%Title: HADRON RADIATION IN TAU PRODUCTION AND
%       THE LEPTONIC Z BOSON DECAY RATE
%Authors: A.H. Hoang, J.H. K\"uhn and T. Teubner,
%published:  <EM>Nucl. Phys.</EM><B> B455 </B><EM>(1995) 3-20.</EM>
% Univ. of Karlsruhe preprint TTP95-17,
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\begin{document}
\noindent
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\begin{flushright}
{\bf TTP95-17}\footnote{The complete postscript file of this
preprint, including figures, is available via anonymous ftp at
ttpux2.physik.uni-karlsruhe.de (129.13.102.139) as
/ttp95-17/ttp95-17.ps or via www at
http://ttpux2.physik.uni-karlsruhe.de/cgi-bin/preprints/ 
Report-no: TTP95-17.}\\ 
{\bf June 1995}\\
hep-ph/9507255\\
\end{flushright}
%
\begin{center}
\begin{bf}
 \begin{large}
 HADRON RADIATION IN \end{large}
 \begin{Large}$\tau$\end{Large}
\end{bf}
 \begin{large}
 PRODUCTION AND\\
 THE LEPTONIC $Z$ BOSON DECAY RATE\footnote{Work 
 supported by BMFT 056 KA 93P.}\\
 \end{large}
%
  \vspace{0.5cm}
  \begin{large}
   A.H. Hoang\footnotemark[4], J.H. K\"uhn\footnote{Present address: 
SLAC, Stanford University, Stanford, CA 94309.} and
   T. Teubner\footnote{e--mails: hoang@ttpux2.physik.uni-karlsruhe.de 
 and tt@ttpux2.physik.uni-karlsruhe.de} 
 \\
  \end{large}
  \vspace{0.1cm}
Institut f\"ur Theoretische Teilchenphysik, Universit\"at Karlsruhe,\\
D--76128 Karlsruhe, Germany \\
  \vspace{0.7cm}
  {\bf Abstract}\\
\vspace{0.3cm}
%
\noindent
\begin{minipage}{15.0cm}
\begin{small}
Secondary radiation of hadrons from a tau pair produced
in electron positron collisions may constitute an
important obstacle for precision measurements of
the production cross section and of branching ratios.
The rate for real and virtual radiation is calculated
and various distributions are presented.
For Z decays a comprehensive analysis is performed which
incorporates real and virtual radiation of
leptons. The corresponding results are also given for
primary electron and muon pairs.
Compact analytical formulae are presented for
entirely leptonic configurations.
Measurements of $Z$ partial decay rates which eliminate
all hadron and lepton radiation are about 0.3\% to 0.4\%
lower than totally inclusive measurements, a consequence of 
the ${\cal O}(\alpha^2)$ negative virtual corrections which 
are enhanced by the third power of a large logarithm. 
\end{small}
\end{minipage}
\end{center}
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%
%
% text
%
\noindent 
{\bf 1.~Introduction}\\
\par
\noindent 
Since the turn on of LEP several years ago experiments have
collected an impressive amount of data, with a combined
sample of more than $10^7$ hadronic and $10^6$ leptonic
events. This enormous event rate has allowed us to measure the
ratio of hadronic versus leptonic events or the leptonic
decay rate with an accuracy of better than two per mill.
Clearly, at this level of precision subtle aspects of
radiative corrections have to be taken into account. One specific 
example is the $Z$ decay rate into a lepton pair: Details of the 
treatment of additional radiation of a pair of soft leptons or of 
a soft hadronic system have a strong impact on the predicted rate. 
Formally the rate of these events is of 
order $(\alpha/\pi)^2\approx 5\cdot 10^{-6}$.
In reality, however, these events are enhanced by the third
power of a large logarithm, which essentially compensates
one factor of $\alpha/\pi$. This positive contribution is to
a large extent cancelled by the corresponding virtual
correction, leaving a small positive shift in the inclusive rate. 
The dominant part of this shift can be incorporated into the 
corrections of order $\alpha/\pi$ by employing the running QED 
coupling $\alpha(Q^2)$, the remainder is truly negligible. 
However, in an experimental analysis which eliminates events with
secondary radiation the large negative virtual corrections must 
be taken into account. 
\par
With this motivation in mind
we present in this paper a comprehensive study of virtual and
real leptonic and hadronic radiation. Particular emphasis is
put on the radiation of hadrons from a primary $\tau$ lepton pair.
This is the most difficult reaction to evaluate, since three
independent scales $s$,  $m_\tau$ and $m_{had}$ enter the
calculation. The results for the radiation of hadrons from
an electron or muon pair are significantly simpler:
The lepton mass is far smaller than the threshold for hadron 
production and can be safely set to zero throughout the calculation.
Results for this latter case have been given in~\cite{KKKS} for 
virtual corrections in the high energy approximation.
The corresponding case for real final state radiation with a
massless primary lepton has been solved in analytic form 
in~\cite{HJKT1,HJKT2}. The virtual and real radiation of a lepton 
pair off a massive primary lepton has been evaluated for
arbitrary $s$ in~\cite{HKT1} for the special case that the mass of the
secondary lepton is far smaller than all other scales of the problem.
In this paper we complete this topic by considering
radiation of hadrons from a primary $\tau$ pair. Since large
data samples of $\tau$ pairs are also collected at lower
energies, e.g. at around 10 GeV, this case also is
scrutinized. However, it is observed that radiation is far
less important in this low energy region, an obvious
consequence of the less prominent role of the ``large'' logarithms.
Hadronic radiation off primary $\tau$ pairs also plays a
special role from a purely practical point of view: $\tau$
leptons decay into hadrons (plus a neutrino) and hence
events with additional hadron radiation might well be
considered experimentally as hadronic final states, since
they have larger hadronic multiplicity and the invariant
mass of the hadronic subsystems exceeds $m_\tau$. By the
same token, the measurement of $\tau$ decays into large
multiplicity final states could in principle be influenced
by inappropriate assignment of the radiative events. Last but 
not least, signals of ``new'' physics could be faked by a
misinterpretation of these Standard Model reactions.
\par
The final state radiation discussed  in this paper is
complementary to the initial state radiation treated in~\cite{KKKS,JSM}.
No interference terms are present between radiation from the initial 
and final state as long as the (dominant) QED reactions are considered
and axial vector couplings from neutral current reactions are ignored.
Final state radiation of lepton (and quark) pairs has also
been treated in~\cite{vdB} in the context
of a Monte  Carlo Program. For hadron radiation this
approach turns out to be inadequate, since the reaction is
dominated by low mass hadronic systems. For lepton radiation our 
results are lower than those of~\cite{vdB} by about 15\%.
An important ingredient for the evaluation of real and virtual 
hadronic radiation is a proper description of $R(s)$, which has to be 
taken from experiment. In our approach we employ the most recent
parametrization of data taken from~\cite{jegerl}.
\par 
The plan of this paper is as follows: In section 2 the
``master formula'' for calculation of the total rate is
presented, which is at the same time a convenient starting
point for a variety of differential distributions. The case
of real radiation with primary $\tau$-leptons and secondary
hadrons is discussed in some detail, considering high
energies ($\sqrt{s}=M_Z$) as well as $\sqrt{s}=10$~GeV 
(characteristic for a $B$ factory) and $4.2$~GeV (relevant 
for a $\tau$-charm factory). 
Predictions for the total 
rate are presented, as well as distributions which are differential
with respect to the mass of the hadronic system and its
energy. The corresponding virtual radiation is calculated 
in section 3 for the same c.m. energies and at the $\tau$ pair 
threshold. Corrections are given for 
the Dirac and Pauli form factor, as well as for the total rate.
Section 4 is concerned with a detailed discussion of $Z$ decays. 
Results for electron, muon and tau as primary
leptons are presented both for hadronic and leptonic
radiation. The corresponding predictions for virtual
radiation are also collected. It is demonstrated that
these virtual corrections reduce the exclusive leptonic rate
of the $Z$ boson by several per mill.
Section 5 contains a summary and \pagebreak[4]
our conclusions.
