%Paper: 
%From: rb@hep.physik.uni-muenchen.de (Ralph Boenisch)
%Date: Mon, 30 Aug 93 14:30:58 MET DST
%Date (revised): Fri, 3 Sep 93 15:12:32 MET DST




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\begin{titlepage}
\begin{quote}\raggedleft  {\bf LMU-14/93 } \\ {\em August 1993}\end{quote}
\vspace{1.5cm}
\begin{center}
{\bf LARGE N RELATIONS BETWEEN MASS MATRIX AND GAUGE GROUP
IN ELECTROWEAK GAUGE EXTENSIONS WITH STRONG COUPLING
\footnote{
to appear in the Proceedings of the 4th Hellenic School on Particle Physics}}
\end{center}
\vspace{1cm}
\begin{center}{\bf Ralf B\"{O}NISCH}

\vspace{1cm}
Ludwig-Maximilians-Universit\"{a}t M\"{u}nchen, Sektion Physik  \\
D -- 80333 M\"{u}nchen, Germany \\
\end{center}
\vspace{1cm}
\begin{abstract}
The Large $N$ expansion in gauge models with an NJL mechanism is discussed
in view of the resulting fermionic mass matrix. A hidden symmetry model is
introduced.
\end{abstract}
\vspace{2cm}
\end{titlepage}
\renewcommand{\thefootnote}{\fnsymbol{footnote}}
% ===============================================================
Electroweak models with an extendend gauge group
$SU(2)_L \times U(1)_Y \times G$ usually underly very strong conditions
like the need to match the
experimental $\rho$-parameter and other precisely known observables.
M. Lindner \cite{mali} reported on top condensate models, which use a
Nambu-Jona-Lasinio mechanism (NJL) to produce a condensate
$\langle \bar{t}t \rangle$ of the order the Fermi scale $G_F$ \cite{tc} .
To make some effort towards the physics beyond the SM,
they have recently been grounded on gauge extensions of the electroweak
sector, which necessarily have a strong coupling to trigger
the breakdown $SU(2)_L \times U(1)_Y \rightarrow U(1)_{em}$ \cite{models}.

Apart from the phenomenological and theoretical constraints which have
to be cared for in every extension of the gauge group, top condensate
models connect the fermionic mass matrix to the gauge group $G$.
Their interplay is strongly determined by the Large $N$ expansion, which
is used to treat the strongly coupled sector.
The parameter $N$ plays a crucial role
in the construction of the model and therefore carries information on the
possible structure beyond the SM.
We briefly recall the features of the Large $N$ expansion, as being
important here, and focus on the consequences for the condensation pattern.
Finally a specific model, which is motivated by substructure, will be
introduced.

In the classical NJL analysis of the interaction
$G  \bar{\psi}_L \psi_R \bar{\psi}_R \psi_L$,
fermion self-energies $\Pi(q^2)$
are summed to the effective scalar propagator as
\be \mi \Delta_{eff}(q^2)= \frac{\mi}{1- G \Pi(q^2)}
\lb{bubblesum},\ee
Fig. \ref{f1}.
The Schwinger-Dyson equation,
\bea
\delta m = m &=&  -G \Pi_{NJL} \nonumber \\
             &=& 2 G m \mi \int \frac{{\mbox d}^4p}{(2 \pi)^4}
                 \frac{1}{p^2 -m^2},
\lb{gap}\eea
is given by the graph in Fig. \ref{f1a}. It
is self-consistent by writing the renormalized mass $m$ also on the
righthand side. This includes graphs like Fig. \ref{f2}.

Restriction to a given order in $1/N$ allows analytical calculations
in limiting the number of conributing graphs \cite{witten}.
The validity of this approximation is the basic assumption of the NJL model.
Starting from an interaction
\be {\cal L}_{NJL}= \sum_{a,b=1}^N \left[ \bar{\psi}^a (\mi \sld -m) \psi_a
+\frac{G}{N} (\bar{\psi}^a_L \psi_{Ra})(\bar{\psi}^b_R \psi_{Lb}) \right],
\lb{sunnjl}\ee
graphs like Fig. \ref{f3} are of order 1/$N$ and suppressed in the limit
of large $N$:
The incoming and outgoing fermions carry the index $a$, it is the
mass counterterm $m_{aa}$.
Lines 1 and 2 in Fig. \ref{f3} have to be labeled $b$.
The summation over $b$ gives a factor $N_b$ for the graph.
Two of the lines 3, 4 and 5 carry an index $c$ and one carries an index
$b$, where the latter is fixed already
from the first vertex. Hence there is one factor $N_c$, but
no more factor $N_b$,
so that with a factor $1/N$ from each vertex the graph is of
order $N^2/N^3$.
Analogously, all
terms with an ingoing index $a$ and an outcoming
index $b$ or vice versa, i.e. mass counterterms $m_{ab}$ with $a \neq b$,
are suppressed in Large $N$.
Hence, starting with a strongly coupled
gauge group $G = SU(N)$ and expanding via this $N$ yields a mass matrix,
which is diagonal in the representation space of $G$.


In the summation eq.(\ref{bubblesum}), Fig. 1, side bubbles which are
connected to the chain by only one vertex are also taken into account by
using the renormalized fermion mass in the self-energy $\Pi(q^2)$,
while more complicated graphes are suppressed by orders of $N$.

