%Title: QCD Corrections to the $e^+e^$ Cross Section and the $Z$ Boson Decay Rate: Concepts and Results
%Authors: K.G. Chetyrkin, J.H. Kuehn, A. Kwiatkowski
%Published: Physics Reports 277 (1996) 189282
% the version as of 25.03.96
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\begin{document}
\renewcommand{\arraystretch}{2}
\begin{titlepage}
\noindent
%
% Datum
%
\mbox{} %{\small hepph/yymmnnn}
\hfill {\small TTP9606}
\mbox{}
March 1996
\hfill {\small MPI/PhT/9619}
\mbox{}
%\mbox{}
\hfill {\small LBL36678} \\
\protect\vspace*{.3cm}
%
%
% Title
%
\vspace{1.0cm}
\begin{center}
\begin{Large}
\begin{bf}
QCD Corrections to the $e^+e^$
Cross Section and the
$Z$ Boson Decay Rate:
Concepts and Results${}^{\dagger}$
\\
\end{bf}
\end{Large}
%
% Author
%
\vspace{1cm}
% \begin{large}
K.G.~Chetyrkin $^{abc}$,
J.H.~K\"uhn$^{a}$,
A.~Kwiatkowski$^{d}$
\\
% \end{large}
\begin{itemize}
\item[$^a$]
Institut f\"ur Theoretische Teilchenphysik,
Universit\"at Karlsruhe \\
D76128 Karlsruhe, Germany
\item[$^b$]{%
MaxPlanckInstitut f\"ur Physik, WernerHeisenbergInstitut, \\
F\"ohringer Ring 6, 80805 Munich, Germany}
\item[$^c$]
Institute for Nuclear Research,
Russian Academy of Sciences, \\
60th October Anniversary Prospect 7a
Moscow 117312, Russia
\item[$^d$]
Theoretical Physics Group,
Lawrence Berkeley Laboratory\\
University of California,
Berkeley, CA. 94720, USA
\end{itemize}
\vspace{0.2cm}
% Abstract
%
\vspace{0.5cm}
{\bf Abstract}
\end{center}
\begin{quotation}
\noindent
{ \small
QCD corrections to the electron positron
annihilation cross section into hadrons
and to the hadronic $Z$ boson decay
rate are reviewed. Formal developments
are introduced in a form particularly
suited for practical applications.
These include
the operator product expansion, the heavy
mass expansion, the decoupling of heavy quarks and
matching conditions.
Exact results for the quark mass
dependence are presented whenever
available, and formulae valid in the
limit of small bottom mass
($m_{\rm b}^2\ll s$) or
of large top mass ($m_{\rm t }^2 \gg s$)
are presented. The differences between
vector and axial vector induced rates
as well the classification of singlet
and nonsinglet rates are discussed.
Handy formulae for all contributions
are collected and their numerical
relevance is investigated.
Prescriptions for the separation of the
total rate into partial rates are
formulated. The applicability of the
results in the low energy region,
relevant for measurements around 10 GeV
and below, is investigated and
numerical predictions are collected for
this energy region. }
\end{quotation}
\vfill
\noindent
emails:
\\
chet@mppmu.mpg.de
\\
johann.kuehn@physik.unikarlsruhe.de
\\
kwiat@theor2.lbl.gov \\
\vfill
\footnoterule
\noindent
$^{\dagger}${\footnotesize
To be published in Physics Reports
\noindent The complete postscript file of this
preprint, including figures, is available via anonymous ftp at \\
wwwttp.physik.unikarlsruhe.de (129.13.102.139) as /ttp9606/ttp9606.ps
or via www at \\
http://wwwttp.physik.unikarlsruhe.de/cgibin/preprints/
}
\end{titlepage}
\tableofcontents
\renewcommand{\arraystretch}{2}
\section{Introduction\label{intro}}
Since experiments at the ${\rm e}^+{\rm e}^$
storage ring LEP started datataking a few
years ago, and by the end of the 1993 run by the four
experiments,
more than seven million
hadronic events
had been collected at the ${Z}$ resonance.
The accuracy of the measurements is
impressive. Numerous parameters of the
standard model can be determined with
high precision, allowing
stringent tests of the standard
model to be performed . Among them: the mass
$M_{{Z}} = (91.1884\pm 0.0022)$ GeV and the
width $\Gamma_{{Z}}=(2.4963\pm 0.0032)$ GeV
of the ${Z}$ boson or the weak mixing angle
$\sin^2\theta_{{\rm eff}}^{{\rm lept}}
=0.23143~\pm~0.00028$ \cite{Olchevski95}.
All experimental results were in remarkable
agreement with theoretical predictions and
%turned into
a triumphant confirmation of the
standard model.
As well as the electroweak sector of the
standard model, LEP provides
an ideal laboratory for the investigation
of strong interactions. Due to their purely
leptonic initial state, events are very
clean from both the theoretical and
experimental point of view and represent
the ideal place for testing
QCD. From crosssection measurements
$\sigma_{{\rm had}}
=(41.488\pm 0.078)$ nbarn
\cite{Olchevski95} as well as from the
analysis of event topologies the strong
coupling constant can be extracted.
Other observables measurable
with very high precision are the (partial)
${Z}$ decay rates into hadrons
$\Gamma_{{\rm had}}/\Gamma_{{\rm e}}=
20.788\pm 0.032$
and bottom quarks
$\Gamma_{{\rm b}\overline{{\rm b}}}/\Gamma_{{\rm had}}=
0.2219\pm 0.0017$.
{}From the line shape analysis of LEP
a value $\alpha_s=0.123\pm 0.004\pm 0.002$
is derived.
The program of experimentation at LEP is
still not complete. The prospect of
an additional increase in the number of
events by a factor of about
two
will further improve the level of
accuracy. This means, for example,
that the relative uncertainty of the
partial decay rate into ${\rm b}$ quarks
$\Delta \Gamma_{{\rm b}}/\Gamma_{{\rm b}}$ falls
significantly below
one percent and that an experimental
error for $\alpha_s$ of 0.003 may be achieved.
Also at lower energies significant
improvements can be expected in the
accuracy of crosssection measurements.
The energy region of around $10$ GeV
just below the ${{\rm B}}{\rm \overline{B}}$ threshold
will be covered with high statistics
at future {B} meson factories. The cross
section between the charm and bottom
thresholds can be measured at the BEPC
storage ring
in Bejing. These measurements could
provide a precise value for $\alpha_s$
and  even more important 
a beautiful proof of the running of the
strong coupling constant.
In view of this experimental situation
theoretical
predictions for the various observables
with comparable or even better
accuracy become mandatory and
higherorder radiative corrections
are required.
It seems appropriate to collect all
presently available calculations and
reliably estimate their theoretical
uncertainties. The aim of this report is
to provide such a review for the QCD
sector of the standard model,
as far as crosssection
measurements are concerned, at the ${Z}$
peak as well as in the `low energy'
region from 5 to 20 GeV.
(Related topics have been also discussed in recent reviews
\cite{reviews}.)
Higherorder QCD
corrections to the ${\rm e}^+{\rm e}^$ annihilation
crosssection into hadrons
will be discussed as well as the
hadronic width of the ${Z}$ boson.
Further interest lies in the partial
rates for the decay of the ${Z}$ boson into specific
quark channels. Of particular importance is the
partial width
$\Gamma(Z\rightarrow {\rm b}\overline{{\rm b}})$, as this
quantity can be measured with high accuracy
and provides important information about
the top quark mass from the $Z{\rm b}\overline{{\rm b}}$
vertex.
However, the decomposition
of $\Gamma_{{\rm had}}$ into partial decay rates of
different quark species is possible
in a simple, straightforward way only
up to corrections of the order of ${\cal O}(\alpha_s)$. Apart from
diagrams where `secondary quarks'
are radiated off the `primary quarks'
one encounters flavour singlet diagrams
that first arise
in order ${\cal O}(\alpha_s^2)$ and lead to a confusion
of different species. They therefore have to be
carefully scrutinized.
For many considerations and experimental conditions
quark masses can be neglected, compared to the
characteristic energy of the problem.
Accordingly, higherorder
QCD corrections to the total crosssection
were first calculated
for massless quarks. At LEP energies this is
certainly a good approximation for ${{\rm u,d,s}}$ and
${\rm c}$ quarks. In view of the accuracy reached at LEP
much effort has been spent in estimating the size of
mass effects of the bottom and the top quark.
Whereas ${\rm b}$ quarks are present as particles in the
final state, top quarks can appear only through
virtual corrections.
A large part of this report
is devoted to these effects.
The application of these formulae and,
if necessary, their numerical
evaluation will also
be covered.
In Part~\ref{general} topics of a
general nature are addressed. In Section~\ref{notations}
the notation is introduced and
the relation between
crosssections and decay rates on the one hand and the
corresponding current correlators
on the other is discussed.
Furthermore, the classification of singlet versus
nonsinglet terms is introduced. The behaviour of coupling
constant, masses, operators and correlators under renormalization
group
transformations is reviewed in Section~\ref{renorm} and the relevant
anomalous dimensions are listed. The decoupling of heavy quarks
and the resulting matching conditions for coupling constant
masses and effective currents are treated
in Section~\ref{decoupling}.
Numerical values of quark masses are discussed in
Section~\ref{Qmasses}.
Part~\ref{calctech} is
concerned with calculational techniques
relevant to the problems at hand. Emphasis is put on the
behaviour of the current correlators at large momenta,
the structure of mass corrections in the small mass limit
and the resummation of large logarithms of $m^2/s$.
And the other extreme, with $s/m^2 \ll 1$
also dealt with in this Part, which concludes with a discussion
of $\gamma_5$ in $D \neq 4 $ dimensions.
The analytical firstorder QCD corrections
to the crosssection are recalled in Part~\ref{exact}.
Approximations in the limits of low and high
energies are given.
Nonsinglet and singlet
contributions to the QCD corrections
are presented in Parts~\ref{nonsinglet} and \ref{singlet},
respectively, and the relevant formulae
for various applications are given.
First, the calculations are reviewed
for massless quarks.
This assumption is evidently not
justified for the heavy top top mass,
which appears as a virtual particle. Top mass
corrections are described in Section~\ref{top}.
The dependence on the mass of the finalstate quarks
is given in Section~\ref{repnsm2}. At low energies not only do the
leading quadratic mass terms have to be taken into
account, but quartic mass terms also become relevant.
They are presented in Section~\ref{repnsm4}.
The influence of
secondary quark production on
determinations of the partial rate is
treated in Section~\ref{nspart}.
Recent resuts for the second order QCD corrections
valid for arbitrary $m^2/s$ are discussed in
Section~\ref{schwinger}.
Flavour singlet contributions are discussed in Part~\ref{singlet}.
They arise for the first time
in second order for the axialinduced rate and in
third order for the vector currentinduced rate.
${\cal O}(\alpha_s^2)$ singlet corrections would
be absent for six massless flavours, but do not
vanish due to the large mass splitting in the
$({\rm b,t})$ doublet.
Massless contributions and
bottommass corrections
from singlet diagrams are covered
in Sections~\ref{singlmassless} and~\ref{singlbottom}
respectively.
The assignment of the
singlet contributions to a partial rate into
a specific quark flavour is explained in Section~\ref{singlpartial}.
and the resulting ambiguity is discussed.
In Part~\ref{numerical} the numerical relevance of the
different contributions are studied. Different sources
of theoretical uncertainties are investigated and their
size estimated.
A collection of formulae is presented
in the Appendix. It provides an
overview and may serve as a quick and
convenient reference for
later use.
%?????????????? I have changed lables below!!!!
\section{General Considerations}\label{general}
\subsection{Notations}\label{notations}
\subsubsection{CrossSections and Decay Rates}\label{cross}
We introduce our notations by casting the
total crosssection for longitudinally polarized
${\rm e}^+{\rm e}^$ into hadrons
in leading order of the electroweak coupling
as:
\begin{eqnarray}
%\begin{array}{l}
%\displaystyle
\sigma_{{{\rm R} \atop {\rm L}}}& =& \frac{4\pi\alpha^2}{3s} \Bigg\{
\frac{(v_{{\rm e}}\mp a_{{\rm e}})^2}{y^2}\left
\frac{s}{sM_{{Z}}^2+iM_{{Z}}
\Gamma_{{Z}}}\right^2
\frac{R^{{\rm V}}+R^{{\rm A}}}{y^2}
\nonumber\\
% \displaystyle
%\hphantom{
%\sigma_{{R \atop L}} =
% \frac{4\pi\alpha^2}{3s}
% }
&&+2Q_{{\rm e}}
\frac{v_{{\rm e}}\mp a_{{\rm e}}}{y} {\rm Re} \left[
\frac{s}{sM_{{Z}}^2+iM_{{Z}}\Gamma_{{Z}}}\right]
\frac{R^{{\rm int}}}{y}+Q_{{\rm e}}^2 R^{{\rm em}} \Bigg\}
{}\, ,
%\\ \displaystyle
%\hphantom{
%\sigma_{{R \atop L}} =
% \frac{4\pi\alpha^2}{3s}
% }
%\end{array}
\end{eqnarray}
with the weak couplings defined through
\begin{equation}
v_{{f}}=2I_3^{{f}}4Q_{{f}}\sin^2
\theta_{{\rm w}}\, ,\;\;\;\;
a_{{f}}=2I_3^{{f}}\, ,\;\;\;\;
y=4\sin\theta_{{\rm w}}\cos\theta_{{\rm w}}
{}\, .
\end{equation}
$R$ and $L$ denote the electron beam polarization
(positrons are assumed to be unpolarized).
The functions $R^{{k}}$ with
$ k={\rm V,A},{\rm em},{\rm int} $
are the natural generalization of the Drell ratio
$R\equiv \sigma_{{\rm had}}/\sigma_{\rm point}=
R^{{\rm em}}$, which is
familiar from purely electromagnetic interactions at lower energies.
They are induced by the vector and axial
couplings of the ${Z}$ boson, the pure QED part
and an interference term.
In the massless parton model they are given by
\begin{equation}\label{not3}
R^{{\rm V}}=3\sum_{{f}}
v_{{f}}^2\, ,\;\;\;\;\;
R^{{\rm A}}=3\sum_{{f}} a_{{f}}^2\, ,
\;\;\;\;\;
R^{{\rm em}}=3\sum_{{f}} Q_{{f}}^2\, ,\;\;\;\;\;
R^{{\rm int}}=3\sum_{{f}} Q_{{f}} v_{{f}} \, .
\end{equation}
Here the sum extends over all flavours $f$.
The hadronic decay rate of the ${Z}$ can be
expressed in a way similar to the
incoherent sum of its vector and axial
vectorinduced parts:
\begin{equation}
\begin{array}{ll}
\displaystyle
\Gamma_{{Z}}^{{\rm had}}
&\displaystyle
= \Gamma^{{\rm V}}+\Gamma^{{\rm A}} \\
& \displaystyle
= \frac{\alpha}{3}\frac{M_{{Z}}}{y^2}(R^{{\rm V}}
+R^{{\rm A}})
{}\, .
\end{array}
\end{equation}
Alternatively one may express $\alpha/y^2$
through the Fermi constant
\begin{equation}
\frac{\alpha}{y^2} = \frac{G_{{\rm F}} M_{{Z}}^2}{8\pi\sqrt{2}}
\end{equation}
and absorb the large logarithms from the
running of QED.
These formulae are equivalent for the
present purpose, where higherorder electroweak
corrections are ignored.
All relevant information needed for the
correction factors $R^{{k}}$
is contained in the
current correlation functions
\begin{equation}
\label{not6}
\begin{array}{rl}
\Pi_{\mu\nu}^{ij}(q) =
%&
%\displaystyle
%i \int {\rm d} x e^{iqx} \sum_{{\rm had}}
%\langle 0 \;
%j_{\mu}^{{i}}(x)\;
%{\rm had}\rangle \; \langle {\rm had}  \;
%j_{\nu}^{j\dagger}(0)\;0 \rangle
%\\
%=
& \displaystyle
i \int {\rm d} x e^{iqx}
\langle 0 \;
T\;j_{\mu}^{{i}}(x)j_{\nu}^{ j }(0)\;0 \rangle
\\
= &
\displaystyle
g_{\mu\nu} \Pi_1^{ij}(q^2) + q_{\mu}q_{\nu} \Pi_2^{ij}(q^2)
{}\, ,
\end{array}
\end{equation}
with $(i,j)={\rm (V,V),(A,A)},({\rm em},{\rm em}),
({\rm em , V})$ for $k={\rm V,A},{\rm em},
{\rm int}$ respectively. The
currents under consideration are defined through
\begin{equation}
j^{{\rm V}}_{\mu}=\sum_{{f}}
v_{{f}} \overline{\psi}_{{f}} \gamma_{\mu}
\psi_{{f}}\, ,\;\;\;\;\;
j^{{\rm A}}_{\mu}=\sum_{{f}} a_{{f}}
\overline{\psi}_{{f}}
\gamma_{\mu}\gamma_5 \psi_{{f}}\, ,\;\;\;\;\;
j^{{\rm em}}_{\mu}=\sum_{{f}} Q_{{f}}
\overline{\psi}_{{f}} \gamma_{\mu} \psi_f
{}\, ,
\end{equation}
where the sum extends over all six flavours.
The relation between the crosssection
$\sigma_{{\rm had}}$ and the corresponding current
correlator is closely connected to the
analytic properties of
$\Pi_{\mu\nu}$.
After the Lorentz decomposition
into the functions $\Pi_1$ and $\Pi_2$,
only $\Pi_1$ enters the
crosssection, since the contraction of
$q_{\mu}q_{\nu}\Pi_2 $ with the lepton tensor is
suppressed by the electron mass.
The threshold energies
for the production of fermion pairs
are branch points of the vacuum
polarization, and
$\Pi_1(s)$
is analytic in the
complex plane cut along the real positive
axis. For energies above the lowestlying
threshold ($s=4m^2$)
the function
$\Pi_1(s)$ is discontinuous when ${s}$
approaches the real axis from above and below.
The optical theorem
relates the inclusive crosssection
and thus the function $R(s)$
to the discontinuity of $\Pi_1$
in the complex plane
\begin{equation} \label{d3}
R(s) = \displaystyle
\, \frac{12\pi}{s} \,{\rm Im}\, \Pi_1(  s i\epsilon)
=
\frac{6\pi i}{s}
[
\Pi_1( s  i\varepsilon)

\Pi_1( s + i\varepsilon)
]
{}\, ,
\end{equation}
where Schwarz's reflection
principle has been employed for the second step.
Conversely, the vacuum polarization
is obtained through a
dispersion relation from
its absorptive part.
Applying Cauchy's
theorem along the integration contour of
Fig.~\ref{contour}
leads to:
\begin{equation}
\label{d4}
\begin{array}{rl} \displaystyle
\Pi_1(s)
=&\displaystyle
\frac{1}{2\pi i} \oint ds' \frac{\Pi_1(s')}{s's}
=
\frac{1}{\pi} \int_0^{\infty} ds'
\frac{{\rm Im}\,
\Pi_1(s^{\prime}i\varepsilon)}
{s^{\prime}s} \;\;
{\rm mod \; sub}
\\
=&\displaystyle
 \, \frac{1}{12\pi^2} \int^{\infty}_0 ds^{\prime}
\frac{s^{\prime}}{s^{\prime}s}R(s^{\prime}) \;\;
{\rm mod \; sub}
{}\, ,
\end{array}
\end{equation}
No subtraction is needed if
$\Pi_1(s)$ vanishes at infinity, since the
large circle does not contribute to the integral
in this case.
If the spectral
function is only bounded by
$s^{n}$ at large
distances, one may apply the dispersion
relation to the function $\Pi_1/s^{n+1}$. This is
achieved by $n+1$ subtractions.
For example, a twicesubtracted
dispersion relation has to be applied
for $\Pi_1(s)$, is given by
\[
\tilde\Pi_1(s)
=
{\Pi}_1(s)

{\Pi}_1(0)

(s) {\Pi}_1'(0)
{}\, .
\]
The absorptive part is
not affected by these subtractions.
For the vector current ${\Pi}_1(0)$
vanishes as a consequence of current
conservation and the second subtraction
corresponds to charge renormalization.
Let us add an additional remark concerning the
applicability of perturbative QCD for the calculation
of radiative corrections to the crosssection
$\sigma_{{\rm had}}$.
Experimental ${\rm e}^+{\rm e}^$ data are taken
in the physical regime of timelike
momentum transfer $q^2>0$. This region is influenced
by threshold and bound state effects which make
the use of perturbative QCD questionable.
However,
perturbative QCD is
strictly applicable for large
spacelike momenta ($q^2=Q^2<0$),
since this region is far away
from nonperturbative
effects due to
hadron thresholds, bound state and
resonance effects \cite{Adl74}.
Therefore,
reliable theoretical predictions can
be made for $\Pi_1(Q^2)$ with $Q^2>0$.
To compare theoretical predictions and
experimental results for timelike momenta,
one has to perform suitable
averaging procedures \cite{PogQuiWei76}.
For large positive ${s}$ one may appeal to the
experimentally observed smoothness of $R$
as a function of ${s}$ and to the absence
of any conceivable nonperturbative
contribution.
For later use it is convenient to
introduce the
Adler function
\begin{equation}
\label{d5}
D(Q^2) = 12\pi^2 Q^2
\frac{d}{dQ^2}\left[
\frac{\Pi_1(Q^2)}{Q^2}
\right]
.\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
{
\begin{figure}[t]
\begin{center}
\hphantom{XXXXXXXXX}
\parbox{3cm}{
\epsfig{file=fig3.eps,width=5.cm,height=5.cm}}
\end{center}
\caption {\label{contour}
Contour integral.}
\end{figure}
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\noindent
It
is related to $R$ through a dispersion relation
which allows a comparison between the
perturbatively calculated Adler function ($Q^2>0$)
and the experiment
if the crosssection $R$ is known
over the full energy scale $s'>0$:
\begin{equation}
\label{d5b}
D(Q^2) = Q^2 \int_0^{\infty}
ds^{\prime}\frac{R(s^{\prime})}
{(s^{\prime}+Q^2)^2}
+ 12\pi^2 \frac{\Pi_1(0)}{Q^2}
{}\, .
\end{equation}
The relation inverse to Eq.~(\ref{d5b}) finally reads
\begin{equation} \label{d6}
R(s) = \frac{1}{2\pi i}
\int_{si\varepsilon}^{s+i\varepsilon}
d Q^2\frac{D(Q^2)}{Q^2}
{}\, .
\end{equation}
Diagrammatically,
current correlators are depicted as
vacuum polarization graphs. Their
absorptive parts are obtained from the sum of
all possible cuts applied to the diagram
(see Fig.~\ref{cuts}).
This means
 according to Cutkosky's rule  that
the absorptive part of a Feynman integral is obtained,
if the substitution
\begin{equation}
\frac{1}{p^2m^2+i\epsilon}\rightarrow
2\pi i \delta(p^2m^2)\theta(p_0)
\end{equation}
is applied to
those propagators associated with to the
cut lines of the corresponding Feynman diagram.
Calculating the twopoint correlator and taking
its absorptive part is equivalent
to the evaluation of the matrix element squared
with a subsequent integration over the phase space of
the finalstate particles.
The former
method has some advantages.
Although the
vacuum polarization graph contains one
loop more than the amplitudes in the
direct calculation of the rate,
the problem is reduced to a propagator type
integral, for which quite elaborate techniques
have been developed and implemented in corresponding
computer packages.
Furthermore, the occurance of infrared divergences
is naturally circumvented, since
virtual and bremsstrahlung corrections
correspond only to different cuts
of the same diagram and hence are combined
in the same amplitude. The
cancellation of infrared divergences is therefore
inherent in each diagram.