\par
\vspace{1cm}\noindent
{\bf 2.~Real radiation}\\
\par 
\noindent 
The calculation of the ratio $\sigma_{\tau^+ \tau^- had}/
 \sigma_{pt}$ (where $\sigma_{pt} = 4\,\pi\,\alpha^2/3\,s$) can be 
reduced to the numerical evaluation of the two dimensional integral 
\begin{eqnarray}
\lefteqn{
 R_{\tau^+ \tau^- had} \equiv 
 \frac{\sigma_{\tau^+ \tau^- had}}
      {\sigma_{pt}} = 
 \frac{1}{3} \, 
 \left(\frac{\alpha}{\pi}\right)^2\,
 \int\limits_{4m_{\pi}^2}^{(\sqrt{s}-2m_{\tau})^2} \frac{{\rm d}s'}{s'} 
 \, R_{had}(s') \, F(s'/s) \,,} \label{masterreal}
 \\[1mm] 
\lefteqn{
F(z) = 4 
 \int\limits_{4m_{\tau}^2/s}^{(1-\sqrt{z})^2} {\rm d}y \, 
 \left\{ 
 - \, 
 \sqrt{1-\frac{4m_{\tau}^2}{s\,y}} \, \Lambda^{1/2}(1,y,z)\,\left[ \, 
  \frac{1}{4} + \frac{\frac{2m_{\tau}^2}{s}+\frac{4m_{\tau}^4}{s^2}+ 
                      \left(1+\frac{2m_{\tau}^2}{s}\right)z}
                     {(1-y+z)^2 - 
                      \left(1-\frac{4m_{\tau}^2}{s\,y}\right) \, 
                     \Lambda(1,y,z)} 
  \, \right]\,\right.}\nonumber\\[1mm]  & & \left. +
 \frac{\frac{2m_{\tau}^4}{s^2}+\frac{m_{\tau}^2}{s}(1-y+z)-\frac{1}{4}
           (1-y+z)^2-\frac{1}{2}(1+z)y}{1-y+z}\, 
  \ln\frac{1-y+z-\sqrt{1-\frac{4m_{\tau}^2}{s\,y}}\,\Lambda^{1/2}(1,y,z)}
          {1-y+z+\sqrt{1-\frac{4m_{\tau}^2}{s\,y}}\,\Lambda^{1/2}(1,y,z)} 
 \mbox{} \,  \right\} \nonumber 
\end{eqnarray}
where
$$
\Lambda(1,y,z)=1+y^2+z^2-2(y+z+y\,z) \,.
$$
$F(s'/s)$ can be interpreted as the normalized cross section 
for $\tau$ pair production with the additional emission of a 
vector boson of mass $\sqrt{s'}$. The integral in (\ref{masterreal}) 
sums up all hadronic contributions of mass $\sqrt{s'}$ weighted 
by $R_{had}(s')$. 
\begin{figure}
\begin{center}
\leavevmode
\epsfxsize=6.5cm
\epsffile[200 360 360 480]{figreal.ps}
\leavevmode
\epsfxsize=6.5cm
\epsffile[200 360 360 480]{figvirt.ps}
\vskip -3mm
\caption[]{{\em Typical diagrams for real and virtual emission of 
hadrons in $\tau$ pair production.}}
\label{figgraphs} 
\end{center}
\end{figure}
%
The numerical 
treatment of this integral allows us to use experimental input for the
function $R_{had}(s')$ and avoids the poor approximations of a parton 
model inspired calculation. 
In contrast to \cite{HKT1} --- there we solved
the integrals analytically in the case where the ``massive'' photon
splits into a pair of fermions with mass far smaller 
than $m_{\tau}$ --- $F(s'/s)$ is calculated numerically.
We would like to emphasize that (\ref{masterreal}) contains no restriction 
on the mass values and is thus also applicable to completely different 
kinematical situations.
In the limit of vanishing external mass $m_{\tau}=0$ we 
recover eq.~(2) of \cite{HJKT1} for $F(s'/s)$. 
\par
The physical situation described by (\ref{masterreal}) consists typically 
of a $\tau$ pair with large invariant mass and a hadronic state with 
low invariant mass (see also the discussion of our results below). 
The hadrons are emitted relatively
collinear with one of the $\tau$ leptons.
The reverse configuration, where a $\tau$ pair is radiated off the 
corresponding quark via a virtual photon could be treated with a 
similar technique. This final state would consist mainly of quite 
soft and collinear virtual photons leading to a $\tau$ pair of low 
invariant mass and is therefore easily distinguished from the 
situation we focus on in this letter.
As far as the total rate is concerned, 
the interference term between these two amplitudes vanishes
if the production is induced 
by the (electromagnetic or neutral) vector current. For axial 
vector induced contributions the interference term is different 
from zero. However, this term does not exhibit a mass singularity 
in the limit of vanishing quark or leptonic mass and
thus is not enhanced by a large logarithm. It can be 
neglected in all cases of interest. In the limit of vanishing 
fermion mass this interference term has been calculated in \cite{KK}. 
\par
For most of this paper we will discuss the total rate for 
secondary radiation. However, eq.(\ref{masterreal}) contains 
implicitly also information on single and double differential 
distributions. The integration variable $s^\prime$ denotes
the squared hadronic mass. The square of the invariant mass
of the primary $\tau^+\tau^-$ system, on the other hand, is given by 
$y\cdot s$, which can be easily related to the energy 
$E_{\scriptsize had}$ of the hadronic system. 
The single differential distribution 
${\rm d} R_{\tau^+ \tau^- had}/{\rm d} s^\prime$
is trivially obtained from~eq.(\ref{masterreal}). The energy 
distribution of the radiated hadronic system is given by 
\begin{eqnarray}
\lefteqn{
\frac{{\rm d} R_{\tau^+ \tau^- had}}{{\rm d} E_{\rm\scriptsize had}}  = } 
  \nonumber\\[1mm]
& & \frac{4}{3}\, \left(\frac{\alpha}{\pi}\right)^2\,
 \int\limits_{s'_{\scriptsize min}}^{E_{\scriptsize had}^2} 
   \frac{{\rm d}s'}{s\,s'} 
 \, R_{had}(s') \, 
 \left\{ - \, 
 \sqrt{1-\frac{4m_{\tau}^2}{s\,y}} \, 
 \sqrt{E_{\scriptsize had}^2-s^\prime} \,\left[ \, 
  1 + \frac{2m_{\tau}^2+\frac{4m_{\tau}^4}{s}+ 
               \left(1+\frac{2m_{\tau}^2}{s}\right)s^\prime}
           {E_{\scriptsize had}^2-
            \left(1-\frac{4m_{\tau}^2}{s\,y}\right) \, 
            (E_{\scriptsize had}^2-s^\prime)} 
  \, \right]\,\right.\nonumber\\[1mm] & & \left.\quad +\,
 \frac{s}{E_{\scriptsize had}}\left(
   \frac{2m_{\tau}^4}{s^2}+
   \frac{2 E_{\scriptsize had} m_{\tau}^2}{s^{3/2}}-
   \frac{E_{\scriptsize had}^2}{s}-\frac{1}{2}(1+\frac{s^\prime}{s})y 
 \right)\, 
  \ln\frac{E_{\scriptsize had}-\sqrt{1-\frac{4m_{\tau}^2}{s\,y}}\,
              \sqrt{E_{\scriptsize had}^2-s^\prime}}
          {E_{\scriptsize had}+\sqrt{1-\frac{4m_{\tau}^2}{s\,y}}\,
              \sqrt{E_{\scriptsize had}^2-s^\prime}}
 \mbox{} \,  \right\}
\label{diffenergyreal}
\end{eqnarray}
with 
\begin{equation}
s\,y = s+s^\prime-2 E_{\scriptsize had} \sqrt{s} \qquad 
\mbox{and}\qquad 
s'_{\scriptsize min} = {\rm max}\left(4 m_\tau^2-s+2 E_{\scriptsize 
had}\,\sqrt{s}\,,\, 4 m_\pi^2\right)\,.
\end{equation}
These formulae will be useful below.
\par
In the numerical evaluation of (\ref{masterreal}) we use experimental data 
for $R_{had}(s')$ as presented in \cite{jegerl}\footnote{We thank 
F.~Jegerlehner and S.~Eidelman for providing their 
compilation of data and discussing its proper treatment.}.
A convenient approach adapted to the experimental procedure is to decompose 
the hadronic final states into continuum and resonance contributions. 
Following \cite{jegerl}, the two pion contribution ($\widehat{=} \rho$) is 
taken into account together with the continuum by a direct integration of 
data points. The most important narrow resonances are added separately 
using the narrow width approximation\footnote{Masses and electronic widths 
are taken from \cite{PDG1994}.}, which is accurate enough for our
purposes. More details about $R_{had}(s')$ are given in the appendix.
\par
In Table~\ref{tab1} 
we display the continuum plus $\rho$ and the resonance 
contributions separately for different center of mass energies 
$\sqrt{s} \le M_Z$. 
