%Title: Corrections of Order ${\cal O}(G_F\alpha_s m_t^2)$ to the Higgs Decay Rate $\Gamma(H\to b\bar{b})$
%Authors: A. Kwiatkowski, M. Steinhauser
%Published: <EM> Phys. Lett. </EM> <B> B338 </B> (1994) 66-70. Erratum-ibid. <B>B342</B> (1995) 455
%hep-ph/9405308
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\newcommand{\api}{\frac{\alpha_s}{\pi}}
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\newcommand{\gm}{\gamma_m}
\newcommand{\G}{\Gamma} 
\newcommand{\g}{\gamma} 
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\begin{document}    

\begin{titlepage}
\noindent                                          
%
% Date
%
\hfill TTP94--7\\ 
\mbox{}   
\hfill May 1994   \\   %\hfill \today \\ 
%
% Title
%
\vspace{0.5cm}
\begin{center}
  \begin{Large}
  \begin{bf} 
Corrections of Order
 ${\cal O}(G_F\alpha_s m_t^2)$
 to the Higgs Decay Rate 
 $\Gamma(H\to b\bar{b})$   
  \end{bf}
  \footnote[1]{A complete postscript file, 
including figures, is available via anonymous ftp
at ttpux2.physik.uni-karlsruhe.de (129.13.102.139)
as /ttp94-07/ttp94-07.ps}
\end{Large}
% 
% Author
%

  \vspace{0.8cm}
  \begin{large} 
 A.Kwiatkowski\footnote[2]{Present address:
   \begin{minipage}[t]{8cm} 
              Lawrence Berkeley Laboratory \\ 
              Bldg. 50 A - 3115, \\
              1 Cyclotron Road \\
              Berkeley, CA. 94720, USA 
   \end{minipage}     }
 M.Steinhauser    \\[3mm] 
    Institut f\"ur Theoretische 
Teilchenphysik\\ 
    Universit\"at Karlsruhe\\ 
    Kaiserstr. 12,  
  Postfach 6980\\[2mm]
   D-76128 Karlsruhe, Germany\\ 
  \end{large}  

%
% Abstract 
%
  \vspace{3cm}
  {\bf Abstract}
\end{center}
\begin{quotation}
\noindent
QCD corrections to the electroweak
 one-loop result for the partial width 
$\Gamma(H\rightarrow b\bar{b})$ are 
studied. For the decay channel into
 bottom quarks the rate is affected by
a virtual top quark through electroweak
interactions. The calculation of QCD
corrections to this quantity is performed
for an intermediate range Higgs mass in
the heavy top mass limit. The leading 
correction
of order ${\cal O}(G_F\alpha_s m_t^2)$
is estimated. Numerically the contribution
is of comparable
 size as the electroweak 
correction, but of opposite sign. 
\end{quotation}                         

\end{titlepage}

\renewcommand{\arraystretch}{2}

\section{Introduction}
At present high energy colliders experimental
data have turned out to be in remarkable
agreement with theoretical predictions 
of the Standard Model. Precision experiments
have covered many aspects of both its 
electroweak and  QCD sector.
Although the formulation of the Standard 
Model as a $SU(3)_C\times 
SU(2)_L\times U(1)_Y$ gauge theory is firmly
established,
experimental evidence
for the physical Higgs boson is still 
missing. Since the Higgs mechanism of 
spontaneous
symmetry breaking and mass generation 
would reveal its structure through the 
properties of the Higgs boson, the high mass $M_H$
of this scalar particle hides
the  Higgs sector
completely from observation at present
 energies.
Future accelerators like LHC and NLC,
which  are dedicated to the search
and the
study  of the Higgs particle, may 
hopefully close the 
energy gap to its production threshold.
 

For intermediate Higgs masses $M_H<2M_W$
the dominant decay channel is the decay
into bottom quarks.
A theoretical prediction for the partial 
width $\Gamma(H\rightarrow b\bar{b})$
is therefore needed for the physics analysis
with high precision.
Consequently  much work has been spent
 in the past on the calculation of 
radiative corrections to this 
quantity. Excellent reviews on the  Higgs
phenomenology can be found for example
in \cite{GunHabKanDaw90,Kni93}.

