%Title: The Tau Lepton: Particle Physics in a Nutshell
%Author: J.H. Kuehn
%Published: *TAU '98 *, Proceedings of the Fifth Workshop on Tau Lepton Physics, Santander, Spain, 1998, edited by A. Pich Zardoya and A. Ruiz Jimeno, *Nucl. Phys. * B (Proc. Suppl.) 76 (1999) 21-44.
%hep-ph/9812399
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\title{\textbf{The Tau Lepton: Particle Physics in a Nutshell}}
\author{Johann K\"uhn
\address{Institut f\"{u}r Theoretische Teilchenphysik,\\
Universit\"{a}t Karlsruhe,
D--76128 Karlsruhe, Germany}}
\begin{document}
\begin{abstract}
\end{abstract}
\maketitle
{\bf Outline}\\
1. Introduction\\
2. Weak Couplings\\
\hspace*{0.5cm}2.1 Charged current interactions\\
\hspace*{0.5cm}2.2 Neutral current interactions\\
\hspace*{0.5cm}2.3 Electric and magnetic dipole moments\\
3. Inclusive Decays, Perturbative QCD and Sum Rules\\
\hspace*{0.5cm}3.1 Inclusive decays and the strong coupling\\
\hspace*{0.5cm} constant\\
\hspace*{0.5cm}3.2 Cabibbo suppressed decays and the\\
\hspace*{0.5cm}Strange Quark Mass\\
4. Exclusive Decays\\
\hspace*{0.5cm}4.1 Form factors and structure functions\\
\hspace*{0.5cm}4.2 Chiral dynamics\\
\hspace*{0.5cm}4.3 Resonances\\
\hspace*{0.5cm}4.4 Isospin and CVC\\
\hspace*{0.5cm}4.5 Hadronic vacuum polarization from \\
\hspace*{0.5cm}$\tau$ decays\\
5. Beyond the Standard Model\\
\hspace*{0.5cm}5.1 CP violation in hadronic $\tau$ decays\\
\hspace*{0.5cm}5.2 ``Forbidden'' decays.
\section {Introduction}
More than 20 years after its discovery, the study of the
tau lepton remains a fascinating field, encompassing particle physics
in its full variety, from strong to electromagnetic and
weak interactions, from resonance physics at long distances to
perturbative short distance physics. The Standard Model in its
large variety of phenomena is of relevance as well as subtle tests
of its validity and searches for physics beyond this well-explored
framework. As a member of the third family with its large mass, more
than three thousand times more massive than the electron, it could be
particularly sensitive to new interactions related to the Higgs
mechanism.
Specifically, as indicated in Fig.1, the production process in
electron-positron collisions allows to explore the lepton
couplings to photon and the Z boson, its charge, magnetic and
electric dipole moment, the vector and axial couplings of both
electron and tau lepton, thus providing one of the most precise
determinations of the weak mixing angle. Its weak decay
gives access to its isospin partner
$\nu_{\tau}$,
providing stringent limits on its mass
$m_{\nu}$
and its helicity
$h_{\nu}$.
Universality of charged current interactions can be tested
in a variety of ways which are sensitive to different scenarios
of physics beyond the Standard Model.
The hadronic decay rate and, more technically, moments of
the spectral function as calculated in perturbative QCD lead to
one of the key measurements of
$\alpha_s$, remarkable
in its precision as well as its theoretical rigor, and
the Cabibbo suppressed transition might well allow for an
accurate determination of the strange quark mass.
Resonances, and finally
$\pi$, K, $\eta$, and $\eta\prime$
are the hadronic decay products, and any theoretical
description of
$\tau$
decays should in the end also aim at an improved understanding
of this final step in the decay process. At low momentum
transfer one may invoke the technology of chiral Lagrangians,
for larger masses of the hadronic system vector dominance
leads to interesting phenomenological constraints. With
increasing multiplicity of the hadronic state the transition
matrix elements of the hadronic current are governed by form
factors of increasing complexity. Bilinear combinations of these
form factors denoted ''structure functions'' determine the angular
distributions of the decay products. In turn, through an appropriate
analysis of angular distributions it is possible to reconstruct
the structure functions and to some extent
form factors.
