%Paper: 
%From: CAMPOSC@vm.ci.uv.es
%Date: Tue, 19 Jul 94 15:53:15 SET

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%Some definitions%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand {\ignore}[1]{}
\newcommand{\nota}[1]{\makebox[0pt]{\,\,\,\,\,/}#1}
\newcommand{\notp}[1]{\makebox[0pt]{\,\,\,\,/}#1}
%\newcommand{\braket}[1]{\mbox{$<$}#1\mbox{$>$}}
\newcommand{\Frac}[2]{\frac{\displaystyle #1}{\displaystyle #2}}
\renewcommand{\arraystretch}{1.5}
\newcommand{\noi}{\noindent}
\newcommand{\bc}{\begin{center}}
\newcommand{\ec}{\end{center}}
\newcommand{\eps}{\epsilon}
\newcommand{\epm}{e^+e^-}
%
\newlength\lah\setlength{\lah}{2\baselineskip}
\newlength\sah\setlength{\sah}{1\baselineskip}
\def\longhookarrow#1{%
\rule{0.1em}{0pt}\rule[-\lah]{0pt}{\lah}\rlap{\raisebox{-\lah}[0pt][0pt]{%
\rule{0.4pt}{\lah}\rule{1.0em}{0.3pt}\raisebox{-0.5ex}{\hspace{-0.1em}
$\rightarrow$}\hbox{#1}}}}
\def\shorthookarrow#1{%
\rule{0.1em}{0pt}\rule[-\sah]{0pt}{\sah}\rlap{\raisebox{-\sah}[0pt][0pt]{%
\rule{0.4pt}{\sah}\rule{0.2em}{0.3pt}\raisebox{-0.5ex}{\hspace{-0.1em}
$\rightarrow$}\hbox{#1}}}}
%
\def\ifmath#1{\relax\ifmmode #1\else $#1$\fi}
% Here are some handy fraction macros
%
\def\half{\ifmath{{\textstyle{1 \over 2}}}}
\def\quarter{\ifmath{{\textstyle{1 \over 4}}}}
\def\3quarter{{\textstyle{3 \over 4}}}
\def\third{\ifmath{{\textstyle{1 \over 3}}}}
\def\twothirds{{\textstyle{2 \over 3}}}
\def\fourth{\ifmath{{\textstyle{1\over 4}}}}
\def\sqrthalf{\ifmath{{\textstyle{1\over\sqrt2}}}}
\def\halfsqrthalf{\ifmath{{\textstyle{1\over2\sqrt2}}}}
\def\nl{\nextline}
\def\cl{\centerline}
\def\vs{\vskip}
\def\hs{\hskip}
\def\ss{\smallskip}
\def\ms{\medskip}
\def\bs{\bigskip}
\def\br{\break}
\def\ra{\rightarrow}
\def\Ra{\Rightarrow}
\def\us{\undertext}
\def\HB{\hfill\break}
\overfullrule 0pt
%\singlespace
\def\lf{\leaders\hbox to 1em{\hss.\hss}\hfill}
\def\ZP{$Z^\prime$ }
\def\21{$SU(2) \ot U(1)$}
%---------------------------This defines fermion and their masses
\def\ne{\hbox{$\nu_e$ }}
\def\nm{\hbox{$\nu_\mu$ }}
\def\nt{\hbox{$\nu_\tau$ }}
\def\ns{\hbox{$\nu_{sterile}$ }}
\def\nx{\hbox{$\nu_x$ }}
\def\Nt{\hbox{$N_\tau$ }}
\def\ns{\hbox{$\nu_S$ }}
\def\nr{\hbox{$\nu_R$ }}
\def\O{\hbox{$\cal O$ }}
\def\L{\hbox{$\cal L$ }}
%--------------------------
\def\mne{\hbox{$m_{\nu_e}$ }}
\def\mnm{\hbox{$m_{\nu_\mu}$ }}
\def\mnt{\hbox{$m_{\nu_\tau}$ }}
\def\mq{\hbox{$m_{q}$}}
\def\ml{\hbox{$m_{l}$}}
\def\mup{\hbox{$m_{u}$}}
\def\md{\hbox{$m_{d}$}}
%---------------------------------------------------------------------
\def\pd{Physics Department\\}
\def\dop{Department of Physics\\}
%
%%%%%%%%%This defines et al., i.e., e.g., cf., etc.
\def\ie{\hbox{\it i.e., }}        \def\etc{\hbox{\it etc. }}
\def\eg{\hbox{\it e.g., }}        \def\cf{\hbox{\it cf.}}
\def\etal{\hbox{\it et al., }}
%%%%%%%%This defines letter endings, etc.
%---------------------------------------
\def\H{\hbox{Higgs }}
\def\nhl{\hbox{neutral heavy lepton }}
\def\NHL{\hbox{Neutral Heavy Lepton }}
\def\nhls{\hbox{neutral heavy leptons }}
\def\NHLs{\hbox{Neutral Heavy Leptons }}
%---------------------------------
\def\Hp{\hbox{Higgs. }}
\def\nhlp{\hbox{neutral heavy lepton. }}
\def\NHLp{\hbox{Neutral Heavy Lepton. }}
\def\nhlsp{\hbox{neutral heavy leptons. }}
\def\NHLsp{\hbox{Neutral Heavy Leptons. }}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def\nv{\hbox{non-vanishing }}
\def\nvp{\hbox{non-vanishing. }}
\def\nvc{\hbox{non-vanishing, }}
\def\fd{\hbox{field }}
\def\Fd{\hbox{Field }}
\def\fds{\hbox{fields }}
\def\eig{\hbox{eigenstate }}
\def\Def{\hbox{Definition }}
\def\defd{\hbox{defined }}
\def\wf{\hbox{wave-function }}
\def\wfs{\hbox{wave-functions }}
\def\wfp{\hbox{wave-function. }}
\def\wfc{\hbox{wave-function, }}
\def\meig{\hbox{mass-eigenstate }}
\def\meigs{\hbox{mass-eigenstates }}
\def\meigc{\hbox{mass-eigenstate, }}
\def\meigp{\hbox{mass-eigenstate. }}
\def\Sst{\hbox{Superstring }}
\def\sst{\hbox{superstring }}
\def\susym{\hbox{supersymmetry }}
\def\Susym{\hbox{Supersymmetry }}
\def\susymp{\hbox{supersymmetry. }}
\def\Susymp{\hbox{Supersymmetry. }}
\def\susymc{\hbox{supersymmetry, }}
\def\Susymc{\hbox{Supersymmetry, }}
\def\susy{\hbox{supersymmetric }}
\def\Susy{\hbox{Supersymmetric }}
\def\susyp{\hbox{supersymmetric. }}
\def\Susyp{\hbox{Supersymmetric. }}
\def\susyc{\hbox{supersymmetric, }}
\def\Susyc{\hbox{Supersymmetric, }}
\def\sym{\hbox{symmetry }}
\def\sym{\hbox{symmetry }}
\def\symp{\hbox{symmetry. }}
\def\symp{\hbox{symmetry. }}
\def\symc{\hbox{symmetry, }}
\def\symc{\hbox{symmetry, }}
\def\sy{\hbox{symmetric }}
\def\sy{\hbox{symmetric }}
\def\syp{\hbox{symmetric. }}
\def\syp{\hbox{symmetric. }}
\def\syc{\hbox{symmetric, }}
\def\syc{\hbox{symmetric, }}
%los left
\def\lh{\hbox{left-handed }}
\def\Lh{\hbox{Left-handed }}
\def\LH{\hbox{Left-Handed }}
\def\lhc{\hbox{left-handed, }}
\def\lhp{\hbox{left-handed. }}
%los right
\def\rh{\hbox{right-handed }}
\def\Rh{\hbox{Right-handed }}
\def\RH{\hbox{Right-Handed }}
\def\Rhc{\hbox{Right-handed, }}
\def\rhp{\hbox{right-handed. }}
\def\sol{\hbox{solution }}
\def\sols{\hbox{solutions }}
\def\solc{\hbox{solution. }}
\def\solp{\hbox{solution. }}
\def\rep{\hbox{representation }}
\def\reps{\hbox{representations }}
\def\repc{\hbox{representation, }}
\def\repp{\hbox{representation. }}
\def\ew{\hbox{electro-weak }}
\def\Ew{\hbox{Electro-weak }}
\def\Em{\hbox{Electromagnetic }}
\def\me{\hbox{matrix element }}
\def\Me{\hbox{Matrix element }}
\def\nad{\hbox{non-adiabatic }}
\def\ad{\hbox{adiabatic }}
\def\Nad{\hbox{Non-adiabatic }}
\def\Ad{\hbox{Adiabatic }}
\def\lfvg{\hbox{lepton flavour violating }}
\def\lfv{\hbox{lepton flavour violation }}
\def\LFV{\hbox{Lepton Flavour Violation }}
\def\IVB{\hbox{Intermediate Vector Boson }}
\def\IVBs{\hbox{Intermediate Vector Bosons }}
\def\ivb{\hbox{intermediate vector boson }}
\def\IGB{\hbox{Intermediate Gauge Boson }}
\def\igb{\hbox{intermediate gauge boson }}
\def\igbs{\hbox{intermediate gauge bosons }}
\def\TLNV{\hbox{Total Lepton Number Violation }}
\def\tlnv{\hbox{total lepton number violation }}
\def\tlnvg{\hbox{total lepton number violating }}
\def\TLN{\hbox{Total Lepton Number }}
\def\tln{\hbox{total lepton number }}
\def\df{\hbox{degrees of freedom }}
\def\Df{\hbox{Degrees of freedom }}
\def\DF{\hbox{Degrees of Freedom }}
\def\gau{\hbox{gauge }}
\def\he{\hbox{high energy }}
\def\HE{\hbox{High Energy }}
\def\sn{\hbox{supernova }}
\def\Sn{\hbox{Supernova }}
\def\Gau{\hbox{Gauge }}
\def\Conf{\hbox{Conference }}
\def\inv{\hbox{invariant }}
\def\stan{\hbox{standard }}
\def\ST{\hbox{Standard Theory }}
\def\st{\hbox{standard theory }}
\def\Lor{\hbox{Lorentz }}
\def\transf{\hbox{transformation }}
\def\expt{\hbox{experiment }}
\def\expts{\hbox{experiments }}
\def\lab{\hbox{laboratory }}
\def\br{\hbox{branching ratio }}
\def\BR{\hbox{Branching Ratio }}
\def\brs{\hbox{branching ratios }}
\def\BRs{\hbox{Branching Ratios }}
\def\Lag{\hbox{Lagrangian }}
\def\wi{\hbox{weak interaction }}
\def\wis{\hbox{weak interactions }}
\def\Dir{\hbox{Dirac }}
\def\Maj{\hbox{Majorana }}
\def\mo{\hbox{model }}
\def\Mo{\hbox{Model }}
\def\cosm{\hbox{cosmology }}
\def\astro{\hbox{astrophysics }}
\def\Cosm{\hbox{Cosmology }}
\def\Astro{\hbox{Astrophysics }}
\def\mos{\hbox{models }}
\def\Mos{\hbox{Models }}
\def\sm{\hbox{standard model }}
\def\SM{\hbox{Standard Model }}
\def\CC{\hbox{Charged Current }}
\def\cc{\hbox{charged current }}
\def\NC{\hbox{Neutral Current }}
\def\neu{\hbox{neutrino }}
\def\sa{\hbox{such as }}
\def\neuless{\hbox{neutrinoless }}
\def\Neuless{\hbox{Neutrinoless }}
\def\Neus{\hbox{Neutrinos }}
\def\neus{\hbox{neutrinos }}
\def\Neu{\hbox{Neutrino }}
\def\phys{\hbox{physics }}
\def\Phys{\hbox{Physics }}
\def\Neuphys{\hbox{Neutrino physics }}
\def\neuphys{\hbox{neutrino physics }}
\def\NeuPhys{\hbox{Neutrino Physics }}
\def\Neuphysp{\hbox{Neutrino physics. }}
\def\Neuphysc{\hbox{Neutrino physics, }}
\def\osc{\hbox{oscillation }}
\def\oscs{\hbox{oscillations }}
\def\Osc{\hbox{Oscillation }}
\def\Oscs{\hbox{Oscillations }}
%
%----------------------------estos son con ,
%
\def\mec{\hbox{matrix element, }}
\def\nadc{\hbox{non-adiabatic, }}
\def\adc{\hbox{adiabatic, }}
\def\Nadc{\hbox{Non-adiabatic, }}
\def\Adc{\hbox{Adiabatic, }}
\def\lfvgc{\hbox{lepton flavour violating, }}
\def\lfvc{\hbox{lepton flavour violation, }}
\def\LFVc{\hbox{Lepton Flavour Violation, }}
\def\IVBc{\hbox{Intermediate Vector Boson, }}
\def\IVBsc{\hbox{Intermediate Vector Bosons, }}
\def\ivbc{\hbox{intermediate vector boson, }}
\def\IGBc{\hbox{Intermediate Gauge Boson, }}
\def\igbc{\hbox{intermediate gauge boson, }}
\def\igbsc{\hbox{intermediate gauge bosons, }}
\def\brc{\hbox{branching ratio, }}