The contribution from a virtual $Z$ as intermediate state remains 
unimportant up to $\sqrt{s} \approx M_Z$. They are strongly 
suppressed for obvious kinematical reasons as can be easily seen 
in Table~\ref{tab1} (see also discussion below). This in turn 
justifies neglecting radiation through the virtual $Z$. 
In Fig.~\ref{scanr}\ $\ R_{\tau^+ \tau^- had}$ 
is plotted for the range $2 m_{\tau} < \sqrt{s} < M_Z$. 
The rise of $R_{\tau^+ \tau^- had}$ and 
$R_{\tau^+ \tau^- l^+ l^-}$ for increasing $\sqrt{s}$ is easily understood 
from the leading term in the high energy limit proportional to the
third power of the logarithm of $s$, i.e. 
$\ln^2(m_{had}^2/s)\ln(m_\tau^2/s)$, $m_{had}$ being 
a characteristic hadronic mass scale of ${\cal O}(300\ {\rm MeV})$. (For 
leptonic radiation these logarithms can be calculated explicitly, 
see eq.~(\ref{mismallreal}).)
\par 
\begin{table}
\begin{center}
\begin{tabular}{|c||c|c|c|} \hline
$\sqrt{s}$ & $4.2$ GeV       & $10$ GeV        & $M_Z$           \\ \hline 
\hline
$\sqrt{s'_{max}}$ 
           & $0.646$ GeV     & $6.446$ GeV     & $87.63$ GeV     \\ \hline 
\hline \hline
Channel    & \multicolumn{3}{|l|}{}                              \\ \hline 
$2 \pi$    & $5.11\cdot 10^{-9}$ & $2.07\cdot 10^{-5}$ & $1.81\cdot 10^{-4}$ 
\\ \hline
$\omega$   & --              & $2.19\cdot 10^{-6}$ & $2.08\cdot 10^{-5}$ \\ 
\hline
$\phi$     & --              & $2.81\cdot 10^{-6}$ & $3.22\cdot 10^{-5}$ \\ 
\hline
$J/\psi$   & --              & $3.54\cdot 10^{-7}$ & $2.82\cdot 10^{-5}$ \\ 
\hline
$\Upsilon$ & --              & --              & $8.17\cdot 10^{-7}$ \\ 
\hline
\multicolumn{4}{|l|}{Continuum (with $R_{had}(s')$ from experiment):} \\ 
\hline 
$0.81$ up to $4.98$ GeV 
           & --              & $1.22\cdot 10^{-5}$ & $2.59\cdot 10^{-4}$ \\ 
$4.98$ up to $10$ GeV 
           & --              & $7.13\cdot 10^{-9}$ & $0.60\cdot 10^{-4}$ \\ 
$10$ up to $25$ GeV 
           & --              & --              & $0.30\cdot 10^{-4}$\\ 
$25$ up to $40$ GeV 
           & --              & --              & $0.03\cdot 10^{-4}$\\ 
\hline 
\multicolumn{4}{|l|}{Continuum (with perturbative $R_{had}(s')$)} \\ 
\hline 
$\sqrt{s'} > 40$ GeV 
           & --              & --              & $3.70\cdot 10^{-7}$ \\ 
\hline \hline 
$R_{\tau^+ \tau^- had}$ 
           & $5.11\cdot 10^{-9}$ & $3.82\cdot 10^{-5}$ & $6.14\cdot 10^{-4}$ 
\\ \hline \hline \hline 
$R_{\tau^+ \tau^- e^+ e^-}$ 
           & $2.75\cdot 10^{-5}$ & $4.05\cdot 10^{-4}$ & $1.82\cdot 10^{-3}$
\\ \hline 
$R_{\tau^+ \tau^- \mu^+ \mu^-}$ 
           & $1.14\cdot 10^{-7}$ & $2.82\cdot 10^{-5}$ & $2.94\cdot 10^{-4}$
\\ \hline 
$R_{\tau^+ \tau^- \tau^+ \tau^-}$ 
           & --                  & $3.07\cdot 10^{-8}$ & $3.13\cdot 10^{-5}$
\\ \hline \hline \hline
   Sum     & $2.76\cdot 10^{-5}$ & $4.71\cdot 10^{-4}$ & $2.76\cdot 10^{-3}$ 
\\ \hline
\end{tabular}
\caption[]{\label{tab1}
{\em Contributions to $R_{\tau^+ \tau^- had}$ and values of 
$R_{\tau^+ \tau^- l^+ l^-}$ ($\,l = e,\,\mu\,,\tau$) 
for center of mass 
energies $\sqrt{s} = 4.2$ GeV/$10$ GeV/$M_Z$. $\sqrt{s'_{max}}$ is the upper 
limit of the phase space integral (\ref{masterreal}). }}
\end{center}
\end{table}
%
\begin{figure}
\begin{center}
\leavevmode
\epsfxsize=12cm
\epsffile[120 300 450 530]{scanr.ps}
\vskip -3mm
\caption[]{{\em Normalized cross section 
$R_{\tau^+ \tau^- had}$.}}
\label{scanr} 
\end{center}
\end{figure}
%
As an interesting exercise one may apply (\ref{masterreal}) to 
$\tau^+ \tau^- q \bar q$ production where the 
quarks are again radiated off the tau leptons leaving $\tau$ pairs of 
large invariant mass\footnote{In this case one just has to use 
$R(s') = N_c \, Q_q^2 \, (1+\frac{2m_q^2}{s'})\,
 \sqrt{1-\frac{4m_q^2}{s'}}$ in the numerical integration.}. 
This leads to a parton model like prediction of 
$R_{\tau^+ \tau^- had}$. Adopting 
the quark mass values of \cite{vdB}, viz.
$m_u=m_d=m_s=300 \ {\rm MeV}$, $m_c=1.5 \ {\rm GeV}$
and $m_b=4.5 \ {\rm GeV}$ and choosing $\sqrt{s} = 10\ {\rm GeV}\,/\,M_Z$ 
one \pagebreak[4] would predict 
\begin{eqnarray}
R_{\tau^+ \tau^- u \bar u}&=&4\,R_{\tau^+ \tau^- d \bar d} = 
                             4\,R_{\tau^+ \tau^- s \bar s} = 
 1.1\cdot 10^{-5}\,/\,2.1\cdot 10^{-4}\,,
 \nonumber\\
R_{\tau^+ \tau^- c \bar c}&=&
 1.2\cdot 10^{-7}\,/\,5.1\cdot 10^{-5}\,,\nonumber\\
R_{\tau^+ \tau^- b \bar b}&=&
 \qquad 0 \quad\ \  / \,2.6\cdot 10^{-6}\,,\nonumber\\[1mm]
R_{\tau^+ \tau^- had}&=&
 1.7\cdot 10^{-5}\,/\,3.7\cdot 10^{-4}\,.
\label{partonmodel}
\end{eqnarray}
It is obvious that this model underestimates the production rate by 
about a factor of two. (The same phenomenon has also been observed 
for hadron radiation off muons and electrons \cite{HJKT1}.) 
For energies near the $\tau^+ \tau^-$ production 
threshold a parton model like prediction is completely impossible 
without artificial tuning of the masses of the light quark flavours.
\par
\begin{figure}
\begin{center}
\leavevmode
\epsfxsize=8.6cm
\epsffile[170 320 400 520]{cutr10.ps}
\leavevmode
\epsfxsize=8.6cm
\epsffile[170 320 400 520]{cutrmz.ps}
\vskip -3mm
\caption[]{{\em Dependence of the normalized cross section 
$R_{\tau^+ \tau^- had}$ from a cut $s'_{cut}$ for 
$\sqrt{s} = 10\ {\rm GeV}$ and $\sqrt{s} = M_Z$.}}
\label{figcut} 
\end{center}
\end{figure}
%
For completeness we also list in Table~\ref{tab1} the rate for the 
radiation of lepton ($e,\,\mu,\,\tau$) pairs. The prediction for the 
radiation of $e^+ e^-$ and $\mu^+ \mu^-$ is based on the analytical 
formula given in \cite{HKT1}, the one for $\tau^+ \tau^- \tau^+ \tau^-$ 
is obtained through numerical integration. An adequate approximation 
(better than 1\%) for 
$R_{\tau^+ \tau^- \tau^+ \tau^-}$ at $\sqrt{s} = M_Z$ is presented in
eq.~(\ref{miequalreal}) at the end of section 4. 
\begin{figure}
\begin{center}
\leavevmode
\epsfxsize=8.6cm
\epsffile[175 335 400 515]{drds10.ps}
\leavevmode
\epsfxsize=8.6cm
\epsffile[175 335 400 515]{drde10.ps}\\
\leavevmode
\epsfxsize=8.6cm
\epsffile[175 335 400 515]{drdsmz.ps}
\leavevmode
\epsfxsize=8.6cm
\epsffile[175 335 400 515]{drdemz.ps}
\vskip -3mm
\caption[]{{\em Differential distributions for real emission of 
hadrons in $\tau$ pair production as described in the text.}}
\label{figtauhaddif} 
\end{center}
\end{figure}
\par
Table~\ref{tab1} 
and Fig.~\ref{scanr} demonstrate that hadronic radiation 
may become important for precision measurements of $\tau$ branching ratios 
on top of the $Z$. Radiation of $e^+ e^-$ or $\mu^+ \mu^-$ may even be of 
relevance in the $10$ GeV region, e.g. at a $B$ meson factory.