Isospin relations put important constraints on these analyses and
allow to interrelate hadronic form factors measured in tau decays
with those measured in electron-positron annihilation.
Obviously, there remains the quest for the unknown, the truly
surprising. CP violation in tau decays, extremely suppressed
in the framework of the Standard Model, is one of these options,
transitions between the tau and the muon or the electron, or even
lepton number violating decays are other exciting possibilities.
This overview also sets the stage for the present talk:
Sect.\ref{S2} will be concerned with the tau as a tool to explore weak
interactions, charged and neutral current phenomenology, in
particular the test for universality, the determination of the
neutrino helicity, and searches for anomalous couplings. Sect.\ref{S3}
will be concerned with perturbative QCD, in particular with recent
results on the beta function, the anomalous dimension and their
impact on current analyses of
$\alpha_{s}$.
\begin{figure*}
\begin{center}
\leavevmode
\epsfxsize=10.cm
\epsffile[80 60 550 380]{kirill.ps}
\hfill
\parbox{14.cm}{
\caption[]{\label{F1}\sloppy
Selected physics topics which can be studied in $\tau$-lepton
production and decay.}}
\end{center}
\end{figure*}
Recent higher-order calculations of the interdependence between
the strange quark mass and the decay rate into Cabibbo
suppressed channels are another important
topic of Sect.\ref{S3}. Predictions for exclusive decays will
be covered in Sect.\ref{S4}. This includes the technique of structure
functions, predictions for form factors based on chiral dynamics,
the inclusion of resonances, and constraints from CVC and
isospin relations. Tau decays in combination with isospin have
been used to measure the pion form factor which in turn is an
important ingredient for the calculation of the anomalous magnetic
moment of the muon and the electromagnetic coupling at the
scale
$M_{Z}$.
The validity of this approach and its limitations will also be
addressed in Sect.\ref{S4}.
A selection from the many speculations on physics
beyond the Standard Model will be presented in Sect.\ref{S5}. This
includes tests for CP violation and a few new suggestions for lepton
number violating decays.
The properties of the tau neutrino, in particular its mass and mixing,
have received considerable attention after the observation of mixing
in atmospheric neutrino studies. This topical subject will be
discussed in a different section of this meeting and is therefore
not included in the present review.
\section {\label {S2} Weak Couplings}
\subsection{\label{Ss21} Charged Current Interactions}
\subsubsection{\label {Sss21} Lepton Universality in Tau Decays}
Universality of weak interactions is one of the cornerstones
of the present theoretical framework. Deviations could
arise from additional gauge interactions which are sensitive
to lepton species, thus providing a clue to the origin of the
triplicate nature of the fermion spectrum, one of the mysteries
of the present theory. Alternatively one might attribute a
deviation from universality to mass dependent
interactions, mediated for example by Higgs exchange in
fairly straightforward extensions of the Standard Model.\\
The different tests of universality as discussed below are
sensitive to different new phenomena. They should therefore
be pursued as important measurements in their own right, and
not simply be judged and compared on the basis of one
''figure of merit''.\\
{\it i}) $\tau \to \nu \pi$\\
The classical test of lepton universality dating back to the
``pre-tau-era'' is based on the comparison between the
decay rates of the pion into electron and muon respectively.
In lowest order this ratio is simply given by the electron and
muon masses, a simple first year's textbook calculation.
A precise prediction must include radiative corrections (Fig.2)
due to virtual photon exchange and real emission
\ba
&& R_{e/\mu} = \frac {\Gamma(\pi \to e \nu)}{\Gamma(\pi \to \mu \nu)}
\nonumber \\
&&= \frac {m_e^2}{m_\mu^2} \left (\frac {m_\pi^2 -
m_e^2}{m_\pi^2-m_\mu^2} \right )^2
\left (1+\delta R_{e/\mu} \right ).
