%Title:   Mass Effects in Higher Order QCD Corrections to the 
	$Z\rightarrow b\bar{b}$ Width
%Authors:  K.G.Chetyrkin, A.Kwiatkowski, and J.H.K\"uhn 
%Published:  <EM>Proceedings of the EPS Conf., Marseille 1993, M. Carr and M. Perrottet, Eds., pp. 463--464.</EM>
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 3 figures appended to this files can be included 
% via the LATEX FEYNMAN package 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% The rest of diagrams will be sent by mail on request
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\documentstyle[12pt]{article}
\setlength{\textwidth}{14.5cm}
\setlength{\textheight}{22.5cm}
%\setlength{\oddsidemargin}{3.cm}
\setlength{\oddsidemargin}{1.cm}
\setlength{\evensidemargin}{1.cm}
\setlength{\headheight}{0.5cm}
\setlength{\headsep}{1.0cm}
\setlength{\topmargin}{0.0cm}
\setlength{\topskip}{0.1cm}
\setlength{\footheight}{0.5cm}
\renewcommand{\baselinestretch}{1.1}
%\setlength{\changebarsep}{0mm}
%%%\setlength{\mathindent}{0mm}
%\setlength{\overfullrule}{5pt}
\setlength{\footskip}{1.5cm}
\frenchspacing
%\pagestyle{myheadings}


\newcommand{\beq}{\begin{equation}}
\newcommand{\eeq}{\end{equation}}
\newcommand{\api}{\frac{\alpha_s}{\pi}}
\newcommand{\apis}{\frac{\dsp\alpha_s(s)}{\dsp \pi}}
\newcommand{\apimu}{\frac{\dsp\alpha_s(\mu)}{\dsp \pi}}
\newcommand{\apim}{\frac{\dsp\alpha_s(m)}{\dsp \pi}}
\newcommand{\ba}{\begin{array}} 
\newcommand{\ea}{\end{array}} 
\newcommand{\ds}{\displaystyle} 
\newcommand{\as}{\alpha_s}
\newcommand{\gm}{\gamma_m}
\newcommand{\G}{\Gamma}
\newcommand{\g}{\gamma}
\newcommand{\dmu}{\mu^2\frac{d}{d\mu^2}}
\newcommand{\msbar}{\overline{\mbox{MS}}}
\newcommand{\ordalp}{{\cal O}(\alpha_s}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\dsp}{\displaystyle}
\newcommand{\EQN}{\label}
\newcommand{\ovl}{\overline}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def\bbuildrel#1_#2^#3%
{\mathrel{\mathop{\kern 0pt#1}\limits_{#2}^{#3}}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\input{fig}

\begin{document}
%\input FEYNMAN.TEX
%\input feynman

\begin{titlepage}
\noindent
%
% Datum
%
\hfill TTP93--15\\
\mbox{}
\hfill  June 93   \\   %                 \hfill \today \\
%
% Title
%
\vspace{0.5cm}
\begin{center}
  \begin{Large}
  \begin{bf}
 Mass Effects in Higher Order QCD Corrections to the 
$Z\rightarrow b\bar{b}$ Width\footnote{Talk to be presented
by A.Kwiatkowski at the 
International Europhysics Conference on High Energy Physics,
Marseille, July 22-28 1993.}
   \\
  \end{bf}
  \end{Large}
%
% Author
%
  \vspace{0.5cm}
  \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, J.H.K\"uhn    \\[3mm] 
    Institut f\"ur Theoretische Teilchenphysik\\   
    Universit\"at Karlsruhe\\  
    Kaiserstr. 12,    Postfach 6980\\[2mm] 
    7500 Karlsruhe 1, Germany\\
  \end{large}
%
% Abstract
%
  \vspace{1.0cm}
  {\bf Abstract}
\end{center}
\begin{quotation}

\noindent
 The partial decay rate \mbox{$\Gamma(Z\rightarrow b\bar{b})$} of
 the $Z$-boson is significantly affected by 
higher order QCD corrections. 
Of particular interest is the influence of the non-\ vanishing bottom quark 
mass. Radiative corrections to the  quadratic mass term 
of the partial $Z$-width are computed in perturbative QCD.
 The calculation of the third order  
${\cal O}(\alpha_s^3 m_b^2/M_Z^2)$ contribution for the
vector induced rate and of the second order 
${\cal O}(\alpha_s^2 m_b^2/M_Z^2)$ correction
for the axial induced rate
is described. The numerical influence of the results and the theoretical 
uncertainties are discussed.  
\end{quotation}
\end{titlepage}


\section{Introduction}
\renewcommand{\arraystretch}{2}                             %
QCD corrections to the partial decay rate $\Gamma(Z\rightarrow b\bar{b})$ 
of the 
$Z$ boson into  bottom quark pairs have been under 
investigation since it became clear that the collision experiments 
at LEP
would provide measurements with  very high precision that make an accuracy
of well below the percent level for theoretical predictions mandatory.
In particular effects of the finite mass of the bottom quark, which enter
those higher order corrections, can no longer be neglected. Their influence
to the $Z$ decay rate is analysed in this work.

 The partial
$Z$ decay rate is composed of the incoherent sum of the 
vector induced and the axial vector induced rates
according  to the chiral structure of the weak neutral current

\beq \EQN{I1}
\ba{rl}
\Gamma(Z\rightarrow b\bar{b}) = 
&
\Gamma_Z^V + \Gamma_Z^A
\\ = &
\dsp \frac{G_F M_Z^3}{24\sqrt{2}\pi}  N_C (v_b^2 R^V + a_b^2 R^A)
.\ea \eeq
Here $N_C$ denotes the number of colour degrees of freedom and the 
vector and axial couplings of the $Z$ boson to fermions are
$v_f=2I^f_3-4Q_f\sin^2_{\theta_W}, a_f=2I^f_3$. 

