%Paper: 
%From: nardi@mich1.physics.lsa.umich.edu
%Date: Tue, 13 Jul 1993 20:57:30 -0400

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%   Process with TEX.
%%   1 PostScript figure (Fig1.eps) appended after \bye
%%   at the end of the file.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\input epsf.tex
%TEX TEMPLATE:
%Please note that this file is a template(instructions)
%   and *not* a style file.
%
%Updated with effect from: 6 Nov. 1991
%World Scientific Publishing Company
%Instructions for Typesetting Camera-ready
%Manuscript Using Computer Software
%Tex version
\headline={\ifnum\pageno=1\firstheadline\else
\ifodd\pageno\rightheadline \else\leftheadline\fi\fi}
\def\firstheadline{\hfil}
\def\rightheadline{\hfil}
\def\leftheadline{\hfil}
        \footline={\ifnum\pageno=1\firstfootline\else\otherfootline\fi}
\def\firstfootline{\rm\hss\folio\hss}
\def\otherfootline{\hfil}
\font\tenbf=cmbx10
\font\tenrm=cmr10
\font\tenit=cmti10
\font\elevenbf=cmbx10 scaled\magstep 1
\font\elevenrm=cmr10 scaled\magstep 1
\font\elevenit=cmti10 scaled\magstep 1
\font\ninebf=cmbx9
\font\ninerm=cmr9
\font\nineit=cmti9
\font\eightbf=cmbx8
\font\eightrm=cmr8
\font\eightit=cmti8
\font\sevenrm=cmr7
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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 \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}
          }

%
\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}

%%%%%% mie  definizioni  %%%%%%%%%%%


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

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


\def\autosez#1#2\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.
\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  ------------------------


\tolerance 100000
\biblskip{+8truept}                        % spaziatura nella bibliografia
\def\hbup{\hfill\break\baselineskip 14pt}  % 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\scal=cmsy5
\font\bcal=cmsy10 scaled \magstep3   %  big calligrafic
 \font\ltitle=cmbx10 scaled \magstep2

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

%%%%%% 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{{\elevenit et.al.}}
\def\ib{{\elevenit ibid.\ }}

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

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

% \autobibliografia    %  per il file BIBL.TEX
\stileincite{}{}     %  Definisce le parentesi nelle citazioni
\stileinbibl{}{.}    %  Definisce le parentesi nella bibliografia

\numerazionedoppia   % avverte che ci sono vari capitoli

%\interlinea
%\standardpage
%\text                %  font per il testo

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%        HERE STARTS THE PAPER : FIT DEF
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\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}}

\def\clesq{\left(c_L^e\right)^2}
\def\clnuesq{\left(c_L^{\nu_e}\right)^2}
\def\clmusq{\left(c_L^\mu\right)^2}
\def\clnumusq{\left(c_L^{\nu_\mu}\right)^2}
\def\slesq{\left(s_L^e\right)^2}
\def\slnuesq{\left(s_L^{\nu_e}\right)^2}
\def\slmusq{\left(s_L^\mu\right)^2}
\def\slnumusq{\left(s_L^{\nu_\mu}\right)^2}
\def\sresq{\left(s_R^e\right)^2}
\def\srnuesq{\left(s_R^{\nu_e}\right)^2}
\def\srmusq{\left(s_R^\mu\right)^2}
\def\srnumusq{\left(s_R^{\nu_\mu}\right)^2}
\def\cllisq{\left(c_L^{\ell_i}\right)^2}
\def\clnuisq{\left(c_L^{\nu_i}\right)^2}
\def\sllisq{\left(s_L^{\ell_i}\right)^2}
\def\slnuisq{\left(s_L^{\nu_i}\right)^2}
\def\srlisq{\left(s_R^{\ell_i}\right)^2}
\def\srnuisq{\left(s_R^{\nu_i}\right)^2}
\def\cli{c_L^i}
\def\sri{s_R^i}
\def\slusq{\left(s_L^u\right)^2}
\def\sldsq{\left(s_L^d\right)^2}
\def\slssq{\left(s_L^s\right)^2}
\def\slcsq{\left(s_L^c\right)^2}
\def\clisq{\left(c_L^i\right)^2}
\def\srisq{\left(s_R^i\right)^2}
\def\clfsq{\left(c_L^f\right)^2}
\def\srfsq{\left(s_R^f\right)^2}
\def\sltausq{\left(s_L^\tau\right)^2}
\def\srtausq{\left(s_R^\tau\right)^2}
\def\srusq{\left(s_R^u\right)^2}
\def\srdsq{\left(s_R^d\right)^2}
\def\srssq{\left(s_R^s\right)^2}
\def\srcsq{\left(s_R^c\right)^2}
\def\slbsq{\left(s_L^b\right)^2}
\def\srbsq{\left(s_R^b\right)^2}
\def\slnutausq{\left(s_L^{\nu_\tau}\right)^2}
\def\srqsq{\left(s_R^q\right)^2}


%    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 \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}}}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def\gweak{SU(2)_L\times U(1)_Y}
\autobibliografia
% \def\adv#1{\global\advance\sect@num by #1}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\TagsOnRight
% \nopagenumbers
\pageno=0
\line{\hfil }
\vglue 1cm
\hsize=6.0truein
\vsize=8.5truein
\parindent=3pc
\baselineskip=10pt
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\null
\rightline{UM-TH 93--14}\qquad\par\noindent
\rightline{hep-ph@xxx/yymmddd}\par\noindent
\vglue 1.0cm
\centerline
{\ltitle REVIEW OF CONSTRAINTS ON FERMION MIXING$^{\displaystyle\dagger}$ }
\vglue 1.5cm
\centerline{\elevenrm ENRICO NARDI }
\baselineskip=14pt
\vglue 0.3cm
\centerline{\elevenit Randall Laboratory of Physics, University of Michigan}
\smallskip
\baselineskip=13pt
\centerline{\elevenit Ann Arbor, MI 48109-1120, U.S.A.}
\vglue 2.0cm
\centerline{\elevenrm ABSTRACT}
\vglue 0.6cm
{\rightskip=1pc
 \leftskip=1pc
 \elevenrm\baselineskip=14pt
 \noindent
The constraints on the mixing angles of the standard fermions with new
heavy particles that can appear in many extensions of the electroweak
theory are reviewed. Some emphasis is put in distinguishing the
effects of a mixing with new states which transform in a non-canonical
way with respect to $SU(2)\times U(1)$ (left-handed singlets or
right-handed doublets), from the effects of a mixing with new states
with standard quantum number assignments. Constraints from flavor
changing neutral current processes, as well as from flavor diagonal
and charged current experimental data are considered. New limits for
lepton flavor violating mixings are presented.
By using the most
recent results on the $\tau$ mass, lifetime and branching ratios,
updated limits on the
mixing parameters of the $\tau$ neutrino are derived.
These limits are improved up to a factor of 3, and no signals
of deviation from the standard model predictions are found.
Limits on the mass of a $Z_1$ gauge boson from E$_6$, from the absence
of muon number violating processes are briefly discussed as well.
\vglue 0.6cm}
\noindent
PACS number(s): 13.10.+q,12.10.Dm,12.15.Ff,14.60.Jj
\vfill
\noindent
--------------------------------------------\phantom{-} \hfil\break
\leftline{$^\dagger$ Invited talk at the
``Workshop on Physics and Experiments at Linear e+e- Colliders'',}
\leftline{\phantom{$^\dagger$} Waikoloa, Hawai, April 26-30, 1993.}
\vglue 0.2cm
\leftline{E-mail: nardi@umiphys.bitnet}
\bigskip
\leftline{UM-TH 93--14}
                   \bigskip
\centerline{July 1993}

\bigskip \bigskip
\eject

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

% \centerline{\tenbf WORLD SCIENTIFIC PUBLISHING COMPANY}
% \vglue 12pt
\phantom{\centerline{\elevenbf REVIEW OF CONSTRAINTS ON FERMION MIXING
}}
%\vglue 5pt
%\centerline{\ninerm (For 20\% Reduction to 8.5 $\times$ 6 in Trim Size)}
\vglue 1.0cm
\phantom{\centerline{\tenrm ENRICO NARDI }}
\baselineskip=13pt
\phantom{
\centerline{\tenit Randall Laboratory of Physics, University of
Michigan}}
\baselineskip=12pt
\phantom{
\centerline{\tenit Ann Arbor, MI 48109-1120, U.S.A.}}
\vglue 0.8cm
\adv{1}
\line{\elevenbf 1. Introduction \hfil}
\bigskip
\baselineskip=14pt
\elevenrm
\noindent
The Standard Model (SM) of the electroweak interactions has achieved a
tremendous success in describing the experimental data within the
range of energies available today,
however too many questions are still left unanswered, and it is
generally believed that the present theory cannot be the whole
story.
Several models which go beyond the SM and
 which do address some of these remaining questions
predict the existence of new particles. In particular, in
models based on gauge groups larger than the SM group
$\G = SU(2)_L \times U(1)_Y \times SU(3)_C$
the new gauge interactions require the presence of new fermions to
insure anomaly cancellation.