Electroweak radiative corrections, 
 which were studied at the one-loop level
 by several
 groups \cite{BarVilKhr91,Kni92,DabHol92},
are  of particular interest for 
the process $H\rightarrow b\bar{b}$,
since they are affected by the mass of the
top quark.  
Virtual top states are possible due to 
bottom-top transitions mediated by 
charged Higgs ghosts 
$\Phi^{\pm}$ or $W^{\pm}$ bosons. The
 leading $m_t^2$ dependence originates from
diagrams with an exchange of $\Phi^{\pm}$
due to
the Yukawa coupling of the Higgs ghosts to
the fermion line. 

In this work QCD corrections
to the electroweak one-loop result 
 are calculated in the heavy top
 mass limit.
For this purpose the 
  hard mass procedure
 \cite{PivTka84,GorLar87,CheSmi87,Smi91}
is employed. It results in a power series
in the inverse heavy mass $1/m_t$, of which
we compute the leading term of order 
 ${\cal O}(G_F\alpha_s m_t^2)$.

It is convenient to consider the 
corresponding scalar current correlator
$\Pi(q^2)$, since its
 absorptive part completely
determines the partial decay rate
\beq \EQN{i1}
\Gamma(H\to b\bar{b})=\frac{1}{M_H}\mbox{Im}\Pi(M_H^2)
.\eeq
The calculation of $\Pi(q^2)$ involves 
multiloop propagator type integrals as well as
massive tadpole integrals. 
Many of the 
computational tools, mostly based on the 
algebraic manipulation language FORM 
\cite{Ver91}, have already been used 
previously for the 
calculation of 
${\cal O}(G_F\alpha_sm_t^2)$ corrections
to the partial width $\Gamma(Z\rightarrow
b\bar{b})$ of the $Z$ boson
 \cite{CheKwiSte93}. 
Nevertheless, the structures of the 
calculations for the processes 
$H\rightarrow b\bar{b}$ and
$Z\rightarrow b\bar{b}$ differ in 
various respects.
First, the generic diagram with a primordial
decay of the Higgs into a pair of charged 
Higgs ghosts and their
 subsequent transition 
into b quarks does not contribute to 
the considered order. This can be seen
 from the  $H\Phi^+\Phi^-$ coupling
 and is explained on dimensional grounds.
 Second, the Yukawa 
couplings of the Higgs to the 
scalar fermion currents are proportional
to the quark masses. Combined with
mass terms from the fermion traces
 the decay rate is proportional to the    
square of the bottom mass 
 and would vanish in the limit $m_b=0$.
Third, the quadratic dependence of the 
correction term in the considered order
on the bottom as well as the top mass
affects the 
renormalization of the scalar correlator.
Wheras for the $Z$ decay the sum of 
the three loop diagrams was finite,
this is not the case for the Higgs rate.
The renormalization of the quark masses
induces contributions from lower order
diagrams which have to be included to 
arrive at a finite result. 

In Section 2.1 the calculation 
of the three loop propagator diagrams is
described. 
Subsequently the relation between 
renormalized and bare quark masses
including  corrections up to the two loop
level is combined with the Born graph 
as well as  pure electroweak and QCD
diagrams. The corresponding contributions
to the order
${\cal O}(G_F\alpha_sm_t^2)$ corrections 
are discussed in Section 2.2.
In Section 3 universal corrections to
$\Gamma(H\rightarrow b\bar{b})$ are added 
to the flavour specific corrections.
The numerical size of the effects is
evaluated.

%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin {figure}
\begin {center}
\begin {tabular}{cc}
\epsfig{file=ttgh.eps,width=4.5cm,height=4.5cm} &
\epsfig{file=segh.eps,width=4.5cm,height=4.5cm}
\end {tabular}
\end {center}
\caption {The electroweak diagrams
 contributing to  
the self energy of the $H$ boson.
Internal dashed line: Higgs ghost,
thin lines: bottom quark, thick lines: 
top quark.}                              
\end{figure} 

\section{Description of the Calculation}
\subsection{Three-Loop  Diagrams}
All diagrams of order 
${\cal O}(G_F\alpha_sm_t^2)$ are obtained from
two different generic graphs (see Figure 1),
to which a gluon has to be attached in all
possible ways. The calculation is performed
with dimensional regularisation
in the on-shell scheme. In 
$D=4-2\epsilon$ dimensions the definition
of the Hermitian, anticommuting 
$\gamma_5$ matrix is used. After combining
pairs of $\gamma_5$ to unity, traces with
single $\gamma_5$ vanish identically. 
 We work in the t'Hooft gauge
with an arbitary QCD gauge
 parameter $\xi_S$. 
The verification of gauge 
invariance serves as 
an internal check of our result. 