\def\BRc{\hbox{Branching Ratio, }}
\def\brsc{\hbox{branching ratios, }}
\def\BRsc{\hbox{Branching Ratios, }}
\def\TLNVc{\hbox{Total Lepton Number Violation, }}
\def\tlnvc{\hbox{total lepton number violation, }}
\def\tlnvgc{\hbox{total lepton number violating, }}
\def\TLNc{\hbox{Total Lepton Number, }}
\def\tlnc{\hbox{total lepton number, }}
\def\dfc{\hbox{degrees of freedom, }}
\def\Dfc{\hbox{Degrees of freedom, }}
\def\DFc{\hbox{Degrees of Freedom, }}
\def\gauc{\hbox{gauge, }}
\def\hec{\hbox{high energy, }}
\def\HEc{\hbox{High Energy, }}
\def\snc{\hbox{supernova, }}
\def\Snc{\hbox{Supernova, }}
\def\Gauc{\hbox{Gauge, }}
\def\Confc{\hbox{Conference, }}
\def\invc{\hbox{invariant, }}
\def\STc{\hbox{Standard Theory, }}
\def\stc{\hbox{standard theory, }}
\def\stanc{\hbox{standard, }}
\def\Stanc{\hbox{Standard, }}
\def\Lorc{\hbox{Lorentz, }}
\def\transfc{\hbox{transformation, }}
\def\exptc{\hbox{experiment, }}
\def\exptsc{\hbox{experiments, }}
\def\labc{\hbox{laboratory, }}
\def\Lagc{\hbox{Lagrangian, }}
\def\wic{\hbox{weak interaction, }}
\def\wisc{\hbox{weak interactions, }}
\def\Dirc{\hbox{Dirac, }}
\def\Majc{\hbox{Majorana, }}
\def\moc{\hbox{model, }}
\def\Moc{\hbox{Model, }}
\def\cosmc{\hbox{cosmology, }}
\def\astroc{\hbox{astrophysics, }}
\def\Cosmc{\hbox{Cosmology, }}
\def\Astroc{\hbox{Astrophysics, }}
\def\mosc{\hbox{models, }}
\def\Mosc{\hbox{Models, }}
\def\smc{\hbox{standard model, }}
\def\SMc{\hbox{Standard Model, }}
\def\CCc{\hbox{Charged Current, }}
\def\NCc{\hbox{Neutral Current, }}
\def\ccc{\hbox{charged current, }}
\def\nc{\hbox{neutral current }}
\def\ncc{\hbox{neutral current, }}
\def\neuc{\hbox{neutrino, }}
\def\physc{\hbox{physics, }}
\def\Physc{\hbox{Physics, }}
\def\oscc{\hbox{oscillation, }}
\def\oscsc{\hbox{oscillations, }}
\def\Oscc{\hbox{Oscillation, }}
\def\Oscsc{\hbox{Oscillations, }}
\def\neusc{\hbox{neutrinos, }}
\def\Neuc{\hbox{Neutrino, }}
\def\Neusc{\hbox{Neutrinos, }}
%
%-------------------------esos son con .
%
\def\mep{\hbox{matrix element. }}
\def\nadp{\hbox{non-adiabatic. }}
\def\adp{\hbox{adiabatic. }}
\def\Nadp{\hbox{Non-adiabatic. }}
\def\Adp{\hbox{Adiabatic. }}
\def\brp{\hbox{branching ratio. }}
\def\BRp{\hbox{Branching Ratio. }}
\def\brsp{\hbox{branching ratios. }}
\def\BRsp{\hbox{Branching Ratios. }}
\def\lfvgp{\hbox{lepton flavour violating. }}
\def\lfvp{\hbox{lepton flavour violation. }}
\def\LFVp{\hbox{Lepton Flavour Violation. }}
\def\IVBp{\hbox{Intermediate Vector Boson. }}
\def\IVBsp{\hbox{Intermediate Vector Bosons. }}
\def\ivbp{\hbox{intermediate vector boson. }}
\def\IGBp{\hbox{Intermediate Gauge Boson. }}
\def\igbp{\hbox{intermediate gauge boson. }}
\def\igbsp{\hbox{intermediate gauge bosons. }}
\def\TLNVp{\hbox{Total Lepton Number Violation. }}
\def\tlnvp{\hbox{total lepton number violation. }}
\def\tlnvgp{\hbox{total lepton number violating. }}
\def\TLNp{\hbox{Total Lepton Number. }}
\def\tlnp{\hbox{total lepton number. }}
\def\dfp{\hbox{degrees of freedom. }}
\def\Dfp{\hbox{Degrees of freedom. }}
\def\DFp{\hbox{Degrees of Freedom. }}
\def\gaup{\hbox{gauge. }}
\def\hep{\hbox{high energy. }}
\def\HEp{\hbox{High Energy. }}
\def\snp{\hbox{supernova. }}
\def\Snp{\hbox{Supernova. }}
\def\Gaup{\hbox{Gauge. }}
\def\Confp{\hbox{Conference. }}
\def\invp{\hbox{invariant. }}
\def\stanp{\hbox{standard. }}
\def\Stanp{\hbox{Standard. }}
\def\Lorp{\hbox{Lorentz. }}
\def\transfp{\hbox{transformation. }}
\def\exptp{\hbox{experiment. }}
\def\exptsp{\hbox{experiments. }}
\def\labp{\hbox{laboratory. }}
\def\Lagp{\hbox{Lagrangian. }}
\def\wip{\hbox{weak interaction. }}
\def\wisp{\hbox{weak interactions. }}
\def\Dirp{\hbox{Dirac. }}
\def\Majp{\hbox{Majorana. }}
\def\mop{\hbox{model. }}
\def\Mop{\hbox{Model. }}
\def\cosmp{\hbox{cosmology. }}
\def\astrop{\hbox{astrophysics. }}
\def\Cosmp{\hbox{Cosmology. }}
\def\Astrop{\hbox{Astrophysics. }}
\def\mosp{\hbox{models. }}
\def\Mosp{\hbox{Models. }}
\def\smp{\hbox{standard model. }}
\def\SMp{\hbox{Standard Model. }}
\def\CCp{\hbox{Charged Current. }}
\def\NCp{\hbox{Neutral Current. }}
\def\NCsp{\hbox{Neutral Currents. }}
\def\ncsp{\hbox{neutral currents. }}
\def\ncs{\hbox{neutral currents }}
\def\neup{\hbox{neutrino. }}
\def\physp{\hbox{physics. }}
\def\Physp{\hbox{Physics. }}
\def\oscp{\hbox{oscillation. }}
\def\oscsp{\hbox{oscillations. }}
\def\Oscp{\hbox{Oscillation. }}
\def\Oscsp{\hbox{Oscillations. }}
\def\neusp{\hbox{neutrinos. }}
\def\Neup{\hbox{Neutrino. }}
\def\Neusp{\hbox{Neutrinos. }}
%-------------------------------------
\def\CPv{\hbox{CP violation }}
\def\CPvp{\hbox{CP violation. }}
\def\CPvc{\hbox{CP violation, }}
%
\def\dash{\hbox{---}}
%       common physics symbols
\def\c{\mathop{\cos \theta }}
\def\s{\mathop{\sin \theta }}
\def\cok{\mathop{\rm cok}}
\def\tr{\mathop{\rm tr}}
\def\Tr{\mathop{\rm Tr}}
\def\Im{\mathop{\rm Im}}
\def\Re{\mathop{\rm Re}}
\def\bR{\mathop{\bf R}}
\def\bC{\mathop{\bf C}}
\def\eq#1{{eq. (\ref{#1})}}
\def\Eq#1{{Eq. (\ref{#1})}}
\def\Eqs#1#2{{Eqs. (\ref{#1}) and (\ref{#2})}}
\def\Eqs#1#2#3{{Eqs. (\ref{#1}), (\ref{#2}) and (\ref{#3})}}
\def\Eqs#1#2#3#4{{Eqs. (\ref{#1}), (\ref{#2}), (\ref{#3}) and (\ref{#4})}}
\def\eqs#1#2{{eqs. (\ref{#1}) and (\ref{#2})}}
\def\eqs#1#2#3{{eqs. (\ref{#1}), (\ref{#2}) and (\ref{#3})}}
\def\eqs#1#2#3#4{{eqs. (\ref{#1}), (\ref{#2}), (\ref{#3}) and (\ref{#4})}}
\def\fig#1{{Fig. (\ref{#1})}}
\def\lie{\hbox{\it \$}} % fancy L for the Lie derivative
\def\partder#1#2{{\partial #1\over\partial #2}}
\def\secder#1#2#3{{\partial^2 #1\over\partial #2 \partial #3}}
\def\bra#1{\left\langle #1\right|}
\def\ket#1{\left| #1\right\rangle}
\def\VEV#1{\left\langle #1\right\rangle}
\let\vev\VEV
\def\gdot#1{\rlap{$#1$}/}
\def\abs#1{\left| #1\right|}
\def\pri#1{#1^\prime}
\def\ltap{\raisebox{-.4ex}{\rlap{$\sim$}} \raisebox{.4ex}{$<$}}
\def\gtap{\raisebox{-.4ex}{\rlap{$\sim$}} \raisebox{.4ex}{$>$}}
\def\lsim{\raise0.3ex\hbox{$\;<$\kern-0.75em\raise-1.1ex\hbox{$\sim\;$}}}
\def\gsim{\raise0.3ex\hbox{$\;>$\kern-0.75em\raise-1.1ex\hbox{$\sim\;$}}}
% \contract is a differential geometry contraction sign _|
\def\contract{\makebox[1.2em][c]{
        \mbox{\rule{.6em}{.01truein}\rule{.01truein}{.6em}}}}
\def\half{{1\over 2}}
\def\bel{\begin{letter}}
\def\eel{\end{letter}}
\def\beq{\begin{equation}}
\def\eeq{\end{equation}}
\def\bef{\begin{figure}}
\def\eef{\end{figure}}
\def\bet{\begin{table}}
\def\eet{\end{table}}
\def\bea{\begin{eqnarray}}
\def\ba{\begin{array}}
\def\ea{\end{array}}
\def\bi{\begin{itemize}}
\def\ei{\end{itemize}}
\def\ben{\begin{enumerate}}
\def\een{\end{enumerate}}
\def\ra{\rightarrow}
\def\ot{\otimes}
% double-headed superior arrow added 9.2.86
\def\lrover#1{
        \raisebox{1.3ex}{\rlap{$\leftrightarrow$}} \raisebox{ 0ex}{$#1$}}
%
% commutator added 11.14.86
\def\com#1#2{
        \left[#1, #2\right]}
%
\def\eea{\end{eqnarray}}
%these written by orlando alvarez
% ************************************************************
%       The following macros were written by Chris Quigg.
%       They create bent arrows and can be used to write
%       decays such as pi --> mu + nu
%                              --> e nu nubar
%
\def\bentarrow{\:\raisebox{1.3ex}{\rlap{$\vert$}}\!\rightarrow}
\def\longbent{\:\raisebox{3.5ex}{\rlap{$\vert$}}\raisebox{1.3ex}%
        {\rlap{$\vert$}}\!\rightarrow}
\def\onedk#1#2{
        \begin{equation}
        \begin{array}{l}
         #1 \\
         \bentarrow #2
        \end{array}
        \end{equation}
                }
\def\dk#1#2#3{
        \begin{equation}
        \begin{array}{r c l}
        #1 & \rightarrow & #2 \\
         & & \bentarrow #3
        \end{array}
        \end{equation}
                }
\def\dkp#1#2#3#4{
        \begin{equation}
        \begin{array}{r c l}
        #1 & \rightarrow & #2#3 \\
         & & \phantom{\; #2}\bentarrow #4
        \end{array}
        \end{equation}
                }
\def\bothdk#1#2#3#4#5{
        \begin{equation}
        \begin{array}{r c l}
        #1 & \rightarrow & #2#3 \\
         & & \:\raisebox{1.3ex}{\rlap{$\vert$}}\raisebox{-0.5ex}{$\vert$}%
        \phantom{#2}\!\bentarrow #4 \\
         & & \bentarrow #5
        \end{array}
        \end{equation}
                }
%
%
%abbreviated journal names
%-----------------------------------------------
%estos que siguen no tienen fecha
%------------------------------------------------------------
\def\ap#1#2#3{           {\it Ann. Phys. (NY) }{\bf #1} (19#2) #3}
\def\arnps#1#2#3{        {\it Ann. Rev. Nucl. Part. Sci. }{\bf #1} (19#2) #3}
\def\cnpp#1#2#3{        {\it Comm. Nucl. Part. Phys. }{\bf #1} (19#2) #3}
\def\apj#1#2#3{          {\it Astrophys. J. }{\bf #1} (19#2) #3}
\def\asr#1#2#3{          {\it Astrophys. Space Rev. }{\bf #1} (19#2) #3}