In the present analysis I will be only interested
in fermions with conventional electric charges, implying that
the new states must be singlets or doublets of weak--isospin.
A rather heterodox exception is that of a gauge triplet of
fermions\upcite{trip}, but this
possibility will not be considered here.
The possibilities for the {\elevenit new} fermions are listed in Table 1.
Vector singlet
(doublet) fermions refer to particles whose $L$ and $R$ components both
transform as singlets (doublets) under $SU(2)_L$.
$\gweak$ singlet Weyl neutrinos can also be present.
A typical example in which these new states
appear is represented by E$_6$ models, that will be briefly discussed in
Section 3.
Mirror fermions are another
type of new fermion, whose transformation properties under $SU(2)_L$ are
opposite those of ordinary fermions, i.e.~L-handed singlets and
R-handed doublets. They appear, for instance, in grand unified
theories which include family unification\upcite{mirror}.
\midinsert
{
$$
\vbox{\hsize= 13.35truecm
{\tenrm \baselineskip=12pt
\noindent
Table 1:
Possible $\gweak$ assignments for new fermions. Pairs
of particles enclosed in parentheses indicate $SU(2)_L$-doublets, otherwise
they are $SU(2)_L$-singlets. $N$ and $E$ refer to leptons of charge $0$ and
$-1$, respectively. $U$ and $D$ are quarks of charge $2/3$ and $-1/3$.}
\vskip -.2truecm}
$$
$$
\vbox{\offinterlineskip
\halign{&\vrule#&
   \strut\quad#\hfil\quad\cr
\noalign{\hrule}
%height2pt&\omit&\cr
%&Sequential Fermions&\cr
%&~~~~~~~${N\choose E^-}_L~~~E_R~,~~~{U\choose D}_L~~~{U_R\atop D_R}$&\cr
%height2pt&\omit&\cr
%\noalign{\hrule}
height2pt&\omit&\cr
%&Non-Canonical $\gweak$ Assignments&\cr
&~~~~~{\elevenit a) Vector Doublets}&\cr
&~~~~~~~${N\choose E^-}_L~~~{N\choose E^-}_R~,
      ~~~{U\choose D}_L~~~{U\choose D}_R$&\cr
height2pt&\omit&\cr
&~~~~~{\elevenit b) Vector Singlets}&\cr
&~~~~~~~$E^-_L~~~E^-_R~,~~~U_L~~~U_R~~~D_L~~~D_R$&\cr
height2pt&\omit&\cr
&~~~~~{\elevenit c) Weyl Neutrinos}&\cr
&~~~~~~~$N_L~~~N_R$&\cr
height2pt&\omit&\cr
&~~~~~{\elevenit d) Mirror Fermions}&\cr
&~~~~~~~$E_L^-~~~{N\choose E^-}_R~,~~~{U_L\atop D_L}~~~{U\choose D}_R$&\cr
height2pt&\omit&\cr
\noalign{\hrule}}}
$$
}
\endinsert
There are two ways to search for signals of new fermions:
directly and indirectly.
Model independent limits from direct production come
from LEP\upcite{pdg92} and set a lower bound on the masses of such
particles of $m_{new}\gtrsim 45~{\rm GeV}\,$,
although
the mass limit on singlet neutrinos can be considerably weaker. As
to indirect signals, one possibility is to look for loop-induced effects in
high precision measurements or in
rare processes. This are model-dependent analyses, depending on the
number, masses and couplings
of the new states, and will not be discussed here.
The other possibility, which is the topic of this brief review,
is to search for the new states by looking
for signals of mixings of the new fermions
with the known ones. This possibility
relies on the fact that the most general
gauge invariant Lagrangian involving the new particles usually
leads to mass eigenstates which correspond to some
superposition of the known and new gauge eigenstates.
These mixings can be analysed in a model independent way.

In Section 2 I will briefly outline the formalism for dealing with
mixing effects for the charged fermions in the
neutral current (NC) sector.
For a more complete treatement including the
charged currents (CC) and
the mixings of the  neutral states I refer to the
original works  by Langacker and London\upcite{ll1}, and
Nardi, Roulet and Tommasini\upccite{fit}{fit6}.
In Section 3 I will review the
constraints which current experimental data place upon the mixings
between fermions with different $SU(2)_L$ transformation properties,
and I will also describe the kind of constraints that,
in the frame of E$_6$ models,  can be
set on the $Z_1$  parameters from an analysis of some
unsuppressed flavor changing (FC) mixings of the known leptons.
\vglue 0.6cm
\adv{1}
\line{\elevenbf 2. Formalism \hfil}    %%%%%%%%%%%%%%%%%%%%
\vglue 0.4cm
\noindent
I will assume an effective low energy gauge group of the form
$\G\times U_1(1)$
where, if the rank of the group is larger than 5, as in E$_6$,
the additional abelian factor
arises as a combination of different Cartan generators, and
corresponds to the lightest additional neutral gauge boson.
Then the neutral current Lagrangian in the gauge basis reads
$$
-{\cal L}_{\rm NC}=eJ^\mu_{\rm em}A_\mu +
g_0 J_0^\mu Z_{0 \mu} + g_1 J_1^\mu Z_{1 \mu}.
\eq{2.1}
$$
The SM neutral gauge boson $Z_0$ couples with strength
$g_0=(4\sqrt{2} G_F M^2_{Z_0})^{1/2}$ to the usual combination of the
neutral isospin and electromagnetic currents
$
J^\mu_0=J^\mu_3-\sin^2\theta_W  J^\mu_{\rm em}.
$
Assuming that the new $U_1(1)$ originates from a GUT based on a
{\elevenit simple} group, and normalizing  the new generator $Q_1$ to the
hypercharge axis, the $Z_1$ couples to the new $J_1$ current with
strength
$g_1 \simeq g_0 \sin\theta_W$.
In general the standard  $Z_0$ is expected to be
mixed with the $Z_1$, however in this short presentation I will
neglect these additional
effects, since due to the  tight limits implied
for the $Z_0$--$Z_1$ mixing by low energy NC and
LEP data ($\phi_{Z_0-Z_1}\lesssim 0.02\>$\upccite{fit6}{zp-new})
they turn out to be less important than the effects due to direct
$Z_1$ exchange\upcite{lfc}.
To ensure the absence of
anomalies for the new gauge current $J_1\,$, {\elevenit new} fermions
must be present in addition to the standard 15 {\elevenit known}
fermions per generation. Here I will assume that some of the
additional new fermions are electrically charged, and that they are
mixed with the known states. Each of the conventional {\elevenit
light} fermion mass eigenstate then corresponds to a superposition of
the known states and the new states. Conservation of the electric and color
charges forbids a mixing between gauge eigenstates with different
$U(1)_{\rm em}$ and $SU(3)_{\rm c}$ quantum numbers, implying in turn
that the corresponding currents are not modified by the presence of
the new states. In contrast the neutral isospin generator $T_3$ and
the new generator $Q_1$ are spontaneously broken, and a mixing between
states with different $t_3$ and $q_1$ eigenvalues is allowed. This
will affect the $J_3$ and $J_1$ currents\upcite{fit6}
and in turn the couplings of
the light mass eigenstates to the $Z_0$ and $Z_1$.
In the gauge currents chirality is conserved too, and it is then
convenient to group the fermions with the same electric charge and
chirality $\alpha=L,R$ in a vector of the known ($\cal K$) and new
($\cal N$) gauge eigenstates  $\Psi^{o}_\alpha=(\Psi^o_{\K},
\Psi^o_{\N})_\alpha^T$. This vector is related to the corresponding vector
of the light ({\elevenit l}$\,$) and heavy ({\elevenit h}) mass eigenstates
$\Psi_\alpha=(\Psi_l,\Psi_h)_\alpha^T$ through a unitary transformation
$$
\pmatrix{\Psi^o_{\K}\cr\Psi^o_{\N}}_\alpha = U_\alpha
\pmatrix{\Psi_l\cr\Psi_h}_\alpha \qquad{\rm where}\qquad
U_\alpha = \pmatrix{A &G\cr F & H}_\alpha ,
\qquad  \alpha=L,R.
\eq{2.2}
$$
The submatrices $A$ and $F$ describe the overlap of the light
eigenstates with the known and the new states respectively, and
the unitarity of $U_\alpha$ implies
$
A^\dagger A+F^\dagger F=A A^\dagger +G G^\dagger =I.
$
Note that there is no extra index to
label the electric charge, nevertheless
$\Psi^o_\alpha$ and $\Psi_\alpha$
will  be treated
as vectors corresponding to a definite value of $q_{\rm em}$.
In terms of the fermion mass eigenstates, the neutral current
corresponding to a (broken) generator ${\cal Q}=T_3,Q_1$ now reads
$$
J^\mu_{\cal Q} =\sum_{\alpha=L,R}
\bar\Psi_{\alpha} \gamma^\mu U^\dagger_\alpha {\cal Q}_\alpha
U_\alpha\Psi_{\alpha},
\eq{2.4}
$$
where ${\cal Q}_\alpha$ represents a generic diagonal matrix of the
charges $q_\alpha=t_3(f_\alpha)$, $q_1(f_\alpha)$ for the chiral
fermion $f_\alpha$. Since we are interested in the indirect effects of
fermion mixings in the couplings of the light mass eigenstates, we
have to project $J^\mu_{\cal Q}$ onto the light components $\Psi_l$.
In the particularly simple case when the mixing is with only
one type of new fermions with the same $q_\alpha^\N$ charges,
by means of the unitarity relations for $U_\alpha$
we easily obtain\upcite{fit6}
$$
J^\mu_{l{\cal Q}}
               =\sum_{\alpha=L,R}
\bar \Psi_{l\alpha} \gamma^\mu \left[ q_\alpha^{\K} I +
(q_\alpha^{\N} - q_\alpha^{\K})
F^\dagger_\alpha F_\alpha \right]\Psi_{l\alpha}.
\eq{2.5}
$$
In Eq. \req{2.5}
$q_\alpha^{\K}I $ represents the coupling of a particular light
fermion in the absence of mixing effects, while the second term
accounts for the modifications due to fermion mixings. The matrix
$F_\alpha^\dagger F_\alpha$ is in general not diagonal, and clearly
whenever the coefficient $(q_\alpha^{\N} - q_\alpha^{\K})$ is nonvanishing,
the off diagonal terms will induce FCNC, while the diagonal terms
will affect the flavor-conserving couplings.
It is useful to parametrize the FC mixing between the light particles $i$ and
$j$ as
$$
{\cal F}_\alpha^{ij} = \left(F_\alpha^\dagger F_\alpha\right)_{ij},
\hskip 1.5truecm   i\ne j,  \qquad  \qquad  \alpha=L,R.
\eq{2.6}
$$
{}From a phenomenological
point of view
we can now distinguish two cases. If the
known states are mixed with
new particles whose $L$ and/or $R$ components
transform in a noncanonical way under $SU(2)_L$
(and hence will be referred to as {\elevenit exotic})
weak--isospin is violated:
$t_3(f_\alpha^{\N}) \ne t_3(f_\alpha^{\K})$. Then the $J_0$ current
is affected and the $Z_0$ interactions will be FC.
If the mixing is with new states which
obey the same transformation
properties than the known  fermions
(i.e.~$L$-handed doublets and $R$-handed singlets, that will be referred
to as  {\elevenit ordinary}) then
$t_3(f_\alpha^{\N}) =
t_3(f_\alpha^{\K})$, and clearly  the $J_0$ current is not
affected and in particular it remains flavor-diagonal.
However, in general we still
have $q_1(f_\alpha^{\N}) \ne q_1(f_\alpha^{\K})$. Then the
isospin-conserving mixings can indeed affect the $J_1$ current,
inducing sizeable FC couplings to the $Z_1$.

