%Paper: 
%From: nardi@mich1.physics.lsa.umich.edu
%Date: Tue, 7 Sep 1993 19:01:45 -0400


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%   Process with plain TEX.  Uses `epsf.tex' to include pictures.
%%   4 PostScript figure (kek1s.ps, kek2s.ps, lep2s.ps, nlcs.ps)
%%   appended after `\bye' at the end of the file.
%%   Strip off and rename kek1s.ps, kek2s.ps, lep2s.ps, nlcs.ps
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\input epsf.tex
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def\pmb#1{\setbox0=\hbox{#1}%
  \hbox{\kern-.025em\copy0\kern-\wd0
  \kern.05em\copy0\kern-\wd0
  \kern-0.025em\raise.0433em\box0} }

\def \mapright#1{\smash{\mathop{\longrightarrow}\limits^{#1}}}
\def\half {{1\over 2}}
\def\eps{\tens{\hbox{\mm \char'42}}_{\hskip -5pt\infty}}
\catcode`@=11
\def\leftrightarrowfill{$\m@th\mathord\leftarrow \mkern-6mu
  \cleaders\hbox{$\mkern-2mu \mathord- \mkern-2mu$}\hfill
  \mkern-6mu \mathord\rightarrow$}
\def\overleftrightarrow#1{\vbox{\ialign{##\crcr
     \leftrightarrowfill\crcr\noalign{\kern-1pt\nointerlineskip}
     $\hfil\displaystyle{#1}\hfil$\crcr}}}
\catcode`@=12
\def\tens{\overleftrightarrow}

\def\approxge{\hbox {\hfil\raise .4ex\hbox{$>$}\kern-.75 em
\lower .7ex\hbox{$\sim$}\hfil}}
\def\approxle{\hbox {\hfil\raise .4ex\hbox{$<$}\kern-.75 em
\lower .7ex\hbox{$\sim$}\hfil}}

%\def \section
%      #1          {\vbox{\smallskip\bigskip\noindent{\bf #1}\medskip}
%                   \nobreak}
\def \abstract#1 {\vskip 0.5truecm\sepline\vskip 0.5truecm
$$\vbox{\hsize=15truecm\noindent #1}$$}
\def \SISSA#1#2 {\vfil\vfil\centerline{Ref. S.I.S.S.A. #1 CM (#2)}}
\def \PACS#1 {\vfil\line{\hfil\hbox to 15truecm{PACS numbers: #1 \hfil}\hfil}}

\def \hfigure
     #1#2#3       {\midinsert \vskip #2 truecm $$\vbox{\hsize=14.5truecm
             \seven\baselineskip=10pt\noindent {\bcp \noindent Figure  #1}.
                   #3 } $$ \vskip -20pt \endinsert }

\def \hfiglin
     #1#2#3       {\midinsert \vskip #2 truecm $$\vbox{\hsize=14.5truecm
              \seven\baselineskip=10pt\noindent {\bcp \hfil\noindent
                   Figure  #1}. #3 \hfil} $$ \vskip -20pt \endinsert }

\def \vfigure
     #1#2#3#4     {\dimen0=\hsize \advance\dimen0 by -#3truecm
                   \midinsert \vbox to #2truecm{ \seven
                   \parshape=1 #3truecm \dimen0 \baselineskip=10pt \vfill
                   \noindent{\bcp Figure #1} \pretolerance=6500#4 \vfill }
                   \endinsert }

\def\hfigureps#1#2#3#4{
\midinsert
      \vskip #2 truecm
      \special{#4}
      $$
         \vbox{
            \hsize=14.5truecm \seven\baselineskip=10pt\noindent
            {\bcp \noindent Figure  #1}. #3
         }
      $$
      \vskip -10pt
\endinsert
}
%
% definizione di
\def \ref
     #1#2         {\smallskip \item{[#1]}#2}
\def \sepline     {\medskip\centerline{\vbox{\hrule width5truecm}} \medskip}
\def \tabrule     {\noalign{\vskip 5truept \hrule\vskip 5truept} }
\def \tabrul2     {\noalign{\vskip 5truept \hrule \vskip 2truept \hrule
                   \vskip 5truept} }

% \def \footnoterule{\kern-3pt \hrule width 0truein \kern2.6pt}

% \pageno=0
\footline={\ifnum\pageno>0 \tenrm \hss \folio \hss \fi }

\def\today
 {\count10=\year\advance\count10 by -1900 \number\day--\ifcase
  \month \or Jan\or Feb\or Mar\or Apr\or May\or Jun\or
             Jul\or Aug\or Sep\or Oct\or Nov\or Dec\fi--\number\count10}

\def\hour{\count10=\time\count11=\count10
\divide\count10 by 60 \count12=\count10
\multiply\count12 by 60 \advance\count11 by -\count12\count12=0
\number\count10 :\ifnum\count11 < 10 \number\count12\fi\number\count11}

\def\draft{
   \baselineskip=20pt
   \def\makeheadline{\vbox to 10pt{\vskip-22.5pt
   \line{\vbox to 8.5pt{}\the\headline}\vss}\nointerlineskip}
   \headline={\hfill \seven {\bcp Draft version}: today is \today\ at \hour
              \hfill}
          }

%
% Max -macros
%

%
\catcode`@=11
%
%------------------------- comandi riservati ---------------------------
%
\def\b@lank{ }

\newif\if@simboli
\newif\if@riferimenti

\newwrite\file@simboli
\def\simboli{
    \immediate\write16{ !!! Genera il file \jobname.SMB }
    \@simbolitrue\immediate\openout\file@simboli=\jobname.smb}

\newwrite\file@ausiliario
\def\riferimentifuturi{
    \immediate\write16{ !!! Genera il file \jobname.AUX }
    \@riferimentitrue\openin1 \jobname.aux
    \ifeof1\relax\else\closein1\relax\input\jobname.aux\fi
    \immediate\openout\file@ausiliario=\jobname.aux}

\newcount\eq@num\global\eq@num=0
\newcount\sect@num\global\sect@num=0

\newif\if@ndoppia
\def\numerazionedoppia{\@ndoppiatrue\gdef\la@sezionecorrente{\the\sect@num}}

\def\se@indefinito#1{\expandafter\ifx\csname#1\endcsname\relax}
\def\spo@glia#1>{} % si applica a \meaning\xxxxx; butta via tutto quello
                   % che produce \meaning fino al carattere >
                   % (v. manuale TeX, pag. 382, \strip#1>{}).

\newif\if@primasezione
\@primasezionetrue

\def\s@ection#1\par{\immediate
    \write16{#1}\if@primasezione\global\@primasezionefalse\else\goodbreak
    \vskip\spaziosoprasez\fi\noindent
    {\bf#1}\nobreak\vskip\spaziosottosez\nobreak\noindent}
%
%------------------------------ a disp. dell'utente:  sezioni -------------

\def\sezpreset#1{\global\sect@num=#1
    \immediate\write16{ !!! sez-preset = #1 }   }

\def\spaziosoprasez{50pt plus 60pt}
\def\spaziosottosez{15pt}

\def\sref#1{\se@indefinito{@s@#1}\immediate\write16{ ??? \string\sref{#1}
    non definita !!!}
    \expandafter\xdef\csname @s@#1\endcsname{??}\fi\csname @s@#1\endcsname}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def\adv#1{\global\advance\sect@num by #1}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\def\autosez#1#2#3\par{
    \global\advance\sect@num by 1\if@ndoppia\global\eq@num=0\fi
    \xdef\la@sezionecorrente{\the\sect@num}
    \def\usa@getta{1}\se@indefinito{@s@#1}\def\usa@getta{2}\fi
    \expandafter\ifx\csname @s@#1\endcsname\la@sezionecorrente\def
    \usa@getta{2}\fi
    \ifodd\usa@getta\immediate\write16
      { ??? possibili riferimenti errati a \string\sref{#1} !!!}\fi
    \expandafter\xdef\csname @s@#1\endcsname{\la@sezionecorrente}
    \immediate\write16{\la@sezionecorrente. #2}
    \if@simboli
      \immediate\write\file@simboli{ }\immediate\write\file@simboli{ }
      \immediate\write\file@simboli{  Sezione
                                  \la@sezionecorrente :   sref.   #1}
      \immediate\write\file@simboli{ } \fi
    \if@riferimenti
      \immediate\write\file@ausiliario{\string\expandafter\string\edef
      \string\csname\b@lank @s@#1\string\endcsname{\la@sezionecorrente}}\fi
    \goodbreak\vskip 48pt plus 60pt
% \noindent{\lltitle\the\sect@num.\quad  #2} % escluso per prD (I.,II.,III)
% \par\nobreak\vskip 15pt \nobreak\noindent}
\centerline{\lltitle #2}                     %  per prD.
\centerline{\lltitle #3}                     %  per prD.
\par\nobreak\vskip 15pt \nobreak\noindent}

\def\semiautosez#1#2\par{
    \gdef\la@sezionecorrente{#1}\if@ndoppia\global\eq@num=0\fi
    \if@simboli
      \immediate\write\file@simboli{ }\immediate\write\file@simboli{ }
      \immediate\write\file@simboli{  Sezione ** : sref.
          \expandafter\spo@glia\meaning\la@sezionecorrente}
      \immediate\write\file@simboli{ }\fi
\noindent\lltitle \s@ection#2 \par}

%------------------------------ a disp. dell'utente:  equazioni -----------

\def\eqpreset#1{\global\eq@num=#1
     \immediate\write16{ !!! eq-preset = #1 }     }

\def\eqref#1{\se@indefinito{@eq@#1}
    \immediate\write16{ ??? \string\eqref{#1} non definita !!!}
    \expandafter\xdef\csname @eq@#1\endcsname{??}
    \fi\csname @eq@#1\endcsname}

\def\eqlabel#1{\global\advance\eq@num by 1
    \if@ndoppia\xdef\il@numero{\la@sezionecorrente.\the\eq@num}
       \else\xdef\il@numero{\the\eq@num}\fi
    \def\usa@getta{1}\se@indefinito{@eq@#1}\def\usa@getta{2}\fi
    \expandafter\ifx\csname @eq@#1\endcsname\il@numero\def\usa@getta{2}\fi
    \ifodd\usa@getta\immediate\write16
       { ??? possibili riferimenti errati a \string\eqref{#1} !!!}\fi
    \expandafter\xdef\csname @eq@#1\endcsname{\il@numero}
    \if@ndoppia
       \def\usa@getta{\expandafter\spo@glia\meaning
       \la@sezionecorrente.\the\eq@num}
       \else\def\usa@getta{\the\eq@num}\fi
    \if@simboli
       \immediate\write\file@simboli{  Equazione
            \usa@getta :  eqref.   #1}\fi
    \if@riferimenti
       \immediate\write\file@ausiliario{\string\expandafter\string\edef
       \string\csname\b@lank @eq@#1\string\endcsname{\usa@getta}}\fi}

\def\autoeqno#1{\eqlabel{#1}\eqno(\csname @eq@#1\endcsname)}
\def\autoleqno#1{\eqlabel{#1}\leqno(\csname @eq@#1\endcsname)}
\def\eqrefp#1{(\eqref{#1})}