Being interested in the leading correction
of the power series with respect to the 
inverse heavy mass $1/m_t$, we apply the 
hard mass procedure as developed and 
elaborated in 
 \cite{PivTka84,GorLar87,CheSmi87,Smi91}.
Those subgraphs are selected, which
comprise all top propagators and become 
one-\-particle-\-irreducible after 
contracting the heavy lines to a point.
They are expanded with respect to 
external  momenta and small masses.
This formal Taylor expansion is then 
inserted as an effective vertex into 
the remaining diagram which has to be
evaluated. In the heavy top limit the 
 bottom mass as well as the 
$W$ mass and the Higgs mass
 are considered small as compared 
to the top mass.
Since the expansion is performed only to the
first nonvanishing order, the Higgs ghost
propagators are always contained in the 
hard subgraphs. For the leading term
$M_W$ is therefore set to zero. As a 
consequence the 
electroweak gauge parameter drops out trivially.
Although the bottom mass
is also a small expansion parameter, it
 cannot be neglected throughout, since that
would result in an identically vanishing
decay rate. The bottom mass is not only 
introduced as an overall factor through 
the Yukawa coupling, but is also present 
in  fermion traces with an odd number 
of Dirac matrices for $m_b=0$. 
Besides the expansion of the hard
 subgraphs as prescribed by the hard mass
procedure also the remaining diagram is
expanded with respect to $m_b^2$. 
After all expansions are performed,
the
 leading piece can be isolated. Its 
contribution $\Delta\Gamma_{\alpha\alpha_s}$
 to the partial Higgs decay rate reads
\beq\EQN{t1}
\ba{ll}\dsp
\Delta\Gamma_{\alpha\alpha_s}
& \dsp
=\Gamma_0m_b^2
 x_t\frac{\alpha_s}{\pi}
\left\{ 
9\frac{1}{\epsilon^2}
+\frac{1}{\epsilon}
\left(
19+12\ln\frac{\mu^2}{m_t^2}
-15\ln\frac{M_H^2}{\mu^2}
\right)
\right.
\\ & \dsp
+\frac{199}{3}-24\zeta(3)
-\frac{23}{6}\pi^2 
+16\ln\frac{\mu^2}{m_t^2}
-41\ln\frac{M_H^2}{\mu^2}
\\ & \dsp
\left.
+9\ln^2\frac{\mu^2}{m_t^2}
+\frac{27}{2}\ln^2\frac{M_H^2}{\mu^2}
-18\ln\frac{\mu^2}{m_t^2}
\ln\frac{M_H^2}{\mu^2}
\right\}
\ea
\eeq       
with
\beq\EQN{t2}
\ba{ll}\dsp
\Gamma_0 
&\dsp
=  \frac{G_F}{4\sqrt{2}\pi}N_CM_H
\\ \dsp
 x_t
& \dsp
=\frac{G_Fm_t^2}{8\sqrt{2}\pi^2}
.\ea\eeq  
According to the conventions of  
the multiloop integration package
MINCER \cite{LarTkaVer91}, which we used for
the calculation of massless integrals,
terms with $\gamma_E$ and $\ln(4\pi)$ 
are suppressed.
In addition for each loop integration
a factor $e^{\zeta(2)\epsilon^2/2}$
is included for convenience.
It is clear, that physical results are not
influenced by this convention, since 
only terms 
of order $\epsilon^2$ are affected.
A consistent convention holds also 
for the calculation of massive
tadpole integrals \cite{Che93}.
Our  result is obtained in the 
limit of large top masses and is
applicable only in the range of intermediate 
Higgs masses, in particular in the regime 
$M_H<2m_t$ below the top threshold.

