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% Numerical values of various quantities
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% \newcommand {\MTval}     {(1776.98    ^{+0.27}_{-0.25})\mathrm{MeV}}   %  BES + DELCO (my average)       ??
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\newcommand {\MTval}     {(1776.96^{+0.18+0.25}_{-0.21-0.17})\mathrm{MeV}}   %  BES only PR D53 20.
\newcommand {\MMval}     {(105.658389 \pm 0.000034)    \mathrm{MeV}}   %  PDG 96   
\newcommand {\MEval}     {(0.51099907 \pm 0.00000015)  \mathrm{MeV}}   %  PDG 96   
\newcommand {\MPval}     {(139.56995  \pm 0.00035)     \mathrm{MeV}}   %  PDG 96   
\newcommand {\MKval}     {(493.677    \pm 0.016)       \mathrm{MeV}}   %  PDG 96   
\newcommand {\MRval}     {(768.5      \pm 0.6)         \mathrm{MeV}}   %  PDG 96   
\newcommand {\MWval}     {(80.33      \pm 0.15)        \mathrm{GeV}}   %  PDG 96   
\newcommand {\MZval}     {(91.187     \pm 0.002)       \mathrm{GeV}}   %  WARD 97   
%
\newcommand {\TAUMval}   {(2.19703    \pm 0.00004)  \mu\mathrm{s}}     %  PDG 96      
\newcommand {\TAUTval}   {(290.55     \pm 1.06)        \mathrm{fs}}    %  LI LP97   
\newcommand {\TAUPval}   {(26.033     \pm 0.005)       \mathrm{ns}}    %  PDG 96      
\newcommand {\TAUKval}   {(12.386     \pm 0.024)       \mathrm{ns}}    %  PDG 96      
%
\newcommand {\BRTEval}   {(17.786     \pm 0.072)       \%}             %  LI LP97
\newcommand {\BRTMval}   {(17.356     \pm 0.064)       \%}             %  LI LP97 
\newcommand {\BRTPval}   {(11.01      \pm 0.11)        \%}             %  PICH 97 review
\newcommand {\BRTKval}   {(0.692      \pm 0.028)       \%}             %  LI LP97
%
\newcommand {\BRMEval}   {100                          \%}             %  PDG 96
\newcommand {\BRPMval}   {(99.98782   \pm 0.00014)     \%}             %  PDG 96               
\newcommand {\BRKMval}   {(63.51      \pm 0.18)        \%}             %  PDG 96               
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\newcommand {\GF}        {G_{\mathrm{F}}}    
\newcommand {\GFe}       {G_{\mathrm{F}}^{\mathrm{e}}}    
\newcommand {\GFmu}      {G_{\mathrm{F}}^\mu}    
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\newcommand {\GFMUval}   {(1.16639  \pm 0.00002) \times 10^{-5}   \mathrm{GeV^{-2}}} % PDG 96
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\newcommand {\GFRATEval} {0.9992\pm0.0055~~(-0.15\sigma)}    % GF (tau) / GF (mu lifetime)
\newcommand {\GFRATMval} {1.0026\pm0.0053~~(+0.48\sigma)}
\newcommand {\GFRATPval} {1.0077\pm0.0110~~(+0.70\sigma)}
\newcommand {\GFRATKval} {0.9650\pm0.0394~~(-0.89\sigma)}
%
%
\newcommand {\DELTAEval} { -0.0008 \pm 0.0055} 
\newcommand {\DELTAMval} { +0.0026 \pm 0.0053}
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%%%%% \newcommand {\DELTAEval} { -0.0008 \pm 0.0055 \mbox{~~~} (-0.15\sigma)} 
%%%%% \newcommand {\DELTAMval} { +0.0026 \pm 0.0053 \mbox{~~~} (+0.48\sigma)}
%%%%% \newcommand {\DELTAPval} { +0.0079 \pm 0.0110 \mbox{~~~} (+0.72\sigma)}
%%%%% \newcommand {\DELTAKval} { -0.0362 \pm 0.0392 \mbox{~~~} (-0.93\sigma)}
%
%
\newcommand {\GFRATval} {1.xxx\pm x.xx~~(+xxx\sigma)}        % the above four channels combined....
%                                                              (done by hand)
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\newcommand {\GFRATMEval} {1.003 \pm0.006~~~(+0.61 \sigma)}  % GF(mu) / GF(e) in tau decays
%
\newcommand {\FPIVUDval} {(127.4 \pm 0.1)  \mathrm{MeV}}     %  
\newcommand {\FKVUSval}  {(35.18 \pm 0.05) \mathrm{MeV}}     %  
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\newcommand {\ETAval}     {0.009 \pm 0.022}     %  central values from fits 
\newcommand {\KAPPAMval}  {0.001 \pm 0.008}     %  plus/minus one sigma
\newcommand {\KAPPAEval}  {0.00  \pm 0.16}     %