%%%%%%%%%%%%%%%%%%%%%    abbreviazioni       %%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\def\eq{\autoeqno}
\def\req{\eqrefp}
\def\chap{\autosez}        % #1 per il numero (scritto)     #2 per il titolo
\def\nochap{\semiautosez}  % #1 per il numero (non scritto) #2 per il titolo


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

%--------------- bibliografia automatica: riservati ----------------------

\newcount\cit@num\global\cit@num=0

\newwrite\file@bibliografia
\newif\if@bibliografia
\@bibliografiafalse

\def\lp@cite{[}
\def\rp@cite{]}
\def\trap@cite#1{\lp@cite #1\rp@cite}
\def\lp@bibl{[}
\def\rp@bibl{]}
\def\trap@bibl#1{\lp@bibl #1\rp@bibl}

\def\refe@renza#1{\if@bibliografia\immediate        % scrive su .BIB
    \write\file@bibliografia{
    \string\item{\trap@bibl{\cref{#1}}}\string
    \bibl@ref{#1}\string\bibl@skip}\fi}

\def\ref@ridefinita#1{\if@bibliografia\immediate\write\file@bibliografia{
    \string\item{?? \trap@bibl{\cref{#1}}} ??? tentativo di ridefinire la
      citazione #1 !!! \string\bibl@skip}\fi}

\def\bibl@ref#1{\se@indefinito{@ref@#1}\immediate
    \write16{ ??? biblitem #1 indefinito !!!}\expandafter\xdef
    \csname @ref@#1\endcsname{ ??}\fi\csname @ref@#1\endcsname}

\def\c@label#1{\global\advance\cit@num by 1\xdef            % assegna il numero
   \la@citazione{\the\cit@num}\expandafter
   \xdef\csname @c@#1\endcsname{\la@citazione}}

\def\bibl@skip{\vskip +4truept}

%------------------------ bibl. automatica: a disp. dell'utente ------------

\def\stileincite#1#2{\global\def\lp@cite{#1}\global   %  Definisce le parentesi
    \def\rp@cite{#2}}                                 %  nelle citazioni
\def\stileinbibl#1#2{\global\def\lp@bibl{#1}\global   %  Definisce le parentesi
    \def\rp@bibl{#2}}                                 %  nella bibliografia

\def\citpreset#1{\global\cit@num=#1
    \immediate\write16{ !!! cit-preset = #1 }    }

\def\autobibliografia{\global\@bibliografiatrue\immediate
    \write16{ !!! Genera il file \jobname.BIB}\immediate
    \openout\file@bibliografia=\jobname.bib}

\def\cref#1{\se@indefinito                  % se indefinito definisce
   {@c@#1}\c@label{#1}\refe@renza{#1}\fi\csname @c@#1\endcsname}

\def\cite#1{\trap@cite{\cref{#1}}}                  %  [5]
\def\ccite#1#2{\trap@cite{\cref{#1},\cref{#2}}}     %  [5,6]
\def\ncite#1#2{\trap@cite{\cref{#1}--\cref{#2}}}    %  [5-8] senza definire
\def\upcite#1{$^{\,\trap@cite{\cref{#1}}}$}               % ^[5]
\def\upccite#1#2{$^{\,\trap@cite{\cref{#1},\cref{#2}}}$}  % ^[5,6]
\def\upncite#1#2{$^{\,\trap@cite{\cref{#1}-\cref{#2}}}$}  % ^[5-8] senza def.
\def\Rcite#1{Ref. \cref{#1}}
\def\Rccite#1#2{Refs. \cref{#1},\cref{#2} }
\def\clabel#1{\se@indefinito{@c@#1}\c@label           % sola definizione
    {#1}\refe@renza{#1}\else\c@label{#1}\ref@ridefinita{#1}\fi}
\def\cclabel#1#2{\clabel{#1}\clabel{#2}}                     % def. doppia
\def\ccclabel#1#2#3{\clabel{#1}\clabel{#2}\clabel{#3}}       % def. tripla

\def\biblskip#1{\def\bibl@skip{\vskip #1}}           % spaziatura nella bibl.

\def\insertbibliografia{\if@bibliografia             % scrive la bibliografia
    \immediate\write\file@bibliografia{ }
    \immediate\closeout\file@bibliografia
    \catcode`@=11\input\jobname.bib\catcode`@=12\fi}

%--------- per comporre il file con la bibliografia --------------

\def\commento#1{\relax}
\def\biblitem#1#2\par{\expandafter\xdef\csname @ref@#1\endcsname{#2}}

% ricordare: una lista in chiaro della bibliografia si
% ottiene eseguendo $ TEX BIBLIST

%------------------------------ F I N E ---------------------------------
\catcode`@=12
%---------------------- P E R  L A  P A G I N A  ------------------------

%  Fissa i parametri della pagina

\magnification=1200
\topskip 20pt
\def\interlinea{\baselineskip=16pt}
% \def\interlinea{\baselineskip=22pt}                     % for proofs
\def\standardpage{\vsize=21.0truecm\voffset=+0.8truecm
                  \hsize=15.truecm\hoffset=+10truemm
%                  \hsize=14.truecm\hoffset=+15truemm   % for proofs
                  \parindent=1.2truecm}

\tolerance 100000
\biblskip{+8truept}                        % spaziatura nella bibliografia
\def\hbup{\hfill\break\baselineskip 16pt}  % spaziatura fra i vari items

%------------------------- P E R  L E  N O T E --------------------------

\global\newcount\notenumber \global\notenumber=0
\def\note #1 {\global\advance\notenumber by1 \baselineskip 10pt
              \footnote{$^{\the\notenumber}$}{\nine #1} \interlinea}
\def\clearnotenumber{\notenumber=0}

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

%  Fonts......

\font\text=cmr10
\font\scal=cmsy5
\font\bcal=cmsy10 scaled \magstep3   %  big calligrafic
\font\it=cmti10
\font\name=cmssi10 scaled \magstep1
\font\lname=cmssi9
\font\btitle=cmbx10 scaled \magstep4     %%%%%%%%%%%
\font\title=cmbx10 scaled \magstep3      %   per titoli normali
% \font\ltitle=cmbx10 scaled \magstep2
\font\ltitle=cmbx12 scaled \magstep1
\font\lltitle=cmbx12
% \font\lltitle=cmbx10 scaled \magstep1    %%%%%%%%%%%
\font\bmtitle=cmmi12                      %%%%%%%%%%%
\font\mtitle=cmmi12 %  scaled \magstep1     %   per titoli in matematico
\font\lmtitle=cmmi12                     %%%%%%%%%%%
\font\itltitle=cmti10 scaled \magstep2   %   per titoli in italico
\font\ittitle=cmti10 scaled \magstep3    %   per titoli in italico
\font\subtitle=cmbx10 scaled \magstep1
\font\abs=cmti10 scaled \magstep1        %    per l' abstract

\font\mb=cmmib10
\font\seven=cmr7                         %  per note e captions
\font\eight=cmr8                         %  per note e captions
\font\nine=cmr9                         %  per note e captions
\font\bcp=cmbx7
\font\icp=cmmi7
\font\bb=cmbx12 scaled \magstep1
\font\bl=cmbx12

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

% NUMERAZIONE AUTOMATICA DELLE FORMULE.
% Istruzioni dell'automacro:

%   \eqlabel{pippo}:     aumenta di 1 il contatore delle equazioni.
%   \eqref{pippo}:       richiama il numero associato a pippo.
%   \autoeqno:           e' definito come
%                        "\eqlabel{pippo}\eqno(\eqref{pippo})".
%   \autosez{mike}:  aumenta di uno il contatore delle sezioni
%                        e azzera il contatore delle equazioni.
%   \sref{mike}:         richiama il numero associato a mike.
%    per cominciare una sezione: \autosez{1} titolo \par

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%    queste istruzioni sono riportate nell' automacro !!!

%    \def\eq{\autoeqno}       % per numerare le equazioni
%    \def\req#1{(\eqref{#1})} % per richiamare le equazioni

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

%%%%%% maggiore-circa,  minore-circa

\def\gtrsim{\ \rlap{\raise 2pt \hbox{$>$}}{\lower 2pt \hbox{$\sim$}}\ }
\def\lesssim{\ \rlap{\raise 2pt \hbox{$<$}}{\lower 2pt \hbox{$\sim$}}\ }

%%%%%% definizioni generali

\def\salta{\vskip 2pt}
\def\mn{\medskip\noindent}
\def\bs{\bigskip}
\def\hb{\hfil\break}
\def\no{\noindent}

\def\scs{\scriptstyle}
\def\scss{\scriptscriptstyle}
\def\ph#1{\phantom#1}
\def\o{\over}

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

%  definizioni per la bibliografia

\def\ea{{\it et.al.}}
\def\ib{{\it ibid.\ }}

\def\npb#1{Nucl. Phys. {\bf B#1},}
\def\plb#1{Phys. Lett. {\bf B#1},}
\def\prd#1{Phys. Rev. {\bf D#1},}
\def\prl#1{Phys. Rev. Lett. {\bf #1},}
\def\ncim#1{Nuo. Cim. {\bf #1},}
\def\zpc#1{Z. Phys. {\bf C#1},}
\def\prep#1{Phys. Rep. {\bf #1},}
\def\rmp{Rev. Mod. Phys. }
\def\ijmpa#1{Int. J. Mod. Phys. {\bf A #1},}

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

% \autobibliografia    %  per il file BIBL.TEX
\stileincite{}{}     %  Definisce le parentesi nelle citazioni
\numerazionedoppia   % avverte che ci sono vari capitoli

\interlinea
\standardpage
\text                %  font per il testo

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

\def\scs#1{{\scriptstyle #1}}
\def\scss#1{{\scriptscriptstyle #1}}
\def\ph#1{\phantom{#1}}
\def\o{\over}
\def\half{{1\over 2}}

%    FITDEF.TEX


\def\clel{c_L^{e}}
\def\slel{s_L^{e}}
\def\crel{c_R^{e}}
\def\srel{s_R^{e}}

\def\clmu{c_L^{\mu}}
\def\slmu{s_L^{\mu}}
\def\crmu{c_R^{\mu}}
\def\srmu{s_R^{\mu}}

\def\cltau{c_L^{\tau}}
\def\sltau{s_L^{\tau}}
\def\crtau{c_R^{\tau}}
\def\srtau{s_R^{\tau}}

\def\clu{c_L^{u}}
\def\slu{s_L^{u}}
\def\cru{c_R^{u}}
\def\sru{s_R^{u}}

\def\cld{c_L^{d}}
\def\sld{s_L^{d}}
\def\crd{c_R^{d}}
\def\srd{s_R^{d}}

\def\cls{c_L^{s}}
\def\sls{s_L^{s}}
\def\crs{c_R^{s}}
\def\srs{s_R^{s}}

\def\clc{c_L^{c}}
\def\slc{s_L^{c}}
\def\crc{c_R^{c}}
\def\src{s_R^{c}}

\def\clb{c_L^{b}}
\def\slb{s_L^{b}}
\def\crb{c_R^{b}}
\def\srb{s_R^{b}}

\def\clnue{c_L^{\nu_e}}
\def\slnue{s_L^{\nu_e}}
\def\crnue{c_R^{\nu_e}}
\def\srnue{s_R^{\nu_e}}