For the calculation of QCD corrections to the $Z$ width it proves to be
  convenient  to  consider the two-point current correlation
functions $\Pi_{\mu\nu}^{V/A}(q)$. They may be decomposed according to 
their Lorentz structure and are given in terms of the spectral functions
$\Pi_1,\Pi_2$:     
\beq \ba{rl}\EQN{I2}
 \Pi_{\mu\nu}^{V/A}(q)  = &\dsp i \int dx e^{iqx} 
\langle 0| \;
T\;j_{\mu}^{V/A}(x)j_{\nu}^{V/A\dagger}(0)\;|0 \rangle 
\\ = & \dsp 
g_{\mu\nu} \Pi_1^{V/A} + q_{\mu}q_{\nu} \Pi_2^{V/A}
. \ea \eeq 
The absorptive parts of $\Pi_1^{V/A}$ are related to the 
 correction factors $R_V,R_A$  through 
 \beq \EQN{I3}
R^{V/A}(s)  =  
 -\frac{4\pi}{s} {\cal Im} \Pi_1^{V/A}(s+i\varepsilon) 
.\eeq
For both the vector and the axial vector contribution
the diagrams representing  two-point correlators may in general be devided
into flavour non singlet and flavour singlet ones. The 
diagrams of the non singlet class
are constructed from the vacuum polarization graph
by attaching virtual
gluons in all possible ways to the fermion loop. As an example 
the diagrams contributing up to order ${\cal O}(\as)$ are shown in Figure 1.
For massless quarks the corrections for the vector and the axial vector 
current correlators are equal, whereas they differ for massive quarks.

Different contributions to vector and axial vector correlators 
arise even in the massless limit  due to the occurence of   flavour singlet 
diagrams . They appear in the axial case at second order ${\cal O}(\as^2)$
and for the vector case at third order ${\cal O}(\as^3)$.
Graphically they are   represented
by ``light-by-light'' type diagrams
 with two  fermion triangles and a purely gluonic 
intermediate state as  illustrated in Figure 2.
At second order no contribution to the vector current correlator is present
 due to Furry's
theorem, whereas a second order singlet contribution arises for the 
axial vector correlator due to the large mass splitting in the
top-bottom doublet.

QCD corrections to the $Z$-width including the complete dependence of 
the final state mass are known to first order ${\cal O}(\as)$
\cite{Sch73,JerLaeZer82,ReiRubYaz85}. However, massive corrections of higher
orders are  much more difficult to compute.
 A convenient way to simplify the 
computation is to make an expansion in $m_b^2/s$, which is certainly a good
approximation at the $Z$-resonance. Sorting the corrections according to their
order in $m_b^2/s$ and $\alpha_s/\pi$ one has
\beq\EQN{I4}
 \ba{ll} \dsp 
R^{V/A}(s) \dsp
 &  =\dsp  
 1 + R_1^{V/A} \frac{\alpha_s}{\pi}  
    + R_2^{V/A}
\left( \frac{\alpha_s}{\pi}\right)^2
    + R_3^{V/A}
\left( \frac{\alpha_s}{\pi}\right)^3
 + \dots \\  
 & \dsp 
  + \frac{\bar{m}_b^2(s)}{s}  
\left[ R_0^{V/A(m)}
 + R_1^{V/A(m)} \frac{\alpha_s}{\pi}  
    + R_2^{V/A(m)}
 \left( \frac{\alpha_s}{\pi}\right)^2    
    + R_3^{V/A(m)}
 \left( \frac{\alpha_s}{\pi}\right)^3 + \dots \right] 
\\ & \dsp
+ {\cal O}(\bar{m}_b^4)    
\ea 
\eeq 
where 
$\bar{m}(s)$ and $\alpha_s(s)$ denote the quark mass and 
coupling constant in the $\overline{MS}$-scheme at the   
renormalization point $\mu^2=s$.
For completeness we briefly recall the massless results
\cite{CheKatTka79,BerWet81,GorKatLar90,SurSam92}
\beq\EQN{I5}
\ba{ll}
R_1^V & = 1
\\
R_2^V & = 1.41 + \Delta_{NS} 
\\
R_3^V & = -12.76 + \Delta^V_{S}
\ea\eeq
where the non-singlet contribution $\Delta_{NS}$ contains the top mass
dependence arising from virtual top loops ($\Delta_{NS}=0.303$
 for $m_t=130$ GeV \cite{BerWet81}). The singlet term   
$\Delta^V_{S}$ was obtained by
\cite{GorKatLar90} for vanishing quark masses ($\Delta^V_{S}=-0.33$).
Whereas the non-singlet terms for the vector and the axial vector
correlators coincide, $R^A_2$ receives an additional singlet contribution
$\Delta^A_S=-3.68$ for $m_t=130$ GeV \cite{KniKue,CheKue93}.

In this work we describe the calculation of the quadratic mass corrections
to third order ${\cal O}(m_b^2\as^3)$ for the vector current induced rate
and to second order ${\cal O}(m_b^2\as^2)$ for the axial vector induced
rate.  