\newcommand {\ETAlim}     {-0.034 < {\eta_{\tau\mu}} < 0.053}     %  range from fits at 95% CL
\newcommand {\KAPPAMlim}  {-0.014 < \kappa < 0.016}     %
\newcommand {\KAPPAElim}  {|\tilde{\kappa}| < 0.26}     %

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% RESULTS FOR A PRIOR DISTRIBUTION FLAT IN M(nu_tau)
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% The limits obtained with all parameters free
% --------------------------------------------
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% Convention is: 
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%              xxxxxA    90% CL with all parameters free
%              xxxxxB    95% CL   "   "      "        "
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\newcommand {\MNUTvalA} {32}    
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\title{Constraints on anomalous charged current couplings, tau
      neutrino mass and fourth generation mixing from tau leptonic
      branching fractions \\
       {\normalsize\em{Invited talk at the TAU'98 Workshop, 14-17 September 1998, Santander, Spain}}}

\author{Maria Teresa Dova\address{Universidad Nacional de La Plata, La Plata, Argentina},
        John Swain and 
        Lucas Taylor\address{Department of Physics,
                             Northeastern University, Boston, MA02115, USA}% 
%%%%%                                                         
%%%%%         \thanks{Footnotes should appear on the first page only to
%%%%%                 indicate your present address (if different from your
%%%%%                 normal address), research grant, sponsoring agency, etc.
%%%%%                 These are obtained with the {\tt\ttbs thanks} command.}
%%%%%         and 
%%%%%         X.-Y. Wang\address{Economics Department, University of Winchester, \\
%%%%%         2 Finch Road, Winchester, Hampshire P3L T19, United Kingdom}%
%%%%%                                                         
}%

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\begin{document}

\begin{abstract}
We use recent experimental measurements of tau branching fractions 
to determine the weak charged current magnetic and electric dipole 
moments of the tau and the Michel parameter $\eta$ with 
unprecedented precision.
These results are then used to constrain the tau compositeness scale 
and the allowed parameter space for Higgs doublet models.
We also present new constraints on the mass of the tau 
neutrino and its mixing with a fourth generation neutrino.
%
\end{abstract}

% typeset front matter (including abstract)
\maketitle
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\section{INTRODUCTION}
%
The tau lepton in the Standard Model is an exact duplicate of
the electron and muon, apart from its greater mass and
separately conserved quantum number.
Its charged current interactions are expected to be mediated by
the $W$ boson with pure $V\!-\!A$ coupling.
%
In this paper we present constraints on anomalous charged current 
couplings of the $\tau$~\cite{ANOMALOUS_COUPLINGS}.
These are derived from an analysis of the branching 
fractions for 
%
$\tau^-\rightarrow\mathrm{e}^-\bar{\nu}_{\mathrm{e}}\nu_\tau$ and
%
$\tau^-\rightarrow\mu^-\bar{\nu}_\mu\nu_\tau$, where charge-conjugate
decays are implied.
%
In particular, we consider derivative terms in the Hamiltonian which 
describe anomalous weak charged current magnetic and 
electric dipole couplings~\cite{RIZZO97A,CHIZHOV96A} and deviations 
from the $V\!-\!A$ structure of the charged current, to which the 
Michel parameter $\eta$ is sensitive~\cite{MICHEL}. 
The results for the $\eta$ parameter are used to constrain
extensions of the Standard Model which contain more than one 
Higgs doublet and hence charged Higgs bosons.

