%Title: Three-loop QCD Corrections to \delta\rho, \DELTA r and \Delta\kappa
%Authors: K.G. Chetyrkin, J.H. Kuehn, M. Steinhauser
%Published:  <EM>Proceedings of the Workshop "Perspectives for Electroweak Interactions in e+e- Collisions", B. A. Kniehl, ed., World Scientific 1995, pp. 97-108</EM>
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%
% own definitions
%
\newcommand{\beq}{\begin{equation}}
\newcommand{\eeq}{\end{equation}}
\newcommand{\beqa}{\begin{eqnarray}}
\newcommand{\eeqa}{\end{eqnarray}}

\newcommand{\al}{\alpha}
\newcommand{\as}{\alpha_s}
\newcommand{\api}{\frac{\alpha_s}{\pi}}
\newcommand{\msbar}{\overline{\mbox{MS}}}
\newcommand{\ep}{\epsilon}
 
\newcommand{\drtb}{\Delta r_{tb}}
\newcommand{\drtbbar}{\Delta \bar{r}_{tb}}
\newcommand{\dktb}{\Delta\kappa_{tb}}
\newcommand{\drho}{\delta\rho}
\newcommand{\sineff}{\sin^2\!\bar{\Theta}}
\newcommand{\mzmt}{\frac{M_Z^2}{M_t^2}}
\newcommand{\lms}{\bar{l}_Z}
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\begin{document}

\begin{flushright}
TTP 95--20 \\
April 1995 \\
\end{flushright}

\vspace{0.5cm}

\centerline{\normalsize\bf THREE-LOOP QCD CORRECTIONS TO $\delta\rho$, 
$\Delta r$ and $\Delta\kappa$}
\baselineskip=22pt
%\centerline{\footnotesize\sf (For subsequent 20\% photoreduction
%to 17.8 $\times$ 11.9 cm text area)\footnote{The \LaTeX\ source
%file for this document may be used as a template for your
%article, and can be requested by e-mailing {\sf
%worldscp@singnet.com.sg}.}}

%\vfill
%\vspace*{0.6cm}
\centerline{\footnotesize K.G. CHETYRKIN, J.H. K\"UHN and M. STEINHAUSER}
\baselineskip=13pt
\centerline{\footnotesize\it 
   Institut f\"ur Theoretische Teilchenphysik,Universit\"at Karlsruhe,}
\baselineskip=12pt
\centerline{\footnotesize\it 
   Kaiserstr. 12, Postfach 6980, D-76128 Karlsruhe, Germany}
%\centerline{\footnotesize E-mail: username@abc.def.gh}
%\vspace*{0.3cm}
%\centerline{\footnotesize and}
%\vspace*{0.3cm}
%\centerline{\footnotesize SECOND AUTHOR'S NAME}
%\baselineskip=13pt
%\centerline{\footnotesize\it Group, Company, Address, City, State ZIP/Zone,
%Country}

%\vfill
\vspace*{0.9cm}
\abstracts{
QCD corrections to electroweak observables are reviewed. Recent results
on contributions from the top-bottom doublet
of ${\cal O}(\as^2)$ to $\drho$, $\Delta r$ and $\Delta\kappa$ are presented.
It is demonstrated that the first three terms
in the expansion in $M_Z^2/M_t^2$ provide an excellent approximation to
the exact result.
Calculational techniques are briefly discussed.
}
 
%\vspace*{0.6cm}
\normalsize\baselineskip=15pt
\setcounter{footnote}{0}
\renewcommand{\thefootnote}{\alph{footnote}}

\section{Introduction}

The precision of measurements of electroweak observables
at LEP, SLC and the TEVATRON has increased steadily during the past years
\cite{Sch94}.
Theoretical predictions for these quantities on the basis of the 
Standard Model (SM) exhibit a strong dependence on the mass of the top quark.
The remarkable agreement between the 
recent value of $M_t$ published by the CDF and D0 
collaborations 
\cite{CDF}
of $M_t=176\pm8\pm10$ GeV and $M_t=199^{+19}_{-21}\pm22$ GeV on one side and
the value deduced indirectly from precision
measurements of $M_t=173^{+12+18}_{-13-20}$ GeV on the other side
can be considered as striking demonstration of the importance of 
quantum corrections and the predictive power of precision measurements
combined with theoretical calculations. Once the top mass is
measured at hadron or electron positron colliders with sufficient
precision, one may subsequently aim towards a determination of the Higgs
mass $M_H$ with similar methods. As a consequence of the relatively
weak $M_H$ dependence fairly precise predictions of the $M_t-M_H$ 
interdependence are required: A shift of $M_t$ of 2 GeV corresponds 
to a shift of about 15 MeV in $M_W$.
Conversely, for $M_W$ and $M_Z$ fixed, the same shift of 2 GeV in
$M_t$ leads to a shift of about 25\% in the prediction for $M_H$.
From these considerations it becomes clear that a relative accuracy
in $\Delta r$ or $\delta \rho$ of about 2 \% will be required
from theory in order to match the anticipated experimental precision.
This justifies the effort to perform a calculation of QCD corrections
to order $\alpha_s^2$, requiring the evaluation of massive three-loop
diagrams.