\section{Quadratic Mass Corrections from Flavour Non-Singlet Diagrams}
In this section we point out that the flavour non-singlet contributions to the
third order
${\cal O}(m_b^2 \as^3)$ correction to $\Gamma^V$ and the
second order
${\cal O}(m_b^2 \as^2)$ correction to $\Gamma^A$ do not require
 an explicit  four-loop and three-loop calculation respectively,
but instead may be obtained by employing renormalization group (RG)
arguments \cite{CheKue90,CheKueKwi92}.
The derivation is based on the RG properties of the two point correlators.
The correlators
$\langle 0| \;T\;j_{\mu}^{V/A}(x)j_{\nu}^{V/A\dagger}(0)\;|0 \rangle$
contain  non-integrable singularities in the vincinity of the 
point $x=0$, which cannot be removed by standard quark mass and coupling 
constant renormalizations, but must be subtracted independently.
As a consequence the relevant RG equation read
\beq \EQN{NS1}
\dmu \Pi^{V/A}_{\mu\nu} = (q_{\mu}q_{\nu}-g_{\mu\nu}q^2)
\gamma_q^{V/A}(\as)\frac{1}{16\pi^2} 
+ (\bar{m}\pm \bar{m}')^2 g_{\mu\nu} \gamma_m^{V/A}(\as) \frac{1}{16\pi^2}
,\eeq
 where
 \beq \EQN{NS2}
 \mu^2\frac{d}{d\mu^2} = \mu^2\frac{\partial}{\partial\mu^2} 
+ \pi\beta(\alpha_s) \frac{\partial}{\partial \alpha_s}
- 2\bar{m}^2 \gm(\alpha_s)\frac{\partial}{\partial \bar{m}^2}
 \eeq
and the  $\beta$-function 
as well as the  quark mass anomalous dimension $\gm$ are defined
in the usual way
\beq \EQN{NS3}
\dmu \left( \frac{\as(\mu)}{\pi} \right) = \beta(\as) \equiv
-\sum_{i\geq0}\beta_i\left(\api\right)^{i+2}
, \eeq
\beq 
\dmu \bar{m}(\mu) = - \bar{m}(\mu)\gm(\as) \equiv 
-\bar{m}\sum_{i\geq0}\gm^i\left(\api\right)^{i+1}
. \eeq
In the vector case no contribution from $\gamma_m^V$ is present
in eq.(\ref{NS1}) for 
equal masses $\bar{m}=\bar{m}'=\bar{m}_b$, which is not the case 
for  $\gamma_m^A$. This different behaviour remains if we 
consider the quadratic mass part of the Adler function
\beq\EQN{NS4}
 D^{V/A(m)} = -4\pi^2 Q^2 
\frac{d}{dQ^2}\left(\frac{\Pi^{V/A(m)}_1}{Q^2}\right)
.\eeq 
While $D^{V(m)}$ is RG invariant, $D^{A(m)}$ satisfies an inhomogenous 
RG equation:
\beq \EQN{NS6} 
\ba{ll}\dsp
\dmu D^{V(m)} &=  0
\\ \dsp
\dmu D^{A(m)} & \dsp = \frac{m^2}{Q^2}\gamma^A_m
 \equiv \frac{m^2}{Q^2} \sum_{i\geq 0} 
\left(\gamma^A_m\right)_i \left(\api\right)^i 
.\ea\eeq
Writing down the general expansions for the Adler functions $D^V,D^A$ 
(with $\ell=\ln\mu^2/Q^2$)
\beq \EQN{NS7} \ba{ll}\dsp
D^V(Q^2)  = 
&\dsp 1 + (a^V_{01}+a^V_{11}\ell)\api +
(a^V_{02}+a^V_{12}\ell+a^V_{22}\ell^2)\left(\api\right)^2 
\\  &
\dsp   - \frac{\bar{m}^2(\mu^2)}{Q^2} \left[ 
b^V_{00} + (b^V_{01}+b^V_{11}\ell)\left(\api \right) \right.  \\
& \ds \hphantom{ - \frac{\bar{m}^2(\mu^2)}{Q^2} } 
+ (b^V_{02}-a^V_{02}+b^V_{12}\ell+b^V_{22}\ell^2)\left(\api\right)^2  \\ 
& \ds \hphantom{ - \frac{\bar{m}^2(\mu^2)}{Q^2} } \left.
+ (b^V_{03}+b^V_{13}\ell+b^V_{23}\ell^2+b^V_{33}\ell^3)\left(\api\right)^3
\right] \\ 
D^A(Q^2)  = 
&\dsp
  1 + (a^A_{01}+a^A_{11}\ell)\api +
(a^A_{02}+a^A_{12}\ell+a^A_{22}\ell^2)\left(\api\right)^2 
\\  &
\dsp   - \frac{\bar{m}^2(\mu^2)}{Q^2} \left[ 
b^A_{00}+b^A_{10}\ell
 + (b^A_{01}+b^A_{11}\ell+b^A_{21}\ell^2)\api\right.  \\
& \ds \hphantom{ - \frac{\bar{m}^2(\mu^2)}{Q^2} } \left.
+ (b^A_{02}+b^A_{12}\ell+b^A_{22}\ell^2+b^A_{32}\ell^2)\left(\api\right)^2  
\right]
, \ea    
\eeq 
the equations (\ref{NS6}) translate into recursion relations
between the coefficients $b^{V/A}_{ij}$:
\beq\EQN{NS8} \ba{ll}      
b^V_{13} & \ds = 2(\beta_0 + \gm^0)(b^V_{02}-a^V_{02})
                 +(\beta_1 + 2\gm^1) b^V_{01} + 2\gm^2 b^V_{00} ,\\
b^V_{23} & \ds = (\beta_0 + \gm^0)b^V_{12}
                 +\frac{1}{2}(\beta_1 + 2\gm^1) b^V_{11} ,\\
b^V_{33} & \ds = \frac{2}{3}(\beta_0 + \gm^0)b^V_{22}, \\
b^A_{12} & \ds = - \left( \gamma^A_m\right)_2 +   (\beta_0 + 2\gm^0)b^A_{01}
                 + 2 \gm^1 b^A_{00} , \\
b^A_{22} & \ds = \frac{1}{2}  (\beta_0 + 2\gm^0) b^A_{11} 
                 +  \gm^1  b^A_{10} ,\\
b^A_{32} & \ds = \frac{1}{3}   (\beta_0+2\gm^0) b^A_{21} 
.\ea \eeq
One observes that a derivation of coefficients of a given order
not only 
requires the knowledge of the corrections of the lower orders, but also
the RG functions $\beta,\gamma_m$ 
 and --- in case of axial induced corrections ---
 the anomalous dimension $\gamma^A_m$ 
 up to that  given order.
The mass corrections to $D^V$ up to order ${\cal O}(\as^2)$ are known from
 \cite{GorKatLar86}, those to $D^A$ to first order were obtained
by several groups \cite{CheKueKwi92,Bro}.
The  RG functions have been calculated in \cite{Tar}.
The second order result for the anomalous dimension $\gamma^A_m$
can be deduced (see \cite{CheKueKwi92})  
with the help of the axial Ward identity
  the result of the work \cite{GorKatLarSur90}.

