmp#1#2#3{{\it Comm.~Math.~Phys.~}{\bf #1} (#2) #3}
\def\jcp#1#2#3{{\it J.~Comp.~Phys.~}{\bf #1} (#2) #3}
\def\ijmpc#1#2#3{{\it Int.~J.~Mod.~Phys.~}{\bf #1} (#2) #3}
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\end{document}