\ea
\begin{figure*}
\begin{center}
\mbox{}\hspace{.5cm}\epsfxsize=7.cm\epsffile[180 340 420 470]{pic1.ps}
\parbox[b]{10cm}{
\parbox[b]{.8cm}{+\vspace{1.5cm}}
\epsfxsize=7.cm\epsffile[180 340 420 470]{pic2.ps}\parbox[b]{2.cm}{+\,\ldots
\vspace{1.5cm}}}\\[2mm]
\quad {{\bf + real emission}}
\caption[]{\label{F2}\sloppy
Radiative corrections affecting the pion decay.
}
\end{center}
\end{figure*}
Short distance corrections cancel in
$R_{e/\mu}$ since it is the same effective Hamiltonian
which governs both numerator and dominator. The
dominant $m_{\ell}$ dependent correction,
\be
R_{e/\mu} = \frac {3\alpha}{\pi} \ln \frac {m_e}{m_\mu} + ...
= -3.72 \%
\ee
comes remarkably close to a more refined prediction
including renormalization group improvement plus an
estimate of the dominant logarithms from hadronic effects
\cite{MS1}
\be
\delta R_{e/\mu}=-(3.76 \pm 0.04) \%.
\ee
A more refined treatment \cite{DF} including a detailed
modeling of hadronic form factors leads to
\be
\delta R_{e/\mu}=-(3.74 \pm 0.01) \% ,
\ee
well consistent with the earlier result.
The reduced theoretical uncertainty can again
be traced to the long distance aspect
of hadronic interactions.
The final prediction and the measurement are well consistent:
\ba
&&R^{\rm theory}_{e/\mu}=(1.2356 \pm 0.0001)\cdot 10^{-4},
\\
&&R^{\rm exp}_{e/\mu}=(1.230 \pm 0.004)\cdot 10^{-4}.
\ea
Universality in this reaction is thus verified at a level of
$3\cdot 10^{-3}$
and further progress is possible by improving the
experimental precision.
The small uncertainty of the theoretical prediction
is a natural consequence of the low momenta probed
in this calculation, with $m_\pi$ far below $m_\rho$, the
typical scale relevant for hadronic form factors.
The situation is drastically different for tau decays.
In lowest order the ratio
\ba
&&R_{\tau/\pi} = \frac {\Gamma(\tau \to \pi \nu)}{\Gamma(\pi \to \mu \nu)}
\nonumber \\
&&= \frac {m_\tau^3}{2m_\pi m_\mu^2} \left (
\frac {1-m_\pi^2/m_\tau^2}{1-m_\mu^2/m_\pi^2} \right )^2
\left (1+\delta R_{\tau/\pi} \right )
\ea
is again fixed by the
masses of the leptons and the pion. The radiative
corrections collected in $\delta R_{\tau/\pi}$
can again be predicted
in leading logarithmic approximation \cite{MS2}
\be
1+\delta R_{\tau/\pi} =
\frac {1+2\frac {\alpha}{\pi}\ln m_Z/m_\tau}
{1+\frac {3}{2} \frac {\alpha}{\pi}\ln m_Z/m_\pi
+\frac {1}{2} \frac {\alpha}{\pi}\ln m_Z/m_\rho}.
\label{e1}
\ee
The short distance piece as given by the
$\ln m_{Z}$ terms cancels again in the ratio, the remainder depends
critically on the guess for the low energy cutoff. The
choice as given in Eq.(\ref{e1}) leads to
\be
\delta R_{\tau/\pi}= -0.96 \%
\label{e2}
\ee
with an uncertainty estimated in \cite{MS2} to be $\pm 0.5 \%$.
A detailed explicit calculation based on a careful separation
between long and short distance contributions predicts \cite{DF}
\be
\delta R_{\tau/\pi} = \left (0.16^{~~+0.09}_{~~-0.14} \right ) \%
\label{e21}
\ee
The difference between these two results is entirely due to the more detailed
treatment of long distance phenomena. The important lesson to be drawn
from this comparison is that estimates of long distance effects for
exclusive channels may well fail at the level of $1 \%$. This uncertainty
of around $1 \%$ might also set the scale for predictions of
$\Gamma(\tau \to \pi^- \pi^0 \nu)$
based on CVC and isospin symmetry.