Now, to give an example of the possible form of the fermion mass matrices
and to discuss the expected magnitude of the mixings between the known and the
new fermions, let us introduce for each fermion family
a vector gauge singlet of new fermions
$({X^o_{\scss E}}_L,{X^o_{\scss O}}_R)_i$
($E$ = exotic, $O$ = ordinary, $i=1,2,3$)
with the same electric and color
charges than the  known fermions $({f^o_{\scss O}}_L,{f^o_{\scss
O}}_R)_i$. Then in the gauge eigenstate basis
the mass term reads
$$
{\cal L}_{\rm mass}=
{(\bar {f^o_{\scss O}}, \bar {X^o_{\scss E}})}_L \>
{\cal M} \> {f^o_{\scss O} \choose  X^o_{\scss O}}_R,
\hskip 1truecm
{\cal M} = \pmatrix{D &D^\pr\cr S^\pr &S},
\eq{2.7}
$$
where e.g. $f^o = (f^o_1,f^o_2,f^o_3)^T$ etc..
The entries
$D$ and $D^\pr$ in the non diagonal mass matrix ${\cal M}$, are
$3\times3$ matrices generated by vacuum expectation values (vevs) of
doublets multiplied by Yukawa couplings, while $S$ and $S^\pr$ are
generated by vevs of singlets.
As a general rule, while the mass terms which couple ordinary
L-fermions to ordinary R-fermions (or exotic L-fermions to exotic
R-fermions) arise from vevs of Higgs doublets, the entries which
couple ordinary fermions to the exotic ones are generated by vevs of
singlets.
Higgs singlets are responsible for the large masses of the
new heavy fermions in vector multiplets and, in most cases, also
contribute to the mass of the new heavy gauge boson; hence it is
natural to assume $S,S^\pr \gg D,D^\pr$.
In the presence of such a hierarchy among the entries in ${\cal M}$ it
is easy to see\upcite{lfc} that for the matrix $U_\alpha$ in
Eq. \req{2.2}
describing the
ordinary--exotic mixings between $L$-states
it is natural to expect that
the submatrices $F$ and $G$ would acquire an overall suppression
factor $\sim S/D \ll 1 $ of the order of the ratio of the light to
heavy mass scale. In contrast such a suppression is not present for
the ordinary-ordinary $F$ and $G$ mixing terms for the $R$-states\upcite{lfc}.
Now, since it is precisely $F^\dagger F$ in Eq. \req{2.5} which affects
the flavor diagonal couplings and also induces  FCNC,
if the mass scale for the exotic fermions is large enough, the suppression
of the ordinary--exotic mixings explains in a natural way the
non--observations of these effects in the $Z_0$ interactions.
On the other hand, for the ordinary--ordinary mixings there is no
reason to expect the elements of $F^\dagger F$ to be particularly
small\upcite{lfc} , and accordingly FC processes can be expected to
occur at a sizeable rate in $Z_1$ interactions.
Then, in the cases in which the FC couplings to the $Z_1$ are not
suppressed, the limits on FC processes point toward a rather heavy
$Z_1$\upccite{lfc}{mue}.

As can be seen
from Table 2, for $t_3(f_\alpha^{\N}) \ne t_3(f_\alpha^{\K})$
very stringent constraints can be derived for
${\cal F}_{L,R}^{ij},~(i\ne j)$
which strongly limits the mixing of ordinary and exotic
fermions. However, it is possible to evade these bounds by considering the
fine-tuned cases in which the mixing matrices
$F_\alpha^\dagger F_\alpha$ are
diagonal. These correspond to those directions in parameter space in
which each known ordinary fermion mixes with its own exotic fermion.
If the strong assumption ${\cal F}_{L,R}^{ij}=0$ for $i\ne j$
is made, one can write
$$
\left(F_\alpha^\dagger F_\alpha\right)_{ij}=
\left(s^i_\alpha\right)^2 \delta_ij \hskip 1truecm \alpha=L,R,
\eq{2.8}
$$
in which $\left(s^i_\alpha\right)^2 \equiv \sin^2\theta_\alpha^i$,
where $\theta_{L(R)}^i$ is the mixing angle of the
$i^{th}$ $L$-handed ($R$-handed) light fermion.
In this case the ordinary-exotic mixing is
parametrized by one angle per each $L$- or $R$-handed charged
fermion. The same angles enter also the expression for
the CC measurables.
We refer to Refs. [\cite{ll1},\cite{fit}] for the extension of the
formalism  to the CC sector.
Even if the limits on FCNC processes are not effective
to constrain these particular fine-tuned mixings,
still all the $\left(s^i_\alpha\right)^2$  can be constrained
by looking at the high precision data involving
CC and flavor-diagonal NC.
The corresponding limits are given in Table 3.

The formalism for the neutral sector is
more complicated, both due to the possibility of
Dirac and Majorana masses, and because there is no empirical evidence
requiring the absence of FCNC between neutrino species. However, due to the
fact that neutrinos are unobserved in experiments, it is possible to
parametrize their mixing by one effective angle for neutrino
flavor\upcite{ll1},
with the addition of an auxiliary
effective parameter $\Lambda$ that
takes into account the type of new neutrinos involved in the mixing.
If the mixing is purely with
new ordinary, exotic singlet or exotic doublet
neutrinos, we have respectively $\Lambda=$0,2,4.
Again we refer to Refs. [\cite{ll1},\cite{fit}] for
a detailed discussion of the mixings in the neutral sector.
\vglue 0.6cm
\adv{1}
\line{\elevenbf 3. Results \hfil}
\bigskip
\line{\elevenit 3.1.
Limits on flavor changing ordinary-exotic fermion mixing \hfil}
\smallskip\noindent
The limits on the
FCNC parameters
${\cal F}^{ij}$ Eq. \req{2.6}
which can be obtained assuming a mixing with
new exotic fermions are listed in Table 2.
The bounds on lepton flavor violating (LFV) NC
from the leptonic decays $\mu\not\to 3e$, $\tau\not\to 3e$, $3\mu$
are taken or adapted from Ref.[~\cite{lfc}].
The bound on ${\cal F}^{\mu e}$ from the non-observation of
$\mu$--$e$ conversion in nuclei is taken from
Ref.[~\cite{mue}].
The limits on $\vert{\cal F}^{\tau e}\vert$ and
$\vert{\cal F}^{\tau \mu}\vert$
from $Z_0\not\to {\tau e}$, ${\tau \mu}$
are new, and have been derived from the experimental results
given in the first paper in Ref. [\cite{lep-lfc}].
It is interesting to note that in all cases the best bounds
are obtained from processes different from leptonic decays.
The limits on hadronic FCNC
have been presented in Ref.~[\cite{london}] and
were taken, updated or adapted from
Ref.~[\cite{Silverman}].
There is no bound on
$\vert{\cal F}^{bd}\vert$ from $B_d^0$-${\overline{B_d^0}}$ mixing because
this mixing can in principle be explained by a non-zero
${\cal F}^{bd}$\upcite{Silverman}.
\midinsert
{
$$
\vbox{\hsize= 13.35truecm
{\tenrm \baselineskip=12pt
\noindent
Table 2:
Limits on the FCNC parameters
${\cal F}^{ij}$ for ordinary-exotic fermion mixing.
The bounds on leptonic FCNC
from leptonic decays are
taken or adapted from Ref.[~\cite{lfc}], the one from
$\mu N \not\to e N$ is taken from Ref.[~\cite{mue}],
and those from $Z_0\not\to \tau e$, $\tau \mu$ are new.
The limits on hadronic FCNC
have been presented in Ref.~[\cite{london}] and
were taken, updated or adapted from
Ref.~[\cite{Silverman}].
The experimental reference is given in the `Source' column.}
\vskip -.2truecm}
$$