The high value of the top quark mass allows to identify observables
which are enhanced by a factor $G_F M_t^2$ or even to go one step further
and perform an expansion in $1/M_t^2$.
Apart of the $Zb\bar{b}$ vertex, the $W$ and $Z$ boson self energies
induce correction terms $\propto M_t^2$. These are collected in the $\rho$
parameter
\beq
\delta\rho = \frac{\Pi_{ZZ}(0)}{M_Z^2}
           - \frac{\Pi_{WW}(0)}{M_W^2}.
\eeq
Recently, the calculation of $\drho$ up to 
${\cal O}(\alpha_s^2)$ was performed independently by two
groups
\cite{AvdFleMikTar94CheKueSte95}.

The $\rho$ parameter governs the leading contribution of two 
important observables, namely $\Delta r$ and the effective 
mixing angle $\sineff$. It has become customary to pa\-ra\-me\-trize the
radiative corrections between the electroweak parameters $M_W$, $M_Z$,
$\alpha$ and $G_F$ through the quantity $\Delta r$
\cite{Sir80}:
\beq
M_W^2 = \frac{M_Z^2}{2} \left(
        1+\sqrt{1-\frac{4\pi\al}{\sqrt{2}M_Z^2 G_F(1-\Delta r)}}
                          \,\,\right).
\label{mweqn}
\eeq
$\Delta r$ can be expressed in terms of the selfenergies 
$\Pi_{WW}(q^2), \Pi_{ZZ}(q^2), \Pi_{\gamma\gamma}(q^2)$ 
and $\Pi_{\gamma Z}(q^2)$.
If we restrict ourselves to the contribution from the (t,b)-doublet, 
$\Delta r$ is given by:
\beq
\drtbbar = \frac{c^2}{s^2}\mbox{Re}\!\!
     \left(\frac{\Pi_{WW}(M_W^2)}{M_W^2}\!
     -\!\frac{\Pi_{ZZ}(M_Z^2)}{M_Z^2}\right)
      + \tilde{\Pi}_{\gamma\gamma}(M_Z^2)
      + \frac{1}{M_W^2}\!\left(\Pi_{WW}(0)\!-\!\mbox{Re}\Pi_{WW}(M_W^2)\right)
\label{deltar}
\eeq
with $\tilde{\Pi}_{\gamma\gamma}(q^2)=\Pi_{\gamma\gamma}(q^2)/q^2$.
The quantity $\drtbbar$ is introduced in order to indicate 
the treatment of the photon polarization:
To circumvent the mass singularities from the approximation of 
vanishing bottom mass $\tilde{\Pi}_{\gamma\gamma}(q^2)$
is evaluated at $q^2=M_Z^2$. 
The difference is accounted for by dispersion relations with input
from the actual measurement of $\sigma(e^+ e^- \to hadrons)$ in
the low energy region
\cite{Jeg95}.

The effective weak mixing angle governs in particular the asymmetries
\cite{Hol}
in $Z$ boson production and decay. It is related to
$\sin^2 \Theta \equiv 1-M_W^2/M_Z^2$ through a correction factor
\beq
\sineff = (1+\Delta\kappa)\sin^2\Theta.
\eeq
$\Delta\kappa$ can be expressed in terms of two point functions as follows
\beq
\dktb = -\frac{c}{s}\frac{\Pi_{\gamma Z}(M_Z^2)}{M_Z^2}
     -\frac{c^2}{s^2}\mbox{Re}
     \left(\frac{\Pi_{WW}(M_W^2)}{M_W^2}-\frac{\Pi_{ZZ}(M_Z^2)}{M_Z^2}\right).
\eeq

It is possible to expand $\Delta r$ and $\sineff$ in the small parameter
$M_Z^2/M_t^2$
by evaluating the Taylor series of the self-energies
using the techniques described in
\cite{AvdFleMikTar94CheKueSte95}.
In principle this expansion can be performed up to arbitrary high powers
in the inverse
top mass and up to order $\alpha_s^2$.
However,
it will be demonstrated that
already the first three terms provide an excellent approximation to the
full answer 
\cite{CheKueSte95}.

In section 2 some technical details to the calculation are given. In section 3
we present the result and discuss briefly the numerical consequences.


\section{Calculational Techniques}

Three different types of integrals arise:
Diagrams involving only light quarks 
lead to massless two-point functions. These are calculated
with the help of the FORM
\cite{Ver91} package MINCER
\cite{LarTkaVer91}.
They give rise to terms depending logarithmically on $M_Z^2/M_t^2$.