The prediction for decays into $\pi$ and similarly into $K$
is well consistent with the experimental result
\be
B_{\pi+K} = (11.79 \pm 0.12)\% .
\ee
The lifetime $\tau_\tau = 290.8 \pm 0.6~{\rm fs}$
which has to be used in this comparison as additional input is
derived from the direct measurement
$\tau_\tau = 290.5 \pm 1.0~{\rm fs}$
and the indirect result
of $291.0 \pm 0.7~{\rm fs}$
which is obtained from the leptonic
branching ratios $B_e = 17.81 \pm 0.06$\% and
$B_\mu = 17.36 \pm 0.06$\%.
The test of universality, ${\rm Exp/Th} = 1.013 \pm 0.010$, is thus only
a factor three less precise than the universality test in pion decay.\
It is interesting to see that experiments with their accuracy of
one percent are already able to discriminate
between the theoretical predictions Eq.(\ref{e2}) and Eq.(\ref{e21}).
The former, based on leading logarithm
considerations only, disagrees by more than
two standard deviations, the latter is consistent with the
measurements. Future, improved measurements of
$\tau \to \nu \pi$ will therefore
be critical tests of our understanding of radiative corrections, with
important implications for the validity of isospin symmetry and CVC
in the prediction of $\tau \to \nu \pi^- \pi^0$
from $e^{+}e^{-}$ data discussed below (Sect. 4).
Although the present result for the mode
$\tau \to \nu K$ with a branching
ratio $0.71 \pm 0.05$\% is not yet competitive it should be
emphasized that this channel is particularly interesting.
It connects quarks and leptons of the second and third generation,
respectively. It is particularly sensitive to exotic mass dependent
effects such as charged Higgs exchange and its rate is
predicted with an accuracy better than one percent.\\
{\it ii})~~Leptonic decays\\
The rates of two of the three purely leptonic decays can be
calculated without theoretical ambiguity if the third one has
been measured. Conventionally one chooses the muon decay rate
for normalization. The electromagnetic corrections are finite
and given separately in the implicit definition of $G_{F}$
\ba
&&\Gamma_{\mu} = \frac {G_F^2 m_\mu^5}{192 \pi^3} \left (
1+\frac {3m_\mu^2}{5m_W^2} \right )
f(\frac {m_\mu^2}{m_W^2})
\left (1+\delta_{\rm QED} \right ),
\nonumber \\
&&
f(x) = 1-8x+8x^3-x^4-12x^2\ln x
\\
&&
\delta_{\rm QED} = \left (\frac {\alpha \left (m_\mu \right)}{\pi}\right )
\left (\frac {25}{8}-\frac
{\pi^2}{2} \right ) + \left (\frac {\alpha}{\pi} \right )^2 6.743.
\nonumber
\ea
The ${\cal O}(\alpha)$ term in $\delta_{QED}$
has been calculated nearly forty years
ago \cite{muon1}, the second term of order $\alpha^{2}$ was evaluated only
recently \cite{Timo}.
Two independent tests of universality are thus accessible: the
comparison of electronic and muonic decay rates of the $\tau$,
and the comparison between one of the tau decay rates and
$\Gamma_\mu$.
Electron-muon universality is best tested in the ratio
$B_{\mu}/B_e$. The experimental result, expressed in terms of the ratio
of couplings \cite{Passaleva}
\be
\frac {g_\mu}{g_e} = 1.0015 \pm 0.0025.
\ee
comes close in precision to the one
achieved in pion decay $(1.0023 \pm 0.0016)$.
Higgs exchange amplitudes could in particular be enhanced in
$\tau \to \mu \nu \nu$ and thus lead to observable differences
in $B_e/B_\mu$ (Fig.3). In pion decays they would be
practically absent.
\begin{figure}
\caption[]{\label{F3}\sloppy Amplitudes mediated by Higgs exchange.
}
\begin{center}
\leavevmode
\epsfxsize=10.cm
\hspace{.5cm}\epsfxsize=7.cm\epsffile[135 335 420 470]{pic3.ps}\quad
\epsfxsize=7.cm\epsffile[180 335 420 470]{pic4.ps}
\hfill
% \parbox{14.cm}{
% \caption[]{\label{test}\sloppy Amplitudes mediated by Higgs exchange.