\par
It is evident that experimental sensitivity and the probability to 
assign $\tau^+ \tau^-$ events with hadronic radiation to exclusive or 
inclusive $\tau$ pair production depends strongly on the invariant mass 
of the hadronic system. As mentioned above the integral (\ref{masterreal}) 
is in fact dominated by small $s'$. In the 
numerical approach this is easily shown by replacing the upper limit 
of the integral in (\ref{masterreal}) by a value 
$s'_{cut} < s'_{max} = (\sqrt{s}-2\,m_{\tau})^2$. In 
Fig.~\ref{figcut} we display our results for 
$R_{\tau^+ \tau^- had}(s,s'_{cut}) = 
\sigma_{\tau^+ \tau^- had}(s,s'_{cut})/\sigma_{pt}$  for 
$\sqrt{s}=10$ GeV/$M_z$. Even for $Z$ decays 86\% of the integral 
originates from $\sqrt{s'} < 5$ GeV and 58\% from 
$\sqrt{s'} < m_{\tau}$. 
In Fig.~\ref{figtauhaddif} we present the differential distributions 
${\rm d} R_{\tau^+ \tau^- had}/{\rm d} s^\prime$ and
${\rm d} R_{\tau^+ \tau^- had}/{\rm d} E_{\scriptsize had}$
for $\sqrt{s} = 10\ {\rm GeV}\,/\,M_Z$ 
using eqs.(\ref{masterreal}) and (\ref{diffenergyreal}). 
Contributions from the narrow resonances are not taken into account 
for the distributions ${\rm d} R_{\tau^+ \tau^- had}/{\rm d} s^\prime$ but 
can easily be read off from Table~\ref{tab1}. 
As expected the distributions are peaked at small $s^\prime$ (already 
evident from Fig.~\ref{figcut}) and at fairly small $E_{\scriptsize had}$ 
illustrating that hadronic radiation is clearly dominated by 
the $\rho$--resonance. 
It is not inconceivable that these events might 
affect precision measurements of multipion decay modes of the~$\tau$.
In view of possible experimental applications we also present in 
Fig.~\ref{figtauedif} the differential distributions 
${\rm d} R_{\tau^+ \tau^- e^+ e^-}/{\rm d} s^\prime$ and
${\rm d} R_{\tau^+ \tau^- e^+ e^-}/{\rm d} E_{e^+ e^-}$ for the 
radiation of electron--positron pairs at $\sqrt{s} = 10\ {\rm GeV}$ 
and at $M_Z$. These are evidently also peaked at 
extremely small values of $s^\prime$ and $E_{e^+ e^-}$. 
\begin{figure}
\begin{center}
\leavevmode
\epsfxsize=8.6cm
\epsffile[175 335 400 515]{rimte10.ps}
\leavevmode
\epsfxsize=8.6cm
\epsffile[175 335 400 515]{rente10.ps}\\
\leavevmode
\epsfxsize=8.6cm
\epsffile[175 335 400 515]{rimtemz.ps}
\leavevmode
\epsfxsize=8.6cm
\epsffile[175 335 400 515]{rentemz.ps}
\vskip -3mm
\caption[]{{\em Differential distributions for real emission of 
$e^+ e^-$ in $\tau$ pair production as described in the text.}}
\label{figtauedif} 
\end{center}
\end{figure}
\par
If one would simply eliminate all events with 
secondary radiation of hadrons and leptons, one would arrive at 
a measurement of $\Gamma(Z \to \tau^+ \tau^-)$ significantly 
below (about 0.4\%) the generally quoted inclusive value. 
Both real and virtual radiation are enhanced by the square of the large 
logarithm $\ln(M_Z^2/m_{had}^2)$ times $\ln(M_Z^2/m_l^2)\,$. 
Because the coefficients of these logarithms only differ by sign, 
real and virtual corrections cancel to a large extent. This leads to a 
small contribution to the inclusive decay rate, as can be seen in the next 
section. Measurements of exclusive $\tau$ pair production, however, are 
strongly affected. With this application in mind we proceed to the 
calculation of the virtual \pagebreak[4] corrections.
\par
\vspace{1cm}\noindent
{\bf 3.~Virtual radiation of hadrons}\\
\par 
\noindent 
Virtual corrections to $\tau$ pair production affect both Dirac and 
Pauli form factors, $F_1$ and $F_2$. Leptonic corrections have been 
calculated in analytic form in \cite{HKT1} for the cases that the 
masses of the radiated leptons are much smaller than or equal to 
the $\tau$ mass. The effect of virtual radiation of hadrons via 
hadronic vacuum polarisation of one exchanged photon, as depicted in 
Fig.\ref{figgraphs}b, is given 
by an expression similar to eq.~(\ref{masterreal}).
For virtual radiation the phase space integration has to be 
extended to infinity corresponding to the dispersion integration of
the absorptive part of the hadronic vacuum polarisation.
The function $F(s',s)$, now representing the normalized cross
section for the production of a $\tau$ pair with an exchanged vector
boson of mass $\sqrt{s'}$, may be easily calculated analytically
and reads
\begin{equation}
F(s',s) =  \left(\frac{\alpha}{\pi}\right)^2\,
\bigg[\,\omega(3-\omega^2)\left[
{\rm Re}\widehat{F}_1(s',s)+{\rm Re}\widehat{F}_2(s',s)
\right] +
\omega^3\,{\rm Re}\widehat{F}_2(s',s)\,\bigg]\,,
\label{fhatvirt}
\end{equation}
where 
\begin{equation}
\omega = \sqrt{1-4m_\tau^2/s}\,.
\label{omega} 
\end{equation} 
The functions ${\rm Re}\widehat{F}_{1,2}$ are
given in~\cite{HKT1} (eqs. (15) and (16)) and denote the real parts of the 
electromagnetic form factors arising from the exchange of a vector 
boson of mass $\sqrt{s'}$. The integral 
\begin{equation}
R_{\tau^+\tau^-\,had}^{virt} = 
 \frac{1}{3} \, 
 \left(\frac{\alpha}{\pi}\right)^2\,
 \int\limits_{4m_{\pi}^2}^{\infty} \frac{{\rm d}s'}{s'} 
 \, R_{had}(s') \, F(s',s) 
\label{mastervirt}
\end{equation}
is calculated numerically. A similar formula applies to 
the form factors individually. 
