%Title: Leading Top Mass Corrections of Order ${\cal O}(\alpha\as m_t^2/M_W^2)$ to the Partial Decay Rate $\Gamma(Z\rightarrow b\bar{b})$
%Authors: K.G. Chetyrkin, A. Kwiatkowski, and M. Steinhauser
%Published: <EM> Mod. Phys. Lett. </EM> <B> A8 </B> (1993) 2785-2792. 
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\def\bbuildrel#1_#2^#3%                                                         
{\mathrel{\mathop{\kern 0pt#1}\limits_{#2}^{#3}}}                               
\newcommand{\beq}{\begin{equation}}                                             
\newcommand{\eeq}{\end{equation}}                                               
\newcommand{\api}{\frac{\alpha_s}{\pi}}                                         
\newcommand{\ba}{\begin{array}}                                                 
\newcommand{\ea}{\end{array}}                                                   
\newcommand{\as}{\alpha_s}                                                      
\newcommand{\gm}{\gamma_m}                                                      
\newcommand{\G}{\Gamma}                                                         
\newcommand{\g}{\gamma}                                                         
\newcommand{\msbar}{\overline{\mbox{MS}}}                                       
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\begin{document}                                                                
                                                                                
\begin{titlepage}                                                               
\noindent                                                                       
%                                                                               
% Datum                                                                         
%                                                                               
\hfill TTP93--12\\                                                              
\mbox{}                                                                         
\hfill May 1993   \\   %                 \hfill \today \\                       
%                                                                               
% Title                                                                         
%                                                                               
\vspace{0.5cm}                                                                  
\begin{center}                                                                  
  \begin{Large}                                                                 
  \begin{bf}                                                                    
Leading Top Mass Corrections of Order ${\cal O}(\alpha\as m_t^2/M_W^2)$         
to the Partial Decay Rate $\Gamma(Z\rightarrow b\bar{b})$                       
  \\                                                                            
  \end{bf}                                                                      
  \end{Large}                                                                   
%                                                                               
% Author                                                                        
%                                                                               
  \vspace{0.8cm}                                                                
  \begin{large}                                                                 
K.G.Chetyrkin  \\[3mm]                                                          
Institute for Nuclear Research\\                                                
Russian Academy of Sciences\\                                                   
60th October Anniversary Prospect 7a\\                                          
 Moscow, 117312, Russia \\[5mm]                                                 
 A.Kwiatkowski, M.Steinhauser    \\[3mm]                                        
    Institut f\"ur Theoretische Teilchenphysik\\                                
    Universit\"at Karlsruhe\\                                                   
    Kaiserstr. 12,    Postfach 6980\\[2mm]                                      
    7500 Karlsruhe 1, Germany\\                                                 
  \end{large}                                                                   
%                                                                               
% Abstract                                                                      
%                                                                               
  \vspace{3cm}                                                                  
  {\bf Abstract}                                                                
\end{center}                                                                    
\begin{quotation}                                                               
\noindent                                                                       
The order ${\cal O} (\alpha\as m_t^2/M_W^2)$ corrections to the                 
partial decay rate of $\Gamma(Z\rightarrow b\bar{b})$ are computed              
using the heavy top mass expansion. They are given by the absorptive            
part of the corresponding three loop $Z$-self energy diagrams.  Our             
method provides an independent way to calculate the QCD corrections             
to the electroweak one loop result for  $\Gamma(Z\rightarrow                    
b\bar{b})$ and our result represents another confirmation of the one            
recently published by Fleischer et al.  \end{quotation}                         
\end{titlepage}                                                                 
                                                                                
\renewcommand{\arraystretch}{2}                             %                   
                                                                                
\section{Introduction}                                                          
Experiments at LEP have limited the top mass $m_t$ to be larger than the mass   
of the $Z$ boson. Many observables measured at LEP nevertheless are affected    
by $m_t$ due to the appearance of the top quark as a virtual state in higher    
order radiative corrections.                                                    
                                                                                