$$
\vbox{\offinterlineskip
\halign{&\vrule#&
   \strut\quad#\hfil\quad\cr
\noalign{\hrule}
height2pt&\omit&&\omit&&\omit&\cr
& Quantity && Upper Limit && Source &\cr
height2pt&\omit&&\omit&&\omit& \cr
\noalign{\hrule}
height2pt&\omit&&\omit&&\omit&\cr
& $\vert{\cal F}^{\mu e}\vert$ && $2.4\times 10^{-6}$ &&
			$\mu\not\to 3e$ \quad[\cite{pdg92}] &\cr
& \omit && $1.0\times 10^{-5}$ &&
                        $\mu N \not\to e N$ \quad[\cite{mue-exp}] &\cr
& $\vert{\cal F}^{\tau e}\vert$ && $1.4\times 10^{-2}$ &&
			$\tau\not\to 3e$ \quad[\cite{pdg92}] &\cr
& \omit && $8.0\times 10^{-3}$ &&
                        $Z_0 \not\to \tau e $ \quad[\cite{lep-lfc}] &\cr
& $\vert{\cal F}^{\tau \mu}\vert$ && $1.1\times 10^{-2}$ &&
			$\tau\not\to 3\mu$ \quad[\cite{pdg92}] &\cr
& \omit && $9.5 \times 10^{-3}$ &&
                        $Z_0 \not\to \tau \mu $ \quad[\cite{lep-lfc}] &\cr
& $\vert{\cal F}^{ds}\vert$ && $6\times 10^{-4}$ &&
			$\Delta m_{K_L K_S}$ \quad[\cite{pdg92}] &\cr
& \omit && $1\times 10^{-5}$ && $K_L\to\mu^+\mu^-$ \quad[\cite{pdg92}] &\cr
& $\vert{\cal F}^{cu}\vert$ && $1\times 10^{-3}$ &&
		$D^0$-${\overline{D^0}}$ mixing \quad[\cite{pdg92}] &\cr
& $\vert{\cal F}^{bd}\vert$, $\vert{\cal F}^{bs}\vert$
     && $2\times 10^{-3}$ && $B\not\to\ell^+\ell^- X$ \quad[\cite{UA1-B}] &\cr
height2pt&\omit&&\omit&&\omit& \cr
\noalign{\hrule}}}
$$
}
\endinsert
\bigskip
\line{\elevenit 3.2.
Limits on flavor diagonal ordinary-exotic fermion mixing \hfil}
\smallskip\noindent
The limits on the flavor diagonal
NC parameters
$\left(s^i_{L,R}\right)^2$ Eq. \req{2.8} are listed in Table 3,
which is taken from Ref.~\cite{fit}.
They correspond to those fine-tuned direction in parameter space
for which the constraints from FCNC processes are evaded.
The results of two different fits are shown. In the
`individual fit' only one mixing angle at a time is allowed to be
non-zero, while in the `joint fit' all mixing angles
are simultaneously present.
In the `Source' column in Table 3 are listed those
observables which are most important for constraining the mixing angles in
the individual fits.
The $\Gamma$'s denote partial widths of the $Z_0$,
the $A$'s are asymmetries, $\nu e$ and $\nu q$ refer to low energy NC
scattering experiments, $e q$ refers to measurements of parity violation
in atoms, $g_\ell$ ($\ell=e$,$\mu$,$\tau$) are derived from lepton universality
in CC  processes, $V^2_{ui}$ refers to the unitarity of the CKM
matrix, while
$s_{eff}^{LEP}$ and $s_{eff}^{NC}$ refer to the weak mixing
angle as extracted in NC measurements at the $Z_0$ peak and at
low energy, respectively.
In the joint fit it is possible to evade the
bounds from these observables through fine-tuned cancellations between
different mixings,
so that the constraints in the joint fit are somewhat weaker than
those in the individual fit.
In this case other observables, which depend on
different combinations of the mixings
and which are denoted by a * in Table 3,
become important.
It should be stressed that the data used to obtain these constraints are
already a bit out of date. For example, only the 1990 LEP data was
used. The inclusion of the 1991-92 LEP data would surely strengthen
most of the bounds somewhat.
The most important new development is in $\tau$-decays.
The value of $(g_\tau/g_e)^2$ that was used in Ref. [\cite{fit}]
was about 1.5 standard deviations away from its SM value,
pointing towards a non-zero mixing for $\nu_\tau$.
In the present analysis the new values for the tau mass
$m_\tau=1776.9\pm 0.5\,$MeV\upcite{taumass},
for the $\tau$ lifetime $T_\tau=295.7\pm3.2\,$fs\upcite{taulife},
and for the $\tau$ leptonic branching fractions\upcite{taubranch}
$B(\tau\to e\bar\nu\nu)=(17.75\pm0.15)\%$ and
$B(\tau\to \mu\bar\nu\nu)=(17.39\pm0.15)\%$
have been used.
As a result any hint for a non-zero mixing of the $\tau$
neutrino has disappeared, while the limits on
$\left(s_L^{\nu_\tau}\right)^{2}$ have been improved up to a factor of 3.
\midinsert
{
$$
\vbox{\hsize= 6.0truein % 13.35truecm
{\tenrm \baselineskip=12pt
\noindent
Table 3:
90\% C.L. upper limits on the ordinary-exotic flavor diagonal
mixing angles for individual fits (one
angle at a time is allowed to vary) and joint fits (all angles allowed to
vary simultaneously)\upcite{fit}. Observables which are most
important for the constraints are shown in the `Source' column.
The different values of the parameter $\Lambda$ correspond to
mixings with different kinds of new neutrinos, as explained in the
text. For the limits on $\slnutausq$
the most recent results on the $\tau$ mass\upcite{taumass},
lifetime\upcite{taulife} and branching
ratios\upcite{taubranch} have been used. }
\vskip -.2truecm}
$$

$$
\vbox{\offinterlineskip
\halign
%\hsize= 6.0truein
{&\vrule#&
   \strut\quad#\hfil\quad\cr
\noalign{\hrule}
height2pt&\omit&&\omit&&\multispan5 &&\omit&\cr
& \omit && \omit\hidewidth Individual \hidewidth &&
				\multispan5 Joint && Source &\cr
height2pt&\omit&&\omit&&\multispan5 &&\omit&\cr
\noalign{\hrule}
height2pt&\omit&&\omit&&\omit&&\omit&&\omit&&\omit&\cr
& \omit && \omit && $\Lambda=2$ && $\Lambda=0$ && $\Lambda=4$ && \omit &\cr
height2pt&\omit&&\omit&&\omit&&\omit&&\omit&&\omit&\cr
\noalign{\hrule}
height2pt&\omit&&\omit&&\omit&&\omit&&\omit&&\omit&\cr
& $\slesq$ && 0.0047 && 0.015 && 0.0090 && 0.015 &&
			$\Gamma_e,M_W^*,A_\mu^{FB*},eq^*,g_e^*$ &\cr
& $\sresq$ && 0.0062 && 0.010 && 0.0082 && 0.010 &&
			$\Gamma_e,A_e^{FB},A_\mu^{FB*},\nu e^*$ &\cr
& $\slmusq$ && 0.0017 && 0.0094 && 0.0090 && 0.011 &&
		$V_{ui}^2,\nu q,g_\mu,\Gamma_\mu,s_{eff}^{LEP*}$ &\cr %
& $\srmusq$ && 0.0086 && 0.014 && 0.014 && 0.013 &&
			$\Gamma_\mu,A_\mu^{FB}$ &\cr
& $\sltausq$ && 0.011 && 0.017 && 0.015 && 0.017 &&
			$\Gamma_\tau,A_\tau^{FB},g_\tau,A_\tau^{pol*}$ &\cr
& $\srtausq$ && 0.011 && 0.012 && 0.014 && 0.012 &&
			$\Gamma_\tau,A_\tau^{pol},A_\tau^{FB},g_\tau^*$ &\cr
& $\slusq$ && 0.0045 && 0.019 && 0.015 && 0.019 &&
			$V_{ui}^2,\Gamma_h,\Gamma_Z,eq,\nu q$ &\cr
& $\srusq$ && 0.018 && 0.024 && 0.025 && 0.024 &&
			$\nu q,\Gamma_h,\Gamma_Z,eq$ &\cr
& $\sldsq$ && 0.0046 && 0.019 && 0.016 && 0.019 &&
			$V_{ui}^2,\Gamma_h,\Gamma_Z,\nu q$ &\cr
& $\srdsq$ && 0.020 && 0.030 && 0.028 && 0.029 &&
			$eq,\Gamma_h,\Gamma_Z,\nu q$ &\cr
& $\slssq$ && 0.011 && 0.038 && 0.039 && 0.041 &&
			$\Gamma_h,\Gamma_Z,V_{ui}^2$ &\cr
& $\left(s_R^s\right)^{2}$ && 0.36 && 0.67 && 0.63 && 0.74 &&
			$\Gamma_h,\Gamma_Z$ &\cr
& $\slcsq$ && 0.013 && 0.040 && 0.042 && 0.042 &&
			$\Gamma_h,\Gamma_Z,\Gamma_c^*,A_c^{\gamma Z*}$ &\cr
& $\srcsq$ && 0.029 && 0.097 && 0.10 && 0.099 &&
		$\Gamma_h,\Gamma_Z,A_c^{\gamma Z*},\Gamma_c^*,A_c^{FB*}$ &\cr
& $\slbsq$ && 0.011 && 0.070 && 0.072 && 0.069 &&
			$\Gamma_h,\Gamma_Z,\Gamma_b,A_b^{FB*}$ &\cr
& $\left(s_R^b\right)^{2}$ && 0.33 && 0.39 && 0.40 && 0.39 &&
		$\Gamma_b,\Gamma_Z,\Gamma_h,A_b^{\gamma Z},A_b^{FB*}$ &\cr
& $\slnuesq$ && 0.0097 && 0.015 && 0.016 && 0.014 &&
			$s_{eff}^{LEP},g_e,s_{eff}^{NC},M_W^*$ &\cr
& $\slnumusq$ && 0.0019 && 0.015 && 0.0087 && 0.011 &&
			$V_{ui}^2,g_\mu,\nu q,s_{eff}^{LEP},M_W^*$ &\cr
& $\left(s_L^{\nu_\tau}\right)^{2}$
	&& 0.023 && 0.033 && 0.034 && 0.026 && $\Gamma_Z,g_\tau$ &\cr
height2pt&\omit&&\omit&&\omit&&\omit&&\omit&&\omit&\cr
\noalign{\hrule}}}
$$
}
\endinsert
\line{\elevenit 3.3.  Limits on a Z$_1$ from E$_6$ from
ordinary-ordinary fermion mixing \hfil}
\smallskip\noindent
E$_6$ GUTs are well known examples of theories where additional
fermions and new neutral gauge bosons are simultaneously present.
For a general
breaking of E$_6$ (rank 6) to the SM (rank 4) it is possible to define
a whole class of $Z_1$ bosons corresponding to a linear combination of the two
additional Cartan generators.
I will parametrize this combination in terms of an
angle $\beta$. Fermions are assigned to the
fundamental {\underbar {\elevenbf 27}}
representation of the group which contains 12 additional states
for each generation, among which we have a vector doublet of new
leptons $(N\> E^-)_L^T$, $(E^+ \> N^c)_L^T$. Non-diagonal mass
terms with the standard $(\nu\> e^-)_L^T$ and $e^c_L$ leptons will
give rise respectively to ordinary-ordinary and ordinary-exotic
mixings, and in particular will induce LFV
{\elevenit L} and {\elevenit R} chiral
couplings between the first and second
\vfil\eject