For the graphs where the massive top quark couples to the photon or the 
$W$ or $Z$ boson massive three-loop tadpole diagrams have to be evaluated.
After the decomposition of the scalar products of the numerator into 
appropriate combinations of the denominator recurrence relations provided by
the the integration-by-parts (IP) method 
\cite{CheTka81}
are used in order to reduce the calculation to a few
simple integrals plus a small set of so-called master integrals.
The IP method was first applied in
\cite{Bro92}
to three-loop tadpole diagrams. There the subclass relevant for the
neutral bosons  was considered. The recurrence relations for the
$W$ graphs were derived in
\cite{AvdFleMikTar94CheKueSte95}.
The three master diagrams are pictured in Fig. \ref{figmaster}.
The corresponding integrals are given by:
\begin{eqnarray*}
\mbox{(a)} 
&=&
  \left(\frac{\mu^2}{M^2}\right)^{3\ep}\left[
        \frac{2}{\ep}\zeta(3) +  6\zeta(3) - 9\zeta(4) + 2 B_4
                       \right]
\\
\mbox{(b)} 
&=& 
    \left(\frac{\mu^2}{M^2}\right)^{3\ep} \left[
                  \frac{2}{\ep}\zeta(3) + D_3 
                                         \right]
\\
\mbox{(c)} 
&=&
   \left(\frac{\mu^2}{M^2}\right)^{3\ep} M^4 \left[
         \frac{1}{\ep^3} 
       + \frac{15}{4\ep^2}
       + \frac{3}{2\ep}\zeta(2) + \frac{65}{8\ep} 
       + \frac{81}{4} S_2 - \zeta(3) + \frac{45}{8}\zeta(2) + \frac{135}{16}
          \right]
\end{eqnarray*}
On the r.h.s. we followed the standard $\msbar$ practice and discarded terms
proportional to $\gamma_E$ and $\log 4\pi$.
The first master integral has been calculated analytically in 
\cite{Bro92}; the results for the last two integrals are given in
\cite{AvdFleMikTar94CheKueSte95} ($D_3$ is presently only known numerically).
The values for the 
constants $B_4$, $S_2$ and $D_3$ are as follows:
\begin{eqnarray}
B_4 &=& 16\, \mbox{Li}_4\left(\frac{1}{2}\right)
      + \frac{2}{3}\log^4 2
      - \frac{2}{3}\pi^2\log^2 2
      - \frac{13}{180}\pi^4
     = -1.76280\ldots 
\nonumber\\
S_2 &=& \frac{4}{9\sqrt{3}} \mbox{Cl}_2\left(\frac{\pi}{3}\right)
     = 0.260434\ldots
\nonumber\\
D_3 &=& -3.02700\ldots.
\nonumber
\end{eqnarray}

\begin{figure}
\vspace*{13pt}
 \begin{center}
 \begin{tabular}{ccc}
   \epsfxsize=4.0cm
   \leavevmode
   \epsffile[180 270 430 520]{masterB4.ps}
&
   \epsfxsize=4.0cm
   \leavevmode
   \epsffile[180 270 430 520]{masterD3.ps}
&
   \epsfxsize=4.0cm
   \leavevmode
   \epsffile[180 270 430 520]{masterS2.ps}
 \end{tabular}
\fcaption{The master diagrams. Solid lines represent massive scalar
          propagators, dashed lines massless ones.}
\label{figmaster}
\end{center}
\end{figure}

For the calculation of $\delta\rho$ the external momentum has been set
to zero. The evaluation of $\drtbbar$ and $\dktb$
require also polarization functions for external momentum $q^2=M_{W/Z}^2$.
We approximate these functions through a Taylor expansion 
in $M_{W/Z}^2/M_t^2$.
Using the d'Alembertian operator the expansion of the transverse part
of the polarisation tensor is given by
\beqa
\lefteqn{\Pi(0)+q^2\Pi^{\prime}(0) =}
\nonumber\\
&&
    \frac{1}{D-1}\Bigg\{
      g_{\mu\nu}\Pi^{\mu\nu}(0) 
    + \frac{q^2}{2D}\left(\Box g_{\mu\nu}\Pi^{\mu\nu}(q)\right)|_{q=0}
\nonumber
\\
&&
    - \frac{1}{q^2}\Bigg[
        \frac{q^2}{2D}\left(\Box q_\mu q_\nu \Pi^{\mu\nu}(q)\right)|_{q=0}
       +\frac{(q^2)^2}{8D(D+2)}
              \left(\Box^2 q_\mu q_\nu \Pi^{\mu\nu}(q)\right)|_{q=0}
                   \Bigg]
                                           \Bigg\}.
\eeqa
A similar expression holds for $\Pi^{\prime\prime}(0)$.

The third class of integrals consists of the double triangle diagram
and the double bubble diagram where the external quark is massless
and the second quark is the top.
In order to get a correct expansion in $1/M_t$ these diagrams have to
be treated with the heavy mass expansion technique
\cite{Smi94}.
Besides the
expansion of the whole diagram in the external momentum also all
hard subgraphs have to be considered. A hard subgraph is by definition
one-particle irreducible w.r.t. the light lines and contains all top 
propagators. These subdiagrams have to be expanded in their external
momentum (and eventually small masses) before the momentum
integration is performed. So the three-loop graphs factorize into
a one-loop times a two-loop integral.
%%%(Formally the diagrams of type two also have such subgraphs but within
%%%the framework of dimensional regularization the result form them 
%%%is zero.)