\def\clnumu{c_L^{\nu_\mu}}
\def\slnumu{s_L^{\nu_\mu}}
\def\crnumu{c_R^{\nu_\mu}}
\def\srnumu{s_R^{\nu_\mu}}

\def\clnutau{c_L^{\nu_\tau}}
\def\slnutau{s_L^{\nu_\tau}}
\def\crnutau{c_R^{\nu_\tau}}
\def\srnutau{s_R^{\nu_\tau}}

\def\kud{\kappa_{ud}}
\def\kus{\kappa_{us}}
\def\kcd{\kappa_{cd}}
\def\kcs{\kappa_{cs}}

%    BOTDEF.TEX

\def\A{{\cal A}}
\def\ww{M^2_W}
\def\zz{M^2_Z}
\def\zzo{M^2_{Z_0}}
\def\zzp{M^2_{Z^\prime}}
\def\ss{s^2_w}
\def\cc{c^2_w}
\def\G{{\cal G_{\rm SM}}}
\def\E{{\rm E}_6}
\def\sb{s_{\beta}}
\def\cb{c_{\beta}}

\def\N{{\hbox{\scal N}}}
\def\K{{\hbox{\scal K}}}

%    FIT6DEF.TEX

\def\pr{\prime}
\def\nubar{{\buildrel (-) \over \nu}}
\def\numubar{{\buildrel (-) \over {\nu_\mu}}}

%    LFC.TEX

\font\mbf=cmmib10  scaled \magstep 2      %  boldface mathematic and italic.
\def\bfchi{{\hbox{\mbf\char'037}}}
\def\bftau{{\hbox{\mbf\char'034}}}
\def\bflambda{{\hbox{\mbf\char'025}}}
\def\bfmu{{\hbox{\mbf\char'026}}}
\def\bfepsilon{{\hbox{\mbf\char'017}}}
\def\bfe{{\hbox{\mbf\char'145}}}
\def\bfu{{\hbox{\mbf\char'165}}}
\def\bfF{{\hbox{\mbf\char'106}}}
\def\bfC{{\hbox{\mbf\char'103}}}
\def\bfS{{\hbox{\mbf\char'123}}}
\def\bfa{{\hbox{\mbf\char'141}}}

\font\bigm=cmmi12            %  big mathematic and italic.
\def\bchi{{\hbox{\bigm\char'037}}}
\def\btau{{\hbox{\bigm\char'034}}}
\def\blambda{{\hbox{\bigm\char'025}}}
\def\bmu{{\hbox{\bigm\char'026}}}
\def\bepsilon{{\hbox{\bigm\char'017}}}
\def\be{{\hbox{\bigm\char'145}}}
\def\bu{{\hbox{\bigm\char'165}}}
\def\bF{{\hbox{\bigm\char'106}}}
\def\bC{{\hbox{\bigm\char'103}}}
\def\bS{{\hbox{\bigm\char'123}}}
\def\ba{{\hbox{\bigm\char'141}}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%        HERE STARTS THE PAPER : FLIP6DEF
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\def\ie{{\it i.e.\ }}
\def\eg{{\it e.g.\ }}

\def\ra{\rangle}
\def\la{\langle}

\def\scs{\scriptstyle }
\def\scss{\scriptscriptstyle }
% \def\scs#1{{\scriptstyle (#1)}}
% \def\scss#1{{\scriptscriptstyle (#1)}}
\def\ph#1{\phantom{(#1)}}
\def\o{\over}
\def\half{{1\over 2}}


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

\def\lu{{H^cQu^c}}
\def\ld{{HQd^c}}
\def\lt{{HLe^c}}
\def\lq{{S^chh^c}}
\def\lc{{hu^ce^c}}
\def\ls{{LQh^c}}
\def\lst{{\nu^chd^c}}
\def\lo{{hQQ}}
\def\ln{{h^cu^cd^c}}
\def\ldi{{H^cL\nu^c}}
\def\luu{{H^cHS^c}}

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

\def\l#1#2#3#4{\lambda^{^{\scriptstyle (#1)}}_{\scriptscriptstyle #2#3#4}\>}
\def\lw#1{\lambda^{^{\scriptstyle (#1)}}\>}


\def\ww{M^2_W}
\def\zz{M^2_Z}
\def\zzo{M^2_{Z_0}}
\def\zzu{M^2_{Z_1}}
\def\zzp{M^2_{Z^\prime}}
\def\G{{\cal G_{\rm SM}}}
\def\E{{\rm E}_6}
\def\sb{s_{\beta}}
\def\cb{c_{\beta}}

\def\pr{\prime}
\def\nubar{{\buildrel (-) \over \nu}}
\def\numubar{{\buildrel (-) \over {\nu_\mu}}}

%    LFC.TEX
\font\mbf=cmmib10  scaled \magstep1      %  boldface mathematic and italic.
\def\bfchi{{\hbox{\mbf\char'037}}}
\def\bftau{{\hbox{\mbf\char'034}\>}}
\def\bflambda{{\hbox{\mbf\char'025}}}
\def\bfmu{{\hbox{\mbf\char'026}}}
\def\bfepsilon{{\hbox{\mbf\char'017}}}
\def\bfe{{\hbox{\mbf\char'145}}}
\def\bfu{{\hbox{\mbf\char'165}}}
\def\bfF{{\hbox{\mbf\char'106}}}
\def\bfC{{\hbox{\mbf\char'103}}}
\def\bfS{{\hbox{\mbf\char'123}}}
\def\bfa{{\hbox{\mbf\char'141}}}

\font\bigm=cmmi12            %  big mathematic and italic.
\def\bchi{{\hbox{\bigm\char'037}}}
\def\btau{{\hbox{\bigm\char'034}\>}}
\def\blambda{{\hbox{\bigm\char'025}}}
\def\bmu{{\hbox{\bigm\char'026}}}
\def\bepsilon{{\hbox{\bigm\char'017}}}
\def\be{{\hbox{\bigm\char'145}}}
\def\bu{{\hbox{\bigm\char'165}}}
\def\bF{{\hbox{\bigm\char'106}}}
\def\bC{{\hbox{\bigm\char'103}}}
\def\bS{{\hbox{\bigm\char'123}}}
\def\ba{{\hbox{\bigm\char'141}}}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%                    HERE STARTS THE PAPER
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\autobibliografia
% \pageno=1                    % for proofs
\pageno=0
\vsize=23.8truecm
\hsize=15.7truecm
\voffset=-1.truecm
\hoffset=+6truemm
\baselineskip 12pt
\rightline{UM-TH 93--19}\par\noindent
\rightline\par\noindent

\bs\bs\bs
\centerline{\ltitle
Signals of Unconventional E$_{\bf 6}$ Models}
\bs
\centerline{\ltitle at \bfe$^{\displaystyle\bf +}$
\bfe$^{\displaystyle\bf -}$ Colliders.}
\medskip
\medskip
\bs\bs                                     %    !!!!!!!!!!!!!!!!!!
                 \centerline{Enrico Nardi}
\bs
\centerline{\it Randall Laboratory of Physics, University of Michigan,
           Ann Arbor, MI 48109--1120}
\vskip 1truecm                             %    !!!!!!!!!!!!!!!!!!!

\medskip
\centerline{\bf {\abs Abstract}}           %    !!!!!!!!!!!!!!!!!!!
\bs

% \baselineskip 20pt                       %    !!!!!!!!!!!!!!!!!!!

\noindent
Generation dependent discrete symmetries
often appear in models derived from superstring theories.
In particular, in the framework of $\E$ models the presence of such
symmetries is required in order to allow for the radiative generation
of naturally small neutrino masses.
Recently it was shown that
by imposing suitable generation dependent
discrete symmetries, a class of models  can be consistently
constructed in which  the three sets of known fermions
in each generation do not have the same assignments with
respect to the {\bf 27} representation of E$_6$.
In this scenario, the different embedding in the gauge group of the
three generations implies in particular
that the known charged leptons couple in a
non--universal way to the new neutral gauge bosons $(Z_\beta)$
present in these models.
We exploit this fact to study the signature of this class of models
at present and future $e^+e^-$ colliders. We show that
some signals of deviation from lepton universality
as well as  some other discrepancies with the standard model predictions
which have been observed at the
TRISTAN collider in the production rate of $\mu$ and $\tau$,
can be accounted for if the $Z_\beta$ mass is not
much heavier than 300 GeV. We also study the discovery limits for
lepton universality violation of this type at LEP-2 and at the 500
GeV $e^+e^-$ Next Linear Collider (NLC). We show that models
predicting unconventional assignments for the leptons will give an
unmistakable signature, when the $Z_\beta$ mass is as heavy as $\sim 800$
GeV (LEP-2) and $\sim 2$ TeV (NLC).
\bs
\noindent
PACS number(s): 12.10.Dm,12.15.Ff,13.15.Jr,14.60.Gh
\vfill
\noindent
--------------------------------------------\phantom{-} \hb
\leftline{Electronic address: nardi@umiphys.bitnet}
\bigskip
\leftline{UM-TH 93--19}
                   \medskip
\centerline{August 1993}

\eject

\standardpage                            %    !!!!!!!!!!!!!!!!!!!!!!!!
\interlinea                              %    !!!!!!!!!!!!!!!!!!!!!!!!

\null
\chap{Int} {I. INTRODUCTION}{}

It was recently shown [\cite{f}] that in the framework of
superstring derived E$_{\bf 6}$ models,
it is possible to implement an
unconventional scenario in which
some of the known fermions of the three families are embedded in the
fundamental {\bf 27} representation of the group in a generation
dependent way, meaning that their gauge quantum numbers do not
replicate throughout the three generations.
It was also argued [\cite{f}] that if
$\E$ models are required to allow for small
neutrino  masses (as is needed for any
particle physics solution of the solar neutrino problem,
and/or for explaining the  atmospheric $\nu_\mu$ deficit via
$\nu_\mu$ oscillations)
the Unconventional Assignments (UA) scenario should
be considered as a natural alternative to the standard schemes.
In fact, in the framework of superstring inspired $\E$ theories
the most  natural way for generating small neutrino masses
is through radiative corrections [\cite{ellis-e6},\cite{MNS}]
since in these models the Higgs representation necessary to implement
a see-saw mechanism [\cite{see-saw}]
is absent.\note{\noindent See however Ref. [\cite{nandi-sarkar}]
for a discussion of a see-saw mechanism induced by gravitational effects.}
In order to implement the generation of $\nu$ masses at the loop
level, a  suitable discrete symmetry must be imposed on
the superpotential to insure that at the lowest order $m_\nu=0$.
Branco and Geng [\cite{branco}] have shown that no
generation-blind symmetry exists that forbids non--vanishing neutrino
masses at the tree level and at the same time allows for their
radiative generation. As a result, in order to implement in a consistent way
the generation of $\nu$ masses via loop diagrams, a symmetry that
does not act in the same way on the three generations is needed.