\def\apjl#1#2#3{         {\it Astrophys. J. Lett. }{\bf #1} (19#2) #3}
\def\ass#1#2#3{          {\it Astrophys. Space Sci. }{\bf #1} (19#2) #3}
\def\ib#1#2#3{           {\it ibid. }{\bf #1} (19#2) #3}
\def\nat#1#2#3{          {\it Nature }{\bf #1} (19#2) #3}
\def\nps#1#2#3{          {\it Nucl. Phys. B (Proc. Suppl.) }
                         {\bf #1} (19#2) #3}
\def\np#1#2#3{           {\it Nucl. Phys. }{\bf #1} (19#2) #3}
\def\pl#1#2#3{           {\it Phys. Lett. }{\bf #1} (19#2) #3}
\def\pr#1#2#3{           {\it Phys. Rev. }{\bf #1} (19#2) #3}
\def\prep#1#2#3{         {\it Phys. Rep. }{\bf #1} (19#2) #3}
\def\prl#1#2#3{          {\it Phys. Rev. Lett. }{\bf #1} (19#2) #3}
\def\pw#1#2#3{          {\it Particle World }{\bf #1} (19#2) #3}
\def\ptp#1#2#3{          {\it Prog. Theor. Phys. }{\bf #1} (19#2) #3}
\def\jppnp#1#2#3{         {\it J. Prog. Part. Nucl. Phys. }{\bf #1} (19#2) #3}