\par
Virtual corrections from different hadronic intermediate states cannot 
be separated experimentally and contribute in a coherent manner. Their 
combined effect on the form factors and the rate are displayed in 
Table~\ref{tab2} 
for four characteristic energies. 
\begin{table} 
\begin{center} 
\begin{tabular}{|c||c|c|c|c|} \hline 
$\sqrt{s}$ 
& $3.4552$ GeV $= 2\,m_{\tau}$ & $4.2$ GeV & $10$ GeV & $M_Z$ \\ \hline \hline 
$F_{1, had}$ & $7.45\cdot 10^{-5}$ & $3.55\cdot 10^{-5}$ & 
               $2.69\cdot 10^{-6}$ & $-2.78\cdot 10^{-4}$ \\ \hline 
$F_{2, had}$ & $9.00\cdot 10^{-6}$ & $4.68\cdot 10^{-6}$ & 
               $-3.21\cdot 10^{-7}$ & $-6.03\cdot 10^{-8}$ \\ \hline 
$R_{\tau^+ \tau^- had}^{virt}$
             & $0$ & $5.89\cdot 10^{-5}$ & 
               $4.45\cdot 10^{-6}$ & $-5.56\cdot 10^{-4}$ \\ \hline \hline 
$F_{1, e^+ e^-}$ & $1.50\cdot 10^{-2}$ & $5.84\cdot 10^{-5}$ & 
                   $-1.82\cdot 10^{-4}$ & $-8.94\cdot 10^{-4}$ \\ \hline 
$F_{2, e^+ e^-}$ & $2.30\cdot 10^{-3}$ & $-4.26\cdot 10^{-6}$ & 
                   $-2.43\cdot 10^{-6}$ & $-8.21\cdot 10^{-8}$ \\ \hline 
$R_{\tau^+ \tau^- e^+ e^-}^{virt}$
                 & $0$ & $7.78\cdot 10^{-5}$ & 
                   $-3.68\cdot 10^{-4}$ & $-1.79\cdot 10^{-3}$ \\ 
\hline \hline 
$F_{1, \mu^+ \mu^-}$ & $5.86\cdot 10^{-5}$ & $1.76\cdot 10^{-5}$ & 
                       $-5.98\cdot 10^{-6}$ & $-1.38\cdot 10^{-4}$ \\ \hline 
$F_{2, \mu^+ \mu^-}$ & $7.74\cdot 10^{-6}$ & $1.74\cdot 10^{-6}$ & 
                       $-3.37\cdot 10^{-7}$ & $-2.57\cdot 10^{-8}$ \\ \hline 
$R_{\tau^+ \tau^- \mu^+ \mu^-}^{virt}$
                     & $0$ & $2.82\cdot 10^{-5}$ & 
                       $-1.28\cdot 10^{-5}$ & $-2.77\cdot 10^{-4}$ \\ 
\hline \hline 
$F_{1, \tau^+ \tau^-}$ & $1.25\cdot 10^{-6}$ & $1.31\cdot 10^{-6}$ & 
                         $2.16\cdot 10^{-6}$ & $-1.08\cdot 10^{-5}$ \\ \hline 
$F_{2, \tau^+ \tau^-}$ & $1.03\cdot 10^{-7}$ & $1.14\cdot 10^{-7}$ & 
                         $6.00\cdot 10^{-8}$ & $-3.73\cdot 10^{-9}$ \\ \hline 
$R_{\tau^+ \tau^- \tau^+ \tau^-}^{virt}$
                       & $0$ & $2.08\cdot 10^{-6}$ & 
                         $4.47\cdot 10^{-6}$ & $-2.16\cdot 10^{-5}$ \\ 
\hline 
\end{tabular} 
\caption[]{\label{tab2} 
{\em Virtual corrections due to hadronic and leptonic vacuum 
polarisation for center of mass energies 
$\sqrt{s} = 2\,m_{\tau}/4.2$ GeV/$10$ GeV/$M_Z$ for form factors and 
the resulting cross section as described in the text.}} 
\end{center} 
\end{table} 
For completeness we also display the contributions from leptonic vacuum 
polarisation. Contributions from virtual electron pairs are obtained 
with analytical formulae derived in the limit $m_e \ll m_{\tau}$ (see 
\cite{HKT1}). 
In the same reference an analytic formula for virtual $\tau$ pairs 
is also at hand. It is evident from Table~\ref{tab2} 
that these corrections must be taken into 
consideration in precision measurements of partial rates.
Close to threshold the virtual corrections give a positive 
contribution to $R$ as is illustrated in Fig.~\ref{scanv}, where 
$R_{\tau^+ \tau^- had}^{virt}$ is plotted for the range 
$2\,m_{\tau} < \sqrt{s} < M_Z$. This is a consequence of the 
Coulomb attraction mediated by the exchange of a vector boson. 
Far above threshold the corrections are negative. They lead to 
a reduction of the exclusive decay rate of $Z \to \tau^+ \tau^-$ by 
nearly three per mill, exceeding the ${\cal O}(\alpha)$ correction 
of $3/4(\alpha/\pi) \approx 1.7 \cdot 10^{-3}$. As mentioned above 
these negative corrections cancel for the most part 
the positive corrections from real emission, leaving a small positive 
overall correction. This remainder can to a large extent be 
absorbed into a running 
electromagnetic coupling constant $\alpha(s)$ as discussed in~\cite{HKT1}. 
%
\begin{figure}
\begin{center}
\leavevmode
\epsfxsize=12cm
\epsffile[120 300 450 530]{scanvrun.ps}
\vskip -3mm
\caption[]{{\em Contribution to $e^+ e^- \to \tau^+ \tau^-$ due to virtual
radiation of hadrons.}}
\label{scanv} 
\end{center}
\end{figure}
%
\par
\vspace{1cm}\noindent
{\bf 4.~Corrections to $Z$ decays into $\mu^+ \mu^-$ and $e^+ e^-$}\\
\par 
\noindent 
In \cite{KKKS,HJKT1} analytical and numerical results were presented for 
real and virtual radiation of massive leptons or hadrons from the primary
production of massless leptons. For completeness it seems appropriate to
present in this paper a brief update of the hadronic radiation in $Z$ 
decays to $\mu^+ \mu^-$ and $e^+ e^-$ together with a complete table
for the radiation of leptons. 
Real and virtual radiation are presented in Tables~\ref{tab3} and 
\ref{tab4} respectively. 
It is evident that 
exclusive determinations of $Z \to e^+ e^-$ or $\mu^+ \mu^-$ will 
become sensitive towards these corrections. 
\par
The results for $R_{e^+ e^- had}$ and 
$R_{\mu^+ \mu^- had}$ are $\approx$6\% larger than our old 
value \cite{HJKT1}. Half of this 
effect is due to the updated description of the continuum and 3\% 
originate from the inclusion of the effective electromagnetic coupling 
constant $\alpha$ (see also the appendix). 
Comparing our result for $R_{e^+ e^- had}^{virt}$ (taking 
$\sqrt{s} = 93$ GeV) with the corresponding one in \cite{KKKS} the 
difference amounts to only 1\%. 
Our results for the radiation of leptons are about 15\% lower than 
those of~\cite{vdB}, essentially a consequence of the different 
choice for $\alpha$. There the running electric coupling constant 
$\alpha(M_Z^2)$ was chosen. However, radiation of leptons, in 
particular of electrons, is dominated by small $s^\prime$. Therefore 
the low energy scale adopted in this work is more appropriate.
\begin{table}
\begin{center}
\begin{tabular}{|c||c|c|c|} \hline
secondary$\backslash$primary 
        & $e$                 & $\mu$               & $\tau$ 
\\ \hline \hline 
$e$     & $2.78\cdot 10^{-3}$ & $2.53\cdot 10^{-3}$ & $1.82\cdot 10^{-3}$ 
\\ \hline 
$\mu$   & $3.32\cdot 10^{-4}$ & $3.31\cdot 10^{-4}$ & $2.94\cdot 10^{-4}$ 
\\ \hline 
$\tau$  & $3.23\cdot 10^{-5}$ & $3.21\cdot 10^{-5}$ & $3.13\cdot 10^{-5}$ 
\\ \hline \hline 
$hadrons$
        & $6.68\cdot 10^{-4}$ & $6.67\cdot 10^{-4}$ & $6.14\cdot 10^{-4}$ 
\\ \hline \hline \hline
   Sum     & $3.81\cdot 10^{-3}$ & $3.56\cdot 10^{-3}$ & $2.76\cdot 10^{-3}$ 
\\ \hline
\end{tabular}
\caption[]{\label{tab3}
{\em Real radiation $R_{primary, secondary}$ for $s = M_Z^2$.}}
\end{center}
\end{table}
\begin{table}
\begin{center}
\begin{tabular}{|c||c|c|c|} \hline
secondary$\backslash$primary 
        & $e$                  & $\mu$                 & $\tau$ 
\\ \hline \hline 
$e$     & $-2.75\cdot 10^{-3}$ & $-2.50\cdot 10^{-3}$ & $-1.79\cdot 10^{-3}$ 
\\ \hline 
$\mu$   & $-3.15\cdot 10^{-4}$ & $-3.14\cdot 10^{-4}$ & $-2.77\cdot 10^{-4}$ 
\\ \hline 
$\tau$  & $-2.24\cdot 10^{-5}$ & $-2.24\cdot 10^{-5}$ & $-2.16\cdot 10^{-5}$ 
\\ \hline \hline 
$hadrons$
        & $-6.09\cdot 10^{-4}$ & $-6.09\cdot 10^{-4}$ & $-5.56\cdot 10^{-4}$ 
\\ \hline \hline \hline
   Sum  & $-3.69\cdot 10^{-3}$ & $-3.44\cdot 10^{-3}$ & $-2.64\cdot 10^{-3}$ 
\\ \hline
\end{tabular}
\caption[]{\label{tab4}
{\em Virtual radiation $R_{primary, secondary}^{virt}$ for $s = M_Z^2$.}}
\end{center}
\end{table}
\par
Below we also list the high energy expansions (neglecting terms of 
order $m_{1/2}^2/s$) of $R_{f_1 \bar f_1 f_2 \bar f_2}$ and 
$R_{f_1 \bar f_1 f_2 \bar f_2}^{virt}$ for the cases 
$m_1 \gg m_2$, $m_1 = m_2$ and $m_1 \ll m_2$, where $m_1$ is
the mass of the primary produced fermion $f_1$ and $m_2$ the mass of the
secondary emitted fermion $f_2$. These formulae provide excellent 
approximations for $\sqrt{s} = M_Z$ (better than one per mill in all 
cases except for $f_2 = \tau$, where the relative difference between 
complete and approximate result amounts to nearly $1\%$) and 
therefore can directly be applied to the corresponding leptonic $Z$ 
partial widths.