\subsection{Mass Renormalization and 
Induced Contributions}
The result of eq.(\ref{t1}) 
is still divergent. Renormalization of the 
quark masses induces additional 
contributions which lead to a finite
expression for the Higgs decay rate.
The relation between the bare mass $m_b$ 
and the renormalized mass $m_b^{OS}$
of the bottom quark in the on-shell scheme
may be written as follows
\beq\EQN{m1}
m_b^{OS} = m_b\left\{
1+C_{\alpha}x_t+C_{\alpha_s}
\frac{\alpha_s}{\pi}
+C_{\alpha\alpha_s}
x_t\frac{\alpha_s}{\pi}
\right\}
\eeq
where only corrections of orders relevant
for our problem are taken into account.
The 
corresponding coefficients
 $C_{\alpha},C_{\alpha_s}$ and 
$C_{\alpha\alpha_s}$ can be derived
from the fact that 
the renormalized (pole) mass 
$m_b^{OS}$ is defined 
through the location of the pole of the 
quark 
propagator
\beq\EQN{m2}
S_F=\frac{1}{i}\frac{1}{m_b-p\hspace{-.45em}/+\Sigma(p)}
.\eeq
 The selfenergy  $\Sigma(p)$ 
of the bottom quark is decomposed 
in the form
\beq\EQN{m3}
\Sigma(p)=m_b\Sigma_2
 \left( \frac{m_b^2}{p^2} \right) 
         +(m_b-p\hspace{-.45em}/ \delta)
\Sigma_1 \left( \frac{m_b^2}{p^2} \right)
.\eeq
The functions $\Sigma_{1,2}$ receive
 contributions of the order 
${\cal O}(\alpha_s)$ from diagram 2a with $\delta=1$,
of order ${\cal O}(G_Fm_t^2)$ from diagram 2b with
 $\delta=(1-\gamma_5)$ and of order
${\cal O}(G_F\alpha_sm_t^2)$ from diagrams
 2c--2e with $\delta=(1-\gamma_5)$.
For the  calculation of the
quark selfenergies we again employ
the  hard mass procedure. 
 As suggested in
\cite{GraBroGraSch90} 
remaining standard scalar integrals
 may be simplified by an expansion in 
$m_b^2/p^2$ around $1$. Neglecting all higher
order terms only integrals  
 on the bare mass shell
need to be evaluated:
\beq\EQN{m4}
m_b^{OS}=m_b\left[1 
        + \Sigma_2(1)
        - \Sigma_2(1) \left(2\Sigma_2
^{\prime}(1)+\Sigma_1(1)\right)
        + \cdots        \right]
\eeq
The derivatives 
$\Sigma_2^{\prime}\equiv \partial
\Sigma_2/\partial (m_b^2/p^2)$
may be conveniently obtained through 
derivations with respect to the bare 
 mass, thus raising the power in the 
denominator of the quark propagator. 
One obtains
\beq\EQN{m5}
\ba{ll}\dsp
C_{\alpha} 
& \dsp
=\frac{3}{2\epsilon}+\frac{5}{4}
+\frac{3}{2}\ln\frac{\mu^2}{m_b^2}
\\ \dsp
C_{\alpha_s} 
& \dsp
=\frac{1}{\epsilon}+\frac{4}{3}
+\ln\frac{\mu^2}{m_b^2}
\\ \dsp
C_{\alpha\alpha_s} 
& \dsp
=3\frac{1}{\epsilon^2}
+\frac{1}{\epsilon}
\left(
\frac{19}{4}+\frac{9}{2}\ln\frac{\mu^2}{m_t^2}
+\frac{3}{2}\ln\frac{\mu^2}{m_b^2}
\right)
\\ & \dsp
\frac{95}{8}+\frac{1}{3}\pi^2
+\frac{25}{4}\ln\frac{\mu^2}{m_t^2}
+\frac{13}{4}\ln\frac{\mu^2}{m_b^2}
\\ & \dsp
+\frac{15}{4}\ln^2\frac{\mu^2}{m_t^2}
+\frac{3}{4}\ln^2\frac{\mu^2}{m_b^2}
+\frac{3}{2}\ln\frac{\mu^2}{m_t^2}
\ln\frac{\mu^2}{m_b^2}
.\ea\eeq
A replacement of the bare bottom and top
masses
through the renormalized ones
according to eq.(\ref{m1})
in the results for the Born graph,
the electroweak  
and the QCD
 corrected diagrams 
leads to induced contributions
to the order ${\cal O}(G_F\alpha_s m_t^2)$.
Combined with  eq.(\ref{t1}) 
one obtains the  finite  result for the 
decay rate.