Depending on the cut,
final states with a different number of particles
are represented by the same diagram, as is shown in
Fig.~\ref{cuts}.
\begin{figure}[t]
\hspace*{0.5cm}
\begin{center}
\begin{tabular}{cccccc}
Im
&
\parbox{3cm}{
\epsfig{file=fig1a.eps,width=3.cm,height=3.cm}
}
&
=
&
\parbox{3cm}{
\epsfig{file=fig1b.eps,width=3.cm,height=3.cm}
}
&
+
&
\parbox{3cm}{
\epsfig{file=fig1c.eps,width=3.cm,height=3.cm}
}
\\
&
&
+
&
\parbox{3cm}{
\epsfig{file=fig1d.eps,width=3.cm,height=3.cm}
}
&
+
&
\parbox{3cm}{
\epsfig{file=fig1e.eps,width=3.cm,height=3.cm}
}
\end{tabular}
\end{center}
\caption[]{\label{cuts}
The absorptive part of a current correlator
is obtained by cutting the diagram in all
possible ways.}
\end{figure}
%
\subsubsection{Classification of Diagrams
\label{classification}}
Higherorder QCD corrections
to ${\rm e}^+{\rm e}^$ annihilation into hadrons
were first
calculated for the electromagnetic case in the
approximation of massless quarks.
Considering the annihilation process through
the ${Z}$ boson,
numerous new features and subtleties
become relevant at the
present level of precision.
The different charge and chiral
structure of electromagnetic and
weak currents respectively has already been addressed
in the previous section:
The functions $R^{{k}}$ as defined above were
classified according to the spacetime
structure of the
currents (vector versus axial vector) and their
electroweak couplings.
Another important distinction, namely `singlet'
versus `nonsinglet' diagrams,
originates from two classes of diagrams
with intrinsically different topology and
resulting charge structure.
The first class of diagrams consists of nonsinglet
contributions with one fermion loop coupled
to the external current. All these amplitudes
are proportional to the charge
structures given in Eq.~(\ref{not3}),
consisting of a sum of terms proportional
to the square of the coupling constant
or the trivial generalization in the
interfence term $R^{{\rm int }}$.
QCD corrections corresponding to these
diagrams contribute
a correction factor independent of the
current under consideration
as long as masses of final state quarks
are neglected.
Singlet contributions
arise from a second class of
diagrams where two currents
are coupled to two
different fermion loops and hence can be cut into two parts
by cutting gluon lines only (see Fig.~\ref{class}).
They cannot be assigned to the contribution from one
individual quark species.
In the axial vector and the vector case the first
contribution of this type arises in order ${\cal O}(\alpha_s^2)$
and ${\cal O}(\alpha_s^3)$ respectively.
Each of them has
a charge structure different
from the one in Eq.~(\ref{not3}).
The lowest order term is therefore ultraviolet finite.
Furthermore, singlet contributions are
separately invariant under renormalization
group transformations.
These diagrams are obviously absent
in chargedcurrentinduced processes
like the $W$ decay.
The functions $R^{{k}}$ are therefore conveniently
decomposed as follows:
\begin{equation}
R^{{\rm V}} = 3\left[ \sum_{{f}} v_{{f}}^2
r^{{\rm V}}_{{\rm NS}}(f)
+ \sum_{f,f'}v_{{f}} v_{f'}
r^{{\rm V}}_{{\rm S}}(f,f')
\right]
{}\, .
\end{equation}
It will be shown below that $r^{{\rm V}}_{{\rm S}}(f,f')$
is independent
of $f$ and $f'$ (meaning the respective
quark masses) up to terms of order
$\alpha_s^4 m_{{\rm q}}^2/s$,
where ${\rm q}$ stands for one of the five light quarks.
Hence
\begin{equation}
R^{{\rm V}}
\approx 3\left[\sum_{{f}} v_{{f}}^2
r^{{\rm V}}_{{\rm NS}}(f) +
(\sum_{{f}} v_{{f}})^2
r^{{\rm V}}_{{\rm S}}\right]
{}\, .
\end{equation}
The functions
$r^{{\rm V}}_{{\rm NS}}$
and $r^{{\rm V}}_{{\rm S}}$ are independent
of the quark charges and arise identically in the
decompositions of $R^{{\rm int}}$ and
$R^{{\rm em}}$:
\begin{equation}
\begin{array}{ll}
\displaystyle
R^{{\rm int}}
& \displaystyle
= 3 \left[\sum_{{f}} v_{{f}}
Q_{{f}} r^{{\rm V}}_{{\rm NS}}(f)
+ \sum_{f,f'}v_{{f}} Q_{f'}
r^{{\rm V}}_{{\rm S}}(f,f')\right]
\\
& \displaystyle
\approx 3 \left[\sum_{{f}} v_{{f}} Q_{{f}}
r^{{\rm V}}_{{\rm NS}}(f)
+ (\sum_{{f}} v_{{f}})(\sum_{f'} Q_{f'})
r^{{\rm V}}_{{\rm S}}\right]
\\
\displaystyle
R^{{\rm em}}
& \displaystyle
= 3 \left[\sum_{{f}} Q_{{f}}^2
r^{{\rm V}}_{{\rm NS}}(f)
+ \sum_{f,f'}Q_{{f}} Q_{f'}
r^{{\rm V}}_{{\rm S}}(f,f')\right]
\\
& \displaystyle
\approx 3 \left[\sum_{{f}}
Q_{{f}}^2 r^{{\rm V}}_{{\rm NS}}(f)
+ (\sum_{{f}} Q_{{f}})^2
r^{{\rm V}}_{{\rm S}}\right]
{}\, .
\end{array}
\end{equation}
A similar decomposition can be derived for $R^{{\rm A}}$:
\begin{equation}
R^{{\rm A}} =
3\left[ \sum_{{f}} a_{{f}}^2 r^{{\rm A}}_{{\rm NS}}(f)
+ \underbrace{\sum_{f,f'}a_{{f}} a_{f'}
r^{{\rm A}}_{{\rm S}}(f,f')}_{r^{{\rm A}}_{{\rm S}}}
\right]
{}\, .
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
{
\begin{figure}[t]
\begin{center}
\begin{tabular}{lll}
\parbox{3cm}{
\epsfig{file=fig2a.eps,width=5.cm,height=5.cm}
}
&
\hphantom{XXXXXXX}
&
\parbox{3cm}{
\epsfig{file=fig2b.eps,width=5.cm,height=5.cm}
}
\end {tabular}
\end{center}
\caption[]{\label{class}
Singlet contribution of order ${\cal O}(\alpha_s^2)$
and ${\cal O}(\alpha_s^3)$.}
\end{figure}
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\noindent
\hspace*{0.3cm}In the limit of massless ${\rm u,d,s}$ and ${\rm c}$
quarks the second term receives contributions from
$f,f'=b$ or ${\rm t}$ only, or  more precisely
 the light
quarks compensate mutually.
The advantage of this decomposition becomes
even more manifest in the limit $m_{{\rm q}}^2/s\rightarrow 0$.
Then the nonsinglet functions
$r^{{\rm V}}_{{\rm NS}}$ and $r^{{\rm A}}_{{\rm NS}}$
are identical and the corrections for nonvanishing, but
small, masses are easily calculated.
%
\subsection{\protect $\beta$ Function and Anomalous Dimensions
\label{renorm} }
In this section several aspects of the
renormalization procedure in QCD are
recalled, which will be of importance
for the subsequent calculations. The
renormalization of various currents and
the corresponding current correlators
will be considered. Green functions
with the insertion of two external
currents require subtractive
renormalization. The corresponding
renormalization constants lead to
anomalous dimensions for the
correlators. The presentation will be
rather short, and for more detail the
reader should consult, for example,
Refs.~\cite{Bog84,JCC84,Zuber80,Yndurain93}.
%[58]
\subsubsection{Renormalization in QCD
\label{QCD}}
The QCD Lagrangian is given by
\begin{eqnarray}
%\begin{array}{rl}
\label{r1}
%\displaystyle
{\cal L }\{{\bf \Phi};\underline{g},\mu\}&= &
%\displaystyle
{\cal L}
\{A_{\mu}^{{a}},\Psi,\overline{\Psi},
C^{{a}},\overline{C}^{{a}};g,{\bf m},\xi,\mu\}
\nonumber\\
\displaystyle
& =& \displaystyle
\frac{1}{4}G^{{a}}_{\mu\nu}G_{{a}}^{\mu\nu}
+\overline{\Psi}(i\not\!\!D {\bf m})\Psi
+ {\cal L}^{\rm GF}
+ {\cal L}^{\rm FP}
{}\, ,
%\end{array}
\end{eqnarray}
\begin{eqnarray}
%\begin{array}{l} \displaystyle
G^{{a}}_{\mu\nu}&=&\partial_\mu A^{{a}}_\nu 
\partial_\nu A^{{a}}_\mu
+
g\mu^\epsilon f^{abc} A^{{\rm b}}_\mu A^{{\rm c}}_\nu \, ,
\nonumber\\
\displaystyle
D_\mu&=&\partial_\mu ig\mu^\epsilon
A^{{a}}_\mu \frac{\lambda^{{a}}}{2}\, ,
\nonumber\\
%\not\!\!D =\gamma_\mu D_\mu
\nabla^{ab}_\mu&=& \delta^{ab}\partial_\mu
 g\mu^\epsilon f^{abc} A^{{\rm c}}
{}\, .
%\end{array}
\label{r1b}
\end{eqnarray}
Here
$f^{abc}$ are the structure constants of the colour
$SU_{{\rm C}}(3)$ group, $\xi$ is the gauge parameter,
$G^{{a}}_{\mu\nu}$
is the
gluon field strength tensor
and $C^{{a}}$ is the ghost fields, with
\begin{equation}
{\cal L}^{\rm GF}
=
\frac{1}{2\xi} (\partial_\nu A_\nu)^2
\ \ \
{\rm and}
\ \ \
{\cal L}^{\rm FP}
=
(\partial_\mu \overline{C})
\nabla_\mu{C}
{}\, .
\label{GFandFP}
\end{equation}
The quark masses are denoted as
${\bf m} = \{m_{{\rm q}}\}$;
$\Psi=\{\Psi_{{\rm q}} q = {\rm u,d,s,c,b,t}\}$
represents the quark fields; while
${\bf \Phi}$ stands for the collection of all
fields, and $\underline{g} = \{g,{\bf m},\xi\}$ for
the `coupling constants'. Anticipating
the use of dimensional regularization,
the unit mass $\mu$ has been introduced in
\re{r1} to keep
the coupling constant $g$ dimensionless even if the
Lagrangian is considered in
$D = 4 2 \epsilon$ dimensions.
A convenient representation of all
(connected) Green functions is provided
by the generating functional
\begin{equation}
Z_{{\rm c}}(I;{\rm\bf \Phi}) =
\left\{
\int [{\rm d}{\bf \Phi}]
\exp \,(
i I + {\bf \Phi}\cdot{\rm\bf \Phi}
)
\right\}_{{\rm c}} \,\, ,
\label{gen.func.def}
\end{equation}
with the normalization condition
$Z_{{\rm c}} (I,0) \equiv 1 $.
Here the action
\begin{equation}
I({\bf \Phi},\underline{g},\mu) =
\int {\cal L}({\bf \Phi},\underline{g}) {\rm d} x
\label{action}
\end{equation}
and
the functional integration
is to be understood in the standard
manner within the perturbation theory
framework.
%\noindent
Finite Green functions
can be constructed from the Lagrangian
Eq.~(\ref{r1}) in three equivalent ways:
The first method is based on the
renormalized Lagrangian,
obtained from the original
one by a rescaling of fields and parameters,
expressing them in terms of renormalized
quantities:
\begin{equation}
{\cal L }_{R}\{{\bf \Phi};\underline{g},\mu\} =
{\cal L } \{Z_3^{\frac{1}{2}}
A_{\mu}^{{a}},Z_2^{\frac{1}{2}}\Psi,
Z_2^{\frac{1}{2}}\overline{\Psi},
\tilde{Z}_3^{\frac{1}{2}}
C^{{a}},\tilde{Z}_3^{\frac{1}{2}}\overline{C}^{{a}},
Z_\xi \xi,
Z_{{g}} g ,{ Z}_{{m}} m\}
\label{r4}
{}\, .
\end{equation}
The explicit form of the
renormalization constants depends on the
renormalization scheme adopted.
The most powerful method, which is
particularly suitable for applications in
QCD, the procedure of dimensional
regularization \cite{dim.rega,dim.regb}
and minimal subtraction
\cite{ms}
is nowadays widely used.
After continuation of the Feynman integrals
to $D=42\epsilon$ spacetime
dimensions divergences reappear as
poles in $\epsilon$.
The renormalization constants
may then be expanded
in the coupling constant
\begin{equation}
\label{r4b}
Z=1+\sum_{i,j}^{0 0$ are neglected.
It is also understood that the effective
Lagrangian without\break\hfill\indent$\,\,\,\,$the
heavy quark field remains
renormalizable. Fortunately, the QCD Lagrangian \re{r1} meets
this\break\hfill\indent$\,\,\,\,$demand. The standard model
however, does not fulfil this requirement.
This leads to the\break\hfill\indent$\,\,\,\,$wellknown deviations
from the
theorem such as the $m_{{\rm t}}^2$ effects in
$\Gamma(Z \to \overline{{\rm b}} {\rm b}) $
or the $\rho$ parameter.}}.
However, a peculiarity of massindependent renormalization schemes
is that the decoupling theorem does not hold
in its na\"{\i}ve form for theories
renormalized in such schemes: the effective QCD
action to appear will not be canonically normalized.
Even worse, large mass logarithms in
general appear when one calculates a
physical observable! (See the example
below.)
\end{itemize}
Fortunately, both problems are controlled once a proper
choice of the expansion parameter is made
and the renormalization group
improvement is performed
\cite{Weinberg80,Ovrut80,Bernreuther82b}
%\[3537].
In order to be specific,
consider QCD with $n'_{f} = n_{{f}} 1$ light quarks
$\psi =
\{
\psi_{{l}} l= 1 \mbox{} n'_{f}
\}
$
with masses
$m =
\{
m_{{l}} l= 1 \mbox{} n'_{f}
\}
$
and one heavy quark $h$ with mass $m_{{\rm h}}$.
The respective action
$I({\bf \Psi},h,\overline{h},g,m,m_{{\rm h}},\mu)$
is determined by
integrating over the spacetime the Lagrangian density
\begin{equation}
\begin{array}{rl}
\displaystyle
{\cal L}
%({{\bf \Psi},h,\overline{h};g,m,m_{{\rm h}}})
= & \frac{1}{4} (G^{{a}}_{\mu\nu})^2
+
\sum_{l}\overline\psi_{{l}}
(i\!\not{\!\! D}  m_{{l}})
{\psi}_{{l}} + \overline h (i\!\not{\!\! D}  m_{{\rm h}}) h
\\
\displaystyle
& + \ \mbox{\rm terms with ghost fields and the gaugefixing
term}.
\end{array}
\label{Lagr}
\end{equation}
%\newpage
In the condensed notation of Section~\ref{renorm}
the collection of fields ${\bf \Phi}$ is now decomposed as
${\bf \Phi} = \{{\bf \Psi},h,\overline{h}\}$
with
${\bf \Psi}=\{\psi,\overline{\psi},A^{{a}}_\mu\}$.
The (renormalized) generating functional
of (connected) Green functions of light fields may now be
written as
\begin{equation}
Z^{{\rm R}}_{{\rm c}}(I;{\rm\bf \Phi},s) =
R_{\overline{{\rm MS}}}
\left\{
\int [{\rm d}{\bf \Psi}{\rm d} h {\rm d} \overline{h}] \exp \,
(i I + {\bf \Psi}\cdot{\rm\bf \Phi}
+ J \cdot s)
\right\}_{{\rm c}} \ .
\label{gen.func}
\end{equation}
%\newpage
Here
$R_{\overline{{\rm MS}}}$
is the ultraviolet $R$operation in
${\overline{{ \mbox{{MS}}}}}$scheme.
${\rm\bf \Phi}$ and ${s}$ are sources for the (light) elementary fields
from ${\bf \Psi}$ and for an external quark
current $J=\overline{\psi}\Gamma\psi$
respectively.
For the sake of notational simplicity we
shall proceed in the Landau gauge
and ignore the ghost field variables.
Integrating out the heavy quark should transform the generating
functional \re{gen.func} to that corresponding to the effective QCD
with $n'_{{f}}$ remaining quark
flavours plus additional higher dimension
interaction terms suppressed by powers of the
inverse heavy mass.
The current $J$ as well as
any other composite operator will generically develop a
nontrivial coefficient function even if one neglects
all powersuppressed terms.
In a more formal language the result
of integrating out the heavy quark may be summarized in the
following master expansion for the generating functional
\re{gen.func}:
\begin{equation}
%\begin{array}{c}
\displaystyle
Z^{{\rm R}}_{{\rm c}}(I;{\rm\bf \Phi},s)
\bbuildrel{=\!=\!=}_{\scriptstyle{m_{{\rm h}}\to\infty}}^{}
R_{\overline{{\rm MS}}}
\left\{
\int [{\rm d}{\bf \Psi}] \exp \,
\left[ i I^{{\rm eff}}({\bf \Psi} , g)
+
{\bf \Psi}'\cdot{\rm\bf \Phi}'
+
\Biggl( J' {\rm z}_{{j}}
+ \sum_{n}\frac{ {\rm z}_{n}}{m_{{\rm h}}^{\delta_{{n}} 3}}
J_{{n}}\Biggl)
\cdot s_\mu\right]
\right\}_c
{}\, ,
%\end{array}
\label{master.exp}
\end{equation}
where the sum is performed over operators
constructed from the light fields,
with the quantum numbers of
those of the initial current $J$ and
of mass dimension $\delta_{{n}}$.
The effective action $I^{{\rm eff}}({\bf \Phi},g)$ can be written as
\begin{equation}
I^{{\rm eff}}({\bf \Psi},g) =
I({\bf \Psi}',g'){{}_{\scriptstyle h=0}}
+
\sum_{{n}}
\int \frac{{\rm z}_{{n}}
\displaystyle O_{{n}}(x)}
{\displaystyle m_{{\rm h}}^{d_{{n}} 4}} {\rm d} x,
\label{effect.action}
\end{equation}
with
\begin{equation}
g' = z_{{g}} g, \ \ \
m_{{\rm q}}' = z_{{m}} m_{{\rm q}}
\label{match1}
\end{equation}
and
${\bf \Psi}' =
\{ \psi'= z_2^{1/2}\psi,(A^{{a}}_\mu)'
= z_3^{1/2} A^{{a}}_\mu\},
\ J' = \overline{\psi'}\Gamma \psi'$.
Here
$\{O_{{n}} \}$ are Lorentz scalars of
dimension
$d_{{n}} > 4$,
again constructed from the (primed) light fields only. At last,
$$
{{\rm\bf \Phi}}' = \{ S_\psi/\sqrt{{\rm z}_2},
S_{\overline{\psi}}/\sqrt{{\rm z}_2},
S_{{\rm A}}/\sqrt{z_3}\}\, ,
%
$$
and
all `normalization constants' $z'$s with various subscripts
are series of the generic form
\begin{equation}
z_? \equiv
\left\{
\begin{array}{ll}
\displaystyle
1 + \sum_{i>0} a^?_{{i}} g^{2 i}
\ \ {\rm if} \ \ ?=\psi,\overline\psi,2,3,g \ \ {\rm or } \ \ m\, ,
\\
\displaystyle
\sum_{i>0} a^?_{{i}} g^{2 i} \ \ {\rm if}
\ \ ? = J,n \, .
\end{array}
\right.
\label{z}
\end{equation}
with {\em finite} coefficients $a_{{n}}$, which are
polynomials in $\ln\, (\mu^2/m_{{\rm h}}^2)$.
The master equation \re{master.exp} requires some comments.
\begin{itemize}
\item
The expansion \re{master.exp} should be understood in the strictly
perturbative sense; once it is performed
it is necessary to utilize the usual
renormalization group methods in order to resum all large logs
of the heavy quark mass (see below).
\item
The master equation \re{master.exp}
governs the $ m_{{\rm h}} \to \infty $ asymptotic behaviour
of {\em all} light Green functions: if one neglects
powersuppressed terms and does not consider extra
current insertions,
then their asymptotic behaviour is completely determined by
a few finite normalization constants. Even more: in the
calculation of physical quantities,
which do not depend on the
normalization of quantum fields,
only two constants remain,
viz. $z_1$ and $z_{{m}}$.
\item
There exist several methods of computing
the finite renormalization constants.
The most advanced approach is based
on the socalled heavy mass expansion
algorithm and will be discussed in
Section \ref{repmtexp}.
\end{itemize}
\subsubsection{Matching Conditions
for $\alpha_s$ and Masses\label{mathcing}}
In this subsection we review the socalled matching conditions
which allow the relating of
the parameters of effective lowenergy
theory without a heavy quark to those of the full theory.
The master equation \re{master.exp}
states that the effective coupling constant $\alpha_s^{\prime} $ and
the (light) quark masses
$m_{{\rm q}}^{\prime}$ are expressed in terms of
those of the full theory, viz.
$\alpha_s $ and $m_{{\rm q}},m_{{\rm h}}$, via Eq. (\ref{match1})
(see \cite{Bernreuther82b})
\begin{eqnarray}
\alpha_s^{\prime}(\mu) &=& \alpha_s(\mu) \, C(\alpha_s(\mu),x)
\label{match2a}
{}\, ,
\\
m_{{\rm q}}^{\prime}(\mu) &=&
m_{{\rm q}}(\mu) \, H(\alpha_s(\mu),x)
{}\, .
\label{match2b}
\end{eqnarray}
Here $x= \ln\,(\overline{m}_{{\rm h}}^2/\mu^2)$ and
the functions $C$ and $H$ exhibit the following
structure:
\begin{eqnarray}
C(\alpha_s,x) &=& 1
+
\sum_{k \geq 1}
C_{k}
\left(
\frac{\alpha_s}{\pi}
\right)^k
\label{alpha2},
\ \
C_{{k}}(x)
=
\sum_{0 \leq i \leq k }
C_{ik} x^{{i}}\, ,
\\
H(\alpha_s(\mu),x)
&=&
1
+
\sum_{k \geq 1}
H_{k}
\left(
\frac{\alpha_s}{\pi}
\right)^{{k}} \, ,
\ \
H_{{k}}(x)
=
\sum_{0 \leq i \leq k }
H_{ik} x^i
\, ,
\label{mass2}
\end{eqnarray}
with $C_{ik}$ and $H_{ik}$ being {\em pure}
numbers.
At present, the functions
$C$ and $H$ are known at twoloop level
\cite{Bernreuther82b,Bernreuther83,timo2}.
%[3739]
and read\footnote{The constant term in $C_2$
is cited according to Ref.~\cite{timo2}, where it has been
recalculated using two\break\hfill\indent$\,\,\,$ different approaches.}
\begin{eqnarray}
C_1 &=& \frac{1}{6}~x\, ,\quad\,
C_2 =
\frac{11}{72} + \frac{11}{24}~x + \frac{1}{36}~x^2\, ,
\label{match3a}
\\
H_1 &=& 0\, , \qquad\,\,
H_2 =
\frac{89}{432} + \frac{5}{36}~x + \frac{1}{12}~x^2
\, \, .
\label{match3b}
\end{eqnarray}
Another useful form of \re{match2a}
is obtained after expressing its r.h.s. in terms of
the pole mass $M_{{\rm h}}$:
\begin{equation}
\alpha'_s(\mu) = \alpha_s(\mu)
\left\{
1
+ \frac{X}{6} \frac{\alpha_s(\mu)}{\pi}
+ \left(\frac{7}{24} + \frac{19 X}{24} + \frac{X^2}{36}\right)
\left[
\frac{\alpha_s(\mu)}{\pi}
\right]^2
\right\}
{}\, ,
\label{match2aPole}
\end{equation}
with $X = \ln\, (M_{{\rm h}}^2/\mu^2)$.
The effective $\alpha'_s$ and the
light quark masses evolve with $\mu$ according to
their own effective RG equations \cite{Ovrut80}.
It is important to
stress that the master equation and hence \re{match2a}
were derived under the requirement that the normalization scale
$\mu$ is much less than $m_{{\rm h}}$. However, once obtained,
Eqs.~(\ref{match2a},~\ref{match2b}), present {\em universal }
relations, valid order
by order in perturbation theory.
This implies that on formal grounds one is free
to choose the {\em matching } value of $\mu= \mu_0$ to
determine the value of, say, $\alpha'_s$ in terms of
the parameters of the full theory. The final result should
not depend on $\mu_0$.
However, in practice, some dependence remains from the
truncation of higherorders.