% }}
\end{center}
\end{figure}
The relative
reduction of the rate in the two Higgs doublet model
$-2m_\mu^2 ({\rm tan}\beta /m_H )^2$
leads to
an interesting limit on $m_H/{\rm tan}\beta$
in the range around or above $1$ GeV \cite{Stahl,Stahltp},
nearly comparable in strength to those from
$b \to s \gamma$, $B \to \tau \nu$, and $Z \to b \bar b$.
Additionally, even stronger limits have been deduced from the decay
spectra, the Michel parameters, to be discussed below.
The ratio $B_{\mu}/B_{e}$ thus leads to a test of
electron-muon universality. The comparison between the leptonic
decay rate of the tau and the muon is in contrast sensitive to
new physics phenomena connected specifically
with the tau lepton.
One example \cite{Guth} is based on enhanced Higgs boson induced vertex
corrections, which leads to a relative reduction of the rate
\be
\frac {\delta \Gamma}{\Gamma} = \frac {G_F m_\tau^2}{16 \pi^2}\;
2\sqrt{2}\; {\rm tan}^2\beta \; {\cal F}\left ( m_{H_i},\alpha_{\rm mix}
\right ),
\ee
where ${\rm tan}\beta$ is given by the ratio of the two vacuum
expectation values and $\alpha_{\rm mix}$ parameterizes the
mixing in the Higgs sector. Depending on the precise
values of these parameters the reduction of the rate might reach
a few per mille. This is indicated in Fig.4 for the specific
choice $m_{H_{1}} = m_{H_{2}} = m_{H_{3}} = m_{H}$ labeling
the curves, ${\rm tan}\beta = 70$, and the charged Higgs mass
varying between $50$ and $1000$ GeV.
\begin{figure}
\caption[]{\label{F4}\sloppy
The ratio $1+\delta\Gamma/\Gamma$ versus the mass
of the charged Higgs boson for
$m_{H_1}=m_{H_2}=m_{H_4}=$
$40-400$ GeV and
${\rm tan}\beta=70$ \cite{Guth}.}
\begin{center}
\leavevmode
\epsfxsize=7cm
\epsffile{plotscan.eps}
\hfill
% \parbox{14.cm}{
% \caption[]{\label{test}\sloppy
%The ratio $1+\delta\Gamma/\Gamma$ versus the mass\\
%of the Higgs boson for
%$m_{H_1}=m_{H_2}=m_{H_4}=$\\
%$40-400$ GeV and
%${\rm tan}\beta=70$ \cite{Guth}.}}
\end{center}
\end{figure}
Larger mass splittings
typically lead to larger deviations. Present experiments
with their sensitivity of a few permille \cite{Passaleva}
\be
\frac {g_\tau}{g_\mu} = 1.0001 \pm 0.0026
\ee
start to approach this interesting region.
\subsubsection{ \label{Sss212}
Lepton Spectra, Michel Parameters, and Neutrino Helicity}
{\it i}) Leptonic decays
\vspace*{0.5cm}
Much of the techniques of
measuring and analyzing the lepton spectrum in tau decays has been
derived from the corresponding muon decay experiments. Starting
from the local four fermion interaction
\be
{\cal M}= \frac {4G_F}{\sqrt{2}}\sum g_i \left (\bar l_{\rm L,R}
\Gamma^N \nu_l \right ) \left (\bar \nu_\tau
\Gamma_N \tau_{\rm L,R} \right )
\label{e4}
\ee
and exploiting either the ``natural'' polarization of $\tau$'s
from $Z$ decays or the ``induced'' polarization from $\tau^{+}\tau^{-}$
decay correlations the four parameters
$\rho,\eta,\xi$ and $\delta$
are determined consistent
with the expectations from V-A interaction $3/4$, $0$, $1$ and $3/4$. The
parameter $\eta$ in particular is sensitive to righthanded
couplings; its measurement can be used to set interesting bounds
on amplitudes induced through charged Higgs exchange
\cite{Stahl,Stahltp} quite comparable to those derived from the rate.