\begin{figure}[t]
 \begin{center}
  \begin{tabular}{cc}
\epsfig{file=selfals.eps,width=4cm,height=4cm}&
\epsfig{file=selfalm.eps,width=4cm,height=4cm}
  \end{tabular}
  \begin{tabular}{ccc}
\epsfig{file=selfil.eps,width=4cm,height=4cm}&
\epsfig{file=selfout.eps,width=4cm,height=4cm}&
\epsfig{file=selfin.eps,width=4cm,height=4cm}
  \end{tabular}
  \caption{\label{figself} Diagrams for the renormalization of the
           bottom quark.}
 \end{center}
\end{figure}

\section{Discussion}             
In the previous section the non-universal corrections to
the partial width $\Gamma(H\rightarrow
 b\bar{b})$ have been discussed. 
Combining all contributions one obtains the following
result for these process dependent corrections:
\beq\EQN{d1}
\Delta\Gamma^{non-univ.}_
{H\to b\bar{b}}=
-6x_t
\Gamma_0 m_b^2
%%%\left\{
   \left[1
     +\frac{\alpha_s}{\pi}
\left(\frac{8}{3}-2\ln\frac{M_H^2}{m_b^2}\right)
\right]
%%%\right\}
\eeq
This result has been confirmed recently in \cite{KniSpi94}.


In addition the partial rate is also 
influenced by universal corrections. 
Process independent terms of order 
${\cal O}(G_F\alpha_sm_t^2)$ originate
from 
 $\Delta r$  due to the use of the
Fermi constant
$G_F$ instead of the electroweak coupling
constant $\alpha$ and 
from the
renormalization of the Higgs vacuum. They are
given by \cite{Kni92}
\beq\EQN{d2}
\Delta\Gamma^{univ.}_
{H\to b\bar{b}}=-\Gamma_0m_b^2
 \left[
            \frac{\Pi^{WW}(0)}{M_W^2}
                             +\mbox{Re}\Pi^{HH\prime}(M_H^2)
                             +\cdots
 \right]
\left( 1+\delta_{QCD}\frac{\alpha_s}{\pi} \right)
\eeq
where
the dots indicate terms of subleading order and the QCD 
correction factor\footnote{We thank the authors of 
\cite{KniSpi94} for pointing out that in the earlier
 version of this paper this factor was missing in 
eq.(\ref{d2}). As a consequence the interference term
 of the 1-loop QCD corrections and the 1-loop universal
electroweak corrections was erraneously not taken into 
account.}
 is given by
$\delta_{QCD}=3-2\ln(M_H^2/m_b^2)$.

For the calculation of the 
 the (unrenormalized)
 selfenergies $\Pi^{WW}$ and $\Pi^{HH}$
again the hard mass procedure
can be  applied. 
Being interested in their real parts the 
leading hard subgraphs are constituted 
by the two loop diagrams themselves with
the $W$ selfenergy to be evaluated for 
zero external momentum. Since the hard mass 
procedure for the  Higgs vacuum
polarization is in fact an expansion of the 
graph with respect to the external momentum,
the derivative Re$\Pi^{HH\prime}$ is readily
obtained from the next to leading contribution to
the power expansion. 
After top mass renormalization the combination
eq.(\ref{d2}) of the $W$-- and the $H$-- selfenergy
is finite in their sum at order 
${\cal O}(G_F\alpha_sm_t^2)$.
The result reads
\beq
\Delta\Gamma^{univ.}_{H\to b\bar{b}}=\Gamma_0m_b^2
   x_t
    \left[7
   -2\frac{\alpha_s}{\pi}\left(3
  +\frac{1}{3}\pi^2\right)
         \right]
\left( 1+\delta_{QCD}\frac{\alpha_s}{\pi} \right)
.\eeq
The electroweak term in the first bracket
 is in agreement with
 \cite{Kni92,DabHol92} and its  QCD contribution
reproduces the result obtained by
\cite{KniSir93} (see also \cite{GroKniWol94,Kni94}).