The main motivation for investigating the UA schemes stems from the
observation that once we chose to build a model based on a symmetry that
distinguishes among the different generations, there is no reason in
principle to expect that this symmetry will result in a set of {\it
light} fermions  (\ie the known states) that will exactly replicate
throughout the three generations [\cite{f}]. This implies the possibility
that the known states belonging to different generations could
have different $\E$ gauge interactions.
Of course, experimentally  we know that the $SU(2)\times U(1)$ gauge
interactions of the known fermions do respect universality with a high
degree of precision. However, since the Standard Model (SM)
gauge group is rank 4, while  $\E$ is rank  6, as many as two
additional massive neutral gauge bosons $(Z_\beta)$ can be present,
possibly with $M_\beta \sim 1\>$TeV or less, and the possibility that
the additional $U_\beta(1)$ interactions could violate universality is
still phenomenologically viable.

Since the fundamental representation of $\E$ is 27 dimensional,
the fermion content of models based on this group
is enlarged with respect to the SM.
In fact, in addition to the standard fermions,
two additional leptonic $SU(2)$-doublets, two $SU(2)$-singlet
neutral states and two color-triplet $SU(2)$-singlet $d$-type quarks
are present. UA models are realised by identifying
in a generation dependent way some of the
known doublets of lepton and/or $d$-type singlet quarks
with the additional fermion multiplets [\cite{f}].
Models based on the UA scenario can be implemented without conflicting
with phenomenological or theoretical constraints. For example the
model described in Ref. [\cite{f}] was shown to be consistent with a
large number of experimental constraints, ranging from the direct and
cosmological limits on the neutrino masses, to the stringent limits on
flavor changing neutral currents (FCNCs). In this model the left (L)
handed lepton doublet ``${\nu_\tau \choose \tau}_L$'' and the right
(R) handed quark singlet ``$b_R$'' of the third
generation are assigned to $SU(2)$ multiplets having a different
embedding in $\E$ with respect to the
corresponding states of the first two generations. The non standard
phenomenology resulting from this model implies in particular that the
``$\tau$" neutrinos have different neutral current
(NC) interactions than to ``$\nu_e$'' and ``$\nu_\mu$''.

In Sec. 2 we will briefly outline the main features of the $\E$ models
based on the UA scenario, and establish our conventions
and notations. A more complete discussion of the theoretical framework
can be found in
Ref. [\cite{f}].  A very
clean signature for the UA models would be the detection of deviations
from universality in neutral current (NC) processes.
Due to the clean experimental environment of $e^+e^-$ annihilation,
such a signature could be more easily detected at $e^+e^-$ colliders
rather than at hadron colliders.
In Sec. 3 we will investigate the phenomenology of UA models
at the present and future $e^+e^-$ machines.
Since
the UA for the known fermions would result in a violation
of universality only in the fermion couplings to the new $Z_\beta$
bosons without affecting the couplings to the standard $Z_0$,
the large amount of data collected
at the $Z_0$ resonance by the Large Electron Positron (LEP) collaborations
are not effective for the search of these kind of effects.
In fact, the contribution of $Z_\beta$--$Z_0$ interference
to the various cross sections and asymmetries measured at LEP-1
vanishes at the peak, and the contribution of pure $Z_\beta$ exchange
is also vanishingly small.
Some effects could still be detected
if the $Z_0$ had a sizeable mixing with $Z_\beta$,
however the existing bounds  on the  $Z_0$--$Z_\beta$
mixing angle are extremely tight [\cite{zp-new}], so that
we will disregard this possibility throughout this paper.

Among presently operating colliders, the one best suited
to reveal the kind of effects we are looking for is
the TRISTAN collider at the KEK laboratories, which is collecting data
at about 60 GeV c.m. energy.
It is intriguing, though certainly not compelling,  that
a few discrepancies between the
TRISTAN data on the total hadronic and leptonic cross sections
and the SM predictions do exist [\cite{kek}], and they
point towards the existence of a $Z_\beta$ at rather low energies
$M_\beta\lesssim 300\,$ GeV [\cite{zp-kek}].
At the same time the data on the rate of production of
$\tau$ and $\mu$ leptons do show a signal of violation of universality
(at the level of 1.6 standard deviations) which could not be explained
by conventional extended gauge models. However, as we will show,
these data can be well accounted for in the framework of models with UA.
We will complete our discussion by analyzing
the discovery limits for UA models at
LEP II and at the 500 GeV Next Linear $e^+e^-$ Collider (NLC).
Finally, in Sec. 4 we will summarize our results and draw the
conclusions.


\chap{flip} {II. UNCONVENTIONAL ASSIGNMENTS}{IN E$_{\bf 6}$ MODELS}

In $\E$ grand unified theories,
matter fields belong to the fundamental {\bf 27}
representation of the group. $\E$ contains $SO(10)\times U_\psi(1)$ as
a maximal subalgebra, and the {\bf 27} branches to
the ${\bf 1} + {\bf 10}
+ {\bf 16}$ representation of $SO(10)$.
In turn $SU(10)$ contains $SU(5)\times U_\chi(1)$.
The $SO(10)$,  $SU(5)$, $U_\psi(1)$ and $U_\chi(1)$ assignments for
the fermions in the {\bf 27} representation
are listed in Table I. Usually the known
particles of the three generations are assigned to the {\bf 16} of
$SO(10)$ that also contains a $SU(2)$ singlet neutrino ``$\nu^c$"
$$
[{\bf 16}]_i = \big[
Q\equiv { % \scs
{u\choose d}}, \,
u^c, \,
e^c, \,
d^c, \,
L\equiv { % \scs
{\nu \choose e}}, \,
\nu^c\big]_i \qquad \qquad i=1,2,3.
\eq{2.1}
$$
\mn
The {\bf 10} and the {\bf 1} of $SO(10)$
contain the new fields
$$
\eqalign{
[{\bf 10}]_i &= \big[
H^c\equiv {  % \scs
{E^c\choose N^c}}, \,
h, \,
H\equiv {  % \scs
{N\choose E}}, \,
h^c\big]_i                                          \cr
[\>{\bf 1}\>]_i&= [S^c]_i   \qquad \qquad \qquad \qquad
\hskip 2.8truecm i=1,2,3. \cr}
\eq{2.2}
$$

%    *************************************************
%            Definizioni per le tavole.
%    *************************************************

\def \tabrule     {\noalign{\vskip 5truept \hrule\vskip 5truept} }
\def \tabrul2     {\noalign{\vskip 5truept \hrule \vskip 2truept \hrule
                   \vskip 5truept} }
\def\om{\omit}

\baselineskip=14pt
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\midinsert
{
\bs
$$
\vbox{\hsize= 13.3truecm
\noindent TABLE I.
{\nine
$SO(10)$, $U_\psi(1)$, $SU(5)$ and $U_\chi(1)$ assignments for the
left-handed fermions of the {\bf 27} fundamental representation of
${\rm E}_6$. The $SU(2)$ doublets $H^c$, $H$, $L$ and $Q$ are
explicitly written in components. The Abelian charges $Q_\psi$ and
$Q_\chi$ can be obtained from the quantum numbers in the
brackets by dividing
by $c_\psi = 6\sqrt{2/5}$ and $c_\chi =
6\sqrt{2/3}$, respectively.
The charges are normalized to the
hypercharge according to: $\sum_{f=1}^{27}(Q_{\psi,\chi}^f)^2 =
\sum_{f=1}^{27}({1 \over 2}Y^f)^2=5$.
}
\vskip -.8truecm
}
$$

$$
\vbox{
\hsize= 13.35truecm
% \baselineskip=12pt
\offinterlineskip
\halign {
% to13.35 truecm{
\vrule#&\strut#&\vrule#&\strut#&\vrule#&\strut\quad#&\vrule#&\strut#&
\vrule#&\strut#&\vrule#&\strut#&\vrule#&\strut\quad#&\vrule#&\strut#&
\vrule#&\strut#&\vrule#&\strut#&\vrule#&\strut#&\vrule#&\strut#&\vrule#\cr
\noalign{\hrule}
height6pt & \om && \om && \om&\om&\om && \om&\om&\om && \om &&
\om&\om&\om && \om&\om&\om&\om&\om &\cr
&\om && $\ S^c\> $ && ${{E^c}\choose{N^c}}\ $&\om&$\ h\ $ && $\
{N\choose{E}}\ $&\om&$\ h^c\ \ $ && $\nu^c\ $ && $\ \>
{\nu\choose{e}}$&\om&$\ d^c\ \>$ && $\ \> e^c\ \>$&\om&$\ u^c\
\>$&\om&$\ {u\choose d}\ $ &\cr
height6pt&\om&&\om&&\om&\om&\om&&\om&\om&\om&&\om&&
\om&\om&\om&&\om&\om&\om&\om&\om&\cr
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\noalign{\hrule}
height6pt&\om&&\om&&\om&\om&\om&\om&\om&\om&\om&&
\om&\om&\om&\om&\om&\om&\om&\om&\om&\om &\om&\cr
& $\ SO(10)\> (c_\psi Q_\psi)\ $ && $\ {\bf 1}\>(4)\ $ &&   % \om&\om&
\multispan7 \hfil ${\bf 10}\> (-2)$\hfil          % &\om&\om&\om&\om
&&                                                % \om&\om&\om&\om&
\multispan{11} \hfil ${\bf 16}\> (1)$ \hfil       % &\om&\om&\om&\om&\om&\om
&\cr
height6pt&\om&&\om&&
\om&\om&\om&\om&\om&\om&\om&&
\om&\om&\om&\om&\om&\om&\om&\om&\om&\om&\om&\cr
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\noalign{\hrule}
height6pt&\om&&\om&&\om&\om&\om&&\om&\om&\om&&
\om&&\om&\om&\om&&\om&\om&\om&\om&\om&\cr
& $\ SU(5)\ (c_\chi Q_\chi)\  $ && $\ {\bf 1}\>(0)\ $ &
&\multispan3\hfil ${\bf 5}\>(2)$ \hfil           %   &\om&\om
&& \multispan3 \hfil ${\bf
\bar 5}\>(-2)$\hfil                  %   &\om&\om
&&\hfil ${\bf 1}\>(-5)\ $ \hfil
&&                                   %  \om&
\multispan3 \hfil ${\bf \bar 5}\>(3)$\hfil        %  &\om
&&                                                % \om&\om&
\multispan5
\hfil ${\bf 10}\> (-1)$\hfil                      %  &\om&\om
&\cr
height6pt & \om && \om && \om&\om&\om && \om&\om&\om && \om &&
\om&\om&\om && \om&\om&\om&\om&\om &\cr
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\noalign{\hrule}}}
$$
\bs
}
\vskip -.6truecm
\endinsert
\interlinea

As it is clear from Table I,
there is an ambiguity in assigning the known states to the {\bf 27}
representation, since under the SM gauge group
$$
\G\equiv SU(3)_c\times SU(2)_L\times U(1)_Y
$$
the ${\bf \bar 5}_{({\bf 10})}$ in
the {\bf 10} of $SO(10)$ has the same field content as the ${\bf
\bar 5}_{({\bf 16})}$ in the {\bf 16}. The same ambiguity is
also present for the two $\G$ singlets, namely
 ${\bf 1}_{({\bf 1})}$ and ${\bf 1}_{({\bf 16})}$.
In the present paper we will concentrate on the consequences of
having different assignments for the known L-handed leptons.
Since these leptons might not correspond to the entries as listed in Table I,
we use quotation marks to denote the known states with
their  conventional labels, while labels not enclosed within quotation
marks will always refer to the fields listed in the table.