With the above described recursion method one cannot obtain the  
coefficients of the constant terms $b^V_{03}$ and
$b^A_{02}$.
They  would require an explicit calculation.

However, the crucial observation is that those coefficients do not enter 
to $R^V$ and $R^A$ in our considered order.
Thus we arrive at  the flavour non-singlet result to the partial
decay rate 
\beq\EQN{NS10}
\ba{ll} \dsp
R^{V(m)}_{NS} 
& \dsp 
= 12\frac{\bar{m}_b^2(s)}{s} \api \left\{ 1 + 8.74 \api
+  45.15 \left( \api\right)^2 \right\}
\\ \dsp
R^{A(m)}_{NS} 
& \dsp 
= - 6\frac{\bar{m}_b^2(s)}{s}\left\{ 1 + \frac{11}{3} \api 
+  14.286  \left( \api\right)^2 \right\}
\ea
\eeq

  
\section{Quadratic Mass Corrections from Flavour Singlet Diagrams}
The mass corrections to the $Z$-decay rate originating
from flavour singlet diagrams require an explicit computation.
In this section we describe the three loop calculation for
 the axial case\footnote{ 
The corresponding
four loop calculation of the mass corrections to the vector current 
correlator has not been performed.
Its contribution is expected to be completely negligible.}.

We consider the axial vector current correlator
(represented by the first graph of Figure 2)
\beq \EQN{S1}
\Pi^A_{\mu\nu}(q) = i \int dx e^{iqx} 
\langle 0|\; 
T\;\Delta_{\mu}(x)\Delta_{\nu}(0) \;|0\rangle 
 =  g_{\mu\nu} 
\Pi_1^A +  q_{\mu}q_{\nu} 
  \Pi_2^A  .
\eeq
with $\Delta_{\mu}=j_{\mu}^{A,t}-j_{\mu}^{A,b}$. 
For convenience of the calculation we split the diagrams into the classes of
so called ``light'' diagrams, which contain two closed $b$-quark loops,
and ``heavy'' diagrams containing one top- and one bottom-loop.
Diagrams with two top-loops do not contribute to $\Gamma^A$ due to the 
heaviness of the top.

Treat the light and the heavy diagrams separately, we deal with the
operators   $j_{\mu}^{A,t}$ and $j_{\mu}^{A,b}$ as single entries,
 whence in the intermediate stage of our discussion 
the regularization of UV divergences becomes necessary, even though
 the difference $\Delta_{\mu}$ is an ultraviolett finite operator.
Using  dimensional regularization
we employ  the definition of the $\gamma_5$ matrix in 
$D=4-2\epsilon$ dimensions  as it is 
 suggested  by 't Hooft and
Veltman \cite{HooVel72}, and formalized by 
Breitenlohner and   Maison \cite{BreMai77}  
with a  modification introduced in
\cite{{AkyDel73}}.

For the light diagrams we expand the $b$-quark propagators with 
respect to the bottom mass up to order ${\cal O}(m_b^2)$, thus ending
up with massless Feynman integrals. These resulting integrals are still
well defined in the region of small loop momenta. The reduction
to massless propagator type integrals is of advantage, since for
these kinds of problems elaborated techniques of computation
have been developed in the past \cite{CheTka81} (for a recent review see also
\cite{Bro92}).
The corresponding algorithms are implemented in computer program packages. 
We have used MINCER \cite{LarTkaVer91} for integrals up to
three loops and LOOPS \cite{SurTka89} for integrals up to two loops.

Our strategy for the calculation of the heavy diagrams is to 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 keep  the leading contribution.
Having done this, the dependence on $m_b$ remains only in two loop
integrals, which can be expanded in $m_b$ and again leads to 
massless integrals only. 
 