\begin{eqnarray}
R_{f_1\bar f_1 f_2\bar f_2}^{\ m_1 \gg m_2} & = &
 \left(\frac{\alpha}{\pi}\right)^2\,
 \bigg[ - \,\ln^2 \frac{m_2^2}{s}\,\frac{1}{6}\,
     \bigg( \ln \frac{m_1^2}{s} + 1 \bigg) \,\nonumber\,
     \\ 
 & & \mbox{}\,\qquad\quad    + 
  \ln \frac{m_2^2}{s}\,\bigg( \frac{1}{6}\,
      \ln^2 \frac{m_1^2}{s} - 
     \frac{13}{18}\,\ln \frac{m_1^2}{s} + 
     \frac{4}{3}\,\zeta(2) - \frac{53}{36} \bigg) \,
   \nonumber\,\\ 
 & & \mbox{}\,\qquad\quad  - 
  \frac{1}{18}\,\ln^3 \frac{m_1^2}{s} + 
  \frac{13}{36}\,\ln^2 \frac{m_1^2}{s} - 
  \left( \frac{133}{108} + \frac{2}{3}\,\zeta(2) \right) \,
   \ln \frac{m_1^2}{s}\,\nonumber\,\\ 
 & & \mbox{}\,\qquad\quad 
   + \frac{5}{3}\,\zeta(3) + \frac{32}{9}\,\zeta(2) - 
  \frac{833}{216} - \frac{3}{4}\,\frac{m_2}{m_1}\,{{\pi }^2} \,\bigg]\,
\label{mismallreal} \\
%
R_{f_1\bar f_1 f_2\bar f_2}^{\ m_1 = m_2} & = &
 \left(\frac{\alpha}{\pi}\right)^2\,
 \bigg[ - \frac{1}{18}\,\ln^3 \frac{m_2^2}{s}  - 
  \frac{19}{36}\,\ln^2 \frac{m_2^2}{s} + 
  \frac{2}{3}\,\left( -\frac{73}{18} + \zeta(2) \right) \,
   \ln \frac{m_2^2}{s}\,\nonumber\,\\ 
 & & \mbox{}\,\qquad\quad 
   + \zeta(3) + \frac{11}{3}\,\zeta(2) - \frac{1829}{216} \,\bigg]\,
\label{miequalreal} \\
%
R_{f_1\bar f_1 f_2\bar f_2}^{\ m_1\ll m_2} & = &
 \left(\frac{\alpha}{\pi}\right)^2\,
 \bigg[ - \frac{1}{18}\,\ln^3 \frac{m_2^2}{s}  - 
  \frac{19}{36}\,\ln^2 \frac{m_2^2}{s} + 
  \frac{2}{3}\,\left( -\frac{73}{18} + \zeta(2) \right) \,
   \ln \frac{m_2^2}{s}\,\nonumber\,\\ 
 & & \mbox{}\,\qquad\quad 
   + \frac{5}{3}\,\zeta(3) + \frac{19}{9}\,\zeta(2) - 
  \frac{2123}{324} \,\bigg]\,
\label{milargereal}
\end{eqnarray}
\pagebreak
\begin{eqnarray}
R_{f_1\bar f_1 f_2\bar f_2}^{\ m_1 \gg m_2, virt} & = &
 \left(\frac{\alpha}{\pi}\right)^2 \,
 \bigg[ \ln^2 \frac{m_2^2}{s}\,\frac{1}{6}\,
   \bigg( \ln \frac{m_1^2}{s} + 1 \bigg) \,\nonumber\,
   \\ 
 & & \mbox{}\,\qquad\quad  + \ln \frac{m_2^2}{s}\,
   \bigg( - \frac{1}{6}\,\ln^2 \frac{m_1^2}{s} 
       + \frac{13}{18}\,\ln \frac{m_1^2}{s} - 
     \frac{4}{3}\,\zeta(2) + \frac{11}{9} \bigg) \,
   \nonumber\,\\ 
 & & \mbox{}\,\qquad\quad  + 
  \frac{1}{18}\,\ln^3 \frac{m_1^2}{s} - 
  \frac{13}{36}\,\ln^2 \frac{m_1^2}{s} + 
  \left( \frac{133}{108} + \frac{2}{3}\,\zeta(2) \right) \,
   \ln \frac{m_1^2}{s}\,\nonumber\,\\ 
 & & \mbox{}\,\qquad\quad 
   - \frac{2}{3}\,\zeta(3) - \frac{32}{9}\,\zeta(2) + 
  \frac{67}{27} + \frac{3}{4}\,\frac{m_2}{m_1}\,{{\pi }^2} \,\bigg]\, 
\label{mismallvirt} \\
%
R_{f_1\bar f_1 f_2\bar f_2}^{\ m_1 = m_2, virt} & = &
 \left(\frac{\alpha}{\pi}\right)^2\,
 \bigg[ \frac{1}{18}\,\ln^3 \frac{m_2^2}{s} + 
  \frac{19}{36}\,\ln^2 \frac{m_2^2}{s} + 
  \frac{2}{3}\,\left( \frac{265}{72} - \zeta(2) \right) \,
   \ln \frac{m_2^2}{s} - \frac{11}{3}\,\zeta(2) + 
  \frac{383}{54} \,\bigg]\,
\label{miequalvirt} \\
%
R_{f_1\bar f_1 f_2\bar f_2}^{\ m_1\ll m_2, virt} & = &
 \left(\frac{\alpha}{\pi}\right)^2\,
 \bigg[ \frac{1}{18}\,\ln^3 \frac{m_2^2}{s} + 
  \frac{19}{36}\,\ln^2 \frac{m_2^2}{s} + 
  \frac{2}{3}\,\left( \frac{265}{72} - \zeta(2) \right) \,
   \ln \frac{m_2^2}{s}\,\nonumber\,\\ 
 & & \mbox{}\,\qquad\quad 
   - \frac{2}{3}\,\zeta(3) - \frac{19}{9}\,\zeta(2) + 
  \frac{3355}{648} \,\bigg]\, 
\label{milargevirt}
%
\end{eqnarray}
Results for virtual corrections were first obtained 
in~\cite{HKT1}, \cite{BURGERS} and \cite{KKKS} for the 
cases $m_1\gg m_2, m_1=m_2$ and 
$m_1\ll m_2$, respectively. Those for real radiation are taken 
from~\cite{HKT1} for $m_1\gg m_2$ and from~\cite{HJKT1,HJKT2} for
$m_1\ll m_2$. The formula for real radiation with $m_1=m_2$ is new. 
The leading cubic logarithms for all these cases can also be found 
in~\cite{BFK}.
Note that in (\ref{mismallreal}) and (\ref{mismallvirt}) linear mass 
corrections $\sim m_2/m_1$ are included. They significantly improve the 
quality of the approximations, especially in the case 
$m_1 = m_{\tau}$, $m_2 = m_{\mu}$. Neglecting these linear mass corrections 
the value of say $R_{\tau^+ \tau^- \mu^+ \mu^-}^{\ m_1 \gg m_2, virt}$ 
for the energy $\sqrt{s} = 20$ GeV differs from the value based on 
numerical integration by 4\%, while including them leads to a 
difference of only three per mill. Of course, for higher energies 
the latter difference decreases and amounts to eight per mill 
vs. half a per mill for $\sqrt{s} = M_Z$. There are no linear mass 
corrections to the other high energy approximations. 