Thus the problem is completely similar to that
discussed in Section~\ref{running}. The correct prescription,
hence, is to solve the matching conditions
(\ref{match2a},~\ref{match2b})
with $\mu$ fixed somewhere in the vicinity of
$m_{{\rm h}}$ to suppress all mass logarithms.
A popular particular choice is to set $\mu = \overline{m}_{{\rm h}}(\mu)$
and thus nullify all mass logarithms.
The mass $m_{{\rm h}} =
\overline{m}_{{\rm h}}(m_{{\rm h}})$ is sometimes
referred to as {\em scale invariant mass}
of the quark $h$. Finally, one should run the effective coupling
constant and quark masses to a lower normalization scale with
the effective renormalization group equations.
\subsubsection{Matching Equations for
Effective Currents \label{mathcing2}}
{}From a fundamental point of view the treatment
of effective currents
does not differ significantly from the one discussed
for the effective
coupling constant and masses. Moreover, for the customary case of
bilinear quark currents it is even easier: in many instances there
exist some extra constraints like Ward identities which help to fix
the constant $z_{{\rm J}}$. Two cases are of particular interest.
\vspace{2mm}
\noindent
{\em Vector current:} This is the most simple
and wellknown case. For
$ J = \overline{\psi_{{\rm q}}}\gamma_\mu \psi_{{\rm q}} $
one derives from the
vector Ward identity\footnote{An explicit derivation
may be found e.g. in Ref.~\cite{me93}.}
that
\begin{equation}
z_{{\rm V}} \equiv
\left\{
\begin{array}{ll}
1 \ \ \mbox{if \ \ q \ \ \ is a light quark}
\\
0 \ \ \mbox{if \ \ q = h }\, .
\end{array}
\right.
\label{zV}
\end{equation}
Thus the functional form of a (light) vector
quark current is unchanged
after integrating out a heavy quark and rewriting it in terms of the
effective (that is properly normalized) light quark fields.
\vspace{2mm}
\noindent
{\em Axial vector current:}
Here the situation is more complicated
due to the famous axial vector
anomaly. A statement similar to \re{zV} may be proved only
for nonsinglet axial vector current constructed from light quark
fields
\cite{Collins78,CK3,CK4}.
%[4143].
Explicitly, if
$J_{{\rm A}} =
\sum_{l,l'} a_{ll'}\overline{\psi}_{{l}}
\gamma_5 \gamma_\mu \psi_{l'}
$ with
a traceless matrix
$\{ a_{ll'} \}$
then the corresponding effective current
reads
\begin{equation}
J'_{{\rm A}} =
\sum_{l,l'} a_{ll'}\overline{\psi}'_{{l}}
\gamma_5 \gamma_\mu \psi'_{l'}\, .
\label{zA}
\end{equation}
It is understood in \re{zA}
that $\gamma_5$ is treated in a way which does not violate the
(nonanomalous) chiral Ward identity. In fact this requirement is
{\em unmet} if the axial vector currents are minimally renormalized
with the {}'t~HooftVeltman
definition of $\gamma_5$. The necessary
modifications are discussed in Section~\ref{gamma5}.
If, however, one has a nonsinglet combination of light
{\em and } heavy diagonal axial vector currents
then there are no simple
formulas like \re{zV} and \re{zA}: the
resulting effective current is
in general
not
a nonsinglet
combination of some light axial vector currents. This case
is discussed in Refs.~\cite{Collins78,CK3}.
\subsubsection{Power Suppressed Corrections\label{power}}
The apparatus of the effective theory also allows
the taking into account of power suppressed corrections.
These can in turn be separated into the corrections
to the effective Lagrangian and an effective
current. Below we list for illustrative purpose
some wellknown results.
\vspace{2mm}
\noindent
{\em QCD Lagrangian}
\noindent
The least power suppressed contribution to the
sum in \re{effect.action} is given by a fourquark
operator of dimension 6 (see Ref.~\cite{NSVZ84}), viz.
\begin{equation}
\frac{\alpha_s'^2}{15m_{{\rm h}}^2}
\sum_{ll'}
(\bar\psi'_{{l}} \gamma_\alpha t^{{a}}\psi'_{l})
\,
(\bar\psi'_{l'} \gamma_\alpha t^{{a}}\psi'_{l'})
{}\, .
\label{4quark}
\end{equation}
Here the colour group generators $t^{{a}}$ are
normalized in the standard way
$
Tr(t^{{a}} t^{{\rm b}}) = \delta^{ab}/2
{}.
$
\vspace{2mm}
\noindent
{\em Vector and axial vector currents}
\noindent
The formulae look almost identical
for vector and axial vector (nonsinglet) currents
(if, of course, the ``correct" treatment of $\gamma_5$ is
employed, see above and Section~\ref{gamma5}).
For the case most useful in
practice, namely that of a massless light quark (axial)
vector current, one obtains \cite{me93}
\begin{equation}
\begin{array}{c}
\displaystyle
\overline\psi_{{l}}\gamma_\mu(\gamma_\mu\gamma_5)\psi_{{l}}
\bbuildrel{=\!=\!=}_{{\scriptstyle{m_{{\rm h}}\to\infty}}}^{}
\overline{\psi'}_{{l}}\gamma_\mu(\gamma_\mu\gamma_5)\psi _{{l}}'
\\
\displaystyle
+
\left\{
\frac{1}{135}\ln\,
\left(
\frac{\mu^2}{m_{{\rm h}}^2}
\right)
\frac{56}{2025}
\right\}
\left(
\frac{\alpha_s'}{\pi}
\right)^2
\frac{\partial^2}{m_{{\rm h}}^2}
[\overline{\psi'_{{l}}}\gamma_\mu(\gamma_\mu\gamma_5)\psi'_{{l}}]
+ O(\alpha_s^3)
+ O(1/m_{{\rm h}}^4)
{}\, .
\end{array}
\label{current.exp}
\end{equation}
\subsubsection{Example\label{example}}
To provide an example of a peculiar realization of
the decoupling theorem in MSlike schemes we now
discuss the evaluation of a `physical'
quantity  the pole mass $M_{{l}}$ of a light
quark $q_{{l}}$ in the full and the effective theories.
First of all we recall that the pole mass is defined as the position
of the pole of the quark propagator computed in
perturbation theory. It is a renormalization scheme and gauge
invariant object \cite{Tarrach81,Nar87}
whose numerical value should
obviously not depend on the theory
 full or effective 
in which it is evaluated in.
The result of the evaluation of $M_{{l}}$ in the
${\overline{{ \mbox{{MS}}}}}$scheme at twoloop
level in the QCD with a heavy quark
reads  the formula below is in fact just an inversion of
\re{mfromM}:
\begin{eqnarray}
%\begin{array}{c}
%\displaystyle
\!\!\!\!\!M_l
&=&
{m_{{l}}(\mu)}\left\{
\rule{0mm}{5mm}
\right.
\!\!1 +
\frac{\displaystyle\alpha_s(\mu)}{\displaystyle \pi}
\left(
\frac{4}{3} + \ln\, \frac{\mu^2}{m_{{l}}^2}
\right)
\label{Mfromm}
\\
\nonumber
\displaystyle
\rule{8mm}{3mm}
&&+
\left[
\frac{\displaystyle\alpha_s(\mu)}{\displaystyle \pi}
\right]^2
\left[
K_{{l}}({\bf m})
\frac{8}{3}
+
\left.
\left(
\frac{173}{24}  \frac{13}{36}~n_f
\right)
\ln\, \frac{\mu^2}{m_{{l}}^2}
+
\left(
\frac{15}{8}  \frac{1}{12}~n_f
\right)
\ln^2 \frac{\mu^2}{m_{{l}}^2}
\right]
%+{\cal O}(\alpha_s^3)
\right\}
{} .
%\end{array}
\end{eqnarray}
If $m_{{\rm h}} \to \infty$
then, according to \re{asymp1},
the function $K_{{l}}({\bf m})$ behaves as
\nonumber
$\ln^2(m_{{\rm h}}/m_{{l}})/3$ and thus the r.h.s. of
Eq.~\re{Mfromm} is {\em not} well defined!
This is, of course, a manifestation of the fact that
in this limit the initial parameters of the full theory
are not adequate to construct the perturbative theory
expansion for a low energy quantity.
However, using the relations
\re{match2a} and \re{match2b} and expressing
the r.h.s of
\re{mfromM} in terms of the {\em effective}
$\alpha'_s$ and ${\bf m}'$, the resulting expression becomes
welldefined at the $m_{{\rm h}} \to \infty$ limit and
reads
\begin{eqnarray}
\displaystyle M_l
&=&
m'_{{l}}(\mu)
\left\{
\rule{0mm}{5mm}
\right.
\!\!1
+
\frac{\alpha'_s}{\pi}
\left[\frac{4}{3}
+
\ln\, \frac{\mu^2}{(m'_{{l}})^2}
\right]
\nonumber
\\
&&+
\left(
\frac{\alpha'_s}{\pi}
\right)^2
\left[
\frac{3049}{288}
+ \frac{2}{3}(2 + \ln\, 2)\zeta(2)

\frac{1}{6} \zeta(3)
 \frac{n_{{f}}'}{3}
\left(
\zeta(2) + \frac{71}{48}
\right)
\right.
%\nonumber
\label{Mfrommprime}
\\
\nonumber
&&+~
\frac{4}{3} \sum_{1 \leq f \leq n_{{f}}'}
\Delta\left(\frac{m_{{f}}}{m'_{{l}}}\right)
+
\Biggl(\frac{173}{24}  \frac{13}{36}~n'_{{f}}\Biggl)\ln\,
\frac{\mu^2}{(m'_{{l}})^2}
+\Biggl(\frac{15}{8}  \frac{1}{12}~n'_{{f}}\Biggl)\ln^2
\frac{\mu^2}{(m'_{{l}})^2}
\Biggl]
\Biggl\}
{}\, .
%\label{Mfrommprime}
\end{eqnarray}
Now it can be easily seen that \re{Mfrommprime}
is nothing but \re{Mfromm} written in the
effective theory with the decoupled heavy quark!
\subsection{\label{Qmasses}Quark Masses}
In this section we briefly discuss
the presently available numeric values of
pole and running quark masses at different
scales. The exposition below serves to explain and
motivate the choice of the input quark masses in the numerical
discussion of Part~\ref{numerical}.
It is not intended to
provide a comprehensive review of this involved
issue (for some recent reviews see, for example, Refs.
\cite{Narison94a,Narison94b}).
\subsubsection{\label{light} Light ${\rm u},{\rm d}$
and ${\rm s}$ Quarks}
For a light quark ${\rm q}= {\rm u,d,s}$
the concept of the pole mass
$M_{{\rm q}}$ is
clearly meaningless, at least in the framework of
the perturbative definition given above.
In contrast, the running
mass $\overline{m}_{{\rm q}}$ is well defined, provided the scale parameter
$\mu$ is not too small. Traditionally, the reference scale
$\mu$ is taken to be 1 GeV.
The latest available values for these masses are
\begin{eqnarray}
\overline{m}_{{\rm u}}(1 \mbox{{\rm GeV}}) + \overline{m}_{{\rm d}}
(1 \mbox{{\rm GeV}}) &=& (12 \pm 2.5) \mbox{{\rm MeV}}, \qquad
\frac{\displaystyle
\overline{m}_{{\rm u}}}{\displaystyle \overline{m}_{{\rm d}}} = 0.4
\pm 0.22~,
\label{ud:quarks}
\\
\overline{m}_s(1 \mbox{{\rm GeV}}) &=& 189 \pm 32 \mbox{{\rm MeV}}~.
\label{s:quark}
\end{eqnarray}
(These values \re{ud:quarks} and \re{s:quark}
are cited according to Refs.
%[4749];
\cite{Rafael94,Jamin94,Karl94};
some earlier
determinations can be found in
%Refs.~[5054]).
Refs.~\cite{Gasser82,Kataev83,Rafael87,Narison89,Paver}).
In all of the applications considered in the present work, it is clearly
more than legible to consider ${\rm u}$ and ${\rm d}$
quarks as massless.
Also, the strange quark mass
will be neglected
everywhere except for
small corrections induced by $m_{{\rm s}}$ in the relation
between pole and running masses of
${\rm c}$ and ${\rm b}$ quarks, respectively,
as discussed below.
\subsubsection{\label{CandB} Charm and Bottom}
Within the effective fourquark theory
the relation between the pole and the running
masses of the charmed quark reads
(it is, in fact, Eq.~\re{Mfromm}
with $n_{{f}}=4 $)
\begin{eqnarray}
\displaystyle
M_c
&=&
{\overline{m}_{{\rm c}}(\mu)}\left\{
\rule{0mm}{5mm}
\right.
\!\!1~+
\frac{\displaystyle\alpha_s(\mu)}{\displaystyle \pi}\Biggl(\frac{4}{3}
+ \ln\, \frac{\mu^2}{\overline{m}_{{\rm c}}^2}\Biggl)
%\nonumber
\label{Mcfrommc}
\\
&&+
\displaystyle
%\rule{8mm}{3mm}
\left[
\frac{\displaystyle\alpha_s(\mu)}{\displaystyle \pi}
\right]^2
\left[
10.319 + \frac{4}{3}
\Delta\left(\frac{\overline{m}_s}{\overline{m}_{{\rm c}}}
\right)
\left.
+
\frac{415}{72}\ln\, \frac{\mu^2}{\overline{m}_{{\rm c}}^2}
+
\frac{37}{24}\ln^2 \frac{\mu^2}{\overline{m}_{{\rm c}}^2}
\right]
\right\}
\, .
\nonumber
\end{eqnarray}
%
This equation may be used in two ways. First,
if one is given a value of $\overline{m}_{{\rm c}}(\mu)$
then \re{Mcfrommc} may be used to construct
$M_{{\rm c}}$ in the following way. One runs
$\overline{m}_{{\rm c}}(\mu)$
(using RG equations in the $n_{{f}}=4$ theory)
to find the scale invariant mass
$m_{{\rm c}} =
\overline{m}_{{\rm c}}(m_{{\rm c}})$, and then evaluates the
r.h.s. of \re{Mcfrommc} with $\mu = m_{{\rm c}}$.
Second, let us suppose that $M_{{\rm c}}$ is known and we would like
to find the running mass $\overline{m}_{{\rm c}}(\mu)$ at some reference
point $\mu$. Even in this case the use of \re{Mcfrommc}
is preferable to that of \re{mfromM}, as the latter
would contain a contribution proportional to the
illdefined pole mass of the strange quark.
In the case of the ${\rm b}$ quark the relation
\re{mfromM} assumes the following form
(all running masses
and the coupling constant are now defined
in the $n_{{f}}=5$ effective
QCD)
\begin{eqnarray}
\displaystyle
\!\!\!\!\!M_b
&=&
{\overline{m}_{{\rm b}}(\mu)}\left\{
\rule{0mm}{5mm}
\right.
\!\!1~
+
\frac{\displaystyle\alpha_s(\mu)}{\displaystyle \pi}\Biggl(\frac{4}{3}
+ \ln\, \frac{\mu^2}{\overline{m}_{{\rm b}}^2}\Biggl)
%\nonumber
\label{Mbfrommb}
\\
\displaystyle
\rule{8mm}{3mm}
&&+
\left[
\frac{\displaystyle\alpha_s(\mu)}{\displaystyle \pi}
\right]^2
\left[
9.278
+ \frac{4}{3}
\Delta\left(\frac{\overline{m}_s}{\overline{m}_{{\rm b}}}\right)
+ \frac{4}{3}
\Delta\left(\frac{\overline{m}_{{\rm c}}}{\overline{m}_{{\rm b}}}
\right)
\left.
+
\frac{389}{72}\ln\, \frac{\mu^2}{\overline{m}_{{\rm b}}^2}
+
\frac{35}{24} \ln^2 \frac{\mu^2}{\overline{m}_{{\rm b}}^2}
\right]
\right\}
\, .
\nonumber
%\label{Mbfrommb}
\end{eqnarray}
This equation is to be used in the same way as \re{Mcfrommc}.
In the literature there is a variety of somewhat different
results for the masses of ${\rm b}$ and ${\rm c}$ quarks.
Also, there exist strong indications that
the very concept of the pole mass is plagued
with severe nonperturbative ambiguities \cite{Shifman94}.
It may well happen that
eventually the most accurate and unambiguous
mass parameter related to a quark will be its running
mass taken at some convenient reference point.
However, for illustrative purposes we will use
the following {\em ansatz} for the pole masses
$M_{{\rm c}}$ and $M_{{\rm b}}$:
\begin{equation}
M_{{\rm c}} = 1.6 \pm 0.10 \mbox{{\rm GeV}} \qquad {\rm {\rm and }}
\qquad M_{{\rm b}} = 4.7 \pm 0.2 \mbox{{\rm GeV}}
{}\, .
\label{cb:mass}
\end{equation}
The central values and uncertainty bars in \re{cb:mass} are in broad
agreement with Refs. \cite{Narison94b,Titard94,Voloshin95} and also
with those used by the Electroweak Precision Calculation Working Group
\cite{DimaBardin}.
%Table~\ref{QmassesTab1} shows the running masses
Table~1 shows the running masses
obtained from (\ref{Mcfrommc},\ref{Mbfrommb}),
and RG equations at various relevant scales in dependence on
$\alpha_s(M_{{Z}})$ with $ M_{{Z}} = 91.188 $ GeV.
\begin{table}
\begin{center}
{\bf Table 1}
\end{center}
\vskip0.3cm
%\caption{ \label{QmassesTab1}
{\small{Values of
$\protect \Lambda_{{\overline{{ \mbox{{\scriptsize MS}}}}}}^{(5)} $,
$\protect \Lambda_{{\overline{{ \mbox{{\scriptsize MS}}}}}}^{(4)} $,
$\protect \overline{m}_{{\rm c}}^{(4)}(M_{{\rm c}})$,
$\protect \overline{m}_{{\rm c}}^{(5)}(M_{{\rm b}})$,
$\protect \overline{m}_{{\rm b}}^{(5)}(M_{{\rm b}})$,
$\protect \overline{m}_{{\rm b}}^{(5)}(M_{{Z}})$
and
$\protect {m}_{{\rm b}}(m_{{\rm b}})$
(in GeVs)
for different values of
$\protect \alpha_s^{(5)}(M_{{Z}})$,
and the default values of
$M_{{\rm c}}$ and $M_{{\rm b}}$ as in
(\protect\ref{cb:mass}).}}
\begin{center}
\vskip0.2cm
\begin{tabular}{cccccccc}
\hline
$\alpha_s^{(5)}(M_{{Z}})$
&
$\Lambda_{{\overline{{ \mbox{{\scriptsize MS}}}}}}^{(5)}$
&
$\Lambda_{{\overline{{ \mbox{{\scriptsize MS}}}}}}^{(4)}$
&
$\overline{m}^{(4)}_{{\rm c}}(M_{{\rm c}})$
&
$\overline{m}^{(5)}_{{\rm c}}(M_{{\rm b}})$
&
$\overline{m}^{(5)}_{{\rm b}}(M_{{\rm b}})$
&
$\overline{m}^{(5)}_{{\rm b}}(M_{{Z}})$
&
${m}_{{\rm b}}(m_{{\rm b}})$
\\ \hline\hline
%
%\input{masstab1.tex}
%
0.11$\phantom{0}$ & 0.129 & 0.188 & 1.27$\phantom{0}$ &
1.03$\phantom{0}$ & 4.13 & 3.01 & 4.20\\ \hline
0.115 & 0.175 & 0.248 & 1.21$\phantom{0}$ &
0.953 & 4.07 & 2.89 & 4.15\\ \hline
0.12$\phantom{0}$ & 0.233 & 0.32$\phantom{0}$ & 1.12$\phantom{0}$ &
0.855 & 3.99 & 2.77 & 4.10\\ \hline
0.125 & 0.302 & 0.403 & 1.01$\phantom{0}$ &
0.734 & 3.91 & 2.64 & 4.04\\ \hline
0.13$\phantom{0}$ & 0.383 & 0.499 & 0.853 &
0.583 & 3.82 & 2.5$\phantom{0}$ & 3.97\\ \hline
\end{tabular}
\end{center}
\end{table}
\subsubsection{\label{topmass} Top}
The top quark mass value as reported by the CDF collaboration
\cite{TOP} is
\begin{equation}
M_{{\rm t}} = 174~\pm 10^{+13}_{12} \mbox{{\rm GeV}}
{}\, .
\label{top:mass}
\end{equation}
In our numerical discussions we shall use a conservative
input value of $M_{{\rm t}} = 174~\pm$\break\hfill 20 GeV.
In order to find the corresponding
running mass we use
the equation below obtained from \re{mfromM}
(we deal now with the fullyfledged
$n_{{f}} =6 $ theory, and
discard completely negligible terms caused by
the masses of ${\rm s}$ and ${\rm c}$ quarks)
\begin{eqnarray}
\displaystyle
\overline{m}_{{\rm t}}(\mu)
&=&
M_{{\rm t}}\left\{
\rule{0mm}{5mm}
\right.
\!\!1  \frac{\displaystyle\alpha_s(\mu)}{\displaystyle \pi}
\left(
\frac{4}{3} + \ln\, \frac{\mu^2}{M_{{\rm t}}^2}
\right)
\label{}
%\nonumber
\\
&&
\displaystyle
%\rule{8mm}{3mm}
\left[
\frac{\displaystyle\alpha_s(\mu)}{\displaystyle \pi}
\right]^2
\left[
9.125
+
\frac{4}{3} \Delta\left(\frac{M_{{\rm b}}}{M_{{\rm t}}}\right)
\left.
+
\frac{35}{8}\ln\, \frac{\mu^2}{M_{{\rm t}}^2}
+
\frac{3}{8} \ln^2 \frac{\mu^2}{M_{{\rm t}}^2}
\right]
\right\}
\, .
\nonumber
\end{eqnarray}
After setting $\mu = M_{{\rm t}}$ and
evaluating $\alpha_s^{(6)}(M_{{\rm t}})$
one finds
\[
\alpha_s^{(6)} (M_{{\rm t}}) = 0.109 , \ \
\overline{m}_{{\rm t}} (M_{{\rm t}}) = 164 \mbox{{\rm GeV}}~,
\]
for our default value
$\alpha_s^{(5)}(M_{{Z}}) = 0.120$ corresponding
to $\Lambda^{(5)}_{{\overline{{ \mbox{{\scriptsize MS}}}}}} =
233 \mbox{{\rm MeV}} $.
\section{Calculational Techniques \label{calctech}}
In this Part we discuss available calculational
techniques to perform small and heavy mass expansions
of twopoint correlators as well as the problem
of $\gamma_5$ in dimensional regularization.
\subsection{Current Correlators at Large Momentum
\label{large}}
A lot of results on higherorder radiative corrections were
derived after neglecting quark masses, originating
from massless diagrams and resulting in a drastic
simplification of calculations. However, problems arise when quark
masses are taken into account, at least in the form of power
corrections. In the simplest cases
the evaluation of, say, a quadratic quark mass correction
may be reduced to the computation of
massless diagrams which are obtained by na\"{\i}vely
expanding the massive propagators in the quark mass.
However, this strategy fails in the general case starting
from quartic mass terms.
The socalled
logarithmic mass singularities appear and render
the simple Taylor expansion meaningless.
In this section the general structure of
nonleading mass corrections will be discussed as well as
some approaches for their evaluation.
%The presentation is mainly based on
%Refs.~[5760].
In investigating the asymptotic behaviour of various correlators at
large momentum transfer, it proves to be very useful to employ
the Wilson expansion in the framework of the MS scheme.
Consider vector current correlator
\begin{equation}
i \int \langle 0
T J_\mu(x) J_\nu(0)
0\rangle
{\rm e}^{iqx} {\rm d} x
= (g_{\mu\nu}  q_\mu q_\nu/q^2) \ \Pi(Q^2)
\, ,
\label{hm1}
\end{equation}
with
$J_\mu=\overline q \gamma_\mu q $.
Here ${{\rm q}}$ is a quark with
mass $m_{{\rm q}}\equiv m$. To simplify the
following discussion
we will consider the second
derivative $\Pi''(Q^2) \equiv d^2\Pi(Q^2)/d(Q^2)^2$, which can be
seen from \re{cc4} to satisfy a homogeneous RG equation,
\begin{equation} \mu^2 \frac{d}{d\mu^2}\Pi''(Q^2) = 0\,.