The large mass expansion
works as follows\footnote{In performing  concrete  
calculations we make use of the so-called method of projectors 
\cite{GorLar87} in a diagram-wise approach  of \cite{PivTka84}.  
The reader interested in the mathematical proofs will find them in  
   \cite{Smi90,Smi91}.}.
In the heavy top mass limit $m_t\rightarrow \infty$ 
 the Feynman integral                                   
$<\Gamma>(q^2)$ corresponding to a graph 
$\Gamma$ 
may be represented in  the form of
 the following asymptotic    expansion: 
\beq\EQN{S2}
<\Gamma>(q^2)\bbuildrel{=\!=\!=}_{{\scriptstyle{m_t\to\infty}}}^{}  
\sum_\g  C^{(t)}_\gamma * <\Gamma/\gamma>^{\mbox{\rm \scriptsize eff}}         
\eeq   
The diagrams   $<\Gamma/\gamma>^{\mbox{\rm \scriptsize eff}}$   
of the effective light theory are by definition composed of the fields of
the light particles (namely the gluon field and the botto  quark field),
  whereas the whole top dependence is contained in the                    
``coefficient functions'' $C^{(t)}_\gamma$. Diagrammatically the latter 
are represented by all possible {\em hard} subgraphs of $\Gamma$, which
by definition,
 comprises         
all {\em heavy} lines od the top quark
and becomes one-particle-irreducible after contracting     
its heavy lines.    Our                   
notation   $\gamma * <\Gamma/\gamma>^{\mbox{\rm \scriptsize eff}} $
means that in the original diagram $\Gamma$ the 
hard subgraph  $\gamma$  is reduced to an effective blob. 
The corresponding Feynman subintegral
 $C^{(t)}_\g$  is then expanded in a (in general multidimensional)   
formal Taylor expansion with respect to its (small) external momenta and
to light masses  eventually contained in $C^{(t)}_{\gamma}$. This Taylor       
expansion is then inserted in the effective blob and the whole        
expression is evaluated.                                                       

The flavour singlet diagrams contain two hard subgraphs, which
could  contribute
to the large mass expansion. They are shown in Figure 3, of which
 the first diagram can be shown to go like
$1/m_t^2$ in the large mass limit and may thus be  safely  neglected.
 
As a result for the complete flavour singlet contribution
to $R^{A(m)}$ including the light as well as the heavy diagrams
we have
\beq \EQN{S3}
R^{A(m)}_{S} = 6\frac{\bar{m}_b^2}{s}\left(\api\right)^2
\left[ 3-\ln\frac{s}{m_t^2}\right]
.\eeq

\section{Numerical Results and Discussion}
The flavour non-singlet and singlet  contributions 
discussed in the previous sections lead in their sum to our final result 
\beq \EQN{N1}
\ba{ll} \dsp
R^{V(\bar{m})} 
& \dsp 
= 12\frac{\bar{m}_b^2(s)}{s} \api \left\{ 1 + 8.74 \api
+  45.15 \left( \api\right)^2 \right\}
\\ \dsp
R^{A(\bar{m})} 
& \dsp 
= - 6\frac{\bar{m}_b^2(s)}{s}\left\{ 1 + \frac{11}{3} \api 
+   \left( \api\right)^2 \left[11.286 + \ln\frac{s}{m_t^2}\right]\right\}
\ea
\eeq
The size of the different contributions to the 
corrections is illustrated graphically in Figures 4--7.
In Figure 4 the  bar chart represents the results of the 
different orders separately  for the massless and the quadratic mass 
contribution
to $R^V$. The massive part of Figure 4 is enlarged in Figure 5.
The mass effects to the vector current induced rate are very small,
they are well below 1 permille. Since the mass term of the 
zeroth order in $\as$ vanishes identically, the mass corrections are 
suppressed by the strong coupling constant.

The corresponding corrections for the axial vector case are shown in
Figures 6 and 7. The difference of the second order massless
contribution as compared to the vector case is due to 
the flavour singlet contribution
to the axial correlator.  The leading order mass corrections
exceed in size the second order massless contribution and amount to
$\simeq -0.5\%$.

   
The numbers presented in the Figures are based on $\Lambda_{QCD}=400$ GeV and 
on the on-shell mass for the $b$-quark
$m_b=4.72$ GeV \cite{DomPav92}, where for the transformation
to $\bar{m}_b(s)$ the following relation is used 
\cite{GraBroGraSch90}:  
\beq \EQN{N2} 
m=\bar{m}(m^2)\left(1+\frac{4}{3}\apim + K_b 
\left(\apim\right)^2 +{\cal O}(\as^3) \right), 
\; K_b\approx 12.4
 \eeq
We have included next-to-next-to leading order terms in the expressions
for the running coupling constant and the running mass. 

It is instructive to compare the above result obtained in the $\msbar$-scheme
with the one coming from the use of the on-shell scheme. In this scheme
only momentum logarithms are summed over through the transition to the
running coupling constant, while bottom quark mass logarithms 
explicitly appear in the result:
\beq \EQN{N3}
\ba{ll} \dsp
R^{V(m)}= 
& \dsp 
 12\frac{m_b^2}{s}\api\left\{ 1 
+  \api \left[ 6.07-2\ln\frac{s}{m_b^2}\right] \right.
\\ & \dsp
+  \left. \left( \api\right)^2 
 \left[ 2.38-24.29\ln\frac{s}{m_b^2}
+0.083\left(\ln\frac{s}{m_b^2}\right)^2\right] 
\right\}
\\ \dsp
R^{A(m)}= 
& \dsp 
 - 6\frac{m_b^2}{s}\left\{ 1
 + \api\left[ 1-2\ln\frac{s}{m_b^2}\right]\right.
\\ &\dsp
+  \left. \left( \api\right)^2 \left[-17.96 + \ln\frac{s}{m_t^2}
-14.14\ln\frac{s}{m_b^2}
+0.083\left(\ln\frac{s}{m_b^2}\right)^2\right]\right\}
\ea
\eeq
The results of both schemes are compared in Figures 8 and 9.
In these diagrams the bars represent the corrections 
up to a respective order  including the lower orders.
One can notice that for the lower order a huge part of the correections
is absorbed in the running mass. 
The difference between the $\msbar$ and the $OS$-scheme level out
if higher orders are also included. The sensitivity on the 
renormalization scheme therefore decreases rapidly  with increasing 
order. For the axial mass corrections of second order the relative
uncertainty due to the choice of the renormalization scheme 
is reduced to  $5\%$. 