It is an interesting fact that the complete results for these linear 
mass corrections, valid for all energies above threshold, are 
remarkably simple:
\begin{eqnarray}
R_{f_1 \bar f_1 f_2 \bar f_2, linear}^{m_1 \gg m_2} &=& 
-\frac{3}{8}\,\omega\,(3-\omega^2)\,\frac{m_2}{m_1}\,\pi^2\,,
\label{linearreal}\\
R_{f_1 \bar f_1 f_2 \bar f_2, linear}^{m_1 \gg m_2, virt} &=& 
\frac{3}{8}\,\frac{1+\omega^2}{\omega}\,\frac{m_2}{m_1}\,\pi^2
\label{linearvirt}
\end{eqnarray}
with $\omega$ defined as in (\ref{omega}). 
They are in very close relation to the non--analytic linear vector boson 
mass terms of the corresponding one loop results denoted by $F$, 
eqs.~(\ref{masterreal}) and (\ref{fhatvirt}). The difference between 
(\ref{linearreal}) and (\ref{linearvirt}) can be reproduced by the 
analogous linear mass correction in the two loop relation between 
pole and running masses, see \cite{Gray}.
\par
Again, concerning the total $Z$ decay rate, it is obvious from
eq.~(\ref{mismallreal})--(\ref{milargevirt}) that by adding the 
corresponding real and virtual contributions the dominant 
logarithmic terms cancel, leaving the result 
\begin{equation}
R_{f_1 \bar f_1 f_2 \bar f_2} + R_{f_1 \bar f_1 f_2 \bar f_2}^{virt} = 
 \left(\frac{\alpha}{\pi}\right)^2\,
 \left[\,\frac{1}{4}\,\ln\frac{s}{m_2^2} + \zeta(3) - \frac{11}{8}\,\right]
\label{rsumrealvirthigh}
\end{equation}
which is universal for all three mass constellations in the high energy 
limit. As mentioned in the previous section the remaining logarithm can be 
absorbed into the running coupling constant $\alpha(s)$, leaving a tiny term 
$\sim (\alpha/\pi)^2$ without logarithm. 
\par
For completeness we also present the prediction for the radiation of 
hadrons from primary electrons and muons. In this case the result can be 
expressed in terms of the moments 
\begin{equation}
R_n = \int_0^1 \frac{{\rm d}x}{x} \frac{\ln^nx}{n!}
\left(R_{had}(4m^2/x)-R_{had}(\infty)\right)
\label{moments}
\end{equation}
leading to the following formulae for real~\cite{HJKT1} and for 
virtual~\cite{KKKS} radiation 
($l=e,\mu$)
\pagebreak
\begin{eqnarray}
\lefteqn{ R_{l^+l^-had}  = }\nonumber\\[1mm]
& & \frac{1}{6} \left(\frac{\alpha}{\pi}\right)^2 \biggl\{
R_{had}(\infty) \left[\, \frac{1}{3} \ln^3\frac{s}{4m_\pi^2} - 
\frac{3}{2} \ln^2\frac{s}{4m_\pi^2} + 
\left(-2\,\zeta(2)+5\right)\ln \frac{s}{4m_\pi^2}+
\left(6\,\zeta(3)+3\,\zeta(2)-\frac{19}{2}\right)
\,\right]\nonumber\\[1mm]
& & \qquad\qquad+\,R_0 \left[\,\ln^2\frac{s}{4m_\pi^2}-
3\ln \frac{s}{4m_\pi^2}-2\,\zeta(2)+5\,\right] +
R_1 \left[\,2\ln \frac{s}{4m_\pi^2}-3\,\right] + 2\, R_2 \biggr\}\,,
\label{momreal} \\ 
\lefteqn{ R_{l^+l^-had}^{virt}  = }\nonumber\\[1mm] 
& & \frac{1}{6} \left(\frac{\alpha}{\pi}\right)^2 \biggl\{
R_{had}(\infty) \left[\, -\frac{1}{3}\ln^3\frac{s}{4m_\pi^2}+\frac{3}{2}
\ln^2\frac{s}{4m_\pi^2}+\left(2\,\zeta(2)-
\frac{7}{2}\right)\ln \frac{s}{4m_\pi^2}-3
\,\zeta(2)+\frac{15}{4}\,\right] \nonumber\\[1mm]
& & \qquad\qquad+\,R_0\left[\,-\ln^2\frac{s}{4m_\pi^2}+
3\ln \frac{s}{4m_\pi^2}+ 2\,\zeta(2)-
\frac{7}{2}\,\right]+R_1 \left[\,-2\ln \frac{s}{4m_\pi^2}+
3\,\right] - 2\,R_2 \biggr\}\,.
\label{momvirt}
\end{eqnarray}
The evaluation of (\ref{moments}) gives
\begin{eqnarray}
R_{had}(\infty) &=& 
%\sum_{f_2=u,d,s,c,b} N_c Q_{f_2}^2 = \frac{11}{3}\,,
 \;\;\;\;\, 4.31\,,\nonumber\\
R_0 &=& -11.15\,,\nonumber\\ 
R_1 &=& \;\;\: 25.05\,,\nonumber\\ 
R_2 &=& -48.00
\label{momnum}
\end{eqnarray}
where we used the dressed $R_{had}(s')$ including the narrow resonances 
and $\sqrt{s'} = 100$ GeV as the integration cutoff. The values of the 
moments individually depend on the choice of this cutoff (and 
correspondingly $R_{had}(\infty)$), but we have checked that the 
results of (\ref{momreal}) and 
(\ref{momvirt}) are identical for all cutoff energies above $40$ GeV. 
\begin{figure}
\begin{center}
\leavevmode
\epsfxsize=8.6cm
\epsffile[175 335 400 515]{drdsmumz.ps}
\leavevmode
\epsfxsize=8.6cm
\epsffile[175 335 400 515]{drdemumz.ps}\\
\leavevmode
\epsfxsize=8.6cm
\epsffile[175 335 400 515]{rimmemz.ps}
\leavevmode
\epsfxsize=8.6cm
\epsffile[175 335 400 515]{renmemz.ps}
\vskip -3mm
\caption[]{{\em Differential distributions for real emission of 
hadrons and of $e^+e^-$ pairs in $\mu$ pair production for $\sqrt{s} = M_Z$.}}
\label{figmuhadedif} 
\end{center}
\end{figure}
\noindent
For the derivation of eqs.~(\ref{momreal}) and (\ref{momvirt}) the 
approximation $m_{l}\ll m_\pi$ was essential. The relative difference 
between the asymptotic formulae and the complete numerical results is at 
most 0.2\% for $\sqrt{s} = M_Z$. For lower energies, of course, the 
difference increases. Similar formulae for
radiation off the primary tau pair provide a poor 
approximation to the true answer for real and 
virtual radiation individually, but are adequate for the sum, which 
is identical to the electron and muon case (see also Tables~\ref{tab3} and 
\ref{tab4}). 
For the sum of real and virtual radiation of leptons and hadrons at 
$\sqrt{s} = M_Z$, which is valid for all three lepton channels 
$f_1 = e, \mu, \tau$, one arrives at
\begin{eqnarray}
\lefteqn{R_{f_1 \bar f_1 had} + R_{f_1 \bar f_1 had}^{virt} +
\sum_{f_2=e, \mu, \tau} \left(\,R_{f_1 \bar f_1 f_2 \bar f_2} +
                   R_{f_1 \bar f_1 f_2 \bar f_2}^{virt}\,\right)} 
\nonumber\\
&=&
 \frac{3}{4}\,\bigg[\,\frac{\alpha(M_Z^2)}{\pi} - \frac{\alpha}{\pi}\,\bigg] 
 + \left(\frac{\alpha}{\pi}\right)^2\,
   \left[\,\zeta(3) - \frac{23}{24}\,\right]\,
 \bigg(\,R_{had}({\infty})+\sum_{f_2=e, \mu, \tau}Q^2_{f_2}\,\bigg) 
\nonumber \\
&\approx& 0.12\cdot 10^{-3}\,,
\label{sumrealplusvirt}
\end{eqnarray}
where the usual definition of the running electromagnetic 
coupling constant
\begin{eqnarray}
\frac{1}{\alpha(s)} - \frac{1}{\alpha} = \Pi(s) 
%- \Pi(0) 
\label{runalphadef}
\end{eqnarray}
was employed. Here the hadronic contributions to the photon vacuum 
polarisation have also been parametrized by the moments defined 
in (\ref{moments}):
\begin{eqnarray}
\Pi(s) 
%- \Pi(0) 
 \stackrel{s \to \infty}{\longrightarrow}
 - \frac{1}{3\,\pi}\,
 \bigg[\,\sum_{i = e, \mu, \tau}\,\bigg(\,\ln\frac{s}{m_i^2} - 
 \frac{5}{3}\,\bigg) + R_{had}(\infty) \ln\frac{s}{4\,m_{\pi}^2} + R_0
\,\bigg]\,.