\section{Results and Discussion}

As already mentioned above the $\rho$ parameter is the leading part of
both $\drtbbar$ and $\dktb$. So we first list
the result for $\delta\rho$. After performing charge and mass
renormalization and setting $C_A=3$, $C_F=4/3$ and $\mu^2=\bar{m}_t^2$
$\delta\rho^{\scriptsize\msbar}$ reads:
\beqa
\lefteqn{\delta\rho_{\scriptsize\msbar} = 3 x_t}
\nonumber
\\ && \Bigg\{ 1
       + \frac{\as}{4\pi} \bigg( 8 - \frac{16}{3} \zeta(2) \bigg)
\nonumber\\ & &
       + \left(\frac{\as}{4\pi}\right)^2\Bigg[
         n_f \bigg(  - \frac{50}{3} - \frac{64}{9} \zeta(3)
               + \frac{112}{9} \zeta(2) \bigg)
\nonumber\\ & &
       +  \frac{26459}{81} + 882 S_2 - \frac{8}{9} D_3
       - \frac{5072}{27} \zeta(3) + \frac{1144}{9} \zeta(4)
       - \frac{25064}{81} \zeta(2) - \frac{16}{9} B_4
\Bigg]
\Bigg\}
\eeqa
with $n_f=6$ denotes the total number of quarks. The variable $x_t$
is defined as $x_t=G_F \bar{m}_t^2/8\sqrt{2}\pi^2$.
In the following we list the results for $\drtbbar$ and $\dktb$ in 
order $\as^0$, $\as^1$ and $\as^2$ including also 
the $M_Z^2/\mbar^2$ terms:
\beqa
\lefteqn{\drtbbar^{\scriptsize\msbar}  =  -\frac{c^2}{s^2} 
          \delta\rho_{\scriptsize\msbar} 
        - 3 \frac{c^2}{s^2} x_t \frac{M_Z^2}{\mbar^2}} 
\nonumber
\\
&&
\Bigg\{            
        \frac{1}{3} + \frac{8}{9} s^2 \lms - \frac{16}{27} s^2 
       -\frac{2}{3} \lms
%
% 1/Mt^2-term:
%
       +\frac{M_Z^2}{\mbar^2}
       \Bigg(  - \frac{2}{5} + \frac{103}{90} s^2 - s^4 \Bigg)
\nonumber\\
&&
       + \frac{\as}{4\pi}\Bigg( 
           \frac{88}{9} + \frac{128}{9} s^2 \zeta(3) 
         - \frac{128}{27} s^2 \zeta(2) 
         + \frac{32}{9} s^2 \lms - \frac{496}{27} s^2 - \frac{32}{3} \zeta(3) 
         + \frac{64}{27} \zeta(2) - \frac{8}{3} \lms 
\nonumber\\
&&
\hphantom{\frac{\as}{4\pi} }
%
% 1/Mt^2-term:
%
       +\frac{M_Z^2}{\mbar^2}
        \Bigg( \frac{4138}{1215} + 8 s^2 \zeta(2) 
         - \frac{40361}{3645} s^2 - \frac{16}{3} s^4 \zeta(2) 
         + \frac{46}{9} s^4 - \frac{8}{3} \zeta(2) \Bigg)
\Bigg)
\nonumber\\
&&
       +\left(\frac{\as}{4\pi}\right)^2 \Bigg[
         n_f \Bigg(  - \frac{4480}{243} + \frac{256}{27} s^2 \zeta(3) \lms 
                     - \frac{2944}{81} s^2 \zeta(3) 
                     + \frac{448}{27} s^2 \zeta(2) 
                     - \frac{352}{27} s^2 \lms 
\nonumber\\
&&
\hphantom{\frac{\as}{4\pi} }
                     + \frac{32}{27} s^2 \lmsz 
          + \frac{2696}{243} s^2
          - \frac{64}{9} \zeta(3) \lms + \frac{2080}{81} \zeta(3) 
          - \frac{176}{27} \zeta(2) + \frac{88}{9} \lms 
          - \frac{8}{9} \lmsz
\nonumber\\
&&
\hphantom{\frac{\as}{4\pi} }
%
% 1/Mt^2-term:
%
       +\frac{M_Z^2}{\mbar^2}
        \Bigg(  - \frac{2095}{1458} 
            + \frac{32}{3} s^2 \zeta(3) - \frac{68}{27} s^2 \zeta(2) 
            + \frac{87587}{21870} s^2 - \frac{64}{9} s^4 \zeta(3) 
\nonumber\\
&&
\hphantom{\frac{\as}{4\pi} }
            + \frac{136}{81} s^4 \zeta(2) 
            - \frac{407}{81} s^4 
            - \frac{32}{9} \zeta(3)
            + \frac{68}{81} \zeta(2) \Bigg)
          \Bigg)
\nonumber\\
&&
\hphantom{\frac{\as}{4\pi} }
         + \frac{94957}{243} + 856 S_2 s^2 - 428 S_2 + \frac{256}{81} D_3 s^2 
         - \frac{128}{81}D_3 - \frac{4480}{27} s^2 \zeta(3) \lms 
\nonumber\\
&&
\hphantom{\frac{\as}{4\pi} }
         + \frac{12016}{27} s^2 \zeta(3) 
         + \frac{12032}{81} s^2 \zeta(4) 
         - \frac{3200}{27} s^2 \zeta(5) - \frac{52816}{243} s^2 \zeta(2) 
         - \frac{256}{27} s^2 B_4 
\nonumber\\
&&
\hphantom{\frac{\as}{4\pi} }
         + \frac{688}{3} s^2 \lms 
         - \frac{560}{27} s^2 \lmsz
          - \frac{24164}{81} s^2 
         + \frac{1120}{9} \zeta(3) \lms 
%\nonumber\\
%&&
%\hphantom{\frac{\as}{4\pi} }
         - \frac{8117}{18} \zeta(3) - \frac{6016}{81} \zeta(4) 
\nonumber\\
&&
\hphantom{\frac{\as}{4\pi} }
         + \frac{800}{9} \zeta(5) 
         + \frac{22736}{243} \zeta(2) 
         + \frac{128}{27} B_4 
         - \frac{1400}{9} \lms + \frac{116}{9} \lmsz 
\nonumber\\
&&
\hphantom{\frac{\as}{4\pi} }
%
% 1/Mt^2-term:
%
       +\frac{M_Z^2}{\mbar^2}
        \Bigg( \frac{1440889423}{17496000} + \frac{1913}{2} S_2 s^2 
         - \frac{1913}{3} S_2 s^4 - \frac{1913}{6} S_2 
         + \frac{8}{3} D_3 s^2 
         + \frac{5632}{6075} s^2 \lms 
\nonumber\\
&&