\phantom{p}
%\medskip
\vskip6.5truecm
\midinsert
{
\vskip -.9truecm
\hskip-1.0truecm\special{psfile=Fig1.eps hscale=60 vscale=25}
$$
\vbox{
\vskip -.5truecm
\hsize= 13.3truecm
{\tenrm \baselineskip=14pt
\noindent
Figure 1:
Limits on the $Z_1$ LFV parameter
$M_{Z_1}\cdot \left({\cal F}_L^{e\mu}\right)^{-1/2}$
from the limits on the $\mu$-$e$ conversion process\upcite{mue-exp},
for a general E$_6$  gauge boson, as a
function of $\sin\beta$.
The mixing term ${\cal F}_L^{e\mu}$ is in units
of $10^{-2}$, vertical units are TeV.
 Limits on the $Z_1$ mass for different values of
${\cal F}_L^{e\mu}$ can be easily read off the figure by
properly rescaling the vertical units.}
}
$$
}
\endinsert
\vskip -.6truecm
\noindent
generation, allowing for LFV processes as
$\mu\rightarrow eee$\upcite{lfc} and
$\mu$-$e$ conversion in nuclei\upcite{mue},
for which very stringent experimental limits exist\upccite{mue-exp}{sindrum}.
Due to the expected suppression of the ordinary-exotic mixings, I have
conservatively neglected
the LFV couplings in the {\elevenit R} sector, and assumed
that the only source of LFV interaction comes from the
ordinary-ordinary mixing in the {\elevenit L} sector\upcite{lfc}.
Since in this case the exchange of the $Z_0$ does not contribute
to the $\mu$-$e$ transition, the relevant parameter is actually
${\cal F}_L^{e\mu}/M_{Z_1}^2$.
Relying on an analogy with the CKM mixings, which  also
conserve weak-isospin and are all $> 10^{-3}$ and noting that the mixing
 between the first and second generation is particularly large,
I have assumed a `natural' range
$10^{-2}$--$10^{-3}$ for the LFV term
${\cal F}_L^{e\mu}$.
The corresponding limits on the $Z_1$ mass
obtained from the limit\upcite{mue-exp}
$B(\mu N\to e N) < 4\times 10^{-12}$
are depicted in Fig.$\, 1$ which is taken from
Ref. [\cite{mue}].
The limits are given as a function of the
parameter $\beta$ that defines the particular E$_6$ boson,
and are more stringent than the corresponding limits implyed
by $\mu \not\to eee$\upcite{lfc}.
It is apparent that these
limits are indeed very strong, however,
since they depend on a specific assumption for the numerical
value of the LFV coupling, they clearly cannot replace
the direct\upcite{zp-direct} bounds or other more
model independent indirect limits\upccite{fit6}{zp-new}.
\vglue 0.6cm
\adv{1}
\line{\elevenbf 4. Acknowledgements \hfil}
\vglue 0.4cm
\noindent
It is a pleasure to thank my friends and collaborators E. Roulet
and D. Tommasini, with whom large part of this work was done.
I would also like to thank D. London for helpful discussions.
\vglue 0.6cm
\line{\elevenbf 5. References \hfil}
\vglue 0.4cm

\biblitem{taumass}
BEPC collaboration, J. Z. Bai \ea, \prl{69} (1992) 3021.\par

\biblitem{taulife}
W. Trischuk, in ``{\elevenit Proc. of the Second Workshop on Tau Lepton
Physics}"
(The Ohio State Univ., Columbus) ed. K. K. Gan (1992) to be published. \par

\biblitem{taubranch}
R. Galik, in Ref. [\cite{taulife}].\par

\biblitem{UA1-B}
UA1 Collaboration, C. Albajar \ea, \plb{262} (1991) 163.\par

\biblitem{mirror}
See, for example, F. Wilczek and A. Zee, \prd{25} (1982) 553;\hbup
J. Bagger \ea, \npb{244} (1984) 247.  \par


\biblitem{lep-lfc}
L3 collaboration, B. Adeva \ea, \plb{271} (1991) 453; \hbup
OPAL Collaboration, M. Akrawy \ea, \plb{254} (1991) 293; \hbup
ALEPH Collaboration, D. Decamp \ea, \prep{216}(1992) 253; \hbup
DELPHI Collaboration, P. Abreu \ea, \plb{298} (1993) 247. \par

\biblitem{fit}
E. Nardi, E. Roulet and D. Tommasini,  \npb{386} (1992) 239. \par

\biblitem{ll1}
P. Langacker and D. London, \prd{38} (1988) 886.\par

\biblitem{ll2}
P. Langacker and D. London, \prd{38} (1988) 907.\par

\biblitem{london}
D. London, Report No. UdeM-LPN-TH131,
to be published in {\elevenit
Precision Tests of the Standard Model}, 1993, ed.\ P. Langacker (World
Scientific).\par

\biblitem{fit6}
E. Nardi, E. Roulet and D. Tommasini, \prd{46} (1992) 3040. \par

\biblitem{lfc}
E. Nardi, \prd{48} (1 August 1993). \par

\biblitem{mue}
J. Bernab\'eu, E. Nardi and D. Tommasini,
UM-TH-93-08 (May 1993).  \par
%and FTUV 93-14

\biblitem{trip}
B.W. Lee, \prd{6} (1972) 1188; \hbup
J. Prentki and B. Zumino, \npb{47} (1972) 99; \par %\hbup
% P. Salati, \plb{253} (1991) 173. \par

\biblitem{Silverman}
D. Silverman, \prd{45} (1992) 1800. \par

\biblitem{pdg92}
Particle Data Group, J.J. Hern\'andez \ea, \prd{45} Part. II (1992). \par

\biblitem{sindrum}
SINDRUM collaboration, U. Bellgardt \ea, \npb{299} (1988) 1. \par


\biblitem{mue-exp}
TRIUMF collaboration,
S. Ahmad \ea, \prd{38} (1988) 2102; \hbup
% \prl{59} (1987) 970;
PSI collaboration, A. Badertscher \ea, J. Phys. {\elevenbf G17} (1991)
S47;\hbup
PSI collaboration, A. Van der Schaaf, Nucl. Phys.
{\elevenbf A546} (1992) 421C.\par

\biblitem{zp-direct}
CDF Collaboration, F. Abe \ea, \prl{67} (1991) 2609; \ib {\elevenbf 68},
(1992) 1463. \par

\biblitem{zp-new}
P. Langacker and M. Luo, \prd{45} (1992) 278; \hbup
J. Layssac, F.M. Renard and C. Verzegnassi, \zpc{53} (1992) 97; \hbup
M.C. Gonzalez Garc\'\i a and J.W.F. Valle; \plb{259} (1991) 365; \hbup
% ======= Improved bounds on extended gauge models from new LEP data
G. Altarelli \ea, \plb{263} (1991) 459; \hbup
% =====  Correlation between M(Z-prime) and m(t) bounds. 1. NC data
% F.del Aguila, J.M. Moreno and M. Quir\'os, \npb{361} (1991) 45; \hbup
% ======  Correlation between M(Z-prime) and m(t) bounds. 2. All data
%F. del Aguila, W. Hollik, J.M. Moreno and M. Quir\'os, \ib {\elevenbf B372}
% 3 (1992). \par
F. del Aguila \ea, \npb{372} (1992) 3. \par