Of particular interest has been the investigation of the partial decay rate     
$\Gamma (Z\rightarrow b\bar{b})$ of the $Z$ boson into a $b$ quark pair.        
On the one hand the present experimental uncertainty of this quantity           
$\Gamma (Z\rightarrow b\bar{b}) = (383\pm 5.6)$ MeV is very likely to be even   
further reduced below 1\% in the future                                         
\cite{Sch93}. On the other hand the $m_t$ dependence                            
enters $\Gamma (Z\rightarrow b\bar{b})$ through non-oblique corrections, which  
helps to disentangle top effects from possible new physics effects.             
                                                                                
The advanced status of the analysis of $\Gamma (Z\rightarrow b\bar{b})$         
is reflected by the number of groups being involved in the calculation of       
radiative corrections to this quantity. The one loop electroweak corrections to 
the partial $Z$ decay rate were obtained in                                     
\cite{AkhBarRie86,BeeHol88,BerPicSan88}. On top of it the knowledge of QCD      
corrections to this result became desirable in view of the high experimental    
precision of $\Gamma (Z\rightarrow b\bar{b})$. This calculation was first       
performed to order ${\cal O}(\alpha\as m_t^2/M_W^2)$ in the heavy top mass      
limit by \cite{FleJegRacTar92}. It involved the computation of two loop vertex  
corrections as well as gluon bremsstrahlung corrections to the one loop         
electroweak vertex correction and found a similar screening of the leading top  
mass effect  as is known from the                                               
universal $\rho$-parameter.                                                     
 The result was confirmed recently by two other calculations                    
\cite{BucBur92,Deg93}, where in the latter a current algebra approach has been  
employed.                                                                       
                                                                                
In this letter we present another independent confirmation of the ${\cal        
O}(\alpha\as m_t^2/M_W^2)$ radiative corrections to                             
$\Gamma (Z\rightarrow b\bar{b})$. Our method is based on the calculation of the 
three loop propagator type (p-)diagrams                                        
\beq \EQN{i1}                                                                   
\Pi_{\mu\nu}(q^2) = g_{\mu\nu}\Pi_1(q^2)+q_{\mu}q_{\nu}\Pi_2(q^2)               
\eeq                                                                            
which are obtained by attaching to the  electroweak                             
diagrams of Figure 1 an internal gluon                                          
line in all possible ways. After summing all diagrams the decay                 
rate is given by the absorptive part of the spectral function $\Pi_1$           
\beq                                                                            
\Gamma (Z\rightarrow b\bar{b})  =                                               
\frac{1}{M_Z}\mbox{\rm Im} \Pi_1(M_Z^2).                                        
\EQN{i2}                                                                        
\eeq                                                                            
There are several advantages of our approach.  The number of diagrams           
is rather small (17 topologically different diagrams).  Each                    
p-diagram is infrared (IR) finite, since it comprises just those                
contributions of vertex corrections and gluon bremsstrahlung for                
which IR cancellation takes place.  The attribution of the problem to           
propagator type diagrams is also convenient as there exist elaborated           
techniques to compute just p-integrals (for a recent review see                 
\cite{Broadhurst92}).                                                           
                                                                                
%with respect to the fact that                                                  
% we can make use of the REDUCE package LOOPS                                   
%\cite{SurTka89} which is designed for the computation of massless              
%p-diagrams.                                                                    
                                                                                
%\ice{                                                                          
\begin {figure}[t]                                                              
\begin {tabular}{ccc}                                                           
\epsfig{file=ttg.eps,width=4.5cm,height=4.5cm} &                                
\epsfig{file=gg.eps,width=4.5cm,height=4.5cm} &                                 
\epsfig{file=seg.eps,width=4.5cm,height=4.5cm}                                  
\end {tabular}                                                                  
\caption {The electroweak diagrams contributing to                              
the self energy of the $Z$ boson.                                               
Dashed lines: Higgs ghost,                                                      
thin lines: bottom quark, thick lines: top quark.}                              
\end{figure}                                                                    
                                                                                