We also derive constraints on the mass of the third generation 
neutrino $\nu_3$ and its mixing with a fourth generation 
neutrino $\nu_4$ of mass $ > M_Z/2$~\cite{MNUTAU,MASS_MIX_UPDATE}.
In particular we constrain the mass $m_{\nu_3}$
and the Cabibbo-like mixing angle $\theta$, where 
the tau neutrino weak eigenstate is given by the
superposition of two mass eigenstates 
$|\nu_\tau\rangle = \cos\theta |\nu_3\rangle 
+ \sin\theta |\nu_4\rangle$.      
We compare the precise measurements of the $\tau$ partial 
widths for the following decays%
\footnote{Throughout this paper the charge-conjugate decays are also implied.
          We denote the branching ratios for these processes as
          $\Be, \Bm, \Bp, \Bk$ respectively;
          $\Bl$ denotes either $\Be$ or $\Bm$ while $\Bh$ denotes either $\Bp$ or $\Bk$.}
:
%
$\tau^-\rightarrow{e}^-\bar{\nu}_{{e}}\nu_\tau$,
%
$\tau^-\rightarrow\mu^-\bar{\nu}_\mu\nu_\tau$,
%
$\tau^-\rightarrow\pi^-\nu_\tau$, and
%
$\tau^-\rightarrow{K}^-\nu_\tau$,
%
with our theoretical predictions, as functions of 
$m_{\nu_3}$ and $\sin^2\theta$ to obtain upper limits on both 
these quantities. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\section{THEORETICAL PREDICTIONS}
%
%------------------------------------------------------
\subsection{Anomalous couplings}
%
The theoretical predictions for the branching fractions $\Bl$ for the 
decay $\tau^-\rightarrow\ell^-\bar{\nu}_{\ell}\nu_\tau (X_{\mathrm{EM}})$, with
$\ell^-=\mathrm{e}^-, \mu^-$ and $X_{\mathrm{EM}} = \gamma,~\gamma\gamma,~e^+e^-,\ldots$, 
are given by:
%
\begin{eqnarray}
%
  \Bl^{\mathrm{th.}}\!\!&\!\!=\!\!&\!\frac {\GF^2 m_\tau^5\tau_\tau}{192\pi^3}
                             \left( 1 -  8x - 12 x^2{\mathrm{ln}}x + 8 x^3 - x^4\right) \nonumber \\                      
              &\!\times\!& \!\!\left[\left(
                                          1 - \frac{\alpha(m_\tau)}{2\pi} 
                                          \left( 
                                                 \pi^2 - \frac{25}{4} 
                                          \right) 
                            \right)  
                           \left(1 + \frac{3}{5} \frac{m_\tau^2}{\mW^2} 
                           \right) \right]                                              \nonumber  \\ 
              &\!\times\!& \left[ 1 + \Delta_\ell \right],
\label{equ:blept}
\end{eqnarray}
%
where 
$\GF    = \GFMUval$ is the Fermi constant~\cite{PDG96SHORT};
$\tau_\tau = \TAUTval$ is the tau lifetime~\cite{LI97A};     
$m_\tau = \MTval$~\cite{MTAUBESNEW} is the tau mass;
and 
$x=m_\ell^2/m_\tau^2$.
%
The first term in square brackets allows for radiative 
corrections\cite{BERMAN58A,KINOSHITA59A,SIRLIN78A,MARCIANO88A}, 
where $\alpha(m_\tau)\simeq 1/133.3$ is the QED coupling constant~\cite{MARCIANO88A} 
and $\mW = 80.400 \pm 0.075$\,GeV is the $W$ mass~\cite{PIC97}.
%
The second term in brackets describes the effects of new physics 
where the various $\Delta_\ell$ we consider are defined below.