The NJL graphs are the first order in $p^2/M^2$ of gauge theory ladder
diagrams, Fig. \ref{f4}, where
$p$ is a typical momentum and $M$ is the regulator of the NJL model or the
gauge boson mass.
The gauge vertex has to be index-changing to yield the dominant graphs of
the interaction eq. (\ref{sunnjl}):
The correct NJL-type interaction is
\be
{\cal L}_{eff} = - \frac{g^2}{2 M_A^2} \sum_{a,b=1}^N
\left(\bar{\psi}^a_L \ga_\mu \psi_{Lb} \right)
\left(\bar{\psi}^b_R \ga_\mu \psi_{Ra} \right),
\lb{njl}\ee
which
(due to the $t$-channel gauge boson exchange)
is Fierz-transformed into
\be {\cal L}_{eff} =   \frac{g^2}{M_A^2}
\sum_{a,b=1}^N
(\bar{\psi}^{a}_L \psi_{Ra})(\bar{\psi}^{b}_R \psi_{Lb}).
\lb{fierznjl}\ee
Hence, to obtain eq. (\ref{fierznjl})
from a gauge theory, one has
to start from the $SU(N)$ vertex
\be
{\cal L}_{gauge}= g \sum_{a,b=1}^N \bar{\psi}^a \ga_\mu \psi_b A^\mu_{ab}.
\lb{sungauge}\ee
In the Large $N$ expansion, the gauge coupling will be redefined
$g \rightarrow g/\sqrt{N}$.

In view of the various standard symmetries
a second fermionic $SU(N)$ degree
of freedom, which is independent from the first, will appear.
If the currents in eqn. (\ref{njl}) and (\ref{sungauge}) are corresponding
singlets,
\be
{\cal L}_{gauge}= g \sum_{a,b=1}^N \sum_{i=1}^M
\bar{\psi}^{a}_i \ga_\mu \psi_{b}^i A^\mu_{ab},
\lb{sungauge2}\ee
the effective scalar interaction takes the form
\be {\cal L}_{eff} =   \frac{g^2}{M_A^2}
\sum_{a,b=1}^N
\sum_{i,j=1}^M
(\bar{\psi}^{a}_{Li} \psi_{Ra}^j)(\bar{\psi}^{b}_{Rj} \psi_{Lb}^i).
\lb{fierznjl2}\ee
Now the Large $N$ expansion (i.e. summing independently over $a$ and $b$)
gives mass counterterms $m_{aa}^{ij}$ (and $m_{bb}^{ij}$), they are
constrained to be singlets only with respect to the Large $N$ index.
In the space of $i$ and $j$ the mass matrix has rank one,
there is only one nonvanishing eigenvalue \cite{rank1}.

What physical quantity should be represented by the parameter $N$?
The SM offers two $N$'s:
The color degree of freedom $N_c$ was used by C. Hill.
It might however be
unconvenient to give color such important meaning in the sector of
electroweak symmetry breaking.
The second candidate is the generation index $N_f$. Counting the number of
identical massless fermions,
it is tightly connected with the spectrum which we want to understand.
Consequently one can construct a strongly coupled horizontal gauge symmetry.
However, the mass matrix will be diagonal in flavor space and
a more complicated construction will be needed to produce a heavy top.
If the horizontal gauge bosons do not carry color,
$a, b$ in eq. (\ref{fierznjl2}) count generations and $i, j$ count color.
Thus, QCD might be broken simultaneously with $SU(2)_L \times U(1)_Y$.
This problem appears for all groups $G$, which do not embed $SU(3)_c$.
QCD corrections are supporting only color-singlet channels.
The strength of
their influence is mainly a question of the scale where the condensation
takes place.
It is not yet clear if and for which values of $M$ this problem can be cured.

One might speculate that fermion masses should not be the consequence
of external quantum numbers, but rather stem from a force which has to be
saturated within the massive objects. The mass hierarchies should then be
understood by inner dynamics.
This force would not directly show up at the Fermi scale and
yield a model with a hidden gauge symmetry, where
the parameter $N$ belongs to a hidden degree of freedom \cite{boekneur}.
We shall now describe such a model.

The breaking of local symmetries goes in two steps:
\be  \begin{array} {lcr}
SU(2)_L \times U(1)_{Y^\prime_L} \times U(1)_{Y^\prime_R} \times G &
\stackrel{spontaneously}{\longrightarrow}& \\
SU(2)_L \times U(1)_{Y_L} \times U(1)_{Y_R} &\stackrel{dynamically}
{\longrightarrow} & U(1)_{em}
\lb{break}\end{array}
\ee
The simplest Higgs-configuration $\Phi$ that works in the first step of eq.
(\ref{break}) transforms like $(0,\frac{1}{2},\frac{1}{2})$ with respect to
$SU(2)_L \times U(1)_{Y^\prime} \times G$.
For one neutral generator in $G$,
this $\Phi$ results in a mixing of the neutral sector:
\begin{equation}
\left( \begin{array}{c} W_{3} \\ v_{\mu}^{0} \\ B \end{array} \right) =
\left( \begin{array}{ccc} 1&0&0 \\ 0&\cos\xi& -\sin\xi \\ 0&\sin\xi&\cos\xi
                                                   \end{array} \right)
\left( \begin{array}{c} \bar{W}_{3} \\ \bar{V}_{3} \\ \bar{B} \end{array}
\right),
\lb{eq414}\end{equation}
where $\xi$ is given by the $U(1)_{Y^\prime}$ and $G$ gauge coupling constants
$g^\prime$ and $g^{\prime \prime}$,
\be \sin \xi = \frac{g^\prime}{\sqrt{g^{\prime \prime 2}+g^{\prime 2}}},
\label{mix} \ee
and $\bar{W_3}$, $\bar{B}$ and $\bar{V_3}$ are primordial neutral
gauge fields of
$SU(2)_L \times U(1)_{Y^\prime} \times G$.
$W_3$ and $B$ are massless standard gauge fields and $v^0$ is a new neutral
massive boson field.
Its mass is related to the VEV $v$ of $\Phi$ via
$m_v = \frac{1}{2} \sqrt{g^{\prime\prime 2}+g^{\pr 2}}$.
The only difference between $U(1)_{Y^\pr}$ and $U(1)_{Y}$
is in the gauge coupling constant (before and after mixing with $G$).
The interaction is given as
\bea {\cal L}_{I} &=& g J^{\mu}_{i} \bar{W}_{\mu i}
                    + g^\pr J^{\mu}_{Y} \bar{B}_{\mu}  \nn \\
                  &=& g J^{\mu}_{i} W_{\mu i}
                    + g^\pr \cos\xi J^{\mu}_{Y} B_{\mu}
                    - g_{v^0} J^{\mu}_{Y} v_{\mu}^{0},
\lb{ww}\eea
where $J_{i}$ ($J_{Y}$) is the standard isospin (hypercharge) current
and $g_{v^0}\equiv g^\pr \sin \xi$.
Charged bosons of $G$ do not couple, because there is no mixing with $SU(2)_L$.