Taking into account all QCD corrections as well as the  effects of 
electroweak corrections (they decrease the QCD corrected rate 
by about 2\%, see
 \cite{BerPicSan91,Nar91}).
 the theoretical prediction for the
partial decay rate $\G(Z\rightarrow b\bar{b})$ reads
\beq  \EQN{N4}
\G(Z\rightarrow b\bar{b})_{theo}   = \left\{
\ba{ll}
375 \mbox{\rm MeV} & \;\;\;(\Lambda_{QCD} = 200 \mbox{\rm MeV}) \\
377 \mbox{\rm MeV} & \;\;\;(\Lambda_{QCD} = 400 \mbox{\rm MeV}) \\
379 \mbox{\rm MeV} & \;\;\;(\Lambda_{QCD} = 650 \mbox{\rm MeV}) 
\ea
\right.
\eeq
The theoretical uncertainty therefore is  dominated by the
 value for the QCD scale.

We finally conclude that QCD bottom mass corrections
to the decay rate of the $Z$ boson into $b$-quark pairs are reliably
estimated. They are small and
will not pose any problems in the high precision tests
of the standard model.


\begin{thebibliography}{99}

\bibitem{Sch73}
J.Schwinger: Particles, Sources and Fields, Vol.2, Addison-Wesley 1973. 

\bibitem{JerLaeZer82}
J.Jersak, E.Laermann, P.M.Zerwas, Phys.\ Rev.\ D 25 (1982) 1218.

\bibitem{ReiRubYaz85}
L.J.Reinders, H.Rubinstein, S.Yazaki, Phys.\ Rep.\  127 (1985) 1.

\bibitem{CheKatTka79}
K.G.Chetyrkin, A.L.Kataev, F.V.Tkachov,  Phys.\ Lett.
\ B 85  (1979) 277;\\
M.Dine, J.Sapirstein, Phys.\ Rev.\ Lett. 43 (1979) 668; \\
W.Celmaster, R.J.Gonsalves, Phys.\ Rev.\ Lett. 44 (1980) 560.

\bibitem{BerWet81}
W.Bernreuther, W.Wetzel, Z.\ Phys.\ C 11 (1981) 113; \\
W.Bernreuther, W.Wetzel, Phys.\ Rev\ D 24 (1982) 2724. \\
K.G.Chetyrkin, Preprint TTP93--5, Karlsruhe 1993.

\bibitem{GorKatLar90}
L.R.Surguladze, M.A.Samuel,  Phys. Rev. Lett. 66 (1990) 560;
S.G.Gorishny, A.L.Kataev, S.A.Larin, Phys.\ Lett. 259 (1991) 144;\\

\bibitem{SurSam92}
L.R.Surguladze, M.A.Samuel,  Preprint FERMILAB-PUB-92/192-T, 1992.

\bibitem{KniKue} 
B.A.Kniehl, J.H.K\"uhn, Phys.\ Lett.\ B 224  (1990) 229; \\
B.A.Kniehl, J.H.K\"uhn, Nucl.\ Phys.\ B 329 (1990) 547.

\bibitem{CheKue93}
K.G.Chetyrkin, J.H.K\"uhn, Preprint TTP93--8, Karlsruhe 1993

\bibitem{CheKue90}
K.G.Chetyrkin, J.H.K\"uhn, Phys.\ Lett.\ B 248 (1992) 359.

\bibitem{CheKueKwi92}
K.G.Chetyrkin, J.H.K\"uhn, Phys.\ Lett.\ B 282 (1992) 221.

\bibitem{GorKatLar86}
S.G.Gorishny, A.L.Kataev, S.A.Larin, Nuovo Cimento 92 (1986) 117.

\bibitem{Bro}
C.Becchi, S.Narison,E. de Rafael, F.J.Yndurain, Z.\ Phys.\ C 8 (1982) 335; \\
D.J.Broadhurst, Phys.\ Lett.\ 101 B (1981) 423; \\
D.J.Broadhurst, S.C.Generalis, Open University Preprint OUT-4102-12 (1984)
(unpublished). \\
S.C.Generalis, J.\ Phys.\ G 15 (1989) L225.

\bibitem{Tar}
O.V.Tarasov, A.A.Vladimirov, A.Yu.Zharkov, Phys.\ Lett.\ B93 (1980) 429;\\
O.V.Tarasov, Preprint JINR P2-82-900 (1982).

\bibitem{GorKatLarSur90}
S.G.Gorishny, A.L.Kataev, S.A.Larin, L.R.Surguladze, Mod.\ Phys.\ Lett.\ 
A 5 (1990) 2703.

\bibitem{HooVel72}
G. 't Hooft, M.Veltman, Nucl.\ Phys.\ B 44 (1972) 189.