Our calulation is performed with dimensional regularization in the              
$\msbar$-scheme. We also transform our final result to the on-shell             
scheme to compare it with the one of                                            
\cite{FleJegRacTar92,BucBur92,Deg93}.  For dimensions $D\neq 4$ we              
use the definition of the hermitean, anticommuting                              
$\gamma_5\;\;(\gamma_5^2=1,\{\gamma_5,\gamma_{\mu}\}=0)$.  We are               
working in the 't Hooft gauge with an arbitrary QCD gauge parameter.            
Explicit verification of QCD gauge invariance serves as an internal             
check of the result.                                                            
                                                                                
In this work the calculation is restricted to the leading $m_t^2$               
dependence of the strong-electroweak corrections. We isolate this               
contribution by using a large top mass expansion and keeping only the           
leading term. The second order term of this series would yield the              
next to leading top mass effect of order ${\cal O}(\alpha\as)$ and              
will be investigated in the future.                                             
%Asymptotic expansions like the                                                 
%large mass expansion have been used sucessfully in the past for                
%calculating radiative corrections involving heavy internal particles           
%\cite{AsyExp}. They represent a powerful approach to problems, for             
%which the exact solution of Feynman integrals is not known                     
%\cite{Smi90,Smi91}.                                                            
                                                                                
A short description of the heavy mass expansion to be  used and                 
further details of the calculation are given in Section 2. Section 3            
contains the discussion of the results and conclusions.                         
                                                                                
\section{Calculation of the Order ${\cal O}(\alpha\as m_t^2/M_W^2)$             
Corrections to $\Gamma(Z\rightarrow b\bar{b})$} The top mass                    
dependence of the electroweak one-loop result for                               
$\Gamma(Z\rightarrow b\bar{b})$ is due to the appearance of the top             
quark as a virtual particle. Intermediate top states originate from             
the $t-b$ transitions, which in general may be induced by the                   
exchange of a charged Higgs ghost $\Phi^{\pm}$ or a $W$ boson. In               
this paper we are interested in the QCD corrections to the leading              
$m_t^2$ term of the ${\cal O}(\alpha)$ corrected partial $Z$ width.             
To this order only diagrams with a Higgs ghost exchange contribute,             
because the Higgs ghost-fermion coupling is proportional to the               
fermion mass, which is not the case for the coupling of the $W$ boson           
to fermions.                                                                    
                                                                                
The method of our calculation is to consider the relevant $Z$ boson             
two point functions and compute their absorptive parts.  Note that              
only the  function $\Pi_1$ is relevant for the $Z$ decay rate                   
(see eq.(\ref{i2})). We may therefore simplify the loop integrals by            
 contracting                                                                    
$\Pi_{\mu\nu}$ with the transversal Lorentz tensor ($D=4-2\epsilon$):           
\beq\EQN{c0}                                                                    
\Pi_1(q^2) = \frac{1}{D-1}\left(g_{\mu\nu} - \frac{q_{\mu}q_{\nu}}{q^2}\right)  
\Pi^{\mu\nu}(q^2)                                                               
\eeq                                                                            
In order to derive the                                                          
${\cal O}(\alpha\as m_t^2/M_W^2)$ corrections to                                
$\Gamma(Z\rightarrow b\bar{b})$ we have to calculate all three loop             
diagrams which can be constructed from those of Figure 1 by inserting           
an additional gluon line in all possible ways.  However, we are                 
working in the limit of large top masses in order to derive the                 
dominant top mass contribution. In this limit  no explicit                      
computation of genuine three loop integrals has to be performed                 
and  the calculation may be drastically simplified in the following             
manner.                                                                         
                                                                                