\label{hm2}
\end{equation}
The high energy behaviour of $\Pi''(Q^2)$ in the deep Euclidean
region may be reliably evaluated in QCD by employing the operator
product expansion:
%%%EQUATION 9
\begin{eqnarray}
& &Q^2\Pi''(Q^2,\alpha_s,m ,\mu)
\bbuildrel{=\!=\!\Longrightarrow}_{\scriptstyle{Q^2\to\infty}}^{}
K_0(Q^2,\alpha_s,m ,\mu)\, {\rm {\bf \large 1}}\nonumber\\
& & + \sum_{n}\frac{1}{(Q^2)^{n/2}}\sum_{dim\, O_{{i}} = n}
K_{{i}}( Q^2,\alpha_s,m ,\mu)
\langle 0 O_{{i}} (\mu)0\rangle\,.
\label{hm9}
\end{eqnarray}
We have explicitly separated the
contribution of the unit operator from
that of the operators with nontrivial
dependence on the field variables. The
coefficient functions $K_0$ and $K_{{i}}$
depend upon the details of the
renormalization prescription for the
composite operators $O_{{i}}$. The usual
procedure of normal ordering for the
composite operators appearing on the
r.h.s. of Eq.~\re{hm9} becomes
physically inconvenient if quark mass
corrections are to be included
\cite{CheSpi88}. From the
calculational viewpoint it also does
not lead to any insight in computing
powersuppressed mass corrections
involving mass logarithms.
Indeed, the coefficient function in
front of the unit operator in \re{hm9}
represents the usual perturbative
contributions and, if normal ordering
is used, it contains in general mass
and momentum logarithms of the form
\begin{equation}
\left(
\frac{m ^2}{Q^2}
\right)^{{n}}
\left(\ln\,\frac{\mu^2}{Q^2}
\right)^{n_1}
\left(
\ln\,\frac{\mu^2}{m ^2}
\right)^{n_2},
\label{hm10}
\end{equation}
with $n, \, n_1$ and $n_2$ being nonnegative integers.
More specifically, one can write
%%%EQUATION 11
\begin{equation}
K^{NO}_0(Q^2,\alpha_s,m ,\mu)
\bbuildrel{=\!=\!\Longrightarrow}_{\scriptstyle{Q^2\to\infty}}^{}
\sum_{n \ge 0,\, l > 0}
\left(\frac{m ^2}{Q^2}
\right)^{n}
\left(\frac{\alpha_s}{\pi}
\right)^{l  1}
F_{nl}(L,M)\,,
\label{hm11}
\end{equation}
where
$L = \ln\,(\mu^2/Q^2)$, $M = \ln\,(\mu^2/m ^2)$, and
the superscript NO is a reminder of the normal ordering prescription
being used.
The function $F_{nl}(L,M)$ corresponds to the contribution of the
$l$loop diagrams, and is a polynomial of degree not higher than $l$,
in both $L$ and $M$. The contributions due to
nontrivial operators  that is containing some dependence on
field variables  are completely decoupled from those of the unit
operator if the normal ordering is
employed, since the vacuum expectation value vanishes
for every nontrivial operator $O$:
\[
\langle 0 O 0\rangle \equiv 0 \, .
\label{hm3}
\]
The situation improves drastically if one abandons the normal
ordering prescription. It was realized some time ago
%Refs.~[5760]
\cite{mto0,BroadGen84,CheSpi88}
that all logarithms of quark masses may be
completely shifted to the vacuum expectation values (VEV) of
nontrivial composite operators appearing on the r.h.s. of \re{hm9}
if the latter are minimally subtracted.
To give a simple example, let us consider the correlator \re{hm1} in
the lowest order oneloop approximation. First, we use the normal
ordering prescription for the composite operators which appear
in the OPE of the time ordered product in \re{hm1}. To determine the
coefficients of the various operators, one possible method is to
sandwich both sides of the OPE between appropriate external states.
By choosing them to be the vacuum, only the unit operator {\bf 1}
will contribute on the r.h.s., {\em if} the normal ordering
prescription is used. This means that the
bare loop of Fig.~\ref{highmom}a
contributes entirely to the coefficient $K_0$ in \re{hm9}. A simple
calculation gives (in the sequel we
neglect all terms of order $1/Q^6$ and higher):
\begin{figure}
\begin{center}
\begin{tabular}{cc}
\subfigure[]{\epsfig{file=fig1la.eps,width=5cm,height=5cm}}
&
\subfigure[$\langle \overline{\Psi}\Psi \rangle $]%
{\epsfig{file=fig1lb.eps,width=5cm,height=5cm}}
\end{tabular}
\end{center}
\caption {
\label{highmom}
(a) Lowest order contribution to the correlator $\Pi''(Q^2)$.
(b) Vacuum diagram contributing to the perturbative VEV of the
operator $\bar q q $.
}
\end{figure}
\begin{equation}
Q^2\Pi''(Q^2)
\bbuildrel{=\!=\!\Longrightarrow}_{\scriptstyle{Q^2\to\infty}}^{}
K_0(Q^2)\, {\rm {\bf \large 1}}
%
+ \frac{4}{Q^4}
\langle 0 m\overline{q} q 0\rangle\, ,
\label{hm11a}
\end{equation}
\begin{equation}
K_0^{NO}(Q^2,m ,\mu) = Q^2\Pi''(Q^2,m ,\alpha_s,\mu)
{{}_{{}_{\scriptstyle \alpha_s =0}}}
=\frac{1}{4\pi^2}
\left[1 + \frac{12 m^4}{Q^4}(1 + L  M)\right].
\label{hm12}
\end{equation}
The coefficient function $K_0^{NO}$
contains mass singularities (the Mterm).
On the other hand, if one does
not follow the normal ordering prescription,
then the operator $m \overline{q} q $
develops a nontrivial vacuum expectation
value even if the quark gluon interaction is turned off by
setting $\alpha_s = 0$.
Indeed, after minimally removing its pole singularity,
the one loop diagram of Fig.~\ref{highmom}b
leads to the following result \cite{Broad81}:
%%%EQUATION 13
\begin{equation}
\langle 0\bar q q0\rangle^{{\rm PT}}=\frac{3m^3}{4\pi^2}
\left(\ln\, \frac{\mu^2}{m^2}+1\right)\,.
\label{hm13}
\end{equation}
By inserting this into \re{hm9}, the new coefficient
function $K_0$ can be
extracted, with the result:
%%%EQUATION 14
\begin{equation}
K_0 = \frac{1}{4\pi^2}
\left[1 + \frac{12 m^4}{Q^4} (2 + L)\right].
\label{hm14}
\end{equation}
The mass logarithms are now completely transferred from the CF $K_0$ to
the VEV of
the quark operator \re{hm13}! The same phenomenon continues to hold
even after the $\alpha_s$ corrections are taken into account for
(pseudo)scalar and pseudovector correlators, independently of their
flavour structure \cite{BroadGen87,Gen89}.
The underlying reason for this was first
found in Ref.~\cite{me82b}.
There it was shown that
no coefficient function can depend on mass logarithms in every
order of perturbation theory if the minimal subtraction procedure is
scrupulously observed\footnote{In particular it also excludes the
normal ordering, as the latter amounts to a specific
nonminimal
\break\hfill\hspace*{0.5cm}
subtraction of diagrams contributing to
VEV's of composite operators.}.
This is true
irrespective of the specific model
and correlator under discussion.
Three important observations
may be made in
this context:
\begin{itemize}
\item
Prior knowledge of the fact that any conceivable correlator
can be expanded in a series of the form (\ref{hm10}) makes
it possible to obtain without calculation important
information on the structure of mass logs as they
appear in various correlators. For example, in QCD any
correlator should contain no mass logarithms in the quadratic
in mass terms \cite{BroadGen84,CheSpi88}.
This holds true because there
does not exist a gaugeinvariant nontrivial operator of
(mass) dimension two in QCD.
\item
From the purely calculational point of view the problem
of computing nonleading mass corrections to current
correlators becomes much simpler. This is due to two facts.
First, all coefficient functions are
expressed in terms of {\em massless } Feynman integrals
while VEV's of composite operators are by definition
represented in terms of some massive integrals without
external momenta (tadpole diagrams).
Second, methods have been elaborated for
computing analytically both types of Feynman integrals.
% (see Refs.~\cite{aip92}).
\item
The abandonment of the normal ordering slightly complicates
the renormalization properties of composite operators. An
instructive example is provided by the `quark mass operator'
$O_2 = m \overline q q$.
The textbook statement (see, for example, Ref.~\cite{JCC84})
that this operator is RG invariant is no longer valid.
Indeed, the vacuum diagram of Fig.~\ref{highmom}b
has a divergent part which
has to be removed by a new counterterm proportional to the
operator $m^4{\rm {\bf \large 1}}$. In other words,
$m\bar qq$ begins to mix with the `operator'
$m^4{\rm {\bf \large 1}}$
\cite{CheSpi88}.
\end{itemize}
To lowest order, the corresponding anomalous dimension matrix reads:
%%%EQUATION 16
\begin{equation}
\mu^2\frac{d}{d\mu^2}
\left(
\begin{array}{c}
m\bar q q\\
m ^4
\end{array}\right)=
\left(\begin{array}{cc}
0 & \frac{\displaystyle 3}{\displaystyle 4\pi^2} \\
0 & 4\frac{\displaystyle \alpha_s}{\displaystyle \pi}
\end{array}\right)
\left(\begin{array}{c}
m\bar q q \\
m^4
\end{array}\right)\,.
\label{hm16}
\end{equation}
The nonvanishing, offdiagonal matrix element describes the mixing of
the two operators under renormalization and was obtained from the
divergent part of the vacuum diagram in Fig.~\ref{highmom}.
The diagonal matrix
elements are just the anomalous dimensions of the respective
operators in the usual normalordering scheme. The lower one is equal
to $4\gamma_{{m}}(\alpha_s)$. Note that the general structure of
the anomalous dimension matrix of all gaugeinvariant operators of
dimension four has been established in Refs.~\cite{Spi84,CheSpi88}.
This information was used recently
\cite{TTP9408} to evaluate the corrections
of order $m_{{\rm q}}^4 \alpha_s^2$ to the
vector current correlator (see Section~\ref{repnsm4}).
\subsection{Top Mass Expansion in $s/m_{{\rm t}}^2$\label{repmtexp}}
Our discussion of the dependence of crosssections and decay rates on
the quark masses has up to now dealt with five flavours
light enough to be produced in ${\rm e}^+{\rm e}^$ collisions.
The top quark, on
the other hand, is too heavy to be present in the final state, even at
LEP energies. Nevertheless it constitutes a virtual particle.
Virtual top loops appear
for the first time in second order $\alpha_s^2$. Massive multiloop
integrals may conveniently be simplified considering the heavy top
limit $m_{{\rm t}} \rightarrow \infty$.
In this effective field theory
approach the top is integrated out from the theory. Then the
Lagrangian of the effective theory contains only light particles.
The effects of the top quark are accounted for through the
introduction of additional operators in the effective Lagrangian. For
the vector current correlator their contributions are suppressed by
inverse powers of the heavy quark mass $s/m_{{\rm t}}^2$.
As we will also
explicitly see,
no decoupling is operative in the case of the
axial vector correlator. A logarithmic top mass dependence signals
the breakdown of anomaly cancellation if the top quark is removed
from the theory.
The heavy mass expansion is constructed as follows
(see Refs.~\cite{GorLar87}\cite{PivTka93};
a rigorous mathematical formulation can be
found in Ref.~\cite{Smi91}):
Let the Feynman integral $\langle
\Gamma\rangle$ of a Feynman graph $\Gamma$
depend on a heavy mass $M$ and some other `light' masses
and external momenta which we will generically denote as
$m$ and $q$ respectively. In the limit $M \to \infty$
with $q$ and $m$ fixed $\langle \Gamma \rangle$
may be represented by
the asymptotic expansion:
\begin{equation}
\langle \Gamma \rangle
\bbuildrel{=}_{{\scriptstyle{m_{{\rm t}} \to\infty}}}^{}
\sum_{\gamma}
C^{(t)}_{\gamma}
\star \langle \Gamma /\gamma\rangle^{{\rm eff}}~.
\label{t1}
\end{equation}
The diagrams $\langle \Gamma /\gamma\rangle^{{\rm eff}}$ of the
effective theory consist of light particles only, whereas the top
mass is only present in the `coefficient functions'
$C^{(t)}_{\gamma}$. The notation $\langle \Gamma
/\gamma\rangle^{{\rm eff}}$ means that
the hard subgraph $\gamma$ of the
original diagram $\Gamma$ is contracted to a blob. By definition a
hard subgraph contains at least all heavy quark lines and becomes one
particle irreducible if each top quark propagator is contracted to a
point. The Feynman integral of the hard subgraph is expanded in a
formal (multidimensional) Taylor expansion with respect to the small
parameters, namely the light masses and the external momenta of $\gamma$.
It should be noted that the set of external momenta for a subgraph
$\gamma$ is defined {\em with respect to} $\gamma$ and thus in general
consists of some genuine external momenta (that is, those shared by
$\gamma $ and the very diagram $ \Gamma $) as
well as momenta flowing through {\em internal} lines of
$ \Gamma $,
which are {\em external ones } of $ \gamma $ (see the
example below). This Taylor series $C^{(t)}_{\gamma}$ is inserted in
the effective blob and the resulting Feynman integral has to be
calculated. All possible hard subgraphs have to be identified and
the corresponding results must be added.
The prescription for the construction of the coefficent function
$C_\gamma^{(t)}$ for a hard subgraph $\gamma$ can be formulated as
follows: Suppose the Feynman integral
$\langle \gamma\rangle (M,{\bf q}^\gamma,{\bf m}^\gamma,\mu)$ corresponds
to a hard subgraph $\gamma$ and
depends on external momenta ${\bf q}^\gamma$
and light masses ${\bf m}^\gamma$ in addition
to the heavy mass $M$. Then
\begin{equation}
C_\gamma^{(t)} =
t_{\{{\bf q}^\gamma, {\bf m}^\gamma\}}
\langle \gamma\rangle
(M,{\bf q}^\gamma,{\bf m}^\gamma,\mu)
\, ,
\label{t2}
\end{equation}
where the operator $t_{\{x_1,x_2\dots\}}$ performs the formal Taylor
expansion according to the rule:
\begin{eqnarray}
t_{\{x_1,x_2\dots\}} &=&
\sum_{n \ge 0}^\infty t_{\{x_1,x_2\dots\}}^{(n)} \, ,
\label{t3a}
\\
t_{\{x_1,x_2\dots\}}^{(n)}
F(x_1,x_2\dots)
&\equiv&
\frac{1}{n!}
\left( \frac{{\rm d}}{{\rm d} \xi} \right)^n
F(\xi x_1,\xi x_2\dots) _{{}_{\scriptstyle \xi = 0}}
{}\,\, .
\label{t3b}
\end{eqnarray}
Here several comments are in order.
\begin{itemize}
\item
The differentiation
with respect to $\xi$ in \re{t3b} may be carried out in
two ways. One could simply differentiate
the Feynman integral, which is a
smooth function of $\xi$ at $\xi\not=0$.
A more practical way is to differentiate the corresponding
{\em integrand.}
\item The operation of setting $\xi$ zero
is to act on the
differentiated integrand.
\item
It may be immediately seen that the
expression
\[
t^{(n)}_{\{{\bf q}^\gamma , {\bf m}^\gamma\}}
\langle \gamma\rangle (M,{\bf q}^\gamma,{\bf m}^\gamma,\mu)
\]
scales with $M$ as $M^{\omega(\gamma)  n}$ where
$\omega(\gamma)$ is the (mass) dimension of the
Feynman integral $\langle \gamma\rangle $
determined without counting any dimensionful
coupling constant as well
the {}'t~Hooft mass $\mu$.
Therefore, in every application of the hard mass expansion
the terms with too high value of $n$ in \re{t3a}
may be dropped.
\item By construction the coefficient function $C_\gamma^{(t)}$
is a polynomial with respect to its external momenta
${\bf q}^\gamma$ and the light masses ${\bf m}^\gamma$.
\end{itemize}
\begin{figure}
\begin{center}
\begin{tabular}{ccccc}
\parbox{3cm}{\epsfig{file=diakc1.eps,width=4cm,height=4cm}}
& \hspace{.5cm} $\bf \longrightarrow$ &
\parbox{3cm}{\epsfig{file=diakc2.eps,width=4cm,height=4cm}}
& \hspace{.5cm} $\bf *$ &
\parbox{3cm}{\epsfig{file=diakc3.eps,width=4cm,height=4cm}} \\
& \hspace{.5cm} $\bf +$ &
\parbox{3cm}{\epsfig{file=diakc4.eps,width=4cm,height=4cm}}
\end{tabular}
\end{center}
\caption{
\label{sdia}
{{Hard Mass Procedure.}}
}
\end{figure}
As an example we consider
the twoloop diagram $\Gamma$ depicted in Fig.~\ref{sdia} which
contributes to the fermion propagator in QED.
The heavy fermion of mass
$M$ is contained in the virtual fermion loop, whereas the open
fermion line corresponds to a propagating light fermion
with mass $m$.
The integral reads (in Feynman gauge):
\begin{equation}
\mu^{4\epsilon}
i^2
\int\frac{d^Dp}{(2\pi)^{{\rm D}}}
\int\frac{d^Dk}{(2\pi)^{{\rm D}}}
\frac{
{\rm {\rm Sp}}\left[
\gamma_\alpha( \FMslash{p} + M)\gamma_\beta
(\FMslash{p} \FMslash{k} + M)
\right]
[
\gamma_\alpha(\FMslash{q}\FMslash{k} + m)\gamma_\beta
]
}{
(k^2)^2 (M^2  p^2)[M^2  (pk)^2] [m^2  (qk)^2]
}
{}\, .
\label{t4}
\end{equation}
The integration momenta are denoted as $k$ and $p$ for the outer and
the inner loops respectively.
Two different integration regions can be identified.
In the first region is
$k \ll M, p \simeq M$. The corresponding hard subgraph
$\gamma_1$
is shown in Fig.~\ref{sdia} and $\langle \gamma_1 \rangle $
has to be expanded
with respect to its only
external momentum, $k$. The second region is characterized by
$k,p \approx M$. The hard subgraph $\gamma_2$ coincides with
$ \Gamma $ and the Feynman integral
$\langle \gamma_2 /\gamma_2\rangle $
reduces to unity. In this case the hard subgraph $\langle
\Gamma\rangle $ must be expanded with respect to the external momentum
$q$ and, in case of a nonvanishing light mass $m$, also with
respect to $m$. The sum of all contributions results in a power series
in the inverse top mass.
Working up to the powersuppressed terms of order $q^2/M^2$, one has
\begin{equation}
C_{\gamma_1}^{(t)} = ( t^{(0)}_{\{k\}} + t^{(1)}_{\{k\}} + t^{(2)}_{\{k\}})
\langle \gamma_1\rangle (M,k,\mu)
\ \
{\rm and}
\ \
C_{\gamma_2}^{(t)} = ( t^{(0)}_{\{q,m\}} +t^{(1)}_{\{q,m\}} )
\langle \gamma_2\rangle (M,q,m,\mu)
{}\, .
\label{t5}
\nonumber
\end{equation}
In explicit form these coefficient functions
are given by the following Feynman integrals
\begin{equation}
C_{\gamma_1}^{(t)} =
\mu^{2\epsilon}
i
\int\frac{d^{{\rm D}} p}{(2\pi)^{{\rm D}}}
\frac{
{\rm {\rm Sp}}\left[
\gamma_\alpha(\FMslash{p} + M)\gamma_\beta
(\FMslash{p} \FMslash{k} + M)
\right]
}{
(M^2  p^2)^2
}
(\, 1 + r + r^2 \, )
%\frac{k^2  2pk}{M^2  p^2} + \frac{4(pk)^2}{(M^2  p^2)^2}
\label{t6}
\, ,
\end{equation}
with
$
\displaystyle
\ \
r = \frac{k^2  2pk}{M^2  p^2}
\ \
$
and
\begin{equation}
%\rule{3mm}{3mm}
C_{\gamma_2}^{(t)} = {}
\mu^{4\epsilon}
i^2
\int\frac{d^Dp}{(2\pi)^{{\rm D}}}
\int\frac{d^Dk}{(2\pi)^{{\rm D}}}
\times
{}
\left[
\frac{
{\rm {\rm Sp}}\left[
\gamma_\alpha(\FMslash{p} + M)\gamma_\beta
(\FMslash{p} \FMslash{k} + M)
\right]
%
\gamma_\alpha (\FMslash{q}  \FMslash{k} + m) \gamma_\beta
%
}{
(k^2)^3 (M^2  p^2)[M^2  (pk)^2]
}
\right]
\, .
\label{t7}
\end{equation}
It is of course understood that the terms of higher than second
order in the expansion parameters are discarded in the integrand of
\re{t6}.
\subsection{Evaluation of Feynman Integrals\label{evaluation}}
In this section we will discuss very briefly the tools
now available to analytically compute massless propagators
and massive tadpoles in higher orders. We limit ourselves
to these rather restricted classes
of Feynman integrals
due to the following reasons:
\begin{enumerate}
\item Practice shows that in many cases
the methods of asymptotic expansions of Feynman
integrals do produce results numerically very well
approximating the exact results when the latter are
available. These methods reduce
initial multiscale Feynman amplitudes to
combinations of massless propagators and massive tadpoles.
\item The bulk of higher order results
discussed in the present work was eventually
obtained applying precisely these methods
to integrals appearing after the small/large
mass/momentum expansions are carried out.
\item Thanks to the intrinsic simplicity of the integrals
under discussion,  they depend on only one
nontrivial scale: an external momentum
or a heavy mass  their analytical
evaluation proves to be feasible in quite high
orders in the coupling constant.
The same simplicity provides the possibility of
constructing {\em regular} algorithms for
evaluating these integrals and for creating dedicated
computer programs allowing to perform the
calculations in a convenient
and automatized way.
\end{enumerate}
\noindent
Following the established practice we shall consider
Feynman integrals in the Euclidean momentum space throughout
this section. To avoid any confusion, all explicit results
cited in this section will be given in the canonical
dimensional regularization; in order to transform these into
those appropriate for the ${\overline{{ \mbox{{MS}}}}}$ scheme
an extra factor
$
\displaystyle
( {\rm e}^{\gamma_{{\rm E}}}/(4\pi))^{\epsilon h}
$
should be introduced
to every diagram with the number of loops equal to $h$
($\gamma_{{\rm E}} \equiv 0.577221566\dots$
is the Euler's constant).
%
\subsubsection{Massless Propagators\label{propagators}}
For brevity massless Feynman integrals
depending on exactly one external momentum will
be denoted by {\em pintegrals}.
At the moment there are tools to analytically
compute arbitrary one two and threeloop pintegrals
(see below).
Fortunately,
in many important cases one is interested only
in the
absorptive part of massless twopoint correlators.
In this case
available theoretical tools are enough to guarantee
at least {\em in principle} the analytical calculability
of absorptive part of an arbitrary 4loop pintegral.
Indeed, as was demonstrated in Ref.~\cite{Gorishny88}
the absorptive part of a fourloop pintegral
is expressible in terms
the corresponding fourloop UV counterterm
along with some threeloop pintegrals.
Next, calculation of UV counterterms is
simpler than that of the very integral.
This is because in MS scheme any UV counterterm is polynomial
in momenta and masses \cite{Collins75}.
This observation was effectively employed in
Ref.~\cite{Vladimirov78}
to simplify considerably the calculation of UV counterterms.
The method was further developed and named
Infrared ReaRrangement (IRR) in Ref.~\cite{me79}.
It essentially amounts to
an appropriate transformation of the IR structure of FI's by setting
zero some external momenta and masses (in some cases after some
differentiation is performed with respect to the latter).
As a result the calculation of UV counterterms is much simplified
by reducing the problem to evaluating massless pintegrals.
The method of IRR was ultimately refined
and freed from unessential qualifications in
Ref.~\cite{me84}. The following statement
has been proven there by the explicit construction of the
corresponding algorithm:
\vglue 0.2cm
\noindent
{\bf
Any UV counterterm for any
(h+1)loop Feynman
integral can be expressed in terms of pole and finite parts
of some appropriately constructed (h)loop pintegrals.