% + {\cal O}\left(\frac{\alpha}{\pi}\right)^2
\label{runalpha}
\end{eqnarray} 
\par
The differential distribution 
${\rm d} R_{l^+l^- had}/{\rm d} s_{\scriptsize had}^\prime$ 
for the case of massless leptons is given in closed form 
(see eqs.~(1) and (2) of~\cite{HJKT1}), 
${\rm d} R_{l^+l^- had}/{\rm d} E_{\scriptsize had}$, 
of course, has to be calculated numerically. The predicted distributions 
for the radiation of hadrons and of $e^+e^-$ pairs off primary produced muon
pairs are shown\footnote{The figure for 
${\rm d} R_{l^+l^- had}/{\rm d} s_{\scriptsize had}^\prime$ does not 
include the narrow resonances. The missing contributions 
to $R_{l^+l^- had}$ 
are $\omega$: $2.33\cdot 10^{-5}$, $\phi$: $3.56\cdot 10^{-5}$, 
$J/\psi$: $2.95\cdot 10^{-5}$ and $\Upsilon$: $8.33\cdot 10^{-7}\,$.} 
in Fig.~\ref{figmuhadedif}. Again 
the dominant configuration 
consists of an $\mu^+\mu^-$ with invariant mass fairly close 
to $M_Z$ and a soft hadronic or $e^+e^-$ system.
\par
\vspace{1cm}\noindent
{\bf 5.~Summary}\\
\par 
\noindent 
A comprehensive treatment of the radiation of hadronic final
states from a primary $\tau$ pair is presented, which is
applicable over the full energy range of interest  in the
foreseeable future. In the low energy region real as well as
virtual radiation is small, typically at the level of
$10^{-4}$ compared to the primary reaction. The situation is
entirely different for a c.m. energy around $90$ GeV. In this
case important differences arise between ``inclusive''  and
``exclusive'' measurements of the $Z\to \tau^+\tau^-$ rate,
in particular if radiation of the various hadronic and
fermionic channels is collected, or eliminated by experimental cuts. 
We have presented differential distributions with respect to
the invariant mass and the energy of the hadronic system, and for
comparison the same quantities for the radiation of
electron--positron pairs. It is evident that all
distributions are peaked at small masses and energies
respectively. The calculation of the rate for hadronic
radiation must necessarily rely on a numerical integration
of experimental data. Leptonic radiation, on the other hand,
can be calculated analytically for the mass assignments of
interest. Handy formulae are presented for different
mass assignments which are applicable in the high energy
limit, and which are valid up to terms of order
$m_{lept}^2/s$. They may be useful also for other
applications.
\par
\vspace{1cm}\noindent
{\em Acknowledgment:}
We would like to thank F.~Jegerlehner and S.~Eidelman for providing 
their compilation of data and discussing its proper treatment. We are 
also grateful to K.~Chetyrkin, W.~Hollik and D.~Schaile for helpful 
discussions. One of the authors (J.H.K.) would like to thank the SLAC 
theory group for hospitality, and the Volkswagen--Stiftung 
grant I/70 453 for generous support. 
\par
\vspace{1cm}\noindent
{\bf Appendix: $R_{had}(s')$}\\
\par 
\noindent 
In our calculation we used experimental data for $R_{had}(s')$ as 
presented in a recent publication by F.~Jegerlehner and 
S.~Eidelman\cite{jegerl}. For a detailed discussion we refer the reader 
to this work.
\par
In the region close to the $\pi^+ \pi^-$ threshold where no experimental 
data are available we use the chiral expansion of the pion form factor 
(eq.~(18) of \cite{jegerl}). 
The continuum-- and $\rho$--contributions are parametrized by weighted 
averages of experimental data from various experiments. From $318$~MeV up 
to $810$~MeV these are given in terms of the pion form 
factor $|F_{\pi}|^2$. The continuum from $810$~MeV up to $40$~GeV 
(above which we use the prediction of perturbation theory) is well 
described by weighted averages of $R_{had}(s)$. At this point two comments 
are in order: The higher resonances $\psi(4040),\ \psi(4160)$\ and 
$\psi(4415)$ are included in the weighted averages of the continuum data 
(as can be seen in Fig.~7 of \cite{jegerl}) whereas the resonances 
$\omega$, $\Phi$, $J/\psi(1S)$, $\psi(2S)$, $\psi(3770)$ and the 
$\Upsilon$--family (six resonances) are treated in the narrow width 
approximation as described above.\\ 
\par
All experimental values of $R_{had}$ (and $|F_{\pi}|^2$) are given 
as ``undressed'' ($\equiv$ lowest order in QED) quantities. In 
order to resum all additional self--energy insertions in the photon 
propagator we 
use for the integration of (\ref{masterreal}), (\ref{diffenergyreal}) 
and (\ref{mastervirt}) the ``dressed'' quantity 
\begin{equation}
R_{had}^{dressed}(s) = 
\left(\frac{\alpha(s)}{\alpha}\right)^2 
R_{had}(s)\,,\quad 
\alpha(s)=\frac{\alpha}{1-\Delta\alpha}\,.
\label{rdressed}
\end{equation}
In the low energy region up to $\sqrt{s'} = 3$\ GeV most experiments 
only used the leptonic contribution of $\Delta\alpha$ to extract 
one--particle--irreducible quantities from the measured cross--sections.
For that reason we take $\Delta\alpha = \Delta\alpha_{lep}$ in 
this region. For all other energies 
$$\Delta\alpha = \Delta\alpha_{lep} + \Delta\alpha_{had}\,.$$
As mentioned above these corrections led to an increase of the 
result by a factor of $\approx 1.03$. 
We also calculated the effect of $\Delta\alpha_{lep}(s)$ on our results, 
comparing the full one loop expression of the fermionic part of the 
photon self energy with the leading logarithmic behaviour (as it appears 
in the renormalization group ``running $\alpha$''). The difference of 
$\approx$0.5\% may be considered as a crude estimate of other effects 
not taken into account via the implicit resummation by 
using $\alpha(s)$ and is smaller than 
the claimed accuracy of our calculation.\\ 
\vskip 1.5cm
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{sloppy}
\begin{raggedright}
\def\app#1#2#3{{\it Act. Phys. Pol. }{\bf B #1} (#2) #3}
\def\apa#1#2#3{{\it Act. Phys. Austr.}{\bf #1} (#2) #3}
\def\lhc{Proc. LHC Workshop, CERN 90-10}
\def\npb#1#2#3{{\it Nucl. Phys. }{\bf B #1} (#2) #3}
\def\plb#1#2#3{{\it Phys. Lett. }{\bf B #1} (#2) #3}
\def\prd#1#2#3{{\it Phys. Rev. }{\bf D #1} (#2) #3}
\def\pR#1#2#3{{\it Phys. Rev. }{\bf #1} (#2) #3}
\def\prl#1#2#3{{\it Phys. Rev. Lett. }{\bf #1} (#2) #3}
\def\prc#1#2#3{{\it Phys. Reports }{\bf #1} (#2) #3}
\def\cpc#1#2#3{{\it Comp. Phys. Commun. }{\bf #1} (#2) #3}
\def\nim#1#2#3{{\it Nucl. Inst. Meth. }{\bf #1} (#2) #3}
\def\pr#1#2#3{{\it Phys. Reports }{\bf #1} (#2) #3}
\def\sovnp#1#2#3{{\it Sov. J. Nucl. Phys. }{\bf #1} (#2) #3}
\def\jl#1#2#3{{\it JETP Lett. }{\bf #1} (#2) #3}
\def\jet#1#2#3{{\it JETP Lett. }{\bf #1} (#2) #3}
\def\zpc#1#2#3{{\it Z. Phys. }{\bf C #1} (#2) #3}
\def\ptp#1#2#3{{\it Prog.~Theor.~Phys.~}{\bf #1} (#2) #3}
\def\nca#1#2#3{{\it Nouvo~Cim.~}{\bf #1A} (#2) #3}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\elevenbf\noindent  References \hfil}
\vglue 0.4cm
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\end{document}