Now we consider the model at low momentum transfers $q^2 \ll m_v^2$.
The effective contact interaction is
\bea
{\cal L}_{4F}=&-&\frac{g^\pr \sin\xi)^{2}}{2M_{v}^{2}}
J_{Yi}^{\mu} J_{Yj\mu}      \\
 =&+&\frac{G^{\alpha\beta}_{LL}}{M_{v}^{2}}
 \left( \bar{\psi}^{i\alpha}_{rL} \gamma^{\mu} \psi_{i\alpha L}^{r} \right)
 \left( \bar{\psi}^{j\beta}_{sL} \gamma_{\mu} \psi_{j \beta L}^{s} \right)
\nn\\
 &+&\frac{G^{\alpha\beta}_{RR}}{M_{v}^{2}}
 \left( \bar{\psi}^{i\alpha}_{rR} \gamma^{\mu} \psi_{i\alpha R}^{r} \right)
 \left( \bar{\psi}^{j\beta}_{sR} \gamma_{\mu} \psi_{j \beta R}^{s} \right)
\nn\\
 &-&\frac{G^{\alpha\beta}_{LR}}{2M_{v}^{2}}
 \left( \bar{\psi}^{i\alpha}_{rL} \gamma^{\mu} \psi_{i\alpha L}^{r} \right)
 \left( \bar{\psi}^{j\beta}_{sR} \gamma_{\mu} \psi_{j \beta R}^{s} \right)
+h.c.,
\label{ia}
\eea
where $i,j=1,2,3$ count generations, $r,s=1,2,3$ count colors and
$\alpha, \beta=u,d$ label isospin components.
The quantum numbers $Y_L$ and $Y_R \equiv Q$ are absorbed into the couplings
$G^{\al\beta}$, they are proportional to
\be Y_{L} \cdot Q= \left\{
\begin{array}{ll}\frac{1}{6}\cdot\frac{2}{3} &\mbox{for $I_{W_3}=1/2$ quarks}
\\
            \frac{1}{6}\cdot(-\frac{1}{3}) &\mbox{for $I_{W_3}=-1/2$ quarks.}
\end{array} \right . \lb{quarkhier}\ee
After Fierz-transformation eq. (\ref{ia}) reads
\bea
{\cal L}_{4F}=&+&\frac{G^{\alpha\beta}_{LL}}{M_{v}^{2}}
 \left( \bar{\psi}^{i\alpha}_{rL} \gamma^{\mu} \psi_{j \beta L}^{s} \right)
 \left( \bar{\psi}^{j\beta}_{sL} \gamma_{\mu} \psi_{i\alpha L}^{r}\right) \nn\\
 &+&\frac{G^{\alpha\beta}_{RR}}{M_{v}^{2}}
 \left( \bar{\psi}^{i\alpha}_{rR} \gamma^{\mu} \psi_{j \beta L}^{s}\right)
 \left( \bar{\psi}^{j\beta}_{sR} \gamma_{\mu} \psi_{i\alpha R}^{r}\right) \nn\\
 &+&\frac{G^{\alpha\beta}_{LR}}{M_{v}^{2}}
 \left( \bar{\psi}^{i\alpha}_{rL}  \psi_{j \beta R}^{s} \right)
 \left( \bar{\psi}^{j\beta}_{sR}   \psi_{i\alpha L}^{r} \right) +h.c.
\label{fierzia}
\eea
As discussed above, the mass matrix is of rank one in generation space and
according to eq. (\ref{quarkhier}) only $I=+1/2$. quarks condense, while
the interaction is repulsive for $I=-1/2$ quarks. Therefore, among quarks
only the top is massive. This mass splitting is a consequence of the
$SU(2)_R$ violation of $U(1)_{Y_R}$ quantum numbers.

We have not adressed the compelling question of quadratic divergencies
and experimental compatibility of the exact value of $m_{top}$ here
\cite{bhj}.

Within the hidden symmetry model, there are several further
questions to be answered. First, there is a problem with the leptons,
because in a three-family scheme the $\tau$ gets very heavy. Going to a fourth
family avoids this problem, but the original idea was to explain the heaviness
of the top, which is extraordinary because it is the only fermion that is not
light on the Fermi scale.