}
\noindent
\vglue 0.2cm
\noindent
{\em Oneloop pintegrals}
We start from a wellknown elementary formula for a generic
one loop pintegral (see Fig.~\ref{Fmethods1}a)
\begin{equation}
\begin{array}{c}
\displaystyle
\int
\frac{{\rm d}^{{\rm D}} \ell }{(2\pi)^{{\rm D}}}
\frac{1}{(q^2)^\alpha (ql)^{2\beta}}
=
\frac{
(q^2)^{2  \epsilon  \alpha  \beta}}%
{(4 \pi)^{2  \epsilon}}
G(\alpha,\beta),
\\
\displaystyle
G(\alpha,\beta) \equiv
\frac{
\Gamma(\alpha + \beta  2 + \epsilon)}{
\Gamma(\alpha)
\Gamma(\beta)
}g
\frac{
\Gamma(2  \alpha  \epsilon)
\Gamma(2  \beta  \epsilon)
}{
\Gamma(4  \alpha  \beta  2 \epsilon)
}
{}\, .
\end{array}
\label{methods1}
\end{equation}
\begin{figure}
\begin{center}
\begin{tabular}{ccc}
\subfigure[]{\epsfig{file=exa.eps,width=4cm,height=4cm}}
&
\subfigure[]{\epsfig{file=exb.eps,width=4cm,height=4cm}}
&
\subfigure[]{\epsfig{file=exc.eps,width=4cm,height=4cm}}
\end{tabular}
\end{center}
\caption{\label{Fmethods1}
Some pintegrals: (a) the generic oneloop pintegral and
(b) an example of primitive fiveloop pintegral;
(c) the master twoloop pintegral.
}
\end{figure}
It is of importance to note that any pintegral
depends homogeneously on its
external momentum. This facts allows the immediate
analytic evaluation of the whole class of
{\em primitive} pintegrals which, by definition, may be
performed by repeated application of the oneloop integration
formula.
For example, the fiveloop scalar
integral of Fig.~\ref{Fmethods1}b is performed by \re{methods1}
with the result
\begin{equation}
\left(
{(q^2)^{\epsilon}}{(4\pi^2)^{2  \epsilon})}
\right)^5
(q^2)^{\epsilon}
(
\Gamma(1,1)
\Gamma(1,\epsilon)
)^2
\Gamma(1+ 2\epsilon,1+2\epsilon)
{}\, .
\label{methods2}
\end{equation}
\vglue 0.2cm
\noindent
{\em Twoloop pintegrals}
Not all pintegrals are primitive ones. One first
encounters nontrivial pintegrals already
at the two loop level.
While oneloop integrals are performed with ease
the evaluation of the master twoloop
diagram (see Fig.~\ref{Fmethods1}c) is not trivial.
The corresponding Feynman integral reads
\begin{equation}
( 4 \pi)^{4  2 \epsilon}
(q^2)^{42\epsilon\sum_{{i}} \alpha_{{i}}}
F(\alpha_1, \dots , \alpha_5 ) \equiv
\int
\frac{ \displaystyle
{\rm d}^{{\rm D}} \ell_1
{\rm d}^{{\rm D}} \ell_2}
{\displaystyle (2\pi)^{2D} }
\frac{\displaystyle 1}{\displaystyle
p_1^{2\alpha_1}
p_2^{2\alpha_2}
p_3^{2\alpha_3}
p_4^{2\alpha_4}
p_5^{2\alpha_5}
}
{}\, ,
\label{methods3}
\end{equation}
with the loop momenta
\[
p_1 = \ell_1, \ \
p_2 = \ell_2, \ \
p_3 = q \ell_2, \ \
p_4 = q \ell_1, \ \
p_5 = \ell_2  \ell_1
{}\, .
\]
In fact, a closed expression for
the function $F(\alpha_1,\dots \alpha_5)$ for generic values of the
arguments is not known. However,
results
do exist for
particular cases. The first
one, valid for a generic spacetime
dimension $D$, was obtained with the
help of the socalled Gegenbauer
polynomial technique in $x$space (GPTX) \cite{me79}.
It reads\footnote{In fact, we put below an
equivalent but simpler formula found in Ref.~\cite{me81b}.}
\begin{equation}
F(\alpha,1,1,\beta,1) =
\frac{G(1,1)}{D 2 \alpha  \beta}
\left\{
\alpha[G(\alpha+1,\beta)  G(\alpha+1,\beta +\epsilon)]
+
(\alpha \leftrightarrow \beta)
\right\}
\label{methods4}
\end{equation}
It has been also shown in Ref.~\cite{me79}
that similar results may be obtained for
the case when the indices $\alpha_2, \alpha_3$ and
$\alpha_5$ are integers while $\alpha_1$ and $\alpha_4$
are arbitrary.
In practice one often needs only a few first terms of the
expansion of $F(\alpha_1 \dots \alpha_5)$ in the Laurent series in
$\epsilon$. This expansion is known for generic values of the
$\alpha_1 \dots \alpha_5$ up to a fixed (quite high) order
(see Refs.~\cite{DKazakov,masterTwoDavid} and references therein).
\vglue 0.2cm
\noindent
{\em Threeloop pintegrals}
{\em In principle} GTPX is also applicable to compute some
nontrivial threeloop pintegrals\footnote{For example, the
basic scalar nonplanar threeloop diagram of Fig.~\ref{Fmethods3}a
was first calculated
via GPTX in Ref.~\cite{me79}.}. However, calculations quickly get
clumsy, especially for diagrams with numerators.
The main breakthrough at the threeloop level happened with
elaborating the method of integration by parts of dimensionally
regularized integrals Refs.~\cite{me81a,me81b}.
The key identity for the method is\footnote{In fact for
twoloop massive integrals a similar identity was used
in the classical work by ${}'$t~Hooft and Veltman Ref.~\cite{dim.rega}.
%the fact obviously being not known to
%the authors of Refs.~\cite{me81a,me81b}.
}
\begin{equation}
\int
{\rm d}^{{\rm D}} \ell
\frac{\displaystyle \partial}{\displaystyle \partial \ell_\mu}
I(\ell,\dots)
\equiv 0
{}\, ,
\label{methods5}
\end{equation}
where $I(\ell,\dots)$
is a Feynman {\em integrand} and $\ell$ is one of its loop
momenta.
The identity reflects the possibility of neglecting the surface terms,
which holds true in dimensional regularization \cite{me83}.
The use of
\re{methods5} along with tricks like
completing momentum squares and
cancelling similar factors in the nominator against
those in the denominator\footnote{
The validity of such operations for {\em divergent}
dimensionally regulated
integrals has been rigorously justified in
Ref.~\protect\cite{me83}.} constitutes the essence of the
approach. The identity depicted in Fig.~\ref{Fmethods2}
is a typical example of relations obtainable with the
help of the integration by parts method.
\begin{figure}
\begin{center}
\begin{tabular}{ccccc}
$\epsilon$
\parbox{3cm}{\epsfig{file=exd.eps,width=4cm,height=4cm}}
&$=$&
\parbox{3cm}{\epsfig{file=exe.eps,width=4cm,height=4cm}}
&$$&
\parbox{3cm}{\epsfig{file=exf.eps,width=4cm,height=4cm}}
\end{tabular}
\end{center}
\caption{\label{Fmethods2}
The exact relation expressing a nonprimitive twoloop
scalar pintegral through primitive integrals;
a dot on a line means a squared scalar propagator.
}
\end{figure}
The general scheme of the use of the
integration by parts method is based on the
exploitation the identities of type \re{methods5}
in the form of recurrence relations with the aim
to express a complicated diagram through the simpler ones.
Unfortunately, there does not exist (at least at the present)
a general method
to study these recurrence relations.
Nevertheless,
all (about a dozen) topologically
different threeloop pintegrals were neatly analyzed
in Ref.~\cite{me81b}
and a concrete calculational algorithm was suggested for every
topology. As a result the algorithm of integration by parts
for threeloop pintegrals was developed.
\begin{figure}
\begin{center}
\begin{tabular}{cc}
\subfigure[]{\epsfig{file=exg.eps,width=4cm,height=4cm}}
&
\subfigure[]{\epsfig{file=exh.eps,width=4cm,height=4cm}}
\end{tabular}
\end{center}
\caption{\label{Fmethods3}
(a), (b) the master threeloop
nonplanar and planar scalar diagrams.
}
\end{figure}
The algorithm constitutes a series
of involved identities which are used
to identically transform any threeloop pintegral
into a sum of primitive oneloop pintegrals and two
basic threeloop pintegrals pictured in Fig.~\ref{Fmethods3}.
\vglue 0.2cm
\noindent
{\em Fourloop and beyond}
At the moment there is no any general algorithm allowing to
analytically compute arbitrary 4loop pintegrals. The problem
of creating such an algorithm seems to be hopelessly difficult
(see Ref.~\cite{aip2} where the point is discussed in some detail).
Considering the comments made above, full control of threeloop
pintegral is sufficient to calculate the absorptive part of
any fourloop pintegral.
\vglue 0.1cm
\subsubsection{Massive Tadpoles\label{tadpoles}}
In this section we discuss another useful class of
Feynman integrals  integrals without external momenta at all.
However massive lines as well as massless ones are
admitted.
It is understood that all the massive
propagators depend on one and the same mass m.
Such integrals 
they will be referred to as {\em mintegrals} 
naturally appear in many problems where the mass m
may be considered as much larger than all other mass scales
involved.
\vglue 0.2cm
\noindent
{\em Oneloop tadpoles}
At oneloop level there is a textbook result which comes
from straightforward integration over Feynman parameters
and reads
\begin{equation}
\begin{array}{c}
\displaystyle
\frac{\displaystyle 1}{\displaystyle (\pi)^{2\epsilon}}
\int \frac{ \displaystyle {\rm d}^{\displaystyle {\rm D}} \ell }%
{(\ell^2 + m^2)^\alpha
(\ell^2)^\beta
}
=
\displaystyle
(m^2)^{D/2 \alpha \beta}
\frac{\displaystyle \Gamma(D/2  \beta)\Gamma(\alpha + \beta  D/2)}%
{\displaystyle
\Gamma(D/2)\Gamma(\alpha)
}
\end{array}
{}\, .
\label{1loop.tadpole}
\end{equation}
\vglue 0.2cm
\noindent
{\em Twoloop tadpoles}
All possible twoloop mintegrals are
pictured in Fig.~\ref{Fmethods4}.
\begin{figure}
\begin{center}
\begin {tabular}{ccc}
\parbox{3cm}{
\epsfig{file=pica.eps,width=3.cm,height=3.cm}
}
&
\parbox{3cm}{
\epsfig{file=picb.eps,width=3.cm,height=3.cm}
}
&
\parbox{3cm}{
\epsfig{file=picc.eps,width=3.cm,height=3.cm}
}
\end {tabular}
\end{center}
\caption {\label{Fmethods4}
Different cases of twoloop mintegrals:
dashed lines are massless; solid lines have mass m.
}
\end{figure}
Those with only one massive line (Fig.~\ref{Fmethods4}a)
may be reduced to the oneloop integral
after first integrating the oneloop psubintegral. Twoloop
mintegrals with more than 1 massive lines
(Fig.~\ref{Fmethods4}b,c) are
more difficult. A simple formula exists for
the integral with two
massive lines \cite{Veltman80}:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{equation}
\frac{\displaystyle 1}{\displaystyle (\pi^2)^{2\epsilon}}
\int \frac{\displaystyle {\rm d}^{{\rm D}} \ell_1\, {\rm d}^{{\rm D}}
\ell_2}%
{\displaystyle (\ell_1^2 + m^2)^\alpha
(\ell_2^2 +m^2)^\beta
((\ell_1 + \ell_2)^2)^{\gamma}}
=
\frac{
(m^2)^{D \alpha  \beta  \gamma }
\Gamma(D/2 \gamma)}{\Gamma(\alpha)\Gamma(\beta)\Gamma(D/2)}
M(\alpha,\beta,\gamma)
{}\, ,
\label{2loop.tadpole}
\end{equation}
with
\begin{equation}
M(\alpha,\beta,\gamma)
=
\frac{\displaystyle %
\Gamma(\alpha + \gamma  {D}/{2})
\Gamma(\beta + \gamma  {D}/{2})
\Gamma(\alpha + \beta + \gamma  D)}
{\displaystyle \Gamma(\alpha + \beta + 2\gamma  D)
}
.
\label{2loop.res}
\end{equation}
\noindent
The case with all lines massive
has been studied e.g. in Ref.~\cite{davyd93}).
\vglue 0.2cm
\noindent
{\em Threeloop tadpoles}
Three loop mintegrals have proved to be also
treatable with the help of recursive relations
stemming from the main identity \re{methods5}
of the integration by parts method
\cite{Broadhurst92,Tarasov94}.
These relations allow one to express a given 3loop
mintegral in terms of
a limited number of master integrals.
The latter need to be evaluated once and for all.
Unlike the situation with threeloop pintegrals
some master integrals are to be evaluated numerically
(see Ref.~\cite{Broadhurst93,Tarasov94} and references therein).
%
%
\subsection{Software Tools}
%
It goes without saying that the calculation of
higher order corrections in gauge theories
is almost impossible without intensive use of
computer algebra methods.
In addition to the old problem of
taking long traces of Dirac $\gamma$
matrixes, the algorithm of integration by parts,
when applied even to a single
threeloop p (or m) integral, generically
produces dozens or even hundreds of
terms. At the moment there exist
essentially three different packages
which implement the algorithm.
For pintegrals they are written
in {\rm SCHOONSCHIP}
\cite{Veltman.2loopTadpole}
(see Refs.~\cite{mincer1a,mincer1b} and in FORM
\cite{Ver91} (see Refs.~ \cite{mincer2,mincer2S}).
%
However, in genuine 4loop calculations the reduction to
threeloop pintegrals is far from being
trivial and also includes a lot of purely algebraic manipulations,
which are difficult to computerize (see a discussion in
Ref.~\cite{aip2}).
For mintegrals the
FORM program SHELL2 has been developed \cite{SHELL2}.
It computes twoloop tadpoles and onshell massive propagators.
A short description of an algorithm to perform threeloop
mintegrals of some particular types
may be found in Ref.~\cite{Broadhurst92}.
Recently the algorithm has been extended to cover all
threeloop mintegrals \cite{Avdeev95}.
\subsection{$\gamma_5$ in $D$ Dimensions
\label{gamma5}
}
Multiloop calculations with dimensional
regularization often encounter the
question of how to treat $\gamma_5$ in $D$
dimensions.
Occasionally the problem can be circumvented
by exploiting chiral symmetry which allows,
for example, the relating of the nonsinglet axial
correlator in the massless limit to the
corresponding vector correlator.
In general, however, a consistent definition
must be formulated. A rigorous choice
is based
on the original definition by 't~Hooft and
Veltman \cite{dim.rega}, and formalized by
Breitenlohner and Maison \cite{BreMai77}
with a modification introduced in
\cite{{AkyDel73}}.
In this selfconsistent approach
$\gamma_5$ is defined as
\cite{dim.rega}
\begin{equation}\label{gam1}
\gamma_5 =
\frac{i}{4!}
\epsilon_{\mu\nu\rho\sigma}
\gamma^{\mu}\gamma^{\nu}\gamma^{\rho}
\gamma^{\sigma}\, ,
\end{equation}
with
$\epsilon_{0123} \equiv 1$.
For our discussion we consider
the cases for both the nonsinglet
axial current $j_{5\mu}^{({\rm NS})a}$ and
the singlet one $j_{5\mu}^{({\rm S})}$,
which are defined with the help of
the antisymmetrized combination
$\gamma^{[\nu\rho\sigma]}=
(\gamma^{\nu}\gamma^{\rho}\gamma^{\sigma}
\gamma^{\sigma}\gamma^{\rho}\gamma^{\nu})/2$
in order to guarantee Hermiticity
for noncommuting $\gamma_5$
\begin{equation}\label{gam2}
\begin{array}{ll}\displaystyle
j_{5\mu}^{({\rm NS})a}
& \displaystyle
= \frac{1}{2}\overline{\Psi}
(\gamma_{\mu}\gamma_5\gamma_5\gamma_{\mu})
t^{{a}}\Psi
\\ & \displaystyle
=
\frac{i}{3!}
\epsilon_{\mu\nu\rho\sigma}
\overline{\Psi}
\gamma^{[\nu\rho\sigma]}
t^{{a}}\Psi
=
\frac{i}{3!}
\epsilon_{\mu\nu\rho\sigma}
A_{{\rm NS}}^{[\nu\rho\sigma]a}
{}\, \, ,
\\
\displaystyle
j_{5\mu}^{({\rm S})}
& \displaystyle
= \frac{1}{2}\overline{\Psi}
(\gamma_{\mu}\gamma_5\gamma_5\gamma_{\mu})
\Psi
\\ & \displaystyle
=
\frac{i}{3!}
\epsilon_{\mu\nu\rho\sigma}
\overline{\Psi}
\gamma^{[\nu\rho\sigma]}
\Psi
=
\frac{i}{3!}
\epsilon_{\mu\nu\rho\sigma}
A_{S}^{[\nu\rho\sigma]}
{}\, \, .
\end{array}
\end{equation}
Here $t^{{a}}$ are the generators of the
$ SU(n_{{f}}) $ flavour group.
The fourdimensional LeviCivita tensor
$\epsilon_{\mu\nu\rho\sigma}$ is kept
outside the renormalization procedure
where all indices can be considered as
four dimensional whereas the calculation
is performed with the generalized
currents $A_{{\rm NS}}^{[\nu\rho\sigma]a},
A_{S}^{[\nu\rho\sigma]}$ in D
dimensions\footnote{
A practical realization of the scheme avoiding the
explicit separation of Lorentz indexes into the 4 and
$(D4)$ dimensional ones was elaborated in
Ref.~\cite{Larin93}.}.
As a consequence of the lost anticommutativity
of $\gamma_5$, standard properties of the
axial current as well as the Ward identities are
violated. In particular, it turns out that
the renormalization constant
$Z^{{\rm NS}}$
of the nonsinglet
current is not one any more. To restore the
correctly normalized nonsinglet axial current an
extra finite renormalization is introduced
with corresponding
finite renormalization constant
$z^{{\rm NS}}$
\cite{JCC84,Trueman79}.
%\cite{JCC84,Trueman79,Larin91,Larin93}.
One thus has for the renormalized nonsinglet
axial current the following
expression:
\begin{equation}
\left(j_{5\mu}^{({\rm NS})a}\right)_{{\rm R}}
\displaystyle
= z^{{\rm NS}} Z^{{\rm NS}}
\left(j_{5\mu}^{({\rm NS})a}\right)_{{\rm B}}
\label{gam3ns}
\end{equation}
with \cite{Larin91,Larin93}
\begin{eqnarray}
%\begin{array}{ll}
%\displaystyle
Z^{{\rm NS}}
%\displaystyle
&=& 1+
a^2
\frac{1}{\epsilon}
\left[
\frac{11}{6}
 \frac{1}{9}~n_{{f}}
\right]
\nonumber\\
%{}
\displaystyle
&&+
~a^3
\frac{1}{\epsilon^2}
\left[
 \frac{121}{36}
+ \frac{11}{27}~n_{{f}}
 \frac{1}{81}~n_{{f}}^2
+ \epsilon
\left(
\frac{391}{72}
 \frac{44}{81}~n_f
+ \frac{1}{486}~n_{{f}}^2
\right)
\right]
%\end{array}
\label{gam4}
\end{eqnarray}
and
\begin{equation}
z^{{\rm NS}} =
1

\frac{4}{3} a
+
a^2
\left( \frac{19}{36} + \frac{1}{54}~n_f
\right)
{} \, .
\label{zns}
\end{equation}
The prescription described above and the use
of the nonsinglet axial current defined according to
Eq.~(\ref{gam3ns})
lead to the same characteristics for nonanomalous
amplitudes as would be
obtained within a na\"{\i}ve approach featuring completely
anticommutating $\gamma_5$.
First, the Ward identity is
recovered. Second, the anomalous
dimension of the nonsinglet axial
current vanishes. For diagrams with an
even number of $\gamma_5$ connected to
the external current it has been
checked that the treatment based on an
anticommutativity of $\gamma_5$ leads
to the same answer \cite{BroadKataev93}.
Similar considerations may be carried
out for the singlet axial vector
current. However, in this case there is
some freedom in defining the
renormalized current. This is due to
the fact that in any physical
application the current never appears
as it is but only in a (virtually
nonsinglet) combination with another
axial vector current. A physically
motivated definition has been
suggested in Ref.~\cite{CK3}, where the
singlet axial vector current has been
defined with the help of the following
limiting procedure:
\begin{equation}
(j^{({\rm S})}_{5\mu})_{{\rm R}}
\bbuildrel{=\!=\!=}_{m_{{\rm T}} \to \infty}^{}
z^{{\rm NS}}Z^{{\rm NS}}
\left[
j^{({\rm S})}_{5\mu}

n_{{f}}\frac{i}{3!}
\epsilon_{\mu\nu\rho\sigma}
\overline{\Psi}_{{\rm T}}
\gamma^{[\nu\rho\sigma]}
\Psi_{{\rm T}}
\right]_{{\rm B}}
{}\, .
\label{limproc}
\end{equation}
Here, $\Psi_{{\rm T}}$ is the field of an auxiliary quark
${\rm T}$
and thus the combination in the squared brackets is
a nonsinglet one (in the extended QCD with $n_{{f}} +1$
flavours!). Due to the asymptotic freedom, the
large $m_{{\rm T}}$ limit of \re{limproc}
does exist and is naturally identified with the
renormalized singlet axial current.
Explicitly, the r.h.s. of \re{limproc} can
be written without any auxiliary fields in the
form (note that the renormalization constant
$Z^{{\rm S}}$ was first found in Ref.~\cite{Larin93})
\begin{equation}
\left(j_{5\mu}^{({\rm S})}\right)_{{\rm R}}
\displaystyle
= z^{{\rm S}} Z^{{\rm S}}
\left(j_{5\mu}^{({\rm S})}\right)_{{\rm B}}
\label{gam3s}
{}\, ,
\end{equation}
with
\begin{eqnarray}
%\begin{array}{ll}\displaystyle
Z^{{\rm S}}
\displaystyle
&=& 1+
a^2
\frac{1}{\epsilon}
\left[
\frac{11}{6}
+ \frac{5}{36}~n_f
\right]
\nonumber\\
\displaystyle
&&+~a^3
\frac{1}{\epsilon^2}
\left[
 \frac{121}{36}
 \frac{11}{216}~n_f
+ \frac{5}{324}~n_{{f}}^2
+\epsilon
\left(
\frac{391}{72}
+ \frac{61}{1296}~n_f
+ \frac{13}{1944}~n_{{f}}^2
\right)
\right]
%\end{array}
\label{zs}
\end{eqnarray}
and
\begin{eqnarray}
%\begin{array}{c}
\displaystyle
\!\!\!z^{{\rm S}}& =&
1
+
a \frac{(  \frac{5}{18}~n_{{f}}  \frac{11}{3})}{\beta_0}
\\
\displaystyle
&&+
~a^2 \left[
\frac{1}{\beta_0^2}
\left(
 \frac{185}{2592}~n_{{f}}^2
+ \frac{391}{864}~n_f
+ \frac{2651}{144}
\right)
\frac{1}{\beta_0}
\left(\frac{13}{1296}~n_{{f}}^2
+ \frac{61}{864}~n_f
+ \frac{391}{48}
\right)
\right],\nonumber
%\end{array}
\label{zs}
\end{eqnarray}
where $\beta_0 = (11  \frac{2}{3}~n_{{f}})/4$.
It should be noted that an equivalent
definition of the singlet axial vector current
is obtained by demanding that it have
a vanishing anomalous dimension.
\section{Exact Result of Order ${\cal O}(\alpha_s)$
\label{exact}}
The exact QCD corrections for arbitrary
quark masses
are
known in order $O(\alpha_s)$. The
result is different for vector and axial current
correlators. Whereas the
former can be taken directly
from QED \cite{Schw73}
the latter have been obtained in Ref.~\cite{Zerwas80}.