\def\rpp#1#2#3{         {\it Rep. on Prog. in Phys. }{\bf #1} (19#2) #3}
\def\ptps#1#2#3{         {\it Prog. Theor. Phys. Suppl. }{\bf #1} (19#2) #3}
\def\rmp#1#2#3{          {\it Rev. Mod. Phys. }{\bf #1} (19#2) #3}
\def\zp#1#2#3{           {\it Zeit. fur Physik }{\bf #1} (19#2) #3}
\def\fp#1#2#3{           {\it Fortschr. Phys. }{\bf #1} (19#2) #3}
\def\Zp#1#2#3{           {\it Z. Physik }{\bf #1} (19#2) #3}
\def\Sci#1#2#3{          {\it Science }{\bf #1} (19#2) #3}
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\begin{document}
\begin{titlepage}
\begin{center}
\hfill{FTUV/94-36, IFIC/94-31}\\
\hfill TIFR/TH/94--25
\vskip 0.3cm
{\Large \bf LIMITS ON ASSOCIATED PRODUCTION OF VISIBLY
AND INVISIBLY DECAYING HIGGS BOSONS FROM Z DECAYS}\\
\vskip 1.0cm
{\large \bf F. DE CAMPOS}
\footnote{E-mail CAMPOSC at vm.ci.uv.es or 16444::CAMPOSC},
{\large \bf M. A. GARCIA-JARE\~NO}
\footnote{E-mail GARCIAMA at vm.ci.uv.es or 16444::GARCIAMA}\\
\vskip 0.5cm
{\large \bf ANJAN S. JOSHIPURA}
\footnote{Permanent address: Theory Group,
 Physical Research Lab., Ahmedabad, India},
{\large \bf J. ROSIEK}
\footnote{E-mail ROSIEK at vm.ci.uv.es or 16444::ROSIEK},
{\large \bf J. W. F. VALLE}
\footnote{E-mail VALLE at vm.ci.uv.es or 16444::VALLE}\\
\vskip .5cm
{\it Instituto de F\'{\i}sica Corpuscular - C.S.I.C.\\
Departament de F\'{\i}sica Te\`orica, Universitat de Val\`encia\\
46100 Burjassot, Val\`encia, SPAIN}\\
\vskip 0.5cm
{\large \bf D. P. ROY}
\footnote{E-mail dproy@theory.tifr.res.in }\\
\vskip .5cm
{\it Theoretical Physics Group,
Tata Institute of Fundamental Research, \\
Homi Bhabha Road, Bombay 400 005, INDIA}
\end{center}

\begin{abstract}
\baselineskip=11pt
{
Many extensions of the standard electroweak model
Higgs sector suggest that the main Higgs decay
channel is "invisible", for example, $h \to J J$
where $J$ denotes the majoron, a weakly interacting
pseudoscalar Goldstone boson associated to the
spontaneous violation of lepton number.
%
In many of these models the Higgs boson may
also be produced in association to a massive
pseudoscalar boson (HA), in addition to the standard
Bjorken mechanism (HZ).
%
We describe a general strategy to determine
limits from LEP data on the masses and couplings
of such Higgs bosons, using the existing data on
acoplanar dijet events as well as
data on four and six $b$ jet event topologies.
For the sake of illustration, we present constraints
that can be obtained for the ALEPH data.
}
\end{abstract}

\end{titlepage}

\section{Introduction}

The Higgs particle remains one of the missing links in the otherwise
well tested standard electroweak model (SM) \cite{HIGGS}. Its mass
is not fixed theoretically and experimental efforts to search for
the higgs boson, notably at LEP, have produced a lower bound
$\sim 63 $ GeV \cite{Aleph92} on its mass. This bound applies
to the standard model (SM) Higgs.

There exist well motivated extensions of the SM which are characterized
by a more complex Higgs structure than that of the SM \cite{hunter,cohen}.
These include ({\sl i}) the minimal supersymmetric standard model
(MSSM) ({\sl ii})
generic two or more Higgs doublet models and ({\sl iii}) the majoron type
models  characterized by a spontaneously broken global symmetry.
This symmetry could either be a lepton number \cite{CMP,fae},
 a combination of family lepton numbers or an R symmetry
in the context of supersymmetry \cite{MASI_pot3}. These extensions
contain one or more parameters in addition to the Higgs mass. Nevertheless,
a considerable region of the parameter space has already been ruled out
in these models using the LEP data \cite{aleph1}.

Among the three extensions mentioned above, the majoron type models are
qualitatively different compared to the other two as well as to the SM.
These models contain a massless goldstone boson, called majoron. There are
many types of majoron models \cite{fae}. Many of such models are
characterized by the spontaneous violation of a global $U(1)$
lepton number symmetry close to the electroweak scale.
This could also have important implications for the
structure of the electroweak phase transition and
the generation of the electroweak baryon asymmetry
\cite{ewbaryo}.

In such models the normal doublet
Higgs is expected to have sizeable invisible decay modes
to the majoron, due to the strong higgs
majoron coupling \cite{bj} - \cite{Joshi92a}.
This can have a significant effect on the Higgs phenomenology at LEP.
In particular, the invisible decay
could contribute to the standard signal looked for at LEP namely two
acoplanar jets and missing momentum. This feature of the majoron model
allows one to strongly constrain the Higgs mass \cite{alfonso,dproy,grivaz}
in spite of the occurrence of extra parameters compared to the standard model.
In particular, the limit on the predominantly doublet Higgs mass
can be seen to be close to the SM limit irrespective of the decay
mode of the Higgs boson \cite{alfonso,dproy,grivaz}.

Apart from giving  an interesting twist to the Higgs search strategy,
the  majoron models are quite well motivated theoretically \cite{fae}.
They allow the possibility of spontaneously generating lepton number violation
and hence neutrino masses. The Higgs couplings to majorons in these models
were discussed at length in \cite{JoshipuraValle92}.

As far as the neutral Higgs sector is concerned, all the models fall
 in two categories.  (A) Minimally extended SM characterized by the
 addition of a singlet Higgs: Such models are typical examples of
 mechanisms where the neutrinos obtain their mass at the tree level
 \cite{CON}.  (B) Models containing two Higgs doublets and a singlet
 under the $SU(2) \times U(1)$ group: These models though more complex
arise naturally in trying to generate the neutrino mass radiatively.
The Zee type \cite{zee} model modified to obtain spontaneous lepton
number violation is a typical example \cite{ewbaryo} in this category.
These models have the virtue of explaining smallness of neutrino masses
without invoking any high scale. Moreover, type B models are different
from the type A ones as far as the Higgs phenomenology is concerned
since they contain a massive pseudoscalar which can be produced in
association with the Higgs as in MSSM. All the existing analysis
of the invisible Higgs search \cite{alfonso,dproy,grivaz}
have concentrated on type A models and hence on the Bjorken process.

In this letter we extend this analysis to include the type (B)
models and particularly the associated production of the pseudoscalar
Higgs boson.
%
We describe a general strategy to determine
limits from LEP data on the H and A masses and
couplings using the existing data on acoplanar
dijet events. We also show how one can set
decay mode-independent limits on H and A masses
and couplings by including data on four and
six b jet events.
%
For the sake of illustration, we display the constraints
that can be obtained for the published ALEPH data.

\section{Higgs Boson Production and Decay}

Let us first recall the salient features of the type (B) models.
The Higgs sector is characterized by two doublets $\phi_{1,2}$ and a
singlet $\sigma$. The part of the scalar potential containing the
neutral Higgs fields is given in this case by
%
\begin{eqnarray}
\label{N2}
V &=&\mu_{i}^2\phi^{\dagger}_i\phi_i+\mu_{\sigma}^2\sigma^{\dagger}
\sigma
+ \lambda_{i}(\phi^{\dagger}_i\phi_i)^2+
   \lambda_{\sigma}
(\sigma^{\dagger}\sigma)^2+\nonumber\\
& &\lambda_{12}(\phi^{\dagger}_1\phi_1)(\phi^{\dagger}_2\phi_2)
+\lambda_{13}(\phi^{\dagger}_1\phi_1)(\sigma^{\dagger}\sigma)+
\lambda_{23}(\phi^{\dagger}_2\phi_2)(\sigma^{\dagger}\sigma)\nonumber\\
& & +\delta(\phi^{\dagger}_1\phi_2)(\phi^{\dagger}_2\phi_1)+
    \frac{1}{2}\kappa[(\phi^{\dagger}_1\phi_2)^2+\;\;h.\;c.]
\end{eqnarray}
where a sum over repeated indices $i$=1,2 is assumed.
Here $\phi_{1,2}$ are the doublet fields and $\sigma$
corresponds to the singlet carrying nonzero lepton number.

In writing down the above equation, we have imposed a discrete
symmetry $\phi_2\rightarrow-\phi_2$ needed to obtain natural
flavour conservation in the presence of more than one Higgs
doublet. For simplicity, we assume all the couplings and VEVS
to be real.

After minimization of the above potential
one can work out the mass matrix for the Higgs fields.
To this end we shift the fields as ($i$=1,2)
\begin{equation}
\label{shift1}
\phi_i=\frac{v_i}{\sqrt{2}}+\frac{R_i+iI_i}{\sqrt{2}} \nonumber
\end{equation}
\begin{equation}
\label{shift2}
\sigma=\frac{v_3}{\sqrt2}+\frac{R_3+iI_3}{\sqrt{2}}.
\end{equation}
%
The masses of the CP even fields $R_a$ ($a$=1...3) are obtained from
%
\begin{equation}
{\cal L}_{mass}=\frac{1}{2}R^T\;M_R^2\;R
\end{equation}
with
\begin{equation}
\label{MR}
M_R^2=\left (
\begin{array}{ccc}
2\lambda_1v_1^2&(\kappa+\lambda_{12}+\delta)v_1v_2&\lambda_{13}v_1v_3\\
(\kappa+\lambda_{12}+\delta)v_1v_2&2\lambda_2v_2^2&\lambda_{23}v_2v_3\\
\lambda_{13}v_1v_3&\lambda_{23}v_2v_3&2\lambda_3v_3^2\\
\end{array}  \right).
\end{equation}
The physical mass eigenstates $H_a$ are related to the corresponding
weak eigenstates as
%
\begin{equation}
\label{me}
H_a=O_{ab}\;R_b
\end{equation}
where,  $O$ is a 3$\times$3 matrix diagonalizing $M_R^2$
%
\begin{equation}
\label{diag}
O\;M_R^2\;O^T=diag \: (M_1^2,M_2^2,M_3^2).
\end{equation}
%
It is convenient to parametrize the matrix $O$ as
%
\begin{equation}
\label{O}
O=\left( \begin{array}{ccc}
     c_{\alpha} c_{\theta}&
     -s_{\alpha}c_{\omega}-c_{\alpha}s_{\omega}s_{\theta}&
     s_{\alpha}s_{\omega}-c_{\alpha}c_{\omega}s_{\theta}\\
s_{\alpha} c_{\theta}&c_{\alpha}c_{\omega}-s_{\alpha}s_{\omega}s_{\theta}&
    - c_{\alpha}s_{\omega}-s_{\alpha}c_{\omega}s_{\theta}\\
 s_{\theta} & s_{\omega} c_{\theta}&
     c_{\omega}c_{\theta}\\
\end{array}  \right)
\end{equation}
%
where $c_{\alpha} \equiv \cos \alpha $ etc.