We employ an effective field theory approach by integrating out the             
heavy top quark from the theory. This results in a power expansion              
with respect to the inverse heavy mass $1/m_t$, from which we obtain            
the leading contribution $\propto m_t^2$.  This large mass expansion            
is constructed as follows\footnote{In performing  concrete                      
calculations we make use of the so-called method of projectors                  
\cite{Gor87} in a diagram-wise approach  of refs. \cite{AsyExp}.                
The reader interested in the mathematical proofs will find them in              
\cite{Smi90,Smi91}.}.                                                     
Let $\G$ be a graph contributing to  the polarization operator                  
$\Pi_{1}$. The corresponding Feynman integral                                   
$<\G>(q^2)$  in the limit of the heavy top quark                                
may be represented in  the form of the following asymptotic                     
expansion:                                                                      
\beq                                                                            
\EQN{c1}                                                                        
<\G>(q^2)                                                                       
\bbuildrel{=\!=\!=}_{{\scriptstyle{m_t\to\infty}}}^{}                           
\sum_\g  C^{(t)}_\g * <\G/\g>^{\mbox{\rm \scriptsize eff}}         
\eeq                                                                            
                                                                                
The diagrams   $<\G/\g>^{\mbox{\rm \scriptsize eff}}$                           
of the effective theory are by definition composed of the fields of             
the light particles (namely the $b$-quark, the Higgs ghosts and the             
gluon), whereas the whole top dependence is contained in the                    
``coefficient functions'' $C^{(t)}_\g$. Diagrammatically the latter             
are represented by all possible {\em hard} subgraphs of $\G$.                   
By definition, a subgraph $\g$ of  $\G$ is hard if, first, it comprises         
all the {\em heavy} lines (i.e. the top quark propagators)                      
and, second, it becomes one-particle-irreducible after contracting              
its heavy lines.  An example is illustrated in Figure 2.  Our                   
notation                                                                        
$                                                                               
\g * <\G/\g>^{\mbox{\rm \scriptsize eff}}                                       
$                                                                               
means that in the original diagram $\G$ the                                     
hard subgraph  $\g$  is reduced to an effective blob.                           
The corresponding Feynman subintegral                                          
$C^{(t)}_\g$  is then expanded in a (in general multidimensional)               
formal Taylor series with respect to its (small) external momenta and           
to light masses  eventually contained in $C^{(t)}_n$. This Taylor               
expansion is then inserted in the effective blob and the whole                  
expression is evaluated.                                                        
                                                                                
                                                                                
%\ice{                                                                          
\begin {figure}[t]                                                              
\begin {tabular}{ccccc}                                                         
\parbox{3cm}{                                                                   
\epsfig{file=ttgvb.eps,width=4.cm,height=4.cm}                                  
            }                                                                   
& $\longrightarrow$ &                                                                         
\parbox{3cm}{                                                                   
\epsfig{file=ttgvb1hs.eps,width=4.cm,height=4.cm}                               
            }                                                                   
& $\star$ &                                                                     
\parbox{3cm}{                                                                   
\epsfig{file=ttgvb1.eps,width=4.cm,height=4.cm}                                 
            }                                                                   
\\                                                                              
& $+$ &                                                                         
\parbox{3cm}{                                                                   
\epsfig{file=ttgvb2hs.eps,width=4.cm,height=4.cm}                               
            }                                                                   
& $\star$ &                                                                     
\parbox{3cm}{                                                                   
\epsfig{file=ttgvb2.eps,width=4.cm,height=4.cm}                                 
       }                                                                        
\end {tabular}                                                                  
\caption {One example for the large mass expansion of a diagram                 
with QCD corrections. Only those hard subgraphs (denoted by the boxes)     
which contribute to order ${\cal O}(\alpha\as m_t^2/M_W^2)$ are depicted.}          
\end{figure}                                                                    
                                                                                