Recently the local ansatz Eq.(\ref{e4})
has been questioned, and a generalization
including momentum dependent vertices has been
introduced \cite{Taylor}. Starting from a coupling of the form
\be
{\cal A} = \frac {g}{\sqrt{2}}\bar \tau \left [
\gamma_\mu + \frac {i}{2m_\tau}\sigma_{\mu \nu}
Q^{\nu} \left ( \kappa - i\tilde \kappa \gamma _5 \right ) \right ]
\nu _{\rm L} W^\mu
\ee
with $Q = P_\tau - P_\nu$ , one
obtains modifications of the distributions in leptonic
and semileptonic decays which are not covered by the Michel parameters.
A limit on the anomalous coupling $\kappa$ obtained this way is easily
converted into a limit on the compositeness scale $\Lambda =m_\tau/\kappa$,
and experiments are getting close to interesting bounds in the
range $\Lambda \sim 100$ GeV.
This topic is closely related to the discussion
of nonlocal neutral current interactions in Sect.2B.
\vspace*{1cm}
{\it ii}) Semileptonic decays and neutrino helicity
\vspace*{0.5cm}
Semileptonic decays offer two distinctly different methods to
determine the relative magnitude of vector vs. axial amplitude in the
$\tau \nu W$ coupling. Given a nonvanishing tau polarisation, the angular
distribution of hadrons relative to the direction of the polarization
is a direct measure of
\be
h_\nu = \frac {2g_Vg_A}{g_v^2+g_A^2},
\ee
with $h_\nu = -1$ in the V-A theory. For the pion decay
for example one obtains, for example:
\be
dN = \frac {1}{2} \left (1 -h_\nu \cos \theta \right ) {\rm d} cos
\theta .
\label{sp}
\ee
For more complicated multi-meson final states the coefficient
accompanying $h_\nu$ is smaller than one, if the hadronic final
state is summed. Full analyzing power is recovered by looking into
suitable angular distributions as discussed below. An alternative
determination \cite{KW} of $h_\nu$, applicable in particular
for unpolarized taus,
is provided by decays into three (or more) mesons. In this case
one can discriminate between hadronic states with helicity $-1$, zero
or $+1$ and thus infer the neutrino helicity from the
analysis of parity violating (!) angular
distributions of the mesons (Fig.\ref{F5}).
\begin{figure}
\caption[]{\label{F5}\sloppy
Three pion mode as a tool for \\the measurement of the neutrino helicity.}
\begin{center}
\leavevmode
\epsfxsize=7cm
\epsffile[70 385 310 450]{pic8.ps}
\hfill
% \parbox{14cm}{
% \caption[]{\label{test}\sloppy
%Three pion mode as a tool for \\the measurement of the neutrino helicity.}}
\end{center}
\end{figure}
As mentioned above, the analyzing power of two- or three-meson
states in a spin one configuration is reduced by a factor
$(m_\tau^2-2 Q^2)/(m_\tau^2+2Q^2)$
which is typically around $0.5$ for two pions
($Q^2 \approx m_\rho^2$) and close to zero
for three pions $Q^2 \approx m_{a_1}^2$.
Full analyzing power of one, equal to the
single pion decay, is recovered if the information on all
meson momenta is retained. Starting from the decay amplitude
\be
{\cal M} = \frac {G_F}{\sqrt{2}} \bar u(\nu) \gamma^\alpha (1-\gamma_5)
u(\tau) J_\alpha
\ee
with $J_\alpha = \langle {\rm had} | V-A| 0 \rangle$
one obtains
\be
{\rm d}\Gamma =\frac {G_F^2}{2m} \left ( \omega
- {\bf H}{\bf S} \right ) {\rm d}PS.
\ee
For the single pion decay ${\bf H}/\omega = h_\nu {\bf n}_\pi$.
In general ${\bf H}$ and $\omega$ are constructed
from a bilinear combination of the hadronic current dependent
on all meson momenta. As shown in \cite{KW,K}
\be
|{\bf H}/\omega | = 1.