(For the nondiagonal current and arbitrary,
different masses the result can be found in
\cite{changgaemers}.)
With $v^2=14m^2/s$ they read:
\begin{equation} \begin{array}{ll}\label{ap1}
r^{{\rm V}}_{{\rm NS}} = & \displaystyle
v\frac{3v^2}{2}\left[1+\frac{4}{3}
\frac{\displaystyle\alpha_s(s)}{\displaystyle \pi} K_{{\rm V}}\right]
{}\, ,
\\ \displaystyle
r^{{\rm A}}_{{\rm NS}} = & \displaystyle
v^3\left[1+\frac{4}{3}
\frac{\displaystyle\alpha_s(s)}{\displaystyle \pi} K_{{\rm A}}\right]
{}\, .
\end{array}
\end{equation}
$K_{{\rm V}}$ and $K_{{\rm A}}$ have been calculated
in Refs.~\cite{Zerwas80,changgaemers,KniKue90b}.
%in Refs.~[7678].
A compact form for the correction can be found
in Ref.~\cite{KniKue90b}:
\begin{equation}
\begin{array}{ll}\label{ap2}
K_{{\rm V}} = & \displaystyle
\frac{1}{v}\left[ A(v) +
\frac{P_{{\rm V}}(v)}{(1v^2/3)}\ln\,\frac{1+v}{1v}
+ \frac{Q_{{\rm V}}(v)}{(1v^2/3)}\right],
\\ \displaystyle
K_{{\rm A}} = & \displaystyle
\frac{1}{v}\left[ A(v)
+ \frac{P_{{\rm A}}(v)}{v^2}\ln\,\frac{1+v}{1v}
+ \frac{Q_{{\rm A}}(v)}{v^2}\right]
{},
\end{array}
\end{equation}
with
\begin{equation}
\begin{array}{ll}\label{ap3} \displaystyle
A(v) =& \displaystyle (1+v^2)
\left[
{\rm Li}_2\left(
\left[\frac{1v}{1+v}\right]^2
\right)
+2{\rm Li}_2\left(\frac{1v}{1+v}\right)
+\ln\,\frac{1+v}{1v}\ln\,\frac{(1+v)^3}{8v^2}
\right]
\\ & \displaystyle
+~3v\ln\,\frac{1v^2}{4v}v\ln\, v
{}\, ,
\end{array}
\end{equation}
\begin{equation}
\begin{array}{ll}
\label{ap4}
\displaystyle
P_{{\rm V}}(v) = \frac{33}{24}+\frac{22}{24}v^2
\frac{7}{24}v^4 \, ,
& \displaystyle
Q_{{\rm V}}(v) = \frac{5}{4}v\frac{3}{4}v^3 \, ,
\\ \displaystyle
P_{{\rm A}}(v) = \frac{21}{32}+\frac{59}{32}v^2
% Below the sign + in front of 19/32 was fixed due to
% A. Hoang suggestion......
+\frac{19}{32}v^4\frac{3}{32}v^6 \, ,
\;\;\;\;& \displaystyle
Q_{{\rm A}}(v) = \frac{21}{16}v+\frac{30}{16}v^3
+\frac{3}{16}v^5
{}\, .
\end{array}
\end{equation}
Convenient parametrizations are
\cite{Kuhn85}:
\begin{equation}
\begin{array}{ll}
\label{n18}
K_{{\rm V}} = & \displaystyle
\frac{\pi^2}{2v}  \frac{3+v}{4}
\left(\frac{\pi^2}{2}\frac{3}{4}\right)
{},
\\
\displaystyle
K_{{\rm A}} = & \displaystyle
\frac{\pi^2}{2v}  \left[\frac{19}{10}
\frac{22}{5}v+\frac{7}{2}v^2\right]
\left(\frac{\pi^2}{2}\frac{3}{4}\right)
{}.
\end{array}
\end{equation}
Let us consider this result in the limit
where ${s}$ approaches the threshold region
($v\rightarrow 0$) as well as the
high energy regime ($v\rightarrow 1$).
For $v\rightarrow 0$ the correction factors simplify
to:
\begin{equation}
\begin{array}{ll} \displaystyle
1+\frac{4}{3}\frac{\displaystyle\alpha_s(s)}{\displaystyle \pi}
K_{{\rm V}} \stackrel{v\rightarrow 0}
{\longrightarrow}
& \displaystyle
\frac{2\pi\alpha_s}{3v}+\left(1\frac{16}{3}~\frac{\alpha_s}{\pi}\right)
{},
\\ \displaystyle
1+\frac{4}{3}\frac{\displaystyle\alpha_s(s)}{\displaystyle \pi}
K_{{\rm A}} \stackrel{v\rightarrow 0}
{\longrightarrow}
& \displaystyle
\frac{2\pi\alpha_s}{3v}+\left(1\frac{8}{3}~\frac{\alpha_s}{\pi}\right)
{}.
\end{array}
\end{equation}
For very small $v$ higherorder contributions
must be taken into consideration.
In QED these can be summed to yield the
Sommerfeld rescattering factor:
\begin{equation}
r_{{\rm QED}} = \frac{3}{2}\; \frac{\pi\alpha}{(1e^{\pi\alpha/v})}
\end{equation}
In QCD the coupling constant $\alpha$ would be
replaced in this formula by $4\alpha_s /3$.
However, the scale of $\alpha_s(Q^2)$ cannot be fixed
with certainty, since subleading logarithms have
not yet been evaluated. It has been argued in
%Ref.~\cite{Kuhn85} that the choice $\alpha_s(2\vec{P}_{{\rm t}})$,
Ref.~\cite{Kuhn85} that the choice $\alpha_s(2{\bf P}_{{\rm t}})$,
combined with
Eq.~(\ref{n18}) allows for an adequate
description of $R$ in
the threshold region
and provides a smooth connection
between resonances and continuum.
This ansatz will be discussed further in Section~\ref{schwinger}.
For top quarks a new element enters
through their large decay rate. Resonance and open
$t\overline{t}$ production merge. An account of the
resulting phenomena is beyond the scope
of this paper and can be found
in Refs.~\cite{Fadin87,Peskin91,Kuhn92,Teubner93,Sum92}.
%in Refs.~[8186???].
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\pictup{3cm}{10cm}{kvexp}%
\begin{figure}
\begin{center}
\epsfxsize=12.0cm
\leavevmode
\epsffile[130 300 460 525]{vector.ps}\\
\epsfxsize=12.0cm
\leavevmode
\epsffile[130 300 460 525]{axial.ps}
\caption{\label{kvexp}{{Comparison between the
complete ${\cal O}(\alpha_s)$ correction
function (solid line) and approximations of increasing order
(dashed lines) in $m^2$ for vector (upper graph)
and axial vector current (lower graph) induced rates.}}}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The behaviour of the result for large $s$
can easily be extracted from the analytic formulae
\cite{KniKue90b,Kuhn90b,CheKue90}.
In Born approximation the leading term
of the vector and the axial vector
correlators are of order
$m^4/s^2$ and $m^2/s$ respectively:
%\newpage
\begin{equation}
\begin{array}{rl} \displaystyle
v\frac{3v^3}{2}
& \displaystyle
\longrightarrow 16\frac{m^4}{s^2}
+{\cal O}(m^6/s^3)
{}\, ,
\\ \displaystyle
v^3
& \displaystyle
\longrightarrow 16\frac{m^2}{s}+6\frac{m^4}{s^2}
+{\cal O}(m^6/s^3)
{}\, .
\end{array}
\end{equation}
Including firstorder QCD corrections leads to:
\begin{equation}\label{f1}
\begin{array}{ll}\displaystyle
r^{{\rm V}}_{{\rm NS}}
\stackrel{v\rightarrow 1}{\longrightarrow}
& \displaystyle
16\frac{m^4}{s^2}
\\ & \displaystyle
+~\frac{\alpha_s}{\pi}
\left[
1+12\, \frac{m^2}{s}+\frac{m^4}{s^2}
\left(
1024 \, \ln\,\frac{m^2}{s}
\right)
\right]
{}\, ,
\\ \displaystyle
r^{{\rm A}}_{{\rm NS}}
\stackrel{v\rightarrow 1}{\longrightarrow}
& \displaystyle
16\frac{m^2}{s}+6\frac{m^4}{s^2}
\\ & \displaystyle
+~\frac{\alpha_s}{\pi}
\left[
1\frac{m^2}{s}
\left(
6+12\, \ln\,\frac{m^2}{s}
\right)
+\frac{m^4}{s^2}
\left(
22+24\, \ln\,\frac{m^2}{s}
\right)
\right]
{}\, .
\end{array}
\end{equation}
The approximations to the correction functions
for the vector and the axial vector current
correlators
(including sucessively higherorders without
the factor $\alpha_s/\pi$)
are compared to the full result
in Fig.~\ref{kvexp}.
As can be seen in this figure,
for high energies  say for $2 m_{{\rm b}}/\sqrt{s}$
below 0.3  an
excellent approximation is provided by the
constant plus the $m^2$ term.
In the region of $2m/\sqrt{s}$ above 0.3
the $m^4$ term becomes
increasingly important. The inclusion of
this term improves the
agreement significantly and leads to an
excellent approximation, even up
to $2m/\sqrt{s}\approx 0.7$ or 0.8. For the
narrow region between 0.6
and 0.8 the agreement is further improved
through the $m^6$ term.
The mass $m$ in this formula is understood
as the physical mass, defined through the
location of the pole of the quark propagator in
complete analogy with the treatment of the electron
mass in QED.
However, if one tried to control fully the
$m^2/s$ and $m^4/s^2$ terms
one might worry about the logarithmically enhanced
coefficient which could invalidate perturbation
theory.
These leading logarithmic terms
may be summed through renormalization group
techniques.
In order $\alpha_s$ this can be trivially achieved by
substituting
\[
m^2=\overline{m}^2 \Biggl[1+\frac{\alpha_s}{\pi}
(8/32\ln\,(\overline{m}^2/s))\Biggl]
\]
which implies (for completeness also $m^6/s^2$
terms from Ref.~\cite{TTP9408} are included)
\begin{equation}\label{f2}
\begin{array}{ll}\displaystyle
r^{{\rm V}}_{{\rm NS}}
\stackrel{v\rightarrow 1}{\longrightarrow}
& \displaystyle
16\frac{\overline{m}^4}{s^2}8\frac{\overline{m}^6}{s^3}
\\ & \displaystyle
+~\frac{\alpha_s}{\pi}
\left[
1+12\frac{\overline{m}^2}{s}22\frac{\overline{m}^4}{s^2}
\frac{16}{27}
\left(
6\ln\,\frac{\overline{m}^2}{s}+155
\right)\frac{\overline{m}^6}{s^3}
\right]
{}\, ,
\\ \displaystyle
r^{{\rm A}}_{{\rm NS}}
\stackrel{v\rightarrow 1}{\longrightarrow}
& \displaystyle
16\frac{\overline{m}^2}{s}+6\frac{\overline{m}^4}{s^2}
+4\frac{\overline{m}^6}{s^3}
\\ & \displaystyle
+~\frac{\alpha_s}{\pi}
\left[
122\frac{\overline{m}^2}{s}+10\frac{\overline{m}^4}{s^2}
+\frac{8}{27}
\left(
39\ln\,\frac{\overline{m}^2}{s}+149
\right)
\frac{\overline{m}^6}{s^3}
\right]
{}\, .
\end{array}
\end{equation}
A systematic discussion of higherorder terms
will be given in the subsequent sections.
\section{Nonsinglet Contributions
\label{nonsinglet}}
\subsection{Massless Limit\label{repnsm0}}
This section will cover those results
obtained in the limit of massless
quarks. As discussed in the previous part,
nonsinglet contributions
exhibit a universal charge
factor which is given by the Born result
and can be trivially factored.
The firstorder correction was derived in the
context of QED some time ago \cite{Schw73}.
The second order coefficient has been calculated
by several groups \cite{CheKatTka79}.
The initial calculation of the
$O(\alpha_s^3)$ term described in
\cite{Gorishny88} was later corrected
by two groups \cite{Gorishny91,Surguladze91}.
An implicit test of the results has recently been
performed \cite{BroadKataev93}.
The full result for the
nonsinglet current reads as follows:
\begin{eqnarray}
\displaystyle
r^{(0)}_{{\rm NS}}(s)& =&
1
+
\frac{\alpha_s(s)}{\pi}
+
\left[\frac{\alpha_s(s)}{4\pi}\right]^2
\left\{
\frac{730}{3}  176~\zeta(3) +
\left[\frac{44}{3} + \frac{32}{3} \zeta(3)\right] n_f
\right\}
\nonumber
\\
\displaystyle
&&+
\left[\frac{\alpha_s(s)}{4\pi}\right]^3
\left\{
\frac{174058}{9}  17648~\zeta(3) + \frac{8800}{3} \zeta(5)
\right.
\label{ns1}
\\
&&+
\left[\frac{62776}{27} + \frac{16768}{9}
\zeta(3)  \frac{1600}{9} \zeta(5)
\right] n_f
\nonumber
\\
\displaystyle
&&+
\left.
\left[ \frac{4832}{81}  \frac{1216}{27} \zeta(3)
\right] n_{{f}}^2
 \left[
\frac{484}{3}  \frac{176}{9}~n_{{f}}
+ \frac{16}{27}~n_{{f}}^2
\right] \pi^2
\right\}
{}\, .
\nonumber
%\label{ns1}
\end{eqnarray}
This leads to the following
numerical result:
\begin{eqnarray}
\displaystyle
r^{(0)}_{{\rm NS}}(s) &=&
1
+
\frac{\displaystyle\alpha_s(s)}{\displaystyle \pi}
+\left[\frac{\displaystyle\alpha_s(s)}{\displaystyle \pi}\right]^2
(
1.9857  0.1153\,~n_f
%1.98570740  0.1152953981\, n_f
)
\nonumber
\\
\displaystyle
&&+
\left[\frac{\displaystyle\alpha_s(s)}{\displaystyle \pi}\right]^3
(
 6.6369
 1.2001 \, n_f
 0.0052 \,~n_{{f}}^2
% 6.63693554  1.200134058 n_{{f}}  .00517835519 n_f
)
{}\, .
\label{ns2}
\end{eqnarray}
Those terms which depend on the number
of quark flavours $n_{{f}}$ are due to virtual
fermion loops with light quarks. They appear
for the first time at second order $\alpha_s^2$.
Mixed QED and QCD corrections can be deduced from
the QCD results in a straightforward manner
\cite{Kat92}.
One obtains
\begin{equation}\label{ns3}
r^{(0)}_{{\rm QED}} = Q_{{f}}^2~\frac{3}{4}
~\frac{\alpha(s)}{\pi}
\left[
1\frac{1}{3}~\frac{\displaystyle\alpha_s(s)}{\displaystyle \pi}
\right]
{}\, .
\end{equation}
Corrections of order $\alpha^2$ are
also given in Ref.~\cite{Kat92}. They are
small and will not be considered
here.
\subsection{Top Mass
Corrections\label{top}}
The top quark is also
present at second order through a virtual quark loop.
The corrections in the
corresponding double bubble diagram
(Fig.~\ref{doubub}) are known in analytical
form, if the masses of the quarks in the
external loop are neglected
\cite{Kniehl90,Hoang94}.
The absorptive part from the cut through
the two (massless) quark lines contributes for
$s>0$ and is calculated in Ref.~\cite{Kniehl90}.
The one from the cut through all four
quark lines contributes for $s>4m_{{\rm t}}^2$ and
can be found in Ref.~\cite{Hoang94}. Only
the former is of relevance for the present
discussion.
Its contribution to
$r_{{\rm NS}}^{(0)}$ reads:
\begin{eqnarray}
\label{eq162}
%\begin{array}{ll}\displaystyle
r_{{\rm NS}}^{(0)} &=&
\left[\frac{\displaystyle\alpha_s(s)}{\displaystyle \pi}\right]^2
\displaystyle
\left\{
\frac{4}{9}\left(16x^2\right)
\left[
{\rm Li}_3(A^2) \zeta(3)2\zeta(2)\ln\, A
+ \frac{2}{3}\ln^3 A
\right]
\right.
\nonumber\\ \displaystyle
&&+~\frac{2}{27}\left(19+46x\right)
\sqrt{1+4x}
\left[
{\rm Li}_2(A^2) \zeta(2)+\ln^2 A
\right]
\\
\displaystyle
&&+~\frac{5}{54}\left(\frac{53}{3}+44x\Biggl)\ln\, x
+\frac{3355}{648}+\frac{119}{9} x
\right\}\, ,\nonumber
%\end{array}\end{equation}
\end{eqnarray}
where
$A=(\sqrt{1+4x}1)/\sqrt{4x}$ with $x=m_{{\rm t}}^2/s$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\begin{center}
\parbox{3cm}{
\mbox{\epsfig{file=fig4.eps,width=5.cm,height=5.cm}}
}
\end{center}
\caption[]{\label{doubub}{{Double Bubble Diagram.}}}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The leading term has also been determined
\cite{me93}
by employing the heavy mass expansion as
described in Section~\ref{decoupling}.
In the heavy top
limit the correction reads:
\begin{equation}
\label{mt2c}
r_{{\rm NS}}^{(0)} = \left[\frac{\displaystyle\alpha_s(s)}{\displaystyle
\pi}\right]^2 \frac{s}{m_{{\rm t}}^2}
\left(\frac{44}{675}+ \frac{2}{135}\ln\,
\frac{m_{{\rm t}}^2}{s}
\right)
{}\, .
\end{equation}
As shown in Fig.~\ref{figadd},
the heavy mass expansion provides an excellent
approximation to the full answer from
$m_{{\rm t}} \gg s$, even down to the threshold
$4m_{{\rm t}}^2=s$.
The result was
derived in the theory with $n_{{f}} =6$, whence
$\alpha_s$ should be taken for the correction term
accordingly. However, since
$\alpha_s\Big_{n_{{f}} =5}=\alpha_s\Big_{n_{{f}} =6}+
{\cal O}(\alpha_s^2)$,
this distinction is irrelevant for the terms under
consideration. Note that the diagrams
of Fig.~\ref{doubub} were
studied in Refs.~\cite{Bernreuther81,Bernreuther82},
where an exact double integral
representation was obtained.
The r.h.s. of (153)
was numerically evaluated in
Ref.~\cite{Soper94}.
It seems appropriate at this point to already here
anticipate the mass corrections
arising from internal loops of quarks
with $m^2/s\ll 1 $. Also, these corrections are
universal. They will be derived in Sections
\ref{repnsm2} and \ref{repnsm4}.
%4.3 and 4.4.
The leading $\alpha_s^2m^2/s$ term is absent.
The first nonvanishing terms are of order
$\alpha_s^3m^2/s$ and $\alpha_s^2m^4/s^2$ and provide
a correction,
\begin{equation}
\begin{array}{ll}\displaystyle
r^{(0)}_{{\rm NS}} =
& \displaystyle
\left[\frac{\displaystyle\alpha_s(s)}{\displaystyle \pi}\right]^3
\left[15+\frac{2}{3}~n_{{f}} \right]
\left[\frac{16}{3}4\zeta(3)\right]
\sum_{{f}} \frac{\overline{m}_{{f}}^2}{s}
\\ & \displaystyle
+\left[\frac{\displaystyle\alpha_s(s)}{\displaystyle \pi}\right]^2
\sum_{{f}} \frac{\overline{m}_{{f}}^4}{s^2}
\left[
\frac{13}{3}\ln\,\frac{\overline{m}_{{f}}^2}{s}
4\zeta(3)
\right]
{}\, .
\end{array}
\label{mt2b}
\end{equation}
These corrections,
as well as those from a heavy top,
apply equally well to
vector and axial correlators.
%\end{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}
\begin{center}
\leavevmode
\mbox{}\epsffile[100 300 500 540]{rhovlx.ps}
\caption{\label{figadd}
The function
$\varrho^{{\rm V}}$ describing virtual corrections in the
range $s/m^2<4$ (solid curve) and
the approximation of the heavy mass expansion
(dashed curve).}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Mass Corrections of Order $m^2/s$
\label{repnsm2}}
In view of the high precision reached in the
crosssection measurements the large size of the
firstorder corrections made the knowledge
of higher order QCD corrections desirable.
Their exact computation for arbitrary
quark masses would be a tremendous task.
Fortunately
for many considerations and experimental
conditions, quark masses can be neglected
in comparison with the characteristic energy of the
problem, or are considered as small parameters.
This holds true for the light
${\rm u,d}$ and ${\rm s}$ quarks, once the CMS energy
exceeds a few GeV, and is equally valid for charm and bottom quarks
at LEP energies of about $90$ GeV.
The problem may therefore be
simplified by performing an expansion in the
small parameter $m^2/s$,
which reduces the calculational effort
to massless propagator integrals.
The leading quadratic terms $m^2/s$ and,
for the case of lower energies, the
quartic mass terms $m^4/s^2$, represent
a very good approximation.
In first order this expansion is trivially
obtained in Eq.~(\ref{f1}) from the exact result.
We already noticed the large
logarithm $\ln\, m^2/s$ which makes the reliability
of perturbation theory questionable.
Its occurrence is connected to the use of the pole
mass as an expansion parameter.
The problem may be overcome
by employing RG techniques and is
conveniently achieved in the ${\overline{{ \mbox{{MS}}}}}$scheme.
In this calculational scheme leading logarithms
$\ln\, m^2/s$ are summed and absorbed in the
${\overline{{ \mbox{{MS}}}}}$ mass $\overline{m}(\mu^2)$ with $\mu$
being the
renormalization scale.
One can then write
\begin{eqnarray}
%\label{mbexp1}
%\begin{array}{ll}
\displaystyle r^{{\rm V}}(f)& =&
\displaystyle
r^{(0)} + \frac{\overline{m}^2}{s}r^{V(1)}
+ \frac{\overline{m}^4}{s^2}r^{V(2)}
+ {\cal O}(m^6/s^3)
{}\, ,
\nonumber\\ \displaystyle
\displaystyle r^{{\rm A}}(f) &=&
\displaystyle
r^{(0)} + \frac{\overline{m}^2}{s}r^{A(1)}
+ \frac{\overline{m}^4}{s^2}r^{A(2)}
+ {\cal O}(m^6/s^3)
\, .
%\end{array}
\end{eqnarray}
The massless results $r^{(0)}$ are identical
for the vector and the axial vector
correlators, whereas the mass corrections
$r^{V(n)}$ differ from $r^{A(n)}$ for $n\geq 1$.
It was found in Ref.~\cite{me82b}
(discussed in some detail in Section \ref{large})
that the ${\overline{{ \mbox{{MS}}}}}$ scheme has the remarkable
property of all coefficient functions for
QCD operators being polynomial in masses and momenta.
{}From this, and the fact that
no nontrivial operators of mass dimension two
exist in QCD, follows that no logarithms
$\ln\, m^2/s$ appear in $r^{V(1)}$ and $r^{A(1)}$.
Therefore $r^{V/A(0)}$ and $r^{V/A(1)}$ can be
written as a perturbation expansion to all orders
in $\alpha_s$ with mass independent coefficients.
Since to first order
$\alpha_s$ only trivial operators (unit operator
times a combination of quark masses) of mass
dimension four exist, logarithms
in $r^{(2)}$ are absent in order $\alpha_s$ and
show up for the first time in second order $\alpha_s^2$.
\subsubsection{VectorInduced Corrections\label{vector}}
In this section we demonstrate that
the mass corrections of
order ${\cal O}(\alpha_s^3)$ to the
flavour nonsinglet contribution of the
vectorinduced decay rate $\Gamma^{{\rm V}}$
can be obtained
from the threeloop vector current
correlator \cite{CheKue90}
without an explicit
fourloop calculation.
The argument
is based on the RG invariance
of the Adler function
{\renewcommand{\arraystretch}{2}
\begin{eqnarray}
\displaystyle
D^{{\rm V}}(Q^2) &=&
12 \pi Q^2 \frac{d}{dQ^2}\left(
\frac{\Pi^{{\rm V}}_1}{Q^2}\right)
\nonumber
\\
&&=
3~\Bigg\{ \left[ 1 + \frac{\displaystyle\alpha_s(\mu)}{\displaystyle \pi} +
\dots\right]
 \frac{\overline{m}^2(\mu)}{Q^2} \left[
b^{{\rm V}}_{00} + (b^{{\rm V}}_{01}+b^{{\rm V}}_{11}\ell)
\left(\frac{\displaystyle\alpha_s(\mu)}{\displaystyle \pi} \right) \right.