\hphantom{\frac{\as}{4\pi} }
         - \frac{16}{9} D_3 s^4 
         - \frac{8}{9} D_3 - \frac{4346699}{7290} s^2 \zeta(3) 
         - \frac{536}{3} s^2 \zeta(4) + \frac{593473}{2430} s^2 \zeta(2) 
         - \frac{128}{1215} s^2 \lmsz 
\nonumber\\
&&
\hphantom{\frac{\as}{4\pi} }
         + \frac{2345403277}{13122000} s^2 
         + \frac{20812}{81} s^4 \zeta(3) + \frac{1072}{9} s^4 \zeta(4) 
         - \frac{591937}{3645} s^4 \zeta(2) 
         + \frac{284077}{5832} s^4 
\nonumber\\
&&
\hphantom{\frac{\as}{4\pi} }
         + \frac{4169129}{51840} \zeta(3) 
         + \frac{536}{9} \zeta(4) 
         - \frac{595393}{7290} \zeta(2) 
         - \frac{7024}{6075} \lms + \frac{32}{405} \lmsz \Bigg)
\Bigg]
\Bigg\},
\\
\lefteqn{\dktb^{\scsi\msbar} =  \frac{c^2}{s^2} 
          \drho_{\scsi\msbar} - 3 \frac{c^2}{s^2} x_t \frac{M_Z^2}{\mbar^2}} 
\nonumber\\
&&
\Bigg\{ 
       - \frac{1}{3} - \frac{4}{9} s^2 \lms + \frac{8}{27} s^2 
       + \frac{2}{3} \lms
%
% 1/Mt^2-term:
%
       +\frac{M_Z^2}{\mbar^2}
        \Bigg( \frac{2}{5} - \frac{37}{45} s^2 + \frac{1}{2} s^4 \Bigg)
\nonumber\\
&&
       + \frac{\as}{4\pi}\Bigg( 
                   - \frac{88}{9} - \frac{64}{9} s^2 \zeta(3) 
                   + \frac{64}{27} s^2 \zeta(2) 
                   - \frac{16}{9} s^2 \lms + \frac{248}{27} s^2 
                   + \frac{32}{3} \zeta(3) - \frac{64}{27} \zeta(2) 
                   + \frac{8}{3} \lms 
\nonumber\\
&&
\hphantom{\frac{\as}{4\pi} }
%
% 1/Mt^2-term:
%
       +\frac{M_Z^2}{\mbar^2}
        \Bigg(  - \frac{4138}{1215} - \frac{16}{3} s^2 \zeta(2) 
         + \frac{24838}{3645} s^2 + \frac{8}{3} s^4 \zeta(2) 
         - \frac{23}{9} s^4 + \frac{8}{3} \zeta(2) \Bigg)
\Bigg)
\nonumber\\
&&
       +\left(\frac{\as}{4\pi}\right)^2 \Bigg[
         n_f  \Bigg( \frac{4480}{243} - \frac{128}{27} s^2 \zeta(3) \lms 
                   + \frac{1472}{81} s^2 \zeta(3) 
                   - \frac{224}{27} s^2 \zeta(2) + \frac{176}{27} s^2 \lms 
\nonumber\\
&&
\hphantom{\frac{\as}{4\pi} }
                   - \frac{16}{27} s^2 \lmsz - \frac{1348}{243} s^2 
                   + \frac{64}{9} \zeta(3) \lms - \frac{2080}{81} \zeta(3) 
                   + \frac{176}{27} \zeta(2) - \frac{88}{9} \lms 
                   + \frac{8}{9} \lmsz
\nonumber\\
&&
\hphantom{\frac{\as}{4\pi} }
%
% 1/Mt^2-term:
%
       +\frac{M_Z^2}{\mbar^2}
        \Bigg( \frac{2095}{1458} - \frac{64}{9} s^2 \zeta(3) 
         + \frac{136}{81} s^2 \zeta(2) 
         - \frac{35633}{10935} s^2 + \frac{32}{9} s^4 \zeta(3) 
         - \frac{68}{81} s^4 \zeta(2) 
\nonumber\\
&&
\hphantom{\frac{\as}{4\pi} }
         + \frac{407}{162} s^4 
         + \frac{32}{9} \zeta(3)
         - \frac{68}{81} \zeta(2) \Bigg)
\Bigg)
\nonumber\\
&&
\hphantom{\frac{\as}{4\pi} }
                 - \frac{94957}{243} - 428 S_2 s^2 + 428 S_2 
                 - \frac{128}{81} D_3 s^2 + \frac{128}{81} D_3 
                 + \frac{2240}{27} s^2 \zeta(3) \lms 
\nonumber\\
&&
\hphantom{\frac{\as}{4\pi} }
                 - \frac{6008}{27} s^2 \zeta(3) - \frac{6016}{81} s^2 \zeta(4) 
                 + \frac{1600}{27} s^2 \zeta(5) 
                 + \frac{26408}{243} s^2 \zeta(2) + \frac{128}{27} s^2 B_4 
\nonumber\\
&&
\hphantom{\frac{\as}{4\pi} }
                 - \frac{344}{3} s^2 \lms 
                 + \frac{280}{27} s^2 \lmsz 
                 + \frac{12082}{81} s^2 
                 - \frac{1120}{9} \zeta(3) \lms 
                 + \frac{8117}{18} \zeta(3) 
                 + \frac{6016}{81} \zeta(4) 
\nonumber\\
&&
\hphantom{\frac{\as}{4\pi} }
                 - \frac{800}{9} \zeta(5) 
                 - \frac{22736}{243} \zeta(2) 
                 - \frac{128}{27} B_4 
                 + \frac{1400}{9} \lms - \frac{116}{9} \lmsz 
\nonumber\\
&&
\hphantom{\frac{\as}{4\pi} }
%
% 1/Mt^2-term:
%
       +\frac{M_Z^2}{\mbar^2}
        \Bigg(  - \frac{1440889423}{17496000} - \frac{1913}{3} S_2 s^2 
         + \frac{1913}{6} S_2 s^4 + \frac{1913}{6} S_2      
         - \frac{16}{9} D_3 s^2 
\nonumber\\
&&
\hphantom{\frac{\as}{4\pi} }
         + \frac{8}{9} D_3 s^4 
         + \frac{8}{9} D_3  
         + \frac{5283239}{14580} s^2 \zeta(3) + \frac{1072}{9} s^2 \zeta(4) 
         - \frac{593089}{3645} s^2 \zeta(2) - \frac{2816}{6075} s^2 \lms 
\nonumber\\
&&
\hphantom{\frac{\as}{4\pi} }
         + \frac{64}{1215} s^2 \lmsz 
         - \frac{506454163}{6561000} s^2 