Eqn. (\ref{mix}) and (\ref{ww}) tell us that for a large $g_{v^0}$,
$g^\pr/g^{\pr\pr}$ must be large.
By the usual running of gauge coupling constants,
this ratio is increasing with increasing energy and the situation
of coming down from a symmetric phase at a high scale looks troublesome,
at least if dim $G >$  1.

The questions of leptons and $SU(3)_c$-conservation also ask for the number $N$
of
hidden degrees of freedom. (Note, that this $N$ has nothing to do with the
dimension of $G$.) Deriving conditions (like e.g. lower bounds) on $N$
potentially gives hints on substructure and/or further physics beyond the
SM.

\vspace{2cm}
{\bf Acknowledgement} \newline
I thank the organizers very much for the invitation to this
communacative and inspiring workshop.

\newpage
\begin{thebibliography}{99}
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\bibitem{tc}
Y. Nambu, G. Jona-Lasinio, Phys. Rev {\bf 122} (1961) 345;
Y. Nambu in {\em New Theories in Phyics}, proceedings of the XI
                 International Symposium on Elementary Particle Physics,
                 Kazimierz, Poland, 1988, Singapore 1989;
W.J. Marciano, Phys. Rev. Lett. {\bf 62} (1989) 2793 and
                                         Phys. Rev. {\bf D41} (1990) 219;
V.A. Miransky, M. Tanabashi and K. Yamawaki, Phys. Lett.
              {\bf B} 221 (1989) 177 and Mod. Phys. Lett. {\bf A4} (1989) 1043;
W.A. Bardeen, C.T. Hill and M. Lindner, Phys. Rev.
              {\bf D41} (1990) 1647;
\bibitem{models}R. B\"{o}nisch, Phys. Lett. {\bf B268} (1991) 394.
C.T. Hill, Phys. Lett. {\bf B 266} (1991) 419;
M. Lindner and D. Ross, Nucl. Phys. {\bf B370} (1992) 30;
S.P. Martin, Phys. Rev. {\bf D45} (1992) 4283;
                              Phys. Rev. {\bf D46} (1992) 2197;
C.E.M. Wagner, Munich preprint MPI-PAE/PTh,
                 {\em Proceedings of Workshop on Phenomenology of Particle
                 Physics}, Munich, November 1991;
S. Siegemund-Broka, Phys. Rev. {\bf D46} (1992) 1141;
S. P. Martin, Phys. Rev. {\bf D46} (1992) 2769;
\bibitem{witten} E. Witten, Nucl. Phys. {\bf B}160, (1979)57
\bibitem{rank1}
P. Kaus und S. Meshkov, Phys. Rev. {\bf D42} 1863(1990)
                             und  Mod. Phys. Lett. {\bf A3} 13(1988), 1251;
H. Fritzsch und J. Plankl, Phys. Lett. {\bf B} 237(1990), 451
R. B\"{o}nisch, Munich LMU-91/03 (unpublished), ;
\bibitem{boekneur} R. B\"onisch and J.-L. Kneur, LMU-92/03
\bibitem{bhj} as recent work see
W.A. Bardeen, C.T. Hill and D.-U. Jungnickel, SSCL-Preprint-490,
FERMI-PUB-93/035-T;
R. B\"onisch and A. Leike, LMU 04/93, DESY-93-111, 
\end{thebibliography}
\newpage
%1
\begin{figure}
\hspace*{0.5cm}
\psfig{figure=f1.eps,height=2cm,xscale=1.1,yscale=1.1}
\psfig{figure=f2.eps,height=2cm,xscale=1.1,yscale=1.1}
\caption{NJL-summation of fermion loops.}
\lb{f1}
\end{figure}

\begin{figure}\hspace*{3.75cm}
\psfig{figure=f3.eps,height=2cm,xscale=1.3,yscale=1.3}
\caption{NJL-self-energy entering eq. (\protect\ref{gap}).}
\lb{f1a}\end{figure}

%2
\begin{figure}\hspace*{3.7cm}
\psfig{figure=f4.eps,height=3cm,xscale=1.3,yscale=1.3}
\caption{By self-consistency of eq. (\protect\ref{gap}) included graph.}
\lb{f2}\end{figure}

%3
\begin{figure}[htb]
\hspace*{5.2cm}
\psfig{figure=f5.eps,height=4cm,xscale=1,yscale=.75}

\hspace*{3.1cm}
\psfig{figure=f6.eps,height=2.5cm,xscale=1,yscale=1}
\caption{Simple graph of order $1/N$.}
\lb{f3}
\end{figure}

%4
\begin{figure}[htb]\hspace*{1.9cm}
\psfig{figure=f7.eps,height=3cm,xscale=1,yscale=1}
\hspace*{1.9cm}
\psfig{figure=f8.eps,height=4.5cm,xscale=1,yscale=1}
\caption{Ladder diagram with notation of the index $N$.}
\lb{f4}\end{figure}

\end{document}

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 end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% psfig.tex following:

% LAST EDIT: Sun Jul  7 19:21:09 1991 by Michael Janich (massa)
% psfigTeX macros
%
% All software, documentation, and related files in this distribution of
% psfig/tex are Copyright (c) 1987 Trevor J. Darrell
%
% Permission is granted for use and non-profit distribution of psfig/tex
% providing that this notice be clearly maintained, but the right to
% distribute any portion of psfig/tex for profit or as part of any commercial
% product is specifically reserved for the author.
%
%
% $Header: psfig.tex,v 1.7 87/01/19 15:55:27 trevor Exp $
%
\catcode`\@=11\relax
\newwrite\@unused
\newdimen\hscale
\newdimen\vscale
\newdimen\@mybbllx
\newdimen\@mybblly
\newdimen\@mybburx
\newdimen\@mybbury
\newdimen\@myrwi % 10 times of xsize, don't know why