In addition, there exists also a massive CP odd state A,
related to the doublet fields as follows
%
\begin{equation}
A=\frac{1}{V}(v_2I_1-v_1I_2).
\end{equation}
Its mass is given by
\begin{equation}
M_A^2=-\kappa V^2
\end{equation}
where $ V=(v_1^2+v_2^2)^{1/2}$.
Thus, after spontaneous $SU(2) \times U(1) \times U(1)_L$
breaking, we have a total of three massive CP even scalars
$H_{i} \:$ (i=1,2,3), plus a massive pseudoscalar $A$ and
the massless majoron $J$, simply given as $J=I_3$.

At an $e^+ e^-$ collider there are two main production mechanisms
that allow the production of the higgs boson through their
couplings to $Z$.
%
The relevant couplings for their production through
Bjorken process (HZ) are given as follows ($a$=1...3)
%
\begin{equation}
\label{HZZ2}
{\cal
L}_{HZZ}=(\sqrt{2}G_F)^{1/2}M_Z^2\;Z_{\mu}Z^{\mu}[\frac{v_1}{V}O_{1a}+
\frac{v_2}{V}O_{2a}]H_a.
\end{equation}
%
As long as the mixing appearing \eq{HZZ2} is $\O$(1), the Higgs
bosons can have significant couplings and hence appreciable
production rates through the Bjorken process.

In addition, the $H_a$ can also be produced in
association with the CP odd field $A$ through the HA coupling
%
\begin{equation}
\label{HAZ2}
{\cal L}_{HAZ}=-\frac{g}{cos\theta_W}Z^{\mu}\left [
\frac{v_2}{V}O_{1a}-\frac{v_1}{V}O_{2a}\right]H_a
\stackrel{\leftrightarrow}{\partial^\mu}\;A.
\end{equation}

In what follows we assume that at LEP only the
the lightest of the CP even Higgs boson, denoted
by $H_1 \equiv H$ can be accessible. Using the
matrix $O$ of \eq{O} in \eq{HZZ2} and \eq{HAZ2}
one gets the couplings of H to ZZ and HA as
%
\beq
\label{HZZ3}
{\cal
L}_{HZZ}=(\sqrt{2}G_F)^{1/2}M_Z^2\;Z_{\mu}Z^{\mu} \cos(\beta-\alpha)
\cos\theta H
\eeq
where $\tan \beta \equiv \frac{v_2}{v_1}$ and
%
\begin{equation}
\label{HAZ3}
{\cal L}_{HAZ}=-\frac{g}{cos\theta_W}Z^{\mu} \sin(\beta-\alpha)
\cos\theta\;H
\stackrel{\leftrightarrow}{\partial^\mu}\;A.
\end{equation}

As expected, one recovers the well known expressions
for the two doublets case in the limit $\theta\rightarrow 0$.
In particular, note that for sizeable value of $\cos \theta$,
one cannot simultaneously
suppress the production of Higgs through \eq{HZZ3} and \eq{HAZ3}
if it is allowed kinematically.

Now we turn to Higgs boson decay. For Higgs boson
masses accessible at LEP energies the main CP even
Higgs decay modes are into bb and JJ. The coupling
of the physical Higgses to $J$ follows from \eq{N2}.
One can express this coupling entirely in terms of the masses
$M_a^2$ and the mixing angles in the matrix $O$
%
\begin{eqnarray}
\label{J2}
{\cal L}_J&=&\frac{1}{2}J^2(2\lambda_3v_3R_3+\lambda_{13}v_1R_1+
           \lambda_{23}v_2v_3R_2)\\
         &=&\frac{J^2}{2v_3}(M_R^2)_{3a}R_a\\
       &=&\frac{1}{2}(\sqrt{2}G_F)^{1/2} \tan \gamma (O^T)_{3a}M_a^2H_a J^2
\end{eqnarray}
%
where tan$\gamma\equiv\frac{V}{v_3}$.
We have made use of \eq{me} and \eq{diag} in writing the last line.

Note from \eq{J2} that the CP even scalar Higgs bosons $H_i$
couple strongly to a pair of majorons leading to the invisible
decay signature. In contrast to $H_i$, the pseudoscalar $A$ does
 not decay into one or three majorons since the couplings
$AJ^3$ or $AHJ$, although possible in general, do not exist
at tree level in our simplest model described above.
%
Because of the form of the scalar potential given in
\eq{N2} only even powers of the majoron field
$J \equiv R_3$ appear once the expansions
in \eq{shift1} and \eq{shift2} are used.
As a result the couplings $AJ^3$ and $AHJ$
are absent from the potential and therefore the
$A$ decays visibly to fermion antifermion pair
%
\footnote{Note that $A$ could decay invisibly at the
tree level through the process $A\rightarrow Z^* H\rightarrow
 \nu \overline{\nu} JJ$. However the branching ratio in this mode
is quite small.}.
%
The branching fraction $B_A$ for $A\rightarrow b \overline{b}$ is nearly one.
Deviation from unity of this branching ratio is model dependent.
If all fermions obtain their mass by coupling to
only one higgs boson, as in the models considered in
ref. \cite{Joshi92,ewbaryo}, then one has
%
\begin{equation}
B_A=\frac{1}{1+r}
\end{equation}
where
\begin{equation}
r \approx \sum_{f} \frac{m_f^2 (1- 4 m_f^2/m_A^2)^{1/2}}
{m_b^2 (1 - 4 m_b^2/m_A^2)^{1/2}}
\end{equation}
%
and the sum is over all the fermions except $b$.
The rate for $H\rightarrow b \overline b$ can be expressed as
%
\begin{equation}
\Gamma(H\rightarrow b \overline b)=\frac{3\sqrt 2 G_F}{8
\pi}M_Hm_b^2(1-4m_b^2/M_H^2)^{3/2} \cos^2 \alpha \cos^2 \theta
\end{equation}

On the other hand the  width for the invisible $H$ decay  can be
parametrized by
%
\begin{equation}
\label{HJJ}
\Gamma(H\rightarrow JJ)=\frac{\sqrt 2 G_F}{32 \pi} M_H^3 \:
(\tan \gamma\;\; O_{13})^2
\end{equation}
For many choices of parameters the invisible decay mode can be
rather important and, in fact, provide the strongest limits.
Finally, note that additional decay modes of $H$ to AA may also exist.

\section{Search Strategy and Analysis}

Since $A$ can decay only visibly, one expects dijets +
missing momentum as a signal of the Higgs production in our
simplest model. This signal arises from three processes -- i.e.
$Z^* \rightarrow  \nu \bar{\nu}$ with
$H \rightarrow b \overline{b}$,
or $Z^* \rightarrow q \overline{q}$ with $H \ra JJ$
or $H\rightarrow JJ$ with $A\rightarrow b \overline{b}$.
%
For each process one has a sizeable missing momentum which is aligned
neither along the beam nor along the jets.  In contrast, for the SM
background, the missing momentum arises from i) jet fluctuation
(including decay $\nu$ from $b,c$ and $\tau$ jets) in which case it is
aligned along the jets, and ii) initial state radiation (ISR) or the
two-photon process $e^+e^- \rightarrow (e^+e^-) \gamma\gamma$ in
which case it is aligned along the beam direction.  This enables one
to eliminate the SM background in the dijet + missing momentum channel
by a suitable combination of kinematic cuts without depleting the
signals seriously.  In fact, this procedure has been extensively used
to search for the SM Higgs signal (first process) in the LEP data and
obtain the corresponding mass limit \cite{Aleph92}.  More recently the
analysis has been extended to the invisibly decaying Higgs signal
(second process) and obtain the corresponding mass limits for the A
type majoron models \cite{alfonso,dproy,grivaz}.  In this section we
shall extend the analysis further to include the Higgs signal from the
third process of associated production (HA) and study the resulting mass
limits for the B type models.

We shall use a parton level Monte Carlo event generator, which has
been shown \cite{dproy} to reproduce the signals obtained with the
full Monte Carlo program \cite{aleph1} quite well.