There are different mass scales involved in our problem. We neglect             
the bottom mass throughout in the calculation. The $W$ boson mass               
$M_W$ enters through the propagator of the Higgs ghost                          
\beq \EQN{c2}                                                                   
D(p) = \frac{1}{i}\frac{1}{\xi_W M_W^2-p^2-i\epsilon}                           
\eeq                                                                            
and is of the same order as the energy scale of the process, i.e. the           
mass of the $Z$ boson. Finally in our approximation the top mass is             
assumed to be very large ($m_t\rightarrow \infty$). Therefore we use            
$M_W$ as an expansion parameter within the hard subgraphs                       
$C^{(t)}_n$, whereas it should be kept finite within the effective              
theory. As a consequence of the evaluation of $C^{(t)}_n$ at $M_W=0$            
the electroweak gauge parameter $\xi_W$ vanishes trivially in the               
leading top mass correction.                                                    
                                                                                
This is different in the case of the QCD gauge. The QCD gauge                   
parameter $\xi$ is introduced through the gluon propagator                      
\beq\EQN{c3}                                                                    
D^{\mu\nu}_{\alpha\beta}(k) = \frac{1}{i} \delta_{\alpha\beta}                  
\frac{\dsp -g^{\mu\nu}+(\xi-1)\frac{k^{\mu}k^{\nu}}{-k^2-i\epsilon}}            
     {-k^2-i\epsilon}                                                           
\eeq                                                                            
and is kept arbitrary for an explicit check of gauge invariance.                
The calculation shows that those diagrams which originate from the same         
electroweak two point function of Figure 1 (characterized by the vertices       
$Zt\bar{t}, Z\Phi^+\Phi^- $ and the quark self energy) already form a gauge     
invariant subset of diagrams.                                                   
                                                                                
The isolation of the leading $m_t^2$ term in the large mass expansion           
allows to discard contributions at three different levels. First,               
only Higgs ghost exchange graphs need to be considered. As was noted            
before, $W$ boson exchange diagrams contribute at most to next to               
leading order (namely $\sim \ln(m_t^2/M_W^2)$. Second, only subgraphs           
with the highest dimension in $m_t$ are taken into account. Finally,            
higher order contributions of the Taylor expansion of the hard                  
subgraphs lead to higher orders in $1/m_t$ and may be discarded.                
                                                                                
The heavy mass expansion described above is a powerful method for the           
calculation of radiative corrections. (As an exercise we have also              
used this method to reproduce the ${\cal O}(\alpha \ln(m_t^2/M_W^2))$           
term in the electroweak result for $\Gamma(Z\rightarrow b\bar{b})$ by           
\cite{AkhBarRie86}.) Due to the application of this procedure the               
original three loop integrals simplify a lot. They factorize  and may           
be reduced to one loop massive tadpoles and two loop massless                   
integrals or to one loop massive tadpoles and two loop massless                 
integrals.  The tadpole integrals are given by the following                    
analytical expressions                                                          
\cite{Veltman.2loopTadpole}:                                                    
\beq\EQN{c4}                                                                    
\ba{rl} \dsp                                                                    
T_1 =                                                                           
& \dsp                                                                          
\int_E \frac{d^Dp}{(M^2+p^2)^a p^{2b}}                                          
\\ & \dsp                                                                       
\left(M^2\right)^{2-\epsilon-a-b-c}                                             
\left(\frac{\mu^2}{M^2}\right)^{\epsilon}                                       
\frac{\Gamma(2-\epsilon-b)\Gamma(a+b+\epsilon-2)}{\Gamma(2-\epsilon)\Gamma(a)}  
\\ \dsp                                                                         
T_2 =                                                                           
& \dsp                                                                          
\int_E \frac{d^Dp d^Dl}{(M^2+p^2)^a (M^2+l^2)^b (l+p)^{2c}}                     
\\ = & \dsp                                                                     
\left(M^2\right)^{4-2\epsilon-a-b}                                              
\left(\frac{\mu^2}{M^2}\right)^{2\epsilon}\cdot                                 
\\ & \dsp                                                                       
\frac{\Gamma(a+c-2+\epsilon)\Gamma(b+c-2+\epsilon)                              
      \Gamma(a+b+c-4+2\epsilon)\Gamma(2-\epsilon-c)}                            
     {\Gamma(2-\epsilon)\Gamma(a)\Gamma(b)\Gamma(a+b+2c-4+2\epsilon)}           
,                                                                               
\ea                                                                             
\eeq                                                                            
where $\mu^2$ is the renormalization scale of dimensional                       
regularization.  For the calculation of the massless integrals we               
have used the program LOOPS \cite{SurTka89}, which employes the                 
algorithm of Ref. \cite{Chet81} to do massless p-integrals.                     
                                                                                