In the sum we arrive at the final result
for the partial decay rate 
\beq
\Gamma_{H\to b\bar{b}}=
\Gamma_0m_b^2
\left\{1+x_t  
      \left[1 +
     \frac{\alpha_s}{\pi}\left(
-1-\frac{2}{3}\pi^2
-2\log\frac{M_H^2}{m_b^2}
       \right)\right]
\right\}
.\eeq
The QCD correction to the electroweak
result is of comparable size as the
electroweak correction itself, but of
opposite sign.
With $m_b=4.7$ GeV, $m_t=174$ GeV
 and $M_H=60$ GeV ($120$ GeV)
the electroweak contribution 
of $0.32\%$ normalized to the Born width 
is combined with
a QCD corrected term of 
$-0.21\%$ ($-0.24\%$).
 
To conclude, in this work 
 we have calculated the QCD corrections
to the electroweak one-loop result for the 
partial Higgs decay rate $\Gamma(H\rightarrow b\bar{b})$
in the heavy top mass limit. 
For intermediate Higgs masses 
electroweak corrections are 
significantly reduced
by the ${\cal O}(G_F\alpha_sm_t^2)$ corrections.
 
\vspace{5ex}                  
{\bf Acknowledgments} 

\noindent
 We would like to 
thank K.G.Chetyrkin and J.H.K\"uhn 
for helpful discussions.
  


%\appendix
%\section{Appendix} 

\begin{thebibliography}{99}

\bibitem{GunHabKanDaw90}
J.F.Gunion, H.E.Haber, G.Kane, S.Dawson,
The Higgs Hunter's Guide, Addison Wesley 1990.

\bibitem{Kni93}
B.Kniehl, Preprint DESY 93--069,
 Hamburg, August 1993. 

\bibitem{BarVilKhr91}
D.Yu.Bardin, B.M.Vilenskii, 
P.Kh.Khristov,
Sov.\ Journ.\ Nucl.\ Phys.\ 53 (1991) 152.

\bibitem{Kni92}
B.A.Kniehl, Nucl.\ Phys.\ B 376 (1992) 3.

\bibitem{DabHol92}
A.Dabelstein, W.Hollik, 
Z.\  Phys.\ C 53 (1992) 507.

\bibitem{PivTka84}
G.B.Pivovarov, F.V.Tkachov,
 Preprint INR P-0370 (1984), Moscow; \\
F.V.Tkachov, Int.\ Journ.\ Mod.\ Phys. 
A 8 (1993) 2047; \\
G.B.Pivovarov, F.V.Tkachov,  Int.\ Journ.\
 Mod.\ Phys. A 8 (1993) 2241.

\bibitem{GorLar87}
S.G.Gorishny, S.A.Larin,
 Nucl.\ Phys. B 287 (1987) 452;\\ 
S.G.Gorishny, Nucl.\ Phys. B 319 (1989) 633.

\bibitem{CheSmi87}
K.G.Chetyrkin,V.A.Smirnov,
 Preprint INR P-518 (1987), Moscow; \\
K.G.Chetyrkin, Preprint MPI-PAE/PTh 
13/91 (1991), 
Munich; \\
V.A.Smirnov, Commun.\ Math.\ Phys.
 134 (1990) 109. 

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

\bibitem {Ver91} J.A.M. Vermaseren,
 Symbolic Manipulation with FORM, 
Version 2, CAN (Amsterdam, 1991).

\bibitem{CheKwiSte93}
K.G.Chetyrkin, A.Kwiatkowski, 
M. Steinhauser, 
Mod. Phys. Lett. A8 (1993) 2785.

\bibitem {LarTkaVer91}
 S.A. Larin, F.V. Tkachov, 
J.A.M. Vermaseren, 
Preprint NIKHEF-H/91-18 (1991). 

\bibitem {Che93}
K.G.Chetyrkin, 
to appear in Proceedings of the III Workshop on
Artificial Intelligence and Expert Systems in HEP,
Oberammergau, October 1993.

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

\bibitem{KniSir93}
B.A.Kniehl, A.Sirlin,
Phys.\ Lett.\ B 318 (1993) 367.

\bibitem{GroKniWol94}
E.Gross, B.A.Kniehl, G.Wolf,
Preprint DESY 94-035, Hamburg (March 1994),
in press Z.Phys.C

\bibitem{Kni94}
B.A.Kniehl, 
Phys.\ Rev.\ D 50 (1994) 3314

\bibitem{KniSpi94}
B.A.Kniehl, M.Spira,
Preprint DESY 94-102, HEP-PH/9410319, Hamburg (October 1994).
\end{thebibliography}

\end{document}