\ee
This observation is also one of the important
ingredients of the Monte Carlo program TAUOLA \cite{Tauola},
which is currently used to simulate tau decays.
It has furthermore been used \cite{Da}
for a simplified
analysis of the tau polarization in terms of one ``optimal variable''.
The complete kinematic information is not immediately available in
these events since the momenta of the two neutrinos from the
decay of $\tau^+$ and $\tau^-$ respectively are not determined
and one is left with a twofold ambiguity,
even in double hadronic events.
This leads to
a reduction of the
analyzing power for two and three meson decays. However, as shown in
\cite{K1}
the measurement of the tracks close to the production point
with vertex detectors allows to resolve the ambiguity. Specifically,
it is the direction of the vector ${\bf d}_{\rm min}$
which characterizes the distance between the tracks of
$\pi^+$ and $\pi^-$ and which provides the missing piece of information
for events where both taus decay into one pion each.
The situation for more complicated modes is discussed in
\cite{KM2}
Only recently, with significant advance in the vertex detectors
this technique has been applied, providing a factor two improvement
compared to the result without reconstruction of the full
kinematics \cite{ALEPH}.
\subsection{\label {Ss2.2} Neutral Current Couplings}
$Z$ decays into tau pairs provide an extremely powerful tool
for the analysis of neutral current couplings. This has been
made possible by the large event rates collected by the four
LEP detectors and the inclusion of most of the tau decay modes
in the analysis. At present these measurements compete with
those from the SLD experiment based on longitudinally polarized
beams, thus providing an important test of lepton universality
in the neutral current sector and an accurate independent
determination of the weak mixing angle. A first important
test is obtained from a comparison of the rate \cite{rate}
\ba
&&R_e=20.757 \pm 0.056;
\nonumber \\
&&R_\mu = 20.783 \pm 0.037;
\nonumber \\
&&R_\tau = 20.823 \pm 0.056,
\ea
which confirms lepton universality of neutral currents at a level
comparable to the charged current result.
The $\tau$ polarization as function of the production angle $\theta$
depends on both electron and tau couplings:
\be
P_\tau (\cos \theta) = -\frac {A_\tau(1+\cos ^2 \theta ) + 2 A_e \cos \theta}
{(1+\cos ^2 \theta ) + 2 A_\tau A_e \cos \theta }
\ee
and the asymmetry coefficients
\be
A_l = \frac {2(1-4\sin^2 \theta_W)}{1+(1-4\sin^2 \theta_W)}
\ee
are extremely sensitive to the effective weak
mixing angle $\sin^2 \theta_W$:
\be
\delta A_l \approx 8 \delta \sin^2 \theta_W.
\ee
%%%%%Tables from Alemany
\begin{table}
\caption[]{\label {T1}Preliminary $A_e$ and $A_\tau$ results from ALEPH
Collaboration
with statistical and systematic
uncertainties for the $1990-1995$
data based on the $\tau$ direction method. From \cite{Alemanytp}.}
\renewcommand{\arraystretch}{1.3}
\begin{center}
{\small
\begin{tabular}{|l|l|l|}
\hline\hline
Channel & $A_\tau(\%)$ & $A_e(\%)$\\
\hline
hadron & $15.49 \pm 1.01 \pm 0.66 $ & $ 17.36 \pm 1.35 \pm 0.13$\\
rho & $13.71 \pm 0.79 \pm 0.57$&
$15.04 \pm 1.06 \pm 0.078$\\
$a_1(3h)$ &$15.01 \pm 1.55 \pm 1.30$& $15.78 \pm 2.07 \pm 0.40$\\
$a_1(h2\pi^0)$& $15.94 \pm 1.73 \pm 1.7$ & $12.65 \pm 2.31 \pm 0.41$\\
electron & $14.98 \pm 2.18 \pm 0,82$& $16.96 \pm 2.92 \pm 0.15$\\
muon & $14.45 \pm 2.13 \pm 1.06$ & $12.05 \pm 2.78 \pm 0.24 $\\
acol. & $13.34 \pm 3.83 \pm 1.8$ & $19.41 \pm 5.02 \pm 0.24$\\
combi. & $14.61 \pm 0.53 \pm 0.37$ & $15.52 \pm 0.71 \pm 0.09$\\
\hline\hline
\end{tabular}
}
\end{center}
\end{table}
\begin{table}[th]
\caption[]{\label{T2}
Summary on $A_\tau$ and $A_e$ measurements
at LEP. The first error is statistical and the second is systematic
(from Ref. \cite{Alemanytp}).