\message{Neues psfig V0.8009}
\def\typeout#1{{\let\protect\string\immediate\write\@unused{#1}}}
\def\psglobal#1{
\immediate\special{psfile=#1 }} % ist ok fuer dvips % was soll
			        % dies hier?
\def\psfiginit{%	\typeout{psfiginit}
%\immediate\psglobal{figtex.pro}
}
%
% @psdo control structure -- similar to Latex @for.
% I redefined these with different names so that psfig can
% be used with TeX as well as LaTeX, and so that it will not
% be vunerable to future changes in LaTeX's internal
% control structure,
%
\def\@nnil{\@nil}
\def\@empty{}
\def\@psdonoop#1\@@#2#3{}
\def\@psdo#1:=#2\do#3{\edef\@psdotmp{#2}\ifx\@psdotmp\@empty \else
    \expandafter\@psdoloop#2,\@nil,\@nil\@@#1{#3}\fi}
\def\@psdoloop#1,#2,#3\@@#4#5{\def#4{#1}\ifx #4\@nnil \else
       #5\def#4{#2}\ifx #4\@nnil \else#5\@ipsdoloop #3\@@#4{#5}\fi\fi}
\def\@ipsdoloop#1,#2\@@#3#4{\def#3{#1}\ifx #3\@nnil
       \let\@nextwhile=\@psdonoop \else
      #4\relax\let\@nextwhile=\@ipsdoloop\fi\@nextwhile#2\@@#3{#4}}
\def\@tpsdo#1:=#2\do#3{\xdef\@psdotmp{#2}\ifx\@psdotmp\@empty \else
    \@tpsdoloop#2\@nil\@nil\@@#1{#3}\fi}
\def\@tpsdoloop#1#2\@@#3#4{\def#3{#1}\ifx #3\@nnil
       \let\@nextwhile=\@psdonoop \else
      #4\relax\let\@nextwhile=\@tpsdoloop\fi\@nextwhile#2\@@#3{#4}}
%
%
\def\psdraft{
	\def\@psdraft{0}
	\typeout{draft level now is \@psdraft \space . }
}
\def\psfull{
	\def\@psdraft{100}
	\typeout{draft level now is \@psdraft \space . }
}
\psfull
\newif\if@prologfile
\newif\if@postlogfile
%%% These are for the option list.
%%% A specification of the form a = b maps to calling \@p@@sa{b}
%Die neuen von M. Janich
\newif\if@xscale
\newif\if@yscale
\newif\if@scale
% und die alten
\newif\if@bbllx
\newif\if@bblly
\newif\if@bburx
\newif\if@bbury
\newif\if@height
\newif\if@width
\newif\if@rheight
\newif\if@rwidth
\newif\if@clip
\newif\ifhfromw
\newif\ifwfromh
\def\@p@@sclip#1{\@cliptrue}
\def\@p@@sfile#1{\typeout{PostScript File #1 wird eingebunden}
		   \def\@p@sfile{#1}
}
\def\@p@@sfigure#1{\def\@p@sfile{#1}}
\def\@p@@sbbllx#1{
		%	\typeout{bbllx is #1}
		\@bbllxtrue
		\dimen100=#1
		\divide\dimen100 by 65536
		\edef\@p@sbbllx{\number\dimen100}
}
\def\@p@@sbblly#1{
		%	\typeout{bblly is #1}
		\@bbllytrue
		\dimen100=#1
		\divide\dimen100 by 65536
		\edef\@p@sbblly{\number\dimen100}
}
\def\@p@@sbburx#1{
		%	\typeout{bburx is #1}
		\@bburxtrue
		\dimen100=#1
		\divide\dimen100 by 65536
		\edef\@p@sbburx{\number\dimen100}
}
\def\@p@@sbbury#1{
		%	\typeout{bbury is #1}
		\@bburytrue
		\dimen100=#1
		\divide\dimen100 by 65536
		\edef\@p@sbbury{\number\dimen100}
}
\def\@p@@sheight#1{
		%	\typeout{Height is #1}
		\@heighttrue
		\dimen100=#1
		\divide\dimen100 by 65536
   		\edef\@p@sheight{\number\dimen100}
		%	\typeout{Height is \@p@sheight [pt]}
}
\def\@p@@swidth#1{
		%	\typeout{Width is #1}
		\@widthtrue
		\dimen100=#1
		\divide\dimen100 by 65536
		\edef\@p@swidth{\number\dimen100}
                %	\typeout{Width is is \@p@swidth [pt]}
}