In the models under investigation
as many as two additional neutral gauge bosons
can be present, corresponding for example, to some linear combinations
of the $U_\chi(1)$ and $U_\psi(1)$ generators.
The interaction of the fermions in the ${\bf \bar 5}$ of $SU(5)$
with these gauge bosons will depend on the specific assignments
to the {\bf 16} or to the {\bf 10} of $SO(10)$.
The
two additional neutral gauge bosons are usually parametrized as
$$
\eqalign{
&Z_\beta^\pr= Z_\psi \sin\beta + Z_\chi \cos\beta  \cr
&Z_\beta^{\pr\pr}= Z_\psi \cos\beta - Z_\chi \sin\beta, \cr}
\eq{2.3}
$$
\mn
where $\beta$ is a model dependent parameter.
In the following we will denote the
lightest of the two new gauge bosons as $Z_\beta$.
In the presence of a `light' $Z_\beta$
different assignments will lead to a different phenomenology.
In contrast, in the limit $M_{\beta}\rightarrow \infty$
the choice of the assignment is irrelevant as long as
we are only concerned with the gauge interactions.  However,
even in
this limit the requirement of $U_{\psi,\chi}(1)$ gauge invariance for the
superpotential, together with the phenomenological constraints on the
absence of FCNCs in the Higgs sector, strongly
constrain the structure of the viable models [\cite{f}].

A model realizing an UA scenario,
in which what we call ``$\tau_L$" corresponds
to the charged component of the $H_3$ weak doublet belonging to
${\bf \bar 5}_{{\bf 10}}$, while the
``$e_L$'' and the ``$\mu_L$'' leptons are  as usual assigned to the
${\bf \bar 5}_{{\bf 16}}$, was recently proposed in Ref. [\cite{f}].
This model is realized by imposing on the superpotential
a particular family-non-blind
$Z_2\times Z_3$ discrete symmetry.
As a result  of such a symmetry,
the masses of the known (light) chiral leptons
are generated by vacuum expectation values (VEVs) of Higgs doublets,
through the terms
$m_\tau E_{3L}e_{3R}$
(with $m_\tau\sim \la\tilde L_3\ra_0$) and
$m_{\alpha\beta}e_{\alpha L}e_{\beta R}$
(with $m_{\alpha\beta} \sim \la\tilde H_2\ra_0$ and
$\alpha,\beta=1,2$).
The remaining charged leptons
$e_{3L}$, $E_{3R}$,
$E_{\alpha L}$, $E_{\alpha R}$ are vectorlike, and acquire large
masses from VEVs of Higgs singlets.

As it was argued in Ref. [\cite{f}], several interesting features
of this model are peculiar to the UA schemes in general,
independently of this particular realization.
For example, in contrast to the conventional $\E$ models  [\cite{ellis-e6}],
in the UA schemes
rank 6 models are not disfavoured with respect to rank 5,
so that the general parametrization of the two additional gauge bosons
given in \req{2.3} is well motivated.
We stress that other assignments, leading to models with a structure
similar to the model proposed in Ref. [\cite{f}], but implying
a different phenomenology, can be easily obtained
by means of some different discrete symmetries.


If some of the
$\nu^c$ and/or $S^c$ $\>SU(2)$ neutral
singlets
are massless or are very light ($m_s\lesssim 1\,$ MeV),
cosmological arguments suggest that the $Z_\beta$
bosons should be heavier than about $\sim 1$-2 TeV
[\cite{nucleo-e6}], thus excluding the possibility
of detecting any signal at TRISTAN and LEP-2.
In fact, though singlet under $\G$, these states do
have $U(1)_\beta$ interactions.
Then, not to conflict with the limit of 3.6 relativistic neutrinos in
thermal equilibrium at the time of nucleosynthesis [\cite{nucleo-nu}]
(which can be derived from the data on the light element abundances) we
have to require this interaction to be weak enough to allow for the
decoupling of the light $\G$-singlets at a sufficiently early time
(for example
before the QCD phase transition) so that their number density can be
safely diluted. Requiring the $U_\beta(1)$
interaction to be ``superweak" results in
the quoted lower bound on the $Z_\beta$ mass [\cite{nucleo-e6}].
We would like to mention that there are two models
corresponding to the particular values of the angle $\beta$
in \req{2.3}
($\tan 2\beta = -{\sqrt{15} \o 7},\> 0$)
in which the
nucleosynthesis constraints on $M_\beta$ can be evaded
even in the presence of light
$SU(2)$ singlets [\cite{f}].
In fact,  for these two values of
$\beta$, respectively
the $\nu^c$ and the $S^c$ degrees of freedom decouple from
the lightest $Z_\beta$, behaving as
`effective singlets' with respect
to all the `light' gauge bosons.
Then they  could play the
role of the helicity partners of the standard
neutrinos, allowing in particular for non zero neutrino Dirac masses,
while at the same time their gauge interactions would not be effective
to keep them in thermal equilibrium in the early Universe.

However, for the sake of generality, in the present analysis we will
assume that all the $\nu^c$ and $S^c\>$ $\G$-singlets
are heavy ($m_s\gg 1\,$ MeV).
In this case,
independently of the value of $\beta$,
the nucleosynthesis constraints on $M_\beta$ are evaded.
Therefore the $Z_\beta$  boson could be as light as
allowed by the present limits from direct searches at colliders
[\cite{zp-direct}] and from the analysis of $Z_\beta$ indirect effects
[\cite{zp-new}], resulting in both cases in $M_\beta \gtrsim
200-300\,$GeV.
As we will see, in the UA schemes
the presence of a $Z_\beta$ with a mass in this range
can give rise to lepton universality violating
effects that could be
detected at the colliders presently in operation.


\chap{phen} {III. SIGNALS OF UNCONVENTIONAL ASSIGNMENTS}
{AT \bfe$^{\displaystyle\bf +}$ \bfe$^{\displaystyle\bf -}$ COLLIDERS}

In the presence of additional neutral gauge bosons, the
lowest order cross section\note{In the numerical computations
we have taken into account the leading one-loop corrections by using an
improved Born approximation [\cite{leprad}].}
for the process $e^+e^-\to l^+l^-$ with $l\neq e$, is
$$
\eqlabel{3.1}
\eqlabel{3.2}
\eqlabel{3.3}
\eqalignno{
\sigma(s)&= {4\o 3}{\pi\alpha^2\o s}\sum_{m,n=0}^N
C(m,n)\chi_m(s)\chi_n^*(s)                      &\req{3.1}  \cr
&\cr
C(m,n)&=
\left[v_m(e)v_n^*(e)+ a_m(e)a_n^*(e) \right]\cdot
\left[v_m(l)v_n^*(l)+ a_m(l)a_n^*(l) \right]    &\req{3.2}   \cr
&\cr
 \chi_m(s)&={g^2_m\o 4\pi\alpha}{s\o s - M^2_m - iM_m\Gamma_m}.
                        &\req{3.3}
 \cr}
$$
\mn
We will henceforth assume
that one of the two new bosons in \req{2.3} is  heavy enough so that
its effects on the low energy physics are negligible. Then
$m,n=0$,$1$,$2$ correspond respectively to the
$\gamma$, $Z_0$ and $Z_\beta$
amplitudes. The couplings in \req{3.2} and \req{3.3} are
$$
\eqlabel{3.4}
\vbox{
\baselineskip=18pt
\settabs \+
$g_1=(\sqrt{2}G_\mu M^2_Z)^{1\o 2}$ \quad
&$v_1(\ell)=T_{3L}^\ell-2Q_{\rm em}^\ell s^2_w$   \quad
&$a_2(\ell)=Q_\beta^\ell+Q_\beta^{\ell^c}$   \quad
&\quad$\ \ell=e,\mu,\tau$
\cr  \+
$g_0=e$
&$v_0(\ell)=Q_{\rm em}^\ell$
&$a_0(\ell)=0$ &
\cr  \+
$g_1=(\sqrt{2}G_\mu M^2_Z)^{1\o 2}$
&$v_1(\ell)=T_{3L}^\ell-2Q_{\rm em}^\ell s^2_w$
&$a_1(\ell)=T_{3L}^\ell$
&  &\phantom{\quad$\ \ell=e,$}\req{3.4}
\cr  \+
$g_2=s_w g_1$
&$v_2(\ell)=Q_\beta^\ell-Q_\beta^{\ell^c}$
&$a_2(\ell)=Q_\beta^\ell+Q_\beta^{\ell^c}$
&\quad$ \ell=e,\mu,\tau$
\cr  }
$$
\interlinea
\mn
where $Q_{\rm em}^\ell=-1$ is the electric charge of the leptons,
$T_{3L}^\ell=-{1\o 2}$ is the third component of the
weak isospin, $s_w\equiv \sin\theta_w$ with $\theta_w$
the weak mixing angle, $Q_\beta^{\ell,\ell^c} =
Q_\psi^{\ell,\ell^c}\sin\beta + Q_\chi^{\ell,\ell^c}\cos\beta\ $
is the lepton coupling to the $Z_\beta$ boson in \req{2.3}.
The new charges
$Q_\psi^{\ell,\ell^c}$, $Q_\chi^{\ell,\ell^c}$ are given in Table I,
and are normalized to the
hypercharge generator $Y/2$. In addition,
in \req{3.4} we have assumed
for the abelian coupling, $g_2$,
a renormalization
group evolution down to the electroweak scale
similar to that of
the hypercharge coupling $g_{\scss Y}\simeq s_w g_1$.

In \req{3.4} the vector and axial-vector couplings,
$v_{0,1}(\ell)$ and $a_{0,1}(\ell)$,
do not  depend on the specific assignments for the leptons, and
are unmodified with respect to the SM.
In contrast, $v_{2}(\ell)$ and $a_{2}(\ell)$
do depend on the particular assignments of the $\ell$ lepton.
With the notations given in \req{2.1} and \req{2.2}, and
referring to the {\bf 16} and to the {\bf 10} representations
of $SO(10)$, the possible assignments for the
L-handed $\ell_1=$``$e_L$'', $\ell_2=$``$\mu_L$'',
$\ell_3=$``$\tau_L$''
charged leptons are
$$
\eqalign{
&``\ell_i" \in L_i \in {\bf 16} \cr
&{\rm or }                       \cr
&``\ell_i" \in H_i \in {\bf 10} \qquad\qquad i=1,2,3.  \cr
}
\eq{3.5}
$$
\mn
With these assignments
three different cross sections for the process
$e^+e^- \to l^+l^-$ ($l\neq e$)
are possible. They are
$\sigma_{16\to 16}$,
$\sigma_{16\to 10} = \sigma_{10\to 16}$ and
$\sigma_{10\to 10}$, where the subscripts refer to
the specific embedding of the
L-handed components of the initial $e^-$ and final
$l^-$ states in the {\bf 16} or in the {\bf 10} of $SO(10)$.
In the following we will give results for the
quantities $R_{16\to 16}$, $R_{16\to 10}$ and $R_{10\to 10}$
corresponding to the different cross sections
normalized to the point-like QED cross section for muon pair
production.