The effects of anomalous weak charged current dipole moment 
couplings at the $\tau\nu_\tau W$ vertex are 
described by the effective Lagrangian
%
\begin{eqnarray}
{\cal{L}}\!&\!=\!&\!\frac{g}{\sqrt{2}}\bar{\tau} 
                            \left[ 
                            \gamma_\mu +
                            \frac{i\sigma_{\mu\nu}q^\nu}{2m_\tau}
                            (\kappa_\tau-i\tilde{\kappa}\gamma_5)
                            \right]
                      P_L \nu_\tau W^\mu \nonumber \\
                      & & + ({\mathrm{Hermitian\ conjugate}}),
\end{eqnarray}
%
where $P_L$ is the left-handed projection 
operator and the parameters $\kappa$ and 
${\tilde{\kappa}}$ are the (CP-conserving) magnetic and (CP-violating) electric dipole form 
factors respectively~\cite{RIZZO97A}.
They are the charged current analogues of the weak neutral current dipole 
moments, measured using $Z\rightarrow\tau^+\tau^-$ events~\cite{PICH97A}, 
and the electromagnetic dipole moments~\cite{BIEBEL96A,TTGNUCPHYSB} 
recently measured by L3 and OPAL using 
$Z\rightarrow\tau^+\tau^-\gamma$ events~\cite{OPALTTG,L3TTG,TAYLOR_TAU98}.
In conjunction with Eq.~\ref{equ:blept}, the effects of non-zero 
values of $\kappa$ and ${\tilde{\kappa}}$ on the tau leptonic 
branching fractions may be described by~\cite{RIZZO97A}
%
\begin{eqnarray}
%
  \Delta_\ell^{\kappa}         & = &  {\kappa}/{2} + {\kappa^2}/{10};     \\         
  \Delta_\ell^{\tilde{\kappa}} & = &  {\tilde{\kappa}^2}/{10}.            
 \label{equ:deltalk}
\end{eqnarray}
%
The dependence of the tau leptonic branching ratios on $\eta$ is given,
in conjunction with Eq.~\ref{equ:blept}, by~\cite{STAHL94A}
%
\begin{eqnarray}
%
  \Delta_\ell^{\eta} & = &  4{\eta_{\tau\ell}} {\sqrt{x}},                  
%                         
\label{equ:deltaln}
\end{eqnarray}
%
where the subscripts on $\eta$ denote the initial and final 
state charged leptons.
%%%%%                 
%%%%%           (N.B. PICH  gives more detailed expression but 
%%%%%           extra terms are irrelevant).
%%%%%                 
%
Both leptonic tau decay modes probe the charged current couplings of 
the transverse $W$, and are sensitive to $\kappa$ and ${\tilde{\kappa}}$.
In contrast, only the $\tau^-\rightarrow\mu^-\bar{\nu}_\mu\nu_\tau$ channel 
is sensitive to $\eta$, due to a relative suppression factor of $m_e/m_\mu$ for 
the $\tau^-\rightarrow\mathrm{e}^-\bar{\nu}_{\mathrm{e}}\nu_\tau$ channel.
Semi-leptonic tau branching fractions are not considered since 
they are insensitive to $\kappa$,  ${\tilde{\kappa}}$, and $\eta$.

%------------------------------------------------------
\subsection{Tau neutrino mass and mixing}

The theoretical predictions for the branching fractions $\Bl$ 
allowing for the $\nu_\tau$ mass and mixing with a fourth lepton
generation are given by~\cite{MNUTAU}:
%
\begin{eqnarray}
%
\Bl^{\mathrm{th.}}\!&\!=\!&\!\frac {\GF^2 m_\tau^5\tau_\tau}{192\pi^3}\left( 1 -  8x - 12 x^2{\mathrm{ln}}x + 8 x^3 - x^4\right)                 \nonumber \\                                      
              &\!\times\!& \!\!\!\left(
                                          1 - \frac{\alpha(m_\tau)}{2\pi} 
                                          \left( 
                                                 \pi^2 - \frac{25}{4} 
                                          \right) 
                            \right)  
                           \left(1 + \frac{3}{5} \frac{m_\tau^2}{\mW^2} 
                           \right)                                              \nonumber  \\
             &\!\times\!&\!\!\!\left[ 1 - \sin^2\theta \right] \left[ 1 - 8y(1-x)^3+\cdots\right]
\label{equ:bleptm}
\end{eqnarray}
%
where the tau mass used is determined by BES from the $\tau^+\tau^-$ 
production rate with no dependence on the tau neutrino mass
and 
$x=m_\ell^2/m_\tau^2$.
%
The first term in square brackets describes mixing with a fourth generation neutrino
which, being kinematically forbidden, causes a suppression of the decay rate.
The second term in brackets  parametrises the suppression 
due to a non-zero mass of $\nu_3$, where $y=m_{\nu_3}^2 / m_\tau^2$
and the ellipsis denotes negligible higher order 
terms~\cite{MNUTAU}.                              