\def\@p@@srheight#1{
		%	\typeout{Reserved height is #1}
		\@rheighttrue
		\dimen100=#1
                \divide\dimen100 by 65536
		\edef\@p@srheight{\number\dimen100}
		%	\typeout{rheight is \@p@srheight [pt]}
}
\def\@p@@srwidth#1{
		%	\typeout{Reserved width is #1}
		\@rwidthtrue
		\dimen100=#1
		\divide\dimen100 by 65536
		\edef\@p@srwidth{\number\dimen100}
		%	\typeout{rwidth is \@p@srwidth [pt]}
}
\def\@p@@sxscale#1{
		%	\typeout{h-Scaling factor is #1}
		\@xscaletrue
		\edef\@p@sxscale{#1}
}
\def\@p@@syscale#1{
		%	\typeout{v-Scaling factor is #1}
		\@yscaletrue
		\edef\@p@syscale{#1}
}
\def\@p@@sscale#1{
		%	\typeout{Scaling factor is #1}
		\@scaletrue
		\edef\@p@sscale{#1}
}
\def\@p@@sprolog#1{\@prologfiletrue\def\@prologfileval{#1}}
\def\@p@@spostlog#1{\@postlogfiletrue\def\@postlogfileval{#1}}
\def\@cs@name#1{\csname #1\endcsname}
\def\@setparms#1=#2,{\@cs@name{@p@@s#1}{#2}}
%
% initialize the defaults (size the size of the figure)
%
\def\ps@init@parms{
		\@bbllxfalse \@bbllyfalse
		\@bburxfalse \@bburyfalse
		\@heightfalse \@widthfalse
		\@rheightfalse \@rwidthfalse
		\@xscalefalse \@yscalefalse \@scalefalse
		\wfromhfalse \hfromwfalse
		\def\@p@sbbllx{}\def\@p@sbblly{}
		\def\@p@sbburx{}\def\@p@sbbury{}
		\def\@p@sheight{}\def\@p@swidth{}
		\def\@p@srheight{}\def\@p@srwidth{}
		\def\@p@sfile{}
		\def\@p@scost{10}
		\def\@sc{}
		\@prologfilefalse
		\@postlogfilefalse
		\@clipfalse
}
%
% Go through the options setting things up.
%
\def\parse@ps@parms#1{
	 	\@psdo\@psfiga:=#1\do
		   {\expandafter\@setparms\@psfiga,}}
%
% Compute bb height and width
%
\newif\ifno@bb
\newif\ifnot@eof
\newread\ps@stream
\newread\serienfile
\def\bb@missing{
	%	\typeout{psfig: searching \@p@sfile \space  for bounding box}
%	\openin\serienfile=\@p@sfile
%	\ifeof\serienfile % File existiert nicht
%        	\typeout{psfig: File \@p@sfile\space not found, type 'x' now!}
%	\fi
	\openin\ps@stream=\@p@sfile
	\no@bbtrue
	\not@eoftrue
	\catcode`\%=12
	\loop
		\read\ps@stream to \line@in
		\global\toks200=\expandafter{\line@in}
		\ifeof\ps@stream \not@eoffalse \fi
		%	\typeout{ looking at :: \the\toks200 }
		\@bbtest{\toks200}
		\if@bbmatch\not@eoffalse\expandafter\bb@cull\the\toks200\fi
	\ifnot@eof \repeat
	\catcode`\%=14
}
\catcode`\%=12
\newif\if@bbmatch
\def\@bbtest#1{\expandafter\@a@\the#1%%BoundingBox:\@bbtest\@a@}
\long\def\@a@#1%%BoundingBox:#2#3\@a@{\ifx\@bbtest#2\@bbmatchfalse
\else\@bbmatchtrue\fi}
\long\def\bb@cull#1 #2 #3 #4 #5 {
	\dimen100=#2 bp\edef\@p@sbbllx{\number\dimen100}
	\divide\dimen100 by 65536
	\edef\@mybbllx{\number\dimen100}
	\dimen100=#3 bp\edef\@p@sbblly{\number\dimen100}
	\divide\dimen100 by 65536
	\edef\@mybblly{\number\dimen100}

	\dimen100=#4 bp\edef\@p@sbburx{\number\dimen100}
	\divide\dimen100 by 65536
	\edef\@mybburx{\number\dimen100}
	\dimen100=#5 bp\edef\@p@sbbury{\number\dimen100}
	\divide\dimen100 by 65536
	\edef\@mybbury{\number\dimen100}
	\no@bbfalse
}
\catcode`\%=14
%
\def\compute@bb{
		\no@bbfalse
		\if@bbllx \else \no@bbtrue \fi
		\if@bblly \else \no@bbtrue \fi
		\if@bburx \else \no@bbtrue \fi
		\if@bbury \else \no@bbtrue \fi
		\ifno@bb \bb@missing \fi
		\ifno@bb \typeout{FATAL ERROR: Keine BoundingBox gefunden}
			\no-bb-error
		\fi
		%
		\count203=\@p@sbburx
		\count204=\@p@sbbury
		\advance\count203 by -\@p@sbbllx
		\advance\count204 by -\@p@sbblly
		\divide\count203 by 65536
		\divide\count204 by 65536