\midinsert
{
\vskip 1truecm
\epsfxsize=14.0truecm %   \hsize
\epsffile{kek1s.ps}
\vskip -10.5truecm
% \hskip-1.0truecm\special{psfile=kek1s.ps hscale=60 vscale=25}
\baselineskip=12pt
$$
\vbox{
% \vskip -4.5truecm
\hsize= 13.3truecm
\noindent
FIG. 1.
{\nine
The TRISTAN total cross sections
 for $\mu$ pair production ($R_{\mu\mu}$) and
$\tau$ pair production ($R_{\tau\tau}$),
normalized to the point-like QED cross section,
compared to the standard model prediction $R^{SM}_{ll}$.
The dotted and dashed lines correspond to the
one standard deviation experimental errors for
$R_{\mu\mu}$ and $R_{\tau\tau}$ respectively.
The shaded areas, enclosed between the solid lines
($M_\beta=200\>$GeV) and the dot-dashed lines
($M_\beta=300\>$GeV)
depict the predictions for lepton pair
production at the TRISTAN c.m. energy $\sqrt{s}=58\>$GeV
in the unconventional assignments $\E$ models.
The results are given
for a general $Z_\beta$ from $\E$, as a function of $\sin\beta$.
$R_{16\to 16}$ refers to the case in which
the L-components of both the initial and final leptons
are assigned to the {\bf 16} representation of $SO(10)$,
and similarly for $R_{16\to 10}$ and $R_{10\to 10}$.
}
}
$$
}
\vskip -.6truecm
\endinsert
\interlinea
\noindent
In Fig. 1 we compare the theoretical values for
$R_{16\to 16}$,
$R_{16\to 10}$ and
$R_{10\to 10}$ at $\sqrt{s}=58\,$ GeV c.m. energy,
with the SM prediction
$R^{SM}_{ll}=1.053$  (heavy solid line),
and with the TRISTAN experimental data
$R_{\tau\tau}=1.026\pm 0.037$ and
$R_{\mu\mu}=0.982\pm 0.036$.
These figures have been obtained by combining the
results of the AMY, TOPAZ and VENUS
collaborations given in Ref. [\cite{kek}].
In deriving the averages we have assigned a common systematic error
of $\pm 0.030$ for the uncertainty in the luminosity.
Fig. 1 shows that the measured values of $R_{\tau\tau}$
and $R_{\mu\mu}$ are both lower than the SM prediction.
However, while the value of
$R_{\tau\tau}$ is consistent with the SM
within one standard deviation,
$R_{\mu\mu}$ is about two standard deviations off the
expected value.
The shaded areas show the predictions for the three ratios
$R_{16\to 16}$,
$R_{16\to 10}$ and
$R_{10\to 10}$
for a $Z_\beta$ mass ranging between 200 GeV and 300 GeV.
This range coincides with the range of the lower bounds
on $M_\beta$ obtained in the framework of conventional E$_6$ models.
For example, for the models usually referred to as $\psi$,
$\eta$ and $\chi$ models which correspond
to the particular values $\sin\beta=-\sqrt{5/8}$, 0, 1
[\cite{rizzo-e6}],
the most conservative direct bounds
are respectively 200, 230 and 280 GeV.
These bounds have been obtained
at hadron colliders
from the limits on the
process $p\bar p\to Z_\beta\to l^+l^-$,
by assuming $Z_\beta$ decay to all allowed
fermions and supersymmetric fermions [\cite{zp-direct}].
Other indirect limits have been obtained from
high precision electroweak data by
analysing the
$Z_\beta$ indirect effects on NC observables.
The indirect bounds also  suggest $M_\beta \gtrsim 200\>$GeV
for all the values of the parameter $\beta$ [\cite{zp-new}].
Clearly the limits derived from analyses based on the conventional
scheme cannot be straightforwardly applied to the
$Z_\beta$ of UA models,
since in the present case a large number of fermion couplings
could be different.
However, we have no reason to expect that by assuming UA
the bounds could be greatly strengthened or relaxed, and
hence we will take the quoted limits as a reasonable guess for
the lower bounds on $M_\beta$ also in the UA schemes.

{}From Fig. 1, it is apparent that
in the presence of a light $Z_\beta$,
the experimental data
on $R_{\mu\mu}$ would be better accounted for by either
assigning both ``$e_L$" and ``$\mu_L$" to the {\bf 16} representation of
$SO(10)$ (as in conventional $\E$ models),
and for values of $\sin \beta$ centered around zero, or by
assigning both these leptons to the {\bf 10}
and for $\sin \beta$ close to unity.
At the same time, for any choice of the assignments
and for any value of $\beta$
the various $R$ are in good agreement
with the
experimental value of $R_{\tau\tau}$. Only a small region
in the vicinity of $\sin \beta \sim 0.6$
is slightly disfavoured if the
assignments
``$e_L$"$\in$  {\bf 16} and  ``$\tau_L$"$\in$  {\bf 10}
are chosen.

\midinsert
{
\vskip 1truecm
\epsfxsize=14.0truecm %   \hsize
\epsffile{kek2s.ps}
\vskip -10.5truecm
% \hskip-1.0truecm\special{psfile=kek2.sps hscale=60 vscale=25}
\baselineskip=14pt
$$
\vbox{
% \vskip -4.5truecm
\hsize= 13.3truecm
\noindent
FIG. 2.
{\nine
The TRISTAN result for the ratio of
$\mu$ to $\tau$ pair productions
$(\sigma_{\mu\mu}/\sigma_{\tau\tau})_{\rm EXP}$
compared to the standard model prediction
$(\sigma_{\mu\mu}/\sigma_{\tau\tau})_{\rm SM}=1$.
The dotted lines correspond to the
one standard deviation experimental error.
The shaded areas, enclosed between the solid lines
($M_\beta=200\>$GeV) and the dot-dashed lines
($M_\beta=300\>$GeV)
depict the predictions for the ratio of pair
productions of two different lepton flavors
at $\sqrt{s}=58\>$GeV,
in the unconventional assignments $\E$ models.
The results are given
for a general $Z_\beta$ from $\E$, as a function of $\sin\beta$.
$\sigma_{16\to 16}$ refers to the case in which
the L-components of both the initial and final leptons
are assigned to the {\bf 16} representation of $SO(10)$,
and similarly for $\sigma_{16\to 10} = \sigma_{10\to 16}$
and $\sigma_{10\to 10}$.
}
}
$$
}
\vskip -.6truecm
\endinsert
\interlinea
% \vskip -.6truecm
\noindent
One of the most spectacular signals of UA models
would be a deviation of the ratio $\rho_{\mu/\tau}\equiv
\sigma (e^+e^- \to \mu^+\mu^-)/ \sigma (e^+e^- \to \tau^+\tau^-)$
from unity.
Since many systematic errors, take for example
the uncertainties in the luminosity measurements,  cancel
in this ratio, the experimental error
is statistically dominated, implying a
very transparent significance for such a measurement.
At the $Z_0$ resonance,
the measured value for this observable
 $\rho_{\mu/\tau}=0.998\pm 0.006$ [\cite{lep-93}],
in striking agreement with $\mu$--$\tau$ universality.
Undoubtedly it would be difficult to accommodate
any large deviation from unity at $\sqrt{s}\ne M_Z$
by means of some mechanism different from the one discussed here.
For this reasons we believe that if a value
$\rho_{\mu/\tau}\ne 1$ is measured off $Z_0$ resonance,
this  would represent
a very clean and almost unmistakable signature of the UA
models.

According to the assignments in \req{3.5}, and without referring to
any specific lepton flavor,
we can write two expressions for
this ratio  which deviate from unity:
$\rho_{16} \equiv {\sigma_{16\to 16}/ \sigma_{16\to 10}}$
and
$\rho_{10}\equiv {\sigma_{10\to 10}/ \sigma_{10\to 16}}$, where the
subscripts label the assignment for the L-electron in the initial
state.
Fig. 2 shows the two ratios $\rho_{16}$
and $\rho_{10}$ compared to the combined TRISTAN
measurement $\rho_{\mu/\tau}=0.957\pm 0.027$.
This value is about 1.6 standard deviation off the value of unity
predicted by any model which assumes lepton universality.
Again it is apparent that the experimental data
can be better accounted for by taking
$$
\eqlabel{3.6}
\eqlabel{3.7}
\eqalignno{
``e_L",``\mu_L" &\in{\bf 16}, \quad ``\tau_L"\in  {\bf 10}
\qquad {\rm and} \qquad -0.5\lesssim\sin\beta\lesssim 0.8 &\req{3.6} \cr
{\rm or} \phantom{``\mu_L" }& & \cr
``e_L",``\mu_L" &\in{\bf 10}, \quad ``\tau_L"\in  {\bf 16}
\qquad {\rm and} \qquad \sin\beta\gtrsim0.4\, &\req{3.7} \cr
}
$$
and assuming $M_\beta \lesssim 300$ GeV.
We note that the  set of assignments in \req{3.6}
coincides with the assignments in the model
discussed in Ref. [\cite{f}].

For the particular value $\sin\beta=-\sqrt{5\o 8}$,
which corresponds to the rank 5 $\eta$ model
[\cite{rizzo-e6}],
the cross section \req{3.1} is invariant with respect to
the different choices of the assignments.
This is apparent from Fig. 1, and in particular
Fig. 2 shows that in this case lepton universality is preserved.
This follows from the fact
that for this value of $\beta$,
the $Q_\eta$ charges for the
the ${\bf \bar 5}_{({\bf 10})}$ and for
the ${\bf \bar 5}_{({\bf 16})}$ are equal [\cite{rizzo-e6},\cite{lfc}]
(this is true also for
 $Q_\eta({\bf 1}_{({\bf 1})})$ and $Q_\eta({\bf 1}_{({\bf 16})})$)
implying that for all the leptons
the couplings to the $Z_\eta$
are the same independently of the UA.

\midinsert
{
\vskip 1truecm
\epsfxsize=14.0truecm %   \hsize
\epsffile{lep2s.ps}
\vskip -10.5truecm
% \hskip-1.0truecm\special{psfile=lep2s.ps hscale=60 vscale=25}
\baselineskip=14pt
$$
\vbox{
% \vskip -4.5truecm
\hsize= 13.3truecm
\noindent
FIG. 3.
{\nine
The predictions for the ratio of pair
productions of two different lepton flavors
in the unconventional assignments $\E$ models at the
LEP-2 c.m. energy $\sqrt{s}=190\>$GeV  (shaded areas),
compared to the standard model prediction (thick solid line).
The dotted lines depict  the expected
one standard deviation experimental error, based
on an integrated luminosity of 500 pb$^{-1}$.
The solid lines enclosing the shaded areas correspond to
$M_\beta=600\>$GeV while the dot-dashed lines correspond to
$M_\beta=800\>$GeV. The results are given for a general $Z_\beta$ from
$\E$, as a function of $\sin\beta$.
}
}
$$
}
\vskip -.6truecm
\endinsert
\interlinea
% \vskip -.6truecm
\noindent
In Fig. 3 we depict the theoretical values of
$\rho_{16}$  and $\rho_{10}$
for $M_\beta$ in the range 600 -- 800 GeV,
and for $\sqrt{s}=190\>$GeV, corresponding to
the c.m. energy at LEP-2.
The dotted lines depict the one standard deviation statistical error
achievable with 500 pb$^{-1}$ of integrated luminosity corresponding to
one year run [\cite{lep2-perkins}]
($\sim 3\times 10^3$ leptonic events per flavor).
It is apparent that the signature of UA models could be
easily recognized for $Z_\beta$ bosons corresponding to
most of the $\sin\beta$  positive values.