}
\renewcommand{\arraystretch}{1.3}
\begin{center}
{\small
\begin{tabular}{|l|l|l|}
\hline\hline
Exp. & $A_\tau(\%)$ & $A_e(\%)$\\
\hline
ALEPH & $14.52 \pm 0.55 \pm 0.27 $ & $ 15.05 \pm 0.69 \pm 0.10$\\
DELPHI & $13.81 \pm 0.79 \pm 0.67$&
$13.53 \pm 1.16 \pm 0.33$\\
L3 &$14.76 \pm 0.88 \pm 0.62$& $16.78 \pm 1.27 \pm 0.30$\\
$OPAL $ & $13.4 \pm 0.9 \pm 1$ & $12.9 \pm 1.4 \pm 0.5$\\
\hline\hline
\end{tabular}
}
%\parbox{14.cm}{\small
%\caption[]{\label{tabfull}
%Summary on $A_\tau$ and $A_e$ measurements
%at LEP. \\The first error is statistical and the second is systematic\\
%(from Ref. \cite{Alemany}).
%}}
\end{center}
\end{table}
The conceptually simplest method to determine $P_\tau$ is based on the
decay rate into a single pion Eq.(\ref{sp}).
However,
despite their large event rates the LEP experiments are still limited
by the statistical error. Therefore it is important to use the maximum
number of decay channels and exploit the full multidimensional
distribution of the two \cite{Tsai,KW,Rouge}
and three \cite{KW,KM,Da} pion decay mode. The
importance of exploiting as many channels as possible becomes evident
from Table \ref{T1}, the improvement from combining the four
LEP experiments is shown in Table \ref{T2}. On the one had this result
can be used for a test of universality:
\be
\frac {A_e}{A_\tau} = 1.03 \pm 0.07,
\ee
on the other hand $A_e$ and $A_\tau$ can be combined to
\be
A_l = (14.52 \pm 0.34 ) \% .
\ee
Converted to a measurement of the effective weak mixing angle
\be
\sin^2 \theta_W = 0.23176 \pm 0.00043,
\ee
the result competes well with the measurement of SLD with
longitudinally polarized beams \cite{Baird}
\be
\sin^2 \theta_W = 0.23110 \pm 0.00029,
\ee
and is evidently an important ingredient in the combined LEP and
SLD Standard Model fit which gives \cite{Karlen}
\be
\sin^2 \theta_W = 0.23155 \pm 0.00019.
\ee
The result from $\tau$ polarization alone is about a factor five
more accurate than anticipated in the original plans for physics at LEP where
an accuracy
$\delta \sin^2 \theta_W \le 0.002$
was considered a reasonable goal \cite{Alt}.
\subsection{\label {Ss2.3} Electric and Magnetic Dipole Moments}
As a member of the third family with a mass drastically larger
than the one of the electron or muon the tau lepton is ideally
suited for speculations about anomalous couplings
to the photon as well as the $Z$ boson. The effects
of the electric or anomalous magnetic moments or their weak
analogues increase with the energy, hence $Z$ decays
are particularly suited for these investigations. Constraints
on the weak couplings are derived from the decay rate or from
$CP$ violating correlations in the decay $Z \to \tau^+ \tau^-$
\cite{Bernreuther}. By searching for an excess of hard noncollinear photons in
the radiative final state $\tau^+ \tau^- \gamma$ of the
$Z$ boson decays
give also access to anomalous electromagnetic couplings.
\begin{table*}[tbh]
\caption[]{\label{T3}
Limits on the electromagnetic and weak dipole moments
of $\tau$ (from Ref. \cite{Garcia})}
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
& $\gamma\tau\bar\tau$ & $Z\tau\bar\tau$\\
\hline
$a_\tau$ & $-0.05