		\edef\@bbw{\number\count203}
		\edef\@bbh{\number\count204}
\typeout{psfig: Bild \@p@sfile, Orig. Hoehe = \@bbh pt, Breite = \@bbw pt}
}
%
% \in@hundreds performs #1 * (#2 / #3) correct to the hundreds,
%	then leaves the result in @result
%
\def\in@hundreds#1#2#3{\count240=#2 \count241=#3
		     \count100=\count240	% 100 is first digit #2/#3
		     \divide\count100 by \count241
		     \count101=\count100 % Hier: verhaeltnis von
					 % hoehe zu breite!
		     \multiply\count101 by \count241
		     \advance\count240 by -\count101
		     \multiply\count240 by 10
		     \count101=\count240	%101 is second digit of #2/#3
		     \divide\count101 by \count241
		     \count102=\count101
		     \multiply\count102 by \count241
		     \advance\count240 by -\count102
		     \multiply\count240 by 10
		     \count102=\count240	% 102 is the third digit
		     \divide\count102 by \count241
		     \count200=#1\count205=0
		     \count201=\count200
			\multiply\count201 by \count100
		 	\advance\count205 by \count201
		     \count201=\count200
			\divide\count201 by 10
			\multiply\count201 by \count101
			\advance\count205 by \count201
			%
		     \count201=\count200
			\divide\count201 by 100
			\multiply\count201 by \count102
			\advance\count205 by \count201
			%
		     \edef\@result{\number\count205}
}
\def\compute@wfromh{
		% computing : width = height * (bbw / bbh)
		\wfromhtrue
		\in@hundreds{\@p@sheight}{\@bbw}{\@bbh}
		%	\typeout{ \@p@sheight * \@bbw / \@bbh, = \@result }
		\edef\@p@swidth{\@result}
		%	\typeout{w from h: width is \@p@swidth}
}
\def\compute@hfromw{
		% computing : height = width * (bbh / bbw)
		\hfromwtrue
		\in@hundreds{\@p@swidth}{\@bbh}{\@bbw}
		%	\typeout{ \@p@swidth * \@bbh / \@bbw = \@result }
		\edef\@p@sheight{\@result}
		%	\typeout{h from w : height is \@p@sheight}
}
\def\compute@handw{
		\if@height
			\if@width
			\else
				\compute@wfromh
			\fi
		\else
			\if@width
				\compute@hfromw
			\else
				\edef\@p@sheight{\@bbh}
				\edef\@p@swidth{\@bbw}
			\fi
		\fi
}
\def\compute@resv{
		\if@rheight \else \edef\@p@srheight{\@p@sheight} \fi
		\if@rwidth \else \edef\@p@srwidth{\@p@swidth} \fi
}
%
% Compute any missing values
\def\compute@sizes{
	\compute@bb
	\compute@handw
	\compute@resv
}
%
% \psfig
% usage : \psfig{file=, height=, width=, bbllx=, bblly=, bburx=, bbury=,
%			rheight=, rwidth=, clip=}
%
% "clip=" is a switch and takes no value, but the `=' must be preset.
\def\psfig#1{\vbox {
	% do a zero width hard space so that a single
	% \psfig in a centering enviornment will behave nicely
	%{\setbox0=\hbox{\ }\ \hskip-\wd0}
	%
	\ps@init@parms
	\parse@ps@parms{#1}
	\compute@sizes
	%
	\ifnum\@p@scost<\@psdraft{
%	\typeout{psfig: including \@p@sfile \space }
%		\special{:\@p@swidth \space
%			\@p@sheight \space
%			\@p@sbbllx \space \@p@sbblly \space
%			\@p@sbburx  \space
%			\@p@sbbury \space startTexFig}
%	\typeout{Freizulassender Raum (H/B): \@p@srheight\space \@p@srwidth}
% Und hier wird der scallierungsfactor berrechnet
	\hscale=\@p@swidth pt
	\divide\hscale by \@bbw
	\vscale=\@p@sheight pt
	\divide\vscale by \@bbh
	\if@xscale
		%	\typeout{X-Scale set to \@p@sxscale 00}
		\hscale=\@p@sxscale pt
	\fi
	\if@yscale
		%	\typeout{Y-Scale set to \@p@syscale 00}
		\vscale=\@p@syscale pt
	\fi
	\if@scale
		%	\typeout{Scale set to \@p@sscale 00}
		\vscale=\@p@sscale pt
		\hscale=\@p@sscale pt
	\fi

	\ifhfromw
		% We calculated hsize from vsize, so we don't
		% need to calculate hscale, we take it from
		% vscale
		\vscale=\hscale
		%	\typeout{hscale=vscale=\number\vscale}
	\fi
	\ifwfromh
		% We calculated vsize from hsize, so we don't
		% need to calculate wscale, we take it from
		% hscale
		\hscale=\vscale
		%	\typeout{vscale=hscale=\number\hscale}
	\fi
	\divide\hscale by 655
	\divide\vscale by 655

	%	\typeout{hscale is \number\hscale \space Prozent}
	%	\typeout{vscale is \number\vscale \space Prozent}
%		\special{:\number\hscale\space\number\vscale\space @massaScale}
% now something stupid: i copied it from epsf.tex: we need 10 times
% of x-scale into rwi, so we do!
	\@myrwi=\@bbw sp
	\multiply \@myrwi by 10
 %\typeout{rwi = 10 hsize = \number\@myrwi}
   \vbox to \@p@srheight true pt{\vfil
 	\hbox to \@p@srwidth true pt{
                    \special{psfile=\@p@sfile\space
        		 hscale=\number\hscale\space
			 vscale=\number\vscale\space
			 llx=\@mybbllx\space
			 lly=\@mybblly\space
			 urx=\@mybburx\space
			 ury=\@mybbury\space
			 rwi=\number\@myrwi}
                   \hfill
                }
   } % lasse freiraum fuer bild

		\if@clip{
			\typeout{(clip), sorry, we can't do clip yet.}
			\special{:\@p@sbbllx \space
				\@p@sbblly \space \@p@sbburx \space
				\@p@sbbury \space doclip}
		}\fi % ifclip
		\if@prologfile
		    \special{psfile=\@prologfileval \space } \fi
%		\special{psfile=\@p@sfile \space }
		\if@postlogfile
		    \special{psfile=\@postlogfileval \space } \fi
%		\special{!endTexFig}
		% Create the vbox to reserve the space for the figure
	}\else{ % draft
		% draft figure, just reserve the space and print the
		% path name.
		\vbox to \@p@srheight true pt{
		\vss
			\hbox to \@p@srwidth true pt{
				\hss
				\@p@sfile\hss
                        }
                \vss
                }
       }\fi
}}