For our illustrative purposes, below we will determine
the corresponding Higgs boson mass limits which can
be obtained from the ALEPH data sample of ref. \cite{Aleph92}
based on a statistics of $\sim  1.23$ million hadronic Z events.
%
A detailed account of the experimental cuts can
be found in \cite{aleph1}.  We shall only summarize the main features.
One starts with a visible mass cut, $M_{jj} < 70~{\rm GeV}$, to ensure a
sizeable missing energy ($E\!\!\!/$) and momentum ($\vec p\!\!\!/$).
A low angle cut, requiring the energy deposit within $12^\circ$ of the
beam axis to be $< 3$ GeV and that beyond $30^\circ$ to be $> 60\%$ of
the visible energy, removes jets close to be beam pipe where
measurement errors can simulate a large $E\!\!\!/$.  An acollinearity
cut, requiring the angle between the two jets to be $< 165^\circ$,
suppresses the $Z \rightarrow q\bar q$ and $\tau^+\tau^-$ background
where the $E\!\!\!/$ can come from jet fluctuation (including escaping
$\nu$).  Moreover an isolation cut on $\vec p\!\!\!/$ removes the
$E\!\!\!/$ background from the fluctuation of any one of the jets.  A
cut on the angle ($\alpha$) of $\vec p\!\!\!/$ with respect to the beam
axis, $\tan \alpha > 0.4$, suppresses the background from ISR and two
photon processes.  The $E\!\!\!/$ background coming from the jet
fluctuations along with an ISR are removed by an acoplanarity cut,
requiring the azimuthal angle between the two jets to be $<
175^\circ$.  An acoplanarity cut for 3-jet like events, requiring the
sum of the 3 dijet angles to be $< 350^\circ$, removes the $E\!\!\!/$
background arising from the fluctuation of these jets.  The remaining
few events are residual two-photon events, which are removed by a
total $p_T$ cut.  The cuts remove all the events from the data sample
\cite{aleph1}, while retaining $\gsim 50\%$ of the signal for each of
the three processes mentioned above.

We denote the number of signal events for the three processes, after
the cuts, by $N_{SM}, N_{JJ}$ and $N_A$ respectively, assuming no
suppression due to the mixing angles or branching fractions in each
case.  Then the expected number of signal events, after incorporating
these effects, is given by
\begin{equation}
N_{expt}=\epsilon_B^2\left[ B N_{J}+(1-B) N_{SM}\right]+\epsilon_A^2
B_A B N_A
\end{equation}
where
\begin{equation}
B = BR (H \ra JJ)
\end{equation}
is the branching fraction for the $H$ decaying into the invisible mode
and $\epsilon_A \equiv \sin (\beta - \alpha) \cos \theta$ and
$\epsilon_B \equiv \cos (\beta - \alpha) \cos \theta$.

The number of signal events for the three processes that pass the cuts
are shown in Fig. 1. As we can see $N_A \gg N_{SM}, \: N_{JJ}$ thus
implying that, if kinematically open and not suppressed by mixing
angles, associated production tends to give the strongest limits.  We
shall now consider the limits which can be obtained by comparing eq.
(22) with the 95\% CL limit of 3 events, corresponding to 0 events in
the data sample \cite{aleph1} after the cuts.


First, we assume that B=0. In this case we get the
limits corresponding to the SM decay mode. Note that this
is the weakest possible limit for the Bjorken
coupling strength $\epsilon_B^2$.
%
This is in complete agreement with the results of ref.
\cite{alfonso,dproy} as seen in Fig. 2.
%
Note that in this case we can set no limits on
$\epsilon_A^2$.

We can also obtain limits assuming B=1. From Fig. 1
one can see that, except for the region very close
to the edge of the phase space, $m_A + m_H \sim m_Z$,
the major contribution comes from the associated production
($N_A$) allowing the corresponding coupling strength parameter
$\epsilon_A^2$ to be strongly constrained, as seen
in Fig. 3. It is easily seen that the limits close to the
edge of the phase space are typically at the level
$\epsilon_A^2 \sim 0.05$. However, very close to
the kinematical limit the Bjorken process becomes
important and could further strengthen the limits
we have obtained.
%
Outside the region where associated production
is kinematically possible, the Bjorken process can
still take place and the limits on $\epsilon_B^2$ are
the same as obtained in ref. \cite{alfonso,dproy,grivaz}.
The limits on the coupling strengths $\epsilon^2_A$ and
$\epsilon^2_B$ can be improved further by using more
recent data from LEP.

In the case B=0 considered earlier one does not get any limit on
$\epsilon_A^2$ since in this case the associated production does not
lead to the dijets + missing momentum signal, we have considered so
far. However one could derive limits on the $\epsilon_A^2$ using data
on four and six b jets topologies, assuming good b identification.
%
As an illustration of what can be achieved we assume some typical
value for the corresponding branching ratios
$BR( e^+ e^- \rightarrow H\:A \rightarrow 4b) \leq \: 5 \times 10^{-4}$.
Using this value we can determine the excluded region for different values
of $\epsilon^2_{A}$ as a function of $m_A$ and $m_H$ from
%
\beq
\label{rich}
BR_{HA}=\frac{1}{2}\:(BR)_{\nu \nu} \lambda^3\:\epsilon^2_{A} \:,
\eeq
%
where ${BR}_{\nu \nu}$ is the Z decay branching ratio into
one generation of neutrinos in the SM, the factor 1/2 refers to the
scalar HA decay mode and $\lambda$ is the corresponding phase space
factor
$$
\lambda(s,m_H,m_{A})=
\frac{\sqrt{(s+m_{H}^{2}-m_{A}^{2})^2-4\:s\:m_{H}^2}}{s}
$$
%
The resulting excluded regions are shown in Fig. 4.
%
On the other hand if H has a mass bigger than
$2m_A$ then \eq{rich} would give a weaker limit
since it should now be multiplied by the branching ratio
for H into 2b. But now additional information can be obtained
from the study of events with six b jet topologies.
One may obtain a similar limit using an illustrative
reference limit for the corresponding branching
%
$BR( e^+ e^- \rightarrow H\:A \rightarrow 6b) \leq 4 \times \:10^{-4}$.


\section{Discussion}

Here we have shown that the production of invisibly decaying
Higgs bosons in Z decays through the associated channel HA
can lead to substantial limits on Higgs boson masses and couplings.
Such invisibly decaying Higgs bosons arise in
a wide class of \21 majoron-type models, such as the majoron
extension of the Zee model for radiatively induced \neu masses,
as well as extensions of the minimal supersymmetric standard
model with spontaneously broken R parity \cite{MASI_pot3}.

Note that it is also possible to derive a truly
decay mode independent limit on the H mass in the
case of associated HA production by allowing B to vary
from 0 to 1 and by combining
the data on dijet + missing momentum, with those from four
and six b jet searches described above.

It is important to update the limits obtained here by
using the full statistics provided by the four LEP
experiments and take them into account in designing the
strategies to search for the Higgs boson at higher
energies, such as at the LHC \cite{kane} and NLC
\cite{EE500}.

Finally we note that a similar analysis may be performed
in the context of the minimal supersymmetric standard model,
where the invisible decay of the Higgs bosons into neutralinos
may take place.
A major difference in the latter case arises from the fact
that in the MSSM case the pseudoscalar boson A is the one
most likely \cite{Zerwas} to decay invisibly, whereas in the
case discussed in the present paper the pseudoscalar
A always decays into $b\bar{b}$.

{\bf Acknowledgements}

We thank Francois Richard for useful discussions
during a visit to LAL at Orsay (F. de C. and J.W.F.V.).
J.W.F.V. also wishes to express gratitude to the
organizers of WHEPP-3 workshop held in Madras
for the hospitality extended to him in January 94.
This visit initiated the collaboration with D.P.R.
%
This work was supported by DGICYT under grant number
PB92-0084 and SAB94-0014 (A. S. J.), as well as a postdoctoral
fellowship (J.R.) and an FPI fellowship (M.A.G.J.). The work
of F. de Campos was supported by CNPq (Brazil).

\newpage

\noi
{\bf Figure Captions}\\
{\bf Figure 1}\\
Numbers of expected
dijet+missing momentum events for the processes $e^+ e^- \rightarrow
Z^*H \rightarrow \nu \bar{\nu}$  $b\bar b$ (N$_{SM}$), $e^+ e^-
\rightarrow Z^*H \rightarrow q \bar q $ JJ (N$_{JJ}$) and $e^+ e^-
\rightarrow H\:A \rightarrow J\: J b\bar{b}$ (N$_{HA}$) after imposing
ALEPH cuts \cite{aleph1,Aleph92}.\\
%
{\bf Figure 2}\\
Illustrative limits on $\epsilon^2_{B}$ for visibly decaying
H (B=0) from the $ Z^* H \rightarrow \nu \nu$ + two jets channel
as a function of $m_H$ using the ALEPH data of ref. \cite{Aleph92}.\\
%
{\bf Figure 3}\\
Limits on $\epsilon^2_{A}$ in the $m_A m_H$ plane, based on the
 $e^+ e^- \rightarrow H \:A \rightarrow J\:J b\bar{b}$ production channel.
We have assumed BR ($H \rightarrow J\:J$) = 100\% and a visibly
decaying A with branching ratio into bb given in eq. (18)\\
%
{\bf Figure 4}\\
Limits on $\epsilon^2_{A}$ for the B=0 case based only on
 $e^+ e^- \rightarrow H\:A \ra b \bar{b} b \bar{b}$ channel.
We have assumed a hypothetical sensitivity for the $4b$ channel
of BR($e^+ e^- \rightarrow H\:A \rightarrow 4b) \leq 5\times\:10^{-4}$.
\\
%
\newpage
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%%Pages: 1
gr gr
%%EOF