The branching fractions for the decays $\tau^-\rightarrow{h}^-\nu_\tau$, 
with ${h}=\pi/{K}$, are given by
%
\begin{eqnarray}  
\Bh^{\mathrm{th.}}\!\!\!\!&\!=\!&\!\!\!\!\left(\frac {\GF^2 m_\tau^3 } {16\pi}\right)\tau_\tau f_{{h}}^2 |V_{\alpha\beta}|^2  
                       \left(1 - x\right)^2                                                          \nonumber \\         
                 &\!\times\!&\!\!\!\!\!\left( 
                                     1 + \frac{2\alpha}{\pi} {\mathrm{ln}} 
                                     \left( \frac {\mZ} {m_\tau} \right)+\cdots 
                       \right) \left[ 1 - \sin^2\theta  \right]                                              \nonumber  \\                                                                                                                                             
             &\!\times\!&\!\!\!\!\!\left[1\!-\!y\left(\frac{2\!+\!x\!-\!y}{1-x}\right)\left(1\!-\!\frac{y(2\!+\!2x\!-\!y)}{(1-x)^2}\right)^{\frac{1}{2}}\right]               
         \label{equ:bhad}
\end{eqnarray}
%
where 
$x=m_{{h}}^2 / m_\tau^2$,
$m_{{h}}$ is the hadron mass, 
$f_{{h}}$ are the hadronic form factors, 
and $V_{\alpha\beta}$ are the CKM matrix elements, 
$V_{{ud}}$ and $V_{{us}}$,
for $\pi^-$ and $\kn$ respectively.
%
From an analysis of
$\pi^-\rightarrow\mu^-\bar{\nu}_\mu$ and
$\kn\rightarrow\mu^-\bar{\nu}_\mu$ decays, 
one obtains
$f_\pi |V_{{ud}}| = \FPIVUDval$ and
$f_{{K}} |V_{{us}}| = \FKVUSval$\cite[and references therein]{MARCIANO92A}. 
%
The ellipsis represents terms, estimated to be ${\cal{O}}(\pm 0.01)$\cite{MARCIANO92A},        % OK
which are neither explicitly treated nor implicitly absorbed into $\GF$,
$f_\pi |V_{{ud}}|$, or $f_{{K}} |V_{{us}}|$.
%
The first term in square brackets describes mixing with a fourth generation neutrino
while the second parametrises the effects of 
a non-zero $m_{\nu_3}$~\cite{MNUTAU}.

The fourth generation neutrino mixing affects all the tau branching
fractions with a common factor whereas a non-zero tau neutrino 
mass affects all channels with different kinematic factors.
Therefore, given sufficient experimental precision, these two effects 
could in principle be separated.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{RESULTS}
%------------------------------------------------------
\subsection{Anomalous couplings}
%
We use the world average values for the measured tau 
branching fractions~\cite{LI97A}:
%
$\Be = \BRTEval$ and 
$\Bm = \BRTMval$.
%
Substituting in Eq.~\ref{equ:blept} for 
these and the other measured quantities we obtain
%
$\Delta_e   = \DELTAEval$ and 
$\Delta_\mu = \DELTAMval$ 
%
where the errors include the effects of the uncertainties on all 
the measured quantities appearing in Eq.~\ref{equ:blept}.
These results are consistent with zero which, assuming that there are 
no fortuitous cancellations, indicates the absence of anomalous 
effects within the experimental precision.

We therefore proceed to derive constraints on $\kappa$, $\tilde{\kappa}$, and 
${\eta_{\tau\mu}}$ from a combined likelihood fit to both tau decay channels. 
The likelihood is constructed numerically following the 
procedure of Ref.~\cite{NIM_LIKELIHOOD_PAPER} by randomly 
sampling all the quantities used according to their errors, 
conservatively assuming for each parameter that the other 
two parameters are zero.

%%%%%
%%%%%------------------------------------------------------
%%%%% WHAT ABOUT PI/K FINAL STATES (FOR CHARGED HIGGS) ???
%%%%%------------------------------------------------------
%%%%%

We determine $\kappa = {\KAPPAMval}$, where the errors correspond 
to one standard deviation, and constrain it to the 
range ${\KAPPAMlim}$ at the 95\% confidence level (C.L.).
This result improves on the 95\% C.L. constraint of 
$|\kappa| < 0.0283$ determined by Rizzo~\cite{RIZZO97A}. 

We determine $\tilde\kappa = {\KAPPAEval}$ and constrain it to the 
range ${\KAPPAElim}$ at the 95\% C.L.
Our constraint, which is the first on this quantity, is considerably 
less stringent than that on $\kappa$ due to the lack of linear terms.
This also means that the likelihood for $\tilde\kappa$
is symmetric by construction.
Were $\tilde\kappa$ to differ significantly from zero, then the likelihood
distribution would have two distinct peaks either side of zero. 
Such structure was not, however, observed.
The decay $W\rightarrow \tau\nu$ is also sensitive to charged current 
dipole terms but, given that the energy scale is $\mW$, the interpretation 
in terms of the static properties $\kappa$ and $\tilde\kappa$ is  
less clear.