\bibitem{BreMai77}
P.Breitenlohner, D.Maison, Comm.\ Math.\ Phys 52 (1977) 11.
 
\bibitem{AkyDel73}
D.A.Akyeampong, R.Delbourgo, Nuovo Cimento 17 A (1973) 578.

\bibitem{CheTka81}
K.G.Chetyrkin, F.V.Tkachov, Nucl.\ Phys.\ B 192 (1981) 159.

\bibitem{Bro92}
D.J.Broadhurst, Talk at AI-92 Conference, January 1992;\\
 Preprint OUT-4102-37.

\bibitem{LarTkaVer91}
S.A.Larin, F.V.Tkachov, J.A.M.Vermaseren, Preprint NIKHEF-H/91-18 (1981).

\bibitem{SurTka89}
L.R.Surguladze, F.V.Tkachov, Comp.\ Phys.\ Comm. 55 (1989) 205.

\bibitem{GorLar87}
S.G.Gorishny, S.A.Larin, Nucl.\ Phys.\ B 287 (198) 452.

\bibitem{PivTka84}
G.B.Pivovarov, F.V.Tkachov, Preprint INR P-0370 (1984); \\
K.G.Chetyrkin, Preprint MPI-PAE/PTh 13/91 (1991).

\bibitem{Smi90}
V.A.Smirnov, Commun.\ Math.\ Phys.\ 134 (1990) 109.

\bibitem{Smi91}
V.A.Smirnov, Remormalization and Asymptotic Expansions, (Birkh\"auser,
Basel, 1991).

\bibitem{Vel67}
M.Veltman, SCHOONSCHIP, A CDF 6600 program for symbolic evaluation
of algebraic expressions, CERN Report, 1967. 

\bibitem{DomPav92}
C.A.Dominguez, N.Paver, Phys.\ Lett. B 293  (1992) 197.

\bibitem{GraBroGraSch90}
N.Gray, D.J.Broadhurst, W.Grafe, K.Schilcher, 
Z.\ Phys.\ C 48 (1990) 673.

\bibitem{BerPicSan91}
J.Bernabeu, A.Pich, A.Santamaria, Nucl.\ Phys.\ B 363 (1991) 326.

\bibitem{Nar91}
S.Narison, Phys.\ Lett.\ B 260 (1991) 411.
\end{thebibliography}

\newpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Figures

\figz
 
{\bf Figure 1:} Flavour non singlet diagrams of order up to ${\cal O}(\as)$. 

\vspace{2ex}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\figy

{\bf Figure 2:} Flavour singlet diagrams to order ${\cal O}(\as^2)$ and
${\cal O}(\as^3)$. 

\vspace{2ex}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\figi

\vspace*{0.5ex}
{\bf Figure 3:}  Diagrams contributing  to the 
coefficient functions $C^{(t)}_n$.