\insertbibliografia

\vfill\eject\bye

%%%%%%%%%%%%%%%%%%%%%%  Fig1.eps   %%%%%%%%%%%%%%%%%%%%%%%%%%

%!PS-Adobe-1.0
%%EndComments
save 50 dict begin /psplot exch def
/StartPSPlot
   {newpath 0 0 moveto 0 setlinewidth 0 setgray 1 setlinecap
    1 setlinejoin 72 300 div dup scale}def
/pending {false} def
/finish {pending {currentpoint stroke moveto /pending false def} if} def
/r {finish newpath moveto} def
/d {lineto /pending true def} def
/l {finish 4 2 roll moveto lineto currentpoint stroke moveto} def
/p {finish newpath moveto currentpoint lineto currentpoint stroke moveto} def
/e {finish gsave showpage grestore newpath 0 0 moveto} def
/lw {finish setlinewidth} def
/lt0 {finish [] 0 setdash} def
/lt1 {finish [3 5] 0 setdash} def
/lt2 {finish [20 10] 0 setdash} def
/lt3 {finish [60 10] 0 setdash} def
/lt4 {finish [3 10 20 10] 0 setdash} def
/lt5 {finish [3 10 60 10] 0 setdash} def
/lt6 {finish [20 10 60 10] 0 setdash} def
/EndPSPlot {clear psplot end restore}def
% end fixed prolog
StartPSPlot
%%EndProlog
%%Page: 1 1
   4 lw lt0  500  500 r 2300  500 d  500  500 r  500  532 d  590  500 r
  590  532 d  680  500 r  680  564 d  770  500 r  770  532 d  860  500 r
  860  532 d  950  500 r  950  532 d 1040  500 r 1040  564 d 1130  500 r
 1130  532 d 1220  500 r 1220  532 d 1310  500 r 1310  532 d 1400  500 r
 1400  564 d 1490  500 r 1490  532 d 1580  500 r 1580  532 d 1670  500 r
 1670  532 d 1760  500 r 1760  564 d 1850  500 r 1850  532 d 1940  500 r
 1940  532 d 2030  500 r 2030  532 d 2120  500 r 2120  564 d 2210  500 r
 2210  532 d 2300  500 r 2300  532 d  634  436 r  670  436 d  684  422 r
  682  420 d  684  418 d  686  420 d  684  422 d  708  460 r  702  458 d
  700  454 d  700  448 d  702  444 d  708  442 d  716  442 d  722  444 d
  724  448 d  724  454 d  722  458 d  716  460 d  708  460 d  704  458 d
  702  454 d  702  448 d  704  444 d  708  442 d  716  442 r  720  444 d
  722  448 d  722  454 d  720  458 d  716  460 d  708  442 r  702  440 d
  700  438 d  698  434 d  698  426 d  700  422 d  702  420 d  708  418 d
  716  418 d  722  420 d  724  422 d  726  426 d  726  434 d  724  438 d
  722  440 d  716  442 d  708  442 r  704  440 d  702  438 d  700  434 d
  700  426 d  702  422 d  704  420 d  708  418 d  716  418 r  720  420 d
  722  422 d  724  426 d  724  434 d  722  438 d  720  440 d  716  442 d
  994  436 r 1030  436 d 1044  422 r 1042  420 d 1044  418 d 1046  420 d
 1044  422 d 1076  456 r 1076  418 d 1078  460 r 1078  418 d 1078  460 r
 1056  430 d 1088  430 d 1070  418 r 1084  418 d 1398  460 r 1392  458 d
 1388  452 d 1386  442 d 1386  436 d 1388  426 d 1392  420 d 1398  418 d
 1402  418 d 1408  420 d 1412  426 d 1414  436 d 1414  442 d 1412  452 d
 1408  458 d 1402  460 d 1398  460 d 1394  458 d 1392  456 d 1390  452 d
 1388  442 d 1388  436 d 1390  426 d 1392  422 d 1394  420 d 1398  418 d
 1402  418 r 1406  420 d 1408  422 d 1410  426 d 1412  436 d 1412  442 d
 1410  452 d 1408  456 d 1406  458 d 1402  460 d 1740  422 r 1738  420 d
 1740  418 d 1742  420 d 1740  422 d 1772  456 r 1772  418 d 1774  460 r
 1774  418 d 1774  460 r 1752  430 d 1784  430 d 1766  418 r 1780  418 d
 2100  422 r 2098  420 d 2100  418 d 2102  420 d 2100  422 d 2124  460 r
 2118  458 d 2116  454 d 2116  448 d 2118  444 d 2124  442 d 2132  442 d
 2138  444 d 2140  448 d 2140  454 d 2138  458 d 2132  460 d 2124  460 d
 2120  458 d 2118  454 d 2118  448 d 2120  444 d 2124  442 d 2132  442 r
 2136  444 d 2138  448 d 2138  454 d 2136  458 d 2132  460 d 2124  442 r
 2118  440 d 2116  438 d 2114  434 d 2114  426 d 2116  422 d 2118  420 d
 2124  418 d 2132  418 d 2138  420 d 2140  422 d 2142  426 d 2142  434 d
 2140  438 d 2138  440 d 2132  442 d 2124  442 r 2120  440 d 2118  438 d
 2116  434 d 2116  426 d 2118  422 d 2120  420 d 2124  418 d 2132  418 r
 2136  420 d 2138  422 d 2140  426 d 2140  434 d 2138  438 d 2136  440 d
 2132  442 d  500 3000 r 2300 3000 d  500 3000 r  500 2968 d  590 3000 r
  590 2968 d  680 3000 r  680 2936 d  770 3000 r  770 2968 d  860 3000 r
  860 2968 d  950 3000 r  950 2968 d 1040 3000 r 1040 2936 d 1130 3000 r
 1130 2968 d 1220 3000 r 1220 2968 d 1310 3000 r 1310 2968 d 1400 3000 r
 1400 2936 d 1490 3000 r 1490 2968 d 1580 3000 r 1580 2968 d 1670 3000 r
 1670 2968 d 1760 3000 r 1760 2936 d 1850 3000 r 1850 2968 d 1940 3000 r
 1940 2968 d 2030 3000 r 2030 2968 d 2120 3000 r 2120 2936 d 2210 3000 r
 2210 2968 d 2300 3000 r 2300 2968 d  500  500 r  500 3000 d  500  594 r
  532  594 d  500  699 r  532  699 d  500  803 r  532  803 d  500  908 r
  564  908 d  500 1013 r  532 1013 d  500 1117 r  532 1117 d  500 1222 r
  532 1222 d  500 1326 r  564 1326 d  500 1431 r  532 1431 d  500 1536 r
  532 1536 d  500 1640 r  532 1640 d  500 1745 r  564 1745 d  500 1849 r
  532 1849 d  500 1954 r  532 1954 d  500 2059 r  532 2059 d  500 2163 r
  564 2163 d  500 2268 r  532 2268 d  500 2372 r  532 2372 d  500 2477 r
  532 2477 d  500 2582 r  564 2582 d  500 2686 r  532 2686 d  500 2791 r
  532 2791 d  500 2895 r  532 2895 d  500 3000 r  564 3000 d  452  928 r
  452  890 d  454  932 r  454  890 d  454  932 r  432  902 d  464  902 d
  446  890 r  460  890 d  444 1350 r  438 1348 d  436 1344 d  436 1338 d
  438 1334 d  444 1332 d  452 1332 d  458 1334 d  460 1338 d  460 1344 d
  458 1348 d  452 1350 d  444 1350 d  440 1348 d  438 1344 d  438 1338 d
  440 1334 d  444 1332 d  452 1332 r  456 1334 d  458 1338 d  458 1344 d
  456 1348 d  452 1350 d  444 1332 r  438 1330 d  436 1328 d  434 1324 d
  434 1316 d  436 1312 d  438 1310 d  444 1308 d  452 1308 d  458 1310 d
  460 1312 d  462 1316 d  462 1324 d  460 1328 d  458 1330 d  452 1332 d
  444 1332 r  440 1330 d  438 1328 d  436 1324 d  436 1316 d  438 1312 d
  440 1310 d  444 1308 d  452 1308 r  456 1310 d  458 1312 d  460 1316 d
  460 1324 d  458 1328 d  456 1330 d  452 1332 d  400 1761 r  404 1763 d
  410 1769 d  410 1727 d  408 1767 r  408 1727 d  400 1727 r  418 1727 d
  436 1761 r  438 1759 d  436 1757 d  434 1759 d  434 1761 d  436 1765 d
  438 1767 d  444 1769 d  452 1769 d  458 1767 d  460 1765 d  462 1761 d
  462 1757 d  460 1753 d  454 1749 d  444 1745 d  440 1743 d  436 1739 d
  434 1733 d  434 1727 d  452 1769 r  456 1767 d  458 1765 d  460 1761 d
  460 1757 d  458 1753 d  452 1749 d  444 1745 d  434 1731 r  436 1733 d
  440 1733 d  450 1729 d  456 1729 d  460 1731 d  462 1733 d  440 1733 r