We determine ${\eta_{\tau\mu}} = {\ETAval}$ and constrain it to the
range ${\ETAlim}$ at the 95\% C.L.
The uncertainty on our measurement of ${\eta_{\tau\mu}}$ 
is significantly smaller than that obtained by Stahl using the same 
technique $({\eta_{\tau\mu}} = 0.01 \pm 0.05)$~\cite{STAHL94A} 
and more recent determinations using the shape of 
momentum spectra of muons from $\tau$ decays 
$({\eta_{\tau\mu}} = -0.04 \pm 0.20)$~\cite{PICH97A}.

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

%------------------------------------------------------
\subsection{Tau neutrino mass and mixing}
%
We use the world average values for the measured tau 
branching fractions~\cite{LI97A}:
%
$\Be$ and $\Bm$ as above, $\Bp = \BRTPval$ and $\Bk=\BRTKval$. 
%
Substituting in equations \ref{equ:bleptm} and \ref{equ:bhad} for 
the measured quantities we find that both 
$m_{\nu_\tau}$ and $\sin^2\theta$ are consistent with zero.
%
We therefore derive constraints on $m_{\nu_\tau}$ and $\sin^2\theta$
from a combined likelihood fit to the four tau decay channels. 
The CLEO measurement of the $\tau$ mass was used to further constrain $m_{\nu_3}$.
From an analysis of 
$\tau^+\tau^-$ 
$\rightarrow$ 
$(\pi^+n\pi^0\bar{\nu}_\tau)$
$(\pi^-m\pi^0\nu_\tau)$ 
events (with $n\leq2, m\leq2, 1\leq n+m\leq3$), CLEO determined the $\tau$ mass to be 
$m_\tau = (1777.8 \pm 0.7 \pm 1.7) + [m_{\nu_3}({\mathrm{MeV}})]^2/1400$ MeV\cite{CLEOWEINSTEIN}.
The likelihood for the CLEO and BES measurements to agree, as a function of 
$m_{\nu_3}$, is included in the global likelihood. 

%------------------------------------------------------
The fit yields upper limits of 
%
\begin{eqnarray}
%
 m_{\nu_3}     & < & \MNUTvalB\,{\mathrm{MeV}}  \\  
 \sin^2\theta  & < & \FMIXvalB                   
%
\end{eqnarray}
%
at the 95\% C.L. or 
%
\begin{eqnarray}
%
 m_{\nu_3}     & < & \MNUTvalA\,{\mathrm{MeV}}  \\ 
 \sin^2\theta  & < & \FMIXvalA                   
%
\end{eqnarray}
%
at the 90\% C.L. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{DISCUSSION}
%
%------------------------------------------------------
\subsection{Compositeness of the tau}
% 
Derivative couplings necessarily involve the introduction of a length
or mass scale. 
Anomalous magnetic moments due to compositeness are expected to be
of order $m_\tau/\Lambda$ where $\Lambda$ is the compositeness 
scale~\cite{BRODSKY80A}.
We can then interpret the 95\% confidence level on $\kappa$, 
the quantity for which we have a more stringent bound, as a statement
that the $\tau$ appears to be a point-like Dirac particle up to an energy
scale of $\Lambda \approx m_\tau/0.016 = 110$\,GeV. 
These results are comparable to those obtained from anomalous weak 
neutral current couplings~\cite{PICH97A} and more stringent than those 
obtained for anomalous electromagnetic couplings~\cite{OPALTTG,L3TTG,TAYLOR_TAU98}.

%------------------------------------------------------
\subsection{Extended Higgs sector models}
% 
Many extensions of the Standard Model, such as supersymmetry (SUSY),
involve an extended Higgs sector with more than one Higgs doublet. 
Such models contain charged Higgs bosons which contribute 
to the weak charged current with couplings which depend on the fermion masses.
Of all the Michel parameters, ${\eta_{\tau\mu}}$ is especially
sensitive to the exchange of a charged Higgs.
Following Stahl~\cite{STAHL94A}, ${\eta_{\tau\mu}}$ can be written as 
%
\begin{equation}
{\eta_{\tau\mu}} = -\left( \frac{m_\tau m_\mu}{2} \right)
        \left( \frac{\tan\beta}{m_H} \right)^2
\end{equation}
%
where $\tan\beta$ is the ratio of vacuum expectation values of the
two Higgs fields, and $m_H$ is the mass of the charged Higgs.
This expression applies to type II extended Higgs sector models 
in which the up-type quarks get their masses from one doublet and 
the down-type quarks get their masses from the other. 