\midinsert
{
\vskip 1truecm
\epsfxsize=14.0truecm %   \hsize
\epsffile{nlcs.ps}
\vskip -10.5truecm
% \hskip-1.0truecm\special{psfile=nlcs.ps hscale=60 vscale=25}
\baselineskip=14pt
$$
\vbox{
% \vskip -4.5truecm
\hsize= 13.3truecm
\noindent
FIG. 4.
{\nine
Same as Fig. 3, at the NLC c.m. energy $\sqrt{s}=500\>$GeV, for
$M_\beta=1500\>$GeV (solid lines) and $M_\beta=2200\>$GeV  (dot-dashed
lines). The dotted lines depict  the expected one standard deviation
experimental error, based on an integrated luminosity of 20 fb$^{-1}$
and including an efficiency cut.
}
}
$$
}
\vskip -.6truecm
\endinsert
\interlinea
% \vskip -.6truecm
\noindent
Fig. 4 depicts the ratios $\rho_{16}$  and $\rho_{10}$
for $M_\beta$ in the range 1500 -- 2200 GeV,
and for $\sqrt{s}=500\>$GeV, corresponding to
the NLC c.m. energy.
The one standard deviation error
corresponds to a statistics of
$\sim 8.6\times 10^3$
leptonic events per flavor,
based on an integrated luminosity of 20 fb$^{-1}$
(one year run) and taking into account the efficiency for
a cut to suppress the two-photon background
[\cite{nlc-djouadi}].
The violation of lepton universality, intrinsic of the
UA models, would produce striking effects for
a $Z_\beta$ as heavy as $\sim 2\>$TeV and for
values of $\sin\beta$ not too close to the $\eta$ model.

We have seen that by measuring
the various quantities $R_{ll}$
and $\rho_{l/l^\pr}$, it will be easy to
detect the effects of the UA.
However, from Figs. 1--4 it is also apparent that
these observables alone would not be sufficient to
determine the exact pattern of lepton assignments, since
different assignments can account for the same set of experimental data
by means of a different choice of the $\beta$ parameter.
Working out a procedure for identifying univocally the correct
pattern of assignments is beyond the scope of the present analysis,
however,
we believe that once signals of UA are detected in the leptonic cross
sections, a measurement of the various leptonic asymmetries
would be quite effective to achieve this result.
Also, we would like to point out that if some mechanism resulting in
generation dependent assignments for the lepton doublets
is effective, quite naturally the same mechanism would imply UA for the
$d$-type $SU(2)$ singlet quarks as well. For example
in the model discussed in Ref. [\cite{f}]
the assignments
$``e_L",``\mu_L" \in{\bf 16}$ and $``\tau_L"\in  {\bf 10}$
did imply, for the consistency of the model,  the UA
$``d^c_L",``s^c_L" \in{\bf 10}$ and $``b^c_L"\in  {\bf 16}$
in the quark sector.
Clearly, due to the experimental difficulties in tagging the quark
flavors,
identifying UA for the $d$-quarks would be a much harder task.


\chap{Conc} {IV. CONCLUSIONS}{}

In conclusion we have examined the possibility of
detecting with the present and future $e^+e^-$
colliders,
signals of models predicting UA for the charged leptons.
We have shown that a class of
models based on the gauge group $\E$, in which the known
$SU(2)$ lepton doublets
are embedded
in the fundamental representation of the group
in a generation dependent way,
would result in a unique
type of violation of lepton universality,
which is induced by the exchange
of new $Z_\beta$ bosons.
In agreement with LEP-1 data, no
observable effects are predicted at the $Z_0$ resonance,
however, some signals could be detected off $Z_0$ resonance.
For example, we have shown that a few anomalies
in the production rate of leptons, as well a
hint of violation of $\mu$--$\tau$ universality which have been
observed at the TRISTAN collider,
could be well accounted for in the UA scenarios.
As we have discussed, though these anomalies are not statistically
compelling, it will not be easy to find an alternative
particle physics mechanism that could simultaneously account for
the LEP-1 and the TRISTAN observations.
However, the mechanism proposed here would be effective
only if the $Z_\beta$ mass is not much heavier than $\sim 300\>$GeV.
Though this value is still consistent with the direct limits from
the $p\bar p$ collider [\cite{zp-direct}],
in the near future the data obtainable
at Tevatron will be able to probe or rule out such an explanation
of the leptonic cross section anomalies
[\cite{zp-direct-new}].

We have also discussed the discovery potential for this class of
models at LEP-2, operating at 190 GeV c.m. energy, and at the NLC,
operating at 500 GeV c.m. energy.
We have shown that at these future colliders,
striking effects of lepton universality violation
resulting from the various UA, could be observed
up to $M_\beta \sim 800$ GeV (LEP-2) and
$M_\beta \sim 2200$ GeV (NLC)
for most of the values of the model dependent parameter $\beta$.

\chap{Ackn} {ACKNOWLEDGEMENTS}{}

It is a pleasure to thank the Theory Group at KEK and the Particle
Physics Group at the Yukawa Institute in Kyoto for their kind
hospitality during the initial stage of this work.
I also thank K. Hagiwara and G. Kane for some useful discussions,
and J. F. Dodge, I. Rothstein, and D. Tommasini,  for a
critical reading of the manuscript.

\vfil\eject

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\null
\centerline{\title References}
\baselineskip 8pt

\vskip .8truecm

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% ========
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% ======  Correlation between M(Z-prime) and m(t) bounds. 2. All data
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% =======
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Rome Report No. 912 (November 1992). \par

\biblitem{KAM92a}
K.S. Hirata  {\it et al.} (Kamiokande-II Collaboration),
\plb{280} 146 (1992). \par

\biblitem{IMB92a}
R. Becker-Szendy {\it et al.} (IMB Collaboration), \prd{46} 3720
(1992); see also D. Casper {\it et al.}, \prl{66} 2561 (1991). \par

\biblitem{KAM92c} E.W. Beier  \ea \ \plb{283} 446 (1992). \par

\biblitem{IMB92b} R. Becker-Szendy {\it et al.} (IMB Collaboration),
\prl{69} 1010 (1992). \par

\biblitem{Baksan91}
Baksan Collaboration,
M.M. Boliev {\it et al.}
in {\it Proceedings of the 3rd International Workshop on
Neutrino Telescopes,} Venice, Italy, 1991, edited by Milla Baldo Ceolin
(Istituto Nazionale di Fisica Nucleare, Padova, 1991), p. 235 \par

\biblitem{leprad}
M. Consoli and W. Hollik, in {\it Z physics at LEP 1} Vol. 1,
edited by G. Altarelli \ea, CERN Report No. 89--08, p. 7;
G. Burgers and F. Jegerlehner, \ib, p. 55; \hbup
% W. Hollik, CERN--TH.5661/90 (FEB. 1990); \hbup
G. Burgers and W. Hollik, in {\it Polarization at LEP} Vol. 1,
edited by G. Alexander \ea, CERN Report No. 88--06, p. 136; \hbup
D.C. Kennedy and B.W. Lynn, \npb{322} (1989) 1. \par