\def\cpsfig#1{\begin{center}\hfill\psfig{#1}\hfill\mbox{}\end{center}}
\catcode`\@=12\relax




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% psfig.sty following:

% LAST EDIT: Wed Jun 26 12:14:22 1991 by TeX Installation (tex)
\input psfig.tex


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% psfonts.tex following:

% psfonts.tex
%
% Frank Jensen (September 30, 1990)

\ifx\psfontsloaded\relax
\else \let\psfontsloaded=\relax

\def\declare@font #1#2#3#4%
   {\new@fontshape{#1}{#2}{#3}{%
      <5>#4 at5pt%
      <6>#4 at6pt%
      <7>#4 at7pt%
      <8>#4 at8pt%
      <9>#4 at9pt%
     <10>#4 at10pt%
     <11>#4 at10.95445pt%
     <12>#4 at12pt%
     <14>#4 at14.4pt%
     <17>#4 at17.28pt%
     <20>#4 at20.736pt%
     <25>#4 at24.8832pt}{}}

% Times Family

\declare@font{times}{m}{n}{ptmr}
\declare@font{times}{b}{n}{ptmb}
\declare@font{times}{b}{it}{ptmbi}
\declare@font{times}{c}{n}{ptmrrn}
\declare@font{times}{m}{sc}{ptmrc}
\declare@font{times}{m}{it}{ptmri}
\declare@font{times}{m}{sl}{ptmro}
\declare@font{times}{x}{n}{ptmrre}

\extra@def{times}{}{}



% Helvetica Family

\declare@font{helvetica}{m}{n}{phvr}
\declare@font{helvetica}{b}{n}{phvb}
\declare@font{helvetica}{b}{it}{phvbo}
\subst@fontshape{helvetica}{b}{sl}{helvetica}{b}{it}
\declare@font{helvetica}{c}{n}{phvrrn}
\declare@font{helvetica}{bc}{n}{phvbrn}
\declare@font{helvetica}{bc}{it}{phvbon}
\subst@fontshape{helvetica}{bc}{sl}{helvetica}{bc}{it}
\declare@font{helvetica}{c}{it}{phvron}
\subst@fontshape{helvetica}{c}{sl}{helvetica}{c}{it}
\declare@font{helvetica}{m}{sc}{phvrc}
\declare@font{helvetica}{m}{it}{phvro}
\subst@fontshape{helvetica}{m}{sl}{helvetica}{m}{it}

\extra@def{helvetica}{}{}



% Bookman Family

\declare@font{bookman}{m}{n}{pbkl}
\declare@font{bookman}{b}{n}{pbkd}
\declare@font{bookman}{b}{it}{pbkdi}
\declare@font{bookman}{m}{sc}{pbklc}
\declare@font{bookman}{m}{it}{pbkli}

\extra@def{bookman}{}{}



% New Century Schoolbook Family

\declare@font{ncs}{m}{n}{pncr}
\declare@font{ncs}{b}{n}{pncb}
\declare@font{ncs}{b}{it}{pncbi}
\declare@font{ncs}{m}{sc}{pncrc}
\declare@font{ncs}{m}{it}{pncri}

\extra@def{ncs}{}{}


% Courier Family

\declare@font{courier}{m}{n}{pcrr}
\declare@font{courier}{b}{n}{pcrb}
\declare@font{courier}{b}{it}{pcrbo}
\declare@font{courier}{m}{it}{pcrro}

\extra@def{courier}{\hyphenchar #1\m@ne}{}



% Palatino Family

\declare@font{palatino}{m}{n}{pplr}
\declare@font{palatino}{b}{n}{pplb}
\declare@font{palatino}{b}{it}{pplbi}
\declare@font{palatino}{b}{u}{pplbu}
\declare@font{palatino}{c}{n}{pplrrn}
\declare@font{palatino}{m}{sc}{pplrc}
\declare@font{palatino}{m}{it}{pplri}
\declare@font{palatino}{m}{sl}{pplro}
\declare@font{palatino}{m}{u}{pplru}
\declare@font{palatino}{x}{n}{pplrre}

\extra@def{palatino}{}{}



% AvantGarde Family

\declare@font{avantgarde}{m}{n}{pagk}
\declare@font{avantgarde}{b}{n}{pagd}
\declare@font{avantgarde}{b}{it}{pagdo}
\subst@fontshape{avantgarde}{b}{sl}{avantgarde}{b}{it}
\declare@font{avantgarde}{m}{sc}{pagkc}
\declare@font{avantgarde}{m}{it}{pagko}
\subst@fontshape{avantgarde}{m}{sl}{avantgarde}{m}{it}

\extra@def{avantgarde}{}{}



% Futura Family

\declare@font{futura}{c}{n}{pfuc}
\declare@font{futura}{bc}{n}{pfubc}
\declare@font{futura}{bc}{sl}{pfuboc}
\subst@fontshape{futura}{b}{it}{futura}{b}{sl}
\declare@font{futura}{ebc}{n}{pfuxc}
\declare@font{futura}{ebc}{sl}{pfuxoc}
\declare@font{futura}{lc}{n}{pfulc}
\subst@fontshape{futura}{m}{it}{futura}{m}{sl}

\extra@def{futura}{}{}



% Default settings

\def\sfdefault{helvetica}
\def\ttdefault{courier}

\def\bfdefault{b}

\fi
\endinput