We determine the one-sided constraint ${\eta_{\tau\mu}} > -0.0186$ 
at the 95\% C.L. which rules out the region
%
    $m_H < (1.86 \tan\beta) \,{\mathrm{GeV}}$ at the 95\% C.L.
%%%%% %
%%%%% %
%%%%% (or $m_H < 2.24 \tan\beta \,{\mathrm{GeV}}$ at the 90\% C.L.)
%%%%% %
%%%%% %
%
as shown in Fig.~\ref{fig:chhiggs}.
%
An almost identical constraint on the high $\tan\beta$ region of 
type II models may be obtained from the process 
$B\rightarrow\tau\nu$~\cite{HOU93A}.
The most stringent constraint, from the L3 experiment,
rules out the region
%
    $m_H < (2.09 \tan\beta)\,{\mathrm{GeV}}$ at the 95\% C.L.~\cite{L3B2TAU}.
%%%%% %
%%%%% %
%%%%% (or $m_H < 2.63 \tan\beta \,{\mathrm{GeV}}$ at the 90\% C.L.)
%%%%% %
%%%%% %
Within the specific framework of the minimal supersymmetric standard 
model, the process $B\rightarrow\tau\nu X$ rules out the region 
%
$m_H < (2.33 \tan\beta)\,{\mathrm{GeV}}$ at the 
95\% C.L.~\cite{COARASA97A}.
%
This limit, however, depends on the value of the Higgsino mixing 
parameter $\mu$ and can be evaded completely for $\mu>0$. 
The non-observation of proton decay also tends to rule out 
the large $\tan\beta$ region but these constraints are 
particularly model-dependent.
%
The very low $\tan\beta$ region is ruled out by measurements of 
the partial width $\Gamma (Z\rightarrow b\bar{b})$. 
For type II models the approximate region excluded is 
$\tan\beta < 0.7$ at the $2.5\sigma$ C.L. for 
any value of $M_H$~\cite{GRANT95A}.  
%
Complementary bounds for the full $\tan\beta$ region are derived 
from the CLEO measurement of 
$BR(b\rightarrow s\gamma) = (2.32\pm0.57\pm 0.35)\times 10^{-4}$  
which rules out, for type II models, the region 
%
$M_H < 244 + 63/{(\tan\beta)}^{1.3}$~\cite{CLEO_BTOSGAMMA}.
%
This constraint can, however, be circumvented in SUSY models
where other particles in the loops can cancel out the effect
of the charged Higgs.
%
Direct searches at LEP II exclude the region $m_H < 54.5$\,GeV for 
all values of $\tan\beta$~\cite{DELPHIMH}.
The CDF search for charged Higgs bosons in the process
$t\rightarrow b H^+$ rules out the region of low $m_H$ and 
high $\tan\beta$~\cite{ABE97A}.

The 95\% C.L. constraints in the $m_H$ {\em{vs.}} $\tan\beta$ plane, 
from this and other analyses, are shown in Fig.~\ref{fig:chhiggs}.
%--------------------------------------------------------------------------
%   START NEW TEXT 
%--------------------------------------------------------------------------
We anticipate that the constraints from $Z\rightarrow b\bar{b}$ 
and $b\rightarrow s \gamma$ will improve significantly in the near future 
due to new measurements from the LEP and CLEO collaborations and 
from refinements in the theoretical treatment~\cite{CIUCHINI97A}. 
Some caution is advised in the interpretation of the large $\tan\beta$ 
regime which becomes non-perturbative for $\tan\beta > O(70)$. 
Future improved measurements of the tau branching fractions and 
lifetime will, however, extend the constraints on $\tan\beta$ towards 
lower values, where perturbative calculations are more applicable.
%--------------------------------------------------------------------------
%   END NEW TEXT 
%--------------------------------------------------------------------------
%--------------------------------------------------------------------------

\begin{figure}[htb]
\epsfig{file=mh_vs_tan_beta.eps,width=0.45\textwidth,clip=}\\[-4mm]
\caption{Constraints on $m_H$ as a function of $\tan\beta$ at the 95\% C.L.,
         from this analysis of $\eta_{\tau\mu}$ and the other analyses 
         described in the text.}\label{fig:chhiggs}
\end{figure}
%