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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  965 1992 d  931 1959 r  941 1959 d  954 1959 r  965 1959 d  980 1969 r
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  978 1961 r  980 1960 d  982 1959 d  983 1958 d  984 1956 d  984 1953 d
  983 1951 d  982 1950 d  980 1949 d  977 1949 d  975 1950 d  974 1951 d
  973 1954 d  980 1960 r  982 1958 d  983 1956 d  983 1953 d  982 1951 d
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 1238 1971 r 1238 1959 d 1231 1971 r 1241 1971 d 1251 1971 r 1270 1971 d
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 1262 1959 d 1267 1960 d 1270 1963 d 1268 1971 r 1268 1976 d 1267 1979 d
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 1255 1960 d 1259 1959 d 1281 1992 r 1292 1959 d 1283 1992 r 1292 1963 d
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   3 lw 1977 2616 r 1969 2578 d 1978 2616 r 1971 2578 d 1971 2616 r 1993 2616 d
 1998 2614 d 1999 2612 d 2000 2608 d 2000 2605 d 1997 2601 d 1995 2599 d
 1989 2598 d 1975 2598 d 1993 2616 r 1996 2614 d 1997 2612 d 1999 2608 d
 1998 2605 d 1995 2601 d 1993 2599 d 1989 2598 d 1964 2578 r 1976 2578 d
 1986 2598 r 1990 2581 d 1992 2578 d 1996 2578 d 1999 2581 d 1986 2598 r
 1988 2594 d 1992 2578 d 2006 2585 r 2008 2586 d 2011 2590 d 2011 2567 d
 2010 2589 r 2010 2567 d 2006 2567 r 2016 2567 d 2031 2590 r 2028 2589 d
 2026 2585 d 2024 2580 d 2024 2577 d 2026 2571 d 2028 2568 d 2031 2567 d
 2033 2567 d 2036 2568 d 2038 2571 d 2040 2577 d 2040 2580 d 2038 2585 d
 2036 2589 d 2033 2590 d 2031 2590 d 2029 2589 d 2028 2588 d 2027 2585 d
 2026 2580 d 2026 2577 d 2027 2571 d 2028 2569 d 2029 2568 d 2031 2567 d
 2033 2567 r 2035 2568 d 2036 2569 d 2037 2571 d 2038 2577 d 2038 2580 d
 2037 2585 d 2036 2588 d 2035 2589 d 2033 2590 d 2063 2579 r 2067 2577 d
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 1244 2524 r 1256 2524 d 1266 2544 r 1270 2527 d 1272 2524 d 1276 2524 d
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 1317 2532 r 1316 2531 d 1317 2530 d 1318 2531 d 1318 2532 d 1317 2535 d
 1315 2536 d 1312 2536 d 1309 2535 d 1307 2532 d 1306 2530 d 1304 2526 d
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 1380 2514 d 1382 2513 d 1384 2513 r 1387 2514 d 1389 2516 d 1390 2519 d
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 1834 2861 r 1827 2823 d 1827 2861 r 1849 2861 d 1854 2859 d 1855 2857 d
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 1849 2861 r 1852 2859 d 1853 2857 d 1855 2853 d 1854 2850 d 1851 2846 d
 1849 2844 d 1845 2843 d 1820 2823 r 1832 2823 d 1842 2843 r 1846 2826 d
 1848 2823 d 1852 2823 d 1855 2826 d 1842 2843 r 1844 2839 d 1848 2823 d
 1862 2830 r 1864 2831 d 1867 2835 d 1867 2812 d 1866 2834 r 1866 2812 d
 1862 2812 r 1872 2812 d 1893 2831 r 1892 2830 d 1893 2829 d 1894 2830 d
 1894 2831 d 1893 2834 d 1891 2835 d 1888 2835 d 1885 2834 d 1883 2831 d
 1882 2829 d 1880 2825 d 1880 2818 d 1882 2815 d 1884 2813 d 1887 2812 d
 1889 2812 d 1892 2813 d 1894 2815 d 1896 2818 d 1896 2820 d 1894 2823 d
 1892 2825 d 1889 2826 d 1888 2826 d 1885 2825 d 1883 2823 d 1882 2820 d
 1888 2835 r 1886 2834 d 1884 2831 d 1883 2829 d 1882 2825 d 1882 2818 d
 1883 2815 d 1885 2813 d 1887 2812 d 1889 2812 r 1891 2813 d 1893 2815 d
 1894 2818 d 1894 2820 d 1893 2823 d 1891 2825 d 1889 2826 d 1919 2824 r
 1923 2822 d 1919 2820 d 1916 2827 r 1921 2822 d 1916 2816 d 1903 2822 r
 1921 2822 d 1933 2830 r 1936 2831 d 1939 2835 d 1939 2812 d 1938 2834 r
 1938 2812 d 1933 2812 r 1943 2812 d 1958 2835 r 1955 2834 d 1953 2830 d
 1952 2825 d 1952 2822 d 1953 2816 d 1955 2813 d 1958 2812 d 1960 2812 d
 1964 2813 d 1966 2816 d 1967 2822 d 1967 2825 d 1966 2830 d 1964 2834 d
 1960 2835 d 1958 2835 d 1956 2834 d 1955 2833 d 1954 2830 d 1953 2825 d
 1953 2822 d 1954 2816 d 1955 2814 d 1956 2813 d 1958 2812 d 1960 2812 r
 1963 2813 d 1964 2814 d 1965 2816 d 1966 2822 d 1966 2825 d 1965 2830 d
 1964 2833 d 1963 2834 d 1960 2835 d 1425 2715 r 1417 2677 d 1426 2715 r
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 1491 2712 d 1471 2735 r 1476 2735 d 1490 2735 r 1494 2735 d 1471 2712 r
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 1466 2667 d 1467 2666 d 1470 2666 d 1472 2668 d 1474 2670 d 1472 2689 r
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 1491 2668 d 1492 2670 d 1491 2689 r 1486 2674 d 1485 2669 d 1485 2667 d
 1486 2666 d 1486 2689 r 1491 2689 d 1905 2194 r 1897 2156 d 1906 2194 r
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 1928 2183 d 1925 2179 d 1923 2177 d 1917 2176 d 1903 2176 d 1921 2194 r
 1924 2192 d 1925 2190 d 1927 2186 d 1926 2183 d 1923 2179 d 1921 2177 d
 1917 2176 d 1892 2156 r 1904 2156 d 1914 2176 r 1918 2159 d 1920 2156 d
 1924 2156 d 1927 2159 d 1914 2176 r 1916 2172 d 1920 2156 d 1934 2160 r
 1928 2137 d 1935 2160 r 1929 2137 d 1934 2156 r 1933 2150 d 1933 2147 d
 1935 2145 d 1937 2145 d 1939 2146 d 1942 2148 d 1944 2153 d 1946 2160 r
 1943 2148 d 1943 2146 d 1945 2145 d 1947 2146 d 1949 2149 d 1947 2160 r
 1944 2148 d 1944 2146 d 1945 2145 d 1958 2160 r 1951 2137 d 1959 2160 r
 1952 2137 d 1958 2156 r 1957 2150 d 1957 2147 d 1959 2145 d 1961 2145 d
 1963 2146 d 1965 2148 d 1968 2153 d 1970 2160 r 1966 2148 d 1966 2146 d
 1969 2145 d 1971 2146 d 1973 2149 d 1971 2160 r 1968 2148 d 1968 2146 d
 1969 2145 d  849 2531 r  841 2493 d  850 2531 r  843 2493 d  843 2531 r
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  791 2923 d  791 2932 d  789 2923 d  798 2923 r  800 2926 d  802 2927 d
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  810 2927 d  812 2927 d  814 2926 d  815 2925 d  815 2922 d  814 2916 d
  811 2904 d  812 2927 r  814 2925 d  814 2922 d  813 2916 d  810 2904 d
  696 2034 r  896 2034 d lt4  696 1973 r  896 1973 d   4 lw lt0  600 2716 r
 1400 2716 d 1520 2716 r 2200 2716 d   3 lw  600 2172 r 1880 2172 d   2 lw
 1960 2172 r 2200 2172 d lt2  600 1896 r 2200 1896 d  600 2448 r 2200 2448 d
 lt0 1912 1896 r 1912 2134 d 1912 2210 r 1912 2448 d 1914 1896 r 1894 1916 d
 1934 1916 d 1914 1896 d 1912 1898 d 1916 1898 d 1908 1902 d 1920 1902 d
 1904 1906 d 1924 1906 d 1900 1910 d 1928 1910 d 1896 1914 d 1932 1914 d
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 1910 1900 d 1918 1900 d 1914 2448 r 1894 2428 d 1934 2428 d 1914 2448 d
 1912 2446 d 1916 2446 d 1908 2442 d 1920 2442 d 1904 2438 d 1924 2438 d
 1900 2434 d 1928 2434 d 1896 2430 d 1932 2430 d 1898 2432 d 1930 2432 d
 1902 2436 d 1926 2436 d 1906 2440 d 1922 2440 d 1910 2444 d 1918 2444 d
  600 2509 r  824 2509 d  904 2509 r 2200 2509 d lt1  600 2226 r 2200 2226 d
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 1776 2554 d 1784 2557 d 1792 2561 d 1800 2564 d 1808 2568 d 1816 2571 d
 1824 2575 d 1832 2578 d 1840 2581 d 1848 2585 d 1856 2589 d 1864 2593 d
 1872 2597 d 1880 2601 d 1888 2604 d 1896 2608 d 1904 2612 d 1912 2616 d
 1920 2621 d 1928 2624 d 1936 2629 d 1944 2633 d 1952 2637 d 1960 2641 d
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  640 2732 d  648 2715 d  648 2729 d  656 2708 d  656 2726 d  664 2701 d
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 1000 2642 d 1008 2510 d 1008 2641 d 1016 2507 d 1016 2640 d 1024 2505 d
 1024 2639 d 1032 2502 d 1032 2638 d 1040 2499 d 1040 2637 d 1048 2497 d
 1048 2636 d 1056 2495 d 1056 2635 d 1064 2492 d 1064 2634 d 1072 2491 d
 1072 2633 d 1080 2489 d 1080 2632 d 1088 2486 d 1088 2631 d 1096 2485 d
 1096 2630 d 1104 2483 d 1104 2630 d 1112 2481 d 1112 2629 d 1120 2479 d
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 1216 2621 d 1224 2463 d 1224 2621 d 1232 2463 d 1232 2621 d 1240 2462 d
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 1808 2643 d 1816 2640 d 1824 2637 d 1832 2633 d 1840 2629 d 1848 2625 d
 1856 2621 d 1864 2617 d 1872 2614 d 1880 2610 d 1888 2606 d 1896 2602 d
 1904 2598 d 1912 2594 d 1920 2590 d 1928 2586 d 1936 2582 d 1944 2578 d
 1952 2574 d 1960 2570 d 1968 2565 d 1976 2561 d 1984 2557 d 1992 2553 d
 2000 2548 d 2008 2544 d 2016 2540 d 2024 2535 d 2032 2532 d 2040 2527 d
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 1800 2699 d 1832 2633 d 1816 2696 d 1848 2625 d 1832 2693 d 1864 2617 d
 1848 2689 d 1880 2610 d 1864 2686 d 1896 2602 d 1880 2683 d 1912 2594 d
 1896 2680 d 1928 2586 d 1912 2676 d 1944 2578 d 1928 2673 d 1960 2570 d
 1944 2669 d 1976 2561 d 1960 2666 d 1992 2553 d 1976 2662 d 2008 2544 d
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 1920 2687 d 1928 2689 d 1936 2691 d 1944 2693 d 1952 2694 d 1960 2696 d
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 2016 2709 d 2024 2712 d 2032 2713 d 2040 2716 d 2048 2717 d 2056 2719 d
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 1800 2795 d 1808 2795 d 1816 2796 d 1824 2796 d 1832 2796 d 1840 2797 d
 1848 2797 d 1856 2797 d 1864 2797 d 1872 2797 d 1880 2798 d 1888 2798 d
 1896 2798 d 1904 2797 d 1912 2797 d 1920 2797 d 1928 2797 d 1936 2797 d
 1944 2796 d 1952 2796 d 1960 2796 d 1968 2795 d 1976 2795 d 1984 2795 d
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 2071 2014 d 2083 2014 r 2092 2014 d   3 lw 1004 2404 r  984 2404 d  979 2402 d
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 1877 2785 d 1879 2781 d 1881 2780 d 1885 2778 d 1888 2778 d 1894 2780 d
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 1888 2778 r 1892 2780 d 1895 2785 d 1897 2790 d 1897 2798 d 1895 2801 d
 1908 2801 d 1923 2781 r 1920 2767 d 1923 2781 r 1921 2767 d 1914 2779 r
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 1931 2781 d 1945 2781 r 1942 2767 d 1945 2781 r 1943 2767 d 1935 2779 r
 1937 2781 d 1941 2782 d 1953 2782 d 1935 2779 r 1937 2780 d 1941 2781 d
 1953 2781 d 1960 2823 r 1964 2819 d 1967 2814 d 1971 2807 d 1973 2798 d
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 1964 2769 d 1997 2788 r 1998 2790 d 1998 2785 d 1997 2788 d 1996 2789 d
 1993 2790 d 1989 2790 d 1985 2789 d 1983 2786 d 1983 2784 d 1984 2782 d
 1985 2781 d 1987 2780 d 1994 2778 d 1996 2777 d 1998 2775 d 1983 2784 r
 1985 2782 d 1987 2781 d 1994 2779 d 1996 2778 d 1997 2777 d 1998 2775 d
 1998 2770 d 1996 2768 d 1993 2767 d 1989 2767 d 1985 2768 d 1984 2769 d
 1983 2771 d 1983 2767 d 1984 2769 d 2007 2790 r 2007 2767 d 2008 2790 r
 2014 2770 d 2007 2790 r 2014 2767 d 2022 2790 r 2014 2767 d 2022 2790 r
 2022 2767 d 2023 2790 r 2023 2767 d 2004 2790 r 2008 2790 d 2022 2790 r
 2026 2790 d 2004 2767 r 2010 2767 d 2019 2767 r 2026 2767 d  820 2303 r
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  811 2904 d  812 2927 r  814 2925 d  814 2922 d  813 2916 d  810 2904 d
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 2200 2758 d  600 2237 r 2200 2237 d   2 lw  760 2564 r  760 1911 d  764 1911 r
  744 1931 d  784 1931 d  764 1911 d  762 1913 d  766 1913 d  758 1917 d
  770 1917 d  754 1921 d  774 1921 d  750 1925 d  778 1925 d  746 1929 d
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 1968 2216 d 1976 2216 d 1984 2215 d 1992 2215 d 2000 2215 d 2008 2215 d
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 2064 2216 d 2072 2216 d 2080 2217 d 2088 2219 d 2096 2220 d 2104 2221 d
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e
%%Trailer
EndPSPlot