\nonumber
\\
%\phantom{MMM}
&&+
~(b^{{\rm V}}_{02}a^{{\rm V}}_{02}+b^{{\rm V}}_{12}\ell
+b^{{\rm V}}_{22}\ell^2)\left(
\frac{\displaystyle\alpha_s(\mu)}{\displaystyle \pi}\right)^2
\label{nsmv1}
\\
%\phantom{MMM}
&&+
\left.
(b^{{\rm V}}_{03}+b^{{\rm V}}_{13}\ell
+ b^{{\rm V}}_{23}\ell^2+b^{{\rm V}}_{33}
\ell^3)\left(\frac{\displaystyle\alpha_s(\mu)}{\displaystyle \pi}
\right)^3
\right] \Bigg\}
\nonumber
{}~,
\end{eqnarray}
}
\noindent
\hspace*{0.1cm}where
$\ell\equiv \ln\,(\mu^2/Q^2)$.
The term $a^{{\rm V}}_{02}$ originates from the
${\rm b}$ quark propagating in an inner
fermion loop.
Mass corrections are therefore also present for
the decay of the ${Z}$boson into massless quarks.
%%\newpage
The coefficients up to and including
the second order ${\cal O}(\alpha_s^2)$
were obtained in
\cite{GorKatLar86}
(the~$a^{{\rm V}}_{02}$~term~was~first~computed~in~
\cite{Bernreuther81}).~
They~read\footnote{
The result
\re{V9} was confirmed by
a direct calculation
in Ref.~\protect\cite{Chetyrkin93}. Hence,
the correction of
the originally
published coefficient
$992$ of $\zeta(3)$ in $b^{{\rm V}}_{02}$
to $1008$ suggested in
Ref.~\protect\cite{Levan89}
and unfortunately used
in Ref.~\cite{CheKue90}
turned out to be an error.
This fact has also been recently acknowledged by the very author
of Ref.~\cite{Levan89} in Ref.~\cite{Levan94}.
Numerically the use of the correct result
leads to only a slight
increase (less than 0.6\%)
in the magnitude of $\lambda_6^{{\rm V}}$ in comparison
with that given in Ref.~\protect\cite{CheKue90}.}
\begin{equation} \begin{array}{l}
b^{{\rm V}}_{00} = 6\, , \\
b^{{\rm V}}_{01} = 28\, , \;\;\;
b^{{\rm V}}_{11} = 12\, , \\
b^{{\rm V}}_{02} = [28799 + 992~\zeta(3)
 8360~\zeta(5)  882~n_f]/72\, , \\
b^{{\rm V}}_{12} = (3303  114~n_f)/18\, , \;\;\;
b^{{\rm V}}_{22} = (513  18~n_f)/18\, , \\
a^{{\rm V}}_{02} = [32  24~\zeta(3)]/3
{}\, \, .
\end{array}
\label{V9}
\end{equation}
The crucial point for the subsequent
calculation is the invariance of
$D^{{\rm V}}$ \re{nsmv1}
under RG transformations
\begin{equation}
\mu^2\frac{d}{d\mu^2} D^{{\rm V}} = 0
{}\, ,
\end{equation}
combined with the absence of $\ln\,{\mu^2/m^2}$
terms in Eq.~(\ref{nsmv1}).
Recursion relations between the
coefficients of the Adler function
allow the calculation of
the order ${\cal O}(\alpha_s^3)$ coefficients $b^{{\rm V}}_{13},
b^{{\rm V}}_{23},b^{{\rm V}}_{33}$ from the
lower order coefficients combined with
those of
anomalous mass dimension and the $\beta$function:
\renewcommand{\arraystretch}{2}
\begin{equation}
\begin{array}{l}
b^{{\rm V}}_{11} \displaystyle = 2b^{{\rm V}}_{00} \gamma_m^0 \, ,\\
b^{{\rm V}}_{12} \displaystyle = (\beta_0 + 2\gamma_m^0) b^{{\rm V}}_{01}
+ 2\gamma_m^1 b^{{\rm V}}_{00}\, , \\
b^{{\rm V}}_{22} \displaystyle = \frac{1}{2} (\beta_0
+ 2\gamma_m^0) b^{{\rm V}}_{11}\, , \\
b^{{\rm V}}_{13} \displaystyle = 2(\beta_0 + \gamma_m^0)(b^{{\rm V}}_{02}
a^{{\rm V}}_{02})
+(\beta_1 + 2\gamma_m^1) b^{{\rm V}}_{01}
+ 2\gamma_m^2 b^{{\rm V}}_{00} \, ,\\
b^{{\rm V}}_{23} \displaystyle = (\beta_0 + \gamma_m^0)b^{{\rm V}}_{12}
+\frac{1}{2}(\beta_1
+ 2\gamma_m^1) b^{{\rm V}}_{11} \, ,\\
b^{{\rm V}}_{33} \displaystyle = \frac{2}{3}(\beta_0
+ \gamma_m^0)b^{{\rm V}}_{22}~.
\label{V10}
\end{array}
\end{equation}
The coefficient $b^{{\rm V}}_{03}$
cannot be obtained via this recursion method.
However, the term proportional to this coefficient
does not contribute to $R^{{\rm V}}$.
The vector contribution
to the decay rate is then written in the
form:
\begin{eqnarray}
\frac{\overline{m}^2}{s} r_{{\rm V}}^{(1)}
&=&\frac{\overline{m}^2(\mu)}{s}
\Bigg\{
\lambda_0^{V}
+\frac{\alpha_s (\mu)}{\pi}
\left[
\lambda_1^{V}
+ \lambda_2^{V} \ln\,\frac{s}{\mu^2}
\right]
%\right.
\label{V10b}
\\
&&
+
%\left.
\left[ \frac{\alpha_s(\mu)}{\pi}\right]^2
\left[
\lambda_3^{V} + \lambda_4^{V} \ln\,\frac{s}{\mu^2}
+ \lambda_5^{V} \ln^2\frac{s}{\mu^2}
\right]
+
\left[ \frac{\alpha_s(\mu)}{\pi}\right]^3
\left[\lambda_6^{V} + \dots
\right]
+
\dots
\Bigg\}
{}\, .
\nonumber
\end{eqnarray}
If we set the normalization point $\mu^2=s$,
the remaining logarithms of
$s/\mu^2$ are absorbed in the running
coupling constant and the running
mass. The coefficients $\lambda$
can be obtained from the expansion
coefficients of the Adler function
by first integrating Eq.~(\ref{nsmv1})
to obtain $\Pi^{{\rm V}}/Q^2$ and subsequently taking
the imaginary part of $\Pi^{{\rm V}}/Q^2$ to arrive at
$r^{{\rm V}}$:
\begin{equation}
\begin{array}{lll}
\lambda^{{\rm V}}_0=0\, ,
&
\lambda^{{\rm V}}_1=b^{{\rm V}}_{11}\, ,
&
\lambda^{{\rm V}}_2=0\, ,
\\
\lambda^{{\rm V}}_3=b^{{\rm V}}_{12}2b^{{\rm V}}_{22}\, ,
&
\lambda^{{\rm V}}_4=2 b^{{\rm V}}_{22}\, ,
&
\lambda^{{\rm V}}_5=0\, ,
\\
\lambda^{{\rm V}}_6=
b^{{\rm V}}_{13}2b^{{\rm V}}_{23}
+(6\pi^2)b^{{\rm V}}_{33}\, ,
&
\lambda^{{\rm V}}_7 = 2b^{{\rm V}}_{23}+6b^{{\rm V}}_{33}\, ,
&
\lambda^{{\rm V}}_8=3b^{{\rm V}}_{33}\, ,
\ \
\lambda^{{\rm V}}_9=0
{}\, .
\end{array}
\label{V11}
\end{equation}
The term $\pi^2$ in $\lambda^{{\rm V}}_6$
is a consequence of the analytical
continuation
from spacelike to timelike momenta
and arises from
the term $\ln^3\mu^2/Q^2 \rightarrow
(\ln\, \mu^2/Q^2 \pm i\pi)^3$.
Explicitly, nonzero entries above read:
\begin{equation}
\begin{array}{l}
\lambda^{{\rm V}}_1 = 12\, ,
\\ \displaystyle
\lambda^{{\rm V}}_3 = ~\frac{13}{3}~n_{{f}}
+\frac{253}{2}\, ,
\\ \displaystyle
\lambda^{{\rm V}}_4 =  57 + 2~n_{{f}}\, ,
\\ \displaystyle
\lambda^{{\rm V}}_6 =
~\frac{1}{9}~n_{{f}}^2 \pi^2
+\frac{125}{54}~n_{{f}}^2
+\frac{17}{3}~n_{{f}} \pi^2
\frac{466}{27}~n_{{f}} \zeta(3)
+\frac{1045}{27}~n_{{f}} \zeta(5)
\\ \displaystyle
\phantom{ \lambda_6^{{\rm V}} = }

\frac{4846}{27}~n_f
\frac{285}{4}~\pi^2
+\frac{490}{3}~\zeta(3)
\frac{5225}{6}~\zeta(5)
+2442\, ,
\\ \displaystyle
\lambda^{{\rm V}}_7 =
~\frac{13}{9}~n_{{f}}^2
+\frac{175}{2}~n_f
\frac{4505}{4}\, ,
\\ \displaystyle
\lambda^{{\rm V}}_8 =
\frac{1}{3}~n_{{f}}^2
17~n_f
+\frac{855}{4}
{}\,\, .
\end{array}
\label{V11B}
\end{equation}
The $a_{02}$ contribution originates from the ${\rm b}$
quark vacuum polarization graphs and is thus also
present for final states with massless quarks.
(More precisely, it originates in this case from
QCD corrections to ${\rm q\overline{q}{\rm b}\overline{{\rm b}}}$
configurations.) The same correction
would arise in $r^{{\rm A}}$.
This term has been anticipated in Eq.~(\ref{mt2b}).
The final answer can still be interpreted
as an incoherent sum
of the contributions from different quark species.
In particular this implies that contributions
from three gluon intermediate final states
(singlet contributions) are absent in the
${\cal O}(\alpha_s^3)$ mass terms. This contrasts with the
corrections for $m=0$, which receive
thirdorder contributions precisely
from this configuration  see Eq.~\re{sm0v1} below.
Numerically, one finds a quite decent decrease in
the terms of successively higherorders, which
supports confidence in the applicability of
these results for predictions of the rate. This
will be studied in more detail in
Part~\ref{numerical}.
\subsubsection{Axial VectorInduced Corrections
\label{axial}}
The situation is more involved if
one wants to apply similar RG arguments
to the axial vectorinduced rate
in order to again compute the corresponding
mass corrections
from nonsinglet diagrams.
The comparison of
the expansion of the Adler function
\begin{equation} \begin{array}{ll} \label{V12} \displaystyle
D^{{\rm A}} = & \displaystyle 12\pi^2 Q^2
\frac{d}{dQ^2}\left(\frac{\Pi^{{\rm A}}_1}{Q^2}\right)
\\ \displaystyle
& = 3 \displaystyle
\Bigg\{ \left( 1 + \frac{\displaystyle\alpha_s(\mu)}{\displaystyle \pi} +
\dots\right)  \frac{\overline{m}^2(\mu)}{Q^2}
\left[
\sum_{\stackrel{\scriptsize i \leq j+1}
{i,j \geq0}} b^{{\rm A}}_{ij}\ell^{{i}}
\left(\frac{\displaystyle\alpha_s(\mu)}{\displaystyle \pi}\right)^{{j}}
\right]
+ {\cal O}(m^4) \Bigg\} {}\, ,
\end{array}\end{equation}
with Eq.~(\ref{nsmv1}) shows that in the
axial case the highest order
term of the power series in $\ell$
within a given order ${\cal O}(\alpha_s^{{j}})$
is proportional to $\ell^{j+1}$,
whereas in the vector case the
$\ell$expansion terminated at $\ell^{{j}}$.
This structure is dictated by
the anomalous dimension
\mbox{$\gamma^{{\rm AA}}_m$,} which vanishes in
the vector case.
\noindent
The expansion of $D^{{\rm V}}$ contained
the second order coefficient $a^{{\rm V}}_{02}$,
originating from an inner
${\rm b}$quark loop.
The same ${\cal O}(\alpha_s^3)$
term is also present in the axial case
for massless as well as massive
external quark lines. The mass correction
of ${\cal O}(\alpha_s^3)$ from external quark loops
has not yet been calculated.
The coefficients of the expansion (\ref{V12}) read:
\begin{equation} \begin{array}{lll} \displaystyle
b^{{\rm A}}_{00} = 12\, ,& b^{{\rm A}}_{10} = 6\, , & \\ \displaystyle
b^{{\rm A}}_{01} = \frac{151}{2}
+ 24\zeta(3)\, ,\qquad&
b^{{\rm A}}_{11} = 34\, ,\qquad& b^{{\rm A}}_{21} = 6
{}\, .
\label{V16}
\end{array}
\end{equation}
The complication in deriving from
these the secondorder
coefficients of the logarithmic terms
arises from the fact that
the mass dependent part of $D^{A}$
obeys the inhomogeneous
RG equation:
\begin{equation}
\mu^2\frac{d}{d\mu^2} D^{{\rm A}} = \frac{\overline{m}^2}{Q^2}
\gamma^{{\rm AA}}_m \equiv
\frac{\overline{m}^2(\mu)}{Q^2} \sum_{i\geq 0}
\left( \gamma^{{\rm AA}}_m\right)_{{i}}
\left[\frac{\displaystyle\alpha_s(\mu)}{\displaystyle \pi}\right]^{{i}}
{}\, .
\label{V14}
\end{equation}
Therefore, recursion relations can
be set up again, although in this case
order ${\cal O}(\alpha_s^{{n}})$
coefficients $b^{{\rm A}}_{kn}\;(k > 0)$
are not only expressed through the
$\{b_{ij}\}$ with $0 \leq i \leq j+1 \leq n$,
but also through the
expansion coefficients of the anomalous
dimension $(\gamma^{{\rm AA}}_m)_{{\rm r}}\;(r\leq n)$.
In fact, the secondorder coefficients
satisfy the relations
{\renewcommand{\arraystretch}{2}
\begin{equation} \begin{array}{ll}
%\rule{0em}{1.5em}
b^{{\rm A}}_{10} & \displaystyle =  \left( \gamma^{{\rm AA}}_m\right)_0
{}\, ,\\
%\rule{0em}{1.5em}
b^{{\rm A}}_{11} & \displaystyle =  \left( \gamma^{{\rm AA}}_m\right)_1
+ 2b^{{\rm A}}_{00} \gamma_m^0 {}\, ,\\
%\rule{0em}{1.5em}
b^{{\rm A}}_{21} & \displaystyle = b^{{\rm A}}_{10} \gamma_m^0 {}\, ,\\
%\rule{0em}{1.5em}
b^{{\rm A}}_{12} & \displaystyle =  \left( \gamma^{{\rm AA}}_m\right)_2 +
b^{{\rm A}}_{01} (\beta_0 + 2\gamma_m^0)
+ 2 b^{{\rm A}}_{00}\gamma_m^1 {}\, , \\
%\rule{0em}{1.5em}
b^{{\rm A}}_{22} & \displaystyle = \frac{1}{2}~ b^{{\rm A}}_{11}
(\beta_0 + 2\gamma_m^0)
+ b^{{\rm A}}_{10}\gamma_m^1 {}\, ,\\
%\rule{0em}{1.5em}
b^{{\rm A}}_{32} & \displaystyle = \frac{1}{3}~
b^{{\rm A}}_{21} (\beta_0+2\gamma_m^0) {}\, .
\label{V15}\end{array} \end{equation}
}
[Note: for the vanishing
anomalous dimension the Eqs.
(\ref{V10}) and (\ref{V15}) coincide.]
Therefore the anomalous dimension $\gamma^{{\rm AA}}_m$
must be known to the same order
to which the decay rate is computed.
The calculation of $\gamma^{{\rm AA}}_m$ is sketched
in Section~\ref{correlators}
and leads to the following
result \cite{CheKueKwi92}
\begin{equation}
\gamma^{{\rm AA}}_m = 6\left\{ 1+\frac{5}{3}~\frac{\alpha_s}{\pi}
+\left(\frac{\alpha_s}{\pi}\right)^2
\left[\frac{455}{72}\frac{1}{3}~n_{{f}}
\frac{1}{2}~\zeta(3)\right]
\right\}\, .
\label{V16b}
\end{equation}
As for the vector case
we write a general expansion
for the axial vectorinduced rate:
\begin{eqnarray}
\displaystyle
\frac{\overline{m}^2}{s}r_{{\rm A}}^{(1)}\displaystyle
& =&
\displaystyle
\frac{\overline{m}^2(\mu)}{s}
\Bigg\{ \lambda_0^{A}
+
\frac{\alpha_s (\mu)}{\pi}
\left[
\lambda_1^{A} + \lambda_2^{A} \ln\,\frac{s}{\mu^2}
\right]
\nonumber
\\
&&+
\left[ \frac{\alpha_s (\mu)}{\pi}\right]^2
\left[
\lambda_3^{A} + \lambda_4^{A} \ln\,\frac{s}{\mu^2}
+ \lambda_5^{A} \ln^2\frac{s}{\mu^2}
\right]
\Bigg\}
{}\, .
\end{eqnarray}
The coefficients
read as follows:
\begin{equation}\begin{array}{lll}
\lambda^{{\rm A}}_0=b^{{\rm A}}_{10}~,
&
\quad \lambda^{{\rm A}}_1=b^{{\rm A}}_{11}2b^{{\rm A}}_{21}~,
&
\quad \lambda^{{\rm A}}_2=2b^{{\rm A}}_{21}~,
\\
\lambda^{{\rm A}}_3=
b^{{\rm A}}_{12}2b^{{\rm A}}_{22}
+(6\pi^2)b^{{\rm A}}_{32}{}\, ,
&\quad\lambda^{{\rm A}}_4=2b^{{\rm A}}_{22}+6b^{{\rm A}}_{32}~,
&
\quad\lambda^{{\rm A}}_5=3b^{{\rm A}}_{32}
{}\, .
\end{array}
\end{equation}
Or, explicitly,
\begin{equation}\begin{array}{l}
\displaystyle
\lambda^{{\rm A}}_0 = 6~,
\\
\displaystyle
\lambda^{{\rm A}}_1 = 22~,
\\
\displaystyle
\lambda^{{\rm A}}_2 = 12~,
\\
\displaystyle
\lambda^{{\rm A}}_3=
~\frac{1}{3}~n_{{f}} \pi^ 2
 4~n_{{f}} \zeta(3) + \frac{151}{12}~n_{{f}}
+ \frac{19}{2}~\pi^2 + 117~\zeta(3)  \frac{8221}{24}\, ,
\\
\displaystyle
\lambda^{{\rm A}}_4 =
~\frac{16}{3}~n_{{f}} + 155~,
\\
\displaystyle
\lambda^{{\rm A}}_5=
n_{{f}}  \frac{57}{2}
{}.
\end{array}
\end{equation}
The discussion in this and the previous
section is
tailored for an external current
coupled to ${\rm b}\overline{{\rm b}}$ and includes
mass corrections
from internal
${\rm b}$ quark loops as well as from the
loops coupled to the external current.
A slightly different situation occurs for a
nonsinglet correlator arising
from massless quarks.
Internal bottom quarks as indicated in
the double bubble graph still induce
$m_{{\rm b}}^2/s$ corrections. However, a slight
generalization
of the arguments presented above
demonstrates that these terms are
again absent in order
${\cal O}(\alpha_s^2)$. From corrections of the
diagrams in Fig.~\ref{doubub} one
obtains the terms of order
${\cal O}(\alpha_s^3)$\,,
which should for convenience
be incorporated into
$r_{{\rm NS}}^{(0)}$ and summed over all
massive quark species, adding the term
\begin{equation}
\begin{array}{rl}\displaystyle
r^{(0)}_{{\rm NS}}\longrightarrow
& \displaystyle
r^{(0)}_{{\rm NS}}+
\left(\frac{\alpha_s}{\pi}\right)^3
\sum_f
\frac{\overline{m}_{{f}}^2}{s}
(2)(\beta_0+\gamma^0_{{m}})a_{02}
\\ \displaystyle
=
& \displaystyle
r^{(0)}_{{\rm NS}}
\left[\frac{\displaystyle\alpha_s(s)}{\displaystyle \pi}\right]^3
\left(15\frac{2}{3}~n_{{f}} \right)
\left[\frac{16}{3}4\zeta(3)\right]
\sum_{{f}}\frac{\overline{m}_{{f}}^2(s)}{s}
{}\, .
\end{array}
\label{V17}
\end{equation}\subsection{Mass Corrections of Order $m^4/s^2$
\label{repnsm4}}
Terms of higherorder in $m^2/s$ are quite
unimportant as far as ${Z}$ decays into bquarks
are concerned. However, at lower energies
these should be taken into account in order to arrive
at an adequate description of the crosssection.
In fact, as shown in Fig.~\ref{kvexp},
the order $\alpha_s$ correction functions
$K_{{\rm V}}$ and $K_{{\rm A}}$ introduced in Eq.~(\ref{ap1})
are well described by the first few terms
of the expansion in $m^2/s$\,, not only at
high energies, but even fairly close to
threshold. Hence one should
arrive at a reliable result to
${\cal O}(\alpha_s^2)$ near the threshold
through the incorporation of the first
terms of the expansion in $\alpha_s^2(m^2/s)^{{n}}$.
The secondorder calculation of quartic
mass corrections
presented below is based on
Ref.~\cite{TTP9408}.
The calculation was performed for
vector and axial vector current
nonsinglet correlators. The first
is of course relevant for electronpositron
annihilation into heavy quarks
at arbitrary energies, the second
for ${Z}$ decays into ${\rm b}$ quarks and for
top production at a future linear
collider.
Quartic mass corrections were already
presented to
order $\alpha_s$ in Part~\ref{exact}, expressed
in terms of the pole mass. In this section
the result in the ${\overline{{ \mbox{{MS}}}}}$ scheme is given
and the second order
$\alpha_s^2$ contribution is discussed.
The calculation is based on
the operator product expansion of
the Tproduct of two vector currents,
$J_\mu=\overline u\gamma_\mu d $ and
$J_\nu^+=\overline d\gamma_\nu u $.
Here ${u}$ and ${d}$ are simply two generically
different quarks with
masses $m_{{\rm u}}$ and $m_{{\rm d}}$.
The operator product expansion includes power
law suppressed terms up to operators of
dimension four induced by
nonvanishing quark masses.
Renormalization group arguments
similar to those already employed in
the previous section
allowed a deduction in the $\alpha_s^2 m^4$ terms.
Quarks which are
not coupled to the external
current will influence the result in order
$\alpha_s^2$ through their
coupling to the gluon field.
The result may be immediately transformed
to the case of the
electromagnetic current of a heavy, say,
${\rm t}$ (or ${\rm b}$) quark.
The asymptotic behaviour of the transverse part
of this
(operator valued) function for
$Q^2 = q^2\to\infty$ is given by
an OPE of the following form
(different powers of $Q^2$ may be studied
separately and only operators of dimension
4 are displayed):
\begin{equation}
i\int T(J_\mu(x) J^+_\nu(0)){\rm e}^{iqx} {\rm d} x
\bbuildrel{=}_{q^2\to\infty}^{} \frac{1}{Q^4}
\sum_n
(q_\mu q_\nu g_{\mu\nu} q^2)
%\fos{{\rm T}}{C_{{n}}}
{C_{{n}}}
(Q^2,\mu^2,\alpha_s)
O_{{n}}+\dots
\label{142}
\end{equation}
Only the gauge invariant operators
$G_{\mu\nu}^2, m_{{i}}\overline{q}_jq_{{j}}$ and a polynomial
of fourth order in the masses contributes to
physical matrix elements.
Employing renormalization group arguments
the vacuum expectation value of
$\sum_{{n}} C_n O_{{n}}$
is under control up to terms of order $\alpha_s$
as far as the constant terms are concerned
and even up to $\alpha_s^2$ for the
logarithmic terms proportional to $\ln\, Q^2/\mu^2$.