  450 1727 d  458 1727 d  460 1729 d  462 1733 d  462 1737 d  400 2179 r
  404 2181 d  410 2187 d  410 2145 d  408 2185 r  408 2145 d  400 2145 r
  418 2145 d  458 2181 r  456 2179 d  458 2177 d  460 2179 d  460 2181 d
  458 2185 d  454 2187 d  448 2187 d  442 2185 d  438 2181 d  436 2177 d
  434 2169 d  434 2157 d  436 2151 d  440 2147 d  446 2145 d  450 2145 d
  456 2147 d  460 2151 d  462 2157 d  462 2159 d  460 2165 d  456 2169 d
  450 2171 d  448 2171 d  442 2169 d  438 2165 d  436 2159 d  448 2187 r
  444 2185 d  440 2181 d  438 2177 d  436 2169 d  436 2157 d  438 2151 d
  442 2147 d  446 2145 d  450 2145 r  454 2147 d  458 2151 d  460 2157 d
  460 2159 d  458 2165 d  454 2169 d  450 2171 d  396 2598 r  398 2596 d
  396 2594 d  394 2596 d  394 2598 d  396 2602 d  398 2604 d  404 2606 d
  412 2606 d  418 2604 d  420 2602 d  422 2598 d  422 2594 d  420 2590 d
  414 2586 d  404 2582 d  400 2580 d  396 2576 d  394 2570 d  394 2564 d
  412 2606 r  416 2604 d  418 2602 d  420 2598 d  420 2594 d  418 2590 d
  412 2586 d  404 2582 d  394 2568 r  396 2570 d  400 2570 d  410 2566 d
  416 2566 d  420 2568 d  422 2570 d  400 2570 r  410 2564 d  418 2564 d
  420 2566 d  422 2570 d  422 2574 d  446 2606 r  440 2604 d  436 2598 d
  434 2588 d  434 2582 d  436 2572 d  440 2566 d  446 2564 d  450 2564 d
  456 2566 d  460 2572 d  462 2582 d  462 2588 d  460 2598 d  456 2604 d
  450 2606 d  446 2606 d  442 2604 d  440 2602 d  438 2598 d  436 2588 d
  436 2582 d  438 2572 d  440 2568 d  442 2566 d  446 2564 d  450 2564 r
  454 2566 d  456 2568 d  458 2572 d  460 2582 d  460 2588 d  458 2598 d
  456 2602 d  454 2604 d  450 2606 d  396 3016 r  398 3014 d  396 3012 d
  394 3014 d  394 3016 d  396 3020 d  398 3022 d  404 3024 d  412 3024 d
  418 3022 d  420 3020 d  422 3016 d  422 3012 d  420 3008 d  414 3004 d
  404 3000 d  400 2998 d  396 2994 d  394 2988 d  394 2982 d  412 3024 r
  416 3022 d  418 3020 d  420 3016 d  420 3012 d  418 3008 d  412 3004 d
  404 3000 d  394 2986 r  396 2988 d  400 2988 d  410 2984 d  416 2984 d
  420 2986 d  422 2988 d  400 2988 r  410 2982 d  418 2982 d  420 2984 d
  422 2988 d  422 2992 d  452 3020 r  452 2982 d  454 3024 r  454 2982 d
  454 3024 r  432 2994 d  464 2994 d  446 2982 r  460 2982 d 2300  500 r
 2300 3000 d 2300  594 r 2268  594 d 2300  699 r 2268  699 d 2300  803 r
 2268  803 d 2300  908 r 2236  908 d 2300 1013 r 2268 1013 d 2300 1117 r
 2268 1117 d 2300 1222 r 2268 1222 d 2300 1326 r 2236 1326 d 2300 1431 r
 2268 1431 d 2300 1536 r 2268 1536 d 2300 1640 r 2268 1640 d 2300 1745 r
 2236 1745 d 2300 1849 r 2268 1849 d 2300 1954 r 2268 1954 d 2300 2059 r
 2268 2059 d 2300 2163 r 2236 2163 d 2300 2268 r 2268 2268 d 2300 2372 r
 2268 2372 d 2300 2477 r 2268 2477 d 2300 2582 r 2236 2582 d 2300 2686 r
 2268 2686 d 2300 2791 r 2268 2791 d 2300 2895 r 2268 2895 d 2300 3000 r
 2236 3000 d   1 lw  717 2903 r  686 2852 d  720 2903 r  689 2852 d  689 2903 r
  686 2891 d  686 2903 d  720 2903 d  686 2852 r  720 2852 d  720 2864 d
  717 2852 d  730 2852 r  733 2857 d  736 2858 d  739 2857 d  739 2854 d
  737 2848 d  734 2838 d  736 2858 r  737 2857 d  737 2854 d  736 2848 d
  733 2838 d  737 2848 r  740 2854 d  743 2857 d  746 2858 d  749 2858 d
  751 2857 d  753 2855 d  753 2851 d  751 2844 d  747 2828 d  749 2858 r
  751 2855 d  751 2851 d  750 2844 d  746 2828 d 1416 2903 r 1385 2852 d
 1419 2903 r 1388 2852 d 1388 2903 r 1385 2891 d 1385 2903 d 1419 2903 d
 1385 2852 r 1419 2852 d 1419 2864 d 1416 2852 d 1429 2858 r 1432 2858 d
 1435 2857 d 1436 2855 d 1446 2831 d 1449 2828 d 1432 2858 r 1435 2855 d
 1445 2831 d 1446 2829 d 1449 2828 d 1452 2828 d 1453 2858 r 1452 2855 d
 1448 2850 d 1433 2837 d 1429 2831 d 1427 2828 d 2203 2903 r 2172 2852 d
 2206 2903 r 2175 2852 d 2175 2903 r 2172 2891 d 2172 2903 d 2206 2903 d
 2172 2852 r 2206 2852 d 2206 2864 d 2203 2852 d 2236 2868 r 2227 2828 d
 2237 2868 r 2226 2828 d 2216 2852 r 2219 2857 d 2222 2858 d 2225 2857 d
 2225 2854 d 2223 2847 d 2223 2844 d 2225 2841 d 2227 2839 d 2232 2839 d
 2235 2841 d 2239 2844 d 2222 2858 r 2223 2857 d 2223 2854 d 2222 2847 d
 2222 2844 d 2223 2841 d 2225 2839 d 2227 2838 d 2230 2838 d 2235 2839 d
 2239 2844 d 2242 2848 d 2245 2854 d 2246 2858 d 2295 2879 r 2302 2874 d
 2295 2869 d 2287 2886 r 2299 2874 d 2287 2862 d 2259 2874 r 2299 2874 d
  522 2879 r  515 2874 d  522 2869 d  529 2886 r  517 2874 d  529 2862 d
  517 2874 r  558 2874 d  606 2903 r  575 2852 d  608 2903 r  577 2852 d
  577 2903 r  575 2891 d  575 2903 d  608 2903 d  575 2852 r  608 2852 d
  608 2864 d  606 2852 d  638 2868 r  630 2828 d  640 2868 r  628 2828 d
  618 2852 r  621 2857 d  624 2858 d  627 2857 d  627 2854 d  625 2847 d
  625 2844 d  627 2841 d  630 2839 d  634 2839 d  637 2841 d  641 2844 d
  624 2858 r  625 2857 d  625 2854 d  624 2847 d  624 2844 d  625 2841 d
  627 2839 d  630 2838 d  633 2838 d  637 2839 d  641 2844 d  644 2848 d
  647 2854 d  649 2858 d 1324  356 r 1327  358 d 1327  351 d 1324  356 d
 1317  358 d 1310  358 d 1303  356 d 1300  353 d 1300  351 d 1303  346 d
 1307  344 d 1319  339 d 1324  336 d 1327  334 d 1300  351 r 1303  348 d
 1307  346 d 1319  341 d 1324  339 d 1327  334 d 1327  329 d 1324  327 d
 1317  324 d 1310  324 d 1303  327 d 1300  332 d 1300  324 d 1303  327 d
 1346  375 r 1343  372 d 1346  370 d 1348  372 d 1346  375 d 1346  358 r
 1346  324 d 1348  358 r 1348  324 d 1339  358 r 1348  358 d 1339  324 r
 1355  324 d 1372  358 r 1372  324 d 1375  358 r 1375  324 d 1375  351 r
 1379  356 d 1387  358 d 1391  358 d 1399  356 d 1401  351 d 1401  324 d
 1391  358 r 1396  356 d 1399  351 d 1399  324 d 1365  358 r 1375  358 d
 1365  324 r 1382  324 d 1391  324 r 1408  324 d 1487  375 r 1480  372 d
 1475  368 d 1471  358 d 1468  351 d 1466  341 d 1463  327 d 1461  308 d
 1487  375 r 1483  372 d 1478  368 d 1473  358 d 1471  351 d 1468  341 d
 1466  327 d 1463  308 d 1487  375 r 1492  375 d 1497  372 d 1499  370 d
 1499  363 d 1497  358 d 1495  356 d 1487  353 d 1480  353 d 1492  375 r
 1497  370 d 1497  363 d 1495  358 d 1492  356 d 1487  353 d 1480  353 r
 1487  351 d 1492  348 d 1495  346 d 1497  341 d 1497  334 d 1495  329 d
 1492  327 d 1485  324 d 1478  324 d 1473  327 d 1471  329 d 1468  336 d
 1487  351 r 1492  346 d 1495  341 d 1495  334 d 1492  329 d 1490  327 d
 1485  324 d   2 lw  291 1234 r  342 1234 d  291 1236 r  334 1251 d  291 1234 r
  342 1251 d  291 1268 r  342 1251 d  291 1268 r  342 1268 d  291 1270 r