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%!PS-Adobe-2.0
%%Title: DISK$DELPHI3:[TOFLHC.SS]EPSB.PS;1 (Portrait A 4)
%%Pages: atend
%%Creator: HIGZ Version 1.19/16
%%CreationDate: 12/07/94   09.30
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%%%%%%%%%%%%%%%%%%%%%Figure 3%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%!PS-Adobe-2.0
%%Title: DISK$DELPHI3:[TOFLHC.SS]INV.PS;1 (Portrait A 4)
%%Pages: atend
%%Creator: HIGZ Version 1.19/16
%%CreationDate: 01/07/94   18.04
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s
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s
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1321
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951
 992 l cl s 976 967 m 951 992 l s 1210 992 m 1223 979 l s 1223 979 m 1235 967 l
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852
 1090 m 852 1090 l cl s 877 1066 m 852 1090 l s 1112 1090 m 1124 1078 l s 1124
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 1239 l s 655 1288 m 655 1288 l cl s 679 1288 m 655 1288 l s 704 1263 m 679
1288
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1325
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1461
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1498
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 1572 l s 630 1572 m 642 1560 l s 284 1609 m 284 1609 l cl s 309 1584 m 284
1609
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358
 710 l s 358 710 m 383 710 l s 383 710 m 408 710 l s 408 710 m 432 710 l s 432
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703
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 720 m 1494 707 l s 1494 707 m 1507 695 l s 1297 730 m 1282 745 l s 1307 720 m
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s
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s
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s
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 1037 1164 m 1050 1152 l s 1050 1152 m 1062 1140 l s 902 1174 m 887 1189 l s
911
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 1273 m 763 1288 l s 788 1263 m 778 1273 l s 914 1288 m 926 1276 l s 926 1276 m
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 1619 l s 568 1634 m 580 1621 l s 580 1621 m 593 1609 l s 383 1644 m 368 1658 l
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 1668 m 343 1683 l s 368 1658 m 358 1668 l s 519 1683 m 531 1671 l s 531 1671 m
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 1881 m 333 1868 l s 333 1868 m 346 1856 l s 296 1905 m 309 1893 l s 309 1893 m
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 1943 l s 259 1943 m 272 1930 l s 235 1967 m 247 1955 l s [4 8] 0 sd 1371 720 m
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1507
 695 l s 1371 720 m 1359 745 l s 1457 745 m 1470 732 l s 1470 732 m 1482 720 l
s
 1346 757 m 1334 769 l s 1359 745 m 1346 757 l s 1433 769 m 1445 757 l s 1445
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 1247 868 m 1247 868 l cl s 1272 843 m 1247 868 l s 1334 868 m 1346 856 l s
1346
 856 m 1359 843 l s 1223 893 m 1223 893 l cl s 1247 868 m 1223 893 l s 1309 893
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1210
 992 m 1223 979 l s 1223 979 m 1235 967 l s 1099 1016 m 1099 1016 l cl s 1124
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 1214 l s 988 1214 m 1000 1201 l s 1000 1201 m 1013 1189 l s 877 1239 m 877
1239
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1288
 m 926 1276 l s 926 1276 m 939 1263 l s 803 1313 m 803 1313 l cl s 827 1288 m
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 1337 l cl s 803 1313 m 778 1337 l s 865 1337 m 877 1325 l s 877 1325 m 889
1313
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 1387 m 827 1374 l s 827 1374 m 840 1362 l s 704 1411 m 704 1411 l cl s 729
1387
 m 704 1411 l s 790 1411 m 803 1399 l s 803 1399 m 815 1387 l s 679 1424 m 667
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 1448 m 766 1436 l s 630 1473 m 618 1486 l s 642 1461 m 630 1473 l s 716 1486 m
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 1572 l s 630 1572 m 642 1560 l s 482 1609 m 482 1609 l cl s 506 1593 m 482
1609
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457
 1634 m 457 1634 l cl s 482 1609 m 457 1634 l s 568 1634 m 580 1621 l s 580
1621
 m 593 1609 l s 432 1658 m 432 1658 l cl s 457 1634 m 432 1658 l s 543 1658 m
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 1720 m 494 1708 l s 333 1745 m 321 1757 l s 358 1733 m 333 1745 l s 445 1757 m
 457 1745 l s 457 1745 m 469 1733 l s 284 1782 m 284 1782 l cl s 309 1770 m 284
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 s 259 1807 m 259 1807 l cl s 284 1782 m 259 1807 l s 395 1807 m 408 1794 l s
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 695 l s 1445 722 m 1422 745 l s 1457 745 m 1470 732 l s 1447 720 m 1445 722 l
s
 1470 732 m 1482 720 l s 1420 747 m 1398 769 l s 1433 769 m 1445 757 l s 1422
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s
 1346 822 m 1325 843 l s 1359 843 m 1371 831 l s 1349 819 m 1346 822 l s 1371
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s
 1284 917 m 1297 905 l s 1276 893 m 1272 897 l s 1297 905 m 1309 893 l s 1247
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s
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 1016 m 1198 1004 l s 1177 992 m 1173 995 l s 1198 1004 m 1210 992 l s 1149
1020
 m 1128 1041 l s 1161 1041 m 1173 1029 l s 1152 1016 m 1149 1020 l s 1173 1029
m
 1186 1016 l s 1124 1045 m 1103 1066 l s 1136 1066 m 1149 1053 l s 1128 1041 m
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939
 1263 m 951 1251 l s 929 1239 m 926 1242 l s 951 1251 m 963 1239 l s 902 1266 m
 879 1288 l s 914 1288 m 926 1276 l s 905 1263 m 902 1266 l s 926 1276 m 939
 1263 l s 877 1290 m 854 1313 l s 889 1313 m 902 1300 l s 879 1288 m 877 1290 l
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 1313 m 852 1315 l s 877 1325 m 889 1313 l s 827 1339 m 804 1362 l s 840 1362 m
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 1387 l cl s 803 1363 m 778 1387 l s 815 1387 m 827 1374 l s 804 1362 m 803
1363
 l s 827 1374 m 840 1362 l s 753 1411 m 753 1411 l cl s 778 1387 m 753 1411 l s
 790 1411 m 803 1399 l s 803 1399 m 815 1387 l s 729 1434 m 727 1436 l s 753
 1411 m 729 1434 l s 766 1436 m 778 1424 l s 778 1424 m 790 1411 l s 704 1457 m
 700 1461 l s 727 1436 m 704 1457 l s 741 1461 m 753 1448 l s 753 1448 m 766
 1436 l s 679 1480 m 674 1486 l s 700 1461 m 679 1480 l s 716 1486 m 729 1473 l
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 1510 m 704 1498 l s 704 1498 m 716 1486 l s 630 1525 m 620 1535 l s 646 1510 m
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 1572 m 642 1560 l s 556 1594 m 541 1609 l s 567 1584 m 556 1594 l s 593 1609 m
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%%Trailer
%%Pages: 1
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%%EOF
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%Figure 4%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%!PS-Adobe-2.0
%%Creator: HIGZ Version 1.12/09
%%CreationDate: 19/07/94   15.43
%%EndComments
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{reencsmalldict begin /newcodesandnames exch def /newfontname exch def
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8#260 /agrave 8#265 /Agrave 8#276 /acircumflex 8#300 /Acircumflex
8#311 /adieresis 8#314 /Adieresis 8#321 /ccedilla 8#322 /Ccedilla
8#323 /eacute 8#324 /Eacute 8#325 /egrave 8#326 /Egrave
8#327 /ecircumflex 8#330 /Ecircumflex 8#331 /edieresis 8#332 /Edieresis
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/Times-Roman /Times-Roman accvec ReEncodeSmall
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/Times-Bold /Times-Bold accvec ReEncodeSmall
/Times-BoldItalic /Times-BoldItalic accvec ReEncodeSmall
/Helvetica /Helvetica accvec ReEncodeSmall
/Helvetica-Oblique /Helvetica-Oblique accvec ReEncodeSmall
/Helvetica-Bold /Helvetica-Bold accvec ReEncodeSmall
/Helvetica-BoldOblique /Helvetica-BoldOblique  accvec ReEncodeSmall
/Courier /Courier accvec ReEncodeSmall
/Courier-Oblique /Courier-Oblique accvec ReEncodeSmall
/Courier-Bold /Courier-Bold accvec ReEncodeSmall
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m
 581 595 l s 606 570 m 595 581 l s 1032 595 m 1046 581 l s 1046 581 m 1057 570
l
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431
 720 l s 456 695 m 445 707 l s 907 720 m 921 707 l s 921 707 m 932 695 l s 420
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720
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871
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m
 782 845 l s 231 871 m 219 882 l s 732 896 m 745 882 l s 745 882 m 757 871 l s
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707
 921 l s 656 971 m 670 957 l s 670 957 m 682 946 l s 631 996 m 645 982 l s 645
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595
 1032 l s 595 1032 m 606 1021 l s 556 1071 m 570 1057 l s 570 1057 m 581 1046 l