Before we describe the renormalization of the strong-\-elec\-tro\-weak          
three loop diagrams, let us briefly recall the situation of the purely          
electroweak case.                                                               
Each two loop graph of Figure 1 develops quadratic $1/\epsilon^2$-poles         
due to its UV divergent subgraphs and its overall divergence. In the sum of all 
diagrams the subdivergences compensate each other and an overall                
$1/\epsilon$-pole remains.   In other words, all counterterms introduced to     
subtract the divergences of the various subgraphs cancel each other.            
This is  known as the generalization of the QED Ward identities and             
 manifests the fact that the fermion-Higgs ghost interaction does not           
contribute to the renormalization of the weak coupling constant.                
                                                                                
In the case of the QCD corrected three loop diagrams a similar cancellation of  
subdivergences takes place and all $1/\epsilon^3$ poles drop out. 
Similarly, all $1/\epsilon^2$-poles cancel if the 
${\cal O}(\alpha\alpha_s m_t^2/M_W^2)$ contribution is taken into account,
which is induced by the order ${\cal O}(\alpha m_t^2/M_W^2)$ through
replacing the bare top mass by the renormalized one:
\beq\EQN{c5}                                                                    
m_t^{bare} = \bar{m}_t\left( 1- \api\frac{1}{\epsilon}\right)                   
\eeq                                                                            
Thus after subtraction of all subdivergences,               
we are left with  a simple overall divergence in the real part of the           
result, whereas its imaginary part is finite and transforms to the               
corrected decay rate $\Gamma(Z\rightarrow b\bar{b})$ according to               
eq.(\ref{i2}).                                                                  
                                                                                