  342 1270 d  291 1227 r  291 1236 d  291 1268 r  291 1277 d  342 1227 r
  342 1241 d  342 1260 r  342 1277 d  326 1305 r  356 1286 d  326 1306 r
  356 1288 d  326 1288 r  333 1286 d  326 1286 d  326 1306 d  356 1286 r
  356 1306 d  349 1306 d  356 1305 d  350 1316 r  349 1318 d  346 1320 d
  365 1320 d  347 1319 r  365 1319 d  365 1316 r  365 1324 d  318 1389 r
  320 1386 d  322 1389 d  320 1391 d  318 1389 d  282 1451 r  358 1451 d
  282 1453 r  358 1453 d  282 1451 r  282 1468 d  358 1451 r  358 1468 d
  306 1501 r  310 1499 d  313 1494 d  313 1489 d  308 1487 d  303 1487 d
  298 1489 d  294 1494 d  291 1504 d  291 1523 d  298 1518 d  303 1516 d
  318 1511 d  330 1506 d  334 1504 d  339 1499 d  342 1492 d  342 1485 d
  339 1480 d  334 1480 d  332 1482 d  332 1485 d  334 1487 d  322 1497 r
  320 1499 d  318 1504 d  318 1516 d  315 1521 d  310 1523 d  325 1518 d
  349 1523 r  347 1528 d  346 1533 d  343 1537 d  340 1538 d  337 1537 d
  336 1534 d  336 1530 d  337 1525 d  342 1523 d  346 1521 d  350 1521 d
  353 1523 d  355 1524 d  356 1527 d  356 1530 d  355 1534 d  352 1537 d
  336 1530 r  337 1527 d  342 1524 d  346 1523 d  352 1523 d  355 1524 d
  336 1551 r  366 1543 d  336 1553 r  366 1544 d  342 1551 r  349 1550 d
  353 1550 d  356 1553 d  356 1556 d  355 1559 d  352 1561 d  346 1564 d
  336 1567 r  352 1563 d  355 1563 d  356 1566 d  355 1569 d  350 1571 d
  336 1569 r  352 1564 d  355 1564 d  356 1566 d  282 1622 r  358 1579 d
  301 1642 r  298 1646 d  291 1654 d  342 1654 d  294 1651 r  342 1651 d
  342 1642 r  342 1663 d  291 1697 r  294 1690 d  301 1685 d  313 1682 d
  320 1682 d  332 1685 d  339 1690 d  342 1697 d  342 1702 d  339 1709 d
  332 1714 d  320 1716 d  313 1716 d  301 1714 d  294 1709 d  291 1702 d
  291 1697 d  294 1692 d  296 1690 d  301 1687 d  313 1685 d  320 1685 d
  332 1687 d  337 1690 d  339 1692 d  342 1697 d  342 1702 r  339 1706 d
  337 1709 d  332 1711 d  320 1714 d  313 1714 d  301 1711 d  296 1709 d
  294 1706 d  291 1702 d  282 1727 r  282 1753 d  270 1763 r  272 1765 d
  273 1763 d  272 1762 d  270 1762 d  267 1763 d  266 1765 d  264 1769 d
  264 1775 d  266 1779 d  267 1781 d  270 1782 d  273 1782 d  276 1781 d
  279 1776 d  282 1769 d  283 1766 d  286 1763 d  290 1762 d  295 1762 d
  264 1775 r  266 1778 d  267 1779 d  270 1781 d  273 1781 d  276 1779 d
  279 1775 d  282 1769 d  292 1762 r  290 1763 d  290 1766 d  293 1774 d
  293 1778 d  292 1781 d  290 1782 d  290 1766 r  295 1774 d  295 1779 d
  293 1781 d  290 1782 d  287 1782 d  282 1808 r  358 1808 d  282 1811 r
  358 1811 d  282 1794 r  282 1811 d  358 1794 r  358 1811 d  282 1824 r
  282 1850 d  270 1863 r  269 1866 d  264 1871 d  295 1871 d  266 1869 r
  295 1869 d  295 1863 r  295 1876 d  259 1912 r  305 1886 d  270 1921 r
  272 1922 d  273 1921 d  272 1919 d  270 1919 d  267 1921 d  266 1922 d
  264 1927 d  264 1932 d  266 1937 d  267 1938 d  270 1940 d  273 1940 d
  276 1938 d  279 1934 d  282 1927 d  283 1924 d  286 1921 d  290 1919 d
  295 1919 d  264 1932 r  266 1935 d  267 1937 d  270 1938 d  273 1938 d
  276 1937 d  279 1932 d  282 1927 d  292 1919 r  290 1921 d  290 1924 d
  293 1931 d  293 1935 d  292 1938 d  290 1940 d  290 1924 r  295 1931 d
  295 1937 d  293 1938 d  290 1940 d  287 1940 d  291 2206 r  342 2206 d
  291 2208 r  342 2208 d  291 2191 r  303 2189 d  291 2189 d  291 2225 d
  303 2225 d  291 2222 d  342 2198 r  342 2215 d  322 2239 r  322 2268 d
  318 2268 d  313 2266 d  310 2263 d  308 2258 d  308 2251 d  310 2244 d
  315 2239 d  322 2237 d  327 2237 d  334 2239 d  339 2244 d  342 2251 d
  342 2256 d  339 2263 d  334 2268 d  322 2266 r  315 2266 d  310 2263 d
  308 2251 r  310 2246 d  315 2242 d  322 2239 d  327 2239 d  334 2242 d
  339 2246 d  342 2251 d  291 2285 r  342 2302 d  291 2287 r  334 2302 d
  291 2318 r  342 2302 d  291 2278 r  291 2294 d  291 2311 r  291 2326 d
   6 lw  500  594 r  509  919 d  518  973 d  527  998 d  536 1011 d  545 1017 d
  554 1016 d  563 1011 d  572 1002 d  581  991 d  590  978 d  599  962 d
  608  943 d  617  922 d  626  898 d  635  871 d  644  840 d  653  805 d
  662  764 d  671  714 d  680  647 d  689  528 d  692  500 d  694  500 r
  698  658 d  707  725 d  716  778 d  725  823 d  734  864 d  743  900 d
  752  934 d  761  967 d  770  996 d  779 1025 d  788 1052 d  797 1079 d
  806 1104 d  815 1128 d  824 1152 d  833 1174 d  842 1197 d  851 1219 d
  860 1239 d  869 1259 d  878 1279 d  887 1298 d  896 1317 d  905 1335 d
  914 1353 d  923 1371 d  932 1388 d  941 1405 d  950 1421 d  959 1437 d
  968 1453 d  977 1468 d  986 1484 d  995 1498 d 1004 1513 d 1013 1528 d
 1022 1541 d 1031 1555 d 1040 1568 d 1049 1582 d 1058 1595 d 1067 1607 d
 1076 1620 d 1085 1632 d 1094 1644 d 1103 1656 d 1112 1667 d 1121 1679 d
 1130 1690 d 1139 1701 d 1148 1712 d 1157 1722 d 1166 1732 d 1175 1742 d
 1184 1752 d 1193 1762 d 1202 1772 d 1211 1781 d 1220 1790 d 1229 1799 d
 1238 1808 d 1247 1817 d 1256 1826 d 1265 1835 d 1274 1841 d 1283 1849 d
 1292 1857 d 1301 1865 d 1310 1872 d 1319 1879 d 1328 1886 d 1337 1893 d
 1346 1900 d 1355 1906 d 1364 1913 d 1373 1919 d 1382 1925 d 1391 1931 d
 1400 1936 d 1409 1942 d 1418 1947 d 1427 1952 d 1436 1957 d 1445 1962 d
 1454 1967 d 1463 1971 d 1472 1976 d 1481 1980 d 1490 1984 d 1499 1988 d
 1508 1993 d 1517 1997 d 1526 2000 d 1535 2002 d 1544 2005 d 1553 2008 d
 1562 2010 d 1571 2013 d 1580 2015 d 1589 2017 d 1598 2019 d 1607 2021 d
 1616 2023 d 1625 2024 d 1634 2025 d 1643 2026 d 1652 2027 d 1661 2028 d
 1670 2029 d 1679 2029 d 1688 2029 d 1697 2029 d 1706 2029 d 1715 2028 d
 1724 2028 d 1733 2027 d 1742 2026 d 1751 2025 d 1760 2023 d 1769 2022 d
 1778 2020 d 1787 2018 d 1796 2016 d 1805 2013 d 1814 2010 d 1823 2007 d
 1832 2004 d 1841 2001 d 1850 1997 d 1859 1994 d 1868 1989 d 1877 1984 d
 1886 1979 d 1895 1974 d 1904 1969 d 1913 1963 d 1922 1958 d 1931 1951 d
 1940 1946 d 1949 1938 d 1958 1931 d 1967 1923 d 1976 1916 d 1985 1907 d
 1994 1899 d 2003 1890 d 2012 1880 d 2021 1870 d 2030 1860 d 2039 1849 d
 2048 1839 d 2057 1828 d 2066 1814 d 2075 1801 d 2084 1788 d 2093 1774 d
 2102 1759 d 2111 1743 d 2120 1727 d 2129 1710 d 2138 1692 d 2147 1673 d
 2156 1654 d 2165 1633 d 2174 1611 d 2183 1587 d 2192 1563 d 2201 1536 d
 2210 1506 d 2219 1475 d 2228 1441 d 2237 1404 d 2246 1363 d 2255 1317 d
 2264 1264 d 2273 1201 d 2282 1122 d 2291 1007 d 2300  594 d
e
%%Trailer
EndPSPlot