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244
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294
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859
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m
 745 483 l s 1132 495 m 1146 481 l s 1146 481 m 1157 470 l s 720 508 m 709 520
l
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533
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520
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533
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695
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m
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445
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845
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m
 333 871 l s 358 845 m 344 859 l s 757 871 m 770 857 l s 770 857 m 782 845 l s
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745
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934
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m
 670 957 l s 670 957 m 682 946 l s 631 996 m 645 982 l s 645 982 m 656 971 l s
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1032
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1383
 244 m 1397 231 l s 1397 231 m 1408 219 l s 1069 244 m 1057 269 l s 1358 269 m
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s
 1044 294 m 1021 317 l s 1308 319 m 1321 306 l s 1321 306 m 1333 294 l s 996
342
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843
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 606 l s 1021 606 m 1032 595 l s 745 631 m 732 645 l s 757 620 m 745 631 l s
982
 645 m 996 631 l s 996 631 m 1007 620 l s 720 668 m 718 670 l s 732 645 m 720
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 670 m 695 693 l s 932 695 m 946 682 l s 946 682 m 957 670 l s 670 718 m 668
720
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m
 631 745 l s 668 720 m 645 732 l s 882 745 m 896 732 l s 896 732 m 907 720 l s
 620 757 m 606 770 l s 631 745 m 620 757 l s 857 770 m 871 757 l s 871 757 m
882
 745 l s 595 782 m 581 795 l s 606 770 m 595 782 l s 832 795 m 845 782 l s 845
 782 m 857 770 l s 545 818 m 543 820 l s 570 807 m 545 818 l s 581 795 m 570
807
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m
 520 843 l s 782 845 m 795 832 l s 795 832 m 807 820 l s 495 868 m 492 871 l s
 518 845 m 495 868 l s 757 871 m 770 857 l s 770 857 m 782 845 l s 470 893 m
467
 896 l s 492 871 m 470 893 l s 732 896 m 745 882 l s 745 882 m 757 871 l s 445
 918 m 442 921 l s 467 896 m 445 918 l s 707 921 m 720 907 l s 720 907 m 732
896
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m
 707 921 l s 395 968 m 392 971 l s 417 946 m 395 968 l s 656 971 m 670 957 l s
 670 957 m 682 946 l s 344 994 m 342 996 l s 370 982 m 344 994 l s 392 971 m
370
 982 l s 631 996 m 645 982 l s 645 982 m 656 971 l s 319 1019 m 317 1021 l s
342
 996 m 319 1019 l s 606 1021 m 620 1007 l s 620 1007 m 631 996 l s 294 1044 m
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 1408 m 231 1397 l s [] 0 sd 1221 234 m 1211 244 l s 1233 219 m 1221 234 l s
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s
 1146 326 m 1132 344 l s 1153 319 m 1146 326 l s 1283 344 m 1296 331 l s 1296
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 385 l s 1233 395 m 1246 381 l s 1246 381 m 1258 370 l s 1071 418 m 1069 420 l
s
 1089 395 m 1071 418 l s 1208 420 m 1221 406 l s 1221 406 m 1233 395 l s 1046
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s
 1146 481 m 1157 470 l s 996 500 m 976 520 l s 1001 495 m 996 500 l s 1107 520
m
 1121 506 l s 1121 506 m 1132 495 l s 971 527 m 957 545 l s 976 520 m 971 527 l
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545
 m 946 556 l s 1057 570 m 1071 556 l s 1071 556 m 1082 545 l s 921 581 m 907
595
 l s 932 570 m 921 581 l s 1032 595 m 1046 581 l s 1046 581 m 1057 570 l s 896
 610 m 886 620 l s 907 595 m 896 610 l s 1007 620 m 1021 606 l s 1021 606 m
1032
 595 l s 871 635 m 861 645 l s 886 620 m 871 635 l s 982 645 m 996 631 l s 996
 631 m 1007 620 l s 845 660 m 836 670 l s 861 645 m 845 660 l s 957 670 m 971
 656 l s 971 656 m 982 645 l s 820 685 m 813 695 l s 836 670 m 820 685 l s 932
 695 m 946 682 l s 946 682 m 957 670 l s 795 713 m 788 720 l s 813 695 m 795
713
 l s 907 720 m 921 707 l s 921 707 m 932 695 l s 770 738 m 763 745 l s 788 720
m
 770 738 l s 882 745 m 896 732 l s 896 732 m 907 720 l s 745 763 m 738 770 l s
 763 745 m 745 763 l s 857 770 m 871 757 l s 871 757 m 882 745 l s 720 788 m
713
 795 l s 738 770 m 720 788 l s 832 795 m 845 782 l s 845 782 m 857 770 l s 695
 813 m 685 820 l s 713 795 m 695 813 l s 807 820 m 820 807 l s 820 807 m 832
795
 l s 670 836 m 660 845 l s 685 820 m 670 836 l s 782 845 m 795 832 l s 795 832
m
 807 820 l s 645 861 m 635 871 l s 660 845 m 645 861 l s 757 871 m 770 857 l s
 770 857 m 782 845 l s 620 886 m 610 896 l s 635 871 m 620 886 l s 732 896 m
745
 882 l s 745 882 m 757 871 l s 595 907 m 581 921 l s 610 896 m 595 907 l s 707
 921 m 720 907 l s 720 907 m 732 896 l s 570 932 m 556 946 l s 581 921 m 570
932
 l s 682 946 m 695 932 l s 695 932 m 707 921 l s 545 957 m 527 971 l s 556 946
m
 545 957 l s 656 971 m 670 957 l s 670 957 m 682 946 l s 520 976 m 500 996 l s
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 475 1021 l s 500 996 m 495 1001 l s 606 1021 m 620 1007 l s 620 1007 m 631 996
 l s 445 1045 m 443 1046 l s 470 1026 m 445 1045 l s 475 1021 m 470 1026 l s
581
 1046 m 595 1032 l s 595 1032 m 606 1021 l s 420 1069 m 418 1071 l s 443 1046 m
 420 1069 l s 556 1071 m 570 1057 l s 570 1057 m 581 1046 l s 395 1089 m 385
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s
 1321 304 m 1331 294 l s 1246 337 m 1240 344 l s 1262 319 m 1246 337 l s 1280
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s
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s
 1125 470 m 1121 474 l s 1146 475 m 1152 470 l s 1096 501 m 1078 520 l s 1101
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s
 1071 550 m 1076 545 l s 1021 580 m 1006 595 l s 1025 595 m 1046 574 l s 1031
 570 m 1021 580 l s 1046 574 m 1050 570 l s 996 606 m 982 620 l s 1000 620 m
 1021 599 l s 1006 595 m 996 606 l s 1021 599 m 1025 595 l s 971 632 m 958 645
l
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 921 683 m 909 695 l s 924 695 m 946 674 l s 933 670 m 921 683 l s 946 674 m
949
 670 l s 896 709 m 884 720 l s 899 720 m 921 698 l s 909 695 m 896 709 l s 921
 698 m 924 695 l s 871 734 m 859 745 l s 874 745 m 896 723 l s 884 720 m 871
734
 l s 896 723 m 899 720 l s 845 760 m 835 770 l s 848 770 m 871 748 l s 859 745
m
 845 760 l s 871 748 m 874 745 l s 820 785 m 810 795 l s 823 795 m 845 773 l s
 835 770 m 820 785 l s 845 773 m 848 770 l s 795 810 m 785 820 l s 798 820 m
820
 798 l s 810 795 m 795 810 l s 820 798 m 823 795 l s 770 835 m 760 845 l s 773
 845 m 795 823 l s 785 820 m 770 835 l s 795 823 m 798 820 l s 745 859 m 734
871
 l s 748 871 m 770 848 l s 760 845 m 745 859 l s 770 848 m 773 845 l s 720 884
m
 709 896 l s 723 896 m 745 874 l s 734 871 m 720 884 l s 745 874 m 748 871 l s
 695 909 m 683 921 l s 698 921 m 720 899 l s 709 896 m 695 909 l s 720 899 m
723
 896 l s 670 933 m 657 946 l s 674 946 m 695 924 l s 683 921 m 670 933 l s 695
 924 m 698 921 l s 645 958 m 632 971 l s 649 971 m 670 949 l s 657 946 m 645
958
 l s 670 949 m 674 946 l s 620 982 m 606 996 l s 624 996 m 645 974 l s 632 971
m
 620 982 l s 645 974 m 649 971 l s 595 1006 m 580 1021 l s 599 1021 m 620 1000
l
 s 606 996 m 595 1006 l s 620 1000 m 624 996 l s 570 1031 m 554 1046 l s 574
 1046 m 595 1025 l s 580 1021 m 570 1031 l s 595 1025 m 599 1021 l s 545 1054 m
 527 1071 l s 550 1071 m 570 1050 l s 554 1046 m 545 1054 l s 570 1050 m 574
 1046 l s 520 1078 m 501 1096 l s 525 1096 m 545 1076 l s 527 1071 m 520 1078 l
 s 545 1076 m 550 1071 l s 495 1102 m 474 1121 l s 500 1121 m 520 1101 l s 501
 1096 m 495 1102 l s 520 1101 m 525 1096 l s 470 1125 m 448 1146 l s 475 1146 m
 495 1127 l s 474 1121 m 470 1125 l s 495 1127 m 500 1121 l s 445 1149 m 421
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 s 395 1195 m 393 1196 l s 420 1173 m 395 1195 l s 426 1196 m 445 1177 l s 421
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 370 1218 l s 402 1221 m 420 1203 l s 420 1203 m 426 1196 l s 344 1240 m 337
 1246 l s 366 1221 m 344 1240 l s 377 1246 m 395 1229 l s 395 1229 m 402 1221 l
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 344 1280 l s 344 1280 m 353 1271 l s 269 1305 m 251 1321 l s 280 1296 m 269
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 s 251 1321 m 244 1327 l s 279 1347 m 294 1331 l s 294 1331 m 304 1321 l s 222
 1347 m 219 1349 l s 255 1372 m 269 1357 l s 269 1357 m 279 1347 l s 231 1397 m
 244 1382 l s 244 1382 m 255 1372 l s 219 1408 m 231 1397 l s 219 219 m 219
1973
 l s 252 219 m 219 219 l s 236 269 m 219 269 l s 236 318 m 219 318 l s 236 367
m
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