\end{document}

**********************************************************
\newcommand{\figz}{
\begin{picture}(60000,10000)
% d1
\drawline\photon[\E\REG](0,5000)[3]
\global\advance\photonbackx by 2000
\put(\photonbackx,\photonbacky){\circle{4000}}
\global\advance\photonbacky by 2500
\put(\photonbackx,\photonbacky){$b$}
\global\advance\photonbacky by -2500
\global\advance\photonbackx by 2000
\drawline\photon[\E\REG](\photonbackx,\photonbacky)[3]
%
\global\advance\photonbackx by 3000
% d2
\drawline\photon[\E\REG](\photonbackx,\photonbacky)[3]
\global\advance\photonbackx by 300
\global\advance\photonbacky by 900
\drawline\gluon[\E\REG](\photonbackx,\photonbacky)[3]
\global\advance\gluonfrontx by -300
\global\advance\gluonfronty by -400
\global\advance\gluonbackx by -100
\global\advance\gluonbacky by -400
\put(\gluonfrontx,\gluonfronty){$\bullet$}
\put(\gluonbackx,\gluonbacky){$\bullet$}
\global\advance\photonbackx by -300
\global\advance\photonbacky by -900
\global\advance\photonbackx by 2000
\put(\photonbackx,\photonbacky){\circle{4000}}
\global\advance\photonbacky by 2500
\put(\photonbackx,\photonbacky){$b$}
\global\advance\photonbacky by -2500
\global\advance\photonbackx by 2000
\drawline\photon[\E\REG](\photonbackx,\photonbacky)[3]
%
\global\advance\photonbackx by 3000
% d3
\drawline\photon[\E\REG](\photonbackx,\photonbacky)[3]
\global\advance\photonbackx by 2000
\put(\photonbackx,\photonbacky){\circle{4000}}
\global\advance\photonbacky by 2500
\put(\photonbackx,\photonbacky){$b$}
\global\advance\photonbacky by -2500
\global\advance\photonbackx by 500
\global\advance\photonbacky by 1700
\drawline\gluon[\S\REG](\photonbackx,\photonbacky)[3]
\global\advance\gluonfrontx by -300
\global\advance\gluonfronty by -200
\put(\gluonfrontx,\gluonfronty){$\bullet$}
\global\advance\gluonbackx by -300
\global\advance\gluonbacky by -500
\put(\gluonbackx,\gluonbacky){$\bullet$}
\global\advance\photonbackx by -500
\global\advance\photonbacky by -1700
\global\advance\photonbackx by 2000
\drawline\photon[\E\REG](\photonbackx,\photonbacky)[3]
\end{picture}
}
\newcommand{\figy}{
\begin{picture}(50000,10000)
% double triangle 2 gluons
\drawline\photon[\E\REG](2000,5000)[4]
\drawline\fermion[\NE\REG](\photonbackx,\photonbacky)[4000]
\drawarrow[\NE\ATTIP](\pmidx,\pmidy)
\drawarrow[\SW\ATTIP](\pmidx,\pmidy)
\global\advance\pmidx by -1500
\global\advance\pmidy by 500
\put(\pmidx,\pmidy){$t,b$}
\drawline\fermion[\SE\REG](\photonbackx,\photonbacky)[4000]
\drawline\gluon[\E\FLIPPEDCURLY](\pbackx,\pbacky)[4]
\drawline\fermion[\N\REG](\fermionbackx,\fermionbacky)[5657]
\drawline\gluon[\E\CURLY](\pbackx,\pbacky)[4]
\drawline\fermion[\S\REG](\pbackx,\pbacky)[5657]
\drawline\fermion[\NE\REG](\pbackx,\pbacky)[4000]
\drawline\fermion[\NW\REG](\pbackx,\pbacky)[4000]
\drawarrow[\SE\ATTIP](\pmidx,\pmidy)
\drawarrow[\NW\ATTIP](\pmidx,\pmidy)
\global\advance\pmidx by 500
\global\advance\pmidy by 500
\put(\pmidx,\pmidy){$t,b$}
\drawline\photon[\E\REG](\fermionfrontx,\fermionfronty)[4]
\global\advance\photonbackx by 3000
% double triangle 3 gluons
\drawline\photon[\E\REG](\photonbackx,\photonbacky)[4]
\drawline\fermion[\NE\REG](\photonbackx,\photonbacky)[4000]
\drawarrow[\NE\ATTIP](\pmidx,\pmidy)
\drawarrow[\SW\ATTIP](\pmidx,\pmidy)
\global\advance\pmidx by -1500
\global\advance\pmidy by 500
\put(\pmidx,\pmidy){$t,b$}
\drawline\fermion[\SE\REG](\photonbackx,\photonbacky)[4000]
\drawline\gluon[\E\FLIPPEDCURLY](\pbackx,\pbacky)[4]
\drawline\fermion[\N\REG](\fermionbackx,\fermionbacky)[2828]
\drawline\gluon[\E\CURLY](\pbackx,\pbacky)[4]
\drawline\fermion[\N\REG](\fermionbackx,\fermionbacky)[2828]
\drawline\gluon[\E\CURLY](\pbackx,\pbacky)[4]
\drawline\fermion[\S\REG](\pbackx,\pbacky)[5657]
\drawline\fermion[\NE\REG](\pbackx,\pbacky)[4000]
\drawline\fermion[\NW\REG](\pbackx,\pbacky)[4000]
\drawarrow[\SE\ATTIP](\pmidx,\pmidy)
\drawarrow[\NW\ATTIP](\pmidx,\pmidy)
\global\advance\pmidx by 500
\global\advance\pmidy by 500
\put(\pmidx,\pmidy){$t,b$}
\drawline\photon[\E\REG](\fermionfrontx,\fermionfronty)[4]
\end{picture}
}
\newcommand{\figi}{
\begin{picture}(60000,10000)
% 1-triangle, 1-loop
\global\seglength = 500
\global\gaplength = 300
\drawline\scalar[\E\REG](0,5000)[4]
\drawline\fermion[\NE\REG](\scalarbackx,\scalarbacky)[4000]
\drawarrow[\NE\ATTIP](\pmidx,\pmidy)
\drawarrow[\SW\ATTIP](\pmidx,\pmidy)
\drawline\fermion[\SE\REG](\scalarbackx,\scalarbacky)[4000]
\global\advance\pmidx by -500
\global\advance\pmidy by -1500
\put(\pmidx,\pmidy){$t$}
\drawline\gluon[\E\FLIPPEDCURLY](\pbackx,\pbacky)[4]
%\global\advance\pmidx by -1000
%\global\advance\pmidy by -1500
%\put(\pmidx,\pmidy){$h_{1g}$}
\drawline\fermion[\N\REG](\fermionbackx,\fermionbacky)[5657]
\drawline\gluon[\E\CURLY](\pbackx,\pbacky)[4]
%
\global\advance\gluonbackx by 5000
\global\advance\gluonbacky by -2828
%
% 1-triangle, 2-loop
\global\seglength = 500
\global\gaplength = 300
\drawline\scalar[\E\REG](\gluonbackx,\gluonbacky)[4]
\drawline\fermion[\NE\REG](\scalarbackx,\scalarbacky)[4000]
\drawarrow[\NE\ATTIP](\pmidx,\pmidy)
\drawarrow[\SW\ATTIP](\pmidx,\pmidy)
\drawline\fermion[\SE\REG](\scalarbackx,\scalarbacky)[4000]
\global\advance\pmidx by -500
\global\advance\pmidy by -1500
\put(\pmidx,\pmidy){$t$}
\drawline\gluon[\E\FLIPPEDCURLY](\pbackx,\pbacky)[4]
%\global\advance\gluonfronty by -1500
%\put(\gluonfrontx,\gluonfronty){$h_2$}
\drawline\fermion[\N\REG](\fermionbackx,\fermionbacky)[5657]
\drawline\gluon[\E\CURLY](\pbackx,\pbacky)[4]
\drawline\fermion[\S\REG](\pbackx,\pbacky)[5657]
\drawline\fermion[\E\REG](\pbackx,\pbacky)[2000]
\global\advance\fermionfronty by 5657
\drawline\fermion[\E\REG](\fermionfrontx,\fermionfronty)[2000]
\end{picture}
}