Only these logarithmic terms contribute to
the absorptive part. Hence one arrives at the
full answer for $\alpha_s^2m^4/s^2$ corrections.
Internal quark loops contribute in this order,
giving rise to the terms proportional
to $m^2 m_{{i}}^2$ and $m_{{i}}^4$ below.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The result reads [below we set for brevity
the $\overline{{\rm \rm MS}}$ normalization
scale $\mu = \sqrt{s}$
and $\overline{m}_{{\rm u}}(s) = \overline{m}_{{\rm d}}(s) =
\overline{m}$]:
\begin{figure}
%\vspace*{+0.5cm}
\begin{center}
\mbox{\epsfig{file=alphasex.eps,width=12.cm,height=9.cm,%
bbllx=0.cm,bblly=0.cm,bburx=18.cm,bbury=22.cm}
}
\end{center}
\caption{\label{alphasexp}{{Contributions
to $R^{{\rm V}}$ from $m^4$ terms
including successively higher
orders in $\alpha_s$ (order $\alpha_s^0$/
$\alpha_s^1$/ $\alpha_s^2$
corresponding to dotted/ dashed/ solid lines)
as functions of
$2m_{\rm pole}/\protect\sqrt{s}$.}}}
\end{figure}
\begin{eqnarray}
\displaystyle
\frac{\overline{m}^4}{s^2}
r_{{\rm V}}^{(2)}
\displaystyle &=&\frac{\overline{m}^4}{s^2}\Bigg\{
\displaystyle 6 22 \frac{\displaystyle\alpha_s(s)}{\displaystyle \pi}
\nonumber
\\
&&
+
\left[\frac{\displaystyle\alpha_s(s)}{\displaystyle \pi}\right]^2 \left[
n_{{f}} \left(
\frac{1}{3} \ln\,\frac{\overline{m}^2}{s}
\frac{2}{3}\pi^2\frac{8}{3}\zeta(3)
+\frac{143}{18}
\right)
\right.
\nonumber
\\
&&
\left.
\displaystyle
~\frac{11}{2} \ln\,\frac{\overline{m}^2}{s}
+ 27\pi^2+112\zeta(3)\frac{3173}{12}
+12 \sum_{{i}} \frac{\overline{m}_{{i}}^2}{\overline{m}^2}
\right.
\nonumber
\\
&&
\left.
\displaystyle
+\left(\frac{13}{3}4\zeta(3)\right)
\sum_{{i}} \frac{\overline{m}_{{i}}^4}{\overline{m}^4}

\sum_{{i}} \frac{\overline{m}_{{i}}^4}{\overline{m}^4}
\ln\,\frac{\overline{m_{{i}}}^2}{s}
\right]\Bigg\}
\, ,
\label{VM4N}
\end{eqnarray}
\newpage
\begin{eqnarray}
\displaystyle
\frac{\overline{m}^4}{s^2}
r_{{\rm A}}^{(2)}
&=&\frac{\overline{m}^4}{s^2}\Bigg\{
\displaystyle 6 +10 \frac{\displaystyle\alpha_s(s)}{\displaystyle \pi}
\nonumber
\\
&&
+
\left[\frac{\displaystyle\alpha_s(s)}{\displaystyle \pi}\right]^2 \left[
n_{{f}} \left(
~\frac{7}{3} \ln\,\frac{\overline{m}^2}{s}
+\frac{2}{3}\pi^2+\frac{16}{3}\zeta(3)
\frac{41}{6}
\right)
\right.
\nonumber
\\
&&
\left.
\displaystyle
+~\frac{77}{2} \ln\,\frac{\overline{m}^2}{s}
 27\pi^2220\zeta(3)+\frac{3533}{12}
12 \sum_{{i}} \frac{\overline{m}_{{i}}^2}{\overline{m}^2}
\right.
\nonumber
\\
&&
\left.
\displaystyle
+\left(\frac{13}{3}4\zeta(3)\right)
\sum_{{i}} \frac{\overline{m}_{{i}}^4}{\overline{m}^4}

\sum_{{i}} \frac{\overline{m}_{{i}}^4}{\overline{m}^4}
\ln\,\frac{\overline{m_{{i}}}^2}{s}
\right]\Bigg\}~.
\label{VM4N1}
\end{eqnarray}
Note that the sum over $i$ includes also the quark
coupled to the external current and with
mass denoted by $m$.
Hence in the case of one heavy quark ${\rm u}$ of
mass $m$ ($d \equiv u)$ one should set
$\sum_{{i}} {\overline{m}_{{i}}^4}/{\overline{m}^4} = 1$
and
$\sum_{{i}}{\overline{m}_{{i}}^2}/{\overline{m}^2} = 1$.
In the opposite case, when one considers
the correlator of light
(massless) quarks the heavy quarks appear only
through their coupling to gluons. There one
finds for the correction term:
\begin{equation}
\displaystyle
r_{{\rm V}} = r_{{\rm A}} =
\left[\frac{\displaystyle\alpha_s(s)}{\displaystyle \pi}\right]^2
\sum_{{f}} \frac{\overline{m}_{{f}}^4(s) }{s^2}
\left[
\frac{13}{3}
\ln\,\frac{\overline{m}^2_{{f}}(s)}{s}
4 \zeta(3)
\right]
{}\, ,
\label{VAM4}
\end{equation}
as anticipated in Eq.~(\ref{mt2b}).
The ${Z}$ decay rate is hardly affected by
the $m^4$ contributions. The
lowest order term in Eq.~\re{VM4N} evaluated with
$\overline{m}=2.6 \ {\rm GeV}$
amounts to
$\pm 6 \overline{m}/s^2=\pm 5\times 10^{6}$
for the vector
(axial vector) current induced
$Z\to {\rm b}\bar{{\rm b}} $ rate.
Terms of increasing order in $\alpha_s$
become successively smaller.
The $m_{{\rm b}}^4$ correction
to $\Gamma(Z\to {\rm q}\bar{{\rm q}})$\,,
which starts in
order $\alpha_s^2$\,,
is evidently even smaller.
It is worth noting, however, that the
corresponding series, evaluated in
the onshell scheme, leads to terms which
are larger by about one order
of magnitude and of oscillatory signs.
{}From these
considerations it is clear that $m^4$
corrections to the ${Z}$ decay
rate are well under control  despite
the still missing singlet piece
 and that they can be neglected for
all practical purposes.
The situation is different in the low
energy region  say
several GeV above the charm or the bottom
threshold. For definiteness
the second case will be considered and
for simplicity
all other masses will be put to
zero.
The contributions to $R^{{\rm V}}$ from $m^4$ terms are
presented in Fig.~\ref{alphasexp}
as functions of $2m/\sqrt{s}$ in the range
from 0.05 to 1.
The input parameters
$M _{\rm pole}=4.70$ GeV and
$\alpha_s(m_{{Z}}^2)=0.12$ have
been chosen.
Corrections of higherorders are added
successively. The prediction is
fairly stable with increasing order in
$\alpha_s$ as a consequence of
the fact that most large logarithms were
absorbed in the running mass.
The relative magnitude of the sequence
of terms from the $m^2$
expansion
is displayed in Fig.~\ref{massexp}. The
curves for $m^0$ and $m^2$ are based on
corrections up to third order in $\alpha_s$\,,
with the $m^2$ term starting
at first order. The $m^4$ curve receives
corrections from order zero to
two.
\begin{figure}
\begin{center}
\mbox{\epsfig{file=massexp.eps,width=10.cm,height=8.cm,%
bbllx=0.cm,bblly=0.cm,bburx=17.cm,bbury=21.cm}
}
\end{center}
\caption{\label{massexp}{{Predictions for
$R^{{\rm V}}$ including successively higher
orders in $m^2$.}}}
\end{figure}
Of course, very close to threshold  say
above 0.75 (corresponding to
$\sqrt{s}$ below\break\hfill 13 GeV)  the approximation
is expected to break down,
as indicated already in Fig.~\ref{kvexp}.
The validity of the approximation and a prediction
valid even closer towards threshold can only be achieved through
a full three loop calculation.
Below the ${\rm b}\bar{{\rm b}}$ threshold, however, one
may decouple the bottom quark and consider
mass corrections from the
charmed quark within the same formalism.
\subsection{
Heavy Quark Vacuum Polarisation to Three Loops
\label{schwinger}}
To order $\alpha_s$ the vacuum polarization was calculated by
K\"all\'en and Sabry in the context of QED a long time ago
\cite{Schw73}. In order $\alpha_s^2$ a variety
of qualitatively different contributions arises.
Diagrams with a massless external and a massive internal fermion loop
can be calculated in analytical form. Their contributions to $R$ are
given by Eq.~(\ref{eq162}) and Eq.~(\ref{apxc10}) in the Appendix 3.
Diagrams with two loops of fermions with
equal mass are straightforward: The two particle cut is known in
analytical form, the four particle cut is expressed
through a twodimensional integral \cite{HoaKueTeu95}.
%
The imaginary part of the ``fermionic contribution''  derived
from diagrams with a massless quark
loop inserted in the gluon propagator  has been calculated in
\cite{HoaKueTeu95}.
All integrals could be performed to the end and the result was expressed
in terms of polylogarithms.
In \cite{us} results for the cross section are presented in
order $\alpha_s^2$.
They are obtained from the vacuum polarisation $\Pi(q^2)$ which was
calculated up to three loops.
Instead of trying to perform the
integrals analytically,
the large $q^2$ behaviour of $\Pi(q^2)$ up to terms
of order $m^2/q^2$ was combined with
its Taylor series around $q^2=0$ was calculated up to terms of
order $q^8$. The leading and nexttoleading singularity is
deduced from the known behaviour of the nonrelativistic Green function
and the twoloop QCD potential.
Altogether eight constraints on $\Pi(q^2)$ are thus available,
four from $q^2=0$, two from $q^2\to\infty$ and two from
the threshold.
The contributions $\sim C_F^2, \sim C_A C_F$ and $\sim C_F T n_l$
have to be treated separately since they differ significantly
in their singularity structure.
For each of the three functions an interpolation was constructed
which incorporates the light and low energy calculations and is based
on conformal mapping and Pad\'e approximation.
Since the result for $C_F T n_l$ is available in closed form the
approximation method can be tested and shown to give excellent
result for this case.
To arrive at the final answer the following steps were
performed:
The contributions from diagrams with $n_l$ light
or one massive
internal fermion loop were denoted
by $C_F T n_l\Pi_{\mbox{\scriptsize\it l}}^{(2)}$ and
$C_F T\Pi_{\mbox{\scriptsize\it F}}^{(2)}$ with the group theoretical
coefficients
factored out. Purely gluonic corrections
are proportional to $C_F^2$ or $C_A C_F$. The former are the only
contributions in an abelian theory, the latter are characteristic for
the nonabelian aspects of QCD. It is essential
to
treat these two classes separately, since they exhibit qualitatively
different behaviour close to threshold. The following decomposition
of $\Pi(q^2)$ (and similarly for $R(s)$)
is therefore adopted
\begin{eqnarray}
\Pi &=& \Pi^{(0)} + \frac{\alpha_s(\mu^2)}{\pi} \Pi^{(1)}
\\&&
+ \left(\frac{\alpha_s(\mu^2)}{\pi}\right)^2
\left[
C_F^2 \Pi_{\mbox{\scriptsize\it A}}^{(2)}
+ C_A C_F \Pi_{\mbox{\scriptsize\it NA}}^{(2)}
+ C_F T n_l \Pi_{\mbox{\scriptsize\it l}}^{(2)}
+ C_F T \Pi_{\mbox{\scriptsize\it F}}^{(2)}
\right].
\end{eqnarray}
All steps described below have been performed seperately for
the first three contributions to $\Pi^{(2)}$.
The high energy behaviour of $\Pi$ provides important
constraints on the complete answer.
In the limit of small $m^2/q^2$ the constant term and the one
proportional to $m^2/q^2$ (modulated by powers of $\ln \mu^2/q^2$)
have been
calculated a long time ago
\cite{GorKatLar86}.
General arguments based on the influence of Coulomb exchange close to
threshold, combined with the information on the perturbative QCD
potential and the running of $\alpha_s$ dictate the singularities
and the structure of the leading cuts close to threshold, that
is for small $v=\sqrt{14m^2/s}$.
The $C_F^2$ term
is directly related to the QED result with internal photon lines only.
The leading $1/v$
singularity and the constant term of $R_A$
can be predicted from the nonrelativistic
Greens function for the Coulomb potential
and the ${\cal O}(\alpha_s)$ calculation.
The nexttoleading
term is determined by the combination of one loop
results again with the Coulomb singularities
One finds
\begin{eqnarray}
R_{\mbox{\scriptsize\it A}}^{(2)} &=&
3\left(\frac{\pi^4}{8v}  3\pi^2 + \ldots\right).
\label{Ra}
\end{eqnarray}
The contributions $\sim C_A C_F$ and $\sim C_F T n_l$ can be treated
in parallel.
The leading $C_AC_F$ and $C_F T n_l$ term in $R$ is proportional
to $\ln v^2$ and is responsible for the evolution of the
coupling constant close to threshold. Also the constant term can
be predicted by the observation, that the leading term in
order $\alpha_s$ is induced by the potential.
The ${\cal O}(\alpha_s)$ result
\begin{eqnarray}
R&=&3\left(\frac{3}{2}v+ C_F\frac{3\pi^2}{4}\frac{\alpha_s}{\pi}+\ldots
\right)
\end{eqnarray}
is employed to predict the logarithmic and constant $C_FC_A$ and
$C_FTn_l$ terms of ${\cal O}(\alpha_s^2)$ by replacing $\alpha_s$
%by $\alpha_V(4\vec{p}\,^2=v^2 s)$
by $\alpha_V(4{\bf p}\,^2=v^2 s)$
governing the QCD potential.
governing the QCD potential.
This implies the following threshold behaviour:
\begin{eqnarray}
R_{\mbox{\scriptsize\it NA}}^{(2)}&=&3\pi^2\left(\frac{31}{48}
\frac{11}{16}\ln v^2
+\frac{11}{16}\ln\frac{\mu^2}{s}
+\ldots \right),
\label{Rna}\\
R_{\mbox{\scriptsize\it l}}^{(2)}&=&3\pi^2\left(\frac{5}{12}
+\frac{1}{4}\ln v^2
+\frac{1}{4}\ln\frac{\mu^2}{s}
+\ldots \right).
\label{Rnl}
\end{eqnarray}
This ansatz can be verified for the $C_F T n_l$ term since in this case
the result is known in analytical form
\cite{HoaKueTeu95}.
Extending the ansatz from the behaviour of the imaginary part
close to the branching point into the complex plane
allows to predict the leading term of
$\Pi(q^2)$ $\sim \ln v$ and $\sim \ln^2 v$.
Important information is contained in the Taylor series of $\Pi(q^2)$
around zero:
\begin{equation}
\Pi^{(2)} =
\frac{3}{16\pi^2}
\sum_{n>0} C_{n} \left(\frac{q^2}{4m^2}\right)^n {}.
\end{equation}
The calculation of the first four nontrivial terms is
based on the evaluation of threeloop tadpole integrals.
The results for $C_n$ up to $n= 4$ can be found in \cite{us}.
The vacuum polarisation function
$\Pi^{(2)}$ is analytic in the complex plane
cut from $q^2=4m^2$ to $+\infty$. The Taylor series in $q^2$ is
thus convergent in the domain $q^2<4m^2$ only. The
conformal mapping
which corresponds to the variable
transformation
\begin{eqnarray}
\omega = \frac{1\sqrt{1q^2/4m^2}}{1+\sqrt{1q^2/4m^2}},\,\,\,\,
&&
\frac{q^2}{4m^2} = \frac{4\omega}{(1+\omega)^2},
\label{omega}
\end{eqnarray}
transforms the cut complex $q^2$ plane into the
interior of the
unit circle. The special points
$q^2=0,4m^2,\infty$ correspond to $\omega=0,1,1$, respectively.
The logarithmic singularities at threshold and large $q^2$
are removed by subtraction, the $1/v$ singularity, which is present
for the $C_F^2$ terms only, by multiplication with $v$ as
suggested in \cite{BaiBro95}.
The imaginary part of the remainder which is actually
approximated by the Pad\'e method is thus smooth in the
entire circle, numerically small and vanishes at
$\omega=1$ and $\omega=1$.
After performing the Pad\'e approximation for the smooth remainder
with $\omega$ as natural variable, the transformation (\ref{omega})
is inverted and the full vacuum polarisation function reconstructed
by reintroducing the threshold and high energy terms. This
procedure provides real and imaginary parts of $\Pi^{(2)}$.
In Figure \ref{figfull} the complete results are shown
for $\mu^2=m^2$ with
$R_{\mbox{\scriptsize\it A}}^{(2)}$, $R_{\mbox{\scriptsize\it NA}}^{(2)}$
and $R_{\mbox{\scriptsize\it l}}^{(2)}$
displayed separately. The solid curves are based on the
$[4/2]$, $[3/2]$ and $[3/2]$ Pad\'e approximants for $A$, $NA$ and $l$,
respectively. The threshold and high energy behaviour
%%%Eq.(\ref{threshold})
is given by the dashed curves. The exact analytical
result which is known for the $R_{\mbox{\scriptsize\it l}}^{(2)}$
contribution
only differs from the approximation curve in Figure \ref{figfull}
by less than the thickness of the line. The quality of the
approximation for $R_{\mbox{\scriptsize\it A}}^{(2)}$ and
$R_{\mbox{\scriptsize\it NA}}^{(2)}$
is confirmed by the comparison of the high energy behaviour
of the approximation with the known asymptotic behaviour
(Figure \ref{fighigh}). The quadratic approximation
(dashdotted line) is incorporated into $R^{(2)}$ by construction,
the quartic approximation shown (dashed line) is known from
\cite{TTP9408}, but is evidently very well recovered by the
method presented here.
\newpage
%\begin{figure}[ht]
\begin{figure}[H]
\begin{center}
\begin{tabular}{c}
\epsfxsize=11.5cm
\leavevmode
\epsffile[110 330 460 520]{picf21.ps}
\\
\epsfxsize=11.5cm
\leavevmode
\epsffile[110 330 460 520]{picacf1.ps}
\\
\epsfxsize=11.5cm
\leavevmode
\epsffile[110 330 460 520]{pinf1.ps}
\end{tabular}
\caption{\label{figfull} Complete results
plotted against
$v=\protect\sqrt{14m^2/s}$. The
high energy approximation includes the $m^4/s^2$
term.}
\end{center}
\end{figure}
%\begin{figure}[ht]
\begin{figure}[H]
\begin{center}
\begin{tabular}{c}
\epsfxsize=11.5cm
\leavevmode
\epsffile[110 330 460 520]{picf22.ps}
\\
\epsfxsize=11.5cm
\leavevmode
\epsffile[110 330 460 520]{picacf2.ps}
\\
\epsfxsize=11.5cm
\leavevmode
\epsffile[110 330 460 520]{pinf2.ps}
\end{tabular}
\caption{\label{fighigh}High energy region ($x=2m/\protect\sqrt{s}$).
The complete results (full line) are
compared to the high energy
approximations including the $m^2/s$
(dashdotted) and the $m^4/s^2$ (dashed) terms.}
\end{center}
\end{figure}
%
\newpage
\subsection{Partial Rates
\label{nspart}}
The formulae for the QCD corrections to the total
rate $\Gamma_{{\rm had}}$
have a simple, unambiguous meaning. The
theoretical predictions for individual
${\rm q}\bar{{\rm q}}$
channels, however, require additional
interpretation. In fact, starting from order
${\cal O}(\alpha_s^2)$ it is no longer possible
to assign all hadronic final states to well
specified ${\rm q}\overline{{\rm q}}$ channels in a unique manner.
The vector and axial vectorinduced rates receive
(nonsinglet) contributions from the diagrams,
where the heavy quark pair is radiated off a
light ${\rm q}\overline{{\rm q}}$
system (see Fig.~\ref{ffermi}a).
The analytical result for this contribution
for arbitrary $m^2/s$ can be found
in Ref.~\cite{Hoang94} and is reproduced in the
Appendix.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}
\begin{center}
\begin{tabular}{lll}
\parbox{3cm}{
\epsfig{file=fig5a.eps,width=5.cm,height=5.cm}
}
&
\hphantom{XXXXX}
&
\parbox{3cm}{
\epsfig{file=fig5b.eps,width=5.cm,height=5.cm}
}
\end{tabular}
\end{center}
\caption[]{\label{ffermi}
{{Nonsinglet ${\cal O} (\alpha_s^2)$
four fermion (a) and virtual (b)
contributions.}}
}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The rate for this
specific contribution to the
${\rm q}\bar{{\rm q}}{\rm b}\overline{{\rm b}}$
final state in the limit of small $m^2/s$
is given by
%??????cite{Hoang93}
\begin{eqnarray}
\label{nspart1}
%\begin{array}{ll}
\displaystyle
R^{{\rm NS}}_{{\rm q\overline{q}}{\rm b}\overline{{\rm b}}}
=\frac{\Gamma^{{\rm NS}}
_{{\rm q\overline{q}}{\rm b}\overline{{\rm b}}}}%
{\Gamma^{{\rm Born}}_{{\rm {q\overline{q}}}}}
\displaystyle
&=&
\left(\frac{\alpha_s}{\pi}
\right)^2
\frac{1}{27}\left\{
\ln^3 \frac{s}{m_{{\rm b}}^2}
\right.

\frac{19}{2} \ln^2 \frac{s}{m_{{\rm b}}^2}
+ \left[\frac{146}{3}12\zeta(2)
\right] \ln\, \frac{s}{m_{{\rm b}}^2}
\\
\nonumber
\displaystyle
&&
\left.
\frac{2123}{18}+38 \zeta(2) + 30\zeta(3)
\right\}
+ O(m_{{\rm b}}^2/s)
{}\, .
\end{eqnarray}
%\end{equation}
Numerically one obtains
\begin{equation}\label{nspart2}
R^{{\rm NS}}_%
{{\rm q}\overline{{\rm q}}{\rm b}\overline{{\rm b}}}=
\left(\frac{\alpha_s}{\pi}
\right)^2
\{
0.922/0.987/1.059
\}~;\;\;\;\;\;
{\rm for}\; m_{{\rm b}}=4.9/4.7/4.5\;{\rm GeV}
\, .
\end{equation}
The contributions from this configuration
to the total rate (in particular the
logarithmic
mass singularities) are nearly
cancelled by those from the corresponding
virtual corrections (see Fig. \ref{rhorv}b).
Despite the fact that ${\rm b}$ quarks are present
in the fourfermion final state,
the natural prescription is to assign
these events to the
${\rm q\overline{q}}$ channel.
They must be subtracted experimentally
from the partial rate $\Gamma_{{\rm b}\overline{{\rm b}}}$. This
should be possible, since their signature is
characterized by a large invariant mass of
the light quark pair and a small invariant mass
of the bottom system, which is emitted
collinear to the light quark momentum.
If this subtraction is not performed,
the inclusive bottom rate
is overestimated (for $m_{{\rm b}}$ =\break\hfill 4.70 GeV
and $\alpha_s$ chosen between 0.12 and 0.18) by
\begin{equation}\label{nspart3}
\Delta \equiv
\frac{\displaystyle \sum_{{\rm q}={\rm u,d,s,c,b}}
\Gamma^{{\rm NS}}%
_{{\rm q}\overline{{\rm q}}{\rm b}\overline{{\rm b}}}}
{\Gamma_{{\rm b}\overline{{\rm b}}}}
\approx 0.007 \; .. \; 0.016~.
\end{equation}
As shown in Fig.~\ref{rhorv}a the
leading contribution well matches the full
calculation (for bottom quarks it is about
10\% above the exact answer) leading to
the analytic result presented in the
Appendix.
A slightly different approach to the evalutaion
of heavy quark multiplicities, which attempts
the resummation of leading logarithms, can be
found in Ref.~\cite{Seymor}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}
\begin{center}
\leavevmode
\mbox{}\epsffile[100 300 500 540]{rhor.ps} \\
\mbox{}\epsffile[100 300 500 540]{rhov.ps}
\caption{\label{rhorv}
{{a)
The function $\varrho^{{\rm R}}$ describing the production of four
fermions in the region $0