         - \frac{10406}{81} s^4 \zeta(3)
         - \frac{536}{9} s^4 \zeta(4) + \frac{591937}{7290} s^4 \zeta(2) 
\nonumber\\
&&
\hphantom{\frac{\as}{4\pi} }
         - \frac{284077}{11664} s^4 
         - \frac{4169129}{51840} \zeta(3) 
         - \frac{536}{9} \zeta(4) + \frac{595393}{7290} \zeta(2) 
         + \frac{7024}{6075} \lms - \frac{32}{405} \lmsz \Bigg)
\Bigg]
\nonumber\\
&&
     + i \pi\Bigg[
            -\frac{4}{9} s^2 + \frac{16}{27} s^4 
   + \frac{\as}{4\pi} \Bigg(-\frac{16}{9} s^2 + \frac{64}{27} s^4\Bigg) 
\nonumber\\
&&
\hphantom{\frac{\as}{4\pi} }
   + \left(\frac{\as}{4\pi}\right)^2  \Bigg(
            n_f \bigg( \frac{176}{27}s^2 - \frac{32}{27}s^2 \lms 
                    - \frac{128}{27}s^2\zeta(3)
               -\frac{704}{81}s^4 + \frac{128}{81}s^4 \lms  
               + \frac{512}{81}s^4 \zeta(3) 
\bigg) 
\nonumber\\
&&
\hphantom{\frac{\as}{4\pi}\frac{\as}{4\pi} }
    -\frac{344}{3}s^2 + \frac{560}{27}s^2 \lms + \frac{2240}{27}s^2\zeta(3)
    +\frac{1376}{9}s^4 - \frac{2240}{81}s^4 \lms - \frac{8960}{81}s^4\zeta(3)
\nonumber\\
&&
\hphantom{\frac{\as}{4\pi}\frac{\as}{4\pi} }
%
% 1/Mt^2-term:
%
       +\frac{M_Z^2}{\mbar^2}
        \Bigg( \frac{128}{1215} s^2 \lms - \frac{2816}{6075} s^2  
          - \frac{512}{3645} s^4 \lms + \frac{11264}{18225} s^4 \Bigg)
             \Bigg)
             \Bigg]
\Bigg\}
%%%\nonumber
\eeqa
with $s^2\equiv 1-M_W^2/M_Z^2$ and $\lms\equiv \ln M_Z^2/\bar{m}_t^2 $.
The OS result can be received
with the help of the relation between the
running and the OS top mass
\cite{BroGraGraSch90}. The final result, after setting $n_f=6$, reads
($\mu^2=M_t^2$):
\beqa
\lefteqn{\drtbbar^{OS} = -\frac{c^2}{s^2} 3 X_t }
%%%\\
&&
\hphantom{\dktb^{OS}=\frac{c^2}{s^2}X_t}
\Bigg\{
        1 + \mzmt \Big(0.3333 - 0.6667 \lmzmt +
        (-0.5926 + 0.8889 \lmzmt) s^2 \Big) 
\nonumber\\
&&
\hphantom{\frac{\as}{4\pi} }
        +\left(\mzmt\right)^2
         \Big(-0.4 + 1.144 s^2  - s^4\Big)  
\nonumber\\
&&
    +\api \Big[-2.8599 + \mzmt \Big(-1.5640 - 0.6667 \lmzmt +
          (0.1022 + 0.8889 \lmzmt) s^2 \Big)
\nonumber\\
&&
\hphantom{\frac{\as}{4\pi} }
         +\left(\mzmt\right)^2 
                     \Big(-1.312 + 3.573 s^2  - 3.582 s^4 \Big)
          \Big] 
\nonumber\\
&&
    +\left(\api\right)^2  
         \Big[-14.594 + \mzmt \Big(-17.224 + 0.08829 \lmzmt 
               + 0.4722 \lmzmtz  
\nonumber\\
&&
\hphantom{\frac{\as}{4\pi} }
         + (22.6367 + 1.2527 \lmzmt - 0.8519 \lmzmtz ) s^2 \Big)
         + \left(\mzmt\right)^2 
            \Big(-7.7781 - 0.07226 \lmzmt 
\nonumber\\
&&
\hphantom{\frac{\as}{4\pi} }
                 + 0.004938 \lmzmtz  
         +(21.497 + 0.05794 \lmzmt - 0.006584 \lmzmtz ) s^2  - 21.0799 s^4 
                   \Big) 
\Big]
\Bigg\} 
\nonumber\\
\lefteqn{\dktb^{OS}  = -\frac{c^2}{s^2} 3 X_t }
%%%\\
&&
\hphantom{\dktb^{OS}=\frac{c^2}{s^2} X_t }
\Bigg\{
     -1 + \mzmt \Big(-0.3333 + 0.6667 \lmzmt +
       (0.2963 - 0.4444 \lmzmt) s^2 \Big) 
\nonumber\\
&&
\hphantom{\frac{\as}{4\pi} }
        +\left(\mzmt\right)^2
         \Big(0.4 - 0.8222 s^2  + 0.5 s^4\Big)  
\nonumber\\
&&
    +\api \Big[2.8599 + \mzmt \Big(1.564 + 0.6667 \lmzmt +
          (-0.051103 - 0.4444 \lmzmt) s^2 \Big)
\nonumber\\
&&
\hphantom{\frac{\as}{4\pi} }
       \left(\mzmt\right)^2  \Big(1.312 - 2.682 s^2  + 1.791 s^4 \Big)
\Big] 
\nonumber\\
&&
    +\left(\api\right)^2 
      \Big[14.594 + \mzmt \Big(17.224 - 0.08829 \lmzmt - 0.47222 \lmzmtz  
\nonumber\\
&&
\hphantom{\frac{\as}{4\pi} }
       +   (-11.3184 - 0.6263 \lmzmt + 0.4259 \lmzmtz ) s^2 \Big)
       + \left(\mzmt\right)^2 \Big(7.7781 + 0.072263 \lmzmt 
\nonumber\\
&&
\hphantom{\frac{\as}{4\pi} }
                              - 0.004938 \lmzmtz   
       +(-16.0186 - 0.02897 \lmzmt + 0.003292 \lmzmtz ) s^2  + 10.54 s^4 \Big)
\Big]
\nonumber\\
&&
        +i \mzmt \Bigg[
              -1.396 s^2  + 1.862 s^4  
        + \api  (-1.396 s^2  + 1.862 s^4 ) 
\nonumber\\
&&
\hphantom{\frac{\as}{4\pi} }
       + \left(\api\right)^2  \Big[(-1.968 + 2.676 \lmzmt) s^2  
          +(2.6235 - 3.5682 \lmzmt) s^4  
\nonumber\\
&&
\hphantom{\frac{\as}{4\pi}\frac{\as}{4\pi} }
       +  \mzmt  \Big((-0.09102 + 0.02069 \lmzmt) s^2  
                             + (0.1214 - 0.02758 \lmzmt) s^4 \Big)\Big]
                 \Bigg]
\Bigg\},
\nonumber
\eeqa
where $\lmzmt=\ln M_Z^2/M_t^2$.

\begin{figure}[ht]
 \begin{center}
 \begin{tabular}{c}
   \epsfxsize=11.5cm
   \leavevmode
   \epsffile[50 290 520 560]{delr1l.ps} \\
   \epsfxsize=11.5cm
   \leavevmode
   \epsffile[50 290 520 560]{delr2l.ps} \\
   \epsfxsize=11.5cm
   \leavevmode
   \epsffile[50 290 520 560]{delr3l.ps} 
 \end{tabular}
 \fcaption{\label{figdr}$\drtbbar$ as function of $M_t$. The dotted, dashed 
         and dash-dotted curves correspond to an increasing number of 
         terms in the approximation. In the figures a value $s^2=0.2321$ 
         is chosen.}
 \end{center}
\end{figure}

\begin{figure}[ht]
 \begin{center}
 \begin{tabular}{c}
   \epsfxsize=11.5cm
   \leavevmode
   \epsffile[50 290 520 560]{delk1l.ps} \\
   \epsfxsize=11.5cm
   \leavevmode
   \epsffile[50 290 520 560]{delk2l.ps} \\
   \epsfxsize=11.5cm
   \leavevmode
   \epsffile[50 290 520 560]{delk3l.ps}
 \end{tabular}
  \fcaption{\label{figdk}$\Delta\kappa_{tb}$ as function of $M_t$. 
         The conventions are the same as for $\drtbbar$. Here only the
         real part is plotted.}
 \end{center}
\end{figure}

Finally we want to discuss the numerical effects.
The exact one- and two-loop
results for $\drtbbar$ and $\dktb$ are known 
\cite{Kni90HalKniSto93}
and pictured in Figs. \ref{figdr} and \ref{figdk} (solid line), respectively,
together with the first three terms of the expansion.
Already the lowest order terms of ${\cal O}(G_F M_t^2)$ (dotted line) and
${\cal O}(G_F M_Z^2)$ (dashed) provide an excellent approximation, at least
for $M_t>150$ GeV.
In this region the addition of the ${\cal O}(G_F M_Z^4/M_t^2)$ terms leads
to a result which is almost indistinguishable from the exact curve.
This consideration justifies the restriction of the ${\cal O}(\alpha_s^2)$  
corrections to the leading and the next-to-leading terms.
Also the next-to-next-to-leading terms $\propto M_Z^2/M_t^2$ are
available. The complete result for $\drtbbar$ and $\sineff$ to 
${\cal O}(\alpha_s^2)$ is therefore at hand.

\begin{table}[ht]
\renewcommand{\arraystretch}{1.2}
\tcaption{Numerical results for the $M_t^2$, the constant plus
                     logarithmic and the $1/M_t^2$ contributions. 
         For the numerical evaluation of $\delta\sineff/\sineff$ only the
         real part of $\dktb$ is taken.
         \label{table1} }
\begin{center}
\small
\begin{tabular}{||l|r|r|r||}\hline\hline
     & $M_t^2$ 
     & $M_t^2$ + const. 
     & $M_t^2$ + const. +$1/M_t^2$           \\
\hline
\hline
$\delta M_W/M_W$ (OS)       & 0.00677& 0.00838 & 0.00825\\
$\delta\sineff/\sineff$ (OS) & -0.01618& -0.01832 & -0.01814\\
\hline
\hline
$\delta M_W/M_W$ ($\msbar$)       & 0.00674& 0.00833 & 0.00821\\
$\delta\sineff/\sineff$ ($\msbar$)& -0.01610& -0.01823 & -0.01804\\
\hline
\hline
\end{tabular}

\end{center}
\end{table}


\begin{table}[th]
\renewcommand{\arraystretch}{1.2}
\tcaption{\label{table2} The change in $M_W$ separated according 
                      to powers in $\alpha_s$ and $M_t$ in the on-shell
                      scheme.}
\begin{center}

\begin{tabular}{||l||r|r|r||}
\hline
\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
\hline
\end{tabular}

\end{center}
\end{table}

The numerical effects on $M_W$ and $\sineff$ are given 
in Table \ref{table1}.
The numbers are obtained with the following input data:
$\as(M_t^2)=0.1092$ (corresponding to $\as^{(5)}=0.120$),
$\mbar=165$ GeV 
($\mbar$ is the solution of the equation $\mu^2=\mbar(\mu^2)$), 
$M_t=175$ GeV,
$M_Z=91.188$ GeV, $M_H=300$ GeV,
$\al=1/137.04$ and
$G_F=1.16639\,10^{-5}$ GeV$^{-2}$.
For $\Delta r$ in Eq. (\ref{mweqn}) we used 
(besides our corrections) the complete ${\cal O}(\alpha)$ result
\cite{BurJeg90}
including the contributions from the bosonic diagrams, 
the vertex and box diagrams 
and also the from the light fermions 
with $\Delta\alpha =0.05940$ \cite{Jeg95}.

In Table \ref{table2} the contributions are listed separately according
to powers of $\alpha_s$ and $M_t$.
In the OS scheme the three-loop result induces a shift in $M_W$ of
about $-13.7$ MeV. The terms of order $M_t^2/M_Z^2$ alone
accounts for
for only $-10.9$ MeV. Inclusion of the next two terms increase the shift
by $-2.8$ MeV.

To summarize: The three-loop corrections of
${\cal O}(\as^2)$ to $\Delta r$ and $\Delta\kappa$ from the
top-bottom doublet have been calculated, using
an expansion in $1/M_t^2$.
Already the first three terms lead to an accurrate prediction.

\vspace{.5cm}

%%%\section{Acknowledgements}
{\bf Acknowledgements}\\
We would like to thank Bernd Kniehl and the Max Planck Society for
the invitation to the Ringberg Workshop and the generous hospitality.
This work has been supported by DFG Contract Ku 502/6-1
and BMFT Contract 056KA93P6.

\vspace{.5cm}

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\bibitem{Smi94}
For a recent review see e.g.:
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Ringberg Castle, Germany, 13-19 Nov 1994. 

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D.J. Broadhurst, W. Grafe, N. Gray and K. Schilcher, 
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\bibitem{Kni90HalKniSto93}
B.A. Kniehl, {\it Nucl. Phys.} {\bf B347} (1990) 65.\\
F. Halzen, B.A. Kniehl and M.L. Stong, {\it Z. Phys.} {\bf C58} (1993) 119.

\bibitem{BurJeg90}
G. Burgers and F. Jegerlehner, PSI-PR-89-28,  
{\it Workshop on Z Physics at LEP},
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\end{thebibliography}

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