%------------------------------------------------------
\subsection{Tau neutrino mass and mixing}

The limit on $m_{\nu_3}$ can be reasonably interpreted as a 
limit on $m_{\nu_\tau}$, since the mixing of 
$m_{\nu_3}$ with lighter neutrinos is also small~\cite{PDG96SHORT}.
%
The best direct experimental constraint on the tau neutrino mass 
is $m_{\nu_\tau} < 18.2$\,MeV at the 95\% confidence level\cite{ALEPH_NUTAU} 
which was obtained using many-body hadronic decays of the $\tau$.
%
While our constraint is less stringent, it is statistically independent.
Moreover, it is insensitive to fortuitous or pathological events 
close to the kinematic limits, the absolute energy scale of the detectors,
and the details of the resonant structure of multi-hadron $\tau$ decays~\cite{TAU98_MCNULTY}.
Since LEP has completed running at the Z it is unlikely that 
significantly improved constraints on $m_{\nu_\tau}$, using
multi-hadron final states, will be forthcoming
in the foreseeable future.

Future improved measurements of the tau branching fractions,
lifetime, and the tau mass from direct reconstruction 
would enable significant improvements to be made in the 
determinations of both $m_{\nu_\tau}$ and $\sin^2\theta$.
If CLEO and the b-factory experiments were to reduce the uncertainties 
on the experimental quantities by a factor of approximately 2, 
then the constraints on $m_{\nu_\tau}$ from the technique 
we have described would become the most competitive.
Were a tau-charm factory to be built, then the determination
of $m_{\nu_\tau}$ by direct reconstruction would again become
the most sensitive technique.  
Our upper limit on $\sin^2\theta$ is already the most stringent 
experimental constraint on mixing of the third and fourth 
neutrino generations.
  


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\section{SUMMARY}
%
From an analysis of tau leptonic branching fractions we determine
%
\begin{eqnarray}
%
\kappa           & = & {\KAPPAMval};          \\
\tilde\kappa     & = &  {\KAPPAEval};         \\
{\eta_{\tau\mu}} & = &  {\ETAval}.            
%
\end{eqnarray}
%
Each of these results is the most precise determination to date.
%
The result for $\kappa$ indicates that the tau is point-like up
to an energy scale of approximately 110\,GeV.
The result for ${\eta_{\tau\mu}}$ constrains the charged Higgs 
of type II two-Higgs doublet models, such that the region 
%
\begin{equation}
%
m_H < (1.86 \tan\beta) \,{\mathrm{GeV}}
%
\end{equation}
%
is excluded at the 95\% C.L.
%
The fit for tau neutrino mass and mixing yields upper limits of 
%
\begin{eqnarray}
%
 m_{\nu_3}     & < & \MNUTvalB\,{\mathrm{MeV}}  \\  
 \sin^2\theta  & < & \FMIXvalB                   
%
\end{eqnarray}
%
at the 95\% confidence level.

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\section*{Acknowledgements}
%
M.T.D. acknowledges the support of CONICET, Argentina. 
J.S. and L.T. would like to thank the Department of Physics,
Universidad Nacional de La Plata for their generous hospitality 
and the National Science Foundation for financial support.
J.S. gratefully acknowledges the support of the International
Centre for Theoretical Physics, Trieste.
   
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%%%%% 
%%%%% \begin{thebibliography}{9}
%%%%% \bibitem{Scho70} S. Scholes, Discuss. Faraday Soc. No. 50 (1970) 222.
%%%%% \bibitem{Mazu84} O.V. Mazurin and E.A. Porai-Koshits (eds.),
%%%%%                  Phase Separation in Glass, North-Holland, Amsterdam, 1984.
%%%%% \bibitem{Dimi75} Y. Dimitriev and E. Kashchieva, 
%%%%%                  J. Mater. Sci. 10 (1975) 1419.
%%%%% \bibitem{Eato75} D.L. Eaton, Porous Glass Support Material,
%%%%%                  US Patent No. 3 904 422 (1975).
%%%%% \end{thebibliography}
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