\section{Results and Discussion}                                                
In order to present the result of our calculation let us write                  
the partial decay rate $\Gamma(Z\rightarrow b\bar{b}) $ as the sum of           
the Born decay rate $\Gamma_{Born}$  and the correction $\Delta\Gamma$          
\beq\EQN{r1}                                                                    
\Gamma(Z\rightarrow b\bar{b}) = \Gamma_{Born} + \Delta\Gamma                    
,\eeq                                                                           
for which we obtain                                                             
\beq\EQN{r2}                                                                    
\ba{ll}\dsp                                                                     
\Gamma_{Born}                                                                   
& \dsp                                                                          
=\frac{G_F M_Z^3}{24\sqrt{2}\pi} N_C(v_b^2+a_b^2)                               
\\ \dsp                                                                         
\Delta\Gamma                                                                    
& \dsp                                                                          
=-\frac{G_F^2M_Z^3}{48\pi^3}\left(1-\frac{2}{3}s_W^2\right)                     
N_C \bar{m}_t^2(\mu^2)\left[ 1-                                                 
\left( \frac{\pi^2-11}{3}-2\ln \frac{\mu^2}{\bar{m}_t^2}\right)                 
\frac{\as(\mu^2)}{\pi}\right]                                                   
\ea\eeq                                                                       
Here $\bar{m}_t(\mu^2)$ and $\alpha_s(\mu^2)$ are the renormalized mass and     
the strong coupling constant respectively at the renormalization scale $\mu^2$. 
Furthermore the vector and axial couplings read $v_b = -1+\frac{4}{3}s_W^2,     
a_b = -1$.                                                                      
We transform this result from the $\msbar$-scheme to the                        
on-shell scheme via                                                             
\beq\EQN{r3}                                                                    
\bar{m}(\mu^2) = m_{OS}\left(1-\api\left[\frac{4}{3}+\ln\frac{\mu^2}{m_{OS}^2}  
\right]\right)                                                                  
\eeq                                                                          
Inserting into eq.(\ref{r2}) with  $\mu^2 = M_Z^2$ gives                        
\beq\EQN{r4}                                                                    
\Delta\Gamma                                                                    
=-\frac{G_F^2M_Z^3}{48\pi^3}\left(1-\frac{2}{3}s_W^2\right)                     
N_C m_t^2\left[ 1-\frac{\pi^2-3}{3}\frac{\as(M_Z^2)}{\pi}\right]                
\eeq                                                                          
As was shown in \cite{AkhBarRie86} one can account for electroweak corrections  
to the decay rate by introducing two form factors $\rho,\kappa$ and             
replacing the Fermi constant and the weak mixing angle by corresponding         
effective parameters $\rho G_F$ and $\kappa \sin^2\theta_W$ in $\Gamma_{Born}$. 
The order ${\cal O}(\alpha\as m_t^2/M_W^2)$ corrections to the decay            
rate may be organized in a similar way. The partial $Z$ decay rate              
is expressed in the form                                                        
\beq\EQN{r5}                                                                    
\Gamma(Z\rightarrow b\bar{b}) =  \frac{G_F M_Z^3}{12\sqrt{2}\pi}                
N_C \rho \left[ 1-\frac{4}{3}\kappa s_W^2 + \frac{8}{9}\kappa^2s^4_W\right]     
\eeq                                                                            
with                                                                            
\beq\EQN{r6}                                                                    
\ba{ll} \dsp                                                                    
\rho                                                                            
& \dsp                                                                          
= 1+\Delta\rho                                                                  
\\ & \dsp                                                                       
= 1-\frac{G_F}{2\sqrt{2}\pi^2}m_t^2\left(1-\frac{\pi^2-3}{3}\api\right)         
\\  \dsp                                                                        
\kappa                                                                          
& \dsp                                                                          
= 1+\Delta\kappa                                                                
\\ & \dsp                                                                       
= 1 - \frac{1}{2}\Delta\rho                                                     
\ea\eeq                                                                       
This result is in agreement with that of Fleischer et al.                       
\cite{FleJegRacTar92}.                                                          
                                                                                
To conclude, we have calculated the QCD corrections to the leading              
top mass contribution  of the electroweak one loop result for the partial       
$Z$ width $\Gamma(Z\rightarrow b\bar{b})$.                                      
For this purpose we have computed corresponding propagator type diagrams        
and extracted their imaginary part. The calulation was performed in the large   
mass limit. The leading $m_t^2$ contribution was obtained through a heavy mass  
expansion in the inverse top mass. This work represents an independent          
way to determine the ${\cal O}(\alpha\as m_t^2/M_W^2)$ corrections and          
confirms the recent results of other groups.                                    
                                                                                
                                                                                
\vspace{5ex}                                                                    
{\bf Acknowledgments}                                                           
                                                                                
\noindent                                                                       
We would like to thank J.H. K\"uhn for his support and many helpful             
suggestions.                                                                    
                                                                                
                                                                                
%\appendix                                                                      
%\section{Appendix}                                                             
\begin{thebibliography}{99}                                                     
                                                                                
\bibitem{Sch93}                                                                 
D.Schaile, talk given at the Ringberg Workshop 1993: Electroweak Results at     
LEP.                                                                            
                                                                                
\bibitem{AkhBarRie86}                                                           
A.Akhundov, D.Bardin, T.Riemann, Nucl.\ Phys.\ B 276 (1986) 1.                  
                                                                                
\bibitem{BeeHol88}                                                              
W.Beenacker, W.Hollik, Z. Phys. C - Particles and Fields 40 (1988) 141.         
                                                                                
\bibitem{BerPicSan88}                                                           
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\end{document}                                                                  
                